# A comparative assessment of different methods for detecting inhomogeneities in Turkish temperature data set

код для вставкиСкачатьINTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 18: 523–539 (1998) PREDICTING RAINFALL STATISTICS IN ENGLAND AND WALES USING ATMOSPHERIC CIRCULATION VARIABLES a C.G. KILSBYa,*, P.S.P. COWPERTWAITa,1, P.E. O’CONNELLa and P.D. JONESb,2 Department of Ci6il Engineering, Cassie Building, Claremont Rd, Uni6ersity of Newcastle, Newcastle upon Tyne, NEI 7RU, UK b Climatic Research Unit, Uni6ersity of East Anglia, NR4 7TJ, UK Recei6ed 3 August 1997 Re6ised 12 No6ember 1997 Accepted 13 No6ember 1997 ABSTRACT Regression models are developed to predict point rainfall statistics, with potential application to downscaling general circulation model (GCM) output for future climates. The models can be used to predict the mean daily rainfall amount and the proportion of dry days for each calendar month at any site in England and Wales, and use the following explanatory variables: (i) geographical (altitude, geographic coordinates, and distance from nearest coast); and (ii) atmospheric circulation variables (mean values of air-flow indices derived from mean sea-level pressure grids). Values predicted by the models, for 10-km grid squares covering the whole of England and Wales, are in reasonable agreement with the 1961–1990 climatology of Barrow et al. (1993). The potential use of the models in hydrological climate change impact studies is discussed. © 1998 Royal Meteorological Society. KEY WORDS: England and Wales; climate change; hydrological modelling; general circulation models; downscaling; rainfall; atmospheric circulation; vorticity; regression models 1. INTRODUCTION The impact of changes in climate on hydrological systems is a problem of crucial and growing importance to society. A capability for producing realistic rainfall statistics or rainfall time series for future climate conditions would be of great utility for a range of environmental impact studies. Hydrological impact assessments often require a suitable rainfall time series to be generated for input to a hydrological model of a system. For example, at the catchment-scale, a daily rainfall time series for a future climate may be generated and used with a rainfall-runoff model to simulate future river flow scenarios. In urban drainage studies, where an even finer time resolution is required, an hourly time series may be generated and used with a hydraulic flow simulation model to predict the frequency of sewer overflow discharges to receiving water courses. At the monthly scale, a rainfall series may be simulated to provide predictions for future reservoir levels. The usual tool for simulating future climates is a general circulation model (GCM) of the atmosphere, assuming prescribed changes in atmospheric composition and forcing (Houghton et al., 1996). However, GCMs are not appropriate for simulating rainfall series at the catchment-scale, because they operate at very coarse spatial scales (typically some 300 km × 300 km). Hence, a key requirement for the production of rainfall scenarios from GCMs is the development of methods to ‘downscale’ GCM output to scales appropriate for catchment models of hydrological systems. In particular, methods are required to relate * Correspondence to: Department of Civil Engineering, Cassie Building, Claremont Rd, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK. e-mail: c.g.kilsby@ncl.ac.uk 1 Department of Statistics, Massey University, Albany, Private Bag 102 904, North Shore, Auckland, New Zealand 2 e-mail: p.jones@uea.ac.uk Contract grant sponsor: EC Environment Research Programme under the POPSICLE project; Contract grant number: EV5V-CT940150, Climatology and Natural Hazards CCC 0899–8418/98/050523 – 17$17.50 © 1998 Royal Meteorological Society 524 C.G. KILSBY ET AL. GCM output to single-site rainfall data. Before these methods may be applied to the future climate, they must first be developed and validated for the present climate. This work sets out to develop a simple downscaling tool for relating point rainfall statistics to atmospheric circulation variables, such as those that are output from GCMs. The downscaling tool is based on two simple regression models. The point rainfall statistics can then be used as the basis for parameterizing a stochastic rainfall model for generating time series of rainfall. There is an extensive literature on simulating rainfall scenarios for future climates: recent examples are Bardossy and Plate (1992), Buishand and Brandsma (1997), Conway and Jones (1997) and Hughes and Guttorp (1994). The models described here differ from those in previous work because they enable the prediction of rainfall statistics for any site in England and Wales, and are therefore applicable at sites where data are limited or do not exist. The paper is divided into sections as follows. In Section 2 the observational data used in the regression analysis are described. Appropriate models are then proposed and fitted in Section 3, and their predicted values compared with the observed values. In Section 4, the models are implemented within a geographical information system (GIS) and are used to predict values over 10-km grid squares covering the whole of England and Wales. A validation compares the predicted values with those of the 1961–1990 climatology of Barrow et al. (1993). The paper concludes with a discussion of possible applications to studies of the impacts of climate change on hydrology. 2. DATA AND VARIABLES 2.1. Explanatory 6ariables The data used comprise daily time-series records (1961–1990) of atmospheric circulation variables (see Jones et al., 1993 for details) and rainfall at 67 sites (Figure 1). The rainfall sites were distributed throughout England and Wales, with two located in the adjacent borders region of Scotland, and were obtained from the UK Meteorological Office. The rainfall sites were chosen for uniform spatial coverage, and as complete temporal coverage as possible. Ten sites at higher elevations (in the Welsh mountains, Pennines and Lake District) were specially selected in order to reduce the bias towards low elevations which is inevitable in operational rainfall measurement networks. Four atmospheric circulation variables are used: mean sea-level pressure (p), total shear vorticity (z), and the zonal and meridional components of geostrophic air-flow, (u and 6). A positive value of u corresponds to air flowing from west to east, whilst a positive value of 6 corresponds to air flowing from south to north. The vorticity (z) is a measure of atmospheric rotation, with positive values corresponding to low pressure (cyclonic weather) and negative values corresponding to high pressure (anti-cyclonic weather). The data are derived from a relatively coarse grid of mean sea-level pressure (5° latitude by 10° longitude) shown in Figure 2. The gridded data set was initially developed by Jenkinson and Collison (1977), and later used by Jones et al. (1993). The vorticity and flow-velocity units are geostrophic and expressed as hPa per 10 ° latitude at 55 ° North (see Jones et al., 1993) and when multiplied by 1.2 are speeds in ms − 1. These variables differ from those used in previous work (Conway and Jones, 1997): they used flow strength and direction instead of u and 6, and considered z alone as a measure of cyclonicity. Monthly mean values of the circulation variables are shown in Figure 3. A seasonal variation is evident in the two air-flow velocities, reflecting the dominance of westerly frontal weather systems during the winter in the UK. Each site is characterised by the variables: altitude (a), Ordnance Survey east grid reference (e), north grid reference (n), and distance from the nearest coast (c). Thus, there are a total of eight explanatory variables available for the analysis: p, u, 6, z, a, e, n, and c. For each year-month in each record of data, the mean daily value was found for each variable to give monthly mean time-series of u, 6, z, p and rainfall amount (Y) at each site (Table I). The proportion of dry days was also found for each year-month to give monthly proportion dry (P) time series at each site. © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 525 PREDICTING RAINFALL STATISTICS A day is deemed dry if less than 0.2 mm of rainfall is recorded. Excluding missing data results in a total of 17 567 site-months available for model fitting. The term ‘point rainfall statistic’ will be used here to refer to either the mean daily rainfall or the proportion of dry days at a single rain gauge or site. 2.2. Temporal aggregation A sampling interval, or aggregation level, of one month was adopted. The use of a daily interval (or n-day mean where n is less than five or so) would be preferable, in order to resolve individual synoptic weather systems. However, we could not obtain a reasonable degree of explained variance at aggregation levels below monthly with the present data set. If a daily level is desired for downscaling, then other approaches may be more suitable (e.g. Conway and Jones, 1997). A monthly time level does have a number of advantages: (i) the work presented here provides point rainfall statistics which can be used in the estimation of the parameters of a stochastic rainfall model. Statistics are required for each calendar month (discussed in Section 5). It is therefore appropriate to use this aggregation level throughout (rather than aggregating predicted daily values, for example). The stochastic model will then perform the sub-monthly disaggregation process, and is able to produce daily (and hourly) rainfall amounts with the correct statistical properties. Figure 1. Location of raingauges used in the regression analysis © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 526 C.G. KILSBY ET AL. Figure 2. Grid points used for calculations of circulation indices (ii) each observed time series can be treated as approximately stationary over one month (a longer sampling interval can result in a non-stationary time series due to seasonal variations or seasonal forcing). (iii) the sample autocorrelation for monthly rainfall series tends to be small so that the data may be treated as approximately independent in time. We have analysed all of the months of the year together rather than performing individual analyses on the seasonal or monthly timescale. This ensures that only the explanatory variables are used for producing seasonal variations, rather than the time of year. Otherwise regression relations would be produced for each fixed calendar month, with the risk that the relationship between climate and time of year may not be maintained in future climates. The seasonal variation can then also be used in validation of the method. 2.3. Spatial aggregation We have analysed the whole of England and Wales together rather than performing regional analyses, for example following the sub-division into five regions used by Wigley et al. (1984) or four areas identified by Mayes (1991). Such a regional analysis would be valuable in better resolving synoptic scale weather systems. However, the coarse resolution of the circulation variables used here does not support such an analysis, which could perhaps be performed using Mayes (1991) as a basis. However, for similar reasons as outlined above for the seasonal case, any given regionalization may not be appropriate in a changed future climate, and a better solution is to use circulation indices at higher spatial resolution. © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 527 PREDICTING RAINFALL STATISTICS 3. REGRESSION MODELS 3.1. Formulation of models The mean daily rainfall at site i, year j, and month k, denoted Yijk, always takes values greater than zero. Hence, it is appropriate to formulate the following regression model: Yijk =exp{a0 +aaai +aeei +anni +acci +auujk + a66jk + azzjk + appjk + oijk } (1) where the a’s are parameters to be estimated and oijk is random or modelling error. When oijk has a normal distribution with mean zero and standard deviation s, the expected value is E(Yijk )= ry × exp{a0 +aaai +aeei +anni +acci + auujk + a66jk + azzjk + appjk }, (2) where ry is a correction ratio to allow for the bias resulting from re-transformation from ln(Y) to Y (see Section 3.2). If the oijk come from a normal distribution, ry = exp{s 2/2}. In practice, however, the oijk come from a skewed distribution; provided the sample is large enough, which it is here, an estimate for ry can be found empirically to give an approximately unbiased estimate of E(Yijk ) in Equation (2). The random variable Pijk, representing the proportion of dry days for site i, year j, month k, always takes values between 0 and 1. Hence, it is appropriate to postulate the following regression model: Figure 3. Mean values of z, u, 6, p over the period 1961– 1990 plotted against month. Units of z, u, 6 are hPa per 10° latitude at 55° North © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 528 C.G. KILSBY ET AL. Table I. Variables used in the regression analysis* Notation Variable name Yijk Pijk ai ei ni ci ujk 6jk pjk zjk oijk jijk Mean daily rainfall (mm) Proportion of days with less than 0.2 mm of rain Altitude (10 m) Easting (100 km) Northing (100 km) Distance from nearest coast (km) West–east component of flow velocity South–north component of flow velocity Pressure (hPa) Vorticity Error in predicting Yijk Error in predicting Pijk * Indices i, j and k refer to site, year, and month, respectively. Pijk = 1/[1 +exp{ − (b0 +baai +beei +bnni + bcci + buujk + b66jk + bzzjk + bppjk + jijk )}], (3) where the b’s are parameters to be estimated and jijk is random or modelling error. After taking the expectation of the Taylor expansion of Pijk =f(jijk ), E(Pijk )=rp /[1 + exp{ − (b0 +baai +beei + bnni + bcci + buujk + b66jk + bzzjk + bppjk )}], (4) where rp is a correction ratio for the retransformation bias (greater than 0 and less than 1). Again, provided the sample size is large enough, this ratio can be found empirically to ensure that an approximately unbiased estimate is obtained for E(Pijk ) in Equation (4). 3.2. Fitted models The transformations ln{Yijk } and ln{(Pijk /(1 − Pijk )} were made to give linear equations in the model parameters. The a’s and b’s were then estimated by linear least squares regression (forced entry method); the resulting estimates are shown in Table II, with the percentage of explained variance (r 2), the standard deviation (S.D.) of the residual errors (oijk and jijk ) and empirical estimates of the correction ratios (r). Three model variants were investigated. The first variant (denoted model ‘Z’) used independent variables u, 6, z, a, e, and c. It was found that northing (n) had negligible effect. The second variant (model ‘P’) replaced vorticity (z) with pressure (p) to characterise the cyclonicity of the circulation. Coefficients from both of these models are shown in Table II, and it can be seen that they perform similarly: differences between the two models are discussed further in Section 3.3. The third variant was to construct seasonal models for mean and proportion dry, i.e. four separate regressions for samples separated according to season; winter (December, January, February (DJF)), spring (MAM), summer (JJA) and autumn (SON). Although this seasonal model produced marginally higher percentages of explained variance, it is not considered a practical model, since it introduces a seasonal variable which cannot be guaranteed to behave in a similar fashion under future climate change. The model coefficients appear to be in general agreement with the observed trends in England and Wales rainfall climatology (see for example Hulme and Barrow, 1997). Both models predict more rain at high altitudes and in the west of England and Wales, and less rain on the coast (Table II). Lower rainfall is predicted in the east of England, because of the rain shadow in the lee of the Welsh and Pennine mountains. Moreover, greater rainfall is predicted for low values of pressure or high values of vorticity, as well as for stronger zonal flow (u), corresponding to westerly-type frontal weather systems. Both models explain approximately half the variability in the data, which is reasonable in view of their simple form. More complex models were also considered and found to slightly improve the r 2 values. For example, models using powers, cross-products, and more detailed sample distributional properties (e.g. the variance and autocorrelation of the daily vorticity) were considered. However, it was decided to retain the © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) © 1998 Royal Meteorological Society Table II. Estimated parameters Model P Model Z Estimated parameter (â) Constant a (altitude) c (distance from coast) e (easting) p (pressure) u (zonal flow) 6 (meridional flow) z (vorticity) r (correction factor) Mean daily rainfall (ln Y) Proportion of dry days (ln{P/(1−p)}) t-ratio Estimated S.E. (b. ) parameter (b. ) t-ratio Estimated parameter (â) S.E. (â) t-ratio Estimated parameter (b. ) S.E. (b. ) t-ratio 80.5 0.01712 −0.00234 0.86 4.9×10−4 1.4×10−4 93.8 35.1 −16.8 −104.1 −0.01787 0.00143 0.89 5.0×10−4 1.4×10−4 −117.3 −35.4 9.9 1.276 0.01732 −0.00233 0.023 4.9×10−4 1.39×10−4 53.7 35.5 −16.8 −0.1437 −0.01813 0.00142 0.025 1.4×10−4 1.4×10−4 −5.8 −35.6 9.8 −0.17303 −0.78090 0.00061 −0.00718 5.10×10−3 8.5×10−4 7.5×10−4 8.8×10−4 −33.9 −92.4 −9.6 0.7 0.11971 0.10247 −0.00745 −0.00598 5.3×10−3 8.7×10−4 7.8×10−4 9.2×10−4 22.7 117.2 −1.0 −6.5 −0.17306 — 0.00993 0.00169 5.09×10−3 — 7.3×10−4 8.8×10−4 −33.9 — 13.6 1.9 0.11972 — −0.02319 0.00751 5.3×10−3 — 7.6×10−4 9.2×10−4 22.5 — −30.5 −8.2 — — — 0.03782 4.07×10−4 1.129 92.9 −0.04901 4.25×10−4 0.996 −115.4 1.129 — 0.994 0.56 0.43 0.58 0.53 — — 0.56 0.44 0.59 0.50 529 Int. J. Climatol. 18: 523 – 539 (1998) Residual S.D. r2 S.E. (â) Proportion of dry days (ln{P/(1−p)}) PREDICTING RAINFALL STATISTICS Mean daily rainfall (ln Y) 530 C.G. KILSBY ET AL. simpler models formulated above because the parameter estimates are easier to interpret, and the small improvements to the r 2 values do not justify the additional complexity. The percentage of explained variance for each month for both models P and Z was calculated and is shown in Table III. This shows a poorer performance in the months April–July inclusive, due to a higher relative error caused by the lower mean rainfall in these months. The model performance could again be improved by formulating a seasonal model; this has not been done for the reasons given above. The use of temperature as an explanatory variable was also considered. Monthly time series of Central England temperature were taken from Manley (1974) and Parker et al. (1992) and the regression models re-fitted with a further variable for temperature included. However, the residual standard deviation of the re-fitted models had a value very close to the fitted models of Table II (the differences were less than 0.001). It was therefore decided to omit the temperature variable for this analysis, because it appears to be of no practical value in the present day case. Clearly, for some geographical regions, with climates different from England and Wales, temperature is likely to be an important explanatory variable and should then be included in the model (e.g. see Brandsma and Buishand, 1997). Also, for future impact studies, there is a strong case for the inclusion of temperature, as a rise in temperature may be the major observed signal of climate change. There is clearly potential for more detailed analyses using other variables, including relative humidity as well as a range of synoptic variables at different heights in the atmosphere (e.g. 500 hPa geopotential height). This is discussed further in Section 6. 3.3. Analysis of model performance The residual errors (observed−predicted values) were plotted against the explanatory variables and against predicted values of the point rainfall statistics. These plots revealed no systematic departures in the model assumptions; for example, the linear relationship for ln(Y) and ln{P/(1− P)} does not appear to be violated. A more stringent test involves plotting the residual errors against month as a test of the models’ representation of seasonal variation. This is particularly important as predictions for each month are likely to be required in applications. Consequently, summary statistics of the predicted and observed values were evaluated by pooling the data for each calendar month and are shown in Figures 4 and 5. Plots of all the residuals against month were also considered, rather than summary statistics, but it was difficult to detect discrepancies in these plots because of the large number of data. Table III. r 2 values for regression models for all England and Wales 10 km grid values validated against climatology Month 1 2 3 4 5 6 7 8 9 10 11 12 Annual Mean daily rainfall Proportion of dry days P Z P Z 0.67 0.68 0.65 0.59 0.60 0.54 0.55 0.60 0.63 0.64 0.65 0.67 0.62 0.67 0.68 0.65 0.59 0.60 0.55 0.55 0.60 0.63 0.64 0.65 0.67 0.62 0.68 0.61 0.65 0.21 0.46 0.49 0.40 0.54 0.62 0.72 0.60 0.69 0.56 0.68 0.61 0.64 0.21 0.47 0.49 0.40 0.54 0.62 0.71 0.60 0.68 0.55 © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) PREDICTING RAINFALL STATISTICS 531 Figure 4. Summary statistics of mean daily rainfall (Y) plotted against month for 67 sites used in regression; (a) mean; (b) standard deviation; (c) skewness The sample means are in reasonable agreement; the mean of the predicted values generally follow the seasonal variation evident from the mean of the observed values (Figure 4(a) and 5 (a)). The major discrepancy to be noted is the overprediction of mean rainfall (and corresponding underprediction of proportion dry) by model ‘Z’ in May. However, model ‘P’ performs adequately throughout the year, and for this reason has been preferred for further use in this analysis despite theoretical support for the use of vorticity rather than pressure. The reasons for the overprediction are not clear, but may be found in the seasonal variation in land-sea temperature differences. Sea surface temperature was not included in these models, but can be important for some wind directions and warrants further study. Some minor discrepancies may also be noted, for example a slight tendency to underpredict from August to December (Figure 4(a)). Greater discrepancies are evident in the plots for the sample standard deviations (Figure 4(b)), where it is clear that the models under-predict these values. The skewness plots (Figure 4(c) and 5(c)) show a reasonable agreement between the predicted and observed values for most months; however, a clear over-prediction is evident for some months (e.g. August in Figure 5(c)). The practical implication of these discrepancies, which clearly depends on the intended application, is discussed in Section 5. © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 532 C.G. KILSBY ET AL. 4. INDEPENDENT VALIDATION 4.1. GIS implementation Barrow et al. (1993) derived a baseline climatology for the period 1961–1990 for mean monthly precipitation and number of rain days. They fitted thin-plate splines to monthly data taken from a large number of sites (2376 for mean, 168 for rain days) scattered throughout Great Britain, interpolated onto a 10-km grid using elevation as an independent variable. Although some of the sites correspond with those in the regression analysis (13 for mean rainfall, and eight for rain days), the climatology is the most comprehensive and accurate available and forms a good basis for a spatial validation of the regression models over England and Wales. The regression models and climatology were implemented in a geographical information system (GIS) for 10-km grid squares covering England and Wales, and identical to that of Barrow et al. (1993). The GIS was first used to calculate mean values of the regression site-variables over each grid square. The regression models and the GIS information were then used to predict the mean monthly rainfall and the proportion of dry days for each month over all the grid squares. Time series of predicted Y and P were calculated for each grid square using the 1961–1990 time series of circulation variables and the grid square geographical information. The values for each calendar month were then averaged to give 1961–1990 mean-monthly rainfall statistics. Figure 5. Summary statistics of proportion of dry days (P) plotted against month for 67 sites used in regression; (a) mean; (b) standard deviation; (c) skewness © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) PREDICTING RAINFALL STATISTICS 533 Figure 6. Summary statistics for mean of all England and Wales 10 km grid square values for (a) mean daily rainfall; (b) proportion of dry days The spatial mean for England and Wales was calculated for each calendar month for both model ‘P’ and model ‘Z’, and are plotted in Figure 6 for comparison with the climatological values. This confirms the poor performance of the vorticity model (model ‘Z’) in May. The annual averages are mapped in Figures 7 and 8 for comparison with the climatology, and an example monthly average (for January) is shown in Figure 9. Figure 7. Maps of mean annual rainfall on a 10 km grid. (a) Climatology of Barrow et al. (1993); (b) predicted values given by regression model P © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 534 C.G. KILSBY ET AL. Figure 8. Maps of mean annual proportion of dry days. (a) Climatology of Barrow et al. (1993); (b) predicted values given by regression model P 4.2. Ele6ation dependence The predicted values are found to be rather sensitive to the elevation value taken for the grid square. It is not obvious what the correct elevation estimates are for use at a 10-km scale; e.g. Barrow et al. (1993) discuss whether the grid-square mean or maximum elevation should be used in interpolation. Elevation data taken from the Digital Chart of the World (Defense Mapping Agency, 1992) at approximately 1 km resolution were used to investigate this problem. The use of the mean of the 1-km elevations (over the same 10-km grid used in Barrow et al., 1993) results in a general under-estimation of mean rainfall, whilst use of the highest 1-km grid square value results in an over-estimation. It was found that the best fit to the climatology was obtained using mean elevation plus one standard deviation of the 1-km grid squares contained in the 10-km grid square. One standard deviation ranges from a few metres in a flat region such as East Anglia, to some 100 m in mountainous areas. Figure 9. Maps of mean monthly rainfall for January. (a) Climatology of Barrow et al. (1993); (b) predicted values given by regression model P © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) PREDICTING RAINFALL STATISTICS 535 4.3. Validation The general reproduction of spatial patterns and amounts is good, given that single regression models are used to predict the point rainfall statistics for the whole of England and Wales. The primary gradients, lee and coastal effects are well reproduced. Discrepancies are apparent, particularly at high altitudes where an underprediction occurs (Figure 7), probably because of insufficient data at these altitudes when fitting the regression models. Note that a more detailed map is produced by the regression model than is available for the climatology for the proportion of dry days (Figure 8). This is due to fewer data being used in the climatology for rain days than for the mean rainfall: only 168 of 7201 sites used for the mean daily rainfall climatology had this information available. Caution must be exercised in the use of both climatologies for high elevation sites, and particularly so for the rain day case in view of the small number and low elevation of the sites employed. Some discrepancies are evident for individual months, the most noticeable being an under prediction in the mean monthly rainfall along the south coast in winter (Figure 9 illustrates this for January). Failure to reproduce this more detailed aspect of spatial patterns is unsurprising given the regional nature of the circulation variables and regression models and the relative sparsity of rain gauges used in the regression analysis. The distribution of mean monthly rainfall and proportion of dry days for the climatology, regression models and gauge records were plotted using values for all the grid squares and for all months, averaged over the period 1961 – 1990 (Figures 8 and 9). Satisfactory agreement was obtained, although an under-prediction of high rainfalls is again evident in the tail of the sample distributions (Figure 8). This corresponds with the under-estimation of rainfall at high altitudes evident in the GIS maps (Figure 5). 4.4. Assessment of model stability If the regression models derived above are to be applicable to future climates, the model parameters must be shown to be robust with respect to climate changes. In other words, the relationship between rainfall statistics and circulation indices must be time invariant (or stationary). Wilby (1997) examined the problem of ‘stationarity’ in UK precipitation time series and concluded that it may be necessary to use other atmospheric variables to reliably reproduce observed time series attributes in downscaling models. Other variables suggested include sea surface temperatures and larger scale atmospheric indices (e.g. the North Atlantic Oscillation Index). As this increased complexity is beyond the scope of the present models, it is necessary to assess how stable the present simple models are under the observed range of climate variability. To test the models’ performance the mean annual England and Wales precipitation was calculated using models P and Z for parts of the period for which circulation indices were available (1881–1993). These were then compared with the observed England and Wales precipitation series constructed by Gregory et al. (1991) and updated by Jones and Conway (1997). To allow for differences in the series averages (mainly due to the elevations of the sites used) the model series have been standardized to the observed 1961–1990 average and the 1961–1990 annual rainfall totals are shown for comparison in Figures 10–12. The period 1940 – 1960 was not used for validation because a different procedure was used for gridding mean sea level pressure (MSLP) from surface analyses in the UK region in the period 1940–1965. Although weather type analysis is generally unaffected, quantitatively different circulation indices are produced (Hulme and Jones, 1991). The procedure involved extra smoothing which has the effect of increasing the MSLP of depressions more than the corresponding decrease for anticyclones, which are generally less intense than depressions. This results in higher average MSLP, smaller values of positive vorticity with generally unchanged values of negative vorticity for this period. The smoothing in the period 1961–1965 will have influenced the regression results, but removal of this period from the regression would not allow the direct use of 1961–1990 climatologies in model validation. The effects of smoothing and the possibility of re-gridding the MSLP fields will be addressed in future work. © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 536 C.G. KILSBY ET AL. Figure 10. Sample distributions of monthly mean rainfalls (mean of period 1961 – 1990) The mean rainfall and correlation coefficients for the calibration period and a validation period from 1881 to 1939 are shown in Table IV. The models generally perform as well in the validation period as in the calibration period, and the mean rainfall for both models is within 5% of the observed value. Note also that, at least at the aggregated annual level shown here, model Z performs better than model P. This validation provides some confidence in the stability of the models under climate variability experienced to date. However, there can be no guarantee that the model parameters are stable to future climate changes. Improvement to the models’ physical basis is therefore desirable by the inclusion of other local variables, and is discussed further in Section 6. 5. FURTHER APPLICATIONS A potential application, which has motivated the work here, is to use the predicted point rainfall statistics to modify the parameters of a Neyman – Scott (N–S) stochastic model of precipitation (e.g. see Cowpertwait et al., 1996) used to produce daily and hourly time series. The proposed method is as follows. Figure 11. Sample distributions of monthly mean proportion of dry days (mean of period 1961 – 1990) © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 537 PREDICTING RAINFALL STATISTICS Figure 12. Annual time series of precipitation totals for England and Wales for the period 1961 – 1990 (i) The N–S model is first fitted to data taken from the present climate. If no site data are available at the catchment of interest, and the application is in the UK, then the regionalised Neyman–Scott models developed by Cowpertwait et al. (1996) or Cowpertwait and O’Connell (1997), can be used. (ii) Future values of the mean daily rainfall and the proportion of dry days are then predicted using the regression models together with future values of circulation variables derived from output from perturbed GCM integrations. (iii) Two N–S parameters are then modified using the predicted point rainfall statistics for future climates. (iv) Hourly rainfall time series are then simulated and used as input to a hydrological catchment model to assess the impact of the future climate on river flows. For larger catchments, or cases where there is significant spatial variability (for example due to orographic effects) a spatial-temporal N-S model (Cowpertwait, 1995) may be used to simulate multi-site hourly rainfall time series for input to the hydrological catchment model. Clearly, the method described above involves some broad assumptions. For example, it is necessary to assume that some of the N – S model parameter estimates remain constant in the future. Consideration of the rainfall model structure indicates that two parameters are directly related to changes in the mean rainfall and proportion of dry days: the rate of storm arrivals and the mean cell intensity. These parameters are therefore the most appropriate for refitting the model. Furthermore, if only a small change occurs in a N–S model parameter, when estimated at sites taken over a large geographical region containing significant climate variation, it is reasonable to suggest that this parameter estimate can be treated as approximately constant in the future as long as no major changes in the dominant precipitation mechanism occur. The regionalised N – S model (Cowpertwait et al., 1996) can be used to provide this evidence, supporting the choice of the parameters identified previously. Table IV. Mean annual rainfall and correlation coefficient (r) for England and Wales model and observed annual rainfall. The period 1961 – 1990 was used for development of the regression models; 1881–1939 is used as a validation period. Figures in parentheses have been standardized by the ratio of observed to model mean annual rainfall for the 1961 – 1990 period 1961–1990 Observed P Z 1881 – 1939 Mean (mm) r Mean (mm) r 915 1014 972 — 0.77 0.76 917 977 (881) 931 (876) — 0.77 0.89 © 1998 Royal Meteorological Society Int. J. Climatol. 18: 523 – 539 (1998) 538 C.G. KILSBY ET AL. A further point is that the method presented here is flexible enough to allow its use with transient GCM simulation output. Previous work in this area has used equilibrium GCM integrations, and the N–S model would have been fitted to statistics derived from the whole simulation period. The latest GCM integrations however are transient (Mitchell et al., 1995), and a number of complications arise. The N–S model may be re-parameterized and a simulation of a time series performed at any multiple of 12 months. At one extreme therefore, this allows the fitting of the model to circulation statistics taken from 1-year time windows of a transient GCM simulation, so that the inter-annual variability is derived from the GCM and direct correspondence with other GCM variables is maintained. At the other extreme, a longer time window may be used, which of course may not be stationary, and will result in inter-annual variability produced by the N – S model itself, and with no direct temporal correspondence with the GCM integration. 6. CONCLUSION AND DISCUSSION The regression models provide a simple and parsimonious method of predicting rainfall statistics across England and Wales, accounting for the effects of large scale circulation, and thus allowing their use in downscaling. The use of rather coarse resolution circulation data means that some important regional climatic effects may be ignored (e.g. Mayes, 1991). Given this limitation, and that very simple models have been formulated, it is not surprising that some discrepancies are evident in the validations. However, in some applications, these discrepancies may be unimportant. For example, if the models are used to predict expected values for each month (obtained by averaging all available data for each month in a long record), then the discrepancies in the sample standard deviation plots (Figure 4(b) and 5(b)) are not important. Even if the models are used to predict monthly time series, with under-prediction of rainfall variability by the models, the model variability could be increased by simulating monthly time series of point rainfall statistics. This could be achieved by adding the random variables o and j to the predicted values (i.e. using Equations (1) and (3) instead of the expected values in Equations (2) and (4)). It is obvious that other explanatory variables may be used in regression analyses such as presented here. As longer duration atmospheric data sets from observation and model re-analysis become available to replace the MSLP data set, it will be possible to improve the physical basis of these models. Such data sets include upper air circulation (e.g. at 750 hPa and 500 hPa), humidity and atmospheric stability. The use of spatial time series of sea-surface temperatures can also provide information on land-sea temperature contrasts which have important effects on rainfall generation in England and Wales. An increased physical basis will in turn allow more confidence in the simulated future rainfall statistics, particularly in regard to possible changes in the relationship between rainfall statistics and circulation. It is hoped that the use of such comprehensive and higher spatial resolution observed data sets will allow improvement on the regression models presented here. 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