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Estimating Snow Water Resources From Space: A Passive Microwave Remote Sensing Data Assimilation Study in the Sierra Nevada, USA

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Estimating snow water resources from space: a passive
microwave remote sensing data assimilation study in
the Sierra Nevada, USA
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of The Ohio State University
By
Dongyue Li, B.S., M.S.
Graduate Program in Geodetic Science
The Ohio State University
2016
Dissertation Committee:
Dr. Michael Durand, Advisor
Dr. Ian Howat
Dr. C.K. Shum
ProQuest Number: 10970747
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DONGYUE LI
2016
Abstract
Snowpack in the Sierra Nevada is a critical water resource and an important
climate indicator; improving our understanding of the spatiotemporal pattern of the
snow water equivalent (SWE) in the Sierra Nevada has both civil and scientific merit.
Although SWE measurements have been carried out in the Sierra Nevada for over
one hundred years, currently available methods to characterize SWE, e.g. in-situ
measurements, snow hydrologic modeling, and remote sensing, all have intrinsic
strengths and limitations in estimating mountainous SWE; their strengths are highly
complementary.
In this study, we carried out an experiment to estimate SWE in the sparsely
vegetated Upper Kern Basin, Sierra Nevada, by assimilating AMSR-E observed
brightness temperatures (Tb) into a coupled hydrology and radiative transfer model
using an ensemble Kalman batch smoother (EnBS). The EnBS merges the
complementary SWE information from modeling and remote sensing observations
to improve SWE estimation. The overall strategy was to ensure optimal utilization of
the Tb measurements, extracting all possible SWE information from the data. To this
end, novel modeling methodologies were developed, and novel techniques were
employed to utilize the native resolution Tb measurements.
ii
In this study, we found the snowpack stratigraphic simplification that has been
widely applied in snow hydrologic modeling was ill adapted for snow radiance
estimation; the stratigraphic simplification leads to a significant bias in snow grain
size, which further introduces a bias of ~20 Kelvin in the Tb simulation. To improve
the modeling accuracy, we developed new snowpack stratigraphic resampling and
grain size estimation modules, which were demonstrated be able to constrain the
overall root mean square error (RMSE) of the accumulation season Tb within 3K. In
order to explicitly characterize the significant spatial variability of SWE and
radiance over the rugged terrain, the modeling in this study was set up at a very
high resolution (90m), with a series of mountain environmental factors
parameterized into each modeling pixel to model the effects of the mountainous
environment on snow distribution and microwave radiative transferring. Modeling
results showed the simulated SWE and Tb captured the spatiotemporal variability of
the SWE and radiance in the mountainous areas.
In order to extract as much SWE information as possible from the radiance
observation, we developed a new passive microwave (PM) data processing
algorithm that directly utilize Tb from the raw PM observations, rather than using
the typical resampled Tb data product, which is ~5 times coarser in resolution than
the raw satellite footprint. We showed that the Tb processed from the raw PM
observations is approximately 3 times more sensitive to SWE than the traditional
gridded data.
iii
By merging the enhanced SWE information from the modeling and the PM
observations, the EnBS effectively increased the accuracy of the SWE estimation in
the accumulation season. The validation against the true SWE measured from the insitu gages shows the overall accumulation season SWE RMSE decreased by 35.2%,
and the bias of the April 1st SWE estimation reduced from -0.17 m to -0.02m. The
accuracy improvement in the large-scale mountain SWE estimate is helpful for the
water management in the Sierra Nevada, and the EnBS has the potential to be
applied in other mountain areas to estimate above treeline SWE. Future work will
be focused on assimilating in-situ measurements and visible band observations to
explore the feasibility of extending the applicability of the EnBS in heavily vegetated
areas.
iv
Dedicated to Rui, Claire, and Bella
for their enormous love, support, sacrifice, and inspiration
v
Acknowledgements
I am lucky to have Michael Durand as my advisor. My PhD wouldn’t have been
made possible without Mike’s enduring support, guidance, and encouragement. I am
greatly indebted to Mike for all his support and care for not only me, but also my
entire family. To me, Mike is a great advisor, an amazing person, and a life-long
example.
I am grateful to NASA for three years financial support through the prestigious
NASA Earth and Space Science Fellowship. I would also like to thank CUAHSI for the
Pathfinder Fellowship that supported my field campaigns.
I extend thanks to the professors and researchers who generously helped and
mentored me at various phases of my graduate study: Steve Margulis, Dennis
Lettenmaier, Kostas Andreadis, Jennifer Adam, Roger Bales, Noah Molotch, Ed Kim,
Doug Alsdorf, C.K. Shum, Ian Howat, Alan Saalfeld, and Alper Yilmaz.
Thanks Ben, Jinmei, Melissa, Rhae Sung, Steve T, Steve C, Yuna, Apoorva, and
Renato for being great labmates and dedicated members of the Friday lunch gang. I
appreciate Ben and Ollie for accompanying me on top of the snowy mountain in the
Sierra Nevada during the coldest month of a year, good times.
vi
Last but certainly not least, it’s a truly blessing to have the whole family going
along with me in this journey. I want to express my gratitude to dad, mom, dad and
mom in-law, Rui, Claire, and Bella for the unfailing love and support. You will always
be at the bottom of my heart.
vii
Vita
Nov. 16 1985 ............................................................... Born, Luoyang, Henan, China
July 2009 ....................................................................... Bachelor of Science
Photogrammetry and Remote Sensing
Information Engineering University
Zhengzhou, Henan, China
July 2011 ....................................................................... Master of Science
Geodetic Science
The Ohio State University
Columbus OH, USA
Publications
Li, D., Durand, M. and Margulis, S.A., (2012), “Potential for hydrologic
characterization of deep mountain snowpack via passive microwave remote
sensing in the Kern River Basin, Sierra Nevada, USA”, Remote Sensing of
Environment, Vol. 125, P34-48
Li, D., Durand, M., Margulis, S. A., (2014), “Large-scale high-resolution modeling of
microwave radiance of a maritime alpine snowpack ”, IEEE Transactions on
Geosciences and Remote Sensing, vol.53, no.5, 2308-2322
Li, D., Durand, M., Margulis, S. A., (2015), “Quantifying spatiotemporal variability of
controls on microwave emission from snow covered mountainous regions”, IEEE
Journal of Selected Topics in Applied Earth Observations and Remote Sensing,
10.1109/JSTARS.2015.2440332
Margulis A., S., Gonzalo, C., Girotto, M., Huning, L., Li, D., Durand, M., “Characterizing
the extreme 2015 snowpack deficit in the Sierra Nevada (USA) and the
implications for drought recovery”, Geophysical Research Letters, 43,
doi:10.1002/2016GL068520.
Cai, S., Li, D., Durand, M., Margulis, S., (2015), “Examination of the interaction among
snow water equivalent, microwave radiance and vegetation effects across Sierra
viii
Nevada, USA”, Remote Sensing of Environment, in revision
Li, D., Durand, M., Margulis, S. A., (2015), “Estimating snow water equivalent at Kern
Basin by radiance assimilating via ensemble Kalman batch smoother”, Water
Resources Research, in revision
Field of study
Major field: Geodetic Science
Specializations: Data assimilation, remote sensing, hydrology, land surface
processes modeling
ix
Table of Contents
Abstract....................................................................................................................................................... ii
Acknowledgements ............................................................................................................................... vi
Vita............................................................................................................................................................ viii
List of figures ........................................................................................................................................ xiii
List of tables ......................................................................................................................................... xvii
Chapter 1. Introduction and background ....................................................................................... 1
1.1 The importance of snow to natural and human systems ............................................. 1
1.2 Current methodologies and challenges in snowpack characterization .................. 5
1.2.1 In-situ measurements ...................................................................................................................... 5
1.2.2 Snow hydrologic modeling............................................................................................................. 6
1.2.3 Remote sensing snow observations ........................................................................................... 7
1.3 Passive microwave remote sensing and its application in SWE retrieval ............. 8
1.4 Estimating snow water equivalent with data assimilation technique .................... 9
1.5 The motivations and strategies for assimilating spaceborne passive microwave
observations to estimate large-scale mountainous SWE .......................................... 12
1.6 The organization of the dissertation ................................................................................ 14
Chapter 2: Improving PM data processing for enhanced SWE information ................... 15
2.1 Motivation of directly processing raw PM observations ........................................... 15
2.2 Study area and data ................................................................................................................ 17
2.3 Methods ....................................................................................................................................... 21
2.3.1 Footprint-based areal weighted (FBAW) basin-scale average...................................... 23
2.3.2 Gaussian Inverse Distance Weight (GIDW) point-scale interpolation ....................... 24
2.4 Results and discussions ......................................................................................................... 26
2.4.1 Potential for characterizing snow accumulation ................................................................ 29
2.4.2 Potential for characterizing basin runoff ............................................................................... 34
2.4.3 Potential for characterizing snow ablation ........................................................................... 36
2.5 Summary ..................................................................................................................................... 49
Chapter 3: Improving the SWE and snow radiative transfer modeling ............................ 52
3.1 Motivation of directly processing raw PM observations ........................................... 52
3.2 Study area and data ................................................................................................................ 53
3.2.1 Study area............................................................................................................................................ 53
3.2.2 Data ........................................................................................................................................................ 54
3.3 Models .......................................................................................................................................... 57
3.3.1 Land Surface Model ......................................................................................................................... 57
3.3.2 Radiative transfer model .............................................................................................................. 59
3.3.3 Layer combination resampling scheme during snowfall events .................................. 61
x
3.4 Modeling experiments ........................................................................................................... 65
3.4.1 Correction of precipitation undercatch .................................................................................. 66
3.4.2 Calibration of dry grain growth rate ........................................................................................ 67
3.4.3 Point-scale modeling with iterative resample scheme..................................................... 67
3.4.4 Parameterizing empirical resampling scheme with point-scale modeling results
............................................................................................................................................................................. 68
3.4.5 Basin-scale modeling with empirical resampling scheme .............................................. 69
3.5 Modeling results and discussion ........................................................................................ 70
3.5.1 Evaluating iterative resampling scheme at point scale .................................................... 70
3.5.2 Full width at half maximum method ........................................................................................ 76
3.5.3 Basin-scale experiments ............................................................................................................... 78
3.5.3.1 Inter-annual high-resolution modeling ................................................................................. 78
3.5.3.2 Comparison between aggregated modeling Tb and AMSR-E observations ............. 81
3.5.4 Analysis of spaceborne PM saturation behavior ................................................................. 85
3.6 Summary ..................................................................................................................................... 92
Chapter 4: The development of the Ensemble Batch Smoother assimilation scheme for
SWE estimation in the Upper Kern basin .................................................................................... 95
4.1 Introduction............................................................................................................................... 95
4.2 Methods ....................................................................................................................................... 96
4.2.1 Prediction ............................................................................................................................................ 97
4.2.2 Updating............................................................................................................................................... 99
4.3 Models ........................................................................................................................................ 102
4.3.1 Forcing data disaggregation model ....................................................................................... 102
4.3.2 Land surface model ...................................................................................................................... 102
4.3.3 Radiative transfer model ........................................................................................................... 104
4.4 Study area, data, and experiment design ...................................................................... 106
4.4.1 Study area......................................................................................................................................... 106
4.4.2 Data ..................................................................................................................................................... 107
4.4.3 Experiment configuration.......................................................................................................... 111
4.4.3.1 Open-loop simulation ................................................................................................................ 111
4.4.3.2 Updating ......................................................................................................................................... 114
4.5 Results and discussions ....................................................................................................... 118
4.5.1 Prior estimates ............................................................................................................................... 118
4.5.2 Assimilation results ...................................................................................................................... 125
4.5.3 Comparing assimilation methodologies .............................................................................. 135
4.5.4 Comparing assimilation methodologies .............................................................................. 142
4.6 Summary ................................................................................................................................... 144
Chapter 5. Conclusion and future work ..................................................................................... 146
5.1 Conclusion and original contribution ............................................................................ 146
5.2 Future work ............................................................................................................................. 150
References ............................................................................................................................................ 152
Appendix A. Calculating viewing geometry of AMSR-E observations .............................. 172
Appendix B. Model calibration and re-development ............................................................ 175
xi
B.1 SSiB3 mass-weighted average resampling scheme................................................... 175
B.2 Inadequacy of the mass-weighted average combination resampling ................ 176
B.3 Inadequacy of the mass-weighted average combination resampling ................ 181
B.3.1 The iterative combination re-sampling scheme .............................................................. 182
B.3.2 The empirical combination re-sampling scheme ............................................................ 182
xii
List of figures
Fig. 1 The fraction of the snowmelt-derived runoff to the total local runoff (fQ, SNOW) over the
western U.S. .................................................................................................................................................. 2
Fig. 2 Vegetation density (a), study area digital elevation model, snow courses and stream
gage (b). ....................................................................................................................................................... 19
Fig. 3 AMSR-E viewing geometry variations are caused by changes in the Aqua satellite
orbit throughout the 16-day cycle. In a), b) and c) the AMSR-E measurement swath is
shown in cyan for three different overpasses; the study area location is indicated by a
cross. In d), e) and f), the orientation of the AMSR-E L2A 37 GHz elliptical footprints
is shown corresponding to the AMSR-E swaths shown in a), b) and c), respectively. 22
Fig. 4 Illustration of the process used to compute TbFBAW for L2A (a) and EASE-Grid (b). The
blue outline indicates the Upper Kern study area, the red outlines indicate the
footprint FWHM extents (a) and the EASE-Grid extents (b), and the red shading
indicates the over- lap of each footprint or EASE-Grid cell with the study area ........... 24
Fig. 5 Upper Kern TbFBAW (a), 7-day moving average of both TbFBAW (blue) and air
temperature (red) from CBT (b) and SWE at CBT (c). ............................................................. 28
Fig. 6 Scatterplots of smoothed TbFBAW vs. SWE at UTY (a), CBT (b), CHP (c), WTM (d), and
CSV (e). ......................................................................................................................................................... 31
Fig. 7 Scatterplots of minimum L2A TbFBAW (blue) and minimum EASE-Grid TbFBAW (red)
for WY 2003–2008 versus the basin average maximum SWE, calculated from the five
snow courses. ............................................................................................................................................ 33
Fig. 8 Scatterplot of TbFBAW and integrated discharge from the Upper Kern Basin (a), and
the hydrographs for total annual discharge of WY2005 (blue) and WY2006 (red) (b).
......................................................................................................................................................................... 35
Fig. 9 Tb in the Upper Kern Basin on 11 March (a), 30 April (b) and 17 May (c) of 2006 ...... 37
Fig. 10 Timeseries of the standard deviation of AMSR-E measurements falling within or
partially within the Upper Kern Basin. ........................................................................................... 38
Fig. 11 Temperature DAV for air temperature (blue) measured at CBT, and for TbGIDW (red)
from AMSR-E.............................................................................................................................................. 40
Fig. 12 Division of the water year into accumulation (light gray), transition (medium gray)
and ablation (dark gray) based on snow course SWE; this example is shown from
CBT for WY2003. ...................................................................................................................................... 41
xiii
Fig. 13 Analysis of nighttime TbGIDW (blue) and TbGIDW DAV (red) timeseries at CBT
processed using L2A (a) and EASE-Grid (b) AMSR-E measurements. Threshold values
used to dis- criminate onset and end of snowmelt are indicated as red (nighttime)
and blue (DAV); note that different thresholds were determined for EASE-Grid and
for L2A. ......................................................................................................................................................... 43
Fig. 14 L2A TbGIDW DAV at CBT is shown, as well as the accumulation, transition, and
ablation periods (as defined in Fig. 12) based on snow course data, for water years
2003 (a), 2004 (b), 2005 (c), 2006 (d) and 2008 (e). The horizontal line at the bottom
of each plot indicates the period meeting the criteria for melt-timing identification. 47
Fig. 15 Scatterplot of onset of melt day-of-year (DOY) from L2A TbGIDW (blue) and EASEGrid TbGIDW (red) and the CBT snow course (a). Scatterplot of the end of melt DOY
from L2A TbGIDW (blue) and EASE-Grid TbGIDW (red) and the CBT snow course (b). . 48
Fig. 16 Forest cover (left) and elevation (middle) of the North Fork of Kern River basin.
Three snow pillows are located in the Upper-Kern basin -- the region delineated by
gray line. ...................................................................................................................................................... 54
Fig. 17 The simulated bottom layer grain size at CBT, CHP and UTY for WY04, WY05 and
WY06. The light grey curve and the dark grey curve are the grain size of the top and
middle snow layer, respectively. The green curve is the bottom-layer grain size using
the mass-weighted resampling scheme, and the red curve is the bottom-layer grain
size using the iterative resampling scheme. ................................................................................. 72
Fig. 18 The simulated Tb at CBT, CHP and UTY for WY04, WY05 and WY06. The iterative
resampling scheme (red) and the mass-weighted resampling scheme (green) are
shown along with the AMSR-E Tb (blue). ....................................................................................... 73
Fig. 19 The results of correcting the bottom layer grain size using the empirical resampling
scheme at the three gages for WY03, WY07 and WY08. The corrected bottom layer
grain size (orange) reduced the grain size underestimation compared with grain size
estimated from mass-weighted resampling scheme (green). The light and dark grey
curves are the grain size of the top and middle snow layer, respectively........................ 77
Fig. 20 The corresponding Tb improvements as a result of the bottom layer grain size
estimate being improved by the empirical resampling scheme. The corrected Tb
(orange) is closer to the observed Tb (blue) than the Tb simulated without correction
(green). ......................................................................................................................................................... 78
Fig. 21 Modeled Tb across the Upper Kern Basin at 90 m resolution on the first day of each
month from February through May (left column to right column) of WY03, WY07 and
WY08 (from top row to bottom row). ............................................................................................. 80
Fig. 22 AMSR-E L2A Tb observations (right column), and the predicted AMSR-E
observations aggregated from the modeling pixels (left column) on Feb 1 of WY03
(1st row), WY07 (2nd row) and WY08 (3rd row). ......................................................................... 82
Fig. 23 The seasonal time-series difference between the daily basin-average modeled Tb and
AMSR-E observed Tb in Upper Kern for WY03 (green), WY07 (red) and WY08 (blue).
Tbpred is the basin-average modeled Tb. .......................................................................................... 84
Fig. 24 Comparing the footprint-aggregated modeled Tb and the AMSR-E observations for
the dry snow season of WY2003, WY2007 and WY2008........................................................ 84
Fig. 25 Comparison between the modeled snow transmissivity and the modeled SWE of the
pixel covering the location of gage UTY for the first day of each month in the dry
xiv
season of WY2005. October 1st was omitted in this calculation as there was no snow
at UTY on that date. The RMSE of the polynomial fit is 0.017. .............................................. 87
Fig. 26 The AMSR-E GIDW Tb and measured SWE at gage UTY on the first days of October to
March of WY2004 (red), WY2005 (green) and WY2006 (blue). On each curve, the six
points from left to right are shown in order from October to March. ................................ 88
Fig. 27 The AMSR-E FBAW basin-average Tb and modeled basin average SWE on the first
day of October to March of WY2003 (red), WY2007 (green) and WY2008 (blue). On
each curve, the six points from left to right are shown in order from October to March
......................................................................................................................................................................... 90
Fig. 28 The basin-scale SWE estimate and the AMSR-E footprints on March 1st WY2008..... 91
Fig. 29 The map of the study area, including the DEM of the Kern basin and the locations of
the snow pillows and snow courses.............................................................................................. 107
Fig. 30 The AMSR-E 36.5 GHz antenna energy distribution model, which illustrates the
weight in the weighted-average aggregation of the pixel-scale radiance estimate to
the footprint-scale. Note the grid cells in this figure are only shown to illustrate the
aggregation process; they are not at the pixel resolution (90 m) in this study .......... 118
Fig. 31 Comparing the WY03 dry season prior SWE (red, top row) and prior Tb (red, bottom
row) estimates with the measurements (blue) at the three snow pillows in the Upper
Kern. ........................................................................................................................................................... 120
Fig. 32 (left) the modeled SWE, (middle) Tb and (right) grain size over the Upper Kern on
March 1st 2003. ...................................................................................................................................... 124
Fig. 33 (left) the observed and (right) the aggregated prior Tb in each observational
footprint over the Upper Kern on March 1st 2003. Both colorbars span 8 Kelvin and
the colorbar for the aggregated Tb is systematically higher than that of the observed
Tb by 4K to show the similar spatial trends of the two. ....................................................... 124
Fig. 34 Comparison between the prior SWE estimate, FP posterior estimate, and measured
SWE at the snow pillows (column 1-3) and snow courses (column 4-8). In each figure
the light red curve is the prior estimate, the black curve or dot represent the
measurement, and the blue is the posterior estimate. The shaded areas are the ranges
of the 80% uncertainty of the prior (light red) and posterior (light blue) estimates.
...................................................................................................................................................................... 128
Fig. 35 Comparing the daily prior (red) and posterior (green) SWE estimates from Oct 1st to
Apr 1st with measured SWE at pillows CBT, CHP, UTY (column 1 to column 3) for
WY03-08, and (column 4) comparing the Feb 1st, Mar 1st, April 1st prior and posterior
SWE at the five snow courses for WY03-08. The top row to bottom row show the
results from FBAW, FP and GIDW method, respectively...................................................... 129
Fig. 36 The comparison between the prior and posterior WY03-08 April 1st SWE at the
snow courses (left column) and the snow pillows (right column). The top row to the
bottom row show the posterior estimates from FBAW, FP and GIDW method,
respectively. ............................................................................................................................................ 130
Fig. 37 The six-year accumulation season average bias and RMSE of the prior estimate and
the three posterior estimates at the three pillows and the five courses........................ 136
Fig. 38 The average six-year Apr 1st SWE error and RMSE of the prior estimates and
posterior estimates at the snow pillows and snow courses. .............................................. 136
Fig. 39 The basin-scale (column 1-4) prior, FBAW posterior, FP posterior, and reanalysis
posterior snow estimates on the April 1st of (row 1-6) WY03-08 .................................... 139
xv
Fig. 40 The six-year average (WY03-08) SWE within each longitudinal interval of the basin.
...................................................................................................................................................................... 140
Fig. 41 The effect of different grain size growth rate on simulating Tb using the prior SWE.
The red line represents the value utilized in this study (3.0E-7 m4/kg), other curves
show that increasing the grain growth rate lead to a faster decrease in the simulated
radiance compared with the measured radiance .................................................................... 144
Fig. 42 Illustration of determination of footprint orientation, and the location of an example
Smin and Smax. ............................................................................................................................................. 173
Fig. 43 (Top) Both the SWE from SSiB3 simulation (red) and from in-situ measurement
(blue) at CBT show an intense snowfall event during late December to early January
of WY2005. (Middle) The intense snowfall led to a significant decrease in the
simulated bottom layer grain size (red); top and middle grain size are also shown
(grey). (Bottom) The decrease in grain size led to a sharp increase in the simulated Tb
(red) during the same period, but the AMSR-E Tb (blue) did not show such a
significant Tb increase. ........................................................................................................................ 178
Fig. 44 (Left) The four-layer snow stratigraphy just after a snowfall, with grain size Pex
values given and illustrated graphically via the density of the black dots in each layer,
and the above-surface Tb are shown. (Right) After mass-weighted re-layering, the
bottom layer grain size has decreased and the modeled Tb increased. .......................... 180
Fig.45 The relationship between the Pex correction ratio and the snowfall intensity. The
black curve is the function fitted from the point-scale correction results from the
iteration method. The more intense a snowfall is, the larger the grain size
underestimate is. Therefore in general the correction ratio is positively correlated
with snowfall magnitude. .................................................................................................................. 184
xvi
List of tables
Table 1 Snow course sites used in this study; agency abbreviations refer to either the
California Department of Water Resources (CA DWR) or to the United States Army
Corps of Engineers (USACE). ........................................................................................................... 18
Table 2 Daytime overpass (DOP) and nighttime overpass (NOP) data availability over
Upper Kern for each day of the 16-day AMSR-E cycle. ......................................................... 20
Table 3 Correlation coefficient between the Tb-DAV based and in-situ measured melt onset
and ending. ............................................................................................................................................. 46
Table 4 RMSE (in days) of the Tb-DAV based melt onset and ending estimation, using L2A
and EASE-Grid data. ............................................................................................................................ 46
Table 5 The elevation, fractional forest coverage and peak SWE for WY03-08 at the three
in-situ gages in the Upper Kern...................................................................................................... 55
Table 6 The relative error correction ratio
of the iterative resampling method................. 75
Table 7 The Tb RMSE simulated during the dry snow season without bottom layer
correction, and with correction using the iterative resampling method. All units are
kelvins....................................................................................................................................................... 75
Table 8 The mean (μ), standard deviation (σ), and coefficient of variation (Cv) of the
modeled SWE within the AMSR-E footprints of March 1st WY2008 observation.
Footprint numbers correspond to those shown in Figure 28. .......................................... 91
Table 9 The location and the environmental conditions of the snow pillow CBT, CHP, UTY,
and the snow course BGH, TND, SDM, GYF, RCR.................................................................. 110
Table 10 The parameters that were perturbed with uncertainties and their uncertainty
quantifications. .................................................................................................................................. 113
Table 11 The configuration of the three assimilation experiments ............................................. 117
Table 12 The RMSE of the daily prior SWE estimates of the accumulation season, the error
of the prior April 1st SWE, and the measured April 1st SWE of each water year..... 122
Table 13 The RMSE of the accumulation season posterior SWE estimates from the three
assimilations in each water year at each gage. The cases where the posterior
estimates have a larger RMSE are underlined. ..................................................................... 133
Table 14 The error of the posterior April 1st SWE estimates from the three assimilations in
each water year at each gage. The cases where the posterior estimates have a
larger April 1st SWE error are underlined. ............................................................................. 134
xvii
Chapter1.Introductionandbackground
1.1Theimportanceofsnowtonaturalandhumansystems
Snowpackstoreswaterinwinterandreleasesitinspring,servingasacritical
waterresource.Snowdominatesthewatercycleinmorethanhalfoftheterrestrial
areasinthenorthernhemisphere;morethanhalfofthestreamflowintheseareas
originatesfromsnowmelt.Snowmeltprovidesfreshwaterforonefifthoftheworld
populationandroughlyonefourthoftheglobalGrossDomesticProduct(GDP)
[Barnettetal.,2005].Regionally,forexample,inthesemi-aridwesternU.S.,
snowmeltrunofffromthemountainrangesiscrucialforagricultureirrigation,
powergeneration,recreation,inadditiontourbanconsumptionfrompopulation
centers;snowpackeffectivelyfunctionsasalargesurfacereservoironwhich
millionsofpeopledependforwatersupply(Fig.1[Lietal.2016]).Also,snowmelt
replenishesgroundwaterandreservoirstoragethatarekeytosustainthewater
supplyinthedrysummerandfall;over2/3ofthereservoirsstorageinthewestern
U.S.issnow-derived[Lietal.2016].Asillustratedbytherecentsnowpackdrought
intheSierraNevadaandthePacificNorthwest,alackofsnowfallinwintercanlead
toseverewaterscarcity,groundwateroverdrafts,treemortality,insectoutbreaks,
andenhancedwildfirerisk,allofwhichcansignificantlyunderminethesocio
1
economicwellbeingoftheregion[Diffenbaughetal.2015];themega-droughtlikely
wouldhaveaprolongedsurfaceandsub-surfacewaterdeficit,evenwithseveral
yearsofcontinuedabove-averageprecipitation[Margulisetal.2016].
Fig.1Thefractionofthesnowmelt-derivedrunofftothetotallocalrunoff(fQ,SNOW)overthe
westernU.S.SnowisacriticalwaterresourcefortheWestandthesnowwaterresourceis
mainlystoredinthemountainareas.
Snowhasalsobeenanimportantpartoftheintegratedecologicalsystem.The
highalbedoofsnowstronglyaffectsthesolarradiationreceptionontheearth
2
surface.Inmanyseasonalsnow-coveredareas,snowpackcontrolstheregional
energybalance,andfurtheraffectsthehydrologiccycleandthecarboncyclethat
areessentiallyinter-dependentwiththeenergycycle[Painteretal.,2009].In
addition,thechangesinlarge-scalesnowpackinthewarmingclimate(e.g.declining
snowaccumulationandearliermelt)alterthegreennessandtheecological
functionsofmountainforestsbyreducingtheintensityoftheirphotosynthesisand
evapotranspiration,whichfurtheraffecttheregionalcarbonandwatercycle
[Trujilloetal.,2012].Lastbutnotleast,thesensitivityofsnowtotemperatureraise
hasmadesnowadesirablemediumforquantifyingtheimpactofclimatechange;
analysesofthestreamflowfromsnowmelthasbeenwidelyusedtoquantifyclimate
changeanditsimpactonvariousaspectsofsociety[Stewartetal.2004].
Mountainrangeisanaturalwatertowerthathassubstantialcapacityin
accumulatingandpreservingsnowwaterresource.Vivirolietal.(2007)estimate
that44%ofglobalmontaneareaplaysaroleinwaterresources.Thehigh-elevation
andtheaccompanyinglow-temperaturepromotealargeportionofthelocal
precipitationfallsassnow.Differingtotherainfall-derivedwaterthatrapidlyruns
off,infiltratesandevaporates,thewaterinmountainoussnowpackcanbe
preservedthroughoutthewinterandbereleasedduringthemeltingeventsinthe
succeedingspring;thisdelayedrunoffiscriticaltocompensatetheeffectsofthe
temporalvariationofprecipitationonwatersupply.Forexample,inlargepartsof
3
thewesternU.S.,mostoftheannualprecipitationfallsassnowinthewinterseason;
precipitationinsummerandfallaretypicallyscarce,butmeanwhilethedry
summerandfallarethepeakwaterconsumptionseasonsofayear.Thesnowmelt
storedandre-distributedbythereservoirandcanalnetworkplaysasignificantrole
inmitigatingthisdilemma,andprovideswaterforirrigationandfisheriesduring
thedryseason[Barnettetal.2005].AcrosstheWest,over60%ofthewatersupply
inthesummerandfallseasonreliesonthesnowmeltearlierintheseason[Lietal.
2016].
Snowwaterequivalent(SWE)isametricthatmeasurestheamountofliquid
watercontainedinasnowpack.SWEisthedepthofwaterthatwouldresultfrom
instantaneousmeltoftheentiresnowpack.SWEisarguablythemostimportant
variableinsnowhydrology.However,accuratelyquantifyingmountainousSWEhas
beenchallenging[Dozieretal.,2011],dueinparttosnowspatialvariabilitycaused
bythecomplexphysiographicandatmosphericconditions[MolotchandBales,2005
and2006].TherearedifferentwaystomeasureSWEinthemountainareas;eachof
themhastheirownstrengthsandweaknesses.
4
1.2Currentmethodologiesandchallengesinsnowpackcharacterization
1.2.1In-situmeasurements
In-situmeasurementsuchassnowcoursesandsnowpillowsisthefundamental
methodforSWEcharacterization.In-situSWEmeasurementhasbeencarriedout
manuallyforoverahundredyears.Intherecentdecades,automatedobservation
networkhavebeensetupinmajorsnowcoveredmountainareasforSWEand
meteorologicalmeasurements;e.g.theSNOwpackTELemetry(SNOTEL)networkin
theUnitedStates[Serrezeetal.,1999].However,inmountainousareas[Balesetal.,
2006],interactionbetweenruggedsurfacetopographyandatmosphericprocesses
leadstosignificantspatialvariabilityinsnowproperties.Point-scalein-situ
measurementsareoftennon-representativeofspatialsnowvariability[e.g.,Molotch
&Bales,2005].Inaddition,thedistributionofin-situmeasurementsisoften
extremelysparseinmountainousregions.Forexample,intheSierraNevada,the
numberofoperationalsnowpillowsyieldsanaveragesamplingdensityof~620
km2persensor[Guanetal.2013].Recently,snowsensornetworksbuiltwitha
near-optimalsensorplacementhaveeffectivelyimprovedthecharacterizationof
basin-scaleSWEestimates[Welchetal.2013],butthedevelopmentofsuchsensor
networkscouldbepotentiallylimitedbysignificanteconomiccostsandthe
inaccessibilityofmanysnow-coveredareas.Geostatisticaltechniqueshavebeen
developedtointerpolateavailableinsitumeasurements,suchaskriging[e.g.
5
Erickson&Williams,2005;Erxlebenetal.,2002]andregressiontrees[e.g.Balk&
Elder,2000;Elderetal.,1998;Erxlebenetal.,2002].However,thebest
interpolationstypicallyexplainonly~50%ofthespatialvariabilityinobserved
snowproperties[Kasuraketal.,2011].
1.2.2Snowhydrologicmodeling
Utilizingterrain,landcover,andmeteorologicalforcingdata,snowhydrologic
modelingcanestimatethesnowcovertime-seriesandspatialpatternsby
simulatingsnowaccumulation,evolutionandablation.Increasinglypowerful
computingresourcesandtheavailabilityofmacro-scalemeteorologicalforcingdata
[e.g.,Rieneckeretal.2011,Rodelletal.,2004]haveenabledglobal-scalesnow
modeling.Forcingdatadownscalingtechniqueseffectivelyincreasemodel
resolutionbothspatially[Sharmaetal.,2007]andtemporally[Margulisand
Entekhabi,2001].Progressinunderstandingsnowevolutionhaspromoted
developmentofincreasinglycomplexphysicalmodels[Jordan,1991,Sunetal.,
1999,Brunetal.1989,BarteltandLehning.2002].However,large-scalemodel
forcingdataaresometimespronetobiases,includingprecipitation[Adamand
Lettenmaier,2003,Panetal.,2003]andsolarradiation[Luoetal.,2003],which
dominatesnowaccumulationandablationprocesses,respectively.The
approximationsandassumptionsmadeforthemodelphysicsandmodelstructure
addtotheuncertaintyofthemodelingresults[Clarketal.,2015].Therefore,large-
6
scalemodelingofmountainSWEisoftencharacterizedbysignificantuncertainty
[Panetal.,2003,Balesetal.,2006].
1.2.3Remotesensingsnowobservations
Theneedforspatiallycontinuoussnowobservationshasmotivatedthe
applicationofremotesensing,whichhasbeenappliedtoobserveseasonal
snowpackpropertiesfromlocaltoglobalscale[Nolin,2010].Effortstoremotely
senseSWEhaveutilizedpassivemicrowave(PM)radiometry[Changetal.,1982
and1987,Stankovetal.,2008,Hardyetal.,2008,Vuyovichetal.,2014],active
microwaveradarscatterometry[Clineetal.,2009,Yuehetal.,2009,Sundströmet
al.,2013],airbornegammacounters[Pecketal.,1971,Carrolletal.,1989,Choquette
etal.,2008],satellitegravityrecovery[Frappartetal.,2006,Niuetal.,2007,Forman
etal.,2012],reconstructionorreanalysisusingvisibleandnearinfrared
measurements[Clineetal.,1998;MolotchandMargulis2008;Molotch2009;Slater
etal.,2013;Guanetal.,2013;Girottoetal.2014aand2014b;Margulisetal.2015],
airborneLiDARaltimetry[Deemsetal.,2006and2013,Harpoldetal.,2014,
Mattmannetal.,2014,theAirborneSnowObservatoryhttp://aso.jpl.nasa.gov/],
syntheticapertureradarinterferometry[Deebetal.,2011],andphotogrammetric
reconstruction[VanderJagtetal.,2015,Nolanetal.,2014].Thesediverseremote
sensingtechniqueshaveaddednewopportunitiestoaddresstheissuesassociated
withthelarge-scalesnowwaterresourcecharacterization.However,mostremote
7
sensingtechniquesdonotcontaindirectinformationofSWE;SWEhastobe
retrievedfromitsempiricalorphysicalrelationshipswiththeseremote-sensed
observations.AsnotedrecentlybyLettenmaieretal.[2015]andFreietal.[2012],
accurateSWEretrievalfromspaceborneplatformshasremainedelusive,especially
inmountainousareas.
1.3PassivemicrowaveremotesensinganditsapplicationinSWEretrieval
Oftheseremotesensingtechnologies,PMisarguablytheonlymeasurementthat
containdirectSWEinformationatlarge-scale.Snowscattersandabsorbsthe
upwellingmicrowaveradiationemittedbytheearth,leadingtotheattenuationof
theobservablebrightnesstemperature(Tb),whichisameasurementthemagnitude
ofthemicrowaveradiance;theinversecorrelationbetweentheSWEandTbforms
thebasisforestimatingtheSWEwithPMremotesensing.PMhasglobalcoverage,
sub-dailytemporalresolution,all-weathercapability,andspacebornePMhasbeen
usedforearthobservationforoverfortyyears,formingoneofthelongestremote
sensingdatastreams.TheabilitytoaccuratelyestimateSWEfromPMwouldthusbe
atremendousadvanceforlarge-scaleSWEestimation.Whereas,currentSWE
estimatesfromPMmeasurementsoftencontainerrorsanduncertainties,mainly
duetothefollowingfactors:(1)Vegetationmodulatesthemicrowaveradiancefrom
snow,reducingthesnowinformationintheobservations.(2)Manysnowpack
8
propertiesinadditiontoSWE(especiallysnowgrainsizeandvolumetricliquid
watercontent),andtheverticallayeringofsnowpacks,affectsnowradiative
transferproperties,andcurrentalgorithmsareinadequatetoaccuratelyresolve
them.(3)ThespatialresolutionofPMiscoarse,ontheorderoftensofkilometers,
whileSWEvariesovermuchshorterspatialscales,(4)PMobservationssaturatein
deepsnow;simpleone-to-onerelationshipsbetweenradianceandSWEtendto
saturateat150–200mm.Traditionally,SWEisretrievedfromPMbyasimple
regressionoftheTbdifferencebetweenthe18GHzand37GHzPMobservations;the
basicprincipleofthisSWEretrievalalgorithmisthat37GHzPMobservationisvery
sensitivetotheaccumulationofSWE,andthe37GHzTbisconsideredtoholdan
inverselinearrelationshipwithSWE.Ontheotherhand,18GHzradianceis
generallyinsensitivetotheSWEaccumulation,andthe18GHzTbisconsideredtobe
invariantwithSWEaccumulation.Therefore,theSWEcanberetrievedfromthe
differencesbetweenthe18GHzand37GHzTbobservations.However,duetothe
aforementionedintrinsiclimitationsofPM,theSWEretrievalresultshave
significantuncertaintiesinmountainareas[Tedescoetal.2010].
1.4Estimatingsnowwaterequivalentwithdataassimilationtechnique
Whileeachindividualmethodhasstrengthsandlimitationsinlarge-scaleSWE
quantification,thesnowinformationincurrentlyavailabledatasetsarequite
9
complementary;mergingthesnowinformationinmultipledatasetshasshown
promisesforimprovingtheaccuracyofthesnowwaterresourceestimate[e.g.
DurandandMargulis,2006,Durandetal.2008].Dataassimilationreferstoasetof
mathematicalframeworksthatmergeobservationsintomodelingestimates.
Kalmantypeassimilationisarguablythemostwidelyappliedapproachin
hydrologicdataassimilationstudies.TheKalmantypedataassimilationisbasedon
theBayesiantheoremthatallowsformerginginformationfromdatasetswithvaried
resolutionanduncertainty;itweighsalldatasetsusingtheirunderlying
uncertainties,andimprovesthequalityofadatasetwiththeaidoftheinformation
fromanotherdatasetthathasloweruncertainties.Forexample,whenassimilating
satelliteobservationsintomodelingtoestimateSWE,themodelfirstsimulatesa
priorestimateofSWE;thepriorSWEestimateservesasafirstguessofthetrue
SWE,butcouldbebiasedanduncertain.Withtheassimilationscheme,theprior
SWEisupdatedbytheSWEinformationintheobservationsthatismoreaccurate
andprecise;theupdatingadjuststhepriorestimateandyieldsamoreaccurate
posteriorSWEestimate.
AseriesofremotesenseddatasetshavebeenincorporatedwithmodeledSWEto
improvethelarge-scaleSWEestimate.Forexample,theterrestrialwaterstorage
derivedfromGRACEgravityobservations[Formanetal.2012],thesnowcoverarea
observedbyMODISandLandsat[Clarketal.,2005,DeLannoyetal.2012,Kumaret
10
al.2015,Girottoetal.2014,Margulisetal.2015and2016],andthePMobservations
fromSSM/IandAMSR-E[Durandetal2006].Also,whileKalman-typeassimilation
istheclassicassimilationmethodinhydrologicresearch,emergingmethodshave
addednewperspectives,e.g.theArtificialNeuralNetwork[Darianeetal.2014,
Tedescoetal.2004,Ganetal.2009]andtheMarkovChainMonteCarlo
[Moradkhanietal.2012,DurandandLiu2012].
SincePMremotesensingcontainsdirectinformationofSWE,thereforePMisan
attractiveobservationtobeassimilatedforSWEestimation.InexistingPMdata
assimilationsnowstudies,snowinformationfromPMobservationsaremainly
assimilatedintwoways:1)thePMobservedmicrowaveradianceareassimilated
intoacoupledlandsurfacemodel(LSM)andmicrowaveradiativetransfermodel
(RTM),e.g.Batenietal.2013and2015,Cheetal.,2014,Dechantetal.,2011,Durand
etal.,2006,2007and2008,Pulliainen2006,Takalaetal.2011,Toureetal.2011,2)
directinsertingthePMsnowdepthorSWEretrievalsintheassimilationframework,
e.g.Andreadisetal.2006,DeLannoyetal.,2010and2012,Fletcheretal.2012,
Formanetal.2013,Liuetal.2013,Slateretal.2005,Suetal.2008and2010,Sunet
al.2004,Yatheendradasetal.2012,ZaitchikandRodell,2008.Theseprevious
studieshavedemonstratedthatthecombinationofthesnowinformationfromPM
andmodelingcouldeffectivelybringmoderatetosignificantimprovementinthe
accuracyoftheSWEestimate.
11
1.5Themotivationsandstrategiesforassimilatingspacebornepassive
microwaveobservationstoestimatelarge-scalemountainousSWE
Whileexistingstudieshavelaidfoundationsforapplyingradiancedata
assimilationtoestimateSWE,theyareeithercarriedoutwithsyntheticPM
observations,orfocusedonpoint-scaleassimilation;therehasbeennostudy
appliedspaceborneradianceobservationstoestimatemountainSWEatlarge-scale,
toourknowledge.However,fromtheprecisewaterresourcesestimationpointof
view,SWEneedstobechracterizedatleastatwatershedscale,andonlythe
spacebornePMobservationhasthespatialandtemporalcoveragethatisrequired
bytheoperationalwatershed-scaleSWEestimationviaradiancedataassimilation.
Forinstance,Reichleetal.[2014]outlinedfourmajorchallengesinassimilating
observationstoimprovewatercyclevariablesestimate,andoneofthemis“howcan
satelliteradiances(ratherthangeophysicalretrievals)beassimilatedtoimprove
estimatesoflandsurfacehydrologicalconditions(e.g.,soilmoistureandsnow)?”
AssimilatingspaceborneradianceobservationstoestimatemountainSWEis
challenging,andthedifficultyoriginatesfromboththeobservationandthe
modeling.Observation-wise,PMhasintrinsiclimitationswhenusedforSWE
measurement,asdiscussedinSection1.2.3;theeffectsoftheselimitationsare
magnifiedinmountainousareas,duetothecomplexmountainterrain,ground
features,andtheequallycomplexatmosphericturbulence;thesefactorsleadtothe
12
significantspatialsnowvariabilityontheorderofmetersinmountainousareas.In
comparison,thetraditionalPMdatasetspatiallyaveragesthespatialradianceto
25km×25kmgrid-cells;thecoarseresolutionmakesthecurrentPMdatasetdifficult
tocapturethedetailsofthesnowdistributioninmountains[Tedescoetal.2010].In
addition,whenassimilatingPMradiancetoestimateSWE,ideallytheobserved
radianceisafunctionofonlySWE.However,infactthesnowmicrostructure(grain
size),snowpackstratigraphy,andvegetationcouldallsimultaneouslyimpactthe
microwaveradiancealongwithSWE,therefore,thesefactorswouldinevitably
perturbtheSWEinformationinboththeobservedandthemodeledradiance,and
theyultimatelyserveasnoiseandbringserrorsanduncertaintiestothe
assimilation.Forexample,therecentglobalGlobSnowproduct(whichisbasedon
passivemicrowavemeasurementsandstationobservations)masksoutSWE
estimatesinmountainregionsinthepublishedversionduetolargeuncertainties
[Takalaetal.,2011].
Here,Ipresentaradianceassimilationstudythatisspecificallyadaptedto
estimatemountainSWE.Thestrategytosolvethechallengesbroughtbythe
mountainousenvironmentontheassimilationisimprovingboththemodeling
frameworkandthePMobservationprocessingtoenhancetheSWEinformationin
bothsides.Sincedataassimilationisaframeworkthatmergesthecomplementary
informationfrommodelingandobservations,thehypothesisisthatwiththe
13
enhancedSWEinformationinbothmodelingandobservation,theassimilation
standsabetterchancetosuccessfullyimprovetheSWEestimationinthe
mountainousenvironments.Ifproven,space-bornePMdataassimilationcould
potentiallybeutilizedinoften-inaccessibleareastoimprovelarge-scalesnowwater
resourceestimates.
1.6Theorganizationofthedissertation
Thefollowingpartofthedissertationcomprisesthreemajortopics,witheach
topicdiscussedinonechapter.Chapter2focusesonthedevelopmentofanewPM
dataprocessingalgorithmthataimstomakePMobservationsexploitmoreSWE
informationforassimilation.Chapter3introducestheevaluationanderroranalysis
ofthecurrentsnowmodel,andpresentsthedevelopmentofnewmodelmodules
thatimprovestheaccuracyofthemodeledsnowandradianceestimation.In
Chapter4,weintroducethedevelopmentofanewPMradiancedataassimilation
frameworkanditsapplicationinestimatingtheSWEinaSierraNevadawatershed.
14
Chapter2:ImprovingPMdataprocessingforenhancedSWEinformation
2.1MotivationofdirectlyprocessingrawPMobservations
MostPM-basedresearchhasutilizedtheEqual-AreaScalableEarthGrid(EASEGrid),wherebyrawsatelliteobservationsareresampledto25km×25kmfixedgrid
cells[Armstrong&Brodzik,1995].Infact,PMspatialfootprintsareellipsesthat
varyasafunctionofmicrowavechannel,i.e.measurementwavelength.At36.5GHz,
thechannelmostfrequentlyusedforestimatingSWE,therawAMSR-Efootprintsize
isanellipsewithmajorandminoraxesof14kmand8km[Kawanishietal.,2003],
respectively,givingaspatialareaof87.9km2,whichis17.9%ofthesizeofanEASEGridcell(i.e.~fivetimeshigherspatialresolution).Kelly[2009]describedhowthe
currentAMSR-ENASAproductutilizesAMSR-Enative(L2A)observationsto
calculatesnowdepth,andthenresamplessnowdepthtotheEASE-Grid.Indeed,the
L2AAMSR-Emeasurementswouldbeexpectedtohaveabetterrelationship(i.e.
lessscatter)withsnowpropertiesmeasuredontheground,ascomparedwiththe
EASE-Grid.
Inthischapter,wepresentevidencethatmicrowavemeasurementscontain
informationaboutsnowpropertiesinmountainousareas,especiallyaboutSWEand
melttiming,andthatsignificantsnowinformationcouldbeextractedbyprocessing
15
PMmeasurementsintheirnativeresolution.Wepresenttwomethodsbywhich
AMSR-Emeasurementsintheirnativeresolutioncanbeeitheraveragedoveran
area(hereweuseadrainagebasin),orinterpolatedtoapoint(hereweusesnow
courses).Ourgoalinthischapteristoshowthateveninmountainregions,
correlationsexistbetweentheTbandsnowpackaccumulation,ablation,aswellas
melttiming.ThiswillillustratethepotentialforcharacterizingmountainsnowhydrologicprocessesusingPM.Asystemtoexploitthesecorrelationsfor
monitoringisdiscussedintheconclusion.Inthischapter,wecomparethePM
timeserieswithinsitumeasurementsofSWE,andwithmeasurementsof
streamflow,intheKernRiverbasin,intheSouthernSierraNevada,California
(Section2.2).NewmethodsarepresentedtoprocessPMdataintheirnative
resolution(Section2.3).Wecharacterizehowwellthemicrowavesignalcorrelates
withametricofsnowaccumulation,andametricofsnowablationtiming(Section
2.4),andtheTbprocessedfromthenewfootprint-basedalgorithmsarealso
comparedwiththeEASE-GridTbtoexplorewhetherprocessingthedataathigher
resolutionwouldmakesthedataexploitsmoreinformationofSWE.Ifthenewly
processeddatasetisproventocontainmoreinformationaboutsnowaccumulation
andablation,potentiallyitcanimprovethecharacterizationofSWEinmountainous
regionsusingpassivemicrowavemeasurementsinfuturestudies,forexample
16
withinthecontextofadataassimilationscheme.Italsocouldpropelawider
applicationofPMinsnowstudies.
2.2Studyareaanddata
OurstudyareaistheKernRiverbasininthesouthernSierraNevada,USA(Fig.
2).SnowmeltrunofffromtheSierraisutilizedforirrigatedagricultureintheCentral
Valley,aswellasforurbanconsumptionfromnearbypopulationcenters,effectively
functioningasalargesurface“reservoir”forthestateofCalifornia[DWR(California
DepartmentofWaterResources),2009].FromFig.2,elevationsintheKernbasin
rangefrom760mto4240m,withastronggradientofincreasingelevationfrom
southtonorth.ThetotalbasinareadrainingthroughtheKernRivertoitsterminus
atLakeIsabellais2770km2.Forthisstudy,weselectedonlyaportionoftheKern
basinasourstudyarea,from36°15′N–36°41.5′N;thestudyareaboundaryis
showninFig.2,andhasatotalareaof1276km2.Thisareawillbereferredtoasthe
“UpperKern”,hereafter;theareawaschoseninpartforitsminimalvegetationat
higherelevations(amapoffractionalvegetationisshowninFig.2[Homeretal.,
2004]andforitshighelevations;theaverageelevationoftheUpperKernis
~3000m.
TherearefivesnowcourseswithinorneartheUpperKernatwhichSWEis
measureddailyviasnowpillow(aswellasmonthlyviasnowsurvey),andhavedata
17
thatfullycoverthestudyperiodfromWaterYear(WY)2003–2008:UpperTyndale
Creek(UTY),ChagoopaPlateau(CHP),CrabtreeMeadow(CBT),WetMeadows
(WTM)andCasaViejaMeadows(CSV);thelocationofthesesnowcoursesitesare
showninFig.2.Dailydatafromthesnowpillowsareutilizedinthisstudy.From
Table1,elevationsofthesnowsurveysitesvariedfrom2530to3475m,andthe
inter-annualaverageoftheannualpeakSWE(calculatedoverallyearsofrecord)
rangesfrom50.8cmto68.5cmamongthefivesnowcourses.TheCSVandWTM
snowcoursesarelocatednearthesouthernborderoftheUpperKern,in
significantlymoreforestedareas,andatlowerelevation(seeFig.2).Air
temperatureismeasuredatallsnowcourses;additionally,precipitationis
measuredatCBT.WealsocomparedTbmeasurementswithstreamflowmeasured
attheUSGSgage11186000,withatotalupstreamareaof2191km2.
Table1Snowcoursesitesusedinthisstudy;agencyabbreviationsrefertoeitherthe
Continued
18
Table1continued
CaliforniaDepartmentofWaterResources(CADWR)ortotheUnitedStatesArmyCorpsof
Engineers(USACE).
Fig.2Vegetationdensity(a),studyareadigitalelevationmodel,snowcoursesandstream
gage(b).
TheAMSR-EinstrumentisaboardtheAquaspaceborneplatform;Aquaisina
near-polarsun-synchronousorbitwithdailyascendinganddescendingrevisitsat
13:30and1:30localtime,respectively[Parkinson,2003].TheAquaorbitvaries
19
slightlyfrompasstopass,withasixteen-dayrepeatcycle;withinthissixteenday
cycle,agivenlocationontheEarthsurfaceisobservednotlessthanonceeverytwo
days.Observationtimesofourstudyareawithinthis16-daycycleareshownin
Table2.AMSR-Emeasurementsareavail-ablefrom2002whenAquawaslaunched
untilfall2011whentheAMSR-Einstrumentceasedfunctioning.
Table2Daytimeoverpass(DOP)andnighttimeoverpass(NOP)dataavailabilityover
UpperKernforeachdayofthe16-dayAMSR-Ecycle.
WeutilizedAMSR-EPMTbdataat36.5GHz,verticalpolarization,fromWY0308[Ashcroft&Wentz,2006].The36.5GHzfrequencywasusedsinceittendstobe
themostsensitivetoSWE.Theverticalpolarization(v-pol)wasusedsinceitwas
foundthath-polismoresensitivetothepresenceoficelenses,addingadditional
complexitytotherelationshipbetweenSWEandTb[e.g.,Reesetal.,2010].We
obtainedrawTbmeasurementsfromtheLevel2Aproduct,correspondingtothe
nativeresolutionof14km×8km.Forcomparison,wealsoobtainedtheTb
20
measurementsresampledtotheEASE-Grid.Weusedbothdaytimeandnighttime
measurementsinouranalysis,describedbelow.
2.3Methods
InthischapterwefocusoncomparingAMSR-Emeasurementsattheirnative
spatialresolutiontoground-basedmeasurementsofSWEandstreamflow.
Comparisonofremotesensingmeasurementstoground-basedmeasurementsis
inherentlyproblematicduetodifferencesinspatialscaleoftheobservation.Inthis
paper,weillustratetwowaysofmakingthiscomparison.First,wedefineanaverage
brightnesstemperatureoveraspatialareaofinterestbasedonafootprint-based
arealweighted(FBAW)average(Section2.3.1).Secondly,wedefineabestestimate
ofthebrightnesstemperatureatapointinspaceviainterpolationusingaGaussian
inversedistance-weighted(GIDW)interpolation(Section2.3.2).Notethat
timeseriesofTbmaybeeasilyobtainedbyrepeatingthesemethodsforeachAMSREdataswath,whichishelpfulforseasonalsnowanalysis.
NotethattheviewinggeometrybetweenaspecificplaceontheEarth(e.g.our
studyarea)andtheAMSR-EinstrumentchangesfrompasstopassoftheAqua
satellite.IntheleftcolumnofFig.3,theKernRiverbasinisrespectivelylocatedat
theleft,middleandrightsideofthreedifferentAMSR-Escanswaths(bluegrid).
Foot-printorientation(i.e.therespectiveorientationofthemajorandminorellipse
axes)canbecalculatedfromfootprintcenterlocationsandswathinformation(see
21
AppendixAfordetails).TherightcolumnofFig.3showsthefootprintorientations
correspondingtothethreeoverpassesshownintheleftcolumn.
Fig.3AMSR-EviewinggeometryvariationsarecausedbychangesintheAquasatellite
orbitthroughoutthe16-daycycle.Ina),b)andc)theAMSR-Emeasurementswathisshown
incyanforthreedifferentoverpasses;thestudyarealocationisindicatedbyacross.Ind),
e)andf),theorientationoftheAMSR-EL2A37GHzellipticalfootprintsisshown
correspondingtotheAMSR-Eswathsshownina),b)andc),respectively.
22
2.3.1Footprint-basedarealweighted(FBAW)basin-scaleaverage
FormakingcomparisonsofintegratedSWEwithinthestudyarea,weideally
wantaTbmeasurementintegratedspatiallyoverthestudyareaitself,ratherthan
using,e.g.,theEASE-Gridspatialresolution:seeFig.4foracomparisonofL2A
footprintsoftheAMSR-E37GHzchannelcomparedwiththeEASE-Gridcells.The
mostintui-tivemethodofcalculatingaspatialaverageoveranarbitrarypoly-gon
fromoversampledmeasurementsj=1...n,isviaanaverageweightedbytheareaAj
ofeachmeasurementfootprintcommonwiththepolygonofinterest.Thisisshown
graphicallyinFig.4,andcanbewrittenas:
!!!"#$
=
!
!!! !! ∙!!,!
!
!!! !!
(1)
where!!!"#$ istheFBAWaverage,nisthenumberoffootprintsthatfallwithinthe
polygon,Ajistheareaofintersectionbetweenthejthfootprintandthepolygon,and
Tb,jistheTbvaluerecordedinthejthfootprint.Itshouldbenotedthatthe
coordinatesoffootprintcenters,footprintboundariesandbasinboundaryare
originallygiveninlatitudeandlongitude;inordertocalculate!!!"#$ ,allthese
coordinatesarefirstlyregisteredtothelocalzoneusingtheUniversalTransverse
Mercator(UTM)projection.
23
Fig.4Illustrationoftheprocessusedtocompute!!!"#$ forL2A(a)andEASE-Grid(b).The
blueoutlineindicatestheUpperKernstudyarea,theredoutlinesindicatethefootprint
FWHMextents(a)andtheEASE-Gridextents(b),andtheredshadingindicatestheover-
lapofeachfootprintorEASE-Gridcellwiththestudyarea
2.3.2GaussianInverseDistanceWeight(GIDW)point-scaleinterpolation
Calculatingtheareal-weightedaverageTbwithinapolygonsuchasourstudy
areaisappropriateformakingbasin-widecomparisons,butcanbeexpectedto
smoothoutsomeofthespatialvariabilityinthesignal.Forexample,inother
southernSierraNevadabasins,snowhasbeenshowntomeltmuchlaterathigher
elevationregionsofthebasin[Riceetal.,2011].Themostintuitivewayto
interpolatethemeasuredTbtoasinglepointinspaceisbyanaverageofallTb
withinthestudyarea,weightedbytheantennasamplingfunction.Here,we
24
assumedthattheantennasamplingfollowedatwo-dimensionalGaussian
distribution,wherethestandarddeviationσofthefunctioninthealong-trackand
cross-trackdirectionsiscalculatedfromthefullwidthathalfmaximum(FWHM)
spatialextentofthefootprint:
!"#$ = 2! 2!"2
(2)
At37GHz,theFWHMvaluesaregivenbyKawanishietal.[2003]as14kmand8
kmforthe(full)majorandminoraxes,respectively.Accordingtothismodel,58%of
thetotalenergyintegratedatthesensororiginatesfromwithinthetwo-dimensional
GaussianwithFWHMdefinedasabove.TheGaussianfunctioncanbeusedtodefine
weightstobeusedinaweightedaverageestimateoftheTbatapointinspace.Ifthe
distancebetweenfootprintJandapointofinterestisS,thentheweightoffootprint
Jismodeledas:
!! =
!
!!"#
!"# − !! !
!
!"#
!
!!"#
+ !"# − !! !
!"#
!
(3)
whereSmajandSminarethesub-lengthsofSinthemajorandminoraxisdirection,
respectively.Foranypointofinterest,itsGIDWaver-ageTbistheweightedaverage
ofallthefootprintsinthestudyarea:
!!!"#$ =
!
!!! !! ∙!!,!
!
!!! !!
25
(4)
whereTb,jistheTbforthejthfootprint.AswiththeFBAWmethod,allthese
computationsareperformedinthelocalUTMzone.
Aswithotherradarandradiometersystems,AMSR-Eantennaside-lobeeffects
reducethefrequency-dependentmain-beamefficiency,whichisthepercentageof
theenergycollectedbythemain-beamrelativetothetotalreceivedenergyata
specificfrequency.Generally,lowside-lobeeffectsandhighmain-beamefficiency
aredesirableforradiometerssuchasAMSR-E.AccordingtoKawanishietal.[2003],
theside-lobeeffectintheAMSR-E3dBbeamwidthisinappreciable:thepre-launch
testshaveshownthatevenwithouttheremovalofside-lobeeffect,themain-beam
efficiencyat37GHzisabout93.9%.Thevalueisexpectedtobeabove90%even
takingthein-orbitdegradationcausedbyfactorssuchasthermaldeformationinto
consideration.
2.4Resultsanddiscussions
Thetimeseriesofthe!!!"#$ overtheUpperKernstudyareafortheAMSR-E
measurementsat37GHz,verticalpolarization,nightpassesonlyisshowninFig.5a.
Brightnesstemperaturesrangefromamaximumof275Ktoaminimumof225K.
Thereisclearlyastrongseasonalcomponenttothe!!!"#$ variation,withminima
usuallyoccurringnearthebeginningofApril.Tofirstorder,the!!!"#$ variations
arecontrolledbythesurfacetemperatureandsurfaceemissivity;scatteringby
snowleadstovariationsinsurfaceemissivity.AirtemperaturesfromtheCBTgage
26
overthesameperiodrangefrom290Kto260K;therangeof!!!"#$ (50K)
comparedwiththerangeofphysicaltemperatures(30K)indicatessignificant
changesinsurfaceemissivity,presumablyduetoscatteringofthemicrowave
radiationbysnow.Inadditiontothisseasonalorlow-temporalfrequencyvariability,
thereissignificantday-to-dayorhigh-frequencyvariability.Thetop-of-atmosphere
satellitemeasurementalsoincludestheeffectsofmechanicalanddigitalnoise,and
theeffectsofcalibrationuncertaintyarealsopresent.Inadditiontothe
contributionsofsoil,snow,andvegetation(minimalforourstudyarea),thesatellite
observedbrightnesstemperaturereflectsthecontributionoftheatmosphere,which
doestendtoexhibithigh-frequencyvariability,butwithrelativelysmall(onthe
orderof5K)magnitude.Whetherduetoatmosphericvariabilityorinstrument
noise,thehighfrequencyvariabilityinthenighttime!!!"#$ isnotlikelyreflective
ofchangingsnowconditions,exceptpossiblyforearlyspring,whenthediurnal
melt–refreezecycleacceleratesthesnowgraingrowthandsubstantiallydecreases
Tb,leadingtoTbfluctuations.Inordertoremovethishighfrequencyvariability,the
raw!!!"#$ wassmoothedwithaseven-daymovingaverage;thesmoothedresultis
showninFig.5b,higherfrequencyTbsignalsatspringisreservedwhiletheyare
removedforothertimes.Forcomparison,SWEmeasuredatthesnowpillowatthe
CBTsnowcourse(seeFig.2)isshowninFig.5c.
27
Fig.5UpperKern!!!"#$ (a),7-daymovingaverageofboth!!!"#$ (blue)andair
temperature(red)fromCBT(b)andSWEatCBT(c).
28
2.4.1Potentialforcharacterizingsnowaccumulation
ComparisonbetweentheCBTSWEandthebrightnesstemperaturetimeseries
showsthat!!!"#$ showsstrongannualcycles:foreachwateryear,!!!"#$ decreasesfromOctober–April,duringwhichtimesnowaccumulates.The!!!"#$ increasesdramaticallyfromApril–June,duringwhichtimesnowtypicallymelts.The
!!!"#$ minimumforeachwateryeargenerallyshowsaninverserelationshipwith
theSWE.Forinstance,inWY05,theminimum!!!"#$ was230K,andthepeakSWE
was75cm,whereasinWY04,theminimum!!!"#$ was238K,andthepeakSWE
was37cm.FromFig.5,WY06seemstobeanexceptiontothis,whichwillbe
discussedfurtherbelow.Notethattheminimum!!!"#$ typicallyoccursafterthe
maximumSWEaccumulation;forexample,inWY05,thepeakaccumulationwason
1April,andtheminimum!!!"#$ wason15April.Onepossibleexplanationforthis
isthatassnowbeginstomeltduringthedaytimeandrefreezeatnight,grainsize
growthbeginsincreasingrapidlyandicelayermaystarttoformonthesurfaceorin
thesnow-pack,bothofwhichleadstoadecreasingofTb[Colbeck,1982].The
resultinglargegrainsizesleadtolowerbrightnesstemperatures[Changetal.,1976;
Tedescoetal.,2006].Thustheminimum!!!"#$ valuesareasuperpositionofboth
metamorphicstateandsnowdepth.
29
SWEfromgagesCHP,UTY,WTM,andCSVshowclearannualcyclesofsnowpack
accumulationandablation,similartothatshowninFig.5forCBT.Ourgoalisto
examinehowwellthe!!!"#$ correlateswiththeannualbasin-widesnowstorage.
FortheperiodfromWY03toWY08,wecomparedtheannualminimum!!!"#$ valuestotheSWEvaluesatthetimewhenminimum!!!"#$ occurs,shownasa
scatterplotinFig.6.Theannualminimum!!!"#$ andSWEshowaninverse
correspondence,whichisespeciallysignificantforgagesUTY,CBTandCHP.The
correlationcoefficientsbetween!!!"#$ andSWEare−0.96,−0.94,and−0.99,
respectivelyforthesethreegages.AgeneralinverserelationshipexistsforCSV,but
withalowercorrelationcoefficientof−0.86.ForWTM,threeofthesixyears(2005,
2007,and2008)showsomeinversecorrespondence,whiletheotherthree(2003,
2004,and2006)donotshowanyrelationship.FromTable1,WTMandCSVare
significantlylowerelevationthantheotherthreegages;forthethreegagesabove
3000melevation,thecorrelationcoefficientisbetterthan−0.95.Riceetal.[2011]
haveshownthatadifferenceof300melevationleadstoachangeoftwo–three
weeksinmelttimingintheSouthernSierra.Apparently,theminimum!!!"#$ over
theupperKernbasinreflectstheamountofSWEstorage;thecorrelationbetween
theminimum!!!"#$ andSWEishighathighelevationareas,wheredeep
snowpackexists.
30
Fig.6Scatterplotsofsmoothed!!!"#$ vs.SWEatUTY(a),CBT(b),CHP(c),WTM(d),and
CSV(e).
The!!!"#$ isanaverageacrossthestudyarea,soideallywewouldliketo
comparewithanestimateofthebasinaverageSWE.Thelatterisunavailableexcept
viatheaverageofSWEatthefivesnowcoursesdiscussedpreviously.Fig.7shows
31
the!!!"#$ comparedtothebasin-averageSWE,obtainedbyaveragingoverthefive
snowcourses.Thebasin-wideSWErangesfrom17cmto90cm,and!!!"#$ ranges
from230to243K.Thereseemstobeanon-lineardependenceof!!!"#$ onthe
SWE,whichisexpectedfromradiativetransfertheory[Tsangetal.,2000].Thus,
thereismoredynamicrangeinTbassnowdepthvariesfrom17cmto40cmthan
thereisfrom40cmto90cm.Thisisconsistentwithobservedsaturationof
brightnesstemperaturewithrespecttoSWE:sensitivityofTbtoSWEdecreases
withincreasingSWE[Armstrongetal.,1993].Notethatthereisapparentlystill
somesensitivityofthe!!!"#$ toSWE,evenforSWEgreaterthan50cm.SWE
saturationvaluesareusuallylessthanthisvalue;thismaybesimplyduetothefact
thattheSCAwasgreaterduringyearswithlargeraccumulation(notshown).We
hypothesizeofthe50Kofthetotalvariabilityinthe!!!"#$ overthestudyperiod,
30Kisduetothephysicaltemperaturevariability,and20Kisduetothesnow
emissivityvariability.Thelattercontainsinformationabouttherelativemagnitude
oftheinterannualsnowaccumulation,basedonFig.7.
32
Fig.7ScatterplotsofminimumL2A!!!"#$ (blue)andminimumEASE-Grid!!!"#$ (red)for
WY2003–2008versusthebasinaveragemaximumSWE,calculatedfromthefivesnow
courses.
Inordertoevaluatehowmuchinformationaboutsnowaccumulationprocesses
isduesimplytoreprocessingtherawL2ATbdatabasedontheFBAWmethod,we
performedtheFBAWcalculationsusingtheEASE-GridTbdata,insteadoftheL2A
Tbdata.Specifically,weassumedthattheEASE-Gridcellrepresentsarectanglewith
dimensionsoftheEASE-Griddataspacing,i.e.,asquarethatis25km×25km(see
Fig.4).WethenperformedthesamearealweightingcalculationusingEq.(1)with
thenineEASE-Gridcellsthatfallwithinthestudyarea.Fig.7showsthecomparison
oftheEASE-GridandL2A!!!"#$ withbasin-wideSWE(thesameSWEabscissa
33
valuesareplottedforbothseries).WhiletheEASE-Grid!!!"#$ doesshowaslight
inversetrendwiththebasin-wideSWE,thedynamicrangeofthesignalisonly5K.
Thus,forthisstudyarea,theL2A!!!"#$ isthreetimesmoresensitivetoseasonal
snowaccumulationthanistheEASE-Grid!!!"#$ .Wehypothesizethatthisisdueto
theFBAWprocessing,whichallowscalculationofanaveragebrightness
temperaturecorrespondingtoonlythebasininquestion.
2.4.2Potentialforcharacterizingbasinrunoff
Theseasonalspringsnowmeltrunoffconstitutesalargefractionofthe
accumulatedSWEinthisarea,dueinparttotheshallowsoilsintheSierraNevada.
DailydischargemeasurementsfromtheUSGSgageclearlyexhibitanannualspring
pulse(seeFig.8).Sincelowflowisrelativelyinsignificantcomparedtothespring
freshet,andisrelativelysimilarfromyeartoyear,theintegratedstreamflow
representsameasureofthetotalSWEstoredinthebasin.Foreachyear,we
integratedthestreamflowfromthebeginningtotheendofthespringpulse.The
scatterplotinFig.8ashowsthatannualminimum!!!"#$ valuescorrespond
inverselytointegratedbasinrunoffforeachwateryear,withtheexceptionofWY06.
WY06hasalargerrunoffthanwouldhavebeenindicatedbythe!!!"#$ value.This
isconsistentwiththe!!!"#$ comparisontobasinSWE:theminimum!!!"#$ measuredduringWY06wason1March,whichisapproximatelyfourweeksearlier
34
thannormal(seeFig.5b).Significantsnowfalloccurredafter1March;forexample,
theSWEatCBTincreased25cmbetween1MarchandthepeakSWEon1April.
Theselatesnowfallsdidnotleadtoadecreaseinthe!!!"#$ asinotheryears,
however,and!!!"#$ after1Marchwasalwaysgreaterthanthaton1March.
Fig.8Scatterplotof!!!"#$ andintegrateddischargefromtheUpperKernBasin(a),andthe
hydrographsfortotalannualdischargeofWY2005(blue)andWY2006(red)(b).
Onepossiblereasonfortheanomalous!!!"#$ inAprilWY06isthatthe
precipitationandairtemperaturedatafromtheCBTsnowcourseindicatethat
WY06hadmorefrequentrainfallduringthemeltingseason.Forexample,therewas
significantrainfallinmid-May,whichisunusualbecauseraindidnotoccurduring
thattimeintheotheryears.Anyliquidwaterpresentinthesnowpack(e.g.atlower
elevations)wouldsignificantlychangethemicrowaveemissivity[Mätzler,1987].
35
ThestreamflowforWY05andWY06isshowninFig.8b.WY05isdominatedbya
singlemajorrunoffpeakat170m3/sthatbeginsinmid-May.WY06consistsofa
firstmajorrun-offpeakat145m3/s(beginninginlateApril),whichdecreasesto
75m3/sinlateMay;asecondmajorpeakat143m3/soccursinearlyJune.Thisis
suggestive:thesecondpeakinrunoffisduetorainfallevents,ratherthansnowmelt
runoff.Suchinvestigationsarebetterunderstoodinthecontextofcoupledmodels
ofthesnowpackphysics(includingmeltstate),emissivity,andrunoff.
2.4.3Potentialforcharacterizingsnowablation
The!!!"#$ estimatesasingleTbvaluetorepresenttheentirebasin.Thisis
reasonableformonitoringSWEaccumulation,butasthesnowbeginstomelt,there
isadistinctdependenceofmelttimingonelevation.Inthiscase,thebasinwouldnot
beexpectedtoactuniformly,sincethereisasignificantsouth–northelevation
gradient(seeFig.2).Fig.9showsaspatialmappingofthenighttimeTboftheUpper
Kernbasin,wherethefootprintofeachobservationismappedasanellipsewhere
theminorandmajoraxesaretherespectiveFWHMlengths,andthecolorofeach
ellipseismappedtoaTbvalue.Threesuchmapsareshown,on11March,30April,
and17Mayof2006.On11March,theTbwasfairlyuniform,withameanof
approximately235K,andarangeofjustafewK.Thisuniformspatialpatterninthe
accumulationseasonunderliesuseoftheFBAWtomonitorsnowaccumulation.On
30April,thereisastrongsouth–northgradientintheTb,whereatthe
36
northernmostextenttheTbislow,around215K,butatthesouthernmostextent,
theTbishigh,around280K.Thehighlyvariablespatialpatternisindicativethat
thebasinisatdifferentpointsofmelting.On17May,theTbisapproximately
uniform,withanaverageofabout280K,indicatingthatthesnowhasdisappeared
frommostofthebasin.Thus,theL2AAMSR-Emeasurementsareofhigh-enough
spatialresolutiontodetectspatialvariationinmelttimingacrosstheUpperKern
basin.Basedonthisdiscussion,theTbspatialvariabilitywithinthebasinisan
indicatorofthestatusofmelt:asmeltbeginsatlowerelevations,thevariability
increases.Oncethewholebasinismelting(orhasfinishedmelting),thespatialTb
patternbecomeshomogenous.Fig.10showsthestandarddeviationofallthe
footprintsintheUpperKernbasinduringthestudyperiod.Thestandarddeviation
istypicallylow(~5K)throughthesummerandfall,andtypicallypeaksinAprilor
Mayatabout15K.
Fig.9TbintheUpperKernBasinon11March(a),30April(b)and17May(c)of2006
37
Fig.10TimeseriesofthestandarddeviationofAMSR-Emeasurementsfallingwithinor
partiallywithintheUpperKernBasin.
ThefactthatwetsnowTbisgreaterthandrysnowTbhasbeenexploitedin
numerousstudiestocharacterizemelttiming.Oneofthemostcommonly-used
metricsistheTbdiurnalamplitudevariability(DAV)asusedinRamageetal.[2007],
whichisthedifferencebetweendaytimeandnighttimeTb.Therationaleisthatmelt
occursduringthedaytimeinspring(leadingtoahighTb)andrefreezesatnight
(leadingtoalowTb);thustheDAVisexpectedtoreachaseasonalmaximumduring
thesemelt–refreezecycles.TheTbmeasurementsatUpperKernvarysignificantly
acrossthebasin,especiallyduringthespring(Figs.9and10).Thus,using!!!"#$ to
38
monitorsnowablationisquestionable.Instead,weusetheGIDWtoresampleor
interpolatetheTbtoapointscale(seeEquation.2,3and4).Weperformedthis
operationforboththenighttimeanddaytimeTbvalues,inordertoobtaina
timeseriesof!!!"#$ DAV.Fig.11showsthe!!!"#$ DAVvaluesatCBToverthe
entirestudyperiod,aswellastheinsitumeasurementsofairtemperature(Ta)
diurnalvariations(TaDAV),whichistheairtemperaturedifferencebetweenthe
timesofdaytimeandnighttimesatelliteoverpassingofaday.FromFig.11,TaDAV
rangedfrom−7.5Kto16K,whereasthe!!!"#$ DAVrangedfrom1Kto41K.There
isaseasonal!!!"#$ DAVexcursioncorrespondingtotheperiodwhensnowiswet
duringthedaytime,butrefrozenatnight.Attimescalesofdays,thereissignificant
hightemporalfrequencycoherencebetweentheTaand!!!"#$ DAVvalues.In
ordertoquantifythiscoherence,thecorrelationbetweentheTaand!!!"#$ DAV
arecalculatedannuallyfromDecember1sttoFebruary28th,duringwhichthe
snowpackisdrymostofthetime.Inaddition,thecalculationisonlycarriedoutat
CBT,CHPandUTY;thehighelevationandlowertemperatureatthesegagesfurther
excludeinfluenceofliquidwaterinthesnowpack.Thesix-yearaveragecorrelation
coefficientsbetweenTaand!!!"#$ DAVforCBT,CHPandUTYare0.7626,0.7188
and0.7304,respectively
39
Fig.11TemperatureDAVforairtemperature(blue)measuredatCBT,andfor!!!"#$ (red)
fromAMSR-E.
Wecanquantitativelyevaluatetheutilityofthe!!!"#$ DAVbycomparingthe
DAVtothenievograph(temporalSWEevolutionoveraseason[Fassnacht&Derry,
2010])atthegagesinourstudyarea.Fig.12showstheWY03nievographforCBT,
whichistypical.FromFig.12,SWEincreasesfromNovembertomid-March.One
coulddividetheremainderoftheseasonalcycleintotwoqualitativelydifferent
periods,basedontherateofablation.Frommid-MarchtoearlyMay,thereissome
marginalablationalongwithscatteredprecipitationevents,withdecreasesinthe
SWEatsometimes,andin-creasesatothertimes.FromearlyMaytolateMay,SWE
40
ablationoccursfairlyrapidly.Foreachnievograph,weidentifiedtheaccumulation,
transition,andablationperiodsforeachyear.Theprimarymeansofdifferentiating
theaccumulationandtransitionperiodswasthedateofthepeakSWE;theprimary
meansofdifferentiatingthetransitionandablationperiodswasthebreakinslope
inthenievograph.Fig.12indicatesthedivisionofWY03intothreeperiods—
accumulation(lightgray),ablation(darkgray)andatransitionperiod(medium
gray).
Fig.12Divisionofthewateryearintoaccumulation(lightgray),transition(mediumgray)
andablation(darkgray)basedonsnowcourseSWE;thisexampleisshownfromCBTfor
WY2003.
ItisinstructivetocomparetheL2AandEASE-Griddataintermsoftheirability
tocharacterizeablationtiming;thenighttime!!!"#$ timeseriesatCBTforL2Ais
41
showninFig.13aandforEASE-GridisshowninFig.13b.TheL2A!!!"#$ data
rangefrom225to270K,whereastheEASE-Grid!!!"#$ datarangefrom247to
277K.TheL2ATbGIDWdynamicrangeis45K,comparedwithadynamicrangeof
30KfortheEASE-Grid;theL2Arangeisthus50%greaterthanEASE-Grid.TheL2A
!!!"#$ DAVrangesfrom0to42KforL2A,andfrom5Kto30KforEASE-Grid;the
L2Arangeisthus55%greaterthanforEASE-Grid.FollowingRamageetal.[2007],
wedeterminedthresholdvaluesinthetimeseriesthatcorrespondtotheonsetof
melttiming.Inourexperiment,the“onset”ofmeltwasdeterminedasthefirsttime
thatthethresholdconditionsweremet,andthe“end”ofmeltwasdeterminedasthe
lasttimethatthethresholdconditionsweremet,eachseason.Duetothe
significantlydifferentdynamicranges,thesethresholdvaluesweredifferentforL2A
thanforEASE-Grid;thethresholdsofeachWYweremanuallycalibratedtobest
matchtheonsetandendofmeltdeterminedvianievographsforallyears,suchas
theexampleshowninFig.12.ThethresholdvaluesforL2Awere240Kfor
nighttime!!!"#$ ,and30Kfor!!!"#$ DAV.Thecorrespondingthresholdvaluesfor
EASE-Gridnighttime!!!"#$ and!!!"#$ DAVwere255Kand20K,respectively;
thresholdvaluesareindicatedashorizontallinesinFig.13.
42
Fig.13Analysisofnighttime!!!"#$ (blue)and!!!"#$ DAV(red)timeseriesatCBT
processedusingL2A(a)andEASE-Grid(b)AMSR-Emeasurements.Thresholdvaluesused
todis-criminateonsetandendofsnowmeltareindicatedasred(nighttime)andblue(DAV);
notethatdifferentthresholdsweredeterminedforEASE-GridandforL2A.
Fig.14showsthe!!!"#$ DAVtimeseriesplottedoverthenievographsforCBT
fortheL2Adata.Thetimeperiodidentifiedviathethresholdsasablationis
indicatedbyhorizontalredbars.TheL2A!!!"#$ DAVsignalissensitiveto
differencesinmelttimingamongdifferentyears.Forinstance,thetransitionperiod
identifiedfromthenievographbeganearlyin2004,around1March;in2005,the
43
transitionperiodbeganaround1April.TheL2A!!!"#$ DAVshowsthisdifferencein
melttimingfairlyclearlyforthesetwoyears.Notethatthetransitionperiodas
identifiedbythenievographsisnotreflectiveofwhetherthesnowiswet,butof
whetheranyablation(i.e.decreaseinSWE)hasbeenobserved.Fig15showsthe
day-of-year(DOY)fortheonsetandendofsnowmeltatCBTasobservedfromthe
nievographscomparedtothatpredictedbythe!!!"#$ ,forbothL2AandEASE-Grid
observations.MeltonsetwasfairlyaccuratelypredictedbyL2A!!!!"# witha
correlationcoefficientof0.94andanRMSEof5.04days.ForEASE-Grid,the
correlationcoefficientwas0.87andtheRMSEwas11.7days.ThustheL2Adata
predictsthemeltonsetmorethantwiceasaccuratelyastheEASE-Griddata.The
endofmeltDOYwaspredictedlessaccurately,withacorrelationcoefficientof0.78
andanRMSEof9.24daysusingL2Adata,andacorrelationcoefficientof0.28and
anRMSEof27.3daysusingEASE-Griddata.TheGIDWprocessing,meltdetection,as
wellascomparisonwithEASE-GriddataarealsocarriedoutattheCHPandUTY
stations.Thesetwostations,alongwithCBT,arelocatedabovetree-lineinourstudy
area;CSVandWTMhavehighervegetationdensity,sowedonotpresentGIDW
resultsfromthosestations.MeltdetectionatCHPandUTYareconductedusingthe
samethresholdsasatCBT,thecorrelationcoefficientsandRMSEofthedetected
onsetandendofmeltforthethreegagesareshowninTables3and4.Becauseof
thesimilarityamongCBT,CHPandUTY,theresultsofCHPandUTYaresimilarwith
44
thoseofCBT.TheRMSofthefootprintTbpredictedonsetrangesfrom5to10days,
whiletheRMSofpredictedonsetrangesfrom10to15days.Thecorrelation
coefficientsbetweenin-situmeltonsetandfootprintTbpredictedmeltonsetranges
from0.94to0.99,whilethecorrelationcoefficientsbetweenin-situmeltonsetand
footprintTbpredictedmeltonsetrangesfrom0.80to0.89.Two-sampleStudent'sttestswereusedtoexaminewhetherornottheL2ARMSEwaslessthantheEASEGridRMSE.UsingtheRMSEvaluesinTable3,thep-valueformeltonsetestimatesis
0.069,whilep-valueforthemeltendingestimatesis0.151;thisindicatesthatwe
have93%and85%confidence,respectively,toclaimthatL2A!!!"#$ yieldsbetter
meltonsetandendingestimatesthanEASE-Grid.
45
Table3CorrelationcoefficientbetweentheTb-DAVbasedandin-situmeasuredmeltonset
andending.
Table4RMSE(indays)oftheTb-DAVbasedmeltonsetandendingestimation,usingL2A
andEASE-Griddata.
46
Fig.14L2A!!!"#$ DAVatCBTisshown,aswellastheaccumulation,transition,and
ablationperiods(asdefinedinFig.12)basedonsnowcoursedata,forwateryears2003(a),
2004(b),2005(c),2006(d)and2008(e).Thehorizontallineatthebottomofeachplot
indicatestheperiodmeetingthecriteriaformelt-timingidentification.
47
Fig.15Scatterplotofonsetofmeltday-of-year(DOY)fromL2A!!!"#$ (blue)andEASEGrid!!!"#$ (red)andtheCBTsnowcourse(a).ScatterplotoftheendofmeltDOYfromL2A
!!!"#$ (blue)andEASE-Grid!!!"#$ (red)andtheCBTsnowcourse(b).
48
NotethatinthestudyofRamageetal.[2007]inthemountainousPellyRiver
basinintheYukonterritory(63°latitude),theoptimalthresholdswereidentifiedas
252KfornighttimeTband18KforDAV,whicharewithin3Kand2K,respectively,
oftheoptimalthresholdsidentifiedherefortheKernbasin.Thus,quitesimilar
thresholdsdescribemeltonsetintwomountainousbasins,thoughthe
environmentalandgeographicalconditionsinPellyRiverbasinandKernbasinare
quitedissimilar.ThisisincontrasttoquitedifferentoptimalthresholdsforL2A
comparedwithEASE-Grid,forthesamebasin.NotethatRamageetal.[2007]
comparedTbDAVtostreamflow,whileherewecompareTbDAVtomeltonset
datesdeter-minedfrominsituSWE.
2.5Summary
TheresultspresentedherehaveconfirmedthatPMmeasurementscontain
informationaboutsnowaccumulationandablationinmountainousareas,andthat
thisinformationisbestextractedbyprocessingtheL2ATbmeasurements.
Comparisonbetween!!!"#$ andSWEmeasurementsfromfivesnowcourses
indicatesthatthe!!!"#$ correspondswithsnowaccumulation,despitethelow
spatialresolutionoftheAMSR-Eobservations.Thecorrelationcoefficientbetween
!!!"#$ andbasinaverageSWEwas−0.94.TheL2A!!!"#$ wasthreetimesmore
sensitivethanEASE-Grid!!!"#$ toSWE.Meltonsetwasfairlyaccuratelypredicted
49
by!!!"#$ withacorrelationcoefficientof0.94andanRMSEof5.04daysbytheL2A
data.TheEase-Grid!!!"#$ datahadtwo-thirdsthedynamicrangeastheL2A!!!"#$ data,andpredictedthemeltonsetdatewithapproximatelytwiceasmucherror.
Meltendwaspredictedmuchlessaccurately;theL2Aend-of-meltpredictionwas
moreaccuratethantheEASE-Gridprediction.TbDAVwascorrelatedwithTaDAV,
withanaveragecorrelationcoefficientof0.72forthethreestationsintheUpper
Kern.
TheEASE-GridTbisappealingbecausenoadditionalprocessingisrequiredfor
use.ComparedwithEASE-Griddata,FBAWandGIDWmethodsaremore
computationallyexpensivebecauseviewinggeometryandweightedaveragemust
becalculatedforeachfootprintinthestudyarea.Thisrelativelymore
computationallydemandingprocessincreasestheinformationthatcanbeextracted
fromtheTbsignal.
ThisstudyhasdemonstratedahighdegreeofcorrelationbetweentheTb
timeseriesandinsitumeasurementsofSWEinmountainousareas.However,
designinganaccuratealgorithmforretrievingSWEandmelt-onsettimingwithout
insitudatawouldbeverydifficultfromanalysisoftheTbsignal.Anavenuefor
futureresearchwouldbetouseadataassimilationsystem(asrecommendedby
[Balesetal.,2006])tomergetheTbmeasurementswithaphysicalmodelofsnow
accumulationandablation.Dataassimilationisanintegrativeframeworkthatcan
50
beusedtomergemultipledatastreams[Reichle,2008].Futureworkwillexplore
assimilationoftheTbmeasurementsforoptimalcharacterizationofsnow
accumulationandablation.
.
51
Chapter3:ImprovingtheSWEandsnowradiativetransfermodeling
3.1MotivationofdirectlyprocessingrawPMobservations
InordertoadvancetheapplicationofPMinmountainareas,itisimportantto
improvetheunderstandingoftherelationshipsamongthemicrowaveradiance,
snow,andthesnow-coveredmountainenvironment.Tothisend,anumberof
laboratoryandplot-scalestudieshavebeencarriedout[e.g.,Hallikainenetal.,1987,
Wiesmannetal.,1998,Macellonietal.,2005,Tedescoetal.,2006,Montpetitetal.,
2013].AbetterinterpretationofthespacebornePMmeasurementiscriticalto
improveourknowledgeofhowthemountainenvironmentimpactsmicrowave
emission.Towardthisend,welinkedalandsurfacemodel(LSM)andaradiative
transfermodel(RTM).Withinthecoupledmodel,theLSMtakesinmeteorological
forcingandsimulatesthesnowstates(e.g.,thedepth,thedensity,thetemperature,
andthegrainsize),whereastheRTMusesthesnowstatesmodeledbytheLSMas
inputsandsimulatesthemicrowaveradianceabovethesnowsurface.Withthis
coupledmodel,wecarriedoutahigh-resolutionradiancemodelingstudyfora
maritimemontanesnowpack,comparingandvalidatingagainstspaceborneTb
measurements.Keyparameterscharacterizingthemountainenvironment,
52
includingelevation,aspect,slope,thefractionalforestcoverage,andthemodeling
estimatedsnowcondition,wereexplicitlyresolvedinthehigh-resolutionmodel.
Inthischapter,weseekanswerstothefollowingquestions:Canweusea
relativelysimpleandcomputationallyefficientmodeltocapturethespatiotemporal
patternsandreplicatetheobservedradianceinmountainranges,andwhatarethe
effectsofthesignificantspatialvariabilityofmaritimemountainsnow-packonthe
saturationofthePMmeasurements?
3.2Studyareaanddata
3.2.1Studyarea
OurstudywascarriedoutintheNorthForkoftheKernRiverBasin,Sierra
Nevada,USA.KernislocatedinthesouthernSierraNevadaandischaracterizedby
high-reliefterrainandmaritimeclimate.Inwinters,coldfrontalsystems(midlatitudecyclones)andthelocalorographygeneratesignificantsnowaccumulation
overrelativelyshortperiodsoftime.Indeed,severallargesnowfalleventstypically
accountforalargefractionoftheannualaccumulation[Serrezeetal.,1999].Fig.16
showsthedigitalelevationmodelandforestcoveroftheNorthForkofKern,with
elevationsincreasingtowardthenorth.Theforestvegetationcoveragefraction
decreasesastheelevationincreases.Inthispaper,ourexperimentswereprimarily
carriedoutintheupperpartofthebasin(UpperKernBasinhereafter),wherethe
53
highelevationcoincideswiththelowvegetationcoverageandlong-lastingseasonal
snow.The511-km2UpperKernhasanaverageelevationof∼3600mandan
averageforestcoverageratiolowerthan5%.
Fig.16Forestcover(left)andelevation(middle)oftheNorthForkofKernRiverbasin.
ThreesnowpillowsarelocatedintheUpper-Kernbasin--theregiondelineatedbygrayline.
3.2.2Data
TherearethreeinsitusnowpillowsCBT,CHP,andUTYlocatedintheUpper
KernBasin.WemodeledsnowandTbforwateryears(WYs)2003–2008;thus,the
dailySWEdatameasuredatthesesnowpillowsduringthisperiodwereusedinour
54
experiments.Theelevation,theforestfraction,andthepeaksnowaccumulationat
eachsiteareshowninTable5.
Elevation
[m]
Forest
fraction
CBT
3261
CHP
UTY
Gage
PeakSWE[m]
2003 2004
2005
2006
2007
2008
12.5%
0.41
0.39
0.77
0.65
0.15
0.45
3139
12.5%
0.38
0.46
0.70
0.67
0.29
0.60
3474
0.0%
0.58
0.53
1.02
0.81
0.25
0.65
Table5Theelevation,fractionalforestcoverageandpeakSWEforWY03-08atthethree
in-situgagesintheUpperKern.
Inordertoforcethesimulationofseasonalsnowevolutionandthemicrowaveradiance,
NorthAmericanLandDataAssimilationSystemphase2(NLDAS-2)meteorologicaldata
wereused.ThehourlyNLDAS-2datausedincludeprecipitation,airtemperature,specific
humidity,airpressure,longwaveandshortwaveradiation,andwindspeed.Theraw
NLDAS-2dataareataspatialresolutionof1/8◦(∼15km),whichistoocoarseto
characterizethespatialsnowvariabilityandgroundfeaturesinthecomplexmountainous
terrain.Therefore,wesetthemodelingresolutioninthispaperat90m;alltheNLDAS-2
forcingdataweredisaggregatedtothe90-mmodelingresolution,asdescribedinGirottoet
al.[2014].InGirottoetal.[2014],meteorologicalforcingsuchasairtemperature,air
55
pressure,humidity,andlong-waveradiationaredisaggregatedusingmethodsthathave
beenpublishedandvalidatedelsewhere;asummaryisprovidedinAppendixAofGirottoet
al[2014].FortheNLDAS-2solarradiationthatisoftenuncertaininmountainousareas
[Panetal.,2003],Girottoetal.[2014]separateditintodirectanddiffusefluxes,eachof
whichwastopographicallycorrected.AprobabilisticSWEreconstructionusingthe
disaggregatedforcingreportedthatthemeanrelativeerrorsofthereconstructedSWEwere
2.5%comparedwiththeinsitusnow-pitSWEmeasurements,andthecorrelation
coefficientswiththeobservedWY1997andWY1999dischargewerefoundtobe0.88and
0.97,respectively.
VegetationdatafromNationalLandCoverDatabase2001[Homeretal.,2004]were
used,includingthevegetationtypeandtheforestfractionalcoverage.Theelevationfor
eachpixelwasobtainedfromtheAdvancedSpaceborneThermalEmissionandRe-flection
RadiometerDigitalElevationModel(ASTERDEM)product(aproductoftheMinistryof
Economy,Trade,andIndustry/NationalAeronauticsandSpaceAdministration);theaspect
andslopeofeachpixelwerecalculatedfromtheDEM.Asaforementioned,theseterrainand
grounddatawerealsoata90-mresolutiontobettercharacterizetheruggedmountain
terrainandthevariabilityofthegroundfeatures.
Inthischapter,theTbobservationsfromtheAdvancedMicrowaveScanning
Radiometer-EarthObservingSystem(AMSR-E)instrumentaboardtheAquasatellitewere
utilizedtovalidatethemodeledradiance[Kawanishietal.,2003].Duringitsoperational
periodfrom2002to2011,AMSR-Eobservedmicrowaveemissionat6.9,10.6,18.7,23.8,
36.5,and89GHzwithdualpolarization.Formostregionsoftheworld,Aquahasdaily
56
ascendinganddescendingrevisitsat13:30and1:30localtime,respectively;thehigh
temporalresolutionisidealformonitoringsnowpackevolution.Inthispaper,weusedthe
rawTbobservationatthe36.5-GHzverticalpolarization(V-pol)fromtheAMSR-ELevel2A
product,withanativefootprintsizeof87.9km2[Kawanishietal.,2003].
3.3Models
3.3.1LandSurfaceModel
Theestimationoftheupwellingsurfacemicrowaveradianceisaccomplishedvia
acoupledLSMandamicrowaveRTM.Inthispaper,theSimplifiedSimpleBiosphere
version3.0(SSiB3,see[Xueetal.,1991])LSMwasusedtopredictbulksnowstates.
SSiB3consistsoftheoriginallandmodelSSiBandtheSimpleSnow–Atmosphere–
SoilTransfer(see[Sunetal.,1999])snowscheme.SSiB3characterizestheland
surfacebiospherewiththreesoillayers,twovegetationlayers,andthreesimplified
snowlayers.SSiB3usesasimplere-layeringschemeduringsnowfallevents
[Durand,2007]tomaintainathree-layerapproximationtothefullsnow
stratigraphy.Thesimulationofthephysicalsnowprocessesonlyusingthreelayers
makesSSiB3computationallyefficient.TheprognosticsnowstatevariablesofSSiB3
includesnowdepth,density,temperature,andvolumetricwatercon-tent.These
snowstatesareestimatedbasedonthenumericalsolutionstothemassandenergy
balanceequationsassociatedwiththesnowprocessessuchasfreshsnow
accumulation,compaction,microstructuremetamorphism,andmelting.Thesnow
57
grainsizeplaysadominantroleinradiancesimulation[MätzlerandWiesmann,
1999].WeintegratedthedynamicgraindiametermodeldevelopedbyJordan
[Jordan1991]intoSSiB3.Indeed,comparedwiththesemi-empiricalJordanmodel,
therearemanyphysicalsnowgrainmetamorphismmodelsthatcouldpotentially
betterestimatethegrainsize(e.g.,theFlanner–Zendermodel[FlannerandZender,
2006]).WeselectedtheJordanmodelbecauseithasthebestcombinationofsnow
processrepresentationandcomputationaldemand.TheJordanmodelcharacterizes
thegrowthofdry-snowgraindiameterdtobeproportionaltothevaporgradient
insidethesnowpackandinverselyrelatedtothegrainsizeitselfasfollows:
∂d
U
= α1 v
∂t
d
(5)
where!! isaparameterfordrysnowgraingrowthrate,witharecommendedvalue
of5.0×10!! m4/kg,and!! isthevapordiffusioninthesnowlayer.Substitutionof
theexpressionfor!! intotheaboveequationillustratesthatgraingrowthis
proportionaltothetemperaturegradientofthesnowpack(!"/!"),thetemperature
(T)andatmosphericpressure(!! )ofthesnowpack:
∂d α 1 De0 s ⎛ 1000 ⎞ ⎛ T ⎞
∂T
=
C
⎜
⎟
KT
∂t
d ⎜⎝ Pa ⎟⎠ ⎝ 273.15 ⎠
∂z 6
(6)
where!!!! istheeffectivediffusioncoefficientforsnowat1000mband0°C,andCKT
isthevariationofequilibriumvapordensitywithrespecttotemperature.The
Jordanmodelhasbeensuccessfullyappliedtoestimategrainsize,andhas
58
outperformedmorecomplexgrainsizeestimationschemesinonecase[Huangetal.,
2012].
3.3.2Radiativetransfermodel
AnRTMmustbeemployedinordertosimulatethemicrowaveradianceabove
thesnowsurface.Inthispaper,wehavechosenthemicrowaveemissionmodelof
layeredsnowpacks(MEMLS[WiesmannandMätzler,1999])toestimateTbatthe
36.5-GHzV-pol,whichexhibitsahighersensitivitytothesnow’smicrowave
scatteringthanotherchannelsinspaceborneradiometerssuchasSpecialSensor
Microwave/ImagerandAMSR-E[Changetal.,1982,1987];inaddition,V-polisless
sensitivetotheexistenceoficelayers[Reesetal.,2010].InthecoupledSSiB3and
MEMLSmodel,MEMLStakesinthesnowstatesthatwereoutputbySSiB3and
simulatesthemicrowaveradianceabovethesnowsurface.MEMLSisasimplified
RTMthatsimulatesthetransmissionofthemicrowaveemittedfromtheEarth
surfaceandtheeffectsofvegetationandatmosphereonthemicrowaveradiance.
Themodelutilizestheroughbaresoilmodelin[WegmüllerandMätzler,1999]to
describesoilemission,anditdescribestheattenuationoftheradianceinsidethe
snowpackandattheinter-facebetweenthesoilandthesnowpack.Specifically,the
roughsoilmodelquantifiesthereflectionoftheupwellingmicrowaveradianceat
thesoil–snowinterfacebyintroducingthesoilroughnesslength,whichisafunction
ofmultiplesoilpropertiesandtheradiancefrequency.Thesoilroughnesslength
59
andthegroundtemperaturecontrolthemicrowaveemissionduringsnow-free
periods.Lakes(lessthan1%oftheoverallarea)existinthestudydomain.AsSSiB3
doesnotaccumulatesnowontopoffrozenopenwatersurfaces,theroughsoil
modelisusedheretosimulateradianceforwaterbodies.Insensitivitytests,this
assumptionmadelittleornoimpactontheresults.Futureworkwillexplorethe
optimalwaytoparameterizesnowaccumulationonfrozenlakes.
Tocharacterizethetransmissionoftheradianceinsidethesnowpack,MEMLS
usesthetransmissivityofeachsnowlayertocharacterizevolumescatteringand
absorption.MEMLSalsosimulatesthereflectivityattheinterfaceoftwoadjacent
snowlayersviatheFresnelequations[WiesmannandMätzler1999].Inaddition,
theeffectsofthevegetationandtheatmospherewerealsotakenintoac-countin
theradiancemodelingusingthemethodsdescribedin[TigerstedtandPulliainen,
1998].Thevegetationemissionandattenuationoftheupwellingmicrowave
radiancewascalculated,asdescribedin[Wegmülleretal.,1995].Themodelingof
theatmosphericradiativetransferisbasedontheworkin[Ulabyetal.,1982],
whichcomputestheabsorptionofmicrowaveradiationbyatmospheregasesfor
clearskyconditions.Inthispaper,weappliedtheBornapproximation[Mätzlerand
Wiesmann,1999]tocharacterizethesnowscatteringofmicrowaveradiation;the
Bornapproximationaccountsfortheinternalvolumescatteringofsnowtobetter
characterizetheeffectsoflargesnowgrainsonthemicrowavescattering.InMEMLS,
60
radiativetransferpropertiesarecalculatedfromthelayerthickness,the
temperature,thegrainsize,thedensity,andtheliquidwatercontent.These
parametersmatchwellwiththeoutputsofSSiB3,exceptthemetricusedtodescribe
thegrainsize,i.e.,theJordanmodelestimatesgraindiameterd,whichisanestimate
ofthegeometricgrainsize,butMEMLSrequiresautocorrelationlengthPexto
describethesnowgrainscatteringeffects;thetwograinsizemetricsmustbe
relatedinordertousethecoupledmodeltosimulateTb.Pexcharacterizesthetwopointautocorrelationfunctionoftheice–airmatrix[WiesmannandMätzler,1999],
andinliterature,ithasbeenfoundtobeproportionaltodasfollows:
!!" = !"
(7)
where!isanempiricalconstant,and! = 0.16hasbeenrecommendedin[Mätzler,
2002]and[Huangetal.,2012]hasbeenusedinotherresearch[e.g.Huangetal.,
2012].Thisrecommended!valueandequation(7)willalsobeusedinthisstudyto
relatedandPex.
3.3.3LayerCombinationResamplingSchemeDuringSnowfallEvents
Snowpackisalayeredmedium[Colbeck1991]asaresultofmultiplesnowfall
eventsandsnowpackmetamorphism,suchastheformationofdepthhoarlayers.
Montanesnowpackcanhavetensorhundredsoflayers,inwhichsnowproperties
(e.g.,thedensity,wetness,andthegrainsize)mayexhibitsignificantvertical
stratification[Colbeck1991].Somemodelsindependentlycharacterizeeach
61
snowpacklayerandtracktensorevenhundredsoflayers[Brunetal.,1992],
[BarteltandLehning,2002],whereassimplermodels[Xueetal.,1991],[Oslenetal.,
2010]approximatethesnowpackstratigraphyusingonetofivelayers,withthe
snowpropertiesineachlayerresampledfromthefullstratigraphytoimitatethe
propertiesoftheoriginalsnowpack[Huangetal.,2012].Simulationsusingalimited
numberofsnowpacklayersaretypicallymorecomputationallyefficient,enabling
theiruseinhigh-resolutionlarge-scalemodeling.
Duringsnowfallevents,thethree-layerSSiB3modelmustcombinenewsnow
withtheexistingsnowpacklayers.Thebulkpropertiesofeachlayermustbethen
recalculatedasanaverageoftheoriginalthreelayersthatexistedpre-snowfalland
thenewsnow.Amethodfordoingthiswaspreviouslypresented[Durand2007]
andwillbereferredtoasthe“mass-weightedresamplingscheme.”Themassweightedresamplingschemerecalculatesarangeofsnowpropertiesforthenew
three-layersnowpack,e.g.,snowdensity,temperature,watercontent,depth,and
grainsize.Indeed,resamplinganyofthesesnowpropertieswouldleadto
correspondingchangesinthepredictedmicrowaveradiancefromthesnow.
However,basedonourtestsatthe36.5-GHzV-Pol,onlythechangesinthesnow-
packstratigraphyandthegrainsizeresultingfromthemass-weightedresample
wouldsignificantlyimpacttheradiativetransferpropertiesofthesnowpackby
addingabiasof10–20Ktothesimulatedradiance,i.e.,thechangesintheother
62
snowpropertieshaveaveryminorinfluenceonthemicrowaveradianceabovethe
snowsurface.Therearetwomajorreasons.First,thescatteringbysnowgrainsis
thedominantprocessmodulatingtheupwelling36.5-GHzradiance;theradianceis
highlysensitivetothechangesinthegrainsize[Chang1982,1987]butisnotvery
sensitivetothechangesinothersnowpropertiesatthisfrequency.Second,the
mass-weightedresamplingschemesignificantlychangedthegrainsizeestimate
(showninAppendixB),whereasthechangesinothersnowpropertiesdueto
resamplingaremuchsmaller.Therefore,ourmodelcalibrationonlytreatsthe
stratigraphyandthegrainsize.
Ifthegrainsizeinthebottomsnowpacklayerbeforesnowfallisdefinedas
beforeandthegrainsizeinthebottomlayeraftersnowfallisdafter,then
incorporatingthenewsnowfallviatherecombinationalgorithmleadsto:
!!"#$% = !!"#$%" − ∆
(8)
where∆> 0isthedropingrainsizeduetoincorporatingthefine-grainednewsnow.
Aswillbeshownlaterinthisstudy,itwasfoundthatthelayercombination
resamplingschemeduringsnowfalleventswasofcriticalimportancetothe
accuracyofthemodeling.Indeed,wefoundthatthesimplemass-weightedaverage
introducedlargepositivebiasesintothesimulatedTbduringthefirstlargesnowfall
eventofeachseason.SinceTbisgenerallyinverselycorrelatedwithd,thesepositive
Tbbiasesareindicativeof∆valuesthataretoolarge.Byexaminingtheseasonal
63
grainsizeestimates,weconfirmedthatthemass-weightedaveragescheme
significantlyunderestimatedgrainsizeduringthefirstlargesnowfallevent,andled
topositively-biasedTbestimates(moredetailsaboutthisTbbiasarepresentedin
section3.5.1).ThebiasesinthemodeleddandTbarepossiblyaresultfromthefact
thatthemass-weightedresamplingschemeinSSiB3mainlyfocusesonpreserving
massandenergybalanceofthesnowpack,withlittleconsiderationofsnowpack
microphysicalcharacteristics(i.e.grainsize)andtheimpactonmicrowaveradiative
transfer;i.e.thebiasesaregeneratedbynon-desirablemodelartifactsintroducedby
thesnowpackstratigraphicresampling.Weexploredtwomethodstomodifythe
modelre-layeringschemeandgrainsizeestimateforbetterradiancesimulations.
Thefirstmethodwillbereferredtoasthe“iterativeresamplingscheme”;inthis
scheme,thebottom-layersnowgrainsizeisadjustedinordertoconstraintheTb
predictionofthethree-layermodeltomatchthatofanun-resampledfour-layer
modelrunusingthepre-snowfallsnowpackplusthenewsnowasthetoplayer.The
iterativeresamplingschemepreservesthebulkradiativepropertiesofthe
snowpack,butiscomputationallyinefficient.Thesecondmethodwillbereferredto
asthe“empiricalresamplingscheme”,hereafter,andusesatunableparameterto
approximatelypreservethebulkradiativesnowpackproperties.Specifically,a
positiveconstantCiscalibratedasafunctionofprecipitationintensity,and
incorporatedintothelayercombinationroutine,suchthat:
64
!!"#$% = !!"#$%" − ∆ 1 − ! (9)
Asdescribedbelow,weapplytheiterativeresamplingschemeinpoint-scale
modelingusingthreeyearsofdata,whereinwederivetheparametersforthe
empiricalmethodbyregressionanalysisoftheresultsfromtheiterativemethod.
ThedetailsoftheSSiB3mass-weightedresamplingscheme,itsimpactonradiance
modelingduringlargesnowevents,andthedetailsoftheiterativeandempirical
resamplingschemesarepresentedinAppendixB.
3.4Modelingexperiments
Inthischapterofthedissertation,weaimtoaccuratelysimulatethemicrowave
radiancefromthemountainsnowpackandexplorethesaturationbehaviorofthe
microwaveradiancefromthemodelingresults.Hourlymodelingexperimentswere
carriedoutata90-mspatialresolutionforWYs2003–2008.Inordertofocusonthe
modelstratigraphyandgrainsizeresampling,wecorrectedfortheundercatchof
theNLDAS-2precipitationdatabycalibratingthebiasedprecipitationagainstthe
snowpillowdatatomakethemodeledSWEagreewiththeinsitumeasurement.In
addition,werefinedthedrygraingrowthrateparameterintheJordangrainsize
modelperrecommendation.Wetestedtheiterativeresamplingschemeandthe
empiricalresamplingschemeagainstthesatellite-observedradianceusingthe
modeledTbforthreeWYsandvalidatedoverthreedifferentWYs.Finally,fromthe
65
high-resolutionmodeledradiance,weexploredthePMsaturationbehaviorforour
studyarea.
3.4.1Correctionofprecipitationundercatch
Inthismodelingexperiment,ourobjectiveistoexplorethequestionofwhether
thecoupledLSMandRTMwouldpredictthecorrectTbgiventhecorrectSWEand
graingrowthparameter.TheNLDAS-2precipitationdataareknowntocontain
systematiclowbiases[Panetal.,2003].Therefore,wecalculatedsite-specific
temporal-invariantprecipitationundercatchcorrectionscalarvaluesusingtheSWE
measurementsforeachgageandforeachyearinthesix-yearstudyperiod.The
correctionsshowedthatthebiasintheNLDAS-2appearedtobesimilarboth
spatiallyandtemporallyintherelativelysmallUpperKern,i.e.,allthecorrection
scalarswerewithintherangefrom1.6to1.9.Therefore,inthethree-yearmodeling
testexperimentatthethreeinsitugages,foreachgage,weusedthemeanofthe
correctionscalarscalculatedforthisgageovertheentirestudyperiodtocorrectthe
biasedprecipitationatthisgage.Forthebasinwidemodelingexperiments,we
correctedthebiasofallthepixelswithintheUpperKernwithasinglescalar
calculatedfromthemeanofthethreescalarsusedforthethreegagesinthepointscaleexperiments.
66
3.4.2CalibrationofDryGrainGrowthRate
IntheJordangrainsizemodel,theadjustablegraingrowthrateα1characterizes
theincreaseinthedry-snowgrainsize.AsintroducedinSection3.3.1,Jordan
recommendedanominalα1valueontheorderof5.0×10−7m4/kgandnotedthat
itshouldbeconsideredaparameterforcalibration[Jordan,1991].Forthisreason,
priortothemodelingexperimentsofWY2005(whichhasthelargestsnow
accumulationduringthesix-yearstudyperiod),wecalibratedα1aftercorrecting
theundercatchintheprecipitationforcing,withthemass-weightedresample
scheme.WefoundthatthemodeledradiancedecreasesfasterthantheAMSR-E
observationsduringthesnowaccumulationseason,whichindicatesthenominalα1
istoolargeforcharacterizingthesnowgrowthintheUpperKern.Wefoundthatα1
=3.0×10−7m4/kgbestreproducedtheobservedAMSR-EradianceforWY2005.
Thisα1wasthentestedatthethreegagesforotheryears;thetestresultsshowed
thatthegeneraltrendsofthemodeledradiancewereconsistentwiththe
observations.Therefore,α1=3.0×10−7m4/kgwasadoptedforallthesnowgrain
sizemodeling.
3.4.3Point-ScaleModelingWithIterativeResampleScheme
Inordertotestandvalidatetheiterativeresamplingschemeusedinestimating
thegrainsizeduringlargesnowfallevents,weranmodelsimulationswiththe
precipitationundercatchcorrectionsatthethreesites,orthe90-mmodelpixels
67
correspondingtothethreesnowpillows(CBT,CHP,andUTY)forWYs2004,2005,
and2006.AsshowninTable5,thesethreeyearshavelargesnowaccumulationand
arethusworstcasesingeneratingthebiasesinthemodeledgrainsizeandTb.For
eachyear,modelingwasconductedhourlyforonlythedry-snowseasonfrom
October1totheendofFebruarybecauseliquidwaterstartedtoappearbeginning
inMarch.Whensnowiswet,modelingTbinvolvesanewsetofprocesses;
moreover,forwetsnow,theTbvaluescontainlittleornoinformationaboutthe
SWE.Thus,wefocusonmodelingdrysnowinthispaper.Foreachsnowpillowand
eachofthethreeyears,werantheiterativeresamplingschemeandcalculatedthe
valueofCineachcase.NotethatnoAMSR-Eobservationdatawereusedinthe
calibrationofC(seeAppendixB3.1forthedetailsoftheiterativeresampling
scheme).WecomparedtheTbsimulatedatthethreesnowpillowswiththeAMSR-E
measurementsinterpolatedusingtheGaussianinversedistanceweights(GIDWs)
definedin[Lietal.,2012].
3.4.4ParameterizingEmpiricalResamplingSchemeWithPoint-Scale
ModelingResults
WiththeCvaluescalculatedfromthepoint-scalemodelingwiththeiterative
resamplingscheme,wefitanempiricalrelationshipbetweenthevaluesofCandthe
precipitationintensity(thedetailsoftheempiricalresamplingschemeare
describedinAppendixB2).Suchanempiricalschemeenablesthedirectcalculation
68
ofgrainsizecorrectionratioCbasedontheprecipitationintensity,thusavoiding
theiterationofthecomputationallyexpensiveiterativeresamplingscheme.As
showninthefollowing,theempiricalresamplingschemewastestedatbothpointscale(seeSection3.5.2)andbasin-scalesimulations(seeSection3.5.3),anditwas
validatedviacomparisonwiththeAMSR-Eobservations(seeSection3.5.3).
3.4.5Basin-ScaleModelingWithEmpiricalResamplingScheme
ThecomparisonofthemodeledandmeasuredTbwasper-formedby
aggregatingthehigh-resolutionmodeltothesatellitescaleusingtheAMSR-E
antennapattern.Usingtheaverageprecipitationcorrectionscalarandtheempirical
resamplingschemecalibratedfromthepoint-scalemodelingfromWYs2004–2006,
weperformedbasin-scalemodelingrunsfortheentireseasonofWYs2003,2007,
and2008,yearsforwhichtheresamplingschemewasnotcalibrated.ForeachWY,
weranthehourlysimulationoverthe56168pixelsacrosstheUpperKern.The
independentapplicationofthemodelingateachpixelmakesthecomputational
effortperfectlyparallelizable;therefore,thebasin-scalemodelingexperimentwas
runinparalleltosavecomputingtime.Using100centralprocessingunitscalledfor
thesimulationovertheUpperKernBasin,eachWYtookaround90mintorun.
Inordertovalidatetheempiricalresamplingschemeandtoshowwhetherwe
areabletocapturethespatiotemporalvariabilityobservedintheAMSR-ETb
measurementsfrommodeling,theradiancemodeledateach90-mpixelwas
69
aggregatedtotheAMSR-E36.5-GHzfootprintscale(14km×8km)forcomparison.
DuringWYs2003,2007,and2008,allAMSR-Efootprintswithatleasthalfofits
groundcoveragefallingwithintheUpperKernBasinwereusedinthecomparison.
Theantennasamplingpattern(thefootprintgeolocationandtheorientation)
[Kawanishietal.,2003]oftheseselectedAMSR-Efootprintswasused,asdescribed
in[Lietal.,2012],toresamplethemodeldatatocalculatethemodeledTbvalue
overthesceneobservedbyAMSR-E.Thisallowsforcomparisonbetweenthehighresolutionmodelingandtherelativelycoarseobservation.However,thereliefofthe
terrainwasnotconsideredinthispaper.Sinceterrainreliefcouldalsoaffectthe
patternoftheobservedradiance[Mätzleretal.,1998],theterraineffectsshouldbe
exploredinfuturestudies.
3.5ModelingresultsandDiscussion
3.5.1EvaluatingIterativeResamplingSchemeatPointScale
Point-scalemodelingwascarriedoutatthethreeinsitugagesforWYs2004,
2005,and2006.Fig.17showsthetimeseriesofthegrainsizeestimatesinSSiB3.
ThecoloredcurvesinFig.17representthebottom-layergrainsizeobtainedfrom
simulationswithnocorrection(green)andwiththeiterativeresamplingscheme
(red).Comparingthetwotime-seriesgrainsizeestimates,wefoundthatthe
correctionreducedthedropinthegrainsizecausedbythefirstsnowfallgreater
70
than2cm/daythattypicallyoccurredfromlateDecembertoearlyJanuary.As
describedintheAppendixes,SSiB3requiresthetoptwolayersofthethree
modelinglayerstobethin,withthetoptwolayersmainlyconsistingoffreshsnow
withsmallsnowgrains.Underthismodelrequirement,thechangesinthegrainsize
ofthetoptwolayersasaresultofthestratigraphicresamplinghaveaveryminor
influenceontheradiancemodeling;rather,thebiasisdominatedbythebottom
layerofthethreesinceitismuchdeeperandhaslargersnowgrainscomparedwith
thetoptwolayers.Thebottomlayeraccountsforthevastmajorityofthevolumetric
scatteringoftheentiresnowpack.Therefore,inordertosavecomputationaltime,
noiterativecorrectionswereconductedforthesetwolayers,andthegrainsizein
thetoptwolayersareshowningrayinFig.17.However,itshouldbenotedthatthe
relativelyinsignificantroleofthetoptwolayersinthecurrentstratigraphicsetting
isanothermodelartifact;inreality,thetopfewlayersofthesnowpackplayan
importantroleintheradiativetransfer,andtheirinfluencesareparticularly
significantwhenthesnowpacksaturatesthegroundemissionandwhenliquid
waterstartstoexistinthesurfaceofthesnowpackattheearlymeltseason.Inthis
study,weonlycalculatedthesnowgrainsizeinthebottomlayerwiththeiterative
resamplescheme.Sincenograinsizemeasurementswereavailableatthese
locations,wemustcomparetheTbsimulationwiththemeasuredTbvaluesto
assesstheeffectivenessoftheiterativeresamplingscheme.
71
Fig.18showsthetimeseriesofthesimulatedTbusingboththemass-weighted
resamplingschemeandtheiterativeresamplingscheme,aswellastheAMSR-ETb
measurementsinterpolatedtoeachsnowpillowlocationviatheGIDW,asdiscussed
previously.WecomparedthesimulatedTbwiththeAMSR-EGIDWTbandfound
thattheradiancesimulatedwiththemass-weightedre-layeringschemewasbiased
atallthethreegages.ThecomparisonbetweenthetwosimulatedseasonalTband
GIDWTb(bluecurves)processedfromtheAMSR-Eobservationsclearlyshowsthat,
asaresultofthereducedunderestimateinthebottom-layergrainsize,thepositive
Tbbiasesduringintensesnowfallweresubstantiallyreduced.
Fig.17ThesimulatedbottomlayergrainsizeatCBT,CHPandUTYforWY04,WY05and
Continued
72
Fig17continued
WY06.Thelightgreycurveandthedarkgreycurvearethegrainsizeofthetopandmiddle
snowlayer,respectively.Thegreencurveisthebottom-layergrainsizeusingthemassweightedresamplingscheme,andtheredcurveisthebottom-layergrainsizeusingthe
iterativeresamplingscheme.
Fig.18ThesimulatedTbatCBT,CHPandUTYforWY04,WY05andWY06.Theiterative
resamplingscheme(red)andthemass-weightedresamplingscheme(green)areshown
alongwiththeAMSR-ETb(blue).
73
Inordertoquantifytheimprovementduetotheiterativere-samplingscheme,
wecalculatedtherelativeerrorratiobetweentheTbvaluesestimatedfromthe
iterativeresamplingschemeandthemass-weightedresamplingschemeattheend
ofthefirstlargesnowfalleventofeachyearintheformof
δ=
Tbcorrected − Tbobs
Tbuncorrected − Tbobs (10)
TheTbvaluesinequation(10)arefromtheendoftheintensesnowfallevent;
therefore!evaluateshowtheiterativeresamplingschemecanreducebiasduring
intensesnowfallevents.WealsocalculatedtheRMSEoftheTbestimatedacrossthe
entiredrysnowseason,togetasenseofhowtheiterativeresamplingschemecould
improvetheaccuracyoftheentiredry-seasonestimate.Table6showsdvalues,as
definedin[Chang1982];allcalculatedratiosinTable6arelessthan1.0,indicating
theabsoluteerrorintheestimatedTbafterthesnowfalleventwasreduced.The
meanoftherelative-errorratiooftheiterationmethodinTable6is0.23.This
indicatesthattheiterativeresamplingschemereducedtheTberrorthatwas
generatedduringthefirstintensesnowfalleventby77%onaverage.Table7shows
theTbRMSEfortheentiredry-season.ThemeanRMSEoftheiterativeresampling
schemeatallthreegagesforallthethreeyearsis3.26K.Comparedwiththemean
RMSEoftheTbestimatedfromthemass-weightedresamplingscheme,whichhasa
RMSEof7.27K,theiterationmethodimprovestheaccuracyby55.2%.
74
Thecostoftheseimprovementsistheextracomputationalburden,asthe
improvementinvolvesiterationoftheradiativetransfercalculations.Thepoint
scalemodelingwascarriedoutathourlybasis;eachyearhas8759modelingsteps.
Forthethreeyearswhenthepoint-scalemodelingwasconducted,eachyearthere
werearound75timestepsinwhichtheprecipitationwaslargerthan2cm/hr.When
usingtheiterationmethod,ittypicallytook3to5iterationsfortheconvergence;in
onecase,38iterationswererequired.
Gage
WY04
WY05
WY06
CBT
0.79
0.23
0.67
CHP
0.05
0.05
0.01
UTY
0.07
0.11
0.13
Table6Therelativeerrorcorrectionratiodoftheiterativeresamplingmethod.
Gage
WY04
WY05
WY06
!"#$!"# !"#$!"#$ !"#$!"# !"#$!"#$ !"#$!"# !"#$!"#$ CBT
6.84
6.14
11.04
3.45
5.68
2.30
CHP
5.18
2.46
8.21
4.23
3.20
2.67
UTY
7.19
2.24
10.53
3.57
7.53
2.29
Table7TheTbRMSEsimulatedduringthedrysnowseasonwithoutbottomlayer
correction,andwithcorrectionusingtheiterativeresamplingmethod.Allunitsarekelvins
75
3.5.2FullWidthatHalfMaximummethod
Withthecorrectionresultsobtainedfromthepoint-scaleexperimentsusingthe
iterativeresamplingscheme,weattemptedtodefineaone-to-onerelationship
betweentheamountofthebottom-layergrainsizecorrectionandtheintensityof
snowfallfromthepoint-scalecorrectionresults.Weobtainedthefollowing
relationshipbetweenthecorrectionstoPexandprecipitationintensityPfollowing
theprocessesofdevelopingtheempiricalresamplingschemeinAppendixB3.2:
C pex = 0.6123 × log 3.6 ( P + 1) (11)
Theempiricalresamplingschemeinequation(11)wastestedatthethreegagesfor
WY2003,WY2007andWY2008beforebeingappliedinthebasin-widemodeling.
ThetestingresultsinFig19andFig20showthattheerrorinthebottomlayergrain
sizeestimateswerereduced,andthecorrespondingTbsimulationbecomecloserto
thesatellitemeasurements:theTberrorsatthethreegagewasreducedbyan
averageof66.3%attheendofthefirstintensesnowfallevent,andthemeanRMSE
ofthedryseasonTbwasreducedfrom7.1Kto3.3K.
76
Fig.19Theresultsofcorrectingthebottomlayergrainsizeusingtheempiricalresampling
schemeatthethreegagesforWY03,WY07andWY08.Thecorrectedbottomlayergrain
size(orange)reducedthegrainsizeunderestimationcomparedwithgrainsizeestimated
frommass-weightedresamplingscheme(green).Thelightanddarkgreycurvesarethe
grainsizeofthetopandmiddlesnowlayer,respectively.
77
Fig.20ThecorrespondingTbimprovementsasaresultofthebottomlayergrainsize
estimatebeingimprovedbytheempiricalresamplingscheme.ThecorrectedTb(orange)is
closertotheobservedTb(blue)thantheTbsimulatedwithoutcorrection(green).
3.5.3Basin-ScaleExperiments
3.5.3.1Inter-annualHigh-ResolutionModeling
TheUpperKernBasin90-mTbmodelingresultsareshowninFig.21.Within
eachyear,theSWEmonotonicallyincreasesfromFebruarytoMarch;theincreasein
78
theSWEleadstothedecreaseinTboverthesameperiod.FromFebruarytoMarch,
drysnowdominatedtheentirebasin,withdeepersnowaccumulationatthehighelevationareasshowingarelativelylowerTbthantheareaswithlowelevation.
Frommid-March,thesurfacelayersofthesnowatthelow-elevationareasstarted
melting,andthemeltingextendedtothehigherelevationareaafteritsonset.
Becausewetsnowemitsalmostasablackbodyevenwithaverysmallvolumetric
watercontent,onApril1,highTbvalueswerepresentinlow-elevationareas,butin
higherareas,Tbwasstilllowbecausethesnowstayeddry.OnMay1,watercontent
appearedinthesnowacrossthewholeUpperKernBasin,leadingtohighTbvalues
intheentirebasin.TheinterannualTbcomparisonalsoreflectsthefactthatWYs
2003,and2008havemoresnowaccumulationthanWY2007,i.e.,TbonMarch1of
WY2007arehigherthantheothertwoyears,indicatinglesssnowaccumulation,as
showninTable5.LesssnowaccumulationalsoledtoanearliermeltonsetinWY
2007thantheothertwoyears,i.e.,theareawithhigherApril1Tbislargerthanthe
othertwoyears,indicatingthat,byApril1,snowhasbeenmeltedoveralargerarea
inWY2007.
Fig.21alsoshowsthesimulatedbasin-scaleradiancecharacterizedthedetailsof
theradiancedistributioninthecomplexterrainintheUpperKernBasin,
demonstratingthathigh-resolutionmodelingwasneededtocapturethespatial
variabilityofthelandscapeanditsinfluencesonthemicrowaveradiance.Significant
79
aspect-relatedvariabilityinthenorthwesternpartofthebasinisvisibleinFig.21,
asthecontrastofTbforsouth-andnorth-facingslopesislarge;theTbvaluesforthe
south-facingslopesarehigher,whichismainlyduetothefactthatthesouth-facing
slopesinSierraNevadagetmoresolarenergyandhaveaslightlyhighersnow
temperature.
Fig.21ModeledTbacrosstheUpperKernBasinat90mresolutiononthefirstdayofeach
monthfromFebruarythroughMay(leftcolumntorightcolumn)ofWY03,WY07andWY08
(fromtoprowtobottomrow).
80
3.5.3.2ComparisonBetweenAggregatedModelingTbandAMSR-EObservations
ThemodeledTbateachUpperKernpixelwasaggregatedtotheAMSR-E
footprintscaletocomparewiththeAMSR-Eobservations.Fig.22showsthe
modeledandmeasuredTbonFebruary1forWYs2003,2007,and2008.The
comparisonshowsthat,whensnowisdry(February1),themodeledTbagreeswell
withthesatelliteobservations.Thethree-yearaverageRMSEoftheaggregated
February1TbinFig.22is1.4Kforthe25footprintsshown.Fig.22alsoshowsan
east–westTbgradientinboththemodelingandtheobservationforallthethree
years;theTbinthewestislowerthanthatintheeast.Thisispossiblyaresultofthe
stormtrack,whichcausestheterraintothewesttogenerallyhavegreatersnow
accumulation,effectivelycreatingasmallprecipitationshadow[Girottoetal.,2014].
ThisleadstotheTbtothewestgenerallybeinglowerthanthattotheeast.
81
Fig.22AMSR-EL2ATbobservations(rightcolumn),andthepredictedAMSR-E
observationsaggregatedfromthemodelingpixels(leftcolumn)onFeb1ofWY03(1strow),
WY07(2ndrow)andWY08(3rdrow).
82
Forallthreevalidationyears(WYs2003,2007,and2008),thedailybasinaveragemodeledandmeasuredTbvalueswerecompared,andthedifferences
betweenthemodelandobservationbasin-averageTbvaluesareshowninFig.23.
Thisfigureshowsthat,duringthedry-snowseason(OctobertoMarch1),thebasinscalemodelTbisconsistentwiththeobservedTb.Fig.23alsoshowsthat,after
March1,whenthesnowstartedmelting,thewatercontentinthesnowpackmadeit
emitthemicrowaveradianceapproximatelyasablackbody,anditdramatically
increasedthemodeledTb,leadingtothemodeledTbvaluesthatwere25–30K
higherthantheobservedTb.AfterallofthesnowmeltedinJuly,theground
propertiesbecomedominantinthemicrowaveradianceemissionfromthesnowfreeground.Duringthesummers,themodeledradiancegenerallyagreeswiththe
observationwell,althoughweobservealowbiasofabout2Kinthesimulated
radiance.ThisTberrorispossiblyduetotheerrorsinthesoilroughnessparameter
estimatedbytheroughsoilmodelduringthethawingofthefrozensoiloraslightly
negative-biasedgroundtemperatureestimatedbytheSSiB3model.Fig.24isa
scatterplotshowingthemodeledandmeasuredTbfromOctobertoMarch1forall
thethreeyears.TheRMSEofthebasin-averagedry-snow-seasonmodeledTbis
2.94,3.74,and2.62KforWYs2003,2007,and2008,respectively.Theaverageof
thebasin-scaledry-seasonRMSEacrossthethreeyearsis3.1K,whichis
comparablewiththemeanRMSEinthepoint-scalecorrectionsusingtheiteration
83
method(3.26K).Thebasin-scaledry-seasonmodelingresultsindicatethatthe
relativelysimpleempiricalresamplingschemeisnotonlycomputation-ally
economical,butitalsosuccessfullyavoidsintroducingTbbiasesgeneratedbymodel
artifactsduringintensesnowfallevents.
Fig.23Theseasonaltime-seriesdifferencebetweenthedailybasin-averagemodeledTband
AMSR-EobservedTbinUpperKernforWY03(green),WY07(red)andWY08(blue).Tbpred
isthebasin-averagemodeledTb.
Fig.24Comparingthefootprint-aggregatedmodeledTbandtheAMSR-Eobservationsfor
thedrysnowseasonofWY2003,WY2007andWY2008.
84
3.5.4AnalysisofSpacebornePMSaturationBehavior
NumerousmodelingandfieldstudiesfocusedonestimatingtheSWEwithPM
havereportedthesaturationofthesnowsignalinsatellitePMmeasurements(e.g.,
seeChang1987,DurandandMargulis2006,AndreadisandLettenmaier2012,Dong
etal.,2005,MätzlerandWiesmann,1999,Kellyetal.,2003,Langloisetal.,2011,
Halletal.,1986,Fosteretal.,2005,Sturmetal.,1993,Andreadisetal.,2008,Liang
etal.,2008).Saturationoccurswhenthesnowpackscattersallthemicrowavewhen
ittransmitsinsidethesnow-pack,leadingtotheradianceobservationsnolonger
decreasinginresponsetoanincreaseinthesnowdepth.Sometimes,when
saturationoccurs,theradianceabovethesnowsurfacemainlycomesfromthe
uppersnowlayerswherethegrainsaresmallandwherethereislessscattering.
ThisphenomenonhasbeenobservedtocauseanincreaseinTbinresponsetothe
depth[RosenfeldandGrody,2000].Indeed,increasesinTbinresponsetothe
snowfallwereobservedintheAMSR-Edatainthispaperandwerereproducedby
themodel(seeFigs.18and20).At37GHz,multiplesaturationdepthvalueshave
beenreportedinpreviouspublications,e.g.,0.08–0.8m[Chang1987],0.15m[Dong
etal.,2005],0.8m[Fosteretal.,2005],0.3m[Sturmetal.,1993],and0.5–1m
[Liangetal.,2008].ThesaturationSWEconvertedfromthesereportedsaturation
depthswithatypicalsnowdensityfrom200–400kg/m3wouldrangefrom∼0.1to
0.3m.Thevariabilityinthereportedsaturationdepthismainlyattributabletothe
85
differencesinthedensity,thegrainsize,andthestratigraphyofthesnowpack
[Chang1987],[Halletal.,1986],[Sturmetal.,1993].
Theinter-annualcomparisonbetweenthesimulatedTbandtheSWEatallthree
gagesinFig.18showsthatthereisanobviousdifferenceintheTbpatternbetween
WY2005andWYs2004/2006,i.e.,whereastheobservedTbvaluesinWYs2004
and2006continuetodecreasewiththeincreaseintheSWEfromOctobertoMarch,
thedecreasingtrendsareinsignificantfromDecembertoMarchofWY2005.Thein
situSWEmeasurementsshowthatthesnowaccumulationfromDecembertoMarch
ofWY2005wassignificantlylargerthanthatforthesameperiodofbothWYs2004
and2006.InordertoinvestigatewhethertherelativelystableTbfromDecemberto
MarchofWY2005isaresultofsaturation,wecomparedthemodeledbottom-layer
transmissivityofthesnowpackwiththemodeledSWEforthefirstdayofeach
monthinthedryseasonofWY2005.Asdiscussedpreviously,thetoptwolayersare
thin,andtheireffectsontheradianceattenuationarenegligiblecomparedwiththe
muchthickerbottomlayer;thus,theyarenotincludedinthiscomparison.The
transmissivityandSWEcomparisonwasconductedonlyatgageUTYbecausetheTb
modeledatUTYagreedbetterwiththeAMSR-Emeasurementsthantheothertwo
gages(astheresultsinFig.18andTable6show).TheresultsinFig.25showthat
thetransmissivityofthemodeledbottomlayerdecreaseswiththeincreaseinthe
SWE.However,thetransmissivitysignificantlydecreasesmoreslowlyafterJanuary
86
1;fromJanuarytoMarch,thetransmissivitydecreasesby0.09correspondingwith
anSWEincreaseof0.5m,andincontrast,fromOctobertoDecember,the
transmissivitydecreasedby0.75asaresultofa0.35-mSWEincrease.Therelatively
smallchangeinthesnowtransmissivityisanindicationofmicrowaveradiance
saturation.
Fig.25ComparisonbetweenthemodeledsnowtransmissivityandthemodeledSWEofthe
pixelcoveringthelocationofgageUTYforthefirstdayofeachmonthinthedryseasonof
WY2005.October1stwasomittedinthiscalculationastherewasnosnowatUTYonthat
date.TheRMSEofthepolynomialfitis0.017.
WealsocomparedtheAMSR-ETbwiththemeasuredSWEatUTY.Theresultsin
Fig.26showthesaturationofTbinWY2005.InFigs.25and26,anSWEof0.3–0.5
mtendstobeaturningpointwheretheeffectoffurtherSWEincreasesbecomes
87
insignificantontheattenuatingmicrowaveradiance.Therefore,weconcludethat
the37-GHzmicrowaveemissionintheUpperKernsaturateswhentheSWEreaches
0.3–0.5m.ConvertingthissaturationSWEtothesaturationdepth(∼1–1.5m),itcan
befoundthatthesaturationdepthintheUpperKernishigherthanthetypical37GHzsaturationdepth.OnepossibleexplanationforthisistheUpperKernmaritime
climate,wherethemildtemperaturesandthedeepsnowpackleadtorelatively
smalltemperaturegradients,whichfurtherleadstoaslowgraingrowth[Colbeck
1982].Thus,duringtheaccumu-lationseason,thesnowgrainsizetendstobe
smaller,whichleadstohighersaturationdepths[Chang1987],[Dongetal.,2005],
[Fosteretal.,2005],[Sturmetal.,1993],[Liangetal.,2008].
Fig.26TheAMSR-EGIDWTbandmeasuredSWEatgageUTYonthefirstdaysofOctoberto
MarchofWY2004(red),WY2005(green)andWY2006(blue).Oneachcurve,thesixpoints
fromlefttorightareshowninorderfromOctobertoMarch.
88
AnotherfactorinfluencingthesaturationbehaviorofthesatellitePM
observationsisthelargespatialvariabilityofalpinesnow.IntheUpperKern,the
ruggedterrainandthelargeelevationgradientleadtosignificantspatialvariability
inthesnowaccumulation.Asaresult,insomeareas,thesnowaccumulationisdeep
enoughtosaturatethelocalmicrowaveradiance,butinotherareas,thesnow
accumulationisnotsufficientlydeepforsaturation.Forexample,Fig.27compares
themeasuredbasin-averageTbwiththemodeledbasin-averageSWEforthefirst
dayofeachmonthinthedry-snowseasonofWYs2003,2007,and2008.ThebasinaverageTbwascalculatedfromrawAMSR-Efootprintsusingthefootprint-based
arealweight(FBAW)methodintroducedin[Lietal.,2012],andthebasin-average
SWEforeachdaywascalculatedusingthemeanmodeledSWEacrossthestudy
area.AsFigs.25and26indicate,thesnowradiancesignalsaturateswhentheSWE
reaches0.3–0.5matthepointscale.Atthebasinscale,asshowninFig.27,the
AMSR-EobservationsstillrespondedtotheSWEincreasesthroughMarch1,despite
thefactthat,forWYs2003and2008,thebasin-averageSWEhasreachedthepointscalesaturationSWEbyFebruary.WediagnosedthesubfootprintSWEvariability
onMarch1ofWY2008.Fig.28showsthat,whereasthebasin-averageSWEon
March1ofWY2008isapproximately0.55m,theeastsideofthebasinhassig-
nificantlyloweraccumulationthanthewestside,asdiscussedinSection3.5.3
[Girottoetal.,2014].Wecalculatedthemean,thestandarddeviation,andthe
89
coefficientofthevariationoftheSWEwithineachfootprintfromthehighresolutionmodelpixels,andwecomparedtheminTable8.ThemeanSWEvalues
withineachfootprintvariedfrom0.34to0.63m,whichisnearlyafactoroftwo.The
SWEstandarddeviationwithineachfootprintrangedfrom0.03to0.09m.The
coefficientofvariationrangedfrom0.06to0.24.InFig.28andTable8,footprint3
hasthegreatestcoefficientofvariation,encompassingbothlargeSWEvaluesalong
ridgelines,alongwiththeKernRivervalleyitself,withmuchlowerelevationsand,
thus,lessSWEaccumulation.Resultsshowthat,whereastheaveragefootprintSWE
inthewestpartofthedomainmaybesaturated,theaverageSWEinsome
footprintstotheeastwasatthelowerboundaryofthesaturationrange;therefore,
thesefootprintswerestillsensitivetoincreasesintheSWE.
Fig.27TheAMSR-EFBAWbasin-averageTbandmodeledbasinaverageSWEonthefirst
dayofOctobertoMarchofWY2003(red),WY2007(green)andWY2008(blue).Oneach
curve,thesixpointsfromlefttorightareshowninorderfromOctobertoMarch
90
Fig.28Thebasin-scaleSWEestimateandtheAMSR-EfootprintsonMarch1stWY2008.
Footprint
1
2
3
4
5
6
7
!(m)
0.53
0.38
0.56
0.55
0.34
0.63
0.61
!(m)
0.07
0.09
0.05
0.03
0.04
0.04
0.06
!"
0.13
0.24
0.09
0.06
0.12
0.06
0.10
Table8Themean(!),standarddeviation(!),andcoefficientofvariation(!")ofthe
modeledSWEwithintheAMSR-EfootprintsofMarch1stWY2008observation.Footprint
numberscorrespondtothoseshowninFigure28.
Inapreviousstudyconductedbytheauthors[Lietal.,2012],astrong
correlationwasdiagnosedbetweentheannualminimumAMSR-ETbobservations
91
intheUpperKernBasinandtheinsituSWEmeasurementstakenonthedayofthe
annualmini-mumfromsnowpillows.TheinsituSWEatthetimeswhentheannual
minimumTboccurredhavebeenfoundtoexceedmostofthereportedsaturation
depths.Indeed,theannualmaximumaccumulationandminimumTbareoftenafter
March1,withameasurablechangeinTbvisiblefortheSWEincreasingfrom50to
80cm.SincethesevariationsareafterMarch1whensnowsometimesbecomeswet,
wehypothesizethatthelargevaluesofthesaturationSWEinthatstudywouldneed
tobeinvestigatedalongwiththesnowmelt–refreezecycles,whichisbeyondthe
scopeofthisstudy.
3.6Summary
Aphysicalsnowmodelwithasimplifiedsnowstratigraphicrepresentationwas
setupata90-mspatialresolutiontoat-tempttoreplicatethespatiotemporal
patternofthe37-GHzmicrowaveradiancefromthemaritimesnowpackinthe
UpperKernBasin,SierraNevada.Anewgrainsizeestimateschemewasproposed
andintegratedintothemodeltohandleintensesnowfallswithrespecttothe
stratigraphyrepresentationandthebrightnesstemperatureprediction.The
precipitationandthecoefficientgoverningthegraingrowthratewerecalibrated
usingWYs2004,2005,and2006.Themodelingframeworkwastestedand
validatedusingWYs2003,2007,and2008,andthesaturationbehaviorofthe
92
microwaveradiancefromthehighlyvariablesnowpackintheUpperKernwas
examinedfromthemodelingresults.
Itwasshownthatsimplisticsnowstratigraphicrepresentationsduringintense
snowfalleventssuchasthoseintheSierraNevadacanleadtomodelartifactsthat
degrademodelradiancesimulationsthroughouttheseasonduetotheir
underestimationofthesnowgrainsize.Theerrorsinthegrainsizeestimateleadto
asystematicpositivebiasinthemodeledmicrowaveradianceupto20K.Anew
stratigraphicresamplingschemewasproposedthatavoidsthemodelartifacts,
leadingtodry-season(October–March)RMSEvaluesof3.5Kcomparedwiththe
AMSR-ETbmeasurementsduringthethreeyearsusedtocalibratetheresampling
scheme.TheresamplingschemewasusedtosimulateTbandcompareitwiththe
AMSR-EmeasurementsacrosstheUpperKernBasinduringanotherthreeyearsnot
usedintheresamplingschemecalibration.TheRMSEoftheTbaggregatedfromthe
basin-scalemodelingisabout3Koverthedry-snowseason,indicatingthatthe
microwaveradianceestimatedbythemodelcloselyreplicatedtheradiance
observations.Modelingattemptswereunsuccessfulaftermeltbegan,witherrors
greaterthan10K.
ThesaturationeffectoftheSWEonthemicrowaveradianceinthemountainous
UpperKernwasassessedbyanalyzingthemeasuredSWEandTb,aswellasthe
modeledsnowpacktransmissivityandthemodeledTb.WefoundSWEsaturation
93
valuesbetween0.3and0.5mfromthesimulationsforWYs2004–2006.Thevalues
aresignificantlygreaterthanthevaluereportedinexistingliterature,e.g.thosein
[Chang1987],[Dongetal.,2005],[Fosteretal.,2005],[Sturmetal.,1993],[Lianget
al.,2008];wehypothesizethatthisisaresultofthesmallersnowgrainsinthe
maritimesnowpack.Thelargespatialvarianceofaccumulationimpactsthe
saturationoflarge-scaleradianceobservationsaswell.Analyzingthesub-footprint
SWEdistributionfromthehigh-resolutionmodelingresultsshowsthat,evenlatein
theaccumulationseasonwhenlargeSWEexists,theradiancesignalcouldremain
unsaturatedinAMSR-Eobservationsduetothefootprintaveragingoverboththe
saturatedandunsaturatedareas;thus,theoverallobservationstillrespondstothe
snowincreaseintheunsaturatedregions.
Theexperimentsinthispaperhavedemonstratedthefeasibilityofapplying
simplifiedmodelstoaccuratelyestimatethemicrowaveradiancefordrysnowin
mountainousareasatahighspatialresolution.Accordingtotheworkin[Durand
andMargulis2007],theerrorsofthesnowmicrowaveradiancemodelingneedsto
bewithintherangeof±5KinorderfortheRAtoyieldbenefitstoSWEestimates.
Therefore,whenapplyingRAtoestimatetheSWE,increasingtheaccuracyofthe
microwaveradianceestimatewillallowfortheassimilationsystemtobetter
translateradiometricobservationsintosnowinformation.
94
Chapter4:ThedevelopmentoftheEnsembleBatchSmootherassimilation
schemeforSWEestimationintheUpperKernbasin
4.1Introduction
WedevelopedandvalidatedanEnsembleKalmanBatchSmoother(EnBS)
schemeinwhichspace-bornePMobservationswereassimilatedintophysicalsnow
modelingforthelarge-scaleSWEestimateinalow-vegetatedmountainous
environment.Indeed,vegetationremainsamajorobstacleforapplyingPMto
estimatesnow;severalrecentstudieshavefoundPMlosealmostallthesnow
informationintheareaswithvegetationdensityhigherthanabout25%[Caietal.
2015,Vander-Jagtetal.2013].Therefore,tostartwith,thisstudyevaluatesthe
applicabilityofassimilatingspace-bornePMtoestimateSWEinlow-vegetated
mountains.Specifically,alandsurfacemodel(LSM)andamicrowaveradiative
transfermodel(MEM)werecoupledtosimulatethepriorSWEandthemicrowave
radiancefromthesnowpack.Weconductedthemodelingatahighspatial
resolutiontoattempttocharacterizethespatialvariabilityoftheSWEandradiance
inmountainenvironments.Rawspace-bornePMobservationswereprocessedto
point-scale,footprint-scale(nativePMobservationresolution),andbasin-scale;
threeassimilationswerecarriedoutindependentwiththeobservationsateachof
95
thethreeresolutions.ThedesignoftheEnBSlettheassimilationframeworkuseall
theobservationstoupdateallthemodelpredictionsinabatchmode.The
assimilationgeneratedatime-seriesofposteriorSWEestimatesthatcontainboth
thefine-scalesnowvariationinformationfromthemodelingandthelarge-scale
totalsnowamountinformationfromPM.
Ourhypothesisisthatassimilatingspace-bornePMobservationsintothe
process-levelsnowandradiancesimulationcanimprovemountainSWEestimate.If
proven,space-bornepassivemicrowaveradiancedataassimilationcouldpotentially
beutilizedinoften-inaccessibleareasthatareabovetree-linetoimprovelarge-scale
snowwaterresourceestimates.
4.2Methods
TheEnBSassimilationincludestwosteps:predictionandupdating.Inthe
prediction,themodelpredictsthepriorSWEandthesnowmicrowaveemission
basedonthemeteorologicalandenvironmentalforcings.Thesepriorestimateswill
generallybeerroneousand/orhighlyuncertainduetothepropagationofthe
forcingdatauncertainty(especiallyprecipitation)inthesimulation.Intheupdating
step,theEnBSsystemconditionsallthepriorSWEestimateswithallthePM
observationsinasinglestep,yieldingaposteriorSWEestimatethatisexpectedto
besuperiortothepriorestimateintermsofaccuracyandprecision.
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4.2.1Prediction
Indataassimilation,forwardmodelingprovidespriorestimatesofthestate
variablesandthemeasurements.Inthisstudy,aLandSurfaceModel(LSM)anda
RTMwerecoupledforforwardmodeling;thecoupledmodelwillbediscussedin
moredetailinthesectionsthatfollow.Generally,theLSMtakesinthetemporally
variantmeteorologicalforcingandthetemporallyinvariantenvironmentaldata(e.g.
terrainandvegetationdensity),andsimulatesatime-seriesofsnowstates.Asa
Kalmantypeassimilationframework,EnBShastworequirementsfortheoptimality
oftheforwardmodeling:(1)theforcingdataareGaussiandistributed,and(2)the
forwardmodelsarelinear.However,neitheroftheserequirementsismetin
assimilatingPMforSWEestimate.AvarietyofmodificationstotheKalmantype
assimilationaddresstheseissues,e.g.theextendedKalmanfilter[Sunetal.2004]
andtheparticlebatchsmoother[Margulisetal.2015].Inthisstudy,weadoptedthe
ensemblemethodtoovercomethenon-Gaussianandnon-linearissues.Inthe
ensemblemethod,theprobabilitydensityfunctions(PDF)ofaseriesofforcingdata
andmodelparameterswereperturbedinordertoaccountfortheiruncertainties.
Thedataassimilationframeworksamplesanensembleoftheseforcingdataand
modelparametersovertheiruncertaintyrangewiththeMonteCarlotechnique
[MacKay,1998],andusestheensembletoimplicitlyapproximatethefullprobability
densityofthevariables.
97
IntheEnBS,theforwardsnowmodelcanbewrittenas:
y p, t , j = A ⎡⎣ y p, t−1, j , µ p, t , j , τ p, j , α j ⎤⎦ , y p, t=0, j = y p, 0, j (12)
where A [⋅] isthespace-timeLSMoperator,whichsimulatesthepriorsnow
estimate y p, t , j ;amongthesubscripts,pisthepixelindexinspatialdomain,t
representsthetthtime-step,andjrepresentsthejthensemblemember(referredto
asthejthreplicatehereafter); y p, t , j isthepriorestimateofthesnowstatesforthejth
replicateofthepthpixelattime-stept, y p, 0, j istheinitialcondition. µ p, t , j isthe
meteorologicalforcing, τ p, j isthetime-invariantenvironmentalforcing,and α j isa
setofspatiotemporal-invariantmodelparameterizationusedformeasuringforcing
uncertainty.
AnRTMmustbecoupledwiththeLSMtoobtainapriorradianceestimateofthe
snowpack,sothatthepriorSWEestimatecouldbecorrelatedwiththePMradiance
observationsfortheSWEupdating.Intheradiancemodeling,themodeledprior
snowpropertiesserveastheinputtotheRTM,andtheRTMsimulatestheupwelling
transmissionoftheearthmicrowaveemissionatthesoil-snowinterface,through
thesnowpack,andwithinthevegetationandtheatmosphere.TheRTMultimately
yieldsapriorestimateofthemagnitudeofthesatelliteobservedradiance.The
radiativetransfermodelingcanbewrittenas
z p, t , j = M ⎡⎣ y p, t , j ⎤⎦ 98
(13)
where M [⋅] istheRTMoperator, z p, t , j isthepriorestimateoftheobservedradiance
forthejthreplicateofthepthpixelattime-stept.
TheforwardmodelingshowninEquation(12)and(13)isconductedforeach
replicatewithperturbedforcingandparameterstogenerateanensembleofprior
SWEandradianceestimates.ThePDFofthepriorSWEandradianceestimatesis
characterizedbythemeanandthevarianceacrosstheentireensemble.
4.2.2Updating
Inthisstudy,theEnBSupdatingincorporatesthePMobservationwiththeprior
estimatesandgeneratesamoreaccurateposteriorSWEestimate.TheEnBSupdates
allpriorestimates(acrosseverypixel,andeveryobservationtime)withall
measurements(allfootprints,andeveryobservationtime)inasinglestep,differing
withtheEnsembleKalmanFilter(EnKF)andtheEnsembleKalmanSmoother
(EnKS)whichsequentiallyupdatethestatevariablewithobservationsfromasingle
observationtime,oroverawindowofobservationtimes,respectively.Intermsof
theupdatingimplementation,EnKFandEnKSpausetheforwardmodelingfor
updatingwhenobservationsbecomeavailable,andcontinuetheforwardmodeling
withtheupdatedstatevariableasthenewinitialcondition;thisprocessrequires
theassimilationframeworktobeembeddedintotheforwardmodels,and
oftentimesleadstoanabruptchangeinthevalueofthestatevariableafterthe
updating.Incomparison,theEnBSupdatingisindependentofthemodeling;the
99
updatingisconductedaftertheforwardmodelinggeneratesthepriorestimates
overthefullretrospectiveassimilationwindow,sotheassimilationframeworkdoes
notneedtobeembeddedintothemodel.Inaddition,wehypothesizethat
performingtheupdatingonthewholetime-seriesofpriorestimateswouldleadto
smootherandmoreoptimalresults,especiallyascomparedwiththeEnKF.Despite
thesestructuraldifferences,theEnBSissimilarwithEnKFandEnKSinthetheory
behindtheupdating:
( )
Y j+ = Y j− + K ⎡ Z obs
+ ω j − S ⎡⎣ M Y j− ⎤⎦ ⎤ ⎣ j
⎦
(14)
wherethe‘+’and‘−’superscriptsindicatetheposteriorandpriorestimates,
respectively. Y j+ denotesthevectorthatconsistsofthetime-seriesposteriorstate
variablesofthejthreplicateatallpixels,and Y j− isthevectorofthejthreplicateof
th
priorSWEestimateensemble.Similarly,theobservationvector Z obs
j isthej replicateofallthePMobservations.InEquation(14), ω j istheerrorvector
( )
−
associatedwiththejthreplicateofobservations Z obs
isthepriorradiance
j ,and M Y j
measurementsforecastedbytheforwardmodeling, S [⋅] isascalingoperatorthat
aggregatesthepixel-scalepriorradianceestimatetothemeasurementscalefor
updating,andthescalingwillbediscussedinmoredetailsintheSection4.4. K isthe
Kalmangain,and:
100
−1
K = CYZ ⋅ (Cv + CZZ ) (15)
Inparticular,
CYZ =
T
1 #
Y −Y ) ⋅ ( Z − Z ) &( (
%
'
N −1 $
(16)
CZZ =
T
1 #
Z − Z ) ⋅ ( Z − Z ) &( (
%
$
'
N −1
(17)
Cv = (σ 2 I m ) × C1 (18)
where CYZ isthecross-covarianceofthepriorSWEandpriorradianceestimates, CZZ
isthecovarianceofthepriorradianceestimates. Y and Z arethematricescontain
theentireensembleofthepriorsnowestimateandtheradianceestimate,
respectively,and Y and Z arethemeanof Y and Z ,respectively. Cv istheerror
covariancematrixofthePMmeasurementswhichhasanuncertaintyof!.C1
representsthefifthorderpolynomialin[GaspariandCohn,1999,Equation4.10]
andisutilizedtomodeltheeffectsoftheatmosphericmoistureonthe
measurementsunderthe‘nonclear-sky’conditions.TheKalmangaincorrelatesthe
priorSWEestimateswiththepredictedradiancemeasurements.Itweighsthe
relativeuncertaintyofthepriorestimatewithrespecttothatofthemeasurements
todeterminetheamountofcorrectiontobeaddedtothepriorestimatesothatan
optimalposteriorestimatecanbeobtained.
101
4.3Models
4.3.1Forcingdatadisaggregationmodel
TheforcingdatadisaggregationmodelintroducedinGirottoetal.,[2014a]was
adoptedinthisstudytodownscalethecoarse-resolutionmeteorologicalforcing
data.TheforcingdatadownscalingiskeytocapturethedetailsofthespatialSWE
andradiancevariabilityinthemountainterrain.Thedisaggregationmodelis
capableofdownscalingarangeofcoarsemeteorologicalforcing(e.g.precipitation,
airpressure,specifichumidity,airtemperature,airpressure,humidity,andlongwaveradiation)toanyfinerresolution.Additionally,theFDMseparatesthe
incomingsolarradiationintodirectdownwellingfluxfromtheatmosphereandthe
diffusivefluxfromthesurroundingterrain,andconductsqualitycheckand
topographiccorrectiontobothpartstopreventthesolarradiationfromcontaining
largebias.TheFDMisdescribedindetailintheAppendixof[Girottoetal.2014a].
4.3.2Landsurfacemodel
TheLSMsimulatesthesurfacemassandenergybalance,includingsnow
dynamicsusingthedownscaledmeteorologicalforcing.Inthisstudy,weusedthe
SimplifiedSimpleBiosphereversion3.0[SSiB3,Xueetal.,1991]LSMtoforecastthe
priorsnowproperties.SSiB3consistsoftheoriginallandmodelSSiBandthesnow
schemeSimpleSnow–Atmosphere–SoilTransfer[SAST,Sunetal.,1999].SSiB3
simulatesthelandsurfacebiospherewiththreesoillayers,twovegetationlayers,
102
andthreesnowlayers.SSiB3numericallysolvesthemassandenergybalance
equationsassociatedwiththephysicalsnowprocessessuchasthefreshsnow
accumulation,compaction,sublimation,microstructuremetamorphism,andsnow
melting;solvingtheseequationsyieldstheestimateofthesnowstatesincluding
snowdepth,density,temperature,andvolumetricwatercontent,and
microstructure.
Snowgrainsizesignificantlyimpactsthesnowpackmicrowaveradiativetransfer
properties[MätzlerandWiesmann1999].Inprinciple,bothgrainsizeandSWE
couldbeestimatedsimultaneously,asdoneinpoint-scaleEnBSexperiments
[Durandetal.2009,Batenietal.2013and2015].However,thiswouldfurther
increasecomputationalcosts,andformanylarge-scaleapplicationsestimatesof
grainsizeareoflessvaluethanSWEestimates.Duringtheaccumulationseason,
graingrowthandsnowfallaffectTbindifferentways;therateofgraingrowth
impactstherateatwhichTbdecreasesinbetweensnowfallevents,whilesnowfall
tendstocreateasystematicchangeinTb[Lietal.,2015aandb].Apragmatic
solutionistousejustasingleparametertogoverngraingrowthrate(foreverypixel
andeveryyear),toestimatevaluesofthegraingrowthrateparameterthatcause
themodeledTbtomatchtherateofchangeofobservedTbtimeseriesinbetween
snowfallevents,andthentreatthisgraingrowthrateasanuncertaininputtothe
assimilationscheme.Inthisstudy,theJordandynamicgrainsizemetamorphism
103
model[Jordan1991]thatincludesacalibratablegrainsizegrowthrateparameter
wasintegratedwithSSiB3tosimulatethesnowmicrostructure.TheJordanmodel
hasbeensuccessfullyappliedinseveralstudiestoestimatesnowgrainsize[e.g.
Huangetal.,2012,Lietal.2015a],andhasoutperformedmorecomplexphysical
grainsizemodelinsomecases[Huangetal.,2012].ThedetailsoftheJordanmodel
canbefoundinsectionIVof[Jordan1991].Thegrainsizegrowthcoefficient
utilizedinthisstudywasthevaluedeterminedinLietal.[2015a]viacalibration;
thegraingrowthcoefficientcanbeestimatedessentiallybycomparingmodeledand
observedTbtimeseries,evenforthepriormodelwhichhasanincorrectSWE.This
approachisdiscussedinsection4.5.4,andthedetailedcalibrationmethodcanbe
foundinLietal.[2015a].Inthisstudy,thesamevaluewasusedforeachwateryear
andeachpixelinthestudyarea.WealsousedthemethoddescribedbyLietal.
[2015a]toallowthemodeltoaccountforerrorsduetolayercombinationschemes
duringlargesnowfallevents.
4.3.3Radiativetransfermodel
TheradiativetransfermodelinginthisstudywascarriedoutwithseveralRTM
sub-modules.TheMicrowaveEmissionModelofLayeredSnowpack(MEMLS)
[WiesmannandMätzler,1999]wasusedasthemainRTMtopredictthepriorsnow
radiance.Inaddition,modulesthatbettercharacterizetheradiativetransfer
propertiesofsoil,vegetation,andatmospherewereintegratedintoMEMLSforthe
104
bestpossiblemodelperformance.Specifically,theroughbaresoilmodel
[WegmüllerandMätzler,1999]simulatesthemicrowaveemissionfromsoilandthe
microwaveattenuationatthesoil-snowpackinterface.Theroughsoilmodel
quantifiesthereflectionoftheupwellingmicrowaveradianceatthesoil–snow
interfacebyintroducingthesoilroughnesslength,whichisafunctionofmultiple
soilpropertiesandtheradiancefrequency.Thevegetationemissionandattenuation
oftheupwellingmicrowaveradiancewascalculatedwiththemethodintroducedin
[Wegmülleretal.1995],andthemodelingoftheatmosphericradiativetransferis
basedontheworkin[Ulabyetal.,1982],whichcomputestheabsorptionof
microwaveradiationbyatmospheregasesforclearskyconditions.
MEMLSwascoupledwithSSiB3andtookthesnowlayerthickness,temperature,
grainsize,density,andliquidwatercontentfromSSiB3asinputstosimulatethe
transmissionofthemicrowavewithinthesoil,snow,vegetation,atmosphere,and
theinterfaceoftheselayers.MEMLSultimatelypredictedtheobservedmicrowave
radianceabovethesnowpacksurface.MoredetailsoftheMEMLSmodelcanbe
foundin[WiesmannandMätzler,1999].Inthesnowradiativetransfermodeling,
weappliedtheimprovedBornapproximation[MätzlerandWiesmann,1999]to
characterizetheinternalvolumetricscatteringofmicrowaveradiationbythesnow,
andweusedaconstant0.16asrecommendedin[Wiesmannetal.2000]and
[Mätzler,2002]toconvertthesnowgraindiameteroutputbytheLSMtothe
105
autocorrelationlength,whichcharacterizesthetwo-pointautocorrelationfunction
oftheice–airmatrix[WiesmannandMätzler,1999]andisthesnowgrainsize
metricusedbyMEMLS.Thisempiricalrelationshiphasbeenusedinotherresearch,
e.g.Huangetal.[2012]andhasachievedsufficientlyaccurateradianceestimatesin
ourpreviousstudies[Lietal.2015a].Weutilizedthesoilroughnesscoefficient
determinedinLietal.[2015a].Thiswasessentiallycalibratedtoforcethemodeled
TbtimeseriestomatchtheTbobservationsinsnow-freecases.
4.4Studyarea,data,andexperimentdesign
TheEnBSassimilationwascarriedoutintheupperpartoftheKernRiverBasin,
SierraNevada,USA(referredtoastheUpperKernbasinhereafter),fromWater
Year(WY)2003to2008.Thissix-yearstudyperiodincludeshigh-,low-andaverage
snowyears,whichenablesassimilationperformanceevaluationunderdifferent
snowaccumulationconditions.
4.4.1Studyarea
Figure29showsthedigitalelevationmodeloftheKernBasininthesouthernSierra
Nevada.TheUpperKernischaracterizedbyhigh-reliefterrainanddeepsnowcoverthat
persistsforalongperiodeachseason(morethan250daysonthepeaks)duetothehigh
elevationandlowtemperature;the511km2UpperKernhasanaverageelevationof3600
mandanaveragevegetationdensitylowerthan5%.ThevegetationintheUpperKernis
mainlyconiferousforest;shortvegetation(e.g.shrubsandscrubs)makesupaverylimited
106
portionofthevegetation,andisoftenburiedbysnowinwinter.Thelonglastingdeepsnow
accumulationandthelowforestdensityintheUpperKernareattractiveforthepurposeof
thisstudy.
Fig.29Themapofthestudyarea,includingtheDEMoftheKernbasinandthelocationsof
thesnowpillowsandsnowcourses.
4.4.2Data
TheforwardmodelinginthisassimilationstudywasforcedbythehourlyNorth
Americanlanddataassimilationsystemphase2(NLDAS-2)meteorologicaldata
107
fromWY2003toWY2008.TheNLDAS-2datafieldsthatwereusedinthisstudy
includeprecipitation,airtemperature,specifichumidity,airpressure,longwaveand
shortwaveradiation,andwindspeed.TherawNLDAS-2datawereataspatial
resolutionof1/8°(∼15km).Severalexistingstudies[e.g.Clarketal.2011,Ramage
andSemmens2012]havefoundthelengthscaleof~100macriticalvalueforsnow
de-correlation;conductingmodelingataresolutionof~100morhigheriskeyto
characterizethedetailsofthesnowvariability.Therefore,themodelingandthe
assimilationinthisstudywereconductedat90m-resolution;therawNLDAS2
forcingandstaticforcingdataweredownscaledto90m-resolutionbyFDM.Inthis
study,wecontinuedtousethesebias-removeddownscaledmeteorologicalforcing
thathasbeenusedinourpreviousstudies[Lietal.,2015aand2015b],exceptfor
theprecipitation.SincethegoalofthisstudyistoexaminetheabilityofthespacebornePMobservationinreducingtheerrorsinthemountainSWEestimation,
therefore,weusedtherawNLDAS2precipitationdatathatarenotbiascorrected.
Themicrowaveradiancemeasurementsinthisstudywerecollectedfromthe
AdvancedMicrowaveScanningRadiometer-EarthObservingSystem(AMSR-E)
aboardtheAquasatellite[Kawanishietal.,2003].FortheSierraNevada,AMSR-E
haddailyascendinganddescendingrevisitsat13:30and1:30localtime,
respectively.Inthisstudy,weusedonlythe36.5GHzverticalpolarization(V-Pol)Tb
observationcollectedduringthenightoverpass,because36.5GHztendstobemost
108
sensitivetoSWEamongallAMSR-Efrequencies,andV-Polislesssensitivetoice
layersthanhorizontalpolarization[Reesetal.,2010].Sinceasmallamountofliquid
waterinsnowpackwouldleadthesnowtoemitmicrowavealmostasablackbody,
andwouldextinguishtheSWEinformationintheobservedradiance,thereforeonly
nightobservationswereusedtoavoidthepotentialmeltingasaresultofthe
relativehightemperatureandpossiblewetsnowintheafternoon.
Weutilizedraw(Level-2A)AMSR-Eobservationsandprocessedtheraw
observationstothreedifferentscalesforthreeassimilationexperiments.Following
thealgorithmsreportedinLietal.[2012],therawAMSR-Eradianceobservations
wereprocessedtopoint-scaleandtheUpperKernbasin-scaleusingtheGIDW
algorithmandtheFBAWalgorithm.Inaddition,wealsoassimilatedtherawTb
observationswiththeirnativefootprintsizeat87.9km2(36.5GHzelliptical
footprintsize:14km×8km).Hereaftertheprocessedradianceobservationatthe
point-,native-andbasin-scalearecalledGIDWTb,footprintTb,andFBAWTb,
respectively,andtheycorrespondtothevariableZobsinEquation(14).
SWEmeasuredatthethreesnowpillowsandfivesnowcourseswithinthe
UpperKernwereusedtovalidatetheassimilationresults.Thedetailedinformation
ofthesein-situgagesissummarizedinTable9.SWEmeasurementsfromWY2003
toWY2008werecollectedfromthesegagesforthisstudy.SWEismeasureddailyat
109
thethreesnowpillows.Atthesnowcourses,monthlySWEismeasuredaroundthe
firstdayofeachwintermonth,typicallyfromJanuarytoApril.
Table9ThelocationandtheenvironmentalconditionsofthesnowpillowCBT,CHP,
UTY,andthesnowcourseBGH,TND,SDM,GYF,RCR.
Vegetationandtopographydatasetswereaveragedtoeach90m-resolution
pixeltomodelthedetailsoftheruggedmountainterrainandthevariabilityofthe
groundfeatures,andultimatelytocharacterizetheimpactofthegroundfeatures
andthelandscapeonthesnowevolutionandmicrowavetransmission.Weusedthe
vegetationdatafromtheNationalLandCoverDatabase2001[Homeretal.,2004],
whichincludesthevegetationtypeandtheforestfractionalcoverage.The
topographydatawereobtainedfromtheAdvancedSpaceborneThermalEmission
andReflectionRadiometerDigitalElevationModel(ASTERDEM)product(a
productofNASAandMETIJapan).ThenativeresolutionofASTERDEMwas30m,so
thedatawereresampledtothe90mmodelingresolution.Spatialanalysiswas
110
conductedinpreviousstudies[Girottoetal.,2014a]tocalculatetheaspect,slope,
topographicshadingandthesky-view-factor[MarksandDozier,1979]foreach
pixel.
4.4.3Experimentconfiguration
Aforwardmodelingwithoutupdating(open-loop)wasfirstconductedateach
90m-resolutionpixelfortheperiodofWY2003toWY2008.Then,theopen-loop
simulationresultsofeachwateryearwereupdatedinabatchmodebytherawPM
observationsofthesamewateryear.
4.4.3.1Open-loopsimulation
Theopen-loopsimulationgeneratedanensembleofhourlypriorSWEandTb
estimatesateachpixelduringWY2003toWY2008.Specifically,theLSMwasdriven
bythehourlyforcingdataandsimulatedthesnowpropertyestimatesforthesixyearstudyperiod,andthesimulatedsnowpropertieswerepassedtotheRTMfor
microwaveradiancesimulation.SinceweonlyusedthePMmeasurementsfromthe
nighttimeoverpassofthesatelliteat~2ameachday,therefore,weonlyconducted
therelativelycomputational-demandingRTMsimulationat2ameachdayfor
radianceobservationestimates.Inthisstudy,weusedanensembleofthirty
replicates,followingtheguidelinereportedinDurandandMargulis[2006],which
showedanensemblesizeof25to50couldbestbalancethemodelingaccuracyand
thecomputingefficiencyinradianceassimilationsnowestimate.
111
Weadoptedtheuncertaintyquantificationthatwasspecificallydevelopedfor
theKernbasininGirottoetal.,[2014aand2014b]inthisstudy.Asaforementioned,
theonlydifferenceintheuncertaintyquantificationbetweenthisstudyandthatin
[Girottoetal.,2014a]isthat,thenominalvalueofthescalarforprecipitationforcing
wassetto2.5tocorrectthelow-biasedNLDAS2precipitationin[Girottoetal.,
2014a],butthisscalarwassetto1.0inthisstudy;usingtherawNLDAS
precipitation,thepriorSWEcouldbebiasedandwecouldtesttheabilityofthe
EnBStoimprovetheSWEestimate.Inaddition,anormallydistributedrandom
fluctuationwithameanofzeroandastandarddeviationof20%ofthedefaultvalue
wasaddedtothedrysnowgraingrowthrateandthesoilroughnesslength,inorder
toaccountfortheuncertaintiesincharacterizingthesnowgrainsizeandthe
microwavescatteringatthesoil-snowinterface,respectively.Theperturbation
methodandthequantificationoftheuncertaintymodelvariablesaresummarizedin
Table10andareexplainedinmoredetailin[Girottoetal.,2014a].
112
Table10Theparametersthatwereperturbedwithuncertaintiesandtheiruncertainty
quantifications.
113
Wegeneratedtheensemblefortheforwardmodelingandassimilationby
Monte-CarlosamplingfromthePDFoftheperturbedvariablesandparameters.For
eachsetofperturbedinputs,open-loopsimulationswereconductedateachpixelto
generateanensembleofsnowstatesandmicrowaveradianceestimatesasprior
estimates.Theprobabilitydensityofthesepriorestimateswerecharacterizedby
themeanandstandarddeviationofthemodelingresultsacrossallthereplicates.
4.4.3.2Updating
Inthisstudy,theEnBSupdatingwascarriedoutonawater-yearbasis.SWEis
thestatevariableintheassimilationandcorrespondstothevariableYinEquaiton
(14).ForeachwateryearfromWY2003toWY2008,weusedthedailynighttimeTb
observationsfromthedrysnowseason(December1sttoFebruary28th)toupdate
theSWEat2ameachdayinthesnowaccumulationseason(October1sttoApril1st).
Weonlyusedthemeasurementsinthedrysnowseasonforupdatingbecausesnow
onlydominatesthePMsignalwhensnowisdryanddeep;thestrongmicrowave
emissionfromliquidwatercontentmasksthesnowmicrowaveradiancewhen
snowiswet,whichoftenoccursbeforeDecemberandafterMarchduetothe
meltingwithintheUpperKern[Lietal.2015b].Inaddition,thestrongmicrowave
emissionfromliquidwatercangenerateunpredictableerrors,whichmay
significantlydegradetheassimilationsystem.Inthisstudy,wesettheobservation
uncertaintyas5Kelvin,whichconsistsoftheAMSR-Einstrumenterrorandthe
114
errorsfromatmosphere.Theuncertaintyforthe36.5GHzAMSR-Eobservationsis
around2Kelvin,whichismainlyattributedtotheinstrumentnoiseandgeo-location
error,withouttakingtheatmosphericinfluenceintoaccount[Kawanishietal.,
2003].TedescoandWang[2006]foundthatatmospherecouldcontributeupto6
KelvintoAMSR-E36.5GHzobservation,withthemeancontributionat~3Kelvin.
The5KerrorthusrepresentstheuncertaintyoftheatmosphericRTMdescribed
above,aswellastheuncertaintyoftheTbobservationsthemselves.
Inthisstudy,weperformthreeassimilationexperimentsinwhichthepriorSWE
estimatesareupdatedbythreedifferentscalesofPMdata.Thepoint-scaleGIDWTb
assimilationisconductedattheeightpixelsinwhichthethreesnowpillowsandthe
fivesnowcoursesarelocated,whiletheFBAWTbandfootprintTbupdatingare
conductedovertheentireUpperKern.AsshowninEquation(14),theprior
radianceestimatesateachpixelwereprocessedwithascalingoperator(! ∙ )to
makethescaleofthepredictedmeasurementstobeconsistentwiththescalesofthe
observations.Specifically,thepriorradianceestimatesinthepoint-scaleGIDW
assimilationexperimentcorrespondtotheradiancesimulatedatthepixelsinwhich
thesnowpillowsandcoursesarelocated.Inthebasin-scaleassimilation,theprior
radianceestimatewascalculatedfromthemeanoftheTbacrossallthepixelsin
UpperKern.Inthefootprint-scaleassimilation,thepriorTbatthe90mmodeling
pixelswereaggregatedtothefootprintscalebyapplyingaweightedaverage
115
functiontotheTbateachpixelbasedonthespatialantennasamplingpattern[Liet
al.,2012].Specifically,asshowninFigure30,theAMSR-Eantennaenergy
distributionat36.5GHzcanbecalculatedwitha2-dimensionalGaussianmodel
basedonthefootprintsizeandthefullwidthathalfmaximumpropertyofthe
footprint[Kawanishietal.,2003,Lietal.,2012].Withtheantennaenergy
distributionandtherelativepositionbetweenthefootprintandthemodelingpixels
(representedbythegrid-cellinFigure30),eachpixelinthestudyareacanbe
assignedaweightwhichevaluatesthecontributionoftheradianceatthispixelto
theobservation;generally,thefurtherapixelisfromthefootprint,theless
contributionithastotheobservedradiance,andthelowerweightitbearsintheTb
aggregationforthisfootprint.Suchaggregationisdonetoeachfootprintinthebasin
withthepriorradiancesimulatedatallpixels.InFPassimilation,thisweighted
averageprocessalsohelpssuppressingthepotentialspuriouscorrelationinCYZ
betweenpixelsandfootprintsthatarefaraway.Noothercorrelationlocalization
wasspecificallydone;becauseofthespatiotemporal-continuousnatureofthe
seasonalsnowaccumulationprocess,thepriorSWEandradianceestimatesare
correlatedduringtheaccumulationseason.Also,duetotherelativelysmallsizeof
theUpperKern,weassumethespatialcorrelationsbetweenthefootprint
observationsinthebasinarenon-spurious.Theconfigurationofthethree
assimilationexperimentsissummarizedinTable11.
116
117
Table11Theconfigurationofthethreeassimilationexperiments
Fig.30TheAMSR-E36.5GHzantennaenergydistributionmodel,whichillustratesthe
weightintheweighted-averageaggregationofthepixel-scaleradianceestimatetothe
footprint-scale.Notethegridcellsinthisfigureareonlyshowntoillustratetheaggregation
process;theyarenotatthepixelresolution(90m)inthisstudy
4.5Resultsanddiscussions
4.5.1Priorestimates
SimulatedensemblesofpriorSWEandradianceestimatesatthethreemodel
pixelscorrespondingtothesnowpillowsforWY03areshowninFigure31.The
ensemblemeanSWEissystematicallylowerthanthemeasurements;thelowbiasis
generatedduringthesnowfallevents.ThislowbiasinthemodeledSWEislargely
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duetotheprecipitationundercatch[Lougeay1976],whichhasbeendemonstrated
tobeespeciallysignificantforprecipitationassnowfallinmanymountainousareas
[AdamandLettenmaier,2003],includingtheSierraNevada[Panetal.,2003].Atall
threesnowpillows,themodeledradiancedecreaseswiththeincreaseofSWE,
indicatingthevolumetricscatteringofthesnowpacktothemicrowaveradiance
intensifiesastheSWEincreases.SinceSWEisthemajorcontroloftheradiance
emissioninlow-vegetatedareas[Lietal.,2015b],therefore,thelow-biasedSWE
accumulationattenuateslessmicrowaveradiancethantherealcase,leadingtoa
positivelybiasedmeanradianceestimate,asshowninFigure31.TheWY03
temperatureandprecipitationdatashowtherewereseveralrainfalleventswithin
theaccumulationseasonatgageCHP,butsuchrainfalleventsdidnotoccuratthe
othertwopillows.Rain-on-snoweventsbringenergytosnowpackandpromote
partialmeltofthesnowpack.AsshowninFigure31,therainfallleadstodecreases
inthemeanSWEatCHPduringtheaccumulationseason,anditmeltsouta
significantportionoftheSWEreplicatesatthelowendoftheensemble.Inradiance
modeling,thesereplicatesshowhighTbvaluesandlittleoverallchangeinthe
accumulationseason,indicatingthemodeledradianceofthesereplicatesare
dominatedbyliquidwaterandbaresoil.
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Fig.31ComparingtheWY03dryseasonpriorSWE(red,toprow)andpriorTb(red,bottom
row)estimateswiththemeasurements(blue)atthethreesnowpillowsintheUpperKern.
ToevaluatetheaccuracyofthepriorSWEestimates,wecalculatedtheRMSEof
theensemblemeanSWEatthesnowpillowsandsnowcourseswherein-situ
measurementsareavailable.WealsocalculatedtheerrorofthepriorApril1stSWE
estimatestoevaluatetheaccuracyofthepeakSWEestimate,asApril1stSWEhas
beenusedhistoricallytoinferthetotalstreamflowintheSierraNevada.Table12
showstheerrorstatisticsofthepriorSWEestimates.InTable12,theaverage
accumulationseasonRMSEofthepriorSWEis0.13m,andtheaverageApril1stSWE
erroris−0.17m.InTable12,WY05hasthelargestaccumulationseasonRMSEand
theApril1stSWEerrorduringthesixwateryears,whileWY07hastheleastofboth.
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ThiscoincideswiththefactthatWY05hasthelargestsnowaccumulationinthesix
years,whileWY07hastheleast.Asdiscussed,theerrorsoftheSWEestimateare
mainlyattributedtothesnowfallundercatchduringthesnowfallevents;larger
snowfalleventsleadtogreaterundercatch,whichfurthertranslatestolargerSWE
errors.
121
122
Table12TheRMSEofthedailypriorSWEestimatesoftheaccumulationseason,theerrorofthepriorApril1stSWE,and
themeasuredApril1stSWEofeachwateryear.
Theopen-loopmodelingwasconductedateverypixelintheUpperKernforprior
estimates.Figure32showsthemapsofthepriorSWE,Tb,andsnowgrainsizeestimates
overtheUpperKernonMarch1st,2003.IntheSWEmap,thewestsideoftheUpperKern
accumulatesmuchmoresnowthantheeastside,whichistheeffectoftheprecipitation
shadowcausedbytheatmosphericflowandtheorographicconditions[Girottoetal.,
2014a].AsshowninFigure32,thehigh-resolutionmodelingprovideddetailsoftheprior
radiancedistributionovertheUpperKern,andthepatternoftheradiancedistributionwas
wellcorrelatedwiththesnowgrainsizeandSWEdistributions.Moredetailsaboutthe
connectionsamongtheSWE,microwaveradiance,snowgrainsize,andtheenvironmental
controlsintheUpperKerncanbefoundin[Lietal.,2015b].Figure33showstheobserved
radianceateachAMSR-E36.5GHzfootprint,andthepriorestimateoftheradianceateach
footprintonMarch1st2003;boththeobservedandthemodeledradiancemaintainaminor
east-westTbgradient;thelowerTbinthewestsideoftheUpperKernismainlyduetothe
deepersnowcoverandlargersnowgrainsize,whichisaresultofalargertemperature
gradientwithinthesnowpackandalongersnowmetamorphismperiodinthewestsideof
thebasin[Lietal.2015b].Figure32alsoshowstheobservedradianceisgenerallylower
thanthemodeledpriorradiance,whichisconsistentwiththesnowpillowresultsshownin
Figure31.
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Fig.32(left)themodeledSWE,(middle)Tband(right)grainsizeovertheUpperKernon
March1st2003.
Fig.33(left)theobservedand(right)theaggregatedpriorTbineachobservational
footprintovertheUpperKernonMarch1st2003.Bothcolorbarsspan8Kelvinandthe
colorbarfortheaggregatedTbissystematicallyhigherthanthatoftheobservedTbby4K
toshowthesimilarspatialtrendsofthetwo.
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4.5.2Assimilationresults
TheassimilationofthesnowinformationfromPMobservationsreducedthe
errorsinthepriorSWEestimatesandledtoamoreaccurateposteriorSWE
estimate.Figure34comparesthepriorSWE,theFPposteriorSWEandthe
measuredSWEfromWY2003toWY2008attheeightin-situgages.InFigure34,the
priorSWEestimatesarealllow-biased,withtheaveragebiasratioatabout−20%.
Aftertheassimilation,theposteriorSWEareclosertothemeasuredSWE,andin
manycasestheuncertaintyoftheposteriorestimatesalsoreduces.Comparedwith
otheryears,WY06’sposteriorestimatesshowinsignificantimprovement;the
posteriorSWEestimatesinseveralsnowcoursesandpillowsaregenerallythesame
asthepriorestimates.ThisisduetotherainfalleventsfromlateDecemberto
JanuaryofWY06[Lundquistetal.2009,2010]thatbroughtliquidwaterintopartof
thesnowpack,whichhasbeenfoundtodecreasethesnowinformationinthe
radianceobservationsofWY06[Lietal.,2012,Caietal.,2015].InFigure34,the
posteriorSWEcurvespreservetheshapeofthepriorSWEwithoutintroducing
abruptchangesintheposteriorestimates,whichisamajoradvantageofEnBSin
snowcharacterization,comparedwithfilteringassimilationframeworks.As
discussed,filteringassimilationpausestheforwardmodeling,updatingtheprior
estimateswithpartofalltheobservations,andusetheposteriorestimatesasthe
newinitialconditionfortheforwardmodelingandupdatingthatcomeafterwards.
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ThisoftenleadstotheartificialdiscontinuityintheSWEestimateatthetimesof
update.WhilethisdiscontinuityinSWEinstantlyreducestheerrorintheestimate,
itisundesirableinseasonalsnowcharacterizationbecausethechangesinSWE
time-seriescontaintheinformationofsnowaccumulationandablation;theabrupt
SWEchangesfromtheupdatingintroduceillusionsofsnowaccumulationor
ablation.Incontrast,theEnBSupdatesallthepriorestimatesinonesinglestep,and
resultsinatemporallysmoothposteriorestimatethatpreservesthesnowfalland
snowablationinformation.TheposteriorestimatesfromtheGIDWassimilationand
footprintassimilationaresimilarwiththosefromtheFBAWassimilation;thetimeseriescomparisonoftheposteriorestimatesfromthesetwoassimilation
experimentswiththegagemeasurementsarenotshowntoavoidredundancy.The
scatterplotsinFigure35illustratetheseimprovementsbycomparingthepriorSWE
estimatesandtheposteriorSWEestimateswiththemeasuredSWE.Specifically,for
eachsnowaccumulationseasoninthesixwateryears,wecomparedthedailyprior
andtheposteriorestimatesfromthethreedifferentscalesofassimilation
experimentswiththemeasuredSWEatthethreesnowpillowsandthefivesnow
courses.AsshowninFigure35,atboththesnowpillowsandthesnowcourses,
priorSWEestimatesarelowerthanthemeasurements,indicatingthelowbiasin
thepriorestimates.Themagnitudesofthebiasinthepriorestimatesvaryfrom
gagesandyears.Theposteriorestimatesagreebetterwiththemeasurements,but
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sincetheamountofthepriorestimatebiasanduncertaintydirectlyaffectthe
posteriorestimates,thedegreeofconsistencybetweentheposteriorestimatesand
themeasurementsalsodiffersacrossgagesandyears.Inadditiontothe
accumulationseasoncomparison,wealsocomparedthepriorandposteriorApril
1stSWEwiththemeasuredApril1stSWEtoevaluatetheperformanceofthe
assimilationinimprovingApril1stSWEestimate.Similarwiththeaccumulation
seasonposteriorSWE,theApril1stposteriorSWEestimatesalsoshowsignificant
improvementinaccuracy.Figure36showsthecomparisonbetweeneachwater
year’sposteriorApril1stSWEestimatesfromthethreeassimilationexperiments
andthepriorSWEestimateswiththemeasuredSWEatthesnowpillowsandsnow
courses.TheposteriorApril1stSWEmaintainsahigherconsistencywiththe
measurements,comparedwiththelow-biasedpriorApril1stSWEestimates.
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Fig.34ComparisonbetweenthepriorSWEestimate,FPposteriorestimate,andmeasured
SWEatthesnowpillows(column1-3)andsnowcourses(column4-8).Ineachfigurethe
lightredcurveisthepriorestimate,theblackcurveordotrepresentthemeasurement,and
theblueistheposteriorestimate.Theshadedareasaretherangesofthe80%uncertainty
oftheprior(lightred)andposterior(lightblue)estimates.
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Fig.35Comparingthedailyprior(red)andposterior(green)SWEestimatesfromOct1stto
Apr1stwithmeasuredSWEatpillowsCBT,CHP,UTY(column1tocolumn3)forWY03-08,
and(column4)comparingtheFeb1st,Mar1st,April1stpriorandposteriorSWEatthefive
snowcoursesforWY03-08.ThetoprowtobottomrowshowtheresultsfromFBAW,FP
andGIDWmethod,respectively.
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Fig.36ThecomparisonbetweenthepriorandposteriorWY03-08April1stSWEatthe
snowcourses(leftcolumn)andthesnowpillows(rightcolumn).Thetoprowtothebottom
rowshowtheposteriorestimatesfromFBAW,FPandGIDWmethod,respectively.
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Weconductedquantitativeanalysistoassesstheaccuracyimprovementfrom
theassimilationexperimentsofthespacebornePMobservations.Foreachofthe
threescalesofassimilation,wequantifiedtheposteriorSWERMSEandthe
posteriorApril1stSWEerrorateachgageforeachwateryear,whichissimilarwith
thecomparisonsforthepriorestimates.Table13andTable14summarizethe
accumulationseasonposteriorSWERMSEandtheposteriorApril1stSWEerrorat
eachgageandeachwateryearforallthethreeassimilationexperiments.Consistent
withFigure35andFigure36,thedatainTable13andTable14showthatinmost
cases,theaccumulationseasonSWERMSEdecreasesaftertheassimilation,
demonstratingtheSWEaccumulationestimatesarecharacterizedwithhigher
accuracy.TheposteriorApril1stSWEerroralsodecreasessignificantlyafterthe
assimilationexperiments.WecalculatedtheaverageposteriorRMSEandthe
averageposteriorApril1stSWEerror,asshowninTable13andTable14;bothof
themconsiderablyreducedincomparisonwiththeerrorstatisticsoftheprior
estimates(asinTable12).However,insomeyearsandatsomeofthegages,the
posteriorestimateRMSEandApril1stSWEareslightlylargerthantheprior
estimates;thecasesinwhichtheposteriorSWEhasalargerRMSEorApril1stSWE
errorareunderlinedinTable13andTable14.AsthedatainTable13andTable14
show,theincreasesintheposteriorSWERMSEandtheposteriorApril1stSWEbias
oftenoccuratthesamegageandthesameyear,indicatingtheassimilation
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experimentsarelessaccurateinthesecases.Toexplorethereasonsfortheaccuracy
degradationinthesecases,wefindthatthegagesandtheyearswithlarger
posteriorSWERMSEandApril1stSWEerrorinTable13and14donotnecessarily
havesignificantlyhigherpriorSWERMSEorbiasinTable12,buttheaccuracyof
theposteriorestimatesdovaryinterannually,andtheseinter-annualvariationsare
likelyduetointer-annualSWEvariability.WefindthatSWERMSEincreaseswith
trueSWE;e.g.,2005hadthehighestSWERMSEvalues(seeTable13),alongwith
thehighestvaluesofSWEaccumulation(seeTable12).Weexploredwhether
relativeerrorsmightbemoreinterannuallyconsistent.Weestimatedtherelative
accuracyoftheSWEestimatesbytheratiooftheSWEposteriorRMSEtothetothe
peakSWEaveragedacrosstheeightinsitustations.FortheFPmethod,relative
SWEerrorrangedfrom7.4%in2005to21.2%in2007,withanaverageof13.1%;
overall,relativeerrordecreaseswithincreasingSWEaccumulation.
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133
Table13TheRMSEoftheaccumulationseasonposteriorSWEestimatesfromthethreeassimilationsineach
wateryearateachgage.ThecaseswheretheposteriorestimateshavealargerRMSEareunderlined.
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Table14TheerroroftheposteriorApril1stSWEestimatesfromthethreeassimilationsineachwateryearateachgage.
ThecaseswheretheposteriorestimateshavealargerApril1stSWEerrorareunderlined.
4.5.3Comparingassimilationmethodologies
Althoughinsomecasestheposteriorestimatesshowincreasederror,in83%of
thetotalcasesshowninTable13andTable14,theassimilationexperimentsraised
theestimationaccuracy,andthemagnitudesoftheseimprovementsfaroutweigh
theaccuracydegradationcases,leadingtoasolidincreaseintheoverallaccuracyof
theposteriorestimatesforallthreeassimilationexperiments.Wequantifiedthe
overallaccuracyoftheposteriorestimatesbycalculatingtheSWERMSEandApril
1stSWEerrorusingtheposteriorestimatesfromalltheyearsandallthegagesfor
eachofthethreeassimilationexperiments.Figure37showstheaveragebiasand
theRMSEofalltheaccumulationseasonSWEestimates.InFigure37,theaverage
biasoftheaccumulationseasonSWEestimatereducesby84.0%,74.6%and89.3%
aftertheFBAW,GIDWandfootprintassimilation,respectively.Also,theRMSEofthe
accumulationseasonSWEestimatereducedby36.9%,24.3%and42.0%,
respectively.WealsoevaluatedtheaccuracyimprovementintheposteriorApril1st
SWEestimateusingalltheApril1stSWEestimatesfromallthegagesandallyears.
AsshowninFigure38,afterFBAW,GIDWandfootprintassimilation,theposterior
April1stSWEerrorreducedby76.4%,74.1%and87.6%,andtheApril1stSWE
RMSEreducedby42.4%,29.2%and60.4%,respectively.
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Fig.37Thesix-yearaccumulationseasonaveragebiasandRMSEofthepriorestimateand
thethreeposteriorestimatesatthethreepillowsandthefivecourses.
Fig.38Theaveragesix-yearApr1stSWEerrorandRMSEofthepriorestimatesand
posteriorestimatesatthesnowpillowsandsnowcourses.
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Figure39presentsthemapsofthepriorSWE,FBAWposteriorSWEand
footprintposteriorSWEovertheUpperKernonApril1stofeachofthesixwater
years.WealsocomparedthepriorandposteriorSWEfromthisstudywiththeSWE
fromarecentSierraNevadasnowpackreanalysis[Margulisetal.,2015],which
estimatedthedailySWEat90mresolutionforthepast31years(1985to2015)
basedontheassimilationofthesnowcoveredareainformationfromLandsat
images(hereafterreferredtoasreanalysisSWE).ComparingtheFBAWassimilation
resultswiththefootprintassimilationresults,wefoundsimilaritiesanddifferences
betweenthem.Inallsixyears,bothassimilationexperimentsincreasetheestimate
ofthetotalSWEwithintheUpperKern,andbothassimilationexperimentsshowthe
east-westSWEgradientintheUpperKern,whichisalsoshowninthereanalysis
SWE.However,comparingthetotalSWEintheUpperKernfromtheseFBAWand
theFPassimilationexperiments,theFBAWposteriorestimatesarelargerthanthe
footprintposteriorestimatesinWY03,WY06,WY07,butsuchcomparisonyieldsthe
oppositeresultfortheotherthreeyears.InordertocomparetheoverallSWE
quantityintheUpperKernfromthetwoposteriorSWEestimates,andtogetmore
insightintotheSWEdistributionsovertheUpperKern,wecalculatedthesix-year
averagelongitudinalSWEdistributionintheUpperKernforthepriorSWE,FBAW
posteriorSWE,footprintposteriorSWE,andthereanalysisSWEestimates,as
showninFigure40.TheUpperKernspansfrom-118.569°to-118.241°inlongitude,
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therefore,theaverageSWEcurvesinFigure40isthe‘SWEcross-sections’ofthe
priorandtheposteriorestimatesacrosstheUpperKern.Figure40showsthe
patternsofthelongitudinalSWEdistributionarequitesimilaramongtheprior
estimates,thetwoposteriorSWEestimatesfromthisstudy,andthereanalysisSWE.
TheyallshowtheclearlylargerSWEstorageinthewestsideofthebasin,andthey
allhavesignificantlylowersnowaccumulationintherivervalleyinthemiddleof
thebasinthanintheotherplacesinthebasin,duetothelowerelevationandthe
highertemperatureinthevalley.Forthesix-yearaveragesnowstorageintheUpper
Kern,thepriorSWE,FBAWposteriorSWE,FPposteriorSWE,andthereanalysis
SWEestimatesare0.1539km3,0.2247km3,0.2468km3,0.2421km3,respectively.
WhiletheaveragesnowstoragevaluesaresimilaramongFBAWposteriorSWE,FP
posteriorSWE,andthereanalysisSWE,thereanalysisSWEshowsmorespatial
detailsintheSWEdistributionthantheFBAWandFPposteriorestimates,andthe
FPassimilationexperimentresultsalsoshowmorespatialfeaturesthantheFBAW
experimentresults.Thisismainlyduetothedifferencesintheresolutionofthe
assimilatedobservations;theobservationsforthereanalysisSWEwereat90m
resolutionandismuchhigherthanthefootprintobservationat14km×8kmand
thebasin-scaleobservationat511km2;thehigherresolutionobservationsresolve
moreinformationofspatialSWEvariability,thusrevealingmorespatialfeaturesin
theposteriorSWEestimates.
138
Fig.39Thebasin-scale(column1-4)prior,FBAWposterior,FPposterior,andreanalysis
posteriorsnowestimatesontheApril1stof(row1-6)WY03-08
139
Fig.40Thesix-yearaverage(WY03-08)SWEwithineachlongitudinalintervalofthebasin.
Inestimatinglarge-scaleSWE,boththeFBAWassimilationmethodandtheFP
assimilationmethodhavestrengthsandlimitations.Themajordifferencebetween
thetwoassimilationmethodsisthattheFBAWassimilationusesasingle(average)
radianceoverthewholedomainastheobservation,whiletheFPmethod
assimilatesmultiple(footprints)radianceobservationsintheupdateandeach
observationcoversonlyaportionofthedomain.InestimatingSWEinareaswith
lowspatialvariation,theaverageradianceoftheentireareacouldgenerally
representthe“local”radiancewithinthisarea;theFBAWmethodisattractivein
thiscase,asitsobservationmatrixincludeslessobservationthantheFPmethod,
whichhelpsreducetheheavycomputationalburdenusuallyassociatedwithlargescaleassimilation[Reichle,2008],andtheFBAWmethodalsoavoidstheneedto
140
calculatethefootprintobservinggeometryandenergydistributionwhichare
necessaryfortheFPassimilation.However,areaswithlargesnowstorageareoften
characterizedbysubstantialvariationinelevation,forestdensity,and
meteorologicalconditions.Inthiscase,theFPassimilationisattractiveasthe
footprintobservationscanbetterrepresentthespatialvariability.Additionally,the
localnatureofthefootprintobservationhelprevealmorespatialdetailsofthe
posteriorSWEestimates,incontrast,theaveragenatureoftheFBAWobservation
leadtoamorespatiallysmoothedposteriorSWE(Figure39).
ItwasfoundthattheFBAWassimilationislesspronetoSWEestimate
degradationinthecasethattheassimilatedradianceobservationisdominatedby
factorsotherthanSWE.Forexample,themicrowaveemissionfromvegetation
masksoutthesnowinformationinspacebornePMobservationswhenthefractional
vegetationcoverageexceeds~30%[Caietal.,2015].Inthesecasestheobserved
radianceismainlyfromvegetationratherthanSWE,andassimilatingthese
observationsmaydegradetheposteriorSWEestimates.ComparedwiththeFBAW
method,FPassimilationtendstobemoresensitivetosuchinstanceswhere
radianceserrorsarenotduetoSWEdifferences.Thisisbecausethesmoothing
natureoftheobservationprocessingintheFBAWassimilationthatreducesthe
effectsofthefalseSWEinformationifthedisturbanceismoderate.However,the
footprintobservationsintheFPassimilationarenotspatiallysmoothed,and
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essentiallyalltheobservationsandpriorestimatesarecorrelatedintheupdating
stepoftheFPassimilation.Thisallowsforthepotentialdegradationofthefull
ensemblethroughcorrelationsbetweenpixels.Severalmethodscouldhelp
increasingtherobustnessoftheFPassimilation.Forexample,developing
physically-basedobservationuncertaintyquantificationmethodstocalculatethe
uncertaintyofeachfootprintobservationbasedonthesnowandlandcover
conditionsoftheareathatthefootprintcovers.Radianceobservationsfromareas
withcomplexsnowconditions(e.g.snowwithlargegrainsize)andgroundfeatures
(e.g.densevegetation)tendstobemoreuncertain,andtheKalmantypedata
assimilationsystemcouldminimizetheeffectsoftheobservationwithlarge
uncertaintiesintheKalmangaincalculation.Anotheroptionistoassignvarying
weightstoeachobservationfootprintbasedonthelocalsnowandgroundfeatures
andthequantificationoftheimpactsofthesefeaturesonthemicrowaveradiance
(e.g.[Druandetal.,2008]).Finally,conductingcorrelationlocalizationcouldhelp
preventdegradationofSWEinagivenpixelfromthoseinneighboringpixels.Future
studieswillfocusonthesemethodstoimprovetheFPassimilationandtoextend
theFPassimilationtobroaderareas.
4.5.4Comparingassimilationmethodologies
Weintroducedamethodtoestimatethegraingrowthrateapriori,before
calculatingthepriorestimateandperformingtheEnBSestimation.InLietal.,
142
[2015a],wefoundthesnowgraingrowthratedominatedtherateatwhichthe
modeledradiancedecreasedinthesnowaccumulationseason,andcalibratedthe
graingrowthratetomakethedecreasingrateofthemodeledradiancetobe
consistentwiththatoftheobservedradiance.Thegraingrowthrateof3.0E-7m4/kg
wasfoundtobetheoptimalvaluefortheUpperKern,andwasadoptedinthisstudy.
Figure40illustratesthisapproachbyshowingthesensitivityofthemodeled
radianceasafunctionofthegraingrowthrateusingtheexperimentresultsatthe
in-situgageCBTinWY2006.AsshowninFigure41,thevalueofthegraingrowth
coefficientutilizedinthisstudyleadstoanoverallchangeofTbbetweenDecember
andMarchthatmatchestheobservedTbchangeoverthisperiod.Usingevena25%
greatergrowthrateleadstoachangethatissignificantlylargerthanthe
observation.Thus,weutilizedthevaluecalibratedinLietal.[2015a],andassigned
thatvalueanuncertaintyof20%ingeneratingtheensemble.Thevalueoriginally
proposedbyJordan[1991]was33%greaterthanthatusedinthisstudy.This
approachtograingrowthestimationispragmatic;estimatinggrainsizestarting
fromanensemblespanningthefullrangeofpossibilitywouldlikelyrequirealarger
ensemblesize
143
Fig.41TheeffectofdifferentgrainsizegrowthrateonsimulatingTbusingthepriorSWE.
Theredlinerepresentsthevalueutilizedinthisstudy(3.0E-7m4/kg),othercurvesshow
thatincreasingthegraingrowthrateleadtoafasterdecreaseinthesimulatedradiance
comparedwiththemeasuredradiance
4.6Summary
Typicalmicrowaveretrievalalgorithmstendtocitesaturationdepthsof150mm,
beyondwhichretrievalisimpossibleorpronetolargeerrorsinretrieval.Thisstudy
hasdemonstratedaccurateSWEestimation(averageRMSEof77.4mmusingtheFP
method)usingtheEnBStoassimilatePMmeasurementsforsixwateryearsinthe
upperKern,asparselyforestedmountainouswatershedintheSierraNevada;
averageinsituSWEwas545mm.Theassimilationusesahigh-resolution(90m)
modelrunasafirst-guessorpriorestimate;thepriormodelestimateisfarless
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accuratewhenrunwithoutmicrowaveassimilation(RMSEof119.7mm).Anew
approachtograingrowthestimationwasproposed,essentiallyusingtherateof
changeofthemicrowavemeasurementsinbetweensnowfalleventstodevelopa
firstguessoftheparametergoverningsnowgraingrowth.TheEnBSalgorithm
utilizingthehigh-resolutionmodelingframeworkallowaccurateSWEestimation
utilizingmicrowavemeasurementsevenforsnowfarbeyondthetechnical
saturationdepth.Thehigh-resolutionmodelingframeworkisessentiallyusedto
downscalethecoarse(87.9km2)microwavemeasurements.Thus,itshouldbe
possibletoutilizethismethodinothermountainousareasthatareabovetree-line,
orsparselyforested.
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Chapter5.Conclusionandfuturework
5.1Conclusionandoriginalcontribution
Thisdissertationdescribesthedevelopmentandtheapplicationofadata
assimilationframework,inwhichthespacebornepassivemicrowaveremote
sensingdatafromAMSR-Ewereassimilatedintoasnowmodelingschemeto
improvetheaccuracyandprecisionoftheSWEestimationintheUpperKernBasin,
SierraNevada.Theassimilationframework,namedEnBS,isspecificallydesignedto
usespaceborneradianceobservationstoestimatelarge-scaleSWEinmountainous
environments,wherelargeamountofseasonalsnowaccumulatesandiscriticalfor
watersupplyinmanyregionsworldwide.Thenewassimilationsystemfurthersthe
existingradiancedataassimilationsystemsthatfocusprimarilyonsmall-scaleor
point-scaleSWEestimation.
SinceradiancedataassimilationestimatesSWEbymergingthecomplementary
SWEinformationfromtheSWEmodelingandtheradianceobservations,therefore,
thestrategyfortheEnBStobeabletoimprovethemodeledSWEeveninthetough
mountainenvironmentistoenhancetheSWEinformationinboththemodelingand
theobservedradiancedataset.Inspecific,wedidthreestudiesandpresentedthem
inthreechaptersinthisdissertation.
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Chapter2focusesontheradianceobservationprocessingimprovement,with
thegoalofenhancingtheSWEinformationinthesnowradianceobservations.Inthe
experiment,wedevelopedanewAMSR-Eradiancedataprocessingalgorithmby
directlyprocessingtherawPMobservationstopoint-scaleandwatershed-scale,
followingtheGIDWalgorithmandtheFBAWalgorithm,respectively.The
experimentresultshaveconfirmedthatthePMdataprocessedfromtherawAMSREfootprintsexploitstheinformationaboutsnowaccumulationandablationin
mountainousareas.Atbasin-scale,theprocessed!!!"#$ ishighlycorrelatedwith
thebasin-averageSWE,withanoverallcorrelationcoefficientof−0.94.The
processedradiancealsocontaininformationaboutsnowmeltonset;themeltonset
timingestimatesfromtheprocessedradianceobservationswerecomparedwiththe
in-situmeasuredmelt-timing,andthemelt-onsettimeestimate,andtheMeltonset
predictedbythe!!!"#$ datahasacorrelationcoefficientof0.94andanRMSEof
5.04days.ComparedwiththetraditionalEASE-GridTbdatathatspatiallyaverage
therawobservationsinto25km×25kmgridcells,theTbdirectlyprocessedfrom
therawfootprintsisthreetimesmoresensitivetoSWEaccumulationandabouttwo
timesmoresensitivetomeltonset,indicatingthatdirectlyprocessingTbfromthe
rawfootprintsandaccountfortheviewinggeometryofthesatellitefootprinthelp
enhancetheSWEinformationintheradianceobservation.
Chapter3focusesonincreasetheaccuracyoftheSWEandsnowradiance
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modelingtoimprovetheunderstandingoftherelationshipsamongthemicrowave
radiance,snow,andthesnow-coveredmountainenvironment.ThemodeledSWE
andsnowradianceprovidepriorinformationtotheassimilation,anddirectly
impacttheaccuracyoftheposteriorSWEestimatesfromtheassimilation.Wesetup
themodelingframeworkatahigh-resolution,andexplicitlyresolvedaseriesofkey
mountainenvironmentparameters(includingelevation,aspect,slope,thefractional
forestcoverage)intoeachmodelingpixeltoaccountforthespatialvariabilityofthe
SWEandradiancecausedbytheruggedmountainterrain.Weoptimizedthe
coefficientgoverningthegraingrowthviamodelingcalibration,andalsodeveloped
anewgrainsizeestimateschemeandintegrateditintothemodeltohandleintense
snowfallswithrespecttothestratigraphyrepresentationandthebrightness
temperatureprediction.ThecomparisonbetweenthemodeledTbwiththesatellite
observedTbshowsthat,afterthemodelimprovements,theRMSEoftheTb
aggregatedfromthebasin-scalemodelingisabout3Koverthedry-snowseason,
indicatingthatthemicrowaveradianceestimatedbythemodelcloselyreplicated
theradianceobservations.
WiththeenhancedSWEinformationfromboththemodelingandthePM
observation,Chapter4focusonthedevelopmentandtheapplicationofthe
assimilationsystemthatmergetheSWEinformationfromthemodelingandthePM
observationtoincreasetheaccuracyofthelarge-scalemountainSWEestimation.
Specifically,wedevelopedaBayesianTheorembasedEnBSdataassimilation
148
framework,andassimilatedthespace-bornepassivemicrowaveradiance
observationsintomodelpredictionstoestimatetheSWEinalow-vegetated
mountainousbasinintheSierraNevada.TheEnBScarriesouttheupdatingina
batchmodebyupdatingallpriorestimateswithallmeasurementsinasinglestep,
whichdiffersfromtheKalmanfilteringandsmoothinginwhichstatevariablesare
sequentiallyupdated.Inthisstudy,theassimilationwasconductedinthesnow
accumulationseasons(October1sttoApril1st)ofeachyearinasix-yearstudy
period,fromwateryear2003to208.Themodelingandtheassimilationwere
carriedoutat90-meterresolutioninattempttocapturethesignificantsnowand
radiancevariabilityinthemountainterrain.Theraw36.5GHzV-polmicrowave
radianceobservationsfromAMSR-Ewereprocessedtopoint-scale,basin-scaleand
originalfootprint-scale;threeindependentassimilationswereconductedwitheach
scaleofobservationtoupdatethepriorSWEestimatesandtoevaluatethe
resolutionoftheobservationontheassimilationresults.Foralltheaccumulation
seasonSWEestimates,theiroverallbiasreducedby84.0%,74.6%and89.3%,and
theirRMSEreducedby36.9%,24.3%and42.0%afterthebasin-scale,point-scale
andthefootprint-scaleassimilation,respectively.April1stSWEhasbeenusedasa
proxyofthepeaksnowstorageintheSierraNevadatoinferthewaterresource
availabilityinthelaterseason.Afterthebasin-scale,point-scaleandthefootprintscaleassimilation,theoverallbiasoftheposteriorApril1stSWEestimatesreduced
by76.4%,74.1%and87.6%,andtheRMSEoftheposteriorApril1stSWEestimate
149
reducedby42.4%,29.2%and60.4%,respectively.TheEnBSassimilation
frameworkshouldpossiblybeutilizedtoimprovetheSWEestimationinother
mountainousareasthatareabovetree-line,orsparselyforested.
5.2Futurework
WeplantoexplorethefeasibilityofextendingtheapplicationoftheEnBS
assimilationtoothermountainrangesinthewesternUnitedStatesthathaslowvegetationcoverage.Asdiscussed,thechallengeforasuccessfulassimilationisto
obtainasufficientlyaccuratepriorestimate,especiallythepriorradianceestimate.
Therefore,thefirststeptoextendthecurrentEnBSframeworktootherwesternU.S.
mountainsistoanalyzethesnowpropertydifferencesbetweenthecontinental
snowpackthatdominatesthesnowcoverinalargeportionofthewesternU.S.and
themaritimesnowthatdominatesthePacificWest;theunderstandingofthese
differencesformsthecriticalbaselinetocalibratethecurrentmodelthathasbeen
optimizedforthemaritimesnowsothatitcouldalsobeadaptedtothecontinental
snow.Inaddition,thisstudyisconductedonlyinthelowvegetationmountains,due
tothesignificantimpactofvegetationontheradianceobservation.Toestimatethe
SWEbeneaththevegetation,theassimilationwillneedtoincludetheassimilationof
additionalobservationsandsnowestimates(e.g.thesnowdepthorSWEreanalysis
results)thatarelittleaffectedbytheexistenceofvegetation,e.g.thein-situ
measurements,thevisiblebandremotesensingobservation,andthesnowdepthor
150
SWEmeasurements;assimilatingmulti-sourcesnowdatasetswouldbeaway
forwardtotheestimateofthesnowwaterresources.
151
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AppendixA.CalculatingviewinggeometryofAMSR-Eobservations
TherearethreepointsofAMSR-Eoperationsworthmentioninginorderto
calculatethefootprintorientation.Thefirstisthatthemajoraxisofafootprint
ellipseisperpendiculartothesatellitescan,sooncethescanorientationis
calculated,thefootprintorientationcanbedetermined.ThesecondisthatAMSR-E
scansareparalleltoeachother;thereisadistanceof10.1kmbetweentwo
contiguousscans.ThethirdisthatAMSR-Ehasa55°offnadirangleand800km
orbitalheight,soafullrotationoftheradiometercorrespondstoacirclewitha
perimeterof5400kmontheground.However,ourstudyareaonlyhasawidthof
severaltensofkilometers,whichisaverysmallpartofthewholeAMSR-Eviewing
circle.Thus,thearcofthelargescancircleinthebasinscaleisnotdiscernible;the
partofascaninthebasinexaminedhereiseffectivelyastraightlineinsteadofan
arc.Whenthebasinislargeenoughsothatthecurvatureofasatellitescaninside
thebasinnolongercanbeneglected,thealgorithmneedstoberevised.Theswath
widthofAMSR-Edatais1445km,sothat,forexample,ifasquare-shapedbasinis
largerthan120kmoneachside,whichcorrespondstoa10ochangeinthecenter
angleofAMSR-Escanningcircle,thecurvatureofthescanneedstobeconsidered.
Tocalculateviewinggeometry,weprojecttheextractedfootprintcentersand
thebasincontourtoUTMZone10S,asshownFig.A-1:reddotsrepresentthe
172
footprintcenterswhiletheblueclosedcurvesignifiesthecontourofthebasin.
DefiningthefirstsampleinthescanwithanindexSmin,andthelastsamplewithan
indexSmax,weestimatethetiltangleofthescans,whichisshowninFig.42asthe
anglethereddashlinemakeswiththegrayline.Thecounterclockwisedirectionis
consideredtobethepositiveangulardirection,thusthetiltangleofascanis
! = !"#$!%
!!!"# !!!
!"#
!!!"# !!!!"#
(A1)
Fig.42Illustrationofdeterminationoffootprintorientation,andthelocationofan
exampleSminandSmax.
Themajoraxisofeachfootprintisalwaysperpendiculartoscans,sothereal
footprintorientationcanbeobtainedthroughrotatingeachfootprintbyθ.Denoting
173
thecentercoordinatesofthefootprintwithindexTas(XST,YSY),accordingtothe
parameterizedequationofellipseandthesizeofthefootprint,thentheequationof
footprinttis
! = !!" + 4!"# !
! = !!" + 7!"# !
(A2)
Aftertherotation,thecoordinatesofthefootprintwithcorrectedorientation
becomes
! ! = !!" + 4!"# ! !"# ! − !!" + 7!"# ! !"# !
! ! = !!" + 4!"# ! !"# ! + !!" + 7!"# ! !"# !
(A3)
Thismethodisapplicableforcalculatingviewinggeometryforbothascending
anddescendingpassesandallsatellite-groundpositions.
174
AppendixB.Modelcalibrationandre-development
B.1SSiB3mass-weightedaverageresamplingscheme
SSiB3usesthreestratigraphiclayers.Thetoptwolayersaredesignedtobe
relativelythinsothatthesnowpacktemperaturecanbesufficientlysensitivetothe
atmosphericchanges.Thetoplayerandmiddlelayercanhaveamaximumdepthof
2cmand20cm,respectively,andthebalanceofthesnowdepthbelongstothe
bottomlayer.Intheseasonaltime-seriesmodeling,ifsnowfalloccursinamodeling
step,thesnowpackjustaftersnowfallhasfourlayers:theinitialthree-layer
snowpackfromthepreviousstep,andthenewlyaccumulatedlayer.SSiB3utilizesa
three-layersnowpackinmodeling,sothefourlayersneedtoberesampled.For
simplicity,hereafterthefour-layersnowpackiscalledpre-resamplesnowpack,and
isdesignatedas!! (!):
D4 (Z ) = ⎡⎣ D4 ( Z1 ) D4 ( Z 2 ) D4 ( Z 3 ) D4 ( Z 4 ) ⎤⎦ T
(B1)
where!isavectoroftheheightofeachlayercenterpointabovethesoiland
!! (!! ) representsthejthlayerwithheight!! .Similarly,wecallthethree-layersnow
resampledfrom!! (!)thepost-resamplesnowpack!! (!):
D3 (Z ) = ⎡⎣ D3 ( Z1 ) D3 ( Z 2 ) D3 ( Z 3 ) ⎤⎦
175
T
(B2)
SSiB3appliesamass-weightedaverageresamplingschemefullydescribedinJordan
[1991]toresample!! (!),whichcanbewrittenas:
D3 ( Z ) = w ⋅ D4 ( Z ) , w ∈R 3×4 (B3)
wherewistheresamplingmatrix.Theelementsof!!" (! ∈ [1,3],! ∈ [1,4])isthe
massratioof!! (!! )tobemergedto!! (!! ),therefore
∑w
ij
= 1, wij ∈[ 0,1] (B4)
j=1:4
and
D3 ( Z i ) =
∑w
ij
( )
⋅ D4 Z j j=1:4
(B5)
showsthemasscontributionfromeachofthepre-resamplesnowpacklayers
(! ∈ [1,4])toeachpost-resamplelayers(! ∈ [1,3]).There-layeringschemehasaset
ofprotocoltodetermine!!" basedonthestratigraphicconditionof!! (!).The
detailsoftheprotocolcanbefoundinJordan1991.
B.2Inadequacyofthemass-weightedaveragecombinationresampling
SignificantpositivebiaswasintroducedtothemodeledTbafterthefirstintense
snowfalleventofeachseason;themaximumbiasreached30K.Wefoundthebiasis
solelyamodelartifactasaresultofthemodelstratigraphicsimplification.Figure43
illustratesthefirstintensesnowfallofWY2005anditseffectsontheradiance
modeling.ThefirstintensesnowfallofWY2005occurredfromlateDecembertomid
176
January;thesnowfallaccumulatedatotalof36cmSWE(Figure43,top),leadingto
anincreaseinsnowdepthfrom41cmto103cmintwoweeks.Therapidlyaccumulatedsnowledtoa~30Kincreaseinthemodeledmicrowaveradianceafter
thesnowfall.However,theAMSR-EobservedTbdoesnotshowsuchalargeincrease
inthesameperiod(Figure43,bottom).Wefoundthepositivebiasinthemodeled
Tbisattributabletotwothings.First,themass-weightedstratigraphicresampling
describedinequation(B1)to(B5)underestimatesthesnowgrainsizeafterthe
intensesnowfall,andtheunderestimatesareamplifiedwhenalargeamountofnew
snowaccumulatesatoparelativelyshallowsnowpack.Second,thesnowgrainsin
thelowerpartofsnowpackdominatetheattenuationofthemicrowaveradiance
becauseoftheirstrongscatteringeffects;theunderestimatedgrainsize(especially
thegrainsinthebottomlayer)hasalowerscatteringcoefficient,leadingtothe
positivebiasinthemodeledTb.
177
Fig.43(Top)BoththeSWEfromSSiB3simulation(red)andfromin-situ
measurement(blue)atCBTshowanintensesnowfalleventduringlateDecemberto
earlyJanuaryofWY2005.(Middle)Theintensesnowfallledtoasignificantdecrease
inthesimulatedbottomlayergrainsize(red);topandmiddlegrainsizearealso
shown(grey).(Bottom)Thedecreaseingrainsizeledtoasharpincreaseinthe
simulatedTb(red)duringthesameperiod,buttheAMSR-ETb(blue)didnotshow
suchasignificantTbincrease.
178
Figure44illustratesthemechanismofthegrainsizeunderestimate.Figure44
(left)showsthepre-resamplesnowpack(!! )aftertheintensesnowfallsendedat
mid-January2005;atotalof62cmlow-densityfreshsnow(withPexequalto
0.05mm)accumulatedontopoftheoldthree-layersnowpack(with41cmdepth).
Figure44(right)showsthepost-resamplesnowpack(!! ).Asmentionedearlier,the
topandmiddlelayerofthepost-resamplesnowcannotbedeeperthan2cmand
20cm,respectively.Therefore,thevastmajorityofthefreshsnowwasmovedtothe
bottomlayerintheresampling;thelow-densityfreshsnowtookupmostofthe
resampledbottomlayer,leadingtoadecreaseintheproportionofthebottomlayer
snowmasstothetotalmassofthesnowpack.Therefore,inthemass-weighted
averagesnowpropertycalculationthebottomlayergrainsizedecreased.The
decreaseinthebottomlayergrainsizefurtherleadstotheincreaseinthe
microwaveradiance:atCBT,Tbmodeledfromthepost-resamplesnowpack(!! )is
~259Kattheendofthefirstintensesnowfall(Figure44right),17Khigherthanthe
AMSR-Eobserved242K.However,theTbsimulatedfromtheun-resampled!! is
~240K(Figure44left),muchclosertotheAMSR-Eobservation.Several
experimentscarriedoutatothersnowpillowsyieldedsimilarresults:thebottom
layergrainsizeshowedanoticeabledecreaseafterthestratigraphicresampling,
andconsiderablepositivebiasemergedinthesimulatedTbfromthepost-resample
snowpack(!! ).However,Tbsimulatedwithpre-resamplesnow(!! )wascloseto
179
theTbobservations,becausenostratigraphicresamplingwascarriedout.These
experimentsdemonstratedthattheSSiB3stratigraphicre-layeringschemeleadsto
thepositivebiasinmodeledTb.
Fig.44(Left)Thefour-layersnowstratigraphyjustafterasnowfall,withgrainsize
Pexvaluesgivenandillustratedgraphicallyviathedensityoftheblackdotsineach
layer,andtheabove-surfaceTbareshown.(Right)Aftermass-weightedre-layering,
thebottomlayergrainsizehasdecreasedandthemodeledTbincreased.
WealsofoundthatthebiasinthemodeledTbwasmostnoticeableinthefirst
intensesnowfalleventofaseason;theeffectsofthesnowfallsatlaterofaseason
180
aregenerallynegligible.Thereasonisthatthefreshsnow“reshuffling”which
causedthemodelartifactwasonlysignificantinthefirstsnowfall.Inthelater
snowfalls,theexistingsnowpackwasdeeprelativetothesnowfall;theeffectofthe
shufflingbecomesinsignificantsotherewasnolargebiasgenerated.
B.3Inadequacyofthemass-weightedaveragecombinationresampling
Wedevelopedtwomethodstocorrecttheunderestimatedbottomlayergrainsize:
oneofthemisaniterativemethod,andtheotheroneisaregressionmethodthat
wascalibratedtomatchtheresultsoftheiterativemethod,butatless
computationalexpense.Weassumethatthegoalofthere-samplingschemefor
grainsizeshouldbetomaketheTbpredictionforthethree-layermodelbeasclose
aspossibletothatofthefour-layermodel.Whencalculatingtheimprovedbottom
layergrainsize,wetriedtominimizenotonlythedifferencebetween!!! andthe
post-resample!!! ,butalsothedifferencebetweenthethree-andfour-layergrain
sizevalues.Conceptually,wetrytosolvetheoptimizationproblem:
Minimize wD4 − D3 subjectto Tb3 − Tb4 < ε (B6)
Inequation(B6),thefirstterm !!! − !!! denotesthediscrepancybetweenthe
four-layerTbandthecalculatedpost-resampleTb,and!issomethreshold.In
181
addition,wealsowantthepost-resamplegrainsizetobeascloseaspossibletothe
pre-resamplegrainsizetoreducetheeffectofthere-layeringongrainsizeestimate.
B.3.1Theiterativecombinationre-samplingscheme
Whensnowisdry,snowgrainsattenuatetheupwellingmicrowavetransmission
insidesnowpackthroughscatteringandabsorption,leadingtoanegative
correlationbetweenTbandgrainsize.Thefirstintensesnowfalloccursinearlyor
midwinter,whensnowpackismostlydry,soitisfeasibletotakeadvantageofthe
negativecorrelationbetweenTbandgrainsizetocalibratethebottomlayergrain
size.Inthismethod,thegrainsizeofthebottomlayerofthepost-resample
snowpackiscalibrateduntilthesimulatedTbiswithineofthereferenceTb.Itwas
foundthatifaprecipitationeventislargerthan2cm/day,there-layeringscheme
willgenerateanoticeabledecreaseinbottomgrainsize.Ifprecipitationisgreater
than2cm/daytheniterationsareperformedtograduallyincreasethemassweightedaveragebottom-layergrainsizeby 0.00001mm/iterationuntilthe
absolutedifferencebetweenTb3andTb4islessthane=0.01kelvin.
B.3.2Theempiricalcombinationre-samplingscheme
Astheiterativeschemeisrathercomputationallyexpensive,itwasdesirableto
deriveaone-to-onerelationshipbetweenthemagnitudeofthebottomlayergrain
sizecorrectionandtheintensityofsnowfall.Specifically,sincetheiterative
resamplingschemehasbeenusedtoreducethebiasatthethreesnowpillows,we
182
usedtheiterativecorrectionresultsandcalculatedtheratioofthebottomlayer
grainsizecorrectionwithrespecttothebias.Weperformedaregressiontothe
precipitationmagnitudecollectedfromtheprecipitationgageateachsnowpillow
andtheratioofthebottomlayergrainsizecorrection,tocorrelatethecorrection
withtheintensityofprecipitation.Sucharelationshipbetweencorrectionand
snowfallsimplifiesthebottomlayercorrectiontoalook-uptableproblemand
enablesthefastcomputationofthegrainsizecorrection,whichisdesirablefor
large-scalehigh-resolutionmodeling.Weappliedalogarithmfunctionforthe
regression:
C pex = a × log b ( P + 1) (B7)
whereCpex(%)isthecorrectionratio,Pistheprecipitationmagnitudeincm/day,
andaandbaretheparametersyieldingthebestfitbetweentheCpexfromthe
iterativeresultsandtheircorrespondingprecipitationintensityP.Alogarithm
functionwasusedbecauseitnotonlyfitthegeneraltrendofthedatapointsbutalso
passesthroughtheorigin,whichisattractivebecauseifnosnowfalloccurs,there
shouldbenocorrection.Weusedequation(B7)tofitthecorrectionresultsobtained
fromthepoint-scaleiterationmethod(Figure45).When! = 0.6123and! = 3.6
yieldthebestfitwheretheRMSEofthefitis0.13.
183
Fig.45.TherelationshipbetweenthePexcorrectionratioandthesnowfallintensity.
Theblackcurveisthefunctionfittedfromthepoint-scalecorrectionresultsfrom
theiterationmethod.Themoreintenseasnowfallis,thelargerthegrainsize
underestimateis.Thereforeingeneralthecorrectionratioispositivelycorrelated
withsnowfallmagnitude.
184
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