Nonstationary models for exploring and mapping monthly precipitation in the United Kingdom

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 3: (21) Published online 16 March 29 in Wiley InterScience ( DOI: 1.12/joc.1892 Nonstationary models for exploring and mapping monthly precipitation in the United Kingdom C. D. Lloyd* School of Geography, Archaeology and Palaeoecology, Queen s University, Belfast BT7 1NN, Northern Ireland, UK. ABSTRACT: Spatial statistical algorithms are used widely for both the exploration and mapping of environmental variables such as precipitation amount. One limitation of standard approaches to characterization and spatial interpolation is the usual assumption of stationarity. In short, spatial variation is assumed to be constant across the region of interest. Much research effort has been extended in developing approaches that allow for local variation in spatial structure. Simple moving window approaches are examples of such developments. The purpose of this paper is to apply selected methods for exploring and mapping monthly precipitation amount in the UK. Global regression, moving window regression (MWR) and geographically weighted regression (GWR) are used to explore the relationship between altitude and precipitation amount. Inverse distance weighting (IDW) is used as a basic approach to spatial prediction. Global and local variogram models are estimated and modelled to assess variation in spatial structure of precipitation amount and to inform spatial prediction using ordinary kriging (OK), kriging with an external drift (KED) and simple kriging with local means (SKlm). In the latter case, local means are derived using global regression, MWR (using ordinary least squares (OLS) and generalized least squares (GLS)) and GWR. The importance of choice of algorithm for the estimation of the variogram and for spatial prediction is assessed. The benefits of local as against global approaches and multivariate as against univariate prediction procedures are considered. It is demonstrated that use of elevation data to inform the prediction process reduces prediction errors and that there is also a small reduction in prediction errors when local variogram models are used. In this study, KED based on local variogram models provides more accurate predictions (as judged by cross-validation statistics) than any other approach. The paper concludes by considering some key issues and possible avenues for future work. Copyright 29 Royal Meteorological Society KEY WORDS precipitation; nonstationarity; moving window regression; geographically weighted regression; kriging Received 24 December 27; Revised 14 January 29; Accepted 9 February Introduction There have been a very large number of studies that assess approaches for analysing spatial variation in meteorological variables such as precipitation amount. In addition, the mapping of precipitation amount from sparse samples (i.e. using discrete observations) has been extensively studied. The multiplicity of studies may be justified in that the most appropriate methods used are a function of several factors including location, topography, duration of observation and numerous other issues. As such, making generalized statements from any one study may be extremely problematic. This paper seeks to add to this body of literature by exploring a combination of issues that are of relevance to those concerned with exploring spatial variation in precipitation or making spatial predictions from sparse samples. The focus is original in that the particular concern is with the differences between results obtained using stationary models (those with parameters that are fixed) and nonstationary models (those with parameters which * Correspondence to: C. D. Lloyd, School of Geography, Archaeology and Palaeoecology, Queen s University, Belfast BT7 1NN, Northern Ireland, UK. c.lloyd@qub.ac.uk vary locally). The utility of various locally based (i.e. moving window) procedures for assessing relationships between variables and for characterizing spatial variation in single variables is also assessed and some potential advantages and disadvantages of such approaches are considered. A variety of methods for exploring variations in relations between variables have been developed and applied in many different contexts. Such approaches have been used to explore the relationship between precipitation amount and other variables. One major source of precipitation in the UK, as well as elsewhere, is atmospheric uplift by hills and mountains; this is termed orographic precipitation (Barrow and Hulme, 1997). Thus, altitude and precipitation amount tend to be related in many areas (at least for precipitation measured over periods of weeks or more) (Goovaerts, 2; Lloyd, 25). One concern in this analysis is to explore this relationship both globally and locally. In the latter case, moving window regression (MWR) and geographically weighted regression (GWR) are applied and assessed (Brunsdon et al., 21 used GWR to explore the relationship between elevation and average annual precipitation amount in Britain). Copyright 29 Royal Meteorological Society

2 MONTHLY PRECIPITATION IN THE UNITED KINGDOM 391 Many studies have compared approaches for generating maps of precipitation amount from rain gauge measurements (e.g., Bastin et al., 1984; Tabios and Salas, 1985; Creutin et al., 1988; Hevesi et al., 1992a; Hevesi et al., 1992b; Daly et al., 1994; Hutchinson 1995; Hay et al., 1998; Pardo-Igúzquiza, 1998; Prudhomme and Reed, 1999; Deraisme et al., 21; Gòmez-Hernàndez et al., 21; Nicolau et al., 22; Pardo-Igúzquiza et al., 25; Hancock and Hutchinson, 26). In previous studies, particular concerns have been with comparing commonly used interpolation procedures and assessing the impact on predictions of using secondary data sources. Use of elevation data has been shown to increase the accuracy of predictions of precipitation. Geostatistical prediction (kriging) has been used widely to generate maps of precipitation amount, as well as other meteorological variables. In various applications, kriging (of different forms) has been shown to provide more accurate predictions than other commonly used approaches such as inverse distance weighting (IDW) (Goovaerts, 2; Lloyd, 25). Kriging makes use of information about spatial structure in the variable of interest through a model fitted to the variogram, and uses the model coefficients to assign weights to observations, rather than using some arbitrary function of distance such as the inverse square of distance. One practical limitation of standard applications of kriging is that the variogram is estimated using all available data and it is assumed to represent spatial variation across the entire region of interest. To overcome this limitation, some form of nonstationary model, which has different parameters at different locations, is needed. The most straightforward approach is to estimate the variogram and fit a model using a moving window (Haas, 199). In this paper, such an approach is employed. The present paper builds on work presented by Lloyd (22, 25) who mapped monthly precipitation amount in Britain. In those studies, a variety of spatial prediction approaches were used, including inverse distance weighting (IDW), MWR, ordinary kriging (OK), simple kriging with locally varying means (SKlm) and kriging with an external drift (KED). In these cases, the variograms used for OK, SKlm and KED were assumed to be representative of spatial variation across the whole of Britain. In this paper, local (moving window) variograms are estimated and models fitted to explore geographical variation in the spatial structure of precipitation amount in the UK. The coefficients of the local variogram models are then used for prediction with OK, KED and SKlm. Many approaches exist for semiautomated and automated fitting of models to variograms. Computer code has been provided by several authors including Jian et al. (1996), Pardo-Igúzquiza (1997) and Pardo-Igúzquiza (1999) The methods used in this paper can be divided into those that make use of all data (global models), those that make use of some local data subset or a geographical weighting scheme (local models), those methods that only make use of a single variable and multivariate methods. The methods used include global regression of elevation against precipitation amount, MWR, GWR, IDW, OK, SKlm, moving window (local) variogram OK (LocOK), KED (LocKED) and SKlm (LocSKlm). In summary, the approaches used allow assessment of: 1. Benefits of local as against global approaches (MWR and GWR vs. global regression, local variogram models vs. global variogram models) for exploration and characterization of spatial variation and for prediction 2. Benefits of using a secondary variable to inform predictions (i.e. use of elevation data for global regression, MWR, GWR, KED and SKlm) The paper will, firstly, summarize the techniques applied. Secondly, the study area and the data used will be detailed. Following this, the analyses and the results will be presented and discussed. Finally, the main issues raised in terms of (1) exploring variation in spatial structure of precipitation amount and its relationship with altitude and (2) use of altitude data and local procedures for spatial prediction will be summarized. 2. Techniques The approaches used in this paper are all defined in easily accessible sources, but they are summarized briefly and appropriate references are given. This section presents in turn techniques for local regression, variogram estimation and modelling and spatial prediction Regression The simplest approach for exploring the relationship between altitude and precipitation is to conduct a global regression. Lloyd (25) presents such an analysis as context for analysis using a local (moving window) regression procedure. The MWR procedure used by Lloyd (25) is conducted as follows: a location is visited, the nearest n neighbours to that location are selected and regression of altitude against precipitation amount is conducted using that local data subset. The local regression coefficients are then stored for that location. Once all locations are visited in the same way there are regression coefficients for each data location and these can be mapped. This approach can be extended by weighting locations according to the proximity to a given observation. Usually, all data are used (not just a local subset) but the kernel size is selected such that only local observations have nonnegligible weights. GWR is detailed by Fotheringham et al. (22). The application of GWR for exploring spatial variation in the relation between elevation and monthly precipitation amount is illustrated by Lloyd (26). Brunsdon et al. (21) used GWR to explore the average altitude-precipitation relationship across Britain. In that study, marked spatial variations in the GWR coefficients were observed. GWR is also used in the present paper. To use GWR it is necessary

3 392 C. D. LLOYD to select some form of kernel (i.e. geographical weighting scheme) and a bandwidth, which determines the size of the kernel. The GWR software (Fotheringham et al. 22) allows selection of an optimal bandwidth according to cross-validation scores and the Akaike Information Criterion (AIC). In this analysis an adaptive bandwidth (i.e. a fixed number of nearest neighbours (NN), with a small bandwidth in areas which are densely sampled and a large bandwidth in areas which are sparsely sampled) was selected using the AIC. The bi-square function (Fotheringham et al. 22) was used for geographical weighting Variograms The variogram characterizes the average degree of spatial dependence between samples as a function of the distance and direction separating the samples. In simple terms, the variogram offers a means of characterizing spatial variation in terms of the magnitude of variation and the frequency or range of spatial variation. Where precipitation amount varies over small areas, this can be termed short-range variation. In contrast, if precipitation amounts vary only over comparatively large areas then the spatial variation may be termed long-range variation. Global variograms were estimated and models fitted using Gstat (Pebesma and Wesseling 1998; Pebesma, 24). Directional variograms were also estimated and the anisotropy modelled as geometric (Deutsch and Journel, 1998). It is well known that when a large-scale trend dominates variability this swamps the variogram (Kitanidis, 1997). For this reason (and for the application of SKlm, as described below) variograms were estimated in Gstat from ordinary least squares (OLS) and generalized least squares (GLS) residuals given a global regression of elevation and precipitation, and thus elevation is considered to explain the average variation in precipitation amount. Variograms estimated from OLS residuals are recognized to be biased and GLS has been proposed as a solution (e.g. Hengl et al., 27); Cressie (1993) that provides an in-depth discussion about this topic. The interactive modelling procedure employed in Gstat was used as it enables the fitting of multiple structures (for which there is strong evidence in the global variograms estimated in this study), and this would be difficult to do automatically. In cases where the variogram does not represent well spatial variation across the whole of the region of interest some approach may be necessary to account for the change in spatial variation locally. In the geostatistical literature, there are several methods presented for estimation of nonstationary variograms. These vary from approaches that estimate and model automatically the variogram in a moving window (this approach is discussed below) to approaches that transform the data such that the transformed data have a stationary variogram. Reviews of some methods are provided by Sampson et al. (21) and Schabenberger and Gotway (25). The estimation and automated modelling of local (moving window) variograms for kriging is one published approach that accounts for nonstationarity in the variogram (Haas, 199) and that approach is employed here. The maximum likelihood (ML) functionality of the MLREML routine provided by Pardo-Igúzquiza (1997) was used to derive variograms models in a moving window. For the application of LocKED and LocSKlm, MLREML was used to determine variogram model coefficients with ML estimation and local regression coefficients through GLS. In MLREML, the simplex method for function minimization (Pardo-Igúzquiza, 1997) was selected. The use of ML for variogram model fitting has been discussed by several authors (e.g., McBratney and Webster 1986; Pardo-Igúzquiza 1997). Fortran 77 code was written to visit each observation in the precipitation dataset and call the MLREML routine to estimate the variogram using the n nearest neighbours (NN; with all data pairs in this subset included in variogram estimation) to each observation with the result that there are as many sets of variogram model coefficients as there are observations in the dataset. For SKlm and LocSKlm, several different regression approaches were used to derive trend-free variograms (where the trend is a function of elevation), as summarized in Table I. The simplest approach is to conduct OLS regression locally and work with the residuals, and that approach, amongst the others detailed in Table I, is applied here Spatial prediction Spatial prediction was conducted using three forms of regression: global regression, MWR and GWR. In all cases, precipitation amount was the dependent variable Table I. Regression and models and variograms used for kriging. All regressions refer to elevation against precipitation amount. Method Local means (or external drift) regression approach Variogram OK Global raw LocOK Local (128 NN) ML LocKED Local (128 NN) Local (128 NN) ML GLS SKlm1 Global OLS Global OLS residuals SKlm1a Global GLS Global GLS residuals SKlm2 Local (128 NN) OLS Global using local OLS regression residuals SKlm3 Local OLS GWR Global using local OLS GWR regression residuals LocSKlm1 Local (128 NN) OLS Local (128NN) ML given OLS regression residuals LocSKlm1a Local (128 NN) Local (128 NN) ML GLS LocSKlm2 Local OLS GWR Local (128NN) ML given OLS GWR regression residuals

4 MONTHLY PRECIPITATION IN THE UNITED KINGDOM 393 and altitude was the independent variable. In the case of global regression and GWR, predictions were made using the regression coefficients obtained using all of the data (local coefficients in the case of GWR). In the case of MWR, predictions were made using local regression coefficients with cross-validation. That is, the coefficients used to predict at location i were obtained after the values at that location were extracted (temporarily) from the data set. Experimentation indicated that including or excluding the paired observations at location i for regression prediction made little difference for MWR and consequently had minimal impact on variograms estimated from regression residuals or on local means for SKlm and LocSKlm (see below for more on this topic). MWR was conducted using purpose-written Fortran 77 code while GWR was conducted using the package developed by Fotheringham et al. (22). IDW was applied as it is such a widely used spatial interpolation procedure (see, e.g. Burrough and McDonnell, 1998; Lloyd, 26, for summary and examples). Fortran 77 code was written to perform IDW and crossvalidation was used as a means of assessing the performance of the technique. OK, the core interpolation approach in geostatistics, was applied because of its widespread use in many contexts (introductions are provided by Armstrong, 1998 and Burrough and McDonnell, 1998; more in-depth accounts are provided by Journel and Huijbregts, 1978; Isaaks and Srivastava, 1989; Cressie, 1993; Goovaerts, 1997; Chilès and Delfiner, 1999; Lloyd, 26; Webster and Oliver, 27.) OK was conducted using the geostatistical software library (GSLIB) routine kt3d (Deutsch and Journel, 1998). With OK the mean of the values is estimated in a moving window. KED provides a means of integrating secondary data into the prediction process. With KED, a secondary variable is used which describes the average shape of the primary variable (Wackernagel, 23) and the trend-free variogram is required (here, the trend being modelled as a function of elevation). With simple kriging, the mean needs to be known prior to the interpolation for all target locations. Where the local mean is estimated as a prior stage, this is known as SKlm (Goovaerts, 1997; Lloyd, 26). In this case SKlm was applied with the local means estimated using global OLS regression and global GLS regression, OLS MWR, GWR and GLS MWR (using MLREML) (see Table I). That is, precipitation amount was predicted using these five approaches and the predicted value taken as the local mean for each location. In this analysis, only one secondary variable is used (i.e. elevation) but a particular strength of SKlm is that complex regression models, which make use of many variables could be utilized to generate local means. Like OK, SKlm was conducted using the GSLIB routine kt3d (Deutsch and Journel, 1998). The variograms used in each case are detailed in Table I, but, for clarity, they are summarized here. OK was conducted using global raw variograms (i.e. with no attempt to account for the elevation-precipitation trend). LocOK was based on variogram model parameters derived through ML using the MLREML routine written by Pardo-Igúzquiza (1997). SKlm was conducted using variograms estimated from residuals obtained using several regression approaches as SKlm is dependent on the regression method both for deriving the local mean and for obtaining the trend-free variogram (estimated from the regression residuals). These were (1) OLS global regression, (2) GLS global regression (using Gstat), (3) OLS MWR and (4) OLS GWR. LocSKlm was conducted using local variograms estimated from (1) OLS MWR, (2) OLS GWR and using (3) local variogram model parameters derived using ML. Hengl (27) stresses the inconsistency when the variogram of a global trend is used for KED with a local prediction neighbourhood, whereby the trend is modelled locally. Lloyd (25) used the variogram in the direction with least trend for KED, but a more satisfactory solution is to use local variograms so that the variogram is based on local (altitudeprecipitation) regression residuals and KED is conducted and the trend modelled locally also. The resulting procedure is termed LocKED and made use of local variogram model parameters derived using ML. The routine kt3d was modified for LocOK, LocKED and LocSKlm to enable the use of locally estimated and modelled variograms. In the local variogram case, a single structured component was sufficient in randomly selected examples, and so a local modelling procedure was practical (unlike the case for the global variograms, where more than one structured component was fitted). In summary, SKlm (with means derived using the five regression approaches detailed above) makes use of a secondary variable, LocOK accounts for variation in spatial structure and LocKED and LocSKlm combine information in a secondary variable and account for local variation in spatial structure. 3. Study area and data The data used in the analysis were ground data measured across the UK under the auspices of the UK Meteorological Office (UKMO) as part of the national rain gauge network. The Isle of Man (not a part of the UK, but rather a Crown dependency) was also included. The data were obtained from the British Atmospheric Data Centre (BADC) web site ( Data for Northern Ireland are referenced using the Irish Grid coordinate system. To analyse these data simultaneously with those from Britain and The Isle of Man it was necessary to convert the coordinates into British Grid coordinates and this was done using ArcGIS. The analysis makes use of data for January and July 26. These two months were selected to allow for assessment of seasonal variation. The observations for January and July are shown in Figure 1 (maps of observations appear almost identical for both months, so the two were combined). For prediction purposes in some areas, for a given number of NN, the most distant neighbouring observations may

5 394 C. D. LLOYD Elevation (m) High : 1326 Low : 1 2 Km Figure 1. Precipitation observations for January and July 26. be a very large distance away (and may be in areas with quite different characteristics) this is particularly true in the case of the islands of Scotland and the Isle of Man. The impact of this factor is assessed through varying the number of NN used in prediction. Summary statistics for both sets of data are given in Table II. As expected, the minimum, maximum and mean are larger for January than for July. In addition, that there is more variation in precipitation amounts in January than in July is indicated by the standard deviations. Given that the relationship between elevation and precipitation amount is explored in this paper, elevation for the UK is given in Figure 2 (the digital elevation model has a spatial resolution of m). Maps of precipitation amount generated using IDW with 16 NN are shown in Figures 3 (January) and 4 (July). The major trends in precipitation amount evident in Figures 3 and 4 are consistent with the descriptions by Barrow and Hulme (1997). The distinction between Table II. Monthly precipitation amount summary statistics. Month January July Number Minimum Maximum (mm) (mm) Copyright 29 Royal Meteorological Society Mean (mm) Standard deviation (mm) Km Figure 2. Elevation in the UK. Data available from U.S. Geological Survey/EROS, Sioux Falls, SD. the precipitation amounts in the east and west of Scotland can be explained by the fact that more active and frequent frontal systems cross Scotland from the west to east and this combines with the influence of mountains and hills resulting in much larger precipitation amounts in the west than in the east (Barrow and Hulme, 1997). In parts of central and eastern England, summer precipitation may make the largest contribution to the annual total precipitation (Barrow and Hulme, 1997) and so the internal relative differences in the maps for January and July in these regions (as well as elsewhere) are not surprising. 4. Analysis The first stage of the analysis entailed exploring the relationship between elevation and precipitation amount for January and July. Figure 5 shows the scatterplot, with the OLS regression line for January while Figure 6 shows the same information for July. The results are similar to those obtained by Lloyd (25) in an analysis of data for 1999, in that the relationship between the two variables is quite weak when all of the data are used (i.e. the regression is global). MWR was used to assess the relation between elevation and precipitation amount in a moving window. In the remainder of the paper, in the interests of space, maps and graphs are given only Int. J. Climatol. 3: (21)

6 MONTHLY PRECIPITATION IN THE UNITED KINGDOM Precipitation (mm) High : 476 Precipitation (mm) High : 221 Low : 2 Low : Km 1 2 Km Figure 3. Precipitation amount for January 26: IDW with 16 NN. Figure 4. Precipitation amount for July 26: IDW with 16 NN. for July. The MWR intercept, slope and coefficient of determination for 128 NN for July are given in Figures 7, 8 and 9. The results for GWR (not shown) were broadly similar, although as would be expected there were small differences in some regions. The omnidirectional variogram for July precipitation is shown in Figure 1. A nugget effect and two spherical components were fitted to the experimental variogram (see McBratney and Webster, 1986, for a definition of the spherical model). The variogram model for January (not shown) has a larger total sill than that for July, indicating that there is greater variation in precipitation amounts for January than for July. The sill of the January variogram is reached at a larger distance than is the case for the July variogram. This suggests that precipitation amounts were more spatially continuous in January than in July. A nugget effect was fitted to the July variogram and not to the January variogram, suggesting that, in July, there was a greater amount of spatial variation in precipitation amount over very short distances than in January. The July IDW-derived map of precipitation amount (Figure 4) clearly demonstrates greater short-range spatial variability than does the map for January (Figure 3), thus supporting the conclusions reached through assessing the variogram model coefficients. Directional variograms were also estimated and the anisotropy was modelled as geometric, with the resulting models used for OK. Variograms were estimated using residuals from a global regression of elevation and precipitation. The variogram of global residuals for July in shown in Figure 11(a) (OLS residuals). The variogram has a slighter smaller sill than that estimated from the raw data, but the two have very similar forms. This is not surprising as the global regressions explain such a small proportion of the variation (January r 2 =.1347; July r 2 =.1196). The variogram for July estimated from GLS residuals [Figure 11(b)]) is similar in structure. Variograms were also estimated from residuals from MWR using OLS for 128 observation neighbourhoods. That for July is shown in Figure 12. The variograms of MWR residuals for both January and July were fitted with a nugget effect and a single spherical component. As expected, these variograms are very different in form than those estimated from the raw data or the residuals from global regression as so much of the variation is explained by local regression at some locations (see Figure 9, the mapped local coefficients of determination). The July variogram of GWR residuals was similar to that derived using MWR. Variograms of GWR residuals for both months have slightly smaller sills than those estimated from MWR residuals suggesting that the GWR model explains more variation. The MLREML routine used to estimate and model local variograms allows the fitting of several different models, but in this analysis a nugget effect and a spherical Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (21)

7 396 C. D. LLOYD model was fitted to all of the variograms estimated from raw data and from MWR and GWR residuals. Visual inspection of local variograms indicated that a spherical model was an appropriate choice in most areas considered. The nugget effects for variograms estimated from 128 NN for July are shown in Figure 13, the spherical model structured components are shown in Figure 14 and the ranges in Figure 15 (note that the values are expressed as integers for ease of visualization). The spherical model range for variograms estimated from MWR residuals (using OLS, for 128 locations) using 128 NN for July is shown in Figure 16. For local variograms of MWR residuals, a small number of models did not converge to a fit and the valid model at the nearest location was used in those cases. Global regression predictions and cross-validation prediction errors for MWR, GWR, IDW, OK, SKlm, LocOK, LocKED and LocSKlm are given in Tables III (January) and IV (July). It is clear from a brief examination of the tables that global regression, MWR, GWR and IDW provide less accurate cross-validation predictions than the kriging-based approaches. Results for SKlm and LocSKlm where the means were derived using different regression approaches based on OLS regression (the SKlm approaches in Tables III and IV being based on GLS regression) are summarized in Tables V (January) and VI (July). The results are discussed below. Precipitation (mm) y =.1773x r 2 = Elevation (m) Figure 5. Elevation against precipitation for January Results and Discussion The global regressions presented in Figures 5 and 6 clearly indicate that there is much variation in the relationship between altitude and precipitation amount, with r 2 values of.1347 for January and.1196 for July. As such, application of a local model seems sensible. The MWR intercepts for July (Figure 7) are largest in Northern Ireland, western Scotland, part of Northwest England and part of Southwest England. In these areas, the MWR intercepts are larger than the global intercept (34.17 mm). The slope values for July (Figure 8) are largest in Northern Ireland, southern Scotland, Northeastern England, mid-eastern England, Northwest Wales and Southwest England. A notable feature of the map of local Table III. January precipitation cross-validation prediction error summary statistics; n = 377. See Table I for kriging method definitions. Method Number of nearest neighbours (NNN) Maximum negative error (mm) Maximum positive error (mm) Mean error (mm) RMSE (mm) IDW IDW IDW IDW IDW Global regression All MWR MWR MWR MWR MWR GWR OK OK OK OK OK LocOK LocOK LocKED LocKED SKlm1a SKlm1a LocSKlm1a LocSKlm1a

8 MONTHLY PRECIPITATION IN THE UNITED KINGDOM 397 Table IV. July precipitation cross-validation prediction error summary statistics; n = See Table I for kriging method definitions. Method Number of nearest neighbours (NNN) Maximum negative error (mm) Maximum positive error (mm) Mean error (mm) RMSE (mm) IDW IDW IDW IDW IDW Global regression All MWR MWR MWR MWR MWR GWR OK OK OK OK OK LocOK LocOK LocKED LocKED SKlm1a SKlm1a LocSKlm1a LocSKlm1a Table V. January precipitation SKlm and LocSKlm cross-validation prediction error summary statistics for different regression approaches; n = 377. See Table I for method definitions. Method Number of nearest neighbours (NNN) Maximum negative error (mm) Maximum positive error (mm) Mean error (mm) RMSE (mm) SKlm SKlm SKlm SKlm SKlm SKlm LocSKlm LocSKlm LocSKlm LocSKlm Precipitation (mm) y =.892x r 2 = Elevation (m) Figure 6. Elevation against precipitation for July 26. slopes is the large values in parts of mid-eastern England. The elevation values in this region are small, as are the r 2 values (Figure 9), and this suggests that the slope is not a reliable indicator of the altitude-precipitation relationship in this case. The relationships are strongest in Northern Ireland and Southwest England and some other areas as indicated by the map of r 2 values. Lloyd (25) applies MWR to explore spatial variation in the relationship between altitude and precipitation in July There are some similar patterns, although there are quite marked differences between the maps presented in that study and those presented here. As the summary statistics for the

9 398 C. D. LLOYD Table VI. July precipitation SKlm and LocSKlm cross-validation prediction error summary statistics for different regression approaches; n = 377. See Table I for method definitions. Method Number of nearest neighbours (NNN) Maximum negative error (mm) Maximum positive error (mm) Mean error (mm) RMSE (mm) SKlm SKlm SKlm SKlm SKlm SKlm LocSKlm LocSKlm LocSKlm LocSKlm Intercept Slope Km 1 2 Km Figure 7. MWR, elevation against precipitation for July 26, 128 observation neighbourhood: intercept. Figure 8. MWR, elevation against precipitation for July 26, 128 observation neighbourhood: slope. two studies suggest, precipitation patterns were quite different in July 1999 (minimum =. mm, maximum = 319. mm,mean = mm and standard deviation = mm) than in July 26 (minimum = 2. mm, maximum = 224. mm, mean = mm and standard deviation = mm). The most obvious areas with a combination of large slope values and large r 2 values are Northern Ireland and Southwest England; in these regions precipitation amount clearly has a strong linear relationship with elevation. Brunsdon et al. (21) explore the GWR intercept for annual average total precipitation (for ) and elevation in Britain. The authors consider the relationship between precipitation amounts in low lying coastal areas and the GWR intercepts and note a mismatch in some areas; similar variations can be observed in the present case. That there was greater variation in precipitation amounts in January than in July is indicated by the sills of the variogram models (that for July is in Figure 1). However, the range coefficients suggest that precipitation amounts were more spatially continuous in January than in July. As noted earlier, the variograms of residuals from global regression [Figure 11(a) for OLS and 11(b) for GLS, for July] are very similar to those estimated

10 MONTHLY PRECIPITATION IN THE UNITED KINGDOM 399 r 2 squared Km Figure 9. MWR, elevation against precipitation for July 26, 128 observation neighbourhood: coefficient of determination. from the raw data. The models fitted to the July variograms of residuals from MWR (Figure 12) and from GWR (not shown) have very short ranges, indicating that much of the structured variation in precipitation amount is explained by its local relationship with altitude. The MWR and GWR residual variogram models for July have shorter ranges and larger relative nugget effects than those for January, supporting the assertion that precipitation amount varied more over short distances in July than in January. The moving window variogram model results for July are given in Figures 13 (nugget effects), 14 (spherical model structured components) and 15 (ranges). The nugget effects tend to be largest in western Scotland and Northwest England, with some large values in the southern English midlands and the Southwest. The spherical model structured components are large in western Scotland, parts of the east coast of southern Scotland and England and some other areas. The ranges are largest in western Scotland, parts of Southeastern Scotland and Northeastern England with small areas of large values elsewhere. Large ranges suggest areas with continuous spatial variation over large areas. Some of the observed variation is undoubtedly due to poor model fits, as discussed below. The variogram ranges estimated from MWR residuals for July are given in Figure 16. Values are largest in northern Scotland, with a few scattered areas of large values elsewhere. The remainder of this section focuses on the crossvalidation prediction errors. For January (see Table III), the smallest maximum absolute cross-validation errors are for LocKED with 64 NN ( mm), followed by SKlm2 with 64 NN ( mm) and SKlm2 with 128 NN (148.31). The mean errors closest to zero are for global regression (.6), followed by MWR with 64 NN (.5 mm) and LocKED with 128 NN (.62 mm). The smallest root mean square error (RMSE) is for LocKED with 128 NN ( mm), and the next smallest is for LocKED with 64 NN (13.15 mm) followed by SKlm3 with 128 NN (13.75 mm). Judging by the maximum absolute errors and the RMSE, the use of elevation data for informing prediction is beneficial in that the magnitude of the errors is smaller than when elevation data are not used. In this case, there is a clear 8 7 Semivariance (mm 2 ) Semivariance Nug () Sph ( ) Sph (13498) Distance (m) Figure 1. Monthly precipitation for July 26: omnidirectional variogram.

11 4 C. D. LLOYD (a) 7 6 Semivariance (mm 2 ) Semivariance Nug () Sph (1989) Sph (13782) Distance (m) (b) 7 6 Semivariance (mm 2 ) Semivariance Nug () Sph (2484.7) Sph (1388) Distance (m) Figure 11. Global regression residuals, monthly precipitation for July 26: omnidirectional variogram. (a) OLS, (b) GLS Semivariance (mm 2 ) Semivariance Nug () Sph (2362.3) Distance (m) Figure 12. MWR residuals for 128 NN, monthly precipitation for July 26: omnidirectional variogram.

12 MONTHLY PRECIPITATION IN THE UNITED KINGDOM 41 c c Km 1 2 Km Figure 13. Nugget effect for July 26, 128 observation neighbourhood. proportional increase in prediction accuracy when using locally computed and modelled variograms. Anisotropy was taken into account for OK and this resulted in a small decrease in the RMSE. In the omnidirectional model case mm (for 8 NN) was the smallest RMSE, whereas, for the directional model case, the smallest RMSE was mm (for 16 NN). Directional models were not used for SKlm and KED as, in these cases, directional variation is assumed to be represented by the trend. Comparison of the SKlm and LocSKlm results in Tables III and V indicates that global regression based on OLS (Table V) and GLS (Table III) and local regression based on OLS (Table V) and GLS (Table III) provide similar results, although in some cases the cross-validation errors for OLS-based SKlm are smaller than those for the more complex procedures. Nonetheless, GLS is conceptually more appropriate than OLS regression in this application. For July (see Tables IV and VI, the latter showing SKlm and LocSKlm results for different regression procedures), the three smallest maximum absolute cross-validation errors are for SKlm3 with 128 NN ( mm; Table VI), SKlm3 with 64 NN (84.72 mm; Table VI) and SKlm2 with 64 NN (86.78 mm; again, Table VI). The mean errors closest to zero are for global regression (.4 mm), SKlm1 with 128 NN (.41 mm; Table VI) and SKlm1a with 128 NN (.42 mm). The smallest RMSE values are for LocKED with 128 NN ( mm), followed by LocKED with Figure 14. Spherical model structured component for July 26, 128 observation neighbourhood. 64 NN ( mm) and LocSKlm2 with 128 NN ( mm). For OK, in the omnidirectional model case mm (for 8 NN) was the smallest RMSE. In the directional model case, the smallest RMSE was mm (for 16 NN); thus, accounting for directional variation resulted in a small increase in the RMSE. For January, the absolute difference between the smallest RMSE for a standard global method ( mm for OK with 16 NN) and for a method that makes used of elevation data and locally estimated variograms ( mm for LocKED with 128 NN) is some 6.5% of the larger of the two RMSE values. For July, the equivalent for a standard global method (13.27 mm for OK with 16 NN) and for a method that makes used of elevation data and locally estimated variograms ( mm for LocKED with 128 NN) is some 5% of the larger of the two RMSE values. For both January and July, the advantages of both making use of elevation data through local regression and computing and modelling variograms locally are apparent, if not marked. In terms of the RMSE, the values for LocKED are not much smaller than those for other approaches, but it is notable that this approach is fully automatable. Of the approaches that do not make use of elevation data, for January, the smallest RMSE is for LocOK with 128 NN ( mm) while for July the smallest RMSE is for LocOK with 128 NN (13.2 mm). Thus, there are small increases in

13 42 C. D. LLOYD a a Km 1 2 Km Figure 15. Spherical model range for July 26, 128 observation neighbourhood. Figure 16. Spherical model range for July 26, 128 observation neighbourhood, residuals from MWR for 128 NN. prediction accuracy when local variogram estimation and modelling is applied, but the use of secondary data has a greater impact. The differences between results for LocOK, LocKED and LocSKlm may be due largely to the strength of the local relationship between altitude and precipitation but an additional factor may be the complexity of the local variograms in the two sets of cases. That is, the global variograms of raw data were fitted with two spherical components whereas those of MWR and GWR residuals were fitted with only one spherical component and were simpler in form than those computed from the raw data. The models fitted automatically to the local variograms comprised only a nugget effect and one spherical component. If the variograms of residuals are simpler in form then it seems likely that the locally fitted model will represent the variograms of residuals better than in the case of the local variograms of raw data. Examination of some local variograms of raw data indicated that, given the extent of the areas over which variograms were estimated, some variograms were dominated by a large-range trend in precipitation amount and a power model, rather than a spherical model, would have been more appropriate. In the cases examined, the local variograms of MWR residuals were not subject to the same problem, as accounting for elevation removed (at least partially) the local trends in values. For both months, global regression provides the least biased predictions, but by other measures (maximum negative predictions and RMSE), it does not perform well. The very large maximum negative prediction values are a function of the fact that the regression lines are pulled towards the more numerous small precipitation amounts and, therefore, some larger values are markedly underpredicted. Similarly, for MWR and GWR, the largest negative prediction errors for large prediction neighbourhoods tend to be larger than the values for other approaches. As expected, there are greater gains in prediction accuracy for KED and SKlm over OK at locations where the local elevation-precipitation relation is strong (i.e. r 2 is large). As an example, the relationship between SKlm absolute prediction errors and r 2 values is weak, but there are few large errors at locations where the r 2 value is greater than.5. For January SKlm with local means derived using OLS MWR (i.e., SKlm2), only 2 out of 42 cross-validation errors greater than 5 mm are for locations with an MWR r 2 of greater than.25 (where 1534 out of a total of 377 locations have r 2 >.25). For July SKlm with local means derived using MWR (SKlm2), only three out of 22 cross-validation errors greater than 5 mm are for locations with an MWR r 2 of greater than.25 (where 696 out of a total of 2924 locations have r 2 >.25).

14 MONTHLY PRECIPITATION IN THE UNITED KINGDOM 43 It might be expected that nonstationary models would provide more accurate predictions than stationary models if there is marked spatial variation, as is the case for precipitation amount and its relationship with altitude. For nonstationary models to provide more accurate predictions, the model must represent spatial variation well (or at least better than the global model). Clearly, local regression (whether MWR or GWR) accounts for variation in the relation between altitude and precipitation that is obscured by a global regression. In terms of local variograms, the results suggest that the local models fit the variograms well in most (or at least many) cases as the prediction errors are smaller when local automatically fitted variogram models are used than when the semiautomatically fitted global variogram models are used. As an additional test, a jackknifing procedure (Deutsch and Journel 1998) was used. The dataset was divided into parts. For both months, 7% of the data were used for prediction to 3% of the locations (for January this resulted in a 237/77 split and for July the split was 2193/731). Predictions were made using OK16 and LocKED128. For OK, the variograms were estimated from the 7% subsets while for LocKED the local variograms were estimated at the 3% subset locations given the 128 nearest observations in the 7% subsets. For January, the OK RMSE was while for LocKED it was For July, the OK RMSE was and for LocKED it was The results support those obtained using cross-validation in demonstrating large differences between the standard approach of OK and LocKED, whereby elevation data are utilized and local variograms estimated. However, note that the RMSEs are larger for July than January (in contrast with those for cross-validation), which is largely a function of the data selected for the 7% sample and the 3% prediction location sample in each case. 6. Conclusions This paper has demonstrated how MWR and GWR can be used to explore spatial variation in the relations between altitude and monthly precipitation amount. In addition, the potential benefits of using these approaches to inform spatial prediction have been illustrated. Finally, local variograms were estimated and automatically modelled. The local model coefficients were interpreted and used for prediction. It is clear from the results that global procedures mask extensive local variation and that, in addition, by using local models the accuracy of predictions can be increased. However, for many applications, kriging based on global variograms may be considered to provide predictions at an acceptable level of accuracy. Also, use of covariates such as elevation may not be considered necessary. Although approaches which do make use of secondary variables may represent the precipitation structure more accurately and this may, for some applications at least, be more important than accuracy of individual predictions. Radar precipitation data, when used in conjunction with rain gauge data, offer great potential (see Creutin et al., 1988 for a relevant study) and the methods used here can easily be used to integrate such data. Clearly, the sample density has a major impact on potential differences in predictions made using different approaches. As Lloyd and Atkinson (22) showed in a different context (mapping of elevations) differences in results generally increase as the average distance between samples increases. Therefore, in the present context, the greatest differences in results obtained using different approaches would be expected in areas where the sample density is lowest. Whether these differences are large enough to justify the use of more complex and time consuming methodologies (e.g., local variograms for kriging) will be an issue for individual researchers, but the present analysis at least suggests some ways in which prediction accuracy may be increased. An additional issue concerns the temporal resolution of the data with monthly measurements there will be less spatial variability than with measurements over shorter time periods and, in the latter case, differences in results obtained between interpolation techniques are likely to be larger. Therefore, altering the periods of observation (i.e., using data for shorter durations) would be very worthwhile to demonstrate the transferability of approaches used to other contexts. In terms of the MWR, GWR and local variogram model outputs, there are many characteristics that could be explored further. Seasonal and spatial variations in the relationship between altitude and precipitation amount are complex, as are variations in the spatial structure of precipitation amount. Methods like those applied in this paper offer a powerful means of exploring these issues and developing understanding of them. The local modelling of variograms as applied here is prohibitively slow (approximately 24 hours to estimate and fit models to 3 variograms on a 2.8 GHz Pentium 4 with 512 MB of RAM). This could certainly be improved upon and other fitting procedures are much less computationally demanding. In any case, estimating and modelling the variogram for all observations is probably excessive. Instead, the variograms could be estimated and modelled on a coarse grid and distance-weighted variogram model coefficients used for prediction at a particular location if that location does not correspond with a node of the grid. Alternatively, if the range of the variogram is not required then a linear model could be fitted to the first few lags with obvious advantages in terms of processing time. An additional area for future exploration concerns the estimation of variograms in the presence of trend (external in this case). In addition to ML as used in this analysis, the MLREML routine also offers the capacity to estimate trend coefficients and determine variogram model coefficients using restricted maximum likelihood (REML) and the potential benefits of REML over ML are discussed by Pardo-Igúzquiza (1997). Lloyd (25) used the variogram for the direction which showed least evidence of trend, and this is a further approach which could be explored.