Order Number 8813315 E stim a tio n o f rainfall rates from p assive m icrow ave rem ote sen sin g Sharma, Awdhesh Kumar, Ph.D. University of Wyoming, 1987 U MI 300 N. Zeeb Rd. Ann Arbor, M 48106 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark V . 1. Glossy photographs or pages_____ 2. Colored illustrations, paper or print______ 3. Photographs with dark background_____ 4. Illustrations are poor copy______ 5. Pages with black marks, not original copy _ 6. Print shows through as there is text on both sides of page______ 7. Indistinct, broken or small print on several pages 8. Print exceeds margin requirements______ 9. Tightly bound copy with print lost in spine_______ _Z 10. Computer printout pages with indistinct print______ 11. Page(s)___________lacking when material received, and not available from school or author. 12. Page(s) 13. Two pages numbered 14. Curling and wrinkled pages______ 15. Dissertation contains pages with print at a slant, filmed as received seem to be missing in numbering only as text follows. . Text follows. 16. Other R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Estimation of Rainfall Rates from Passive Microwave Remote S ensing by Awdhesh Kumar Sharma A D i s s ertation Submitted to the Department of Physics a nd Ast r o n o m y and The Graduate School of the Unive r s i t y of Wyoming in Partial Fulfillment of Requirements for the Degree of Doctor of P h i l o s o p h y University of W y o m i n g Laramie, W y o m i n g December, 1987 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. This thesis, having been approved by the special Faculty Committee, is accepted by the Graduate School of the University of Wyoming in partial fulfillm ent of the requirements for the degree of ._EQCfcQE-Qf-EMl-QS-QpbLy-. Dean of the Graduate School Date...I R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. S h a r m a , Awdhesh K ., ESTIMATION OF RAINFALL RATES FROM PASSIVE MICROWAVE REMOTE SENSING. Ph.D., Department of Physics and Astronomy, December, 1987. Rainfall rates have been estimated microwave and visible/infrared D a t a of September 14, Mi c rowave 1978 from Radiometer (SMMR) Scanning board SEA V isible and Infrared Spin Scan Radiometer G O ES-W the passive remote sensing techniques. the on using Multichannel SAT-A and the (VISSR) on board (Geostationary Operational Environmental Satellite - West) w a s obtained and analyzed for rainfall rate retrieval. Mic r o w a v e brightness temperatures the mi c r o w a v e radiative of rates of rainfall due Gamma to drop and a from precipitating clouds and water. Microwave size distributions. Microwave oxygen a nd water vapor are based on the schemes given by Rosenkranz, s cat t e r i n g (MRTM) due to ice and liquid water are calculated using M i e - t h e o r y and a bso r p t i o n model using These MBT were computed as w h i c h are in a combined phase of ice exti n c t i o n are simulated, transfer atmospheric scattering models. function (MBT) a nd Barrett and Chung. The phase matrix involved in the MRTM is found using Eddington's two-stream approximation. The surface effects R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. due to winds and foam are included through the ocean surface e miss i v i t y model. Rainfall o p t i mization linear rates are technique regression data the has been estimating estimates. to inverted from MBT u s i n g the and a Bounds" and multiple relationship b e t w e e n the This relationship has been oceanic rainfall rates from SMMR data. inverted Griffith's scheme. of "Leaps leading rainfall rates and MBT. infer then for the rainfall used to The VISSR rates using This scheme provides an independent means rainfall rates for cross checking SMMR The inferred rainfall rates from both techniques have been plotted on a w o r l d m a p for comparison. A resonably good correlation has been obtained between the estimates. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. two AC K N O W LED G M EN TS The author wishes to express sincere thanks to the faculty and staff of the Department of Physics and Astronomy for their assistance. and constributions to the graduate program Special a p p r eciation is due to Dr. D.J. Dr. R.K. Kakar contributed their (NASA Headquarter, Washington, time, and knowledge, d i r ecting exposition of the thesis. and Hofmann D.C.) who support while I would like to thank my colleagues Arun V. Kulkarni a n d Robert F. Gutmaker for their contributions in proof reading and editing the manuscript. wish to thank B.H. L a m b r i g h t s e n for providing the SMMR and v aluable (JPL, Pasadena, VISSR and California) for his many suggestions in locating the earth co-ordinates for a given IR image of GOES satellite. C.G. data I Griffith (ERL, NOAA, Special Boulder) thanks for to Dr. the trouble she u n d e r t o o k to explain to me the details of her algorithm for rainfall estimation from IR data. This research w as m a inly supported by a grant from the National Aeronautics and Space Administration contract (Project # 956954) (NASA) under from JPL to the Department of Physics and Astronomy U n i v e r s i t y of Wyoming. Finally, I wou l d like to thank the Meteorology R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. iii Division, Space Applications R esearch Organization, Centre Ahmedabad, (SAC), INDIA for Indian Space granting me leave to conduct this research. Appreciation Manish, is given to my wife, and Vikash for their support, Sunita, love, and our sons and patience throughout this period of research. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. TABLE OF CONTENTS CHAPTER I Page INTRODUCTION 1.1 ................................... 1 Statement of the P r o b l e m .............. 4 Review of Past S t u d i e s ................ 6 1.2 II Rainfall Measurements from Space . . . RAINFALL MEASUREMENTS 2.1 2.3 ............................ Remote Sensing ofRainfall .................. 12 .................. 13 Passive Remote Sensing of Precipitation III 10 Limitations of Satellite Remote Sensing in Rainfall Studies 2.5 ............. 9 Advantages of Satellite Remote Sensing in Rainfall Studies 2.4 9 In Situ Techniques and Their Limitations 2.2 ........................ 6 .......................... MICROWAVE RADIATIVE TRANSFER MODEL ........... 14 19 D E F I N I T I O N S ........................ 19 3.1 Intensity and F l u x ....................... 19 3.2 Brightness Temperature 3.3 Polarization and Stoke's Vector 3.4 Microwave Radiometer 3.5 Emission and E x t i n c t i o n ........... 28 3.6 Mie Scattering and the Phase Matrix ................. . . . ................... . 21 23 26 30 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. V THE RADIATIVE TRANSFER EQUATION 3.7 . .. 33 Microwave Interaction with the A t m o s p h e r e ................................33 3.8 Microwave Interaction with the Ocean-Surface IV ......................... 41 MICROWAVE EMISSION AND ABSORPTION BY THE SURFACE AND THE A T M O S P H E R E .................... 50 4.1 Microwave Emission of the Model Ocean S u r f a c e ................................... 50 A Dielectric Constant of Sea Water . . . 50 B Microwave Emission by a Calm Sea . . . 52 C The Effect of Sea Surface Wind and F o a m ....................................... 53 4.2 Microwave Absor p t i o n b y Atmospheric G a s e s ......................................63 A Oxygen Absorption Coefficient B Water Vapor A b s o r p t i o n Coefficient 4.3 . . 67 ............................ 74 Absorption Coefficient due to non-raining Clouds B 63 Microwave Extinction by Liquid Hydrometeors A . . . . ..................... 75 Extinction Cross-Section of a R a i n - d r o p ................................. 77 C Drop Size Distributions of Rain . . . 81 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. D V VI Extinction Coefficient of Rain Volume 87 MODEL ATMOSPHERES AND SIMULATION OF BRIGHTNESS TEMPERATURES ..................... 107 5.1 Atmospheric Models ...................... 107 5.2 Brightness Temperature Simulation . . 112 STATISTICAL METHODS AND OPTIMIZATION T E C H N I Q U E S ....................................... 131 6.1 P r ecipitation using Microwave (SMMR) Data from SEA SAT S a t e l l i t e ........136 6.2 Precipi t a t i o n using Vis/IR (VISSR) Data from GOES S a t e l l i t e ................ 141 VII RESULTS A ND D I S C U S S I O N .........................155 7.1 Maps of Rainfall Rates using M i c r o w a v e D a t a ........................... 155 7.2 Maps of Rainfall Rates using Infrared D a t a ........................ 159 7.3 C o mparison of Rainfall Rates Inferred from Microwave (MW) and Infrared (IR) D a t a ................ 7.4 VIII 160 Sensitivity Study of Rainfall Rates . 175 R E T R O S P E C T .......................................181 C onclusions and Comments BIBLIOGRAPHY ..................... 183 ................................... 188 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. LIST OF FIGURES Figure Page 3.1 Geometry of specific intensity of radiation 3 .2 Geometry of a Polarization Ellipse 3 .3 Geometry of a directional microwave antenna 3.4 Geometry of reflection and refraction at the air-sea interface 4 .1 . ......... 21 . 24 . .............................. 27 43 Plots of Ocean Horizontal and Vertical E missivity versus Frequency and Angles for S a l i n i t y = 3 5 .,S u r f .T e m p . = 300.,Wind=0. . . . 57 4 .2 Plots of Ocean Horizontal and Vertical Emissivity versus Frequency and Angles for S a l i n i t y = 3 5 .,S u r f .T e m p . = 300.,Wind=5.0 . . 4.3 58 Plots of Ocean Horizontal and Vertical E m issivity versus Frequency and Angles for S a l i n i t y = 3 5 . ,S urf.Temp.= 3 0 0 . ,Wind=10.0 4.4 . . 59 Plots of Ocean Horizontal and Vertical Emissivity versus Frequency and Angles for S a l i n i t y = 3 5 . ,Surf.Temp.= 3 0 0 . ,Wind=15.0 4 .5 . . 60 Plots of Ocean Horizontal and Vertical E missivity versus Frequency and Angles for S a l i n i t y = 3 5 . ,S u rf.Temp.= 3 0 0 . ,Wind=20.0 4 .6 . . 61 Plots of Ocean Horizontal and Vertical Emissivity versus Frequency and Angles R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. v iii for S a l i n i t y = 3 5 S u r f .T e m p .=300.,Wind=25.0 . . 62 4.7 Plots of Integrated O x ygen and Water Vapor Absorption Coefficients versus Frequency 4.8 ... 69 . . . . 83 Plots of Extinction Coefficients at SMMR Frequencies versus Rainfall Rates using Marshal1-Palmer Drop size Distribution 4.9 Total number of Drops for Gamma Distribution versus Rainfall Rates ......................... 86 4.10 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate u s i n g the Shape Parameters p=-3 .42 , 6=. 80 , €=.013, and N g = 1 . 2 9 ............ 89 4.11 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate u s i n g the Shape Parameters jj= - 1 .79 , 6 =. 35 , €=.095, and N Q= 9 1 . 3 ............ 90 4.12 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate u s i n g the Shape Parameters U=~l .34, 6=.30, €=.069, and N Q=1310 ......... 91 4.13 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate usi n g the Shape Parameters jj=-0 .01, 6=. 22, €=.081, and N Q=.109E+06 . . . 92 4.14 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate u s i n g the Shape Parameters jli= 1. 63 , 6= .16 , €=.106, and N Q=.754E+07 . . . . R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 93 ix 4.15 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate us i n g the Shape Parameters jj= 5 .04 , 6= .10 , €=. 129 , and N Q=.920E+11 . . . . 94 4.16 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate vising the Shape Parameters jli= — .79 , 6=. 26 , €=.077, and N Q=.724E+04 . . . . 95 4.17 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate usi n g the Shape Parameters y = 0 .18, 6=.21, €=.082, and N Q=.196E+06 . . . . 96 4.18 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate usi n g the Shape Parameters jj= 1 .01, 6=. 18 , €=.110, and N 0=.753E+06 . . . . 97 4.19 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate usi n g the Shape Parameters y =4.65, 6=.11, €=.114, and N Q= .6 4 0 E + 1 1 . . . . 98 4.20 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate usi n g the Shape Parameters jj= 0 .40 , 6=. 20 , € = .118, and N Q=.705E+05 . . . . 99 4.21 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate us i n g the Shape Parameters y = l .01, 6=. 18 , €=.090, and N Q=.246E+07 . . . . 100 4.22 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate us i n g the Shape Parameters jj= 1. 01, 6=. 18 , e= .101, and N Q=.124E+07 . . . . R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 101 X 4.23 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate using the Shape Parameters = 1 .63 , 6=. 16 , €=.130, jj and N Q=.205E+07 . . . . 102 4.24 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate using the Shape Parameters y = - l .39, 6=.31, €=.031, and N 0=.159E+05 . . . 103 4.25 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate using the Shape Parameters jj = - 1 .03, 6 =. 28, €=.055, and Ng=.982E+04 . . . 104 4.26 Extinction Coefficient at SMMR Frequencies versus Rainfall Rate using the Shape Parameters jj= - 5.1 .27, 6 =. 23, € = .080, and N Q=.427E+05 . . . . 105 Frequency Distribution of Sea Surface Temperature (K) of 600 Model A t m o s p h e r e s ................. 113 5.2 Frequency D i s t ribution of Sea Surface W i n d (m/sec) of 600 Model A t m o s p h e r e s ..................... 113 5.3 Frequency D i s t ribution of Water Vapor (gm/cm 3 ) of 600 Model A t m o s p h e r e s ..................... 114 5.4 Frequency D i s t ribution of Liquid Water Content 2 (kg/m ) of 600 Model A t m o s p h e r e s ............ 114 5.5 Frequency Distribution of Rain Water 2 (gm/m ) of 600 Model A t m o s p h e r e s ..................... 115 5.6 Frequency Distribution of Ice Water 2 (kg/m ) of 600 Model A t m o s p h e r e s ..................... 115 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. xi 5.7 Frequency Distribution of Rainfall Rate (mm/hr) of 600 Model A t m o s p h e r e s ......................... 116 5.8 Flow Chart of Microwave Radiative Transfer Model for Brightness Temperature Simu l a t i o n . . . . 5.9 122 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) for a Cloud Model of Liquid Water Content LW = .33 ..................... 5 .10 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) for a Cloud Model of Liquid Water Content LW = .53 ..................... . . . . . 124 5 .11 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) for a Cloud Model of Liquid Water Content LW = 1.04 ................... 5 .12 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) LW = .42 for a Cloud Model of Liquid Water Content ..................... 5 .13 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) for a Cloud Model of Liquid Water Content LW = .40 ..................... 5 . 14 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) LW = .28 for a Cloud Model of Liquid Water Content ..................... R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 5.15 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) for a Cloud Model of Liquid Water Content LW = . 3 2 ....................................... 129 5.16 SMMR Brightness Temperatures versus Rainfall Rate (mm/hr) for a Cloud Model of Liquid Water Content LW = . 5 9 ....................................... 6.1 Frequency Distribution of Estimated Rainfall Rate using four SMMR 6.2 best channels s u b s e t ........... 142 Frequency Distribution of Estimated Rainfall Rate using five SMMR 6.3 130 best channels s u b s e t ........... 142 Frequency Distribution of Estimated Rainfall Rate using six SMMR best channels s u b s e t ............ 143 6.4 Frequency Distribution of Estimated Rainfall Rate using seven SMMR best channels subset 6.5 143 Frequency Distribution of Estimated Rainfall Rate using eight SMMR best channels subset 6.6 . . . . . . . . 144 Frequency Distribution of Estimated Rainfall Rate using nine SMMR best channels s u b s e t ........... 144 6.7 Frequency Distribution of Estimated Rainfall Rate using full ten SMMR channels s e t ................145 6.8 Frequency Distribution of Estimated Rainfall Rate from IR image of 14 September 6.9 1978 at time T.^ 152 Frequency Distribution of Estimated Rainfall Rate from IR image of 14 September 1978 at time T 2 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 152 xiii 6.8 Frequency Distribution of Estimated Rainfall Rate from IR image of 14 September 1978 at time T 7.1 World Map of Rainfall Rates using SMMR Data from Lat.=-50° 7.2 Long.=-180° to -80°157 to +50°and Long.=120° to 180°158 to *-50°and Long.=-180° to -80°161 World Map of Rainfall Rates using IR Data from L a t .=-50° 7.5 +50°and World Map of Rainfall Rates using IR Data from Lat.=-50° 7.4 to World Map of Rainfall Rates using SMMR Data from Lat.=-50° 7.3 153 Plot of jj to +50°and Long.=120° versus Simulated Rainfall Rates to 180°162 (mm/hr) using best five channels of SMMR Frequencies . 177 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. LIST OF TABLES Table 4.1 Page Resonant Frequencies of Molecular Oxygen and Amplitude Factors ................................ 4.2 Atmosphere Model after Cole et a l . (1965) 4.3 Coefficient used in Water Vapor Absor p t i o n 5.1 Cloud Statistics of 600 Model Atmosphere 5.2 Statistics of atmospheric parameters of 600 66 . . . . . 68 . 73 . 110 model a t m o s p h e r e s ................................. 110 5.3 Cloud M o d e l s ........................................121 6.1 Result of Leaps and Bounds Technique for Selecting 5 Best Subsets of Size 4 to 9 C h a n n e l s ............................................135 6.2 Multiple Linear Regression Coefficients for Rainfall Rate Estimation using 600 Simulated Brightness Temperatures at SMMR Frequencies. 6.3 Statistics of Estimated Rainfall Rates from SMMR Data using Different sets of Channels 6.4 . . 140 Constants for the Empirical W e i ghting Coefficients 6.5 . 137 ..................................... 148 The Statistics of Inferred Rainfall Rates from IR Data at times T , T 1 , T ^ ......................151 U R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. ;:v 7.1 Measures of difference between the microwave rainfall estimates estimates 7.2 (IRR) (MWR) and Infrared rainfall for a sample of size N . . . . 164 Measure of Difference B etween M W and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 5 ° ....................... 166 7.3 Measure of Difference Between MW and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 1 ° ....................... 167 7.4 Measure of Difference Between M W and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 0 5 ° ....................... 168 7.5 Measure of Difference Between MW and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 5 ° .........................170 7.6 Measure of Difference Between M W and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 1 ° ....................... 171 7.7 Measure of Difference Between M W and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 0 5 ° ....................... 172 7.8 Measure of Difference Between MW and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0 . 5 ° ....................... 173 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. xvi 7.9 Measure of Difference Between MW and IR Rainfall Estimates Whose Saptial Difference is less than Equal to 0.1° ................ R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 174 CHAPTER (I) INTRODUCTION Precipitation plays a very significant dynamics of the earth's atmosphere. role in the The latent heat released by precipitation is a major source of atmospheric heating in the tropics. therefore, Understanding demands precipitation. the high However, earth's quality atmosphere, measurements of precipitation is highly variable in both space and time. Conventional methods for estimating the rainfall rates inadequate regions of are most desired, be values, considered based on calibration problem. in most political, Over the vast ocean surfaces, where there p recipitation at all. can inaccessible the earth's surface due to economic, and terrain factors. is and are no Island data to radar be close direct on to measurements measurements precipitation, it of whic h the oceanic rainfall suffer from the In practice the calibration of radar is done w i t h the help of rain gauges which are within the sight of the radar. In the absence of gauge rainfall measurements ever the oceanic regions the calibration of difficult problem. In the face radar poses a of increasing demand for rainfall estimates over the land and oceans surfaces, R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. remote 2 sensing of precipitation commands a great deal of attention. Applications of remote sensing data of infrared sensors to the problem of rain estimations on synoptic or climatological scales have shown very encouraging results in the past. But the point measurements like rain gauges and satellite estimates of rainfall averaged over an hour or half an hour are very poorly correlated or not correlated at all. This is due to measurements whereas the values of the fact that rain gauges are point satellite estimates are average rainfall signature over the area of the sensor's field of view. M a n y schemes using visible from both and infrared observations geostationary and polar orbiting satellites have been applied to the precipitation estimation problem and Thiele, 1981; Barrett and Martin, (Atlas 1981). These techniques make use of information obtained from the cloud's t o p - s u r f a c e s . They must be tuned for specific locations and are the thus difficult to apply globally. a c c u m ulated rainfall obtained from such g ood agreement However, techniques are in with the ground measurements, when averaged over a time scale greater than 3-6 hours and a space greater x longitude) than Griffith measurable or (1987). equal The parameters if the scale is reduced. to l°xl° (latitude relationship between scale satellite and rainfall is weakened drastically Therefore the results obtained from R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 3 these techniques are good e n ough to be used only for the study of climatological or synoptic scale problems. Problems like weather flood control etc. are increasingly rainfall measurements. rates from a forecasting, river management, demanding which can passive microwave technique appears to be a through i n t ensity of rainfall Pa s s i v e microwave a thick through techniques cloud and absorption are a nd thus measure and very ones the scattering. useful over ocean surfaces because the ocean emissivity lies 0.6 Despite their passive microwave techniques are the only see term Estimations of instantaneous rainfall v i able source which can fulfill these demands. limitations, short between 0.4 to provides a cold background for detecting any physical changes of atmospheric conditions lying above it. Over land areas, passive microwave results are substantially less quantitative because the land emissivity varies between 0.8 to 1.0 and surface emission mi c r o w a v e spheres) provides as scattering allows background, but the direct due the same order of magnitude of atmosphere to rain particles observation of and it. rainfall over than that in the case of absorption. ice The (water and ice the relationship to the rain rate radiances are more physically related (water above any because the scattering is primarily due to the ice spheres aloft, less does the spheres) a nd to thereby the is Microwave hydrometeors represent R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. a great 4 p otential for improved satellite observations of precipitation. (1.1) STATEMENT OF THE PROBLEM The thermal m i c rowave emis s i o n of the atmosphere due to a tmospheric gases have been empirical a nd observational thorough i nvestigations theoretical and computational rainfall and liquid water well u n d e r s t o o d in the case of no rain from past theoretical, study, (oxygen a nd water vapor) rates using have schemes microwave M u l t ichannel Microwave Radiometer) microwave brightness studies. for e s timating at SMMR frequencies. the (Scanning Theoretical have b e e n calculated at these frequencies u s i n g the radiative transfer equat i o n a this been carried out for data temperatures In for given model atmosphere. A number of model atmospheres are g e n erated from the m e t e o r o l o g i c a l observations rawinsonde data) by parameters. m i c rowave The in t r o d u c i n g rain emission a nd (radiosonde / other surface of the ocean has been s tu d i e d in detail and its d ependence on surface temperature, surface salinity, and surface roughness has been calculated. The ocean surface e m i s s i v i t y model developed b y Kakar (1982) (Hollinger, from the 1971; Strogryn, m o d i f i e d by Kakar has been results of previous 1972; Wilheit 1979) used. The Pandey and studies and later on radiative transfer R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 5 problem is solved for the case of scattering atmosphere, the values of microwave brightness temperatures at all SMMR frequencies are obtained for each of the model The effects of multiple scattering on atmospheres. the transfer of thermal microwave radiation in rain are evaluated using Mie theory and particles. the The brightness Gamma relationships temperatures Bounds" and between the microwave and rainfall rates have been found statistical "Multiple Linear these Regression". techniques are inversion The then methods, results employed by Leaps such as obtained usi n g the measurements to estimate the rainfall rates. Finally, rainfall rates from SMMR these are compared w i t h those inferred from VISSR (Visible and Infrared Spin Scan Radiometer) algorithm developed collected onboard Environmental the drop size distribution of rain usi n g optimization techniques such as "Regression and and b y Griffith et al.(1978). the GOES satellite) algorithm, are essentially microwave data and data using The IR data, (Geostationary satellite, different in the a nd nature Operational Griffith's from the the a l g orithm and therefore provide an independent source for comparison. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 6 R E V I E W OF PAST STUDIES : (1.2) Rainfall Measurements from Space The present state-of-art of rainfall measurements space is Thiele (1981). Basically, limited described to by Atlas et a l . (1982); identifying cover. pr i marily for convective rainfall latitude). Microwave and Atlas and visible and infrared cloud rainfall at methods They low can be latitudes estimates from over are used (<30° oceans are limited to the sensor's field of view. Several d e v eloped visible during and the GARP infrared techniques (Global have Atmospheric Research Program) under the GATE (Arkin, 1979; Augustine et a l ., 1981; H u dlow a nd Patterson, 1979; Stout Richards and et a l ., Arkin (GARP Atlantic Tropical been 1979; (1981) and have relationship between p r ecipitation quantities. Satellite-techniques Woodley analyzed and have a l ., in 1979). Although none of these applied all (Griffith et al., methods ocean, borne can be types of precipitation and to all regions, they can be improved usi n g complementary estimates space observed b e e n extended to an 1981; Wylie, from 1980). detail the satellite empirical estimation of rainfall over land to et Experiment) passive microwave obtained radiometers over the and suitable radar systems over the land or ocean. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 7 Rodgers and Ad l e r (1981) have analyzed tropical cyclone rainfall characteristics data data. Earlier, the usi n g estimation microwave of rainfall rates over the oceans was p e r formed b y Wilheit et al. (Electrically Radiometer) Scanning Nimbus-5 satellite. assumed to be M i c rowave a ssumption is usi n g The not true p a r ticles transparency at ESMR onboard the a n d the Marshall-Palmer drop size d i s t ribution was used. particles (1977) In their scheme the ice transparent radiometric higher were (1948) of ice microwave f requencies and Marshall-Palmer drop size dist r i b u t i o n over estimates the number of drops at low rainfall rates. Savage a nd Wei n m a n technique for rain w h i c h w as later technique was (1975) developed a p a s s i v e microwave measurements tested by over land u s i n g 37.0 GHz Rodgers et al. be marginal microwave it proved The s c attering m e c hanism is enhan c e d frequencies fact was u s e d by Wilheit et tropical and for mapping rain and could not provide the rainfall intensity. higher This based on the scattering of m i c r o w a v e s by the hydrometers near the top of the rain column, to (1979). sto r m data. 92 GHz and 183 GHz, a nd this al. (1982) in analyzing and Lewis (1986) in synoptic scale features of Nor t h Their results the A linear relationship b e t w e e n rainfall rate and the brightness temperature at 37.0 GHz was used Katsaros at analyzing Pacific m e soscale weather by and systems. were in good agreement with the coastal rain R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 8 gauge measurements, when the terrain effects were for. Spencer et al. (1983-a) compared accounted SMMR brightness temperatures to the radar derived rain rates over of Mexico. in freezing the 18-37 level in distribution. None GHz the of frequency cloud the and past extinction range on studies varia t i o n s in these quantities in detail. st u d y to microwave extinction due rain by depend the rain on the drop size have considered In the drops present and different drop size distributions are considered in S c attering Gulf What all the studies of rainfall rate retrieval from microwave data have shown is that drops the the detail. due to hydrometeors is taken into account in the mi c r o w a v e radiative transfer calculations u s i n g Mie a n d the Gamma drop size distribution. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. theory CHAPTER (II) RAINFALL MEASUREMENTS (2.1) In situ Techniques and Their Limitations Rain gauges are most commonly utilized in determining rainfall over the land surfaces. Measurements of rain gauges are generally topography, located affected site, on by wind, an island the and interrelated gauge are design. not likely representative values of rainfall over the vicinity. Since factors Rain to ocean of gauges give in the their ships do not provide a stable platform for rain gauges either, no direct source of rainfall measurement exists over the oceans. Another instrument employed in rainfall monitoring over land and coastal instrument suffers regions from is the problems scattered microwave energy to the partial filling of the radar beam, beam by intervening drops, ground weather of drop radar. relating size the This back distribution, attenuation of the radar absorption and reflection by (anomalous p r o p a g a t i o n ) , and signal calibration. calibration of a radar requires another source measurements. In the absence of of oceanic the The rainfall rainfall measurements by any other mea n s the radar measurements of R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 10 rainfall data over the ocean surfaces can not be assumed to be 100% reliable. However, gauges of p r o viding a the radar h as an spatially advantage continuous view. e mployed extensively in support of rain gauge data. the cost of a radar and technical and engineering the requirement support, the over It is Due to of continual operational world w i d e u se of radar for rainfall m onitoring is limited. In situ measurements of rainfall b y conventional means are deficient in m a n y areas, are required in near real parti c u l a r l y if rainfall data time for weather forecasting, river management a nd flood control. A l s o the rainfall data o btained from conventional mea n s are too inadequate in space and too infrequent in time to be useful to satisfy the large amount of requirements of users. de m a n d i n g situation for Therefore in this extremely rainfall s ensing from satellite p l ateforms is method by measurements, potentially remote the only w h i c h the n e c c e s s a r y measurements can be made in space and t i m e . (2.2) Remote Sensing of Rainfall Remote sensing of rainfall rate measurements beg a n w i t h the development of meteorological radar after w o r l d (Barrett and Martin war II (1981)). Measurements from space were not possible until the Environmental Satellites were R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. placed 11 in orbits. These satellites have been well equipped with s ens ors capable of yiel d i n g data for rainfall measurements. Th e s e sate l l i t e s have been c ategorized into two types namely (i) L o w Earth-Orbiting, The first type and (ii) Geosta t i o n a r y satellites. of satellites o c cupy low-level orbits (usually b e t w e e n 500-1500 km above the earth's surface), u s u a l l y p a s s i n g over the earth's poles w i t h the time periods of the order of 100 minutes. Therefore, are r e q u i r e d for each satellite to cover per day. over the 14-15 orbits entire globe U s u a l l y such satellites v i e w one-half of each orbit the night time side of the globe, globe is thus v i ewed twice per day, once about at night time. The current once in every area on the day light operational and series of A m e r i c a n p o l a r - orbiting environmental satellites are NIMBUS, TIROS, and N 0 A A series. Program which (DMSP) The Defence Meteorological Satellite is the operational m i l i t a r y satellite system, comprises two or more polar-or b i t i n g satellites c a r r y i n g meteorological sensors. The s e c o n d type of satellites are p l a c e d into orbit at ap p r o x i m a t e l y 35400 k m high above the earth's center. orbital p l anes are in the plane of the earth's equator and they move in the same earth. Their d i r ection as the rotation This type of orbit is called geosynchronous, of the in whi c h the satellite kee p i n g pace w i t h the rotation of the earth on its polar axis, a nd a satellite occupying it appears to be R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 12 s t ationary w i t h respect to the earth's m o tion and is ge o s t a t i o n a r y satellite. Such a satellite constant geographic field of v i e w through out called provides its a motion, and the data frequently o b t a i n e d from this has been utilized in rainfall study. The current geosta t i o n a r y meteorological satellite systems are METEOSAT, (2.3) Advan t a g e s of Satellite G M S , GOES, R e mote and INSAT. Sensing in Rainfall satellites in rainfall Studies The advantages of usi n g e a r t h study are following: (a) S a t ellite systems p rovide global coverage of data, a nd thus provide access to remote regions. (b) S a t ellite imaging data, co n t r a s t i n g systems with those yield spatially continuous o b t a i n e d from the irregular netwo r k s of surface wea t h e r stations. (c) Satellite o b s e rvations are distributed more h o m o g e n e o u s l y than in situ observations. (d) Geostationary satellites can provide information more frequently than is commonly o b t a i n e d from surface and upper air we a t h e r stations. (e) S a t ellite sensors me a s u r e the radiances emerging from the u n d e r l y i n g atmosphere and the surface therefore contains the i n f o r mation on the para m e t e r s integrated over R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. the 13 atmospheric column and the surface. These radiances are then inverted to retrieve the information contained in them. (f) Satellite or near real data can be obtai n e d for large area, time for many uses in in real meteorological and oceanographic studies. (2.4) Limitations of Satellite Remote Sensing in Rainfall Studies Despite the great potential for use of in atmospheric and oceanographic satellite studies, data these investigations are often limited in practice by a number problems as described below (a) Satellite data are of : mostly synoptic. Thus there are difficulties in correcting and/or calibrating the data for a number of factors which m ay influence them. (b) Transformation of images from satellite co-ordinate frame to earth locating frame is not straight forward. (c) Conversion desired of information satellite-observed requires the radiances knowledge into the of the relationship between the two. (d) Processing of remotely sensed raw data may eliminate some of its important features and thus may not reproduce all the information contained in the raw data. (e) Extracting the data of interest from a hugh satellite R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 14 data base is technically difficult. (f) Developing the detailed interpretation, and use of methods for satellite the analysis, data for specified operational applications have been difficult due to numbers of parameters required b y the methods. (g) been The types and sufficient resolutions of available data have not for most of the atmospheric and oceanographic studies. (h) A multichannel-system operating at different frequencies (SMMR observing at different frequencies) instantaneous field of view (IFOV) can have different therefore it is very cumbersome to extract the data of all channels corresponding to the same geographic location for their simultaneous use. (i) Retrieval of atmospheric and/or oceanographic parameters from satellite observations require a realistic model of the atmosphere and the ocean and a very efficient inversion technique. (j) Existing operational procedures require changes if n e w data types are to be employed. (2.5) Passive Remote Sensing of Precipitation Satellite estimated rainfall rates using data provided Spin-scan radiometers. by have been (a) Scanning radiometers, derived and (b) The w avelengths most commonly used in R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 15 these instruments for rainfall studies are y m - 0.7 ym) ; (ii) 12.5 y m ) ; a nd Infrared, (i) Visible, (3.5 y m - 4.2 ym) (iii) Microwave, n a t urally (0.5 or (10.5 ym - emitted radiation (0.3 cm - 4.5 c m ) . (a) A scanning radiometer on board p o lar-orbiting satellite consi s t s of a revolving mirror, filters. As the revolving satellite mirror the advances along its the radiances which are r e volution of the m i rror adjacent to the one before, Some of these Resolution visible, e.g. are (A V H R R ) near infrared, (6.6, 10.69, o p e r a t i n g at two polarizations capable o n T iros-n been derived track. of having (including 18.0, The SMMR consists 21.0, and 37.0 of of GHz) (horizontal and v e r t i c a l ) . rates over the u s i n g SMMR data of SEASAT-A. S M M R is a dual p o l arized co n i c a l l y s canned radiometer angle Very and water vapor channels), In this study estimations of rainfall have wavelengths. the four channels A d v a n c e d Radiometer infrared, channels ocean passed produces a n e w scan line, a n d the SMMR on Nimbus-7 a nd SEASAT-A. five then across the sub-satellite radiometers multis p e c t r a l systems, High the beam splitter a n d then the spectral filters to give the intensity of r a d iation at the desired Each orbit scans the target across the sub-satellite track a nd collects thr o u g h a b e a m splitter and spectral incidence approximately 48.8° The which mak e s an nadir. The SMMR has been d e s c r i b e d in detail by Gloer s e n and R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. from the 16 B a rath (b) (1977) a nd Njoku et a l . (1980). Spln-scan radiometers, have b e e n flown on satellites and consist of a scan n i n g device, IR/VIS (reflected light) sensors. these instruments consists of a m e c h anically to provide spinn i n g motion of scanning. Thus the an a telescope and The scanning system in mirror whi c h is advanced n o r t h to south viewing, while the satellite provides west to east image of the entire v isible disc of the earth is built u p over a p e riod of about 20 which geostationary minutes, after the mirror is returned to its initial p o s i t i o n and is ready for another image. VIS S R of GOES (Visible a nd Infrared Spin Scan Radiometer) satellite have been data obtained for e stimating the rainfall rates using the al g o r i t h m given b y Grif f i t h et (1976). This rainfall rate with the algorithm provides an independent source of measurement w h i c h can be used for comparing estimated rainfall rates obtained from SMMR data. The V I S S R instrument has b e e n de s c r i b e d in detail b y (1984). clouds are r e latively bright in cloud tops at d i f ferent altitudes br ightness temperatures in the IR region, interpreted in ice, visible observations are the r a d iation temperatures of Since Gibson Visible data is most strongly related to the albedo of the target i.e. higher reflective surfaces of and al. snow image. the IR target. have different this data may be terms of clo u d top temperatures or heights. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 17 From the cloud top temperatures determined using an rainfall empirical Griffith et a l . (1976), Martin, rates have been relationship obtained by Stout and Sikdar (1975) and many o t h e r s . In the visible images, the probability rainfall of rainfall relationship is as cloud brightness increases, increases, strongly this brightness- time dependent. correspondence between cloud brightness and rainfall at earth's surface to frequency, time. relate cloud intensity and Sikdar brightness to any precipitation and extent must consider changes w i t h This scheme has Stout the is better w h e n a cloud system is young and vigorous than when it is old and decaying. Consequently, attempt The been (1975) successfully used by Martin, for deep convective clouds in the tropics. In the IR relationship satellite between images, the the cloud's simple empirical top temperature rainfall has been obtained by Scofield and Oliver and (1977). Their scheme can be used for estimating rain from convective storms. The scheme and tuning for local terrain Microwave radiometers data the estimates from it need a fine effects and for the climate. observed It satellite-borne have been shown to reveal not clouds, cases of VIS/IR schemes, but clouds. from is in this rain region areas that as in the embedded rain in the has been most R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 18 directly evidenced from satellite data available until Schemes involving V I S/IR data depend now. on less physically direct relationships between clouds and rain. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. CHAPTER (III) MICROWAVE RADIATIVE TRANSFER MODEL In the study theory of microwave radiative (MRT) and its applications to remote sensing one generally encounters a few special physical terms. applicability of these terms (MRS) transfer in The definitions and microwave remote sensing are described here. DEFINITIONS (3.1) : Intensity and Flux The specific intensity spectral intensity), (also known as monochromatic I , or is the flux of energy in a given direction per second per unit frequency range per unit solid angle per unit area normal to the given direction M. (Goody, R. 1964) . Consider the flow through a point P (3.1), surrounded by a small element in space, Figure 2 of area d A s (m ), •—+ normal to the direction of s. From each point on d A s a of solid angle bundle of rays, d O s (steradians) originating on df>s , transports in time dt (sec) cone is drawn about the s. The dAs , and contained within and in the frequency range R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 20 v to v+dv (hertz), the e n e r g y is given by dE^fP) where Iy(p,s) = IV (P,S) d A g d O g dv dt is the specific intensity (3.1) (watt m —2 st —1 hz —1 ) at the point P in the s-direction. The flux in the direction of d, F is defined as j(p)* 1r / Cl at a point the total e n e r g y flowing per second, unit area normal to d, per unit frequency interval dFv , d (P) = dEv (P) / ^ d dv dt P, across (dv») (3'2) where dA s = d A , cos e d (3.3) e is the angle between s and d dA^, . The energy flux across integrated over all s - d i r e c t i o n is given by V P' S> Js F v,d = where the integral extends system of polar cos 0 d0s over co-ordinates all solid with direction of the o utward normal to d A (3.4) s the angles. z-axis In in , the solid angle a the do s is defined as dO s = sin e de d$ (3.5) ' ' and the expression for the net flux is r2iT Fv (0,e) = 0 'e I cos e 1 sin e 1 d e ' d ^ 1 0 v (3.6) or F i;(0,e) = nr I s i n 2e where 1^ is the m e a n value of b etween zenith angles 0 and (3.7) , averaged over the e. surface A black body or lambert R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 21 surface radiates isotropically (i.e. is independent of and <f>) and has the flux of emission F v (0, tt/ 2) or ttI e in the V upper hemisphere. d0„ cJA 5 Geometry of specific intensity of rodiotion Figure (3.1) (3.2) Brightness Temperature Instead of intensity, used in thermal brightness microwave temperature studies. equivalent temperature of a black body amount of energy same frequency band. temperature". The as (BT) is It is defined as the radiating the same that received by a radiometer in the This is also known as "radiometric relationship between BT and intensity is derived from the properties of a black body whose thermal e miss i o n is given by Planck's law, mathematically stated as B (v ,T) = w he r e T (2 h v 3/C2 ) is the temperature (1/{e x p (hi// k T )-1}) (in degree kelvin) (3.8) of the black R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 22 __QI body, h is Planck's constant (6.623 x 10~ Joule sec), C is Q the speed of light in vacuum Boltzman's constant (2.99793 x 10 (1.38 x 10 -23 meter/sec), k is Joule/Kelvin), and v is the frequency in hertz of electromagnetic radiation. At microwave frequency and terrestrial temperature the quantity hv/kT << 1, therefore the Planck's formula reduces to the Rayleigh-Jeans law, w h i c h is given by B(v,T) = Thus, at a (2 k v 2/C2 ) T given frequency interval, proportional frequency the energy flux per unit B(v,T) to the (3.9) (Watt m _o temperature. radiation of a given frequency, v, sec), For is linearly electromagnetic and known intensity, 1^, one can associate a unique temperature known as BT such that B (v,T)|_ = I . Thus equation (3.9) c an be wr i t t e n as B Iv A = (2 k v 2/C2 ) Tb (v) microwave radiometer detects po l a r i z e d components at a time, incident (3.10) only one of the two therefore only half of the unpolarized radiation is received by a radiometer. Hence p = T a (v,p) = where p denotes s ubsituting which in I/2 the (C2/ kv2 ) I (p=v) gives (3.11) V ,p vertical (3.10) or horizontal polarization of radiation. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (p=h) 23 (3.3) Polarization and Stoke's Vector The vertical or horizontal plane of polarization is the plane parallel or perpendicular to that containing the local normal vector (normal to the plane of the (line detector) and Poynting of s i g h t ) . The electric vector of a polarized electromagnetic wave lies in its plane of polarization. term polarized component refers total stream of radiation for to The one component of the electric field vectors oscillate in one given plane of polarization, on the other hand the which the term p o l arized wave is meant to imply that the wave is polarized w i t h fixed phase any two orthogonal components of its electric field vector. Four parameters known as 1960; Vandehulst, between are required to describe the state of an electromagnetic wave. 1852, differences They were introduced S t o k e 1s vector, 1957; Kerker, and m ay be 1969) by Stoke in (Chandrasekhar, expressed as I = {I,Q , U ,V) (3.12) I = Iv + Ih (3.13) Q = Iv - 1^ U = Q tan 2x (3.15) V = I sin 2 1 (3.16) where = I cos 2 Y cos 2x (3.14) 1^ and Iv are the horizontal a nd vertical intensities of the two polarized components of the beam, x is the angle between the vertical axis and major axis is of the ellipse. i R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. the 24 angle of w h i c h the tangent is the ratio of the minor axis to the major axis of the ellipse as shown in the Figure At a ny point, Ip is proportional to EE* where E* (3.2). is c o m p l e x conjugate of E. Thus o ne obtains (3.17) xh = C ' E hE h i I v <= C * E E v v (3.18) LI-HHIS I SINS H -n illS H H FIGURE ( 3 .2 ) GEOMETRY OF A POLARIZATION ELLIPSE R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. the 25 E x pressing the Stoke's vector in terms of electric field vectors I = c' [Ev E* + EhE*] (3.19) Q = c' [EvE ^ - Eh E*] (3.20) U = C* [E E* +E.E*] = 2 C ' L v h h vJ V = ic'[E E* - E.E* ] = L v h h vJ Re (E E*) ' v h' -2C' Re (E E*) 4 v h' w h e r e C 1 is the constant of p r o p o r t i o n a l i t y and the bar electric fields. The Stokes v e ctor due to scattering, c an (3.22) ' i=-F-l, and indicates that time a verage is taken over the time intervals m u c h larger than the p e r i o d of respective (3.21) ' ' v i b ration change reflection of the occurring in the and refraction be d e t e r m i n e d from the change occurring in the electric field components E.. and E. . v h The most important p r o p e r t y of the the additivity beam. However, the of S t o k e 1s vector is its comp o n e n t s for incoherent polarized the state of p o l a r i z a t i o n may be processes of reflection, altered by t r a n s m i s s i o n and scattering. p a r t i a l l y p o l a r i z e d beam of r a d i a t i o n is obtained by A the s u p e r p o s i t i o n of incoherent be a m s a nd can be decomposed into a fully polarized a nd unpolarized (natural) parts. Stokes vector for fully p o l a r i z e d and u n p o l arized parts (>T(Q2+U 2+ V 2 ) , Q, U, V) and (I, 0, 0, 0} respectively. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. The are 26 (3.4) Microwave Radiometer Microwave radiometer is a device used microwave radiation radiometer uses power, an P(e,4>),and a gi v e n antenna the factor known as gain, specific in direction whi c h to measure direction. receives or A the typical transmits pattern of this power is given by a G(e,$), whe r e e and $ define the and a mathematical relationship between g a i n and power is given by (3.23) 0 ( e . ♦) = --- ;— i f Jo" JS 8' <58 d<t> Remote sensing antennas are designed to have beam width (main lobe) defined as the is of G(0,0), linearly Beam width angle subtended at the antenna b etween lines that intersect G(e,4>), where valu e narrow so that unwanted contributions from side lobes in the measurements can be minimized. is a the as shown in the Figure proportional to its gain is half the (3.3). A ntenna gain effective apperture, A e (e,$), at a given w avelength X. A e (6 ,<t>) = ( X 2 / 4 tt) G (e ,<t>) (3.24) The total power per unit frequency interval incident on the antenna is related to the effective apperture by 2 kT r— A e (e,4>) sin e de d<l> (3.25) d P in - "I A where 1/2 is introduced because linearly polarized antennas T2tr rtt R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 27 receive only half the incident randomly polarized radiation, k is Boltzman's constant, A is blackbody the center wavelength of the incident radiation and dv is the frequency wi d t h of the radiation. In T^, the is measured and practice, incident antenna power temperature, received radiometer is then computed from the equations The antenna temperature, a hypothetical by a (3.23-3.25). is defined to be the temperature of resistor, and is related to the incident power as given by Tft = (dPin/kdy) (3.26) y Antenna Effective Aperture Ae (9t4>) Antenna Power Pattern 6 (8,<f>) Side iobes Main lobe c35. ■30 20; Intensity Scale (d B ) Beamwidth — 'o'"' Ocean's Surface G e o m e t r y of a directional m i c r o w a v e antenna F igure (3.3) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 28 (3.5) Emission and Extinction The interaction of radiation w i t h matter processes of emission and extinction. r a d iation travels through a m e d i u m weakened due respectively. to emiss i o n it or involves When is the a pencil of strengthened extinction or processes These processes are governed b y Lambert's law w h i c h is m a t h e m a t i c a l l y s t ated as dl dl w he r e the V v (extinction) = - B (emission) = constants e x tinction e x tinction t ravels the process through (3.27) ds (3.28) T^ e in an<* • in in due in physical (temperature.pressure.composition) .absorption (energy good matter. is matter is held constant. (3.27) to order as the radiation ds(m), a nd sign volume emission proportionality state a p p r o ximate the volume change r everse distance, are negative differential is a . J emi v ®e x t^m ^emi^”1 ^ ’ that ds V proportionality coefficient, coefficient, indicates of a 6X1 only This when the of the Extin c t i o n is defined as the sum of c o n verted into heat) and scattering (energy redistributed in different d irections without change in f r e q u e n c y ) . Since all e x t i n c t i o n processes are linear, it can be defined as J3 ^ = ext where a a b s (m and B SGcl (m *) . abs ct + B ) *s t*16 (3.29) 1 sea volume a bsorption coefficient is the v o lume s c attering coefficient. More R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. 29 conveniently a parameter is d efined as optical path-length, d T , w h i c h is ex p r e s s e d as dT = The - B . ds ext negative sign (3.30) ' in (3.30) is due to convention because the optical path-length is defined to be a definite is zero where quant i t y ' positive whose va l u e at the top of the atmosphere (t = 0 ) . In the case of a plane parallel atmosphere, z is defined along the z e n i t h direction, the distance traveled by radiation in the d i r e c t i o n e degree from the nadir is given by ds = sec e dz The source function, the units of intensity, (3.31) , giv e n in equation (3.28) h as is d efined as the r a d iation emerging in the d i r ection specified by the z e nith angle, e, and the a zimuth angle, $, due to s c attering and emission processes. For a plane parallel horizontally homogeneous scattering atmosphere the source function is given by w Jv= 4 ^ [2tt JO Ptt V * 6 '*'6 ''*') s i n e 1 de'd<t>' +(1- cov ) The first term in (3.32) the right hand Bv (3.32) side of equation defines the radiation emer g i n g due to scattering a n d the second term is a P (e ,<J>,e 1 ,$ 1) is contribution due to self emission. the phase m a trix characterized b y a pencil of radiation incident in the d i r e c t i o n (e1, ^ 1) and scattered R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 30 in the direction e miss i o n B^(T), whose (e,4>). Stokes By is vector the unpolarized is given by thermal [1/2 By (T), 1/2 0, 0], where B y (T) is the Planck function at a frequency, v, and at scattering albedo, u^, scattering is the (e',$') given temperature, is coefficient 1^(0'#$') di r ection a defined to incident the as the T. The single ratio extinction radiation given of the coefficient. emerging in the at the point of scattering. (3.6) Mie Scattering and the Phase Matrix Mie scattering theory has been used to ab sorption determined the and scattering properties of homogeneous spheres of water having the same optical properties. Drops of liquid wat e r in the form of clouds and rain, known as hydrometeors, affect the propagation of microwaves in the atmosphere. intensity of incoming e x p ( - B ext ds) ds, where The radiation is reduced by a factor of in traversing the medium through a distance the extinction coefficient due to rain particles B e x t , will be computed later in chapter IV using Mie theory. The solution of conditions by Maxwell this for a boundary theory (1908) and is known as Mie theory. one can obtain the relationship between incident and scattered electric field vectors the given a sphere of radius r and index of refraction m, was treated by Mie in Usi n g equations intensity components. in terms of This relationship is defined by a R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 31 4x4 transformation m a trix kn o w n as the phase matrix. The elements of the phase 16 matrix are functions of scattering angle e, the index of refraction m, the particle radius r, and the incident wavelength A. For isotropic and homogeneous p a r ticles like hydrometeors these 16 elements are reduced to 4 independent elements. This reduction is possible by the virtue of the principles of reciprocity, spherical symmetry. mirror symmetry and The reduced phase matrix for a single homogeneous sphere is then given by P(e) P.. P, a 0 o = P.* 0 0 P, , 0 0 0 p,, - p s< 0 P 3< P 33 (3.33) where P lf = (A2 / 2TT2r 2 Q s c a ) (i, + ia ) (3.34) P ia = (A2 / 2TT2r 2 Q s c a ) (ia - i.) (3.35) P 33 = (A2 / 2Tr2r 2 Q s c a ) (i3 .+ i4 ) (3.36) P 34 = (A2 / 2Tt2r 2 Q s c a ) (i3 - i4 ) (3.37) i, = S, S* = I |s, I2 (3.38) ia = S 2S* = |S2 |2 (3.39) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 32 x i, = S 2S (3.40) X i., — Sj S (3.41) a (i, Q sea + i2 ) sin e de wh e r e stars denote complex conjugation and are amplitude functions. Since sufficiently far from each other, the the it is S, and particles possible to S2 are study scattering by one particle wi t h o u t reference to others. Consequently, intensities sc a t t e r e d b y various particles m a y be added without regard waves. for (3.42) 0 to the phases of the scattered The independent elements of the phase m a t r i x obtained a sample of particles in the particle range (r1#r2 ) are then given by (3.43) sea (3.44) sea (3.45) 2 2 tt Q sea R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 33 (3.46) The size distribution of the particles, the amplitude functions, dn(r)/dr, and s, , and sa , will be described in the next chapter IV. THE RADIATIVE TRANSFER EQUATION Mi c rowave understanding remote of the : s ensing problems interaction atmo s p h e r e and the earth's surface. of In require m i c rowave this an with the section these interactions will be studied. (3.7) Microwave Interaction w i t h the Atmosphere The total re s u l t i n g from differential the equal to the sum of change Equation radiative the (3.27) a n d (3.28), + dl^(emission) is known as S c h w a r z s c h i l d 1s equation transfer considered (the emission). For in energy a (3.47) (3.43) (I (3.48) intensity interaction of matter and radiation is dl y = d l y (extinction) dl in which lost by plane-parallel energy conservation exti n c t i o n reappears of is as horizontally homogeneous s c a t t e r i n g atmosphere the equation of radiative transfer R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 34 using dl v ,p (3.32) m a y be expressed as (t ,e ' ,<!>)' w VpCr.e,*) dT polarized and components v P (e ,<t>,e 1 ,<t>') 0 - I„(T,e' ,<t>') d (cos e ') d<J> * - where p=h or v; h 2vr '1 (T) (1 - denote wv (t )) B v (T(t )) horizontal respectively, the (3.49) and bar vertical denotes average over an ensemble of particles of the quantity, v the frequency of scattering albedo (B depth given by SCcl observations, w v (t) /B the .), t 6Xt is is the vertical the is single optical (3.30), B (T(t)) is the unpolarized thermal microwave emission given by (3.9), e and 4> are the nadir and azimuth angles of the radiometer is the diffuse radiance respectively. I (t ,e1,$1) averaged over both polarizations incident on the scattering volume from zenith angle azimuth angle e1 and P(e ,<J>,e ' ,$ 1) is the phase function of the particles that describe scattering from zenith angle e 1 to e and azimuth angle $ 1 to 4>. This can be approximated by P (e ,<t>,e ' ,<J>1) = 1 + 3g(T)(cos e cos e 1 + sin e sin e 1 cos w h e r e g(T) (<t>-<t>')} (3.50) is the as y m m e t r y factor of the phase function for s c a t t e r i n g particles. For the purpose of this work, tobe plane-parallel, the atmosphere is assumed horizontally homogeneous R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. and 35 vertically non-isotheraial. Therefore transfer of microwave radiation in the atmosphere will be azimuthally and the independent intensity will be only a function of optical depth and zenith angle. Using Eddington's streams of radiances (Liou, 1980) approximation of two- the intensity is expanded into its components as Iv (T,e ',<J>') = Iq (t) + I 1 (t) cos e' (3.51) where IQ and 3^ are the two components of the intensity (3.50) Substituting source term of 0 TT = w v (t) (3.49), yields [ Iq(t) + g(-r) and equation f i (T'e) of jj I^t) (3.52) cos e 1] (3.49) can be rewritten as =I(t,9) - w (t ) [ - {1 - w where in the scattering part of 1_ P (e ,<J>,e 1 ,$ 1) I (T.e1, ^ 1) d(cos e 1) d<t> n v 2 tt 4 (3.51) and . I0 (t) + 5(t) I ^ t ) » ] (t )> B{T( t )> (3.53) = cos e 1 a nd radiances are assumed to frequency and p o l a r i z a t i o n emission explicitly now and radiances, albedo, depend on functions (not shown explicitly now and h e r e a f t e r ) . Also the single scattering microwave be the hereafter). It albedo frequency may be and thermal (not shown noted that a s y mmetry factor and temperature of the R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 36 atmosphere all are functions of optical depth be transformed to another convenient vertical height (z) of the atmosphere. (t ), which can variable known as The transformation is given by d-r(z) = - B e x t (z) sec e dz t J" fie x t (z (z ) = - sec e (3.54) ) dz (3.55) where B g x t (z) is the total volume extinction coefficient of the atmospheric layer of thickness dz and at a height z from the earth surface. This total volume extinction coefficient is the sum of the coefficients of absorption due to water vapor, liquid water of clouds and of extinction due to rain. These wh i c h dominate frequencies. oxygen, are the main absorption constituents of the atmosphere a nd scattering at microwave Thus the total volume extinction coefficient at an altitude z is given by B e x t (Z) = “o. (Z) + “rH . O (Z) + “ c d ' 2 ’ + B r a i n |Z) where aQ is the oxygen absorption, absorption, clouds and B is ram the liquid Q water is the water vapor absorption frequency in the is the extinction coefficient of rain. All these absorption and extinction coefficients of (3'56) and the atmospheric are functions parameters pressure, temperature and humi d i t y at the height z. Inserting explicit (3.51) dependence and (3.54) into (3.53) and on altitude in the parameters for the R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. deleting 37 subsequent discussion, yields (1 - »> B e xtB(T) In order to separate 1^ (3.57), integrating and (3.57) I components of I in (3.57) w i t h respect to y dy and dy over -l<y<l, yields (3.58) dz dl 1 dz 3 B e x t (1 - w) where B . the extinction ext particles, The (3.59) (I0" B(T)) coefficient is c a lculated from of an ensemble of (3.56). bound a r y conditions are d e t e r m i n e d by the downward flux at the top of the cloud and u p w a r d flux at of the cloud as given by W u a nd W e i n m a n the bottom (1984). (3.60) and where R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 38 and €v are the horizontal and vertical emissivities of the surface and Tg is the surface temperature. Equation (3.58) and (3.59) are differential equations in IQ a nd I first order coupled w h i c h can be transformed into second order uncoupled differential equations for IQ and I 1 . d2lo ri r i __ dl 0 ° + B _ „ ( l - w g ) 5 - [=■ 1 ext w M ' dz , 5 - dz L BB e x -t (1 a " - w « g) g] dz L . 2 - = 3 B^x t (l - Z g) - (1 - Z) (3.63) ,2 . + B e x t (1 " “ »> Hz dz text'1 3 B e x t (1 " The Z) solutions equations (3.63) because the parameters B e x t » w height water. w) -1 d f {T) + 3 B e x t (1 ~ numerical differential ~ during the Therefore, phase equations w 5) 11 of these and (3.64) become unstable and g s e cond vary abrup t l y transitions from (3.58), (3.59) and using a finite difference m e t h o d order with ice to liquid numerically w i t h the boundary conditions given by (3.61) (3.64) are solved (3.60) and (Wu and Weinman, 1984) . Mu l t i p l y i n g (3.49) 2 2 by a factor C /kv and u s i n g and the relative component of given by (3.9), (3.11) one obtains R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 39 the expression terms of for the brightness horizontally equation of radiative transfer in temperature homogeneous and for a plane-parallel axially symmetric atmosphere whi c h is ^ TB v (t 'P) w = TB v (T'*J'p) " d T ~ (t ) fi P(T,y,y') - 2 ~ - (1 - where p denotes temperature, and the T the atmosphere at optical depth, T Dl/ and is wv (t )) T( t ) of ambient t (T,y,p) dy1 Bv polarization is T 0 (3.65) the brightness temperature . The brightness of the temperature, a function of frequency, optical depth, nadir angle polarization. The equation (3.65) m ay be directly integrated to solve for Tg^ r T Bv (t ,y ,p) = eT [ " | Jv (T',y,p) The constant of integration initial condition that is at j (3.66) calculated by e T dT1 + C C is t = 0 , the temperature is given by T _ ( 0 , y , p ) and therefore, T B v (T,y,p) = T g y (0,y,p) eT - eT J v (T',y,p) aV Rearranging T g y (0 , y » P) (3.67) = the brightness e “T 'd r ' (3.67) in the right order T b v (t 'jj,p) e T + Jo e_T ' dT 1 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. (3.68) 40 where J „(T'f*i,p> = (1 - wv (r')) T( t ') P( t ',p ,p ') Equation (t 1,p 1,p) dp (3.69) (3.68).states that at a point P, outside the medium the intensity of radiant energy is the sum of emissions all points upstream of P reduced by a factor e for extinction by the intervening medium. purposes, where radiometer, the equation —T * at to account For remote sensing up w e l l i n g radiances are detected by a (3.68) is simplified and rewritten in an appropriate form. The upward radiances of polarization p (generally h or v) received b y a radiometer at an altitude H, viewing at angle e from nadir is given by where T = exp dz' ] (3.71) exp dz'] dz exp d z 1 ] dz (3.72) (3.73) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. an 41 J(z,e) T s = {1 - w(z)}T(z)+w(z)[i(z)+g(z)I(z) and € (e) represent surface p respectively. equation that The (3.70) is three temperature terms represent on the (3.74) and emissivity the right hand side of (i) the emission the surface attenuated by the intervening atmosphere, (ii) the downward emission of the atmosphere, by cos e] surface, with attenuated in its atmosphere, r , and atmosphere, T . a will , that is reflected reflectivity upwelling (iii) The the surface by (1- path (€ by u p ward the (e)), intervening emission brightness and by the temperature is given by the product of its temperature and emissivity. The surface emissivity has been calculated in the next section. (3.8) Microwave Interaction w i t h the Ocean-Surface When microwave interface, and some Snell's equation. s ea of it by the air-sea is transmitted according to The transmitted radiation gets absorbed b y wa t e r because sea wat e r has a large imaginary part of the dielectric constant complex encounters it is partially r e f lected according to Fresnel's equations, the radiation at microwave Fresnel reflection coefficients (Kerker, frequencies. and are giv e n 1969) Rv = { cos e, - m cos e* } / { cos e, Rh = ( m cos e t - cos ea } / { m cos e, The + m cos es } (3.75) + cos e 2 } (3.76) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 42 w h e r e e, and e a respectively are as angles shown in of the incidence Figure and refraction (3.4) and m is the complex index of refraction of sea water which is given by the square root of its dielectric constant, k. whe r e m = m 1 - i m" (3.77) k - k 1 - i k" (3.78) m = (3.79) i=>T-l; -f k and m ! , k 1 are the real parts and - m " , k" are the imaginary parts dielectric of constant complex index respectively. The of refraction and angles are related t hrough Snell's law as Sin e, = m Sin e a (3.80) Reflection of microwaves from the ocean surface can understood in r e f lected waves components. The d enoted by the terms of S t o k e 's vectors of the incident and by finding the relation between subscript i and r respectively a nd I. (3.81) I (3.82) r their incident and reflected Stoke's vectors are d e s c r i b e d as l be R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. are \ ! / '/: '/ : /■■■ / / /• nil” ••<;:; \ ;: 5 e 3 : W 3 t S r • • •>.. • ■' ’ v . ■i • ■ 2 : ' i v! Nacrr Figure 3.4 Geometry cf reflection one refraction ai the air-sea interface R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 44 where * E. E. + IV IV E ihE ih 3 (3.83} * E. E. IV IV E ihE ih ] (3.84) * * (E. E., + E -. E . ah iv ' av lh (3.85) * <E ivE ih * E .. E . ah iv (3.86) = [ I + I , ] = C ' [ e E* + E ,E*. ] L rv rh J L rv rv rh rh J (3.87) * !. E.. ) i v I r r L •4c Q = U = 2 C 1 Re r lh ~ I V = -2C' r rv In «4c - I u 1= c 1 r E E - E . E . l rh J L rv rv rh rh J (3.88) v ' * * * (E E . ) = C 1 (E E . + E . E ) ' rv rh' ' rv rh rh rv' (3.89) 7 ' (E E* ) = i c 1 (E E . - E . E ) ' rv rh' ' rv rh rh rv' where C 1 ds the constant of proportionality EE* at the interface. (3.90) ' I between Re (E rh ) > Im are respectively. Re (E r v ) are related to the components of the incident radiation at the interface, and and The components of the electric-field vector of the reflected radiation at the interface, and ' real and R e ( E i v ) and imaginary parts ^E i h ^ ' of the where Ra quantity These components are related by R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 45 E rv (3.91) R v E iv (3.92) E rh “ R h E ih The relations between the components reflected (3.83) I Stoke's to vectors are of the determined incident from equations (3.92) which are expressed as =1. R R* + I..R.R* iv v v ah h h (3.93) Q = I . R R* - I .. R, R* ^r av v v ah h h (3.94) r U r = U i Re < R vR h > + V i Im V r ° V i Re ( RvR h ) - u i ^ Equations r Jr 1 and to (3. 93) ' ^r Ur V r R1 R2 R2 0 R1 0 0 0 (3.96) < RvR h > > (3.95) (3.96) can be 0 0 0 0 R3 R4 ui ~R 4 R3 vi r 11 (3.97) or t3.98) where R_ = R„ = (Rv Rv + R h R h )/2 (Rv Rv " R h R h )/2 R„ = Re (Rv R*) (3.99) (3.100) (3 .101) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 46 R 4 = Im The bar h as (Rv R*) been independent dropped quantities. (3.102) because The R^, and R^ are time reflection matrix [R] for de t e r m i n i n g the Stoke's vector of reflected radiation is the same for the radiation originating inside the water as it is for the radiation originating in the air so long of incidence l aw (3.80). incident Since the energy is intensities of 1^, r e f lected ra d iation must be radiation, I , and r of transmitted (ocean-surface) frequency that (3.103) according is also a good emitter to Kirchoff's law. The arrives at the top of the air-sea interface gets absor b e d b y a few centimeters of the top depth) of equal to the sum of the = Ir + 1 t good absorber at the same conserved, the intensity 1^. Ii A angles and refraction are those described by Snell's radiation, radiation, as layer (skin- the sea. This is then re-emitted from sea to air after bei n g abso r b e d by the skin-depth. The intensity of the ra d iation e mitted b y this top layer of the sea must be equal to the intensity of emission of a blackbody having the temperature same as that of the layer. At the air-sea interface, a portion of the r a d iation which arrives from below is reflected back into the sea and the rest is transmitted into air, and is given by It = I. - [ R ] I. = ( [ I ] - [ R ] ) I. R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (3.104) 47 where [I] is an identity 4x4 m a t r i x given by [13 1 0 0 0 = 0 1 0 0 0 0 1 0 0 0 0 1 (3.105) The transmission m a t r i x can be defined as [ T ] = [ I ] - [ R ] 1 - R, [ T ] = - R, (3.106) 0 0 0 - R2 1 " R1 0 0 1 - R, 0 0 (3.107) 0 1 - R and It = [ T ] Ii (3.108) The general transmission m a t r i x [T] may be used for ra d i a t i o n h a ving ar b i t r a r y polarization emerging from either side of the air-sea interface provided that the Fresnel's coefficients used in (3.102) (3.107) are those giv e n by (3.99) to and radiation travelling along the path dictated by the angles e, and e 2 given by (3.80). The Stoke's vector d e f i n e d for the intensity of thermal radiation emerging from b e l o w the air-sea interface is Id = and the { B(v,Ts ) , 0 , 0 , 0 } (3.109) transmitted thermal emission into the air from the sea is R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 48 (1 - R x) B ( v ,T g ) - R 0 B(v,T ) (3.110) s In a local thermodynamic equilibrium, emissivity water. the sea surface is derived from reflective properties of the sea The emissivity of a body is defined as the ratio of a polarized component of the radiation emerging from the body to that of a blackbody whose temperature is the same as that of the body. The sea surface emissivities of horizontal and vertical polarizations, eh and €v , respectively are defined as €h = 1 Rh Rh (3.111) = { It h <v.Ts ) } / { B h (v,Ts ) } e V = 1 - R R* V V (3.112) = { I. (V ,T ) } / { B (V ,T ) } tv ' S v s Expressing the various intensities in equivalent brightness temperatures given terms by corresponding microwave emission of a blackbody, vector for the temperature, Tg B where the of their (3.11) the and Stokes emission of the sea in terms of brightness , can be expressed as < <t b v + t b h >' vertical and <t b v - t b h >' °' horizontal ° > components (3.113) of the brightness temperature of the sea are given by T = € T aBV v s (3.114) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. These represent components the of thermal ocean brightness emission of the ocean given by the first term in the right hand side of m ult i p l i e d temperature (3.70), which is by a factor r the so called transmittance of the under lying atmosphere. Thus the surface contribution to the total radiances received by a radiometer is the transmittance calculated from product and radiative by (3.115). The polarization of radiances enter the transfer w h i c h involves the specific of (3.71) and the corresponding component of thermal emission of the ocean surface given (3.114) then calculations calculation polarization. This through the surface term, of surface calculation emissivity of emissivity will be dealt w i thin the next chapter. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. of surface CHAPTER (IV) MICROWAVE EMISSION AND A B S O R P T I O N BY THE SURFACE AND THE A T MOSPHERE Theoretical the formulation of m i c rowave interaction with ocean and the atmosphere was considered in Chapter III. However the numerical c a l c ulations of the parameters used in the theory was deferred. will be performed In this chapter these using specific ocean calculations surface and atmospheric models. (4.1) Microwave Emission of the Model Ocean Surface M icrowave emission of the ocean surface depenus on e m issivity surface and temperature. (specular surface) The its emissivity of a calm sea is a function of its dielectric constant which in turn is a function of sea surface salinity and t e m p e r a t u r e , (A) Dielectric Constant of Sea Water The theory of dielectric constant of pure liquid has b e e n derived by Debye in 1929. The real water (K1) and the R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 51 imaginary (K") parts of the dielectric constant (K) are given by K = K + ---------------------------- (4.1) (1 + w t ) and qt (K - K ) K" = ------- §---------------------------- (4.2) (1 + w wh e r e K is s dielectric the constant to infinity and static static t T ) dielectric constant, as the angular frequency is the r e l a x a t i o n time de termined form is « the (u=2IIv) tends (sec). Both the dielectric constant and the relaxation time for pure w a t e r are functions of temperature, water. K value of the T. term The is 4.9 for pure liquid W h e n pure water is repla c e d b y sea of the dependence may m o d i f i c a t i o n in the form of k" e x p e r i mentally be water, a ssumed with 0- (4.3) w €q because the conductivity of impure water the impurity. conductivity of the medium temperature, T, and salinity, slight + The second term on the right hand side of with a same : U T (K C - K tf ) K" = ------3 v - - (1 + increases the which S,and is added (containing Here <r is a c.^ (4.3) is is the function salt) ionic of the perm i t t i v i t y R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 52 of vacuum represents (8.854x10 the -12 damping Farad/meter). of The We ppt have chosen the values of K , s on the measurements made by Saxton and these measurements Hollinger, <t / u € q the electric field vector. Sea w at e r salinity is usually between 30 to 40 thousand) . term Lane (parts per and <r based (1952) . Using t (1970) obtained the following empirical equations; Kg = [88.0 - 4.339X10-01 S + 1.71xl0_° 3 S2 - 4.035xl0_01 T + 8.065X10-04 T 2 + 6.170X10-03 S T - 8.910xl0-05 S 2 T - 6.934X10-05 S T 2 + 1.439X10-06 S2 T 2 ] T = (4.4) [18.70 - 7.924X10-02 S + 6.35xl0“°4 S2 - 5.489xl0_01 T + 5.758X10-03 T 2 + 1.889X10-03 S T - - 5.299x10 -07 ST 2 -07 2 - 2.101x10 u/ S^ T 7.209xl0-06 S2 T 2 -12 ] 10 ^ (4.5) or = [7.788xl0~03 S - 1.672xl0_06 S2 - 8.570xl0_15 T + 2.996 Xl0-16 T 2 + 4.059X10-04 S T - 3.215xl0-06 S 2 T - 1.423 X10-06 S T 2 + 3.229X10-08 S 2 T 2 ] 1011 where the frequency, parts per thousands v, is in hertz, (4.6) the salinity, and the temperature, S, is in T, is in degrees centigrade. (B) Microwave Emission by a Calm Sea Microwave emission of a calm sea can be the theory understood by developed in chapter III for the reflection and transmission of electromagnetic radiation by an air-sea R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 53 interface. sea The polarized components of microwave emission of are given by the sea surface emissivity of (3.114) and (3.115) whi c h are dependent on temperature the ocean. a nd horizontal or vertical The s ea surface emissivities are functions of Fresnel's coefficients whi c h in turn depend on the index of refraction of sea water and the zenith angle of the transmitted radiation. The complex index of refraction of sea water is a function of its temperature, mi c rowave frequency. The horizontal emissivities of sDecular sea surface, ca lculated using Fresnel's e ., sh equations salinity and e vertical have sv which and relate been the specular emissivities to the dielectric constant as given by €gk = 1 - [{Cos e - >r(K-Sin2 e)}/{Cos e + -J"(K-Sin2 e )} ] 2 (4.7) and €sv = 1 - [{K Cos e - >T(K-Sin2 e)}/{K Cos e + >T(K-Sin2 e )} ] 2 The complex dielectric constant, the relation given by (4.1) to K, (4.8) is calculated from (4.6) and e is the zenith angle. (C) The Effect of Sea Surface W i n d and Foam The affected ocean emissivity at all microwave frequencies by the sea foam and surface waves is which are R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 54 g e n e r a t e d by the surface winds. This effect can be explained in terms of the surface distribution of the slopes of the whi c h is a function of sea surface temperature, salinity, wind stress, fetch and duration of wind. of microwave wavelength, surface radiation. order the surface can not be treated as a (specular surface) for that w a velength of The exact effect of surface winds and foam on the s ea microwave emission is not known so far because too environmental conditions d u r a t i o n of w i n d speed, have not sea Since the radius of curvature of any sea surface w a v e is of the smooth sea been such as atmospheric sea temperature, and many stability, sea salinity adequately known c o n c urrently over an ocean surface. The d i s t ribution of the sea slopes suggested by Cox a nd M u n k (1954a ,''54b) was u s e d in the study of microwave e miss i o n by the ;,ea of Stogryn (1967). The effect of ocean surface roughness and foam is taken into account by coupling the expressions of specular sea s urface emissivities w i t h the empirical by Pandey and Kakar p o l a r i z e d emissivity, (1982) as e (e), of the relations modified rough by sea obtained Kakar. The surface is hr given b y «p(e) - « sp + 4 € p (4.9) where p = h or v This can be d e composed into two parts the first part is the emissivity due to the foamless rough water surface, R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 55 r e p r e sented by € , and the s e cond part is due to the foam * Jr covered sea surface, denoted by € fp / ant* the total polarized emis s i v i t y can be wr i t t e n as € p ( e) whe r e = (1 - F) €rp + F € fp (4.10) F is the fractional foam cover whi c h ranges from zero for foam free surface to one for 100% foam cover. The foamless rough sea water e missivity is considered to be e n = rp Subs t i t u t i n g e^ + A € sp rp (4.11) into (4.11) (4.10) and comparing with (4.9), yields A£p = A£rp + F <£ f p - £ rp> Equation (4.12) e m issivity is states due to that two the '4 ' 12> total reasons. in the The first of which is change in the foamless rough sea w a t e r second change e m issivity and the being the difference of the foam emissivity from the rough wat e r emissivity. This difference is superimposed only over the foam covered area. The change in two polarized components of foamless rough sea surface emissivities are given by A £ rh = ( W ^ / T g ) (!• 15xl0_01 + 3.80X10-05 e 2 ) A€rv = (WV/T (4.13) and ) (1.17x10 -01 - 2.09x10 E x p ( 7 . 32x10 whe r e -02 -09 ©)) (4.14) W is the surface wind speed in m/s, t emperature in degree kelvins, a nd v is Tg is the surface the frequency R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. in 56 GHz. The two polarized components of foam emissivity are giv e n by €fh = {(208+1.29v)/Tg } (1 - 1.748X10-03 0 - 7.336X10-05 0 2 — + 1.044x10 efv = 07 ^ ) 0 ( (208+1.29v)/Ts ) (1 - 9.946 X10 —04 9 + 3.218x10 - 1.187X10-06 e 3 The (4.15) fraction of foam — 0*1 + 7.Oxio-20 cover empirical relation obtained by Wu, is 9 6 0 10) calculated and Fung the relations (4.7) to from an (1972) F = 7.751X10-06 W 3 *231 Using (4.16) (4.17) (4.17), both polarized components of rough sea water emissivity can be obtained. The graphs of horizontal and vertical and €v of ocean surface verses frequency ten zenith angles 0, 30, d e gree are plotted 35, 40, in Figures sp e e d v arying from 0 to 25 a nd at a salinity surface 45, (4.1) 50, to (1 to 300 GHz) at 55, 85 60, 65, for the wind (m/s) w i t h a interval of 5 temperature 300 and (m/s) a fixed value of (35 0 /0 0 ). These Figures show that the emissivities temperature. At the values than nadir, linear functions nadir looking case, e=0, both the horizontal and vertical components Their €, n (4.6) are non-linear functions of frequency but of emissivities and are equal. increase w i t h the w i n d speed. At angles other increases and ev decreases non linearly with R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. OCEAN EMISSIVITY VS FREQUENCY AND ANGLES SAUNITY=35.,SURF.TEMP.=300.0,WIND = .0 1.0 .9 ,8 E> CO tn s w .7 < y 6 L - 5 0 .0 V a ca > .5 Q Z < ►J .4 Z o K o W N M .3 .2 1 0 0 20 40 60 80 100 120 140 160 1B0 200 220 240 260 280 300 FREQUENCY (GHZ) USING S.T. WU MODEL FOR FOAM Figure (4.1) R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. OCEAN EMISSIVITY VS FREQUENCY AND ANGLES SALLNTTY=35.,SURF. TEMP.=300.0,WIND = 5.0 1.0 tn tn L - SO.0 V 2 w ►j < u H K W R - 6 5 .0 V > 1 - 6 5 .0 Q 2 < iJ < E- 2 O N M K O X 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 FREQUENCY (GHZ) USING S.T. WU MODEL FOR FOAM Figure (4.2) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. V OCEAN EMISSIVITY VS FREQUENCY AND ANGLES SALINITY=35.,SURF.TEMP.=300.0, WIND =10.0 1.0 .9 ■J*; C - 3 0 .0 M .8 >• E- > tn co .7 i—i s w < .6 > .5 u H K Cd Q Z <! .4 <! .4 EZ o N 5 O K .3 .2 .1 0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 FREQUENCY (GHZ) USING S.T. WU MODEL FOR FOAM Figure (4.3) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 60 OCEAN EMISSIVITY VS FREQUENCY AND ANGLES SALINITY=35.,3URF.TEMP.=300.0, WIND =15.0 EMISSIVITY 1.0 VERTICAL L- 60.0 7 HORIZONTAL AND 0 * 6 5 .0 H 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280"300 FREQUENCY (GHZ) USING S.T. WU MODEL FOR FOAM Figure (4.4) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 61 OCEAN EMISSIVITY VS FREQUENCY AND ANGLES SALINITY=35.,SURF.TEMP.=300.01WIND =20.0 1.0 EMISSIVITY C- 30.0 H 0- 30.0 V I- -4S.0 H L- 60.0 V VERTICAL N- 66.0 V 0- 65.0 H R- 65.0 V HORIZONTAL AND 65.0 V FREQUENCY (GHZ) USING S.T. WU MODEL FOR FOAM Figure (4.5) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. OCEAN EMISSIVITY VS FREQUENCY AND ANGLES SALINITY=35.,SURF.TEMP.=300.0,WIND =25.0 1.0 ANGLES L - 50.0 V 3 w 5S .0 V o - so.o h U H w ;.o v > Q $ Z o N M K O X o 20 40 80 80 100 120 140 160 180 200 220 240 260 280 300 FREQUENCY (GHZ) USING S.T. WU UODEL FOR FOAM Figure (4.6) R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. 63 w i n d speed whereas difference of both decrease with temperature. vertical and horizontal emissivities The (e -€. ) ' v h' increases with angle at all frequencies but at higher angles like 85° speed, this difference starts to decrease. ev and values of wind both get saturated at higher frequencies. The behaviour of the difference the At higher the ocean (sv _€h) was analysed from emissivity calculated from this model. The Quantity (e -€. ) decreases with 1 v h' wind SDeed and increases with temperature for all frequencies and angles up to 65 degrees. (4.2) Microwaves Absorption by Atmospheric Gases Oxygen microwaves do not and water absorption scatter vapor significantly in the atmosphere. microwave r a d iation dominate Since these gases significantly, the emissive properties are directly related to their absorption properties, the v o lume absorption coefficients of the gases are to be calculated as functions of the variables of state. (A) Oxygen A b sorption Coefficient There is a complex band of lines in stron g l y absorbing oxygen the atmosphere centered around 60 GHz. The oxygen molec u l e has a permanent magnetic dipole moment. The changes R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. 64 in the orientation of the electronic spin orientation magnetic of the dipole transition at molecular transitions relative near 60 GHz and with and angular momentum J = N frequencies at (4.1). zero oxygen vN+ In addition, by b r oadening there is using Rosenkranz the w o r k of Gorden (1975)), of states a b sorption and are listed in non-resonant absorption Microwave absorption coefficient due to calculated s u g gested line (Rosenkranz frequency. is single J = N ± 1. The transitions between these states permit resonant Table a 118.75 GHz. Coupl i n g of electronic spin w i t h triplet at the rotation produces a band of the rotational angular momentum forms a total to (1967) due the computational scheme (1975) whi c h relies principally on and Van Vleck (1947). Pressure to interactions among atmospheric gas m o l ecules and temperature dependence of this line broadening are taken into account relat i n g to the means of R o s e n k r a n z 1s o x y g e n absorption coefficient frequency temperature by v T (GHz), pressure P a formula (Neper/Km) (millibars), and ( K ) . The a bsorption coefficient is thus given by cr0 j (v) = (CP2 v 2 /T2 )(Z n +N [fJ(v) + f N (_v) + f N (v) + fN (-v)] + where C = 0.330 is (.70 W b /tv2 + a (PWb )2 ])} constant. N is (4.18) the fractional p opu l a t i o n of state N at temperature T R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 65 <t>N = { ( 2 N + D / 0 . 7 2 5 T) exp (-2 .0685 N(N+1)/T) (4.19) The shape factor for the transition line is given by f*(W = <WH < 4 )2+(‘'-V H t )2-Y N ) / <<v - v n ± )2+(PWn )2 > where (4-20) vN+ are the resonant frequencies given in Table (4.1) a nd the summation is over odd rotational states N from 1 to 39. The amplitudes of positive and negative transition lines at v* and v N are given by d* = [N(2 N + 3 ) / (N+l)(2 N + 1 )]° ’5 (4.21) d^ = [(N+l)(2N-1)/N(2N+1)]0,5 (4.22) The non-resonant line width is given by W b = 0.48X10-03 (4.23) (300/T)° *89 GHz/mb and the resonant line broadening half width is computed from W N = 1.16X10-03 The interference (4.24) (300/T)0 '85 GHz/mb coefficient for coupling between near states due to molecular collisions is given by Y N = dN ^ 2dN+2 “ W N (up)^/{v,N “ VN+2* + {2dl-2 V where the line dn)}/{vN - v * _ 2 ) " <W b /V N } " {Wb /(VN + 6 0 *>>J (4.25) widths coupling for the collisional are computed in a sequence from the relations W N (dn) = W b ~ W N " W N (up) (4*26) W N - 2 (UP) = W N (dn) (4.27) and Microwave absorption (4’n /4>N - 2 ) above 40 Km was considered R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. TABLE (4.1) R e s o n a n t F r e q u e n c i e s of M o l e c u l a r Ox y g e n a n d A m p l i t u d e Frequencies N + VN (GHZ) ,'n Factors Amplitude Factors dN dN 1 56.2648 118.7503 0.9129 0.8165 3 58.4466 62.4863 0. 9 8 2 0 0.9759 5 59.5910 60 . 3 0 6 1 0.9924 0.9909 7 60.4348 5 0 .1642 0. 9 9 5 8 0.9952 9 61.1506 58.3239 0.9974 0.9971 11 61.8002 57.6125 0. 9 9 8 2 0.9980 13 62.4112 5 6 . 9682 0. 9 9 8 7 0.9986 15 62.9980 56.3634 0.9990 0.9989 17 63.5685 5 5 . 7838 0. 9 9 9 2 0.9992 19 64.1278 55.2214 0 .9994 0.9993 21 64.6789 5 4 . 6711 0 .9995 0.9994 23 65.2241 54 . 1300 0 .9996 0.9995 25 65.7647 53.5957 0.9996 0.9996 27 66.3020 53.0668 0.9997 0.9997 29 66.8367 5 2 . 5422 0 .9997 0.9997 31 67.3694 5 2 . 0212 0.9998 0.9997 33 67.9007 51 . 5 0 3 0 0. 9 9 9 8 0.9998 35 68.4308 50 . 9873 0.9998 0.9998 37 68.9601 50.4736 0.9998 0.9998 39 69.4887 49 .9618 0.9998 0.9998 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 67 negligible. Zeeman splitting effects were ne g l e c t e d below this altitude where the temperatures are too low. The integrated atmosphere (Table frequencies oxygen absorption for a representative 4.2) between has 1 been to 300 calculated GHz. The for all variation integrated oxygen absorption versus frequency is shown by solid curve in Figure (4.7). absorption between 50 and 70 GHz, GHz. The frequencies This curve and between shows 115 of a strong and 12 2 be t w e e n 50 and 70 GHz have different we ightings of oxygen abso r p t i o n at different altitude in the atmosphere. The concentration w e ightings and the are determined by oxygen temperature dependance of the cross section. (B) Water Vapor Absorption Coefficient The water molecule dipole moment which region. It has a single, possesses a permanent electric causes an absorption in the microwave weak, pure rotational resonance line w h i c h arises from the transition 5_ -6,_ at 22.235 GHz. 23 lo The next resonance frequency is at 183.3 GHz. The resonance frequency 22.235 GHz has been extensively studied by Becker and Autler (1946) and Townes and Shallow (1955). For the frequencies below 170 GHz ab sorption has been the water given b y Barrett and Chung vapor (1962) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. and 68 TA3LE (4.2) Atsosphere Model after Cole et al. (1355, p.2-3) -I- Height (Ka> Pressure (3b) 3.0 1313.25 -I- TeisperaturelDew Point Illat. Vapor ( K) ITeap. ( K) ICen.(gn/nf’) ------------------------------ T ------------------------------- J ------------------------------- 330.4 I ------------------ 3.5543 353.3 237.1 ------------------ 1.3324 300.3 234.4 ------------------ 1•\iLLu 353.3 231.5 235.5 I 232.4 j I 283.7 833.8 283.' .5 753.3 235.3 I 237.3 I -I- I 5.5224 I 275.4 »------- 1----278.2 I 263.7 I 3.6835 ----- j--------1----231.3 3.7611 3 I 11.7712 -I- - - - I 3.3135 -I------ I 1 3.1532 1 I 13.8641 J------------------- T------------------- 1 2.5331 I 15.2322 ------------------- ------------------- -I 2.0373 I 13.4353 1------------------- j------------------ 3 « 533. C ,1 * A** I I T '“t £/ *t• 4 1 iQ't.i 1 i..7u£u inrift -------- T----------- 1-------- 18 273.3 11 .3 6.6542 12 450.3 1I 257.2 1- 2.3 I !- 1-------------- 243.7 253.3 1--- 13 7.5545 14 8.5434 OCC ^ 403.3 i.Jj.4 243.3 9.6432 330.0 1 u,533B -I- T i 0 ot'Anrc TuOO 1 ftj.SA T 1---------I-------- 1 237.8 1--- 15 I 1.4112 !- - - - I 3.3341 241.7 I 233.2 I 0.2337 T _ "i------- i i/iiibJ 1------------------ 1----------- 16 18.3168 233.1 250.3 I 213.2 I 3.3353 1------------------ 1----------- 17 f n 10 12.4354 I 1 4 .2 2 5 1 233.3 I 1 5 3 .3 222.7 I 153.3 I4.3174E-33 - - - - - 1- - - - - - - j- - - I I 1 •-VOQ •”> L U i.L co n iJ O .U 1 3 . 12 G 2 E -C 3 |C ,1 o I \J«J « O I3 .4 S 3 3 E -G 3 « en .■» l diii'iJ Tc ♦ •*n n r nn i O « 't O O c C ” U O T 19 t 1 «r f n « ■> iC iC iO li 23 I 2 3 .6 1 3 4 — ? 1 1 3 3 .0 I 1 9 6 .2 1 33.e I 195.2 -T I I -I- R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. too WATER VAPOR ABSORPTION 90 OXYGEN ABSORPTION 80 70 i — z U HJ O M t i 50 ll. lii o o 2 40 o M J 0. 0 1 o to CD < 20 O O — O O O O O O O O Q C s J C 0 r i / 3 ( D h - C 0 0 ) O o o o o o o o o o o o o o o o o o o o -cvjcoTiiiior^coojo- c j c o ’T i n t o r ^ c o o ) C 'J C M O J C M C v J C V J C N J C 'J 300 to FREOUENCY CGHZ> Figure (4.7) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 70 u s e d by Staelin 1-centimeter (1966) to interpret the microwave data wavelength. The water vapor near absorption at microwave frequency below 170 GHz is a„ _(v) = ,2P105 [(3.24x 10-04/T3 ‘125) n2U e x p (-644./T)(P+0.147PT) ( 1 / { ( v - v 0 )2 + a v 2 > + 1 / { ( v + v 0 )2 + a v 2 } ) +2.55 x 10_ 0 8 (a v /T1 *5 ) ] where water Nepers/Km vapor at a absorption frequency v (4.28) coefficient GHz, P is aH Q T is the temperature in degree kelvin, v 0 resonant frequency GHz and p in the pressure in millibar, 22.235 is is the is the water vapor density in gm/m3 . The line width a v is given by & v = 2 .58x l 0 ~ ° 3 [P+0.0 1 4 7 P T ] ( T / 3 1 8 )“ ° ’625 (4.29) Water vapor absorption for all microwave frequencies as described below w as found by Kakar (Kakar, R.K.; personal communication 1986). 0 (v ) = [Resonance+{(1+n)/n)10 9 p1 , 1 5 (P/P0 ) (31 8 / T ) (a+b)V 2 ]103 (4.30) where a = 1.5 P Q = 1013.25 n = L (mb) (number of terms between L, and L a ) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 71 b = A d Resonance = TXP "2 A it A O ■— ■- — -a (L) / n 2 ,, ' ■ ---- SP(L) T, Z_a Li 2 (v S T ( L ) DEL FACT vR (L) i 2 - vR (L) ) + 4 2 v"1 DEL (4.31) 2 Del = C 1 (L)(P/P0 )(318/T)TXP(L) (1 + C 2 (L)(PT/P)) If {(1.43879/T) (4.32) (VR (L)/29.9793) < 0.1} then Fact = V_.(L) (1.43879/(30 T) } K exp ( - l . 43879 TE1(L)/T) (4.33) otherwise Fact = exp(-TE1(L) 1.43879/T) - e x p [-(TE1(L)+v_(L)/29.9793) K (1.43879/T)] (4.34) The coefficients used in the equations (4.30) are given in Table (4.3) where L, to (4.34) and L2 are calculated from the relations given by L, = L - 3 (1 + P / P 0 ) (4.35) La = L + 3 (1 + P / P 0 ) (4.36) where L defines the number of terms in the given arrangement satisfying the condition that the frequency of v, is less frequency, than or equal a particular resonance v D , w h e n the resonant frequencies are arranged in £\ a monotonically increasing (4.3). to observation, When L, order as shown in the Table is less than 1 it is considered 1 and if L 2 is greater than 9 it is considered 9. The integrated R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. water 72 vapor abso r p t i o n for a representative atmosphere (Table 4.1) has b e e n calculated at all frequencies 1 between and 300 G H z . The v a r iation of this integrated water vapor absorption with frequency is shown in Figure (4.7) by a dashed curve. The total a b sorption of microwaves by the atmospheric is d o m inated b y water vapor over a large range of microwave frequencies. atmosphere The is distribution highly variable of water quantity interest for meterological research. study this quant i t y then surface ph e n o m e n a frequencies w h i c h resonant bands. 22.235 one are and and If one On the would far away other require from Staelin et a l . (1976), GHz vapor and Duff 31.4 would the of most like to hand to to study choose water and Grody the vapor (1976) have GHz to infer total water vapor over oceans. (1979) derived the total liquid water content of the atmosphere from Nimbus-6 Spectrometer) is these content of the atmosphere u s i n g Nimbus-5 data Liou in one must select the frequencies near 22.235 GHz or 183.3 GHz. used gases SCAMS (Scanning Microwave data over land. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. CO CM rH rH c-« rH CD O CM CD O CO to • o O CO CO • o o CD CM o to CO CM CO CO O o O CM O CM O rH to to rH CM CM CO m CM . CM in CO CM CM o • CM O o in • O rH rH to m o in ** • • rH CM O O • CO o rH G) rH CO to rH • to tD CM CO • o CM O CO CO o C"« r* • T-t | ' CM O (4.3) rH m -— . TABLE rH rH tl ■ — ■* DU x Eh o rH K 10 1 — rH CM • in CM in rH w Eh J CO r• to to CO • CO CO CO CO ■ CM CO CM rH 10 o» • CM rH o S' — ■ Oh w o • CO ,— .. cn . #5 N ^ « to '— * to to o 0) • ** CM to CD • rH rH o CO CO « CM ID CD CM • CM • CM •M* CO • CM o O o O • rH t^ CO • rH CO CD CO CO o • o rH CO iO CM CO CM rH to to o • o o rH o o rH • CO CO CD o LO o • CM CO CO CD C** o • o rH CM • o o CO to CO CM • CM to • CM o rH H -— to . rH rH • rH • o rH • CM • CO CO o • CM to o rH • CM O • rH CO o • to o m CM o in CO CM CM o . to r*« cn • rH • rH CM CO CM CM • • CO • O rH in « . o • CO CO CO rH • CM O . CO CO o • o CO CD O • o CO o • o CO o in o • o CO CM UO to CO CD to m o LQ rH o CO rH rH rH • rH o o o rH • rH CD to CO • rH • CO rH to o o CO CM CO CM tH CO CO CO CD CO CM CO to to • to o . in • CO . CO • rH • CD CO CO CO cn rH • o . CO o rH R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 74 (4.3) Microwave Extinction by Liquid Hydrometeors The liquid water drops in clouds and rain are known hydrometeors which absorb radiation. In 1908, Mie scattering properties and scatter described thermal microwave the absorption and of a homogeneous spherical particle. His theory may be used to calculate and as the volume absorption scattering coefficients of a polydispersive medium like rain and clouds if the particle size distribution is Since extinction particles are individual processes randomly extinction are linear distributed coefficient in can in the nature and medium, the be summed over the particle size distribution in order to calculate volume extinction coefficient calculation assumes that distances the of particles known. the are the total medium. This separated by large compared to their respective diameters. The extinction properties of cloud particles smaller than 0.1 mm in diameter are determined us i n g the Rayleigh in approximation which the scattering processes are neglected compared to the absorption processes. For this case Goldstein (1951) has derived absorption a coefficient content. calculate simple of a expression cloud Gunn and East for the volume as a function of its liquid water (1954) have applied Mie theory the attenuation of microwaves by hydrometeors. to In all of these studies regression equations have been obtained R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 75 relat i n g the atmospheric volume e x tinction parameters. properties considered these of to coefficient medium the the calculate as (S.S.), the have not been In this study exact Mie the volume extinction us i n g three different particle size distributions for the medium. known medium calculations. theory has been used of to The distinction between absorption and scattering in coefficient (i) Marshall Palmer These (M.P.), distributions are (ii) Sekhon Srivastava and (iii) Gamma distribution. (A) Absorption Coefficient due to non-raining Clouds Clouds are assumed to be a collection dielectric spheres (me t e r ) . These described by per unit of pure spheres dn(a)/da volume, N(a), are liquid of water distributed homogeneous, of in —4 (m ). The total number between diameters 0 to a N(a) - I ? f i ,a) diameter a the of particles is given b y da The extinction cross-section, medium ‘ 4 - 37> o ^ e x ^.ia ) (m ) , of a drop of diameter a is the sum of its absorption cross-section, o o’ t^(a) (m )> and its scatt e r i n g cross-section, cr (a) aos sea (m2 ) . " e x t 131 = In a tra b s (a> + non-rainin g r sca'a > cloud whose maximum <4 '3 8 > drop diameter, R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 76 amax' *s of t*ie order of -0001 (m) , the scattering cross- s ection is negligible compared to the absorption coefficient w h i c h is given by cra b s (a) = w he r e (ir2v a 3 /C) Im [ (1-m2 )/ (2+m2 ) ] (4.39) Im represents the imaginary part of the bracket term, m is the complex index of r e fraction of pure water, s pe e d of light, r efractive 3.79) and and v is the index for frequency (4.4-4.6). radiation. pure wat e r is calculated from (4.1-4.6) where the salinity, to zero in of C is the The (3.78- S, was as s u m e d equal Therefore the refractive index of pure w at e r is a function of frequency a nd temperature. Thus the volume abso r p t i o n coefficient of a non-raining cloud is given by amax Q “ cld S u b s tituting (4.39) , . dn(a) , abs( di da into a cld = (4.40) (6Trv/Pc > . ... (4‘40) and simplifying one obtains [ (1-m2 )/ ( 2 + m 2 )] M c Im (4.41) where M famax c = J 0— tt g p 3 a N(a) __ (4.42) da is the mass of liquid water content of the v o lume 3 (gm/m ) and Si m p l i f y i n g the P is the e x pression final per of imaginary part of and expression evaluating for the unit 3 (gm/m ). the d ensity of liquid water terms of dielectric constant factor, cloud (4.41) in constant absorption coefficient R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 77 (Np/Kra) due to n on-raining clouds is given by aC where K 1 j.Vui and dielectric = 0.0629 K" the constant [3K'/ { ( 2 + K 1)2+ K " 2 }] real of a nd pure (4.2) and M c is calculated from M imaginary C (4.43) v parts wat e r are given by of the (4.1) and (4.42). (B) Extinction Cross-section of a Rain-drop In the study of theoretical absorption p roperties of atmosphere, two important facts should B eca use a re of polydispersive media, and such be as rain in the considered. the finite size of the particles, assumed characteristics spherical, can not be the (i) even if these angular approximated scattering scattering w i t h sufficient a c c u r a c y by asymptotic e xpressions based on geometric optics or Green's function approximations for the internal field. T he complete Mie-series must be u s e d for each particle, T he (ii) size distribution function u s e d for computing the total extin c t i o n cross-section must be v e r y close to the real size spect r u m of the particles in the medium. Mie-theory, as a Before applying the it is convenient to define few drop size parameter, Q e x t * a bsorption e fficiency e f f i c i e n c y factor, Q parameters such X, e xtinction efficiency factor, factor, Qabs ' and scattering , which are related as follows: SCcl R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 78 X = c ircumference of the sphere w a velength of the radiation ext <ext tt r tt 0 , xabs w h e r e r is the ^ X r =0 - ext radius S ^ 1 (2n+1) 0 (1957) scattering sec 6 9.22) s a n = m X J Yn [ Am i z,+ _ sX_ ]J r m A (z ) n = m z = m X = tt w h e r e m is the complex diameter a (m). -i X -j J J tz; — + — m a v and (Abramowitz and Stegun v „ ? the (Van- (X) - Y n _ 1 (X) (4.48) V x) i (X) n4 ' (4.49) S , (X) n - 1' ' (1972) index of sphere of a are Riccati Bessel functions sec 6 10.3) (X) = X j (X) 1& is X (4.50) refractive Yn sphere, n [ [ m a (4.46) as follows: L L b |bn |2 > an and bn are given [ _ V 2)+ — n_ ] T L (4.45) (4.47) w a v e l e n g t h of incident plane wave, de Hulst (|an |2+ (an +bn ) " “ sea of 44> X S ^ 1 (2n+1) Re ” ‘ X“ sea sea _ 2 Trr (4.51) 11 ?n (X) = T n (X) + ixn (X) (4.52) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 79 where x n (X) = - X y n (X) Jn (x) and Y n (x) are (4.53) spherical Bessel initial two terms of these functions are The given by (X) = Sin (X) (4.54) x Q (X) = Cos (X) (4.55) y q All functions. Y 1 (X) = (1/X) y Q (X) - x Q (X) (4.56) x ^(X) = (1/X) x Q (X) + y Q (X) (4.57) the higher terms of the functions are calculated using the recurrence relations given as fn + l (X) = C ( 2 n + D / X ] ?n (X) - fn _ 1 (X) (4.58) T [Y (X) X n n+1 (4.59) .(X) = n+1 (X) The logarithmic derivative function, An , used in (4.48) and -1 ] of Yn /x(X) n is (4.49) prime value of denotes the significant (4.60) the derivative w i t h respect to Z. The function A n isobtained digits using Lentz method. N, used in the Mie- series theoretical grounds a and is given by An (z) = Y n (z) / Y n (z) where denoted by (4.45) depends on and to at least 5 The number of terms, (4.46), calculated on the size parameter in the following manner 1 + X + N = 2 + X + 2 + X + Once the 4 X 1/3 4.05 X 1/3 4 X 1/3 if .02 < X < 8 if 8 < X < 4200 (4.61) if 4200 < X < 20000 extinction and scattering cross-sections, cr ext R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 80 and o- S C a . are known from e xtinction 13 __ SCa and scattering (per unit length) , these (4.45) coefficients (4.46) a nd coefficients may over be the then the total of rain, obtained by complete range and integrating of the size distribution = l a " 3"1 <re x t (a-” >v)di | a) min da (4’62) Bs c a (m' v) = Ja” 3* <rs c a ( a -m' * ' )dM a> da (4 ' 63> m m where am . and m m a are max the mi n i m u m and diameters of rain drops reaching the surface. and scattering diameter, index of a, cross-sections frequency of refraction of are radiation, The extinction and of the drop complex pure water, m, w h i c h in turn, are functions of temperature, S C a maximum functions v, function of temperature and frequency. B the Therefore, frequency, B is a . and and drop size distribution. The size determined by of precipitation by the p rocesses of collision humid i t y prevail in the atmosphere distance before is p a rtly the strength of the up draught producing the cloud and The particles in the subcloud and coalescence layer. among drops also influence the particle The that size. w h i c h a drop can fall through unsaturated air complectly e v a p o rating increases rapidly R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. with 31 i ncresing drop size; in an atmosphere of 90% humidity drops of 100 pm and 1 m m would fall respectively. at least (Mason 150 m and 40 km Since the bases of dense clouds generally lie a few hundred meters above the ground a radius of 100 ym may be regarded as a lower precip i t a t i o n elements. a The (1975)) . = .0001 man max i m u m drop ( m m/h), and was for the size of Hence (m) (4.64) diameter given limit by depends St e p h e n on the (1962) rain rate, from R empirical studies to be o 913 amax = ° - 0023 R The rainfall coefficient" (m) rate, calculations m a x i m u m drop diameter R, through (4.65) enters the the extinction definition of the (4.65). (C) Drop Size Distributions of Rain T he distribution most commonly used in of rainfall (1948) study is that giv e n (here after called M.P. exponential of drop different the literature by Marshall and Palmer distribution). This has an form obtained e m p i r ically from the measurements diameters at the surface for intensities and time durations. steady rains Mathematically it is e xpressed by dn(a)/da = NQ exp(-^a) of (4.66) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 82 The quantity dn(a)/da is the d iameter (m —4 a rate, R of drops (m) and 8.0 the x 10 6 —4 (m ), variation of ^ a is the b e e n c a lculated from and rainfall (4.67) SMMR (4.62) u s i n g M.P. frequencies distribution given by rates are shown in the Figure (18.0,21.0,37.0 (4.8). GHz) The variation is significant due to scattering processes. GHz) the extinction coefficient can be assumed to be to the not absorption At lower frequencies coefficient which s i g n i f i c a n t l y w i t h the rainfall rate. been have (4.67). The variations of these coefficients with at higher frequencies s u c c essfully used orig i n a t e s from snow flakes over space and time. fails drop (mm h 1 ) , is d e termined from The extinction coefficients at has a (m- 1 ) with rainfall ^ = 4100 R - 0 '21 (4.66) of per unit volume per unit drop-diameter interval ). N q is given by diameter number for a nd does The M.P. vary rain, which sufficiently On the other hand the M.P. equal distribution subtropical is (6.6,10.7 averaged distribution to reproduce the drop size distribution for w a r m rain in the tropics. The vari a t i o n s in N Q for different rainfall types were found to be independent of those that occurred in /s. Therefore, to describe the drop size distribution more p r e c i s e l y Sekhon and S rivastava (1970) found it necessary to s p e c i f y both N Q and ^ as independent variables of size distribution. Using the data of the drop Gunn and Marshall R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.0 HHU- .00 0.6GHZ J0.7GHZ l«.OSHZ 21 .CCHZ OCL- .00 5.5 37.0CH2 5.0 4.5 4.0 H 2 3.5 a u E fe. w o u 3.0 Z 2.5 o 2.0 •B-4 0 10 20 30 40 50 60 70 80 90 100 RAIN FALL RATE (MM/HR) USING M.P. DISTRIBUTION Figure (4.8) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 84 (1958) and others, which were found they provided the to fit a parameters and NQ large number of observations adequately, — ^ = 2290. R o 46 (4.68) N q = 2.5 x 106 R - 0 '94 (4.69) For ice, above freezing level in the atmosphere, parameters take the values given by Gunn and Marshall the (1958) as ^ = 2550. R ~ ° ' 48 (4.70) N Q = 3.8 x 106 R - 0 *87 (4.71) Ulbrich and Atlas rainfall rate distribution (1984) have shown that improvement in estimation can be achieved if the drop size (DSD) is assumed to be a gamma distribution gi v e n by dn(a)/da = The N ay exp U coefficient (-^a) n ow has ; 0 < a < a the (4.72) ulaX unit m 4 y and the exponent y can have any positive or negative value given Ulbrich and Atlas (1984). The r e have been large and sudden changes in N Q from moment to m o ment with type as noticed by Waldvogel These changes were found to be occurred facts, in ^ Ulbrich ^ = by given rainfall (1974) and Donnadieu independent from moment to moment. (1983) a of (1982). those In response to these gave v a lues of the parameters (3.67 + y + 10- 0 *3 (^+ 9 )) / Dq N Q = 6.0 x 1 0 6 e x p (3.2 y ) that (4.73) (4.74) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 85 Inserting (4.74) into (4.72) one obtains dn(a)/da = 6.0 x 106 ay exp (3 .2y-/*a) The m e dium volume diameter, (4.74-a) (m), has been found from an empirical result obtained b y Ulbrich (1983). He has also shown that €, 6, and D Q are expressible in terms of rainfall rate and the parameter y , according to DQ = € R 6 € = (4.75) [(3.67+y) x 102 / (33.31 N Q T l / 6 ) 6 ] (4.76) 6 = 1/(4.67+y) Thus from for (4.73) a ~-,-„ min to a given val u e of y , /* and N Q can be obtained and (4.74). The drop diameter, (4.62) m ax fixed and is given by droos lying and (4.64) rainfall rate as given by of (4.77) (4.63). v ' where (4.65). varies from The value of a . is man as a, a varies max with Therefore the total number be t w e e n the diameters a . to a for the min max g a m m a drop size d i s t ribution is given by N(y,R) ro.0023 R 0 *213 a^e x p (3.2y-^a) J O . 0001 independent param e t e r s y a nd R = 6.0 x 10 Two evalu a t e the v a r i a t i o n of total this number total of number drops of da (4.78) are from drops, needed (4.78). N(A), to The verses rainfall rate for different v a l u e s of y usi n g the gamma drop size F i gure (1983) dist r i b u t i o n (as d e s c r i b e d by (4.9). The different v a l u e s of have been used (4.78)) y is shown in the given by Ulbrich for the curves shown in the Figure R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. GAMMA DROP SIZE DISTRIBUTION VERSES RAINFALL RATES -.27 5.04 -1.70 2.00 o 10- 0 10 20 30 40 50 60 70 80 90 100 RAINFALL RATES (M U /H R ) USING CARLTON W.ULBRICH PAPER F ig u r e (4.9) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (4.9). These curves represent various types of rain ranging from a very light rain to a heavy thunder storm. For a large positive value of y, the total number of drops, with rainfall rate, R, N(A), varies very rapidly u p to 25 mm/h. For a smaller positive value or negative value of y , the variation of N(A) with R is only up to 5 mm/h after which it does v a r y much. N(A) For a fixed value of rainfall rate, with y is very fast, as y increases positive value) for N(A), the the N(A) negative the change in (from negative to decreases very rapidly. value of Therefore, y, the total number of drops, is larger than that for the positive value of sources of different not types of rain and parameters y, €, 6 and N Q are given b y Ulbrich y. The corresponding (1983) but he did not report any extinction coefficients for the different values of those parameters. (D) Extinction Coefficient of Rain Volume The volume extinction coefficient of rain for DSD is obtained by subsituting (4.45) and a gamma (4.74-a) into (4.62), which results in a max ( R ) a 2+yexp (3. 2y-/^a) (4.79) a . Qe x t (m'a 'v) da m m where a . m m and a^_, max are given by 1 (4.64) and R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (4.65) 88 respectively. from Q ext is given by (4.73-4.77). Qext (4.45), volume function of radiation, extinction rainfall is calculated depends on index of refraction, whi c h is a function of temperature Therefore a nd ^ rate, within the coefficient atmosphere. of temperature, rain for a gamma DSD using from in Figures different the Showers, and 6 and N Q are c alculated from (4.72) to (4.77) and (4.62) rainfall (4.10) sets bottom to (4.26) against rainfall rate of these (4.15) Figures. n o n - linearly negative or curves is or positive. that the coefficient linearly Another (iii) beyond which varies according fact (i) Thunderstorm, with Figures Showers, rainfall to whether p is observed from these extinction coefficient increases with frequency and p, and attains a GHz, shown Four types of rain correspond to the first category, in whi c h the e x tinction rate for of values p, €, 6 and N Q . These values were (ii) Wide spread or Stratiform, to rate 1 to 100 mm/h. These coefficients are plotted (iv) Orographic are categorized usi n g the data. (4.10) of for the 17 observed by a number of investigators whose names are at a and p. of values of p, e, var y i n g is frequency Extinction coefficients at SMMR frequencies, sets m, no value 6.5 at calculations were performed. case of Orographic rain (Blanchard, e xtinction is coefficient (Np/Km) 6.9 1953), (Np/Km) the 37.0 In the value at 37.0 GHz. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. of In the MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 -V'- A- O.OGKZ Z B- I0.7GHZ .65 c- ii.omz D- 21.0GHZ C- 37.0GHZ .60 .50 .45 .40 .35 .30 .25 .20 .10 .05 100 RAIN FALL RATE (M M /HR) USING GAMMA DISTRIBUTION SHOWERS HIGGS (1952) Figure (4.10) R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 2.0 g.esHZ KHU— 1.78 do.- .a s I0.7GHZ It.OCHZ 2I.0GKZ 37.0CH2 1.4 E- 1.2 2 td CJ Ed 1.0 O U 2 O E-> U 2 EX Ed 100 RAIN FALL RATE (M M /HR) USING GAMMA DISTRIBUTION SHOWERS FOOTE (1966) Figure (4.11) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ 7ITH AN INTERVAL OF GHZ, ANGLE =48.8 4.5 4.0 3.5 /—N 2 z \ a, z y 2.5 Cs, W o u z o 100 RAIN FALL RATE (MM/HRt USING GAMMA DISTRIBUTION SHOWERS MUCHNIK (1961) Figure (4.12) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ TOTH AN INTERVAL OF GHZ, ANGLE =48.3 6.5 A« S.6GHZ 3- I0.7GH2 6.0 I>21.COC C- 37.0GHZ 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 :go RAIN FALL RATE (M U/HR ) USING GAMMA DISTRIBUTION SHOWERS IMAI (1960) Figure (4.13) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.5 6.0 E- 37.0GHZ 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 100 RAIN FALL RATE (M M /HR) USING GAMMA DISTRIBUTION SHOWERS FUJIWAHA (1965) Figure (4.14) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.5 rlNU- 5.04 A- C.6GXZ 3- I0.7GHZ 6.0 D- 2I.0GHZ 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 100 RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION SHOWERS JONES (1956) Figure (4.15) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6TO37.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE = 48.8 5.0 DU- 4.5 4.0 £ 3.0 o 2.0 1.0 100 RAIN FALL RATE (M U/HR) USING GALIMA DISTRIBUTION WIDESPREAD(STRATIF0RM RAIN) U. P. (1948) Figure (4.16) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =46.8 6.5 6.0 E- 37.0GHZ 5.5 5.0 r— . 2 2 \ 4.5 PU 2 E-* 4.0 2 Cd 5 3.5 fc. 2 W 3 30 2 I —i P E- 2.5 U 2 P X H 2.0 1.5 1.0 100 RAIN FALL RATE (M U/HR ) USING GAMMA DISTRIBUTION WIDESPREAD(STBATIFORM RA1N)FUJIWARA, 1965 Figure (4.17) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.0 HHU- I.01 »« O.OGHZ 5.5 _ o la.GCHT E- 37.0GHZ 5.0 4.5 4.0 3.5 3.0 2.5 2.0 100 RAIN FALL RATE (M U/HR ) USING GAMMA DISTRIBUTION WIDESPREAD(STRATIFORM RAIN) ATLAS (1957) F ig u r e (4 .1 8 ) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.0 KHU- 4.65 A- 6.6GHZ 8- I0.7GHZ 5.5 D“ 21.OGKZ E- 37.0GHZ 5.0 4.5 4.0 3.5 y fc. 3.0 w o u 2 o H U 2.5 2 2.0 1.5 - r*— 100 RAIN FALL RATE (M U /H R ) USING GAMMA DISTRIBUTION WIDESPREAD (STRATIFORM RAIN) JONES (1956) Figure (4.19) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =40.8 5.0 A- o.oxz 0.7CHZ s .ochz 4.5 I.OGMZ 7.0GHZ 4.0 2 3.5 a. COEFFICIENT 3.0 EXTINCTION Z 2.0 2.5 1.0 lao RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION THUNDERSTORM RAIN FUJIWaRA (1965) Figure (4.20) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. MIE CALCULATIONS FROM 8.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.0 B A“ fl.GCHZ B" I0.7SXZ 7 E- 37.0GHZ 6 w 5 4 3 2 1 •A— 0 100 RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION THUNDERSTORM RAIN SAVARAMAKRISHNAN(1956) Figure (4.21) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 A- 8.0GHZ 8- I0.7GHZ 6.0 f- 5.5 5.0 4.5 H 4.0 Z I—I H U z H 2.0 X H 1.0 ■A- 100 RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION THUNDERSTORM RAIN BLANCHARD (1953) Figure (4.22) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 5.5 A- 6.6GHZ B- 10.7GHZ 5.0 _ 0- 21.OGHZ - E- 37.OGHZ 4.5 4.0 3.5 2.5 2.0 100 RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION THUNDERSTORM RAIN JONES (1956) Figure (4.23) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. M E CALCULATIONS FROM 5.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 7.0 0.6GHZ I0.7GKZ 18.0042 6.5 .OGHZ ’.CCHZ 6.0 5.5 5.0 4.0 y fa 3.5 fa o u 2 3.0 o t—« 1 2-5 H X 2.0 Dd 1.5 1.0 100 RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION OROGRAPHIC RAIN BLANCHARD (1953) F ig u r e (4 .2 4 ) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.0 A- 6.6GHZ 8- 10.7GHZ 5.5 E- 37.0CHZ 5.0 _ 4.5 2 a fa 4.0 2 B2 fa y 3.5 k, fa 3.0 fa o u 2 2.5 § 2.0 o S u H X fa IQO RAIN FALL RATE (MM/HR) USING GAMMA DISTRIBUTION OROGRAPHIC RAIN RAMANA,MURTY,GUPTA(1959) Figure (4.25) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. MIE CALCULATIONS FROM 6.6T037.0GHZ FREQ WITH AN INTERVAL OF GHZ, ANGLE =48.8 6.0 6.QGHZ 0.7GHZ 5.5 i.o o a I .o tx z 7.0GKZ 5.0 4.5 2 2 \ ft, 2 4.0 3.5 y Cz* fe. 3.0 w o u 2 2.5 o 2.0 1.5 100 RAIN FALL RATE (MM/HR) USING GAMiiA DISTRIBUTION OROGRAPHIC RAIN WEXLER (1948) F ig u r e (4.26) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 106 radiative transfer the volume extinction coefficient of rain is calculated for the value of y which was randomly selected from the values given by Ulbrich y (1983). This randomness in will generate the all possible types of rain found in the atmospheres. Volume extinction using coefficient of ice is the distribution given by Gunn and Marshall shown in equations 3.8 4 B I c e (R'm,v) (4.70-4.71) tt (1958) as and is given by max 1q6 r~0.87 a The altitude, calculated min 2 a exp(-Aa) Qe x t (m'a 'v) da (4.80) z, enters the calculation of Qe x ^. through the definition of the temperature of the atmosphere at that altitude. Whenever the cloud is found above the 0° isotherm, the volume extinction coefficient has been used in the ice radiative transfer calculations. The atmospheric models used in the simulations of microwave brightness discussed in the next chapter where, temperature the are computational procedure will also be described in detail. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. CHAPTER (V) M O D E L ATMOSPHERES AND SIMULATION OF BRIGHTNESS TEMPERATURES In order to simulate m i c rowave brightness was d i s cussed in the preceding chapters, surface and the atmospheric models. o ce a n surface model are surface surface wind. one has to know the Inputs needed for the temperature, salinity and The atmospheric model requires the knowledge of vertical distribution of its temperature, temperatures water clouds and rain, the ice clouds. vapor parameters density, like pressure, liquid water density in and water content in form of ice present in H o w the values of these parameters were o b t a i n e d is described in the next section. (5.1) Atmospheric Models The humidity vertical are profiles given of pressure, temperature by r a d iosonde/rawinsonde measurements. These m e a s urements are routinely obtained located all over by stations whi c h stations, whi c h are distributed in latitude w ere selected. are Also in selecting the data, the mon t h were varied, and meterological the globe. and Several longitude, both the year and so as to have a representative amount R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 108 of climatology. Six years of data in (from J anuary to December) all season (from 1966 to 1972) v a ried and in all region (by including the stations sprea d e d all over the globe) were c ollected and a careful s e l ection was measurements. profi l e s of levels These meas u r e m e n t s temperature (1013.25, and 1000.0, 620.0, 570.0, 300.0, 250.0, 200.0, 150.0, 135.0, 60.0, 50.0, 30.0, 25.0, 20.0, 3.0, 2.0, 1.5, 1.0, 0.5, 475.0, 0.2, only 430.0, 15.0, 600 up to 20 lower levels. 780.0, 100.0, 0.1 millibar). but pressure 700.0, 400.0, 10.0, 350.0, 85.0, 70.0, 7.0, 5.0, 4.0, The temperature the humidity was The humidity was then e xtr a p o l a t e d for upper 21 v a l u e s from its values. select fixed 850.0, 115.0, w a s observed at all pressure levels measured at 920.0, 670.0, to describe by the vertical humidity 950.0, 500.0, made last 5 measured F r o m the knowledge of vertical profiles of pressure +•v» a n d temperature one can find the altitude of the (i+1) layer in k i lometers using the relat i o n H i - H i+1 = l o g ( P . + 1 / P i ) [(Ti+ T i + 1 )/68.2831] The liquid water in these profiles was introduced from the knowledge of its clouds. (5.1) The distribution distribution of in liquid different types water in a cloud w as a s s u m e d to be uniform from its base to the top. Only in a tmosp h e r e s were clouds and rain introduced; t hem were assumed clear. of 200 and the rest of These 200 cloudy atmospheres w e r e c a t e g o r i z e d into 8 cloud model types of 25 atmospheres each, R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 109 according to their base and top altitudes and content as shown in Table liquid water contents of the Table water content (5.1). (ILWC) liquid water Their integrated columnar are shown in the fourth column (5.1). This ILWC is the product of the liquid (LWC) and the cloud thickness (CL(2)- C L (1)). The rainfall rates for these clouds were calculated from the relation obtained by Paris R = 18.05 Aft e r calculating (1971) (ILWC)1 ’19 (5.2) the rainfall rate from this equation, w as then super imposed by a random number between generated by a random number generator. of a random number is equivalent error in the rainfall rate. to 0 it and 5 This superposition adding a statistical The relative humidity was set equal to 100% in the clouds or rain and the drop temperature was assumed to vicinity. The altostratus, shown in mentioned in the most temperature probable of the air thickness last 4 cumuli form the clouds literature. may go extrapolated from the lowest The its of altocumulus, m o dels of the Table km, as (5.1). However, up to 8 km as The sea surface temperature corresponding to the surface pressure level measurements. in and stratocumulus clouds was assumed 2 the thickness of be 2 values of 1013.25 the (mb) w as radiosonde sea surface wind was randomly introduced in the model between 0 and 25 atmospheres contained a (m/s). certain Each amount of these model of water v a p o r , R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 110 TABLE (5.1) Cloud Statistics of 600 Model Atmospheres I Number of IAtmospheres I I 1 1 I I 1 1 I 400 25 25 25 25 25 25 25 25 Cloud Base (km) CLC1) Cloud Top (km) CL(2) 0.0 1.0 1.0 0.0 7.0 1.0 1.0 2.0 6.0 0.0 2.0 2.0 8.0 8.0 3.0 3.0 4.0 8.0 Liq. Hater Content 1 of Cloud (gra/ar') 1 LHC I 0.0 0.1 0.3 0.1 0.2 0.04 0.08 0.02 0.2 1 I I 1 I I I I I TABLE (5.2) Statistics of atmospheric parameters of 600 model atmospheres IStatistics Surface I Surface Iof ataosp- temperaturl Uind ( K) I (m/s) Ieric models Rainfall Ilntegrated Ilntegrated {Integrated Ilntegrated I IWat. Vapor ILiq. Hater IRain Hater lice Hater I Rate (mo/h) I (gm/cm3 ) I (kg/tf) I (kg/nr) I (kg/a1) I 2BB.43 I 12.67 2.03 I 1.76 1 0.61 I 0.22 I 0.01 I I Kedian 230.80 I 12.76 0.25 I 1.56 I 0.06 I 0.04 I 0.005 I I 233.0 I 8.0 0.0 1 1.0 1 0.0 I 0.0 I 0.0 1 1 Minimum 270.0 1 0.0 8.0 I 0.0 1 0.0 I 0.0 I 0.0 I I Maximum 305.0 I 25.0 18.0 I 6.0 I 18.0 I 6.0 I 1.0 I I 1055.0 I 368.0 I 133.0 I 6.0 I I I Kean Kode Sum 172480.0 I 7600.0 1218.0 I Std.Error 0.42 I 0.23 0.16 I 0.06 I 0.10 I 0.04 I 0.304 I I Std.DEv 10.24 I 7.10 3.85 1 1.43 I 2.32 I 0.86 I 0.100 I I Skeuness -0.311 I -0.004 2.441 I 0.573 I 4.623 I 4.337 I 3.874 I I Kurtosis -1.402 I -1.133 6.140 I -0.411 I 95.817 I I 23.363 I 13.325 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Ill liquid water content in the clouds and rain water in the form of ice clouds. was considered form, and ice The ice water of the clouds to be the equivalent amount of liquid water that exists in the form of ice. All phases of water existing in the atmosphere can be expressed in depths of terms of equivalent liquid wat e r denoted by WV, LW, R W and IW. WV is the equivalent depth of the total amount of water the atmosphere, vapor LW is the total equivalent depth of liquid w at e r contained in the cloud, RW depth of liquid water present in the form of rain, the total clouds. equivalent depth is of total areal , and 400 w e r e clear. content a .524 N q m ax of liquid water. a is given by (4.73) (4.65). _2 (RW) was (1983) and (4.74) The value of respectively, p was randomly (1983). The e quivalent depth of liquid water in the form of rain, —2 ), was obtained 1.5 (5.3) selected from the v a lues gi v e n by Ulbrich m The in terms The rain water 3+ jj . . , exp(-^a) da where ^ and N Q are given by am ax 18, 6 and One m m corresponds to one km m d eter m i n e d from the relation given by Ulbrich and and IW is water present in ice LW, R W a nd IW are from 0 to 60, respectively. RW = equivalent There were a total of 600 model atmospheres obtained ranges of WV, of the ice in w h i c h 200 were cloudy and rainy (mm) in from integrating total RW (kg (5.3) over surface to R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 112 zero degree Isotherm. The frequency distribution of sea surface SST (°K), vapor, W V cloud, sea (gm cm LW (km m — 2 , RW (kg m and rainfall (5.7). -2 -2 -2 ), wind, SSW integrated liquid integrated water water content rate ). Out of 200 (mm/h) are shown IW (kg in Figures m cloudy which atmospheres contained of clouds and rainfall rainfall rain, rate rate, were ), to 141 only (kg 45 more than 0.5 liquid water and ice water of clouds are h i g h l y skewed towards small values. all there —2 (5.1) liquid water content was less than 0.5 (mm/h). The distributions of content of ), integrated liquid water content of rain ), integrated ice water content, the a tmospheres for (m/s), In 541 atmospheres there was no ice cloud and in atmospheres m surface temperature, The complete statistics 600 model atmospheres is given in the Table (5.2). These model atmospheres were then employed by the microwave radiative transfer equation to calculate the microwave brightness temperatures at SMMR frequencies. The algorithm u s e d for the calculations is giv e n in the next section. (5.2) Brightness Temperature Simulation A microwave program), p ola r i z a t i o n which radiative takes effect into of transfer account model the non-isothermal (computer M i e-scattering clouds R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. in 113 Frequency Distribution of SST o f 600 m odel atm ospheres 110 100 - 90 60 70 60 SO - 40 - 30 - 2010 - 270 2 7 2 2 7 4 2 7 6 2 7 6 280 2 8 2 2B4 2 8 6 288 290 292* 294- 296 2 9 8 3 00 302 3 0 4 SST (degrees Kelvin) Figure (5.1) Frequency Distribution of SSW o f 600 m odel atm ospheres 60 50 - 40 - 30 - 20 - 10 - 2 4 6 8 10 12 14 16 18 20 22 SSW (m /s e c ) Figure (5.2) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 114 F re q u en cy D istribution of W ater Vapor o f 600 model atmoaoherea 160 140 - 130 - (N ) 120 of Cases 100 Number - 110 - 90 80 70 - 50 - 30 20 - 10 - 1 2 3 4 6 5 Water Vapor (gm/cn?) Figure (5.3) Frequency D istribution o f LWC oi 600 model atmospheres 12 11 10 9 8 - 7 - 6 5 4 2 1 0 11 I !Z 3 - 21 & 6 7 B 9 10 11 12 13 14 15 16 17 13 Liquid Water Content (kg/m*-) Fimire Ia .4 1 R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 115 F re q u e n cy D istrib u tio n o f Rain Water of 600 modol atmoaoheres 20 - 4 - 87- 4 - 3 2 1 5 4 6 R a in W o t e r ( k g / r r T ) Figure (5.5) F re q u en cy D istrib u tio n o f Ice Water o f 600 model atmospheres 26 24 - 22 - Number of Coaes (N ) 20 4 - 0.5 1 1.5 2 Ice Water above freezing level ( k g / r r f ) Figure (5.6) R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w itho ut perm ission 116 F re q u en cy D istrib u tio n o f R ainfall Rate of 600 model ctmosoherea 40 25 - 20 - N um ber of Coses (N ) 35 - 10 -*7-. 1 2 3 4 5 6 7 B 9 10 11 12 13 14 15 16 17 18 Rainfall Rate ( m m /h ) Figure (5.7) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 117 inhomogeneous atmospheres has b e e n developed. general enough to compute the b r ightness given frequency at any h e ight The program is temperature at a of a given atmosphere. It assumes the Rayleigh s c attering phase function no n - r a i n i n g cloudy atmosphere. into a number of layers; pressure, temperature The case of atmosphere is divided each of which is and humidity. in characterized by This atmospheric model (radiosonde/rawinsonde data together w i t h the cloud and rain model) is required by the computer program as an input. geophysical parameters required b y the computer program are the surface temperature and wind. (GHz), and (degree) and the look angle The microwave frequency of model requires the variation of at the surface, R surface emissivity cloud and rainfall ( m m / h ) . An important part of the p r o g r a m is to calculate the s urface o ce a n e liquid water content w i t h altitude w i thin the cloud extent rate v observation from nadir, are also needed as input by the program. The rain The is emissivity derived from e (e). P The the Fresnel's refl e c t i o n coefficients as d e s c r i b e d earlier in chapters III a n d IV, and is a function of surface temperature, surface wind, salinity, frequency and angle of observation. sea The atmospheric transmittance of e a c h layer of the atmosphere is calcu l a t e d clo u d (3.71). from abso r p t i o n The oxygen absorption, and rain water vapor absorption, e x tinction us i n g (3.56) and upward and d o w n w a r d emission of the atmosphere R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 118 known as T (3.73) by and atm and T sky (3.72) respectively are calculated in whi c h the source term J(z,e) from is given (3.74). The calculation knowledge of the asymmetry factor particles g(z) , of the source single of the the scattering phase scattering requires albedo function the w(z) , of the scattering intensity components IQ (z) and I ^ z ) , the temperature profile T(z) single term and the look angle 9. The albedo is defined as the ratio of volume scattering coefficient of rain at an altitude z to the total volume extinction coefficient altitude. described The by (3.56), coefficients extinction total due coefficients of calculated from volume is the atmosphere extinction composed to oxygen, coefficient of of of rain. The oxygen, water vapor, (4.18), (4.28), and (4.72) and respectively. The are calculated volume asymmetry absorption and absorption clouds (4.41) respectively. from factor (4.62) of the involve and ensemble a g(z) = 2 + jj amax exp(-^a) 0.00013 (5.4) da R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. are The the (4.63) particles is given by max 0.0001 as clouds and volume extinction and scattering coefficients of rain DSD that coefficient, volume water vapor, at of 119 where Is obtained from * (4.73) and the asymmetry factor of individual drops, g ( a ) , is calculated from the Mie-theory. The Kerker g(a) asymmetry (1969) factor of individual drops is given by to be = (4/X2 Q s c a }2:C{n(n+2)/(n+l)}Re (ana*+1+ b nb* + 1 ) + { (2n+l)/n(n+l)}Re (an b * ) ] where an and b n are given by and of Q_„ is calculated from S C d (5.5) which the summation over is given by (5.5) (4.48) and (4.46). (4.49) respectively In the right hand side n takes the value from 0 to N (4.61), a nd X is size parameter given in (4.44). The intensity components 1^ and 1^ are the solutions of first order coupled differential equations (3.58) and (3.59). These coupled differential equations are then solved numerically (3.60) and Weinman, w i t h the boundary conditions given by equations (3.61) using a finite difference method 1984). in the specific direction making an angle e from the nadir, and is then u s e d in and Ts k y ' The u p w elling the radiances calculations received radiometer at an altitude H a nd at a looking angle nadir calculated from is then transmittance and and The source term is calculated at each layer of the atmosphere, T atm (Wu (3.69) which uses ( D , the downward and upward of by e a from the total radiances Tg-tm' T , sky polarized component of emissivity e (e) and P surface temperature (T ). The various steps in this scheme R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 120 are clearly illustrated in a flow chart shown in Figure (5.8). The SMMR brightness temperatures are calculated model atmosphere given in Table by Ulbrich (1983). jj, (5.16) and 6 Figures (5.9) for different cloud models given in Table (5.3). The surface parameters wind, held constant. high frequencies variation NQ , € The variation of these brightness temperatures with rainfall rate are shown in to a (4.2) us i n g Gamma drop size d i s t ribution and a particular set of values given for at These for low temperature and variations small frequency salinity were are h i g h l y non-linear at liquid water (6.6 GHz) content. The is linear for small liquid water density but becomes non-linear for both large den s i t y and high altitude clouds. The simulated b rightness temperatures are then inverted to retrieve the rainfall rates used i nversion optimization technique and a technique uses an in the models. This statistical m e thod wh i c h is e x p lained in the next chapter. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 121 TABLE (5.3) Cloud Models 1 Bottoa I (ka) TOP (ka) Density (ga/o3) Hax.Liquid Hat. (ng/go2) 1 I I 1 2 0.1 10 I I 1 2 0.3 30 I I 0 8 0.1 80 I I 7 8 3.2 20 1 I 1 3 0.04 8 I I 1 3 0.08 16 I I 2 4 0.02 4 1 I 6 8 0.2 40 I R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 122 CLOUD & RAIN MODELS ATMOSPHERIC RADIOMETER GEOPHYSICAL SYSTEM MODEL MODELS ^ ^ Polarization SST . SW , and S X 7 CL 1M A T 0L 0 6 I C A L DA TA P , T. and H MICROWAVE RADIATIVE TRANSFER MODEL FOR BRIGHTNESS TEMPERATURES SIMULATION T * ( V O z x ------------ z x ----- OCEAN SURFACE UP and DOWN WELLING EMISSIVITY MODEL ATMOSPHERIC RADIANCES CALCULATIONS OXYGEN ABSORPTION 7^7 WATER VAPOR ABSORPTION LIQUID WATER ABSORPTION RAIN DROP SIZE DISTRIBUTION EXTINCTION ','T.' MIE THEORY Figure (5.3) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 123 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll W=0.0, SST=300.4, WV.= 5.01 . LW.= .33 300 6 .8 H 280 0- 10.7 V £• 16.0 H F- 16.0 V 260 J- 37.0 V ~ 240 - y j j g 220 - / > 5 -7 5 / | 200 - / 05 I • 7 i 180 ^ K u i— . y A| 160 I - / — 'a‘ 05 m 140 120 100 100 RAIN FALL RATE (MM/HR) USING GAMUA DROP SIZE DISTRIBUTION F ig u r e ( 5 . 9) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 124 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, E?S=.114. D E L = .ll W=0.0, SST=3Q0.4, WV.= 5.01 . LW.= .53 300 s.s H 6.6 V 10.7 H 280 10.7 V E- 16.0 H 16.0 V 21.0 260 H 21.0 V 07.0 n 37.0 V 240 g 200 m 180 S 160 140 120 100 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION Figure (5.10) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll W=0.0, SST=300.4. WV.= 6.96 . LW.= 1.04 300 0.6 H 280 e- la.o h r- la.o v O 21.0 H 260 *>• 37.0 v 240 g 220 S « 200 2 wh E w 180 (0 w a u 160 K G 140 1201- 100 0 10 20 30 40 50 60 70 80 90 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION Figure (5.11) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 126 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, EPS=.114, DEL*. 11 W=0.0, SST=300.4, WV.= 4.67 , LW.= .42 300 6.0 H 6.6 V 10.7 H 280 10.7 V E- 18.0 H 18.0 V 21.0 H 260 21.0 V 37.0 n 37.0 V 240 a K 220 D H < K £ 200 En 180 w a E X o 160 140 120 0 10 20 30 40 50 60 70 BO 90 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION F ig u r e (5.12) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 127 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll W=0.0, SST=300.4, WV.= 5.40 , LW.= .40 300 o.e h :=£!= 280 E- 16.0 H O 21.0M 260 H- 21.0 V 240 w H « 220 Eh < X £ 200 a t E] ot w w z 180 S3 160 o X 01 140 120 100 0 10 20 30 40 50 60 70 80 90 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION Figure (5.13) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 123 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll W=0.0, SST=300.4, WV.= 5.36 , LW.= .28 300 280 D* 10.7 V 260 r- i o .o v 0- 21.0 M H- 21.0 V 1- 07.0 M 240 Ed « 220 Zj < K £ 200 2 Ed En 180 to Ed ' Z trj 160 - o t—4 K m 140 - 120 100 0 10 20 30 40 50 60 70 80 90 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION Figure (5.14) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 129 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MKU=4.65, N0=6.4E10, EPS=.114, D E L = .ll W=0.0, SST=300.4, WV.= 5.40 , LW.= .32 300 S.S H 280 0- 10.7 V E- ia.0 h r - is . o v 260 240 W g 220 H < K £ 200 2 W tm 180 w S a K 160 u K m 140 120 100 - 0 10 20 30 40 50 60 Iu an 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION Figure (5.15) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 130 BRIGHTNESS TEMPERATURE VS RAINFALL RATES MHU=4.65, N0=6.4E10, EPS=.114, D E L = .ll W=0.0, SST=300.4, OT.= 4.77 , LW.= .59 300 S.SH 200 D- 10.7 V E> IS.O H 260 O 21.0 M J- 37.0 V 240 w K 220 H < -M K g 200 2 a en 100 to « S O « n 160 »-4 140 120 100 RAIN FALL RATE (MM/HR) USING GAMMA DROP SIZE DISTRIBUTION Figure (5.16) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. CHAPTER (V I) STATISTICAL METHODS AND OPTIMIZATION TECHNIQUES In order to retrieve rainfall rates from the brightness temperatures (TBs), the optimization technique "Regressions by Leaps and Bounds", Furnival and Wilson wh i c h was developed by (1974), has been used for selecting the radiometric channels, Linear simulated Regression" and the statistical has been used method "Multiple for the inversion. The Leaps and Bounds technique has been successfully applied Pandey and Kakar (1983) temperature and by Kakar in and by the retrieval of sea surface Lambrightsen (1984) in the retrieval of atmospheric water vapor profiles. The problem of selecting the best subset of predictor variables in a linear tedious. the regression model is usually quite The number of subsets increases exponentially with number of (multiplications channels and N. The divisions) number of required to operations invert the moments matrix associated with each subset is of the order 3 N . The computer time required for p erforming such task is enormous even for a moderate value of N. past studies, However, in the the best possible subset of a given size has been determined using a forward selection method. Stepwise R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. 132 regression is a forward selection method which has been commonly used in selecting the best subset of a from a set of channels. the residual have been sums and size In the technique of leaps and bounds of squares for all possible regressions computed operations given with then the minimum best w ithout examining all possible number of arithmetic subset has been determined subsets. The advantage of this optimization technique is in the reduction of number of operations by several other techniques. that it provides in of magnitude as compared to Another privilege information channels subsets of useful orders a desired on using to technique several which is is next best extremely deciding the optimum set of channels for quality For example, one would retrieve nearly the same values of rainfall rate different temperatures subsets of corresponding SMMR scatter measured brightness to the same geographic location and time. B ad radiometric measurements large this the size, control of radiometric measurements. expect of will give rise to in these retrieved rainfall rates and may be rejected on the basis of this large radiometric identified channel can be scatter. on Also the a bad basis of consistent b ad retrievals and the process of elimination. A two step statistical inversion algorithm has been adopted for retrieving the rainfall rates from SMMR data. the first step, the In best radiometric channels subsets of R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 133 different sizes have been obtained from the Leaps and Bounds technique using temperatures a and simulated data rainfall rates. base brightness In the second step, subsets have been used to invert the Multiple of rainfall rates these using Linear Regre ssion method. This method provides the relationship between rainfall brightness temperatures. rate Each a nd of the the corresponding subsets yield regression equation wh i c h has been employed to retrieve rainfall rates temperatures. from These values of measu r e d SMMR rainfall rates, which from compared. This comparison provides a self consistency the regression the brightness obtained on different the a equations, are are then check SMMR measurements and is used for rejecting the bad measurements. If the values of given regressi on equation, by a rainfall rates, which are are consistently bad then that particular subset has been dropped from the rainfall rates retrieval scheme. The criterion for selecting the best possible subset can be made from any of the three criteria viz adjusted R 2 , and penalty. When the search continues other two (ii) (iii) Mallows c(p) with F r a n e 1s variable first until criterion, the R 2 , desired is criteria, adjusted R 2 employed number regressions are found for each size of subset. the o (i) R , of Whereas and Mallows c(p), search terminates after locating the desired number of R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. the best for the best 134 regressions regardless of subset size. the best subset were based on the R 2 All selections for criterion which is d efined as R 2 = 100 where R the ) (6.1) is the residual sum of squares of the dependent sss variable ( 1 - R sss / T ss (rainfall rate) for a subset of size s and ' ' total variable. T is ss sum of squares about m e a n value of the dependent The maximum value that R 2 can a t tain in principle is 100. The results obtained p r e sented in the Table with four to (6.1) from Leaps and Bounds in w h i c h the best five are subsets nine predictor channels are shown along w i t h the corresponding R 2 values. The value of R 2 increases with the number of channels used in the subset from four to nine. However, the the change in the R subset of size six. 2 value is not significant after The value of R 2 for the full ten channels set is 90.62 which is not much larger the R o value for subset of size six. than This implies that adding a channel after six is not very advantageous retrieval of channels (6.6H, be optimum the rainfall rates retrieval. obtained 6.6V, set rate. 18.0V, 89.36, in the Therefore subset of size six 21.0V, 37.OH, 37.0V) seems to whi c h is most appropriate in rainfall The next best subset of size six from can be the last best subset by replacing 21.0V by O 21.OH where the R “ does not change significantly but this R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 135 TABLE (6.1) Result of Leaos ano Bounds Technique for Selecting 5 Best Subsets of Size 4 to 3 SMMR Channels Size of I the best I subsets V I H V n i. T n « X I 1 X T I A I H 1 i I 4 T T 5 1 X X X X X X A A X 7 3 * 5 I 1 GHZ I 21.3 GHZ I 37.3 GHZ V 6 I H V T A T 3 A I X I I 1 T 1 : T T 1 i I T i I T 1 X X I I X X I I I I T i A X i A X I I X X i V I X X I I I X I T i T X X X X X I X X I X X I X X I X X I X X I X X I X I X T 1 T 1 X I t 1 V I I t X X T X V X I X X I ,x X i X X I X T 1 :< A X X A T T 1 37 75 37.51 n-r uI no u -j. nr •n lo :*.n jO r,n X A A X X T X i PQ *? c UO X A T 1 •JU nr, Ju <n I T X X I u j. it, 00 UU• 0 U.-1 *T I X A I 33.73 X X I 23.45 I nn * 1 X 1 A nr J?. 1w X X A I X X I T 1 33.64 » 1 i r X v A A 1 X I X X I X I X I X X I 39.43 00 nn JJiuU I • X X X X T 1 T I X X I X X :< X I J X A 1 v T X X X I X X I I - *> 1 X X i x a i A 1 I I :< T T A X I I X nn nn u 0 UO A i I r A 1 T 1 X nr tn 0J • i J n .♦ n C't . rd n « nn G 7 .0 0 n i ,i i A I T * I I T "-Square A A I I I I X X A X X X A T X A I X X T X X X X X x I X X £ 1 I X J V I 13 I A X i i I I I I X T I T I / 1 H 0 V i X i 6 1 H T I I «n n W . *J 6.6 GHZ 113. 53 GHZ I X ;c iT A X 1 V X X * T ' A A •/ X I A iT T ■< A X i T I * • A I X X 35,34 nr. ,-.n 03, 3l nn nn o3, w w 00 01 jJ i/i; r.n I vu, 13 00 0^ jiu ij I S3.35 1 nn m u 2 . JiL 00 07 -• J * w/ X A 39.43 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 136 set can be useful when 21.0V has an erroneous behavior. A set of retrieval equations of different sizes using the best subset is generated from the multiple linear regression model. The coefficients rainfall rate of obtained summarized in Table the retrieval from (6.2) the for best equations simulated data base are channels of through nine and for the full ten channels set. coefficients are linearly for related to size four These the brightness temperatures in the following form R(s) = A Q + A. T B (vi ) (6.2) where s is the size of the subset, A Q is the A^ 1s are the intercept and regression coefficients corresponding to the brightness temperatures T_(v.) D X at frequencies v .. 3. R(s) is the estimated rainfall rate corresponding to the size s. (6.1) Precipitation using Microwave (SMMR) Data from SEA- SAT Satellite The rainfall values of rates are from the measured SMMR brightness temperatures onboard SEA-SAT and using the relationship given by rates, inferred corresponding to the (6.2). Different rainfall same geographic location and time, are obtained for different set of channels size) as given in Table (varying in (6.2). The discrepancy found in R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE (6.2) Multiple inear Regression Coefficients for Rainfall Rate Essi'.iation Using 00 Simulated Brightness Temperatures at SiiflE Frequencies. - t - -I- -I- Keqressici Coefficients and Associated Standard Errors Degree oil Intersect! multiple ; ■fIS Error Poly- I end Std. I- - - - - - 1- - - - - 1-. . . . . — ... I- - - - - 1- - - - - 1- - - - - - 1- - - - - !. -— I- - - - - - - ICorr.' IS) 1 in Pain F- Test nonial I Error of I 6.5 H I 6.6 V 1 10.69 10.63 V I18.0 H I IS.0V I 21.0 H I 21.0 V 1 27.0 H I37.0 V land adj. 1 fall (p) I AO ! At A4 I A5 I A6 I A7 I A3 I A3 I A10 I R-2 ! - - - - - r- - - - - - r-1 -I-I-I-I-II -4.07506 I I 0.03626 1-0.10373 I 30.320 I ! 0.05533 I I I I I I I x x I I I I 1.5330 1(0.00431)1 1(0.00651)1(0.00753)1(31.453) I 1(0.00563)1 1(1.80772) I j----------- j -I- - - - - 1-I-I-I--------- j------------I--I1 1-0.07714 I I 0.13520 1-0.13554 I 33.624 1-12.33633 1-0.16541 1 0.32433 I 344 :< I I I I I I I I I I 1.37GB 1(0.00413)1 1(1.761625 1(0.01377)1(0.02013)1 --- 1------ j. I. . . . . . r-— -I-I-I-I-I1-0.01257 1 1-0.03356 I 0.16317 1-0.16652 1 33.603 I I -4.2337 1-0.17326 I 0.31333 I 639 6 1 I I 1 x I I I I I 1.3321 I (0.00531) 1(0.00637)I(0.00735)1(37.431) I 1(0.00333)1 I (1.8230) 1(0.01434)1(0.02003)1 -i1-I-I-!- - - - - - I-I-I•I-I-II 0.20721 1-0.20043 I 33.703 I [-0.01360 I 0.00424 1-0.07320 I I -11.7313 1-0.16254 I 0.32333 1 609 7 1 I I 1 x I I I I x I I I I x I(0.01054)1<0.01133)I(37.669) I 1(0.01443)1(0.01570)1(0.00446)1 1(1.35257) 1(0.01561)1(0.02116)1 -I-I-I-I-I-I-I-I-I-II 0.20571 1-0.20710 ! 33.773 I 1-0.00715 1-0.00223 1-0.07373 I 1-11.52130 1-0.15443 1 0.32654 1-0.00173 53B 1 I I I 1.3666 x I 1 I 1 I 1 I I 1(0.01051)1(0.01136)1(37.770) 1 1(1.34315) 1(0.01557)1(0.02111)1(0.00070) 1(0.01461)[(0.01536)I(0.00444)I 1-I-I-I-I-i-I-I-II 0.13337 1-0.13610 I 33.302 1 0.04630 1 0.04373 1-0.05336 [-0.07331 I 1-10.23431 1-0.15313 1 0.31377 [-0.05133 480 I I I I 1.3646 I I I I I 1 I I 1(0.01272)1(0.01313)1(37.606) 1 1(1.33233) 1(0.01701)1(0.02143)1(0.03073) (0.02334)I(0.03773)I(0.03363)I(0.00540)1 i------ r r_ .. 1 . -I-I-I-I-I-II -3.73243 1-0.14523 1 0.23661 1-0.04714 0.0-1133 1-0.01025 I 0.00146 1 0.00332 1-0.03242 I 0. 13057 1-0.13227 1 33.313 I 440 I I I I 1 I I ! 1.3533 10 I I 1’ I 1i15.76156:1(0.00023)I(0.00051)1(0.00033 5 0.00073) I(0.00175) 1(0.00184)I(0.00063)I(0.00033) 1(0.00016)1(.00017) 1(37.310) 1 ------1.... — i......i-..... i— -... i... -— i... -— I..... - - -I- 133 estimated rainfall rates are due to the different functions in the channels contributing to the of the atmosphere various rainfall regression rates equations should have used. However, the same set of and time). Therefore a careful channels m ay reduce this discrepancy in the measurements. selection case of threshold, then that measurement is rejected. value of F-test was obtained for the channels. error subset a preset The maximum of size five This subset gives the multiple correlation 93.62% w h i c h is as good as the full set five-channels subset (6.6H, 93.91% 6.6V, . rainfall rates was employed in m easu r e d retrieval of the SMMR data. (Julian day 257) of this channels at SMMR frequencies. estimating values Therefore 21.OH, 3 7.OH, 37.0V) the best optimum subset with least number 1973 of If the difference in the rainfall rates d erived from the best two subsets is greater than 14, SEA-SAT (SMMR brightness temperatures corresponding to the same location free these nearly the same numerical value because they have been d erived from data response rainfall is for This subset rates from the This SMMR data of September obtained over the Pacific ocean corresponds to the satellite pass at starting time 17 hours, 0 minutes and 9 seconds, m inutes and 55 seconds. occur r e d Figures along (7.1) and the ending A total of 2285 five (7.2). and sub time 18 hours, measurement satellite Each event tracks consisted of R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 52 events shown in radiance 139 detection at five frequencies and two polarizations, total of ten channels. r eferred to as "cases". e s t imated us i n g Table These (6.2). A events rainfall be different regression equations given complete statistics of these estimated Negative estimates rates were (6.3). omitted in the c alculations and are listed as wild cases Mi c r o w a v e henceforth Rainfall rates for these cases were rainfall rates is given in Table of shall for a rest in of Table in the (6.3). estimated rainfall rates for different footprints along the subsatellite track are plotted on the world map in F igures that (7.1) and all These high represent (7.2). F r o m these figures is evident rainfall rates higher than 3.5 mm/hr lie on land. rainfall the actual rates however rainfall estimation over land. do rates technique is not desig n e d to work rate it very not accurately since the microwave well for rainfall The maximum estimated rainfall rate over the ocean for this data set is 3.5 mm/hr at longitude and 10° -155° latitude location. To get a qualitative picture of the microwave estimates the full data set corresponding to points over land and ocean have been grouped into frequencies for different class intervals in rainfall rate. to 14 mm/hr. The classes range from The class interval is 2 mm/hr. 0 Rainfall rates wh i c h correspond to the lower limit of a class interval not included in that class interval. Figures mm/hr (6.1) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. to are (6.7) 140 TABLE (5.3) Statistics o f Estimated Rainfall Ratesfroa SilMR Data llsir.q Different Sets of Channels 12(5) Stati sties I R(4) of Rainfall I I Rates 1 (aa/h) I (aa/h) 1Hean I aZ. r nn »i -I-IMedian I 3.357 I 3. -Iilude I 9 1--Minimum 1 3 . -II 3(7) I 2(3) I vmiaJ n) I (aiii/h) I (raij/’h) -i-1 R(o) •iu*T 3.513 3 . ^ ? i 4 »ni i ■*r£1 -I .18 11 3.257 I 3.273 1.321 : 1.234 . . . . . I. . . . . -II -3.174 I I -3.323 -I-i...... Mild I 13 I -I-I-IIValiu casesi 2275 I 2357 I 2273 I 2127 I. . . . . . . I-IjrvcWitebs (riei/ii) I I -I- n (na/fi) R CIS) 3.333 I 3.333 Maxi mum I 13 -ISun I 5313 td.Error I 3.33 -IStd.Dev. I 4.232 R(3) ■1773 4327 4637 I 3.378 .373 3.35 I 1.331 *i•'\-l u*i -I- 3.331 I 143 71 I 2137 -I. . . . R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 141 show the frequency distributions of these rainfall rates for estimations using four to ten channel represents sets. The number N the number of cases where the estimated rainfall rates were less than or equal to zero. Two observations from these histograms can be made viz (i) The distributions are highly skewed values, (ii) towards the small There are two peaks in all the distributions, one at the lower end between 0 mm/hr and 2 mm/hr, and at It must be the higher end between 8 mm/hr and 10 mm/hr. noted that the data corresponding to the first peak other is the data over ocean while higher rainfall rates belonging to the second peak m ay be regarded represent rainfall over land. as unimportant The fact that the since they different histograms resemble each other implies that there is some consistency between estimations that use different channel sets. (6.2) Precipitation using Vis/IR (VISSR) Data from GOES Satellite The estimation of precipitation from data is (1978). based on the In their scheme, temperatures. satellite scheme provided by Griffith et a l . thermal infrared the VISSR from GOES was used to infer cloud-top Vis/IR (10-12 ym) rainfall data of rates from Initially a technique was developed R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Four SMMR Channels R(4) 400 350 0 f o r N « 1327 250 - 200 - 150 - 100 - 50 2 4 8 6 10 14 12 Figure (6.1) FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Five SMMR Channels R(5) 260 240 - for N = 220 - 200 - 1285 180 160 -? 140 120 - 100 - eo 60 40 - 20 2 6 8 10 12 H Figure (6.2) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 143 FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Six SMMR Channels R(6) 6CQ 1320 400 - 300 - Number of C a sas 500 200 - 100 VZ£ 2 4 6 8 10 12 14 Figure (6.3) FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Seven SMMR Channels R(7) 260 240 220 0 for N = 1245 200 of Coaoa 160 - N um ber 180 - . 120 - 100 - 140 - 80 60 40 20 2 4 6 8 10 12 14 Figure (6.4) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Eight SMHR Channels R(8) 260 240 - 220 0 for N = 1131 - 200 180 160 140 120 - 100 - 80 60 40 - f 20 2 4 6 8 10 12 Figure (5.5) FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Nine SMMR Channels R(9) 450 400 - 350 - 300 ~ 250 200 - 150 100 - 50 - 2 4 6 8 10 m . 12 Figure (6.6) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FREQUENCY DISTRIBUTION OF RAINFALL RATE Using Ten SMMR Channels R(10) 800 700 €00 O m o O 9 i. o a £ 3 Z ECO-' 400 “ , 300 - , 200 100 2 4 6 8 10 12 14 Rgure (6.7) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 146 w h i c h required a sequence of images to determine cloud h i s tories life and the change in the cloud area with time. Later the life history aspect of the technique was by p a ssed estimates were streamlined technique. This technique uses a digital array temperatures made based firstly on to single produce volumetric output of the convection, rainfall rates. image an of infrared estimate and secondly and of to the infer Raining convective clouds are identified by / the threshold temperature of -20®C threshold was chosen to (degree centigrade). maximize the This determination of precipitating clouds, while m i n i m i z i n g the inclusion of n o n r aining The inferred rainfall, expressed as either Q total volumetric output (m ) or area averaged rain depth (mm), clouds. is calculated as a function of both areal extent of the storm at -20°C as well as the fractional coverage of the storm by satellite v o lume colder temperatures. Rainfall rates for each pixel are derived b y apportioning this calculated over the temperatures. storm For as a tropical function storms the of cloud-top computation rainfall rate for each satellite picture element (pixel) of in the grid is expressed by the relationship j = b i5 R v 10"3 / R ij = 0.557 2.g± . (Am / g...) Zb (6.3) (b... Z a ^ where R^ . is the inferred rainfall rate pixel of the grid, b^ is the / Zb) (mm/h) empirical (6.4) in the (i,j) we i g h t i n g R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 147 coefficient of the volumetric (i,j) pixel, output is the area Rv is the satellite-inferred 2 (km ) of the (i,j) pixel. the denominator of the right h a n d side The factor of 2 in of equation aries from the rainfall apportionment scheme. factor 10 _3 un i t s from m measured in the numerator of equation 3 to mm km from 3 -1 (m h ), rate of the storm for one image the 2 . A image a nd is The conversion (6.3) converts the 2 (km ) of is the area m (6.3) the ' storm the sum of areas of the p i xels whose temperature is less than or equal to -20°C, a^ is the fraction of the storm c overed by the kth temperature, is the empirical w e i g h t i n g coefficient corresponding to the kth temperature, and Z a b runs over all pixels K ££ in the storm that are -20°C or colder. The factor of storm rainfall rate fractional (0.0667) 0.557 is (16.7x10 coverage of 3 the product of the tropical 3 —1 2 m hr km ) echo area and for the inferred tropical storms multiplied by the product of the apportionment c onv e r s i o n constant (5x10 -4 ). a nd The weighting coefficient b enters in the calculations by which the inferred rainfall is r e l a t e d to the colder temperatures of the storms. exponential function function is 0 and 255 of temperature. and is an inverse The coefficient is of the form b = exp (^ an of the GOES digital count wh i c h takes on integer values between exponential It + c 2 V) / 11.1249 (6.5) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 148 wh e r e V is either GOES digital count or temperature (°C), the value of b ranges from 1 at -20°C to 4.55 at -110°C. For temperatures The warmer than value of the constants -20°C, and C 2 the values are zero. are given in the Table (6.4) . TABLE (6.4) Constants for the Empirical Weighting Coefficients Digital Counts 154 < D C < 176 Temperature (DC) 177<DC<255 -20° C<T<-310C (T) -32° C<T< — 110° C Cl 0.026667 0. 11537 1.784059 2.278682 C2 0.01547 0.01494 -0.03094 -1.1494 Altho u g h the total area of the storm canopy at -20° C is used in the rain computation of equation necessarily rain does not fall from the entire canopy. Rain is assumed to fall only from distributed (6.4), so the coldest that half of the canopy and is half of the rainfall from the coldest 10% of the canopy area, w i t h the remaining half occurring in the next coldest 40% of the canopy. 50/40-50 rain, apportionment etc.) (10% of This is called the 10- storm area has 50% of the and is based on radar studies intra cloud rain rate distributions made by W oodley et a l . (1980). The factor of 2 in the denominator of equation required to halve the rain volume, (6.3) is therefore Rv . The digital counts of the pixels are converted to b values through equation R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. (6.5). 149 The fractional coverage of the (a^) is determined and storm of the two values. each (bk ) . These products are ( S a ^ b ^ . The sum of b's (Zb) assumes one The decis i o n criteria for the appropriate value are based on a ranking of the b values. values for the storm (which define the top 10 determined. For example, 250th to correspond and 50% b values. are temperature, to the top 10% of the b values the coldest 10% of the storm. Similarly, coldest half of the the The b values in the top 50%, but not in the top 10%, top 10% interest, u s e d in 10%, two Because the b values are inversely are summed and are referred to as Z< 0 0. b. The the values) b if there were 500 pixels comprising top 50% of the b values comprise the storm. the the break points w o u l d be defined by the 50th and largest related After are ranked in descending order, breakpoints the storm, temperature is then multiplied b y the b value that corresponds to the temperature subsequently summed at are summed for if b.. is in the top iJ (6.4). I!« o ox ''0 b S l00, '0 b For (i.e .,coldest) values the in pixel of 10%, 2 t0o b is SQ If b^. is in the top 50%, is used. but not the If b. . is not in the top 50%, iJ top R. . is ij set to zero. A rain computation begins with the navigational digital data from which the clouds are isolated according to 20°C isotherm. Then the rain volume the for every cloud is computed from the component relationship of the technique, R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 150 and an empirical algorithm (the rainfall apportionment) subsequently maps volumetric rainfall on g rid squares constituting the cloud, of rain depths. as the to the satellite resulting in an array The resolutions of the array data spatial scale of are same the navigated satellite digital imagery. The VISSR data of 14th September 1978 (Julian day 257), is obtained for three images starting at different times (17 hours, hours, 47 minutes, 47 minutes, hours, 47 24 seconds, 20 seconds, minutes, 540 m i l l i s e c o n d s ) , 620 milliseconds), and T_ 17seconds, 980 milliseconds). minutes and 35 seconds. (21 v Each of these images are constructed from the data of time 13 (20 interval All three images are then merged t o g e t h e r , in order to have the maximum number of overlap w i t h the SMMR derived rainfall rates. The statistics of rainfall rates corresponding to these images Table at starting times T^ (6.5). The combined IR these times F igures T 2 and T 3 are shown in the rainfall (7.3) and for The this to for (7.4). Ag a i n some data points are found to maximum IR data longitude and 10° close estimates T , T 2> and T g are plotted on the w o r l d map in be on land where the rainfall rates mm/hr. rate set was were as high as 6.7 estimated rainfall rate over the ocean found to latitude location. be 6.1 mm/hr at -152° This location is very that at which the microwave technique estimates a R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 151 TABLE (5.5) The Statistics of Inferred Rainfall Rates froa IR Data at tines Tl, T2, and 13 IStatistics of IR I Rainfall Rates I I I At tine T1 R1 (oim/h) I I I At tine T2 R2 (mn/h) I I I At tine T3 R3 (na/h) I I I I Mean I 2.898 I 2.884 I 2.885 I I Median I 1.938 I 1.915 I 1.945 I I Mode I 2.00 I 2.00 I 2.00 I I Minicun I 0.00 I 0.00 I 0.00 I I Maxinua I 50 I 49.00 I 31 I I Sum I 200113.0 I 255921.0 I 199923.0 I I Std.Error I 0.010 I 0.010 I 0.010 I I Std.Dev. I 2.747 I 2.377 I 2.531 I I Skeuness I 1.877 I 2.806 I 1.667 I I Kurtosis 1 4.973 I 18.333 I 1.344 I I Mild I 4 I 89 I 3 I I Valid cases I 59287 I 88748 I 69290 I R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FREQUENCY DISTRIBUTION o f RAINFALL RATE Using (IR) data at Time T t , R(1) 60 50 - 40 - / n^ o« O tt c 50 *£ EC 20 10 - 2 4 6 10 8 16 12 Figure (6.8) FREQUENCY DISTRIBUTION of RAINFALL RATE Using (IR) data a t Tim© T2, R(2) 60 2629 0 for N 50 - /r r. •*o© s-s a « O TT _ c o u° w U3 oo tt 50 20 - 10 - 2 m 8 10 12 14 16 Figure (6.9) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. FREQUENCY DISTRIBUTION o f RAINFALL RATE Using (IR ) data at Time T3, R(3) 50 45 40 35 30 25 20 -, 10-, E72L 2 4 6 8 10 12 14 16 Figure (6.10) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 154 maxima for the rainfall rate over the ocean. is better than close, spatial resolution longitude) in because these This agreement owing to the limit on the figures (2° latitude X 2° a 2° difference in latitude or longitude might as well correspond to the same position on the earth's surface. Frequency versus class interval histograms have been constructed in a fashion analogous to that of the data for (6.10)) the three sets R 2 , and R 3 microwave (Figures (6.8) - which are the rainfall rates for entire images taken at times T^, T^, and T g respectively. from 0 mm/hr to 16 mm/hr. rainfall rates that classes range The class interval is 2 mm/hr. The are limit of an interval are The less not than or equal to the lower included in that interval. there are two peaks in all histograms just as in the case of microwave data. The first peak is between 0 mm/hr and 2 mm/hr and the second peak is between 8 mm/hr and The resemblance 10 mm/hr. between the microwave and IR distributions is good. Conclusions drawn from this resemblance must be limited to only the first peak since the second peak in the microwave data corresponds to points on land where microwave estimates based on ocean emissivity are next chapter not good. In the VII comparisons are made between the rainfall rates derived from infrared and microwave data. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. CHAPTER (VII) RESULTS AND DISCUSSION The estimated rainfall rates obtained sensors and September, VISSR techniques are described 1978, from the SMMR onboard GOES-W w as Corresponding ship data Atmosphere Data Set, Boulder, Colorado). of using here. onboard Data SEASAT-A obtained COADS different of 14 and and the analyzed. (Comprehensive Ocean- Release 1) w as obtained from NOAA (ERL, Although this data set did not provide the rainfall rate measurements which could be used as ground truth for comparison it did provide the weather condition at the time of observation. was coded in qualitative success a variable comparison. because two data sets The weather condition at that time na m e d PW w h i c h was employed for This attempt did not meet any the space and time differences between the (ship and microwave) were found to be larger than that optimal for rainfall rate comparison. (7.1) Maps of Rainfall Rates Using Microwave Data The processed data of SMMR w as This provided P ropulsion Laboratory). contained observation, the latitude and the longitude of by JPL the the R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. time (Jet of sensors 1 56 field of view, and the ten sets of brightness temperatures at SMMR channels. these The rainfall radiometric rates were estimated temperatures us i n g equation (6.2). estimated rainfall rates were then averaged over a 2° latitude x 2° longitude. The rainfall generated from these averaged values are (7.1) and shown and the second part from 120° world These grid rates in of maps Figures (7.2). The world map was divided into two parts, the first part exhibits the longitude from the from map from -80° -180° to -80°, to 180°. The remaining part of to 120° is not shown because the SMMR data over this region did not exist in the data set. It is evident from these maps that rainfall over v a ried up to 3.5 mm/hr. Rainfall rates higher than 3.5 mm/hr were scattered land. The data over these areas will be referred to as contaminated over data. land areas or regions very close to be expected contaminated data. from land are to true the model estimates for land Only rainfall rates for points far away considered give emissivity as true estimates and these are used for comparison with rainfall rate techniques. land Since radiative transfer calculations in this model involv o n l y ocean surface cannot ocean estimates of other For this data set therefore microwave e s t imated rainfall rates higher than 3.5 mm/hr are irrelevant and are not considered for comparison purposes. In Figure (7.1) estimated rainfall rates R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. for five 157 MAP OF RAIN FALL RATES USING SMMR DATA - A Tm t — -I —7—Z r--" 1= .3 2s .6 3= 1.0 4= 1.3 5s 1.6 6= 1.9 7= 2 .2 8= 2 .5 9= 2 .9 n= 3 .2 2 3 2 3= 3 .5 L_ C= 3 .8 0= 4 .1 E= 4 .5 n F= 4 .0 G= S . l 4 P 4 fl S H= S. 4 1= 5 .7 J= 6 .1 K= 6 .4 L - 6 .7 H= 7 .0 Z 212 I I N= 7 .3 0= 7 .6 P= 8 .0 0= 8 .3 R= 8.6 S= 0 .9 4 4 4 3 4 3 414 3 4 3 3 m i , T= 9 .2 U=9.6 V= 9 .9 W=I0.2 3343 ttet X -1 0 .5 Y=1Q.B Z=ll.l -=ll.S ■ = 11.8 * = 12.1 == 12.4 0= 1 2 .7 LRT=-5Q.Q TQ 5Q.Q . L0NC=-18Q.OT0 -6 0 . Q NUM 0F P01NT5= 319. URT LPN PV=RRGE= 2 MIN = .0 0 MRX= 1 2 .7 4 INCREMENT* .3 2 Figure (7.1) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 153 MAP OF FAIN FALL RATES ■ USING SMMR DATA /■ \- .3 2= .5 3= 1 .0 4= 1.3 5= 1 .5 6 - 1 .9 lL 1 1 J f' r ' / i i vI M L i • i " i 6= 2 .5 9 - 2 .3 ■f fi= 3 .2 B= 3 .5 ' —r l U - l J - h J — fi i im 7= 2 .2 r | «?> \ \ ,/■■v>i l II C= 3.8 D= 4 .1 E= 4 .5 F= 4 .8 r r ; i , i i■i A H Ay~ArfA 4:-A-7-J i 1“ 11 - A l l ' r n A l „ . — , .. "12II 1 ~ F "T G= 5 .1 H= 5 .4 1= 5 .7 J= 6 .1 h1 K= 6 .4 L= 6 .7 Ms 7 .0 K- 7 .3 0= 7 .5 P= 8 .0 D= 8 .3 l J - l l M ,W \ A \ \ \ A_- ' I \ Q . i 4_ _ _ I ' y a _ i 83 i, I II41 i i ^ L 7 ; / / 7 / / / / 1 L -U J ./ R= 8 .6 S= 8 .9 T= 9 .2 U= 9 .6 V= 9 .9 H=!Q-2 X =10.5 7 = 1 0 .8 X \X & T M U 'j L u ? L flT = -5 0 .O T0 SO.O . L J ‘iu= 120.0TA IBO.O MUM 0F P31NTS= 13S. LflT L0N flVERRGE= 2 M1N= .0 0 MflX= 1 2 .7 4 INCREMENTS .3 2 Figure (7.2) R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 2= 11.1 -=11.S i= 1 1 .8 ♦ = 12.1 = = 12 .4 0 = 1 2 .7 159 orbits of SMMR data are seperated by shown. Each of these orbits is a fixed distance on the globe and is observed at different times of the day. It can be observed from this figure that maximum rainfall over the ocean is found at -155° at longitude, -138° 10° latitude in the northern hemisphere and longitude, -36® latitude in the southern hemisphere. In Figure orbits of (7.2) m i c rowave estimated SMMR rainfall latitude and at 164° in the longitude, southern 32° latitude. longitude, Most of 30° the h e misphere is land contaminated and therefore irrelevant for comparison purposes. figures two passes is shown. Maximum rainfall over the ocea n in northern hemisphere is found at 146° d ata for In both the no rainfall is found at the equator wh i l e rain rate is found to increase w i t h latitude in both the hemispheres. (7.2) Maps of Rainfall Rates Usi n g Infrared Data The inferred rainfall rates from d e s cribed in latitude x 2° plo t t e d on chapter longitude. VI, are These VI S S R averaged over a averaged values grid of 2° are then The w o r l d m ap has again been divided into two parts in whi c h the longitudes vary from from as a wor l d map in the same w ay as des c r i b e d in the case of SMMR data. and data, 120° to 180° -180° to -80° and IR estimated rainfall rates for R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 160 these regions have been pl o t t e d in Figures In Figure whi c h m i c rowave data. is an at agreement -152° inferences. In in Figure -36° agreement with IR estimates in the land contaminated areas they are not useful for invalid microwave inferred rainfall rates. In the nothern hemisphere IR estimates far 10° (7.4) and although they are valid results, with longitude, reasonable s outhern hemisphere correspond to points longitude, In the southern hemisphere IR estimates over latitude which fact is also comparison (7.4). w i t h that inferred from the ocean indicate a maximum at -132° m i c rowave and (7.3) maximum rainfall rate over the ocean in the northern hemisphere is found latitude, (7.3) from are found only at the SMMR subsatellite track and therefore cannot be compared. (7.3) C o mparison of Rainfall Rates Inferred (MW) and Infrared of and M W data defining the Microwave (IR) Data The coincident rainfall (VISSR) from rates derived from (SMMR) h a v e been compared. coincident data The criterion rainfall rates depends on the distance between the two data points of IR and MW, if IR that is the differences between the latitudes a nd the longitudes of a IR and a M W data point are less p r e d e f i n e d threshold, than or equal to a then these two data points are grouped R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 161 MAP OF RAIN FALL RATES USING IR DATA 1= .S 2= .5 3= .3 4= .3 5= t . l r / I u r ^ \ > v 6= 1.3 7= 1.4 0 = 1 .6 ~ i ,— JG [ 3 ' 1 I— /SC U r. ~HC Bi MpP^ I r N B!M)G T 3 1 , U I p 1 i i C.K i M iHu 0a 9 / L o 9 f E L 5 S - l E 3 R 0 pig 9= 1 .7 A i .1 F JM.K3 9 I ViP • 10 0 9 C L LIQ i uFr Jj nK ^gst' ,riT A 3 3 IK 2 [3 .1 I B= 2 .1 C* 2 .2 2 0 3 ,0 b 9 J C _ € ,3 S K 9 J J H fi F E Y*B >3 R I C H G L B s u M H Gl7 7 3 5 I K S O E 3 -6 0 2 C S P 8 9 B_ I 9 2 V j is H [ 0 ?\r F C 3 2, 1T2‘- E lfJ F B R M J W Q M H - »K J R C C(6 3 5 ' 7 7 S 1.0 fl F 0 C /6 0 0 2. J fl B 5 2 1 R= 1.3 I M £ J ia E K 8 13 G 19 P ,0 H 0 0 914 6 I 3 2iC 6 D= 2 .4 E= 2 .5 F= 2 .7 C= 2 .3 H= 3 .0 1= 3 .2 J= 3 .3 K= 3 .5 -1 ! _L L= 3 .7 K= 3 .8 N= 4 .0 0= 4 .1 P= 4 .3 3 . OE 1 3 1. 2 0= 4 .5 2 2 R= 4 .6 fl}HR I Z 2\D __2 D I f i’c D ? rlfi8 2 2 B NF 9 I P C C i B 0 5'.4 G I 1 \ 9C 9 0‘f H 6 9 _ R JE 19 2 1 \ f l 9 K EV* 2 9 T273P5T B S C fl \ S= 4 .8 2 1 2RI23H 1 ‘ T= 4 .9 U= 5 .1 . , F * I J G 7 |3 2 2 1 I C iI H F Q C j3 _ f l 2 V= 5 .3 N J ?F 2~t\l ITDT R/TTB 9 2 10 M K K O il G H 8 L M= 5 .4 Xs 5 .6 G 0 3 7 S S I GO H R 4 0/9 B 0 R 213 0 P H Mfc 3 E f l 7 E Y= S. 7 LhCib Z= 5 .9 -= 6 .1 i = 5 .2 += 6 .4 == 6 .5 0= 5 .7 “ 7 ^ LRT=-5G.O T0 5 0 .0 . L0NG=-18O.OT0 - 3 0 .0 NMH Bp POINTS: 569. LRT LBN RVERR3E= 2 h iN = .3 0 0 HflX= 6 .6 9 7 IHCREKENTs .160 Figure (7.3) R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 together and considered for comparison. A common longitudes data and set, which predefined threshold latitude and in longitude. a sub the latitudes, inferred rainfall from both IR and MW data, was obtained for a value contained data set equal to 0.5° in Then by decreasing this threshold was constructed. This sub data set contained the observations w h i c h are more closely spaced and are therefore expected to be better correlated. The frequency distribution general Figures not a normal (6.1-6.10). of rainfall distribution rate is in as already noticed in The measures of difference that are used to describe the discrepancy between M W and IR estimates listed in Table (7.1). The sample ratio, w hi c h indicates that the accumulated M W sample is larger (Rg>l) or accumulated IR estimates for the sample. sample ratio, estimates sample (MWR) RM , are greater than 1 which the than the mean (IRR) averaged over the measure, a (MWR/IRR), which can offset R ^ from those values which ED , is are defined effect of the ratio greater than one and is called the "factor always the Similarly the The values of this ratio Therefore another eliminates (Rg <l) for average of the ratio of the M W to the IR estimates of size N. less than one. is R g , is a measure estimates smaller are of difference" whose value is less than one. The relationship between ER and R^ is analogous to that between |x| and x. The mean error R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. of a 164 TABLE (7.1) Measures of difference between the microwave rainfall estimates (MWR) and Infrared rainfall estimates (IRR) for a sample of size N D ifference Definition Measure Sample Ratio (?-g ) Mean Sample Ratio ZM W R/ZIRR (R^) Factor of Difference (E ) ri Z(MWR/IRR)/N (ZR/N) [R = MWR/IRR if M W R < IRR R = IRR/MWR if IRR<MWR] Average Error (p ) [ (MWR-IRR)2/ N ] 1/2 Root-Mean-Square Error (Ep_M S ) Normalized Root-Mean-Square Error Z(MWR-IRR)/N (Norm. [Z{(MWR-IRR)/lRR}2/ N ] 1/2 E_,M S ) Normalized Bias (Norm.3) Normalized Standard Deviation (Norra.STDEV) Correlation C o e f f i c i e n t ^ 0 } ( 1 / I R R ) [(Z(MWR-IRR)}2/ { N ( N - l ) } ] 1/2 (1 / I R R ) [(Z(MWR-IRR)2R ) 2 N ( M W R - I R R ) 2/ ( N - l ) } ] 1 / 2 (MWR.IRR - MWR . IRR) / [(MWR" - M W R 2 )(IRR2 - IRR2 )]1/2 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 165 sample of size N is calculated from the M W and IR estimates. m easures the The root-mean-square absolute difference estimates of M W and IR pairs, error (norm. er m s ^ measures bias be the the E_.„0 , RMS rainfr.ll rate normalized absolute The of the sample and Comparisons the square rate. of error the squared the sampling are made for the linear quantities (bias a nd standard deviation) rainfall RMS difference as mean variance be t w e e n error, decomposed into two squared quantities, difference. IR (RMS) between whereas n o r m a l i z e d by the IR rainfall rate. can difference All n ormalized by a sample average the measures of difference are calcu l a t e d for the data of M W and IR images corresponding to the star t i n g times T^, T 2 , and T g . C o m bining these three images of IR data a combined data set measures of difference the present study d i fference as these well was obtained. are defined in Griffith measures depend on These (1987). the In spatial as the temporal difference of the data points. I nitially the time dependence was the spatial differences 0.5°, 0.1° and 0.05° considered and (the differences of latitudes and longitudes of M W and IR data) to not whi c h were less than or equal were considered for calculating the m eas u r e s of difference as sho w n in the Tables best co r r e l a t i o n between the M W and IR (7.2-7.4). inferred The rainfall rates is obtained for the spatial difference equal to 0.05°. R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 1 66 TABLE (7.2) Maasurs of Difference Between ffiJ and IR Rainfall Estimates Whose spatial Difference is less than equal to 0.5 I. . . . . . . - ■ ! - - - - - - - - - - !- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - - I Parameter I AT Tine T1 I AT Tine T2 I AT Tine T3 I Combined j---------- j__..........r-----------1---------- 1--------I Humber of data I 9S4 I 940 I 372 I 2736 I Sample (N) I I I I J----------------- T------------------- J------------------ J------------------ J--------------- JAverage IR Rain T I fall'Estisate I I (IR) I i l I 3.15 I 3.13 I I T-------------------T-------------------- T------------------- J------------------- J---------------lAvsraoe HH Rain I I I ! I fall Estimate I 1.32 I 1.33 I 2.03 I 1.33 I (Mi) I I I ! 3.2 I----------------- I I l 3.05 r------------------ :------------------ t --------------- I Sample Ratio I 0.5 ! 0.53 I 0.55 I 0.53 I (Rs) I I I I i— -......... i...........i---------- 1--------I Hean Sample I 0.34 • 0.30 I 0.S5 I 0.90 I Ratio (Res) I I I I I- - - - - - - - - - 1- - - - - - - - - - !- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - Factor of Biff. I 0.47 I 0.34 I 0.51 I 0.51 I (Er) I I I I I- - - - - - - - - - 1- - - - - - - - - - !- - I- - - - - - - - - - 1. . . . . . . . I Averaae Error I -1.23 ! -1.07 I -1.12 I -1,15 I (E) I I I I T---------- j----------- ;-----------j-----------t--------I RHS-Error I 3.7051 I 2,3734 I 2.322 I 3.1733 I (Eras) I I I I J -----------------------------------1-------------------------------------- T------------------------------------ T------------------------------------ r.............................. .. INorn. RHS-Error I 1.0131 I 1.1325 I 0.B135 I 1.0241 I (Horn.Eras) I I I I I- - - - - - - - - 1- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - - - - 1- - - - - - - I Norn. Bias I 0.4 I 0.35 I 0.35 I 0.37 I (Norn. 3) I I I I J ---------------------------------- T-------------------------------------- T------------------------------------ T------------------------------------ r __________________ I Horn. Sid.Dev. I I (NORM. STDEV) I 1.0331 I I 0.3752 I I 0.S233 I I 0.3449 T----------------- T------------------- i------------------ 1------------------ J--------------- I Correlation ! -0.1725 I coeff. (f) I I I- - - - - - - - - - 1- - - - - ! 0.1439 I 0.2312 I ! !- - - - - - - - - - 1. . . . . . I 0.043S 1 - ...... R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 167 TABLE (7.2) Measure if Difference Between Til! and IE Rainfall Sstisates T'hcss spatial Difference is less than equal to 0.1 Parameter AT Tiae T1 I 1- AT Tins T2 I 1- - Nuaber of data I Saapie (N) ! Average IR Rain I fall Estiaaie I f'l'i r 4. OS AT Tine T3 I I37 I Ccibined 119 .S3 Average ill* Rain I faH'Sstizate ! (Mil) I 2.31 2.03 Saapie Ratio I CEs) I 0.43 0.59 (lean Saapie Ratio (Ez) I I 0.93 0.73 Factor of Diff. 1 (Er) I 0.29 Average Error I ’(E) I on -o.se -1.57 4.359S 1.9542 2.532S 3.5339 1.1046 0.5461 0.5208 0.3037 0.57 0.S7 0.3? 0,53 0.5! --------------- j_ RMS-Errcr (Eras) Mora. RHS-Error I ((forz.Erzs) I 1Nora. 31 as I (((ora. 3) ! (fora. Std.Dev. I I (NORM. STDEV) I -----------------------------T I------I Correlation I I coeff, (f) I 0.41 1.0925 0.5197 0.5343 0.3721 -0.4707 O.aiJV 0.6/Ql 0.131 R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 168 TABLE (7.4) 'ieaaure of Diff=rsnce Between ;!W 2nd IR Rainfall Estimates Whose spatial Difference is lass than equal to 0.05 -------r... ....... J- Faraneter )T Time T1 I AT Time 12 I AT Time T3 I Co,shined --------------------------- T- 75 I Himbar of data I I Sample (?!) r IIAverage IR Rain I fall i-jtiaata I (IS) n co J 1 uU lAverags !!H Rain I fall Estimate 2.02 I---------I Sample Ratio I CEs) I --- I Hean Sample I Ratio (Ra) IFactor 0 ? Biff. I fEr) I. . . . . . . . . . I Average Error I (E) I---------I SHS-Error I (Eras) T-------IKora. RUS-Error I (Nora.Eras) r---------- 2.37 0.57 0.31 0.33 - 0.7 0.45 0.55 1.00 -0.44 3.3545 0.57 -1.32 a Li 1.1373 -I- - - - - - I 0.4301 I 0.9513 r ro o i \ 7nnn V* / VUi. 0.34 I (Nora. 2) I ----I liara. Std.Dev, r i /Mi-'r.M ilUlttl. 1.2003 0.48S5 ornc»M -JIUL.V 7__________ I Ccrrslaticn I ccsff. (f) -0.241! R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 169 The M W rainfall rates are directly correlated with rainfall T3 rates at starting times T2 and the but IR anti c orrelated with the IR image at starting time T 1 . Time is investigated by considering time differences of 1 hour, hours, 3 hours, 2 and 5.5 hours between M W and IR estimates. The exact time was calculated for each pixel from the starting time of the image. (7.7), lag in the IR image In the Tables (7.5) to the measures of difference are given as functions of time difference and for spatial differences 0.5°, 0.05° respectively. All data points c o mparison of oceanic rainfall rates in 0.1°, and considered for these Tables were not constrained in any way. Since the IR technique is tuned for estimating the tropical rain a constrained data set is obtained from the full data set based on the c o n dition that the latitude of IR and MW data should lie w i t h i n the 30° c omparisons are made for d ifference are calculated. g i v e n in the Tables 0.5° and 0.1° (7.8) data set 3 hours, set coincident not spatial measures of These measures of difference are and (7.9) for spatial and 5.5 hours. data points In this were d i fference less than or equal to 0.05°. are and The differences respectively for the time differences 1 hour, 2 hours, no this from the equator. constrained found for data spatial U n fortunately there enough coincident data points even for the case of differences 0.5° and 0.1° wh i c h can give R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. any 170 TABLE (7.5) Measure of Difference Between Mr) end IR Rainfall Estimates Nhose spatial Difference is less t'.ian equal to 3.SB I I Parameter ITime difference Time difference ITime difference ITime difference I 2 hours I 3 hours i 1 hour I 5.5 hours I I Number of data I I Sa£ipl& (N) I ‘37 374 I 1331 I 2376 I 2.'35 3.12 ? 1 n «a 0. i L I 3.14 I 3.33 1.32 I 1.37 I 1.35 I 0.3 3.52 I 0.65 I 3.62 I 3.63 3.35 I 0.37 I 0.3 I IFactor of Diff. I I (Er) I 3.52 0.52 t i n cn U .J j I 3.51 I I Average Error I I (E) I -2.37 -1.2 I -1.14 I -1.13 I 5.443 3.1334 j 4 0» ii u4 .u I 3.2317 I 3.7224 Iliorn. RMS-Error I I (Nora.Eros) I t _ __ _______________ r _____ i — ------------------- 1------I Nora. Bias I 3.7 I (Nora. B) I n nn « c tJ.O Oi J I 1.0211 I 1.3181 I 3.33 T T 1 fl 70 t l i JU T 1 I Nora. Std.Dev. I I (NORM. STDEV) I n IAverage IR Rain I I fall Estimate I I (IR) I r. — — T— — IAverage MU Rain I I fall Estimate I I (P!W) I I Sample Ratio I I (Rs) I IT-_ — — — — —— T j . -. — I Mean Saspie I Ratio (Rn) I I T --— - - - - - - - - —-T --— r t I I RMS-Error (Eras) I I t _____________________r I Correlation I coeff. ( f ) ______ I I 1.7233 -3.3337 fl 0 .3 7 /0 I 0.3113 T 1 r» ^ 7 A n ii« Z l 30 ? i n «n n r 0 .1000 I 0.1614 t 1 a O .U O // Ti. n n o R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 171 TABLE (7.S) Measure of Difference Between KW and IR Rainfall Estimates Ufiose spatial Difference is less than equal to S.IQ I - - - - - - - - - - 1- - - - - - - - - - I Parameter ITima Difference ITima Difference Tiae Difference ITine Difference I I 1 Hour I 2 Hours 3 Hours I 5.5 Hours I INumber of data I I Sample (If) I I I 43 73 I 123 I IAverage IF: Sain Ifall Estimate I (IR) 1r I I 1r I I 3.7 nu« nc vlU T1 hj.DO rn T1 IAverage MW Rain Ifall Estimate I (HU) 1f I I Ti I I 2.13 2.14 I 2.37 I I 1 T I Mean Sample I Tl I Ratio (Rs) T1 IFactor of Diff. I I (Er) I I I T 2 Tt, T i I I 0.53 3.64 I 0.57 I 3.£7 0.73 I 3.83 I 3.53 3.53 I 9.51 I I Sample Ratio I (Rs) ___ IAverage Error I '(E) I TI I T1 -1.51 -1.22 I -1.56 I I RMS-Error I (Eras) I I I I 2.5 2.23 I 3.52 I lilora. RMS-Error 1 I (Nora.Eras) I 1 I 3.52 0.53 I 9.81 I I Norm. Bias I (Nora. 3) I T1 I iT 3.41 3.37 I 3.43 1 INorm. Std.Dev. I I (KORN. STDEV) I I I 3.54 n a .ju Ti U$UI Ti I Correlation I coeff. (f) I T 3.53 CC u. uO I 0.13 I I T i en I R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission. Measure of Difference 3etueen MH and IR Rainfall Estimates Hhose spatial Difference is less than equal to 0.35 T I I T Parameter T ITioe difference I t hour r T _ T T 'iBe difference ITime difference ITime difference I I 5.5 hours 1 2 hours I 3 hours T "li I Number of data I I Sample (N) I I 23 I 29 I *3 cn 0 1 uo I 2.31 I 2.37 I 1.31 1 1.88 I 1.93 I I Sample Ratio I I (Rs) I 3.52 I 3.67 I 3.57 I I Mean Sample I Ratio (Rni) 1 I 3.73 I 3.78 I 3.85 I IFactor of Diff. I I (Er) I 3. £3 I 3.53 I 3.54 I I Average Error I I (E) I -1.32 I -0.92 I -0.95 I I I I I 3.2124 I 2.1232 I 2.5533 I INorm. RMS-Error I I (Norm.Erms) I 3.4331 I 0.5433 I 3.7C32 c* cJ.u*T T i U ig - T T 1 iJ .w l'T I Norm. Std.Cev. I I (NORM. STBEV) I 3.7377 I '3.7015 T fl n n r * « I Correlation I coeff. (rt 3.1125 I T I 3.4733 I IAverage IR Rain I I fall Estimate I I (IR) I T ___________________ T ____ IAverage MU Rain I I fall Estimate I I (MU) I I I RMS-Error (Eras) Norm. Bias (Norm. 3) I I n o,i L i I I A T T l r U .O O tll ** - - - - - - 0.159 1 - - ^ 1 T r R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 173 TABLE (7.3) Measure of Difference Between MH and IR Rainfall Estimates Hhose spatial Difference is less than equal to 2.58 I Parameter I Nuaber of data I Sample (H) I- - - - - - - - - Time difference ITiae difference ITime difference ITime difference 1 hour I 2 hours I 3 hours I 5.5 hours I 66 I 157 I 167 T IAverage If: Rain I fall Estimate I (IR) I 2.17 I 2.65 i I j *> £5 IAverage MU Rain I fall Estimate I (MU) I 1.43 I 1.42 I 1.42 I Sample Ratio I (Rs) I 8.63 I 3.53 I Mean Sample I Ratio (Ra) I 8.33 I 8.75 T I f* ir U. /b IFactor of Diff. I (Er) I 3.65 I 3.64 I 8.64 I Average Error I (E) I -0.58 I -1.24 I -1.24 1 I RMS-Error (Er sis) I 1.7443 T a nn«n I 2.3313 INorn. RMS-Error I (Horn.Eros) I 0.5631 I 8.5111 I 3.5111 I I I 3.32 I 3.47 I 8.47 I Norm. Std.Dev. I (NORM. STDEV) I 3.746 I 8.7637 I 8.7537 I Correlation I coeff. (p) i I 3.3733 I 3.8453 I 3.3458 Norm. Bias (Norm. B) J «•* j j 8.53 T I r R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission. 174 TABLE (7.9) Measure of Difference Between MW and IE Rainfall Estimates Hhose spatial Difference is less than equal to 3.13 I I Paraaeter ITiae difference Tiae difference ITiae difference ITiae difference I I 3 hours I 5.5 hours I I 1 hour 2 hours I Nuaber of data I I Saaple (N) I 7 I I 11 I 11 I IAverage IR Rain I I fall Estiaate I I (IR) I 1.84 I i 1.S4 1 1.64 I 1.44 I 1.44 I IAverage MU Rain I I fall Estiaate I I (MU) I I................!................ I Saaple Ratio I I (Rs) I 1.48 i ! 3.89 1 I 3.83 I 3.83 I I I 3.S3 I 1 0.33 I 3.83 I IFactor of Diff. I I (Er) I 3.71 1 I 8.78 I 3.76 I I Average Error I I (E) I -3.13 I 1 -3.28 I -3.23 I I I I I 3.5239 1 I 0.4679 I 3.4679 I IMora. RMS-Error I I (Nora.Eras) I 0.3213 I 1 0.2359 I 3.2359 I 0.12 1 1 8.13 I 3.13 I 0.3233 I I 0.272 I 3.272 I 1.83 I 1 3.6181 I 3.8131 I I Mean Saaple I Ratio (Ra) I I RMS-Error (Eras) Nona. Bias (Nora. 8) I I I Nora. Sid.Dev. I I (NORM. STDEV) I I Correlation I coeff. (/) I I Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 175 definite information regarding the time lag between IR estimates. However, MW and in the absence of data for the time difference 1 hour the best correlation is obtained spatial difference equal to 0.1° for the and for the time difference equal to 2 hours. (7.4) Sensitivity Study of Rainfall Rate Rainfall rates have been simulated for various types of atmospheres to study the sensitivity of rainfall rate to the variation of parameters such as am i n , am a x , p, atmospheric a temperature. The drop and parameter refractive p. index, Since is the very mb spectrum is defined by . , a and the shape of the distribution is nun m ax r the 500 microphysics, temperature decided by especially the sensitive, the rainfall rate variation w i t h the 500 mb temperature has been investigated. Microwave brightness temperatures at 10 SMMR channels have been simulated for 48 different types In atmospheres. these simulations one parameter at a time was varied and the rest of them were assumed constant. for of each of the 10 SMMR brightness Therefore 48 temperatures values were obtained to study the effect of variation in each parameter. Rainfall rates temperatures were using calculated the from relation these (6.2) brightness where Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. the 176 corr esponding regression channels given are coefficients in the Table for the best 5 (6.2). From the detailed a nalysis of this study following inferences are drawn. (a) It was observed that the variation of a . from 50 pm to v ' man 100 pm produces a change in rainfall rate of 3.0 X the fact _2 10 (mm/hr) which is insignificant. that at the lower end of the the order of This is due to spectrum the e xtinction and scattering cross-sections which are functions of the drop diameter are extremely small. (b) The Darameter relation (4.65) incremented a given in was calculated from the empirical max r by 5 steps. Stephens It was then In each step the increment was 10% of the calculated value. For a rainfall (1962). the _3 changes by a factor of the order of 9 X 10 rate 50 % change in a max (mm/hr). The explanation for this lies in the fact that total number of drops decreases with the the diameter and therefore the extinction and scattering coefficients do not change significantly with am a x * (c) The variation of the shape parameter p in the drop size distribution was found to change in produce as much as 3 (mm/hr) rainfall rate. The bulk of this change was found to occur between the values of p from -2.0 to -1.0 as in the Figure parameter orographic p for rain (7.5). Ulbrich different shown (1983) has categorized the types of rain; p < 0 for (indicating a broad DSD with large numbers Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 177 MI IU VERSES RAINFALL RATES FOR ATMOSPHERE NO. 1 P. 5 HO - 1.0 0 .2 .4 .0 .0 1.0 1.2 1.4 1.0 1.0 2 .0 2 .2 2.4 2 .0 2 .0 3 .0 3 .2 RAINFALL HA'IBS (M M/M R) USING FIVE SMMR CHANNELS P lg u re ( 7 . 5 ) Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 3 .4 3.0 178 of small drops) (corresponding and to 0 a < < jj 2 narrower for thunderstorm rain DSD w i t h reduced numbers of small d r o p s ) . For wides p r e a d or stratiform rain the of For showers are more variable but tend to be positive. jj it is not possible to define exact value of p. values However, U l b r i c h has shown from analysis of past studies that m o s t l y within the range -2 < The jj simulation of b rightness temperatures requires the values of jj particular values of known and perhaps jj may not be fixed. reduce the rain rainfall rate and the Nevertheless this is the best that present ranges circumstances. are not Random v a r i a t i o n of type was used. jj This accu r a c y of the simulated brightness temperatures and consequently of the the for given rain types over the known ranges for each to Although for different rain types are reas o n a b l y well known, tends lies < 4. k n o wledge of jj for various rain types. of jj the relationship brightness one can be t w e e n temperatures. do under the If the values of jj were to be kno w n more precisely then the inversion of rainfall rates w o u l d be more accurate. however, data In the absence of such precise kn o w l e d g e of any c o r roborating measurement that yields jj as jj a input must of course m a t e r i a l l y improve the reliabilty of estimations of the model because it will reduce the range of u n c e rtainity in jj . (d) At 500 mb and temperatures below 0°C only ice exists. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. A 179 variation of the temperature at this altitude from 0°C to - 40°C results only in the There change is no change of phase. of the ice temperature. The effect of this temperature variation on the rainfall rate was found to be of the of 3.0 X 10 —2 mm/hr order which may be considered negligible. Therefore one concludes that ice temperatures do not rainfall rate estimations significantly. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. affect CHAPTER (VIII) RETROSPECT In this study a microwave which takes effect into for the microwave precipitation basic physical raaiometery been calculated program clouds atmosphere has been developed. and have been as a function the of an concepts developed. and in In order to do mathematical emission and absorption from the ocean have transfer the Mie-scattering polarization nonisothermal inhomogeneous this account radiation of Microwave atmosphere the appropriate variables of state using classical theories and experimental data obtained by particular, microwave different several ocean investigators emissivity frequencies surface has ranging conditions been from (see Extinction properties of microwaves by have 1 the past. In calculated at all to 300 Figures liquid GHz for (4.1-4.6)). hydrometeors been considered in detail using the Mie-theory and the Gamma drop size distribution. the in Gamma drop size rate and a size parameter graphically in Figure The total number of drops in distribution depends on the rainfall (4.9). jj . This dependence is shown Extinction coefficients at SMMR frequencies for liquid hydrometeors have been calculated for different values of the size parameters jj , N Q> €, and S. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 181 These size parameters decide the distribution extinction curve. The coefficient graphically transfer program has microwave 600 scattering, model rainfall (4.10-4.26). been brightness used the rate temperatures for both 5.16). (6.6 theoretical homogeneous, coefficient of liquid water as and functions vertical of to 0.1 mm was GHz) increase where temperatures as linearly at high polarizations were rainfall rate in Figures The brightness temperatures clouds, shown The brightness temperatures at SMMR frequencies horizontal presented size at SMMR frequencies for h o r i zontally Scattering is calculate drops whose diameters were less than equal negelected. drop The microwave radiation to plane-parallel, atmospheres. of variation of liquid h y d r o m e t e o r s 1 with in Figures shape at with low SMMR rainfall frequencies (5.9- frequency rate for low the brightness are non-linear functions of rainfall rate. The rate of change of brightness temperatures w i t h rainfall rate were positive initially and their saturation points. then SMMR frequency 37.0 brightness temperatures at the two with rainfall rate. effects reaching rainfall GHz. The rate for difference polarizations the in decreases This can be explained to be due to the underlying surface being obscured polarization after The saturation point varies with frequency and has minimum value of largest negative are by eliminated. hea v y rain so that The ice hydrometeors Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 182 present in the form of ice clouds scatter more h igh frequencies temperatures. hydrometeors frequency. On and result the other increases brightness which induces darkening b rightening at low frequencies. frequency radiances respond bottom the presence of of the ice temperature at low These results can be explained to be due scattering at in decreasing the brightness hand the strongly to the at high frequency and It was observed that the low more to water drops at the clouds while the high frequencies radiances were more sensitive to ice near the cloud top. It must be particles m e n tioned were that in assumed spherical this study, and emission and extinction coefficients any changes due to particle density and shape were not included. between the two phases their extinction refractive index ice the in the variable The difference (ice and rain) was considered through coefficients and particle which size are functions distribution. ice and Gamma drop size of The particle size distribution given by Marshall and Gunn for and (1958) distribution for rain were assumed. The best possible subset of SMMR channels for retrivals was obtained (6.6H, 6.6V, 21.OH, rainfall 37.OH, 37.0V) using the optimization technique "Leaps and Bounds" given by Furnival and temperatures Wilson of these (1974). The frequencies computed brightness were then inverted for Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. rain 183 rainfall rates using multiple linear regression method. rainfall rates September, were inferred from the SMMR data of 14 1978 over the Pacific ocean. These estimates were m a p p e d on the World map for comparing the rainfall elsewhere. Another means of of rainfall (7.4). rainfall World GOES satellite. al. (1976). The The computer maps are shown in Figures The comparisons made o btained the from VIS/IR data are generated from the scheme given by Griffith et generated obtained estimating rainfall rate was from infrared data obtained from estimates The for estimated rainfall (7.1)rates u s i n g these two different types of data have shown promising results Tables (7.2-7.9). Conclusions and Comments The main ingredients in rainfall rate estimation through microwave remote sensing are the microwave radiative transfer surface. model Most scattering and models for the atmosphere and the ocean radiative and hence e stimation problem. large rainfall transfer When rates or models are not the radiative do suitable not include for rainfall tranfer .involves higher microwave frequencies the scattering plays a dominant role and cannot The to a three dimensional inhomogeneity of space leads be neglected. radiative transfer equation w h i c h is very difficult to solve and consequently requires an exhorbitant amount of computer Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 184 time. It is impractical to deal with such an equation even though this would be ideal for treating the problem in an exact way. If remote one assumes that the space inhomogeneity lies only in the vertical direction solution of the radiative sensing transfer equation greatly in both conception and application. This then the simplifies assumption does not severly affect the solution of the problem at hand. The polarized ocean surface emits whose intensity is zenith-angle dependent horizontally homogeneous; is considered intensity of the emergent considered azimuthally the second phase validity two-streams order scattering of radiances independent. matrix radiative transfer equation. Eddington's radiances only atmosphere approximate microwave can This in atmospheric one needed This assumption is It the therefore enables elements approximation. and the be to in the called the involves only particles. The this approximation has already been tested for rainy atmosphere and is severely limited only when there are big hailstones rough. The or the polarized surface reflectors components of are extremely microwave ocean emissivity are calculated from the Fresnel equations and the Debye equations for dielectric constant. For surface, various temperature, factors such as a surface the frequency of observation and the calm ocean salinity, angle Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. of 185 incidence influence the surface wind generates surface whose ocean surface emissivity. foam emission and produces a known, rough The physics of (emission and reflection of microwaves) and only empirical estimations have estimation ocean and reflection properties are very different from those of the specular surface. the foam The sea is not well been used. The of rough ocean surface emissivity can perhaps be improved by a composite model in which rough given by a two-scale scattering emission is given water theory surface and the is foam emission by from a the layered dielectric theory. The theoretical basis for estimating the rainfall rates depends upon particles, attenuation (i) (ii) the the size, index properties shape of of the and phase of the rain refraction and radiation. (iii) The estimation becomes better depending upon how accurately the signature is e s timation of different drop seperated the size distributions used, latter rain DSD. drop the size distributions rain is background signature. provided (DSD). Of Marshall Palmer and the Gamma The by the the two DSD, the has been found to closely simulate the actual DSD in different types of rain. types from the can be The behaviour for various parametrized by the exponent jj rainfall in the Gamma Ice particle distributions for various rainfall are not kno w n and are perhaps not unique. However, better knowledge Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 186 about ice the ice particle distribution and their shapes in the clouds can improve the estimation of microwave ex tinction coefficient of rain. The passive method for estimating rainfall rate over the ocean at SMMR frequencies relies relationship between m i c rowave rainfall rate. At high through upon the fundamental brightness temperature and frequencies the penetration depth the precipitation becomes shorter due to scattering and the rain becomes opaque even at very low rain rate. can be observed in 37.0 gets saturated d e creasing w i t h the spatial GHz at very brightness low rate rain rate v e r y rapidly. On which and then starts the other hand resolution of a radiometer for a given antenna size increases with frequency. two rain temperature This competing influences A compromise has between these to be found which will have better resolution and estimation. Microwave rainfall estimates are infrared estimates when time interval and within differences. with comparable 0.1° of latitude and longitude Correlation bet w e e n these two estimates reduces increase microwave in the time a nd spatial differences. underestimates the c omparison w i t h the IR. The d iscrepancy estimates the they were found w i t h i n 2 hours of estimates needs to be tuned for local effects. time with can be attributed to Most rainfall between various The IR of rates the in these two factors. The Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 187 principle employed different in in nature. these two methods directly related Microwave estimates filling, (ii) of the sensors, completely In IR technique the rainfall is based on the cloud's top temperature whereas are are to suffer the from microwave hydrometeors problems like radiances themselves. (i) different instantaneous field of view (iii) height of the rain column and beam (IFOV) (iv) inhomogeneities within a radiometer's IFOV. If one were to ask w h i c h technique might best estimate the rainfall rate, it is required. the answer to this w o u l d depend on where If one needs to know it over the ocean then passive microwave would be the best choice. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY Abramowitz, M. Mathematical and I.A. Functions Mathematical Tables. Arkin, P.A. coverage Stegun high The cloud D., parameter C.W. Atlas, D. and O.W. Space. Thiele Workshop Center Greenbelt, MD. fractional Mon. Wea. R e v ., 107, (1982): 1382-1387. The multi drop-size parameters. A m e r . Meteor. J.A. (1981): Precipitation Measurements NASA/Goddard Space Flight 20771. Chemla Augustine, and and rainfall accumulations during report, , and A.C. Boston, of remote measurement of r a i n f a l l . NASA Tech. Memo., 128 pp. of Graphs, between Ulbrich and R. Meneghini 83971, from Handbook New York. relationship GATE over the B-scale array. Atlas, Formulas, Dover Publication, (1979): of with (1972): (1957) : Soc., Physical-synoptic variations P r o c . Sixth Weather Radar C o n f ., 21-30. et a l . (1981): Insights into errors of inferred GATE Convective rainfall. J. A o p l . Meteor., Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 20, SMS- 189 509-520 Barrett, A.H. and V.K. Chung (1962): A method for the determination of high-altitude water-vapor abundance from ground-based microwave observations. (11), J. Geoohys. Res., 67 4259-4266. Barrett, Eric C. and D.W. M a r t i n (1981): The use of satellite data in rainfall monitoring. Academic Press Inc, London Becker, G.E. and S.H. Autler (1946): Water vapor absorption of electromagnetic radiation in the centimeter wave-length range. P h v s . R e v ., 70, Blanchard, D.C. Hawaiian rains. Chandrasekhar, Publication, Cox, 300-307. (1953): Raindrop size distribution in J . M e t e o r ., 10, 457-473. S. (1960): Radiative Transfer, Dover New York. C. and W. Munk (1954-a): Measurements of the roughness of the sea surface from photographs of the sun's glitter. O p t . Soc. A m . , 44, 838-850. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. J. 190 Cox, C. and W. Munk (1954-b): d erived from sun glitter. Debye.. P. (1929): Statistics of the sea surface J. Marine Res., Polar Molecules. 13, 198-227. Dover Publication, New York. Deirmendjian, D. (1969): Electromagnetic scattering on spherical p o l y d i s p e r s i o n s . American Elsevier Publishing Company I N C , N e w Y o r k . Donnadieu, G.(1982): Observation de deux changements das spectras des gouttes de pluie daus une averse de nuages s t r a t i f o r m e s . J. Rech. Atmos., Foote, G.B. (1966): A Z-R relation for mount a i n thunderstorms. Fujiwara, M. J. A p p l . Meteor., 2, 229-231. ( 1 965):Raindrop-size distribution from individual storms. F u r n i v a l , G.M. and Bounds. 14, 439-455. J. A t m o s . S c i ., 22, and R.W. W i lson Technometrics, Gibson Jannie (1984): 16, (1974): 585-591. Regressions by Leaps 499-511. GOES D A T A USER'S GUIDE. NOAA. N C D C , satellite data service division. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. NESDIS, 191 Gloersen, P. and F.T. B a rath (1977): A scanning multichannel m i c rowave radiometer for Nimbus-G and SEASAT-A. Ocean. Sngr., Goldstein, IEEE J. O E - 2 , 172-178. H. (1951): At t e n u a t i o n by condensed water. P r o p a g a t i o n of Short Radio Waves, McGraw Hill Book Co., 671- 692, N e w York. Goody, R.M. (1964): At mospheric Radiation. o n Meteorology, Gorden, R.G. Oxford Univ e r s i t y Press Pub., (1967): m ulti p l e spectra. Griffith, C.G. Oxford Monographs Oxford. On the pressure broadening of molecular J. Cham. (1987): Phys, 46, 448-455. The Estimation from Satellite Imagery of Summertime Rainfall Over V a r i e d Space and Time Scales. N O A A Technical Memor a n d u m ERL ESG-25. --------- , et a l . (1981): U.S. high plains. J . A p p l . M e t e o r ., 20, --------- , et a l . (1978): satellite imagery; R e v . , 106, Satellite rain estimation in the 53-66. Rain estimation from geosynchronous visible and infrared studies. Mon. Wea. 1153-1171. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 192 --------- , et a l . (1976): Rainfall Estimation from Geosynchronous Satellite Imagery during Daylight Hours. NOAA Technical Report, Grody, N,C. ERL 356-WMPO-7. (1976): Remote sensing of atmospheric water content from satellites using microwave radiometry. Trans. Gunn, Ant. Proo. K.L.S. AP 24, and T.W.R. East 155-162. (1954): The microwave properties of precipitation particles. M e t e o r . S o c ., 80, Quart. J. Roy. 522-545. ------ , and J.S. Marshall (1958): of aggregate snowflakes. J . M e t e o r ., 15, 452-461. Higgs, A.J. IEEE The distribution with size (1952): The measurement of precipitation by r a d a r . Proc. Third Weather Radar C o n f ., Boston, A m e r . M e t e o r . S o c ., D49-D50. Hollinger, J.P. the sea surface. (1970): Passive microwave measurements of J. Geophvs. Res., 75, 5209-5213. ------ , (1971): Passive microwave measurements of sea surface roughness. IEEE Trans. Geosci. Electron, GE-9, 165-169. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 193 Hudlow, M.D. and V.L. Patterson A T L A S . NOAA Special Report, (1979): U.S. Dept. GATE RADAR RAINFALL Commerce, (Stock No. 003-019-00046-2). I m a l , I. (1960): relationships. Raindrop size distributions and Z-R Proc. Eight Weather Radar Conf., Boston, Amer. Meteor. Soc., Jones, (1956): Raindrop-size distribution and radar D.M.A. 211-218. reflectivity. Res. Rep. Meteor. Urbana, Kakar, Lab., R.K. No. 6, Illinois State Water Survey, 20. and B.H. Lambrightsen (1984): A statistical Correlation M e thod for the Retrieval of Atmospheric Moisture Profiles br Microwave Radiometry. Meteorology, Kakar, R.K. Katsaros, 23, J. of Climate and Applied 1110-1114. (1986): Personal communication. K.B. and Robert M. Lewis (1986): Meso scale and synoptic scale features of North Pacific weather systems o bserved with the Scanning Multichannels Microwave Radiometer on Nimbus-7. J . G e o p h v s . R e s ., 91, 2321-2330. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 194 Kerker, M. (1969): The scttering of light and other Electromagnetic radiation. Academic Press. Lents, W.J. (1976): N e w York. Generating Bessel functions in Mie scattering calculations using continued fractions. Appl. O p t ., 15, 668-674. Liebe, H.J., G.G. Gimmested and J.D. Hopponen, (1977): Atmospheric oxygen microwave spectrum experiment versus theory. Liou, IEEE Trans. K.N. and A.D. Ant, and Prop., AP-25 (3), 327-335. Duff (1979): Atmospheric liquid water content derived from parameterization of Nimbus 6 Scanning Microwave Spectrometer data. J. Appl. Meteor. Liou, K.N. 18. 99-103. (1980): An introduction to atmospheric radiation. Academic Press Pub. New York. Marshall, J.S. and W. Mck. Palmer (1948): The distribution of raindrops with size. J. Meteor., Martin, and D.N. D.W., J. Stout, 5, Sikdar 165-166. (1975): rainfall estimation from satellite images. Grant 04-5-158-47, Madison, GATE area Report to NOAA, Sapce Science and Engineering Center, Wisconsin. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 195 ------, and D.N. Sikdar satellite images. (1974): Rainfall estimation from Report to NOAA, Contract 03-4-022-22, Space Science and Engineering Center, Madison, Mason, Univ. Mie, B.J. (1975): Clouds, Wisconsin. Rain and Rainmakinq. Cambridge Press. G. (1908) Ann. P h v s i k . , 25, Muchnik, V.M. measurements. Njoku, E.G. 377-445. (1961): The accur a c y of radar rain intensity Meteor. Gidrol., 2, 44-47. et a l . (1980): The SEASAT Scanning Multichannel Mi c r o w a v e Radiometer (SMMR): ant e n n a pattern corrections development and implementation. IEEE J. Oean E n g r ., 0E-5(2), 125-137. Pandey, Prem C. and R.K. Kakar (1982): An empirical microwave emissivity model for a foam-covered sea. IEEE J. Oceanic E n g r .,O E - 7 ,135-140. Pandey, Prem C. and R.K. Kakar (1983): A two step statistical technique for retrieval of geophysical parameters from microwave radiometic data. IEEE Trans. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 196 G e o s c l . Remote Sensing,T - G E ,21(2),208-214. Paris, J.F. (1971): Transfer of Thermal Microwaves in the Atmosphere. A Ph.D. Thesis submitted to Texas A&M Univ. 253 pp. R a m a n a Murty, B.V., and S.C. Gupta (1959): Precipitation characteristics based on raindrop size measurements at Delhi a nd Khandala during southwest monsoon. 18A, J. Sci. Ind. Res. 352-371. Richards, F. and P. Arkin (1981): On the relationship b e t w e e n satellite-observed cloud cover and precipitation. Mon. Wea. Rodgers, Rev., E.B. 109, 1081-1093. and R.F. Adlar (1981): Tropical cyclone rainfall characteristics as determined from a satellite pas s i v e microwave radiometer. Rodgers, E.H. M o n . W e a . R e v ., 107, et a l . (1979): A statistical 585-598. technique for d etermining rainfall over land employing Nimbus 6 ESMR measurements. Rosenkranz, J . A p p l . M e t ., 13, 978-991. P.W. the Atmosphere. (1975): Shape of the 5 mm Oxygen Band in IEEE Trans. Ant, and Prop., AP-23, 498-505. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 197 Savage, R.C. and J.A. W einman (1975): Preliminary calculations of the upwelling radiance from rain clouds at 37.0 and 19.35 GHZ. Bull. Savaramakrishanan, M.V. Amer. Meteor. Soc., 55, 1272-1274. (1961): Studies of raindrop size characteristics in different types of tropical rain using a simple raindrop recorder. 12, Indian J. Meteor. Hydro. Geophvs., 189-217. Saxton, J.A. sea water: and J.A. Lane (1952): Electric properties of reflection and attenuation characteristics at v.h.f. Wireless Engineer, ,and ---- 29, 269-275. (1952): Dielectric dispersion in pure polar liquids at very high radio-frequencies. II. Relation of experimental results to theory. Proc. Roy. S o c .(London), A213, 473-492. Scofield, R .A . and V.J. Oliver estimating convective rainfall Tech. Memo. Sekhon, R.S. (1977): A scheme for from satellite imagery. NOAA NESS-86, W a s h i n t o n D.C. and R.C. Srivastava and radar reflectivity. (1970): Snow size spectra J . A t m o s . S c i ., 27, 299-307. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 198 Spencer, R.W. et a l . (1983): Satellite microwave radiances correlated w i t h radar rain rates over land. N a t u r e , 304, 141-143. Staelin, D.H. (1966): Measurements and Interpretation of the Microwave Spectrum of the Terrestrial Atmosphere near centimeter Wavelength. J. Geoohys. Res., 71, 1- 2875-2881. , et a l . (1976): Remote sensing of atmospheric water vapor and liquid water with the Nimbus 5 microwave spectrometer. Stephens, J . A p p l . M e t e o r ., 15, J.J. 1204-1214. (1962): Radar characteristics of an exponential drop-size distribution with application to a dual-frequency system. Austin, The univ. Stogryn, of Tex. A. frequencies. Stogryn, A. Tex., Elect. Eng. Res. Lab., 31 pp. (1972): The emissivity of sea foam at microwave J. Geophys. Res.. 77, 1658-1666. (1967): The apparent temperature of the sea at microwave frequencies. IEEE Trans. Ant. Prop., AP-15, 286. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 278- 199 Stout, J. et a l . (1979): Estimating GATE rainfall with geosynchronous satellite images. Mon. Wea. R e v . . 107, 585- 598. Townes, C.H. and A.L. Schallow (1955): Microwave Spectroscopy. M c - G r a w Hill Book Co. New York. Ulbrich, (1983): Natural variations in the analytical C.W. form of the raindrop size distribution. A p p l i e d Meteorology. ------ , and D. Atlas 22(10), J. Climate and 1764-1775. (1984): Assessment of the contribution of differential p o l a rization to improved rainfall measurements. Vandehulst, Radio Sci., H.C. (1957): 19(1), 49-57. Light Scattering by small particles, Vol I, John W i l e y and sons Pub. V a n vleck, J.H. Oxygen. P h y s . R e v ., 71, Waldvogel, A. The absorption of microwaves by 413-424. (1974): The N Q jump of raindrop spectra. A t m o s . S c i ., 31, Water, (1947): Co. N e w York. J. 1067-1078. J.W. et a l . (1975): Remote sensing of atmospheric Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 2 0 0 temperature profiles with the Nimbus 5 microwave spectrometer. Wexler, R. J. Atmos. (1345): Sci., 32, 1953-1969. Rainintensities by radar. J . M e t e o r ., 5, 171-173. Wilheit, near T.T. 19.35, storm Cora. ----- , (1982): Microwave radiometeric observations 92 and 183 GHZ of precipitation in tropical J . A p p l . M e t e o r ., 21, 1137-1145. (1979): A model for the microwave e m i ssvity of the ocean's surface as a function of w i n d speed. G e o s c i . E l e c t ., GE-17(4), IEEE Trans, on 244-249. ----- , et a l . (1977): A satellite technique for quantitative m ap p i n g rainfall rates over the oceans. 16, J. A p p I . Meteor., 551-560. Woodley, W.L. et a l . (1980): convective rainfall The inference of GATE from SMS-1 imagery. 19, 388-408.' Wu, R. and J.A. Weinman J. Appl. (1984): Microwave radiances from precipitation clouds containing aspherical phase, Meteor., and liquid hydrometeors. ice, combined J . G e o p h y s . R e s ., 89(D5), Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 201 7170-7178. Wu, S.T. and A.K. Fung (1972): A noncoherent model for microwave emissions and backscattering from the sea surface. J . G e o p h v s . R e s ., 77, 5917-5929. Wylie, D.P. (1979): An application of a geostationary satellite rain estimation technique to an extratropical area. J . A p p l . M e t e o r ., 18, 1640-1648. Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission.

1/--страниц