A comparison of spatial interpolation methods to estimate continuous wind speed surfaces using irregularly distributed data from England and Wales

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 28: (2008) Published online 2 August 2007 in Wiley InterScience ( A comparison of spatial interpolation methods to estimate continuous wind speed surfaces using irregularly distributed data from England and Wales W. Luo, M. C. Taylor and S. R. Parker Agri-Environment & Crop Management, Central Science Laboratory, Sand Hutton, York, YO41 1LZ, UK ABSTRACT: Seven methods of spatial interpolation were compared to determine their suitability for estimating daily mean wind speed surfaces, from data recorded at nearly 190 locations across England and Wales. The eventual purpose of producing such surfaces is to help estimate the daily spread of pathogens causing crop diseases as they move across regions. The interpolation techniques included four deterministic and three geostatistical methods. Quantitative assessment of the continuous surfaces showed that there was a large difference between the accuracy of the seven interpolation methods and that the geostatistical methods were superior to deterministic methods. Further analyses, testing the reliability of the results, showed that measurement accuracy, density, distribution and spatial variability had a substantial influence on the accuracy of the interpolation methods. Independent wind speed data from ten other dates were used to confirm the robustness of the best interpolation methods. Crown copyright Reproduced with the permission of Her Majesty s Stationery Office. KEY WORDS spatial interpolation; geostatistics; wind speed; weather stations Received 24 April 2006; Revised 12 March 2007; Accepted 27 May Introduction Weather data are generally recorded at point locations, so estimating data values at other locations requires some form of spatial interpolation. A variety of deterministic and geostatistical interpolation methods are available to estimate variables at unsampled locations but, depending on the spatial attributes of the data, accuracies vary widely among methods. The final use of any interpolated variable surface must also be taken into account because different methods result in different surfaces (Willmott, 1984). Spatial interpolation is more worthwhile if a sufficient density of weather stations is available across the study area. The density of the network required depends upon the variable to be estimated. Wind speed, for example, is more variable over shorter distances than temperature or relative humidity, and hence would be expected to require a more dense network of monitoring sites to achieve accurate and precise interpolated surfaces. The UK Meteorological Office (UKMO) operates many weather monitoring stations for the purpose of producing localized forecasts and to inform long-term climate change research. While there have been comparisons of interpolation methods for temperature and precipitation, (Phillips et al., 1992; Collins and Bolstad, 1996; Goovaerts, 2000; Price Correspondence to: M. C. Taylor, Agri-Environment & Crop Management, Central Science Laboratory, Sand Hutton, York, YO41 1LZ, UK. moray.taylor@csl.gov.uk et al., 2000; Jarvis and Stuart, 2001; Vicente-Serrano et al., 2003) few research efforts have been directed towards comparing the effectiveness of different spatial interpolators in predicting wind speed. In this study seven spatial interpolation techniques trend surface analysis (TSA), inverse distance weighting (IDW), local polynomial (LP), thin plate spline (TPS), ordinary kriging, universal kriging and ordinary cokriging were compared. The purpose was to determine which method created the best representation of reality for wind speed data recorded across England and Wales. Estimates were tested against data collected independently and the benefits and limitations of these methods are discussed. In our field of interest, there is a need to estimate the risks posed by the spread of existing and invading non-indigenous pathogens or insects across the landscape. For example, sudden oak death (causal pathogen Phytophthora ramorum), the Horse Chestnut leaf miner (Cameraria ohridella), Colorado Potato Beetle (Leptinotarsa decemlineata) and the peach-potato aphid (Myzus persicae) have a distribution that can be altered by wind speed events. Studies on the risks posed by these pathogens and pests need to estimate how frequently wind speeds above various thresholds, could be expected to push them between hosts, which could be separated by many kilometres. Rather than use the point value from the nearest recording station, which may be many kilometres distant, these attempts were made to interpolate values that would cover crops and woodlands of interest. Crown copyright Reproduced with the permission of Her Majesty s Stationery Office.

2 948 W. LUO ET AL. 2. Data and methods 2.1. Dataset Wind speed measurements were obtained for the period 1 Jan Oct 2002 at approximately 560 separate locations across the UK. Of these, about 320 stations only measured the wind speed once during the day (usually at 0900 h), approximately 110 stations had infrequent records. Around 140 stations had a continuous hourly record of 24 readings each day (Figure 1). The density of the network was almost constant until 2000 but declined in the latter half of the observation period (Figure 2). The two separated lines are evident in Figure 2 because approximately 50 wind stations record on 5 days each week, so the lower line represents the weekends over the 4 year period. In addition, there was a large decline in station numbers at the end of each year, probably due to the Christmas holiday, when manual records were not collected. A substantial number of stations do not record each hour during a day, so in order to gain the most representative estimate of the daily mean wind speed, only stations with at least 12 hourly observations for a particular No. of stations >24 Records per day Figure 1. Histogram of average daily records and number of stations No. of stations Jan98 Jul98 Jan99 Jul99 Jan00 Jul00 Jan0 Jul01 Jan02 Jul02 Date Figure 2. Scatter plot for number of stations recording wind data from 1998 to 2002.

3 WIND SPEED INTERPOLATION FROM IRREGULAR NETWORKS 949 Table I. Number of stations making wind speed observations at three frequencies over either 1 or 4 years. Time period Records per day More than 1 year More than 4 years day were used. This process reduced the number of data available for interpolation substantially (Table I). To add information about regions that were not covered well with long-period stations, the study also made use of data from stations with shorter continuous runs and intermittent records where they were available and had sufficient hourly records for dates of interest. The particular day of 27 Mar 2001 was chosen as the focus date for this paper because the number of valid stations was amongst the highest on that date. Data from 189 wind stations were collected which included some stations in southern Scotland in order to provide more accurate estimates for the northern part of England. The majority of the stations were located in the southeast and north central parts of England but sparsely located in the southwest and Wales. The elevations of the wind stations ranged from below sea level (Mepal, Cambridgeshire, 2 m) to above 800 m (Great Dun Fell, 847 m), but most were below 300 m (Figure 3). In general, daily mean wind speed increases with elevation and proximity to the coast. The actual range of wind speed on this day was m/s Interpolation methods This section briefly introduces the different interpolation methods used in this study. The spatial interpolation methods differ in their assumptions, local or global perspective, and deterministic or stochastic nature. Interested readers could refer to Lam (1983) for detailed criteria to distinguish spatial interpolators. The interpolation techniques were performed by the geostatistical analyst Figure 3. Location of UK Met stations with elevation information, 27 Mar 2001.

4 950 W. LUO ET AL. extension within the geographical information system ArcGIS (Version 8.3, ESRI Inc., Redlands, California, USA) Deterministic methods Trend surface analysis (TSA) Under TSA, the mapped data are approximated by a polynomial expansion of the geographic coordinates of the control points. The coefficients of the polynomial function are found by the method of least squares, insuring that the sum of the squared deviations from the trend surface is minimized. Each original observation is considered to be the sum of a deterministic polynomial function of the geographic coordinates plus a random error term. Wind speed values at unsampled locations were estimated using the mathematical relationship between the locational variables (longitude, latitude) and the regionalized wind speed. Although the polynomial can be expanded to any desired degree, in this study wind speed was fitted to first, second, third and fourth order polynomials which were assumed to be sufficient to capture regional wind speed variations. The unknown coefficients are found by solving a set of simultaneous linear equations. Once the coefficients have been estimated, the polynomial function can be evaluated at any point within the study area Inverse distance weighting (IDW) IDW interpolation combines the idea of proximity espoused by Thiessen polygons (Thiessen, 1911) with the gradual change of a trend surface. Those measured values closest to the prediction location will have more influence on the predicted value than those farther away. This distance-decay approach has been applied widely to interpolate climatic data (Legates and Willmott, 1990; Stallings et al., 1992). IDW assumes that each measured point has a local influence that diminishes with distance. The usual expression is, [ N ]/[ N ] Ẑ(s 0 ) = w(d i )Z(s i ) w(d i ) i=1 where Ẑ(s 0 ), Z(s i ) represent the predicted and observed value at location s 0, s i, N is the number of measured sample points used in the prediction, w(d) is the weighting function and d i is the distance from s 0 to s i. Based on the structure of IDW expression, the choice of weighting function can significantly affect the interpolation results. The comparative merits of various weighting functions are discussed in detail by Lancaster and Salkauskas (1986). The IDW parameters specified in ArcGIS are the power option, search shape, search radius and number of points. A circle with radius of 100 km for search shape with minimum and maximum numbers of points of 10 and 15 were specified for IDW. The power was optimized automatically by ArcGIS. i= Local polynomial interpolation (LP) LP interpolation is similar to TSA, except that it uses data within localized windows rather than using all of the data. The window can be moved around and the surface value at the centre of the window is estimated. LP is a moderately quick deterministic interpolator that is smooth (inexact). It is more flexible than TSA but there are more parameter decisions. When the input dataset exhibits short-range variation, LP interpolation can be a good method to capture finer detail (Akima, 1970) Thin plate spline (TPS) Splines are a deterministic interpolation technique which represent two-dimensional curves on three-dimensional surfaces (Eckstein, 1989; Hutchinson and Gessler, 1994). It is conceptually similar to fitting a rubber membrane through the measured sample values while minimizing the total curvature of the surface. A special class of spline, thin plate, was developed principally by Wahba and Wendelberger (1980). TPS is frequently used for interpolating elevation to create digital elevation models because of the relatively straightforward calculations. After further methodological developments, applications to climate interpolation have been presented by Hutchinson (1991, 1995). The general definition of TPS is given by a linear combination of the basis function (Basis functions are a special class of conical function such as Gaussian or elliptical. Their characteristic feature is that their response decreases (or increases) monotonically with distance from a central point). Ẑ(s 0 ) = p(s i ) + n w i φ(r) where φ(r) = r 2 ln(r) i=1 The term p(s i ) represents a polynomial of degree at most k, which is estimated by least squares, and the second term represents its proximity or fidelity to the data. Here, φ(r) is a basis function, r = s i s 0 is Euclidean distance between the prediction location s 0 and each data location s i. The weights {w i : i = 1, 2,...,n} are estimated from the data values. When the prediction is moved to a location with a measured value, the data value is predicted exactly (forms N equations in N unknowns can be solved uniquely) Geostatistical methods Kriging and cokriging Kriging (Krige, 1966) is a stochastic technique similar to IDW, in that it uses a linear combination of weights at known points to estimate the value at an unknown point. In contrast with deterministic methods, kriging provides a solution to the problem of estimation of the surface by taking account of the spatial correlation. The spatial correlation between the measurement points can be quantified by means of the semi-variance function: γ(h) = 1 N(h) [Z(s i ) Z(s i + h)] 2 2N(h) i=1

5 WIND SPEED INTERPOLATION FROM IRREGULAR NETWORKS 951 where N(h) is the number of pairs of measurement points with distance h apart. The semi-variance can be a function of both distance and direction, and so it can account for direction-dependent variability (anisotropic spatial pattern). A parametric function is used to model the semi-variance for different values of h (Cressie, 1985). Within various variogram models, the spherical model is the most widely used and often preferred when the nugget variance is important and there is a clear range and sill effect (Cressie, 1993; Burrough and McDonnell, 1998). In this study, the spherical model was used to determine the weights for the nearby supporting data to compute the interpolated values. The experimental semivariogram of average wind speed which is fitted by a spherical model in two different directions (Figure 4). The spatial continuity in the direction NE SW is stronger than in the opposite direction (NW SE). In the NE SW direction, with an angle of 49, the semi-variogram levels off to the sill when it reached a distance of 93 km. In the NW SE direction (340 ) the spatial correlation for the semi-variogram is similar except it levels off earlier at 75 km. Therefore, to account for these directional influences in the final surface two spherical models were calculated to increase the precision of the prediction. In recent years, kriging has been gaining more frequent use for ecological applications (e.g. Robertson, 1987; Weisz et al., 1995; Fleischer et al., 1999). These studies conclude that the best quantitative and accurate results are obtained by kriging when compared to inverse distance power functions, thin plate tension splines, trend surfaces and Voronoi (Thiessen) polygons. Varieties of kriging have been developed such as ordinary, universal, simple and indicator but only the first two were used in this study. Ordinary kriging which assumes the mean is unknown, focuses on the spatial component and uses only the samples in the local neighbourhood for the estimate. Universal kriging is similar to that of ordinary kriging, but assumes the presence of a trend in average values across the study area. A detailed account of kriging methods can be found in Isaaks and Srivastava, 1989; Cressie, 1993; Webster and Oliver, Collocated cokriging is defined as a multivariate version of kriging that uses additional covariates, ideally sampled at the same location as the estimated variable, to assist in prediction. When spatial correlation between a covariate and the variable of interest is high, cokriging gives better results for estimates. To apply cokriging one needs to model the relationship between the prediction variable and the covariate. This could be done by fitting a model through the cross semi-variogram. While the directional sample semi-variograms showed some indications of anisotropy for distances above 70 km in certain directions, the number of neighbours used in cokriging would primarily occur within about 60 km of the point being estimated, anisotropy at these distances was not strongly evident regardless of the angle examined. For simplicity of modelling and cokriging, the spatial variability Semivariogram Semivariogram NESW direction Distance (km) NWSE direction Distance (km) Figure 4. Experimental (points) and theoretical spherical (continuous line) semi-variograms of average wind speed for NE SW and NW SE direction. is assumed to be identical in all directions. The semivariograms have been modelled by the same spherical structure in order to apply the linear model of coregionalization in the cokriging process (Journel and Huijbregts, 1978). Parameter values for the model fitted to the wind speed differ substantially from those optimized to the semi-variogram for elevation and the cross semivariogram (Figure 5). For example, the nugget value for wind speed is large ( 0.5) compared with the other two which are close to zero. Cokriging involves more complicated algebraic calculations than kriging and the detailed principles are well explained by Myers (1982) and Cressie (1993) Assessment of interpolation outputs Cross-validation was used to evaluate the performance of each interpolation method. This was achieved by taking each observation in turn out of the sample and estimating it from the remaining observations. This process allowed the mean error (ME) and root mean square error (RMSE) test statistics to be calculated for each interpolation

6 952 W. LUO ET AL. (a) Semivariogram (b) Semivariogram (c) Cross semivariogram Distance (km) Distance (km) Distance (km) Figure 5. Experimental (points) and model (continuous line) for: (a) average wind speed, (b) elevation and (c) cross semi-variogram between average wind speed and elevation. method considered in the study. ME = 1 N N [ẑ(s i ) z(s i )] i=1 RMSE = 1 N N [ẑ(s i ) z(s i )] 2 i=1 where ẑ(s i ) is the predicted value and z(s i ) is the observed value. The ME was used to detect bias, it should ideally be zero if the predictions are unbiased, i.e. centred on the measurement values. The RMSE was used to compare different methods by seeing how closely predicted values match the measured values, the smaller the RMSE the better. 3. Results All the predicted mean wind speed surfaces were calculated for 27 Mar 2001 on a regular grid of 5 5km resolution across the England and Wales land surface area Surface characteristics and surface validation Trend surface analysis (TSA) Comparing all these in terms of the minimum and maximum interpolated values, showed that global trend surfaces of orders lower/higher than three were unreliable (Table II). Thus, a third order polynomial was assumed to be sufficient to capture the regional wind speed variations (Figure 6(a)). The result tended to capture broad regional trends but, due to bias introduced by multi-collinearity, these trends were suspect. For example, along the northwest coast of England, where the wind speed was relatively high, TSA estimated an implausible wind speed. In addition, TSA was not representative of the original data range since the surface is highly susceptible to edge effects where points near the edge of the dataset can have too large an influence over the surface as it nears the edge of the study area. Essentially waving the edges to fit these points can lead to inaccurate values within a short distance. The lowest and highest values obtained by TSA are respectively much greater and much less than the original data Inverse distance weighting (IDW) IDW gave a consistent but poor performance for the daily mean wind speed (Figure 6(b)). Where data are Table II. Summary comparison of mean wind speed surfaces created by the first four orders of TSA. Order of TSA Min-value Max-value ME RMSE

7 WIND SPEED INTERPOLATION FROM IRREGULAR NETWORKS 953 Figure 6. Surface of daily mean wind speed on 27 March 2001 by (a) TSA, (b) IDW, (c) LP and (d) TPS, respectively. unevenly distributed or sparse, as in southwest England, the IDW result was inappropriate because the weight assigned to points will be influenced by neighbouring points when they are more clustered. However, isolated points will exert an influence in all directions which diminishes evenly leading to characteristic bulls eye patterns in the estimated wind surface Local polynomial Wind speed was fitted to a third order polynomial model of the longitude and latitude (Figure 6(c)) giving a consistently better result than TSA. For example, it showed a comparatively high wind speed area in the northwest of England correctly. In addition, broad trends were also captured by LP when they were significant. However, the output wind surface had a lower minimum speed (0.15 m/s) than the observed data Thin plate spline The smooth wind speed surface created by TPS provides the correct general wind speed trends for the data (Figure 6(d)). TPS is useful for quickly obtaining a clear map showing the main features of the wind speed. In the southwest of England and Wales TPS generated a relatively accurate surface that retained small features from only a few sampled points. However, interpolation by TPS was inaccurate for the coastal areas and the mountainous regions where wind speeds exhibit abrupt changes over a small distance (e.g. North and East of England). In addition, TPS interpolated wind speeds well beyond the original data range and, as no account is taken of anisotropy, the most extreme wind speeds were smoothed and hence underestimated.

8 954 W. LUO ET AL Ordinary kriging The surface estimated by ordinary kriging provided a more accurate representation of the variation in wind speed within the north and southwest of England than achieved by TPS or IDW (Figure 7(a)). The ordinary kriging surface also had high values on the northwest/northeast coastal side, which were more consistent with the input values. The minimum and maximum wind speed estimates were broadly representative of the original data range. The standard deviations for ordinary kriging (Figure 7(b)) were larger for the locations near the Welsh border and for locations in the mountainous regions. In general, the kriging standard deviations for the southern part of England were smaller than for the northern mountainous part. This was owing to the lack of meteorological stations in the mountainous region where the daily mean speeds are relatively high. The standard deviations for ordinary kriging were smallest in the neighbourhood of the measurement locations, and largest for the regions with the least measurement locations. It was also large in some regions near the coast, e.g. southwest Wales Universal kriging A fairly smooth pattern was seen for Universal kriging (Figure 7(c)). Visually, the surface obtained by universal kriging was almost identical to the surface produced by ordinary kriging except that it was more spotty. The minimum and maximum wind speed estimated were shrunken above and below the recorded values. The universal kriging standard deviations ranged from 1.3 to 2.1 which was generally larger than those for ordinary kriging (Figure 7(d)). In particular, the standard deviation remained small for all the areas with abundant stations but Figure 7. The daily mean and standard deviation wind speed surfaces for ordinary kriging (a), (b) and universal kriging (c), (d) for 27 Mar 2001.

9 WIND SPEED INTERPOLATION FROM IRREGULAR NETWORKS 955 increased rapidly near the boundaries of north England and Wales. Therefore, the predicted surface appears less accurate than that obtained by ordinary kriging Ordinary collocated cokriging Kriging does not explicitly account for the influence of elevation on wind speed in mountainous areas. If there is a large difference in the elevation between the estimated point and the neighbouring stations, the estimate is likely to be unreliable. To address this problem, elevation was used as covariate to aid the estimation of wind speed. Visual inspection of the wind speed surface showed that cokriging represents changes in wind speed more closely than kriging when the correlation between elevation and wind speed was significant (Figure 8(a)). There was a positive correlation of between the recorded daily mean wind speed and the mean elevation across a 5-km square centred on the monitoring location. When elevation and wind speed were not correlated, the output from cokriging had a strong resemblance to kriging. Apart from the correlation between elevation and wind speed, it is also important to look at their pattern of spatial continuity (Goovaerts, 1997). Elevations that are moderately correlated with wind speed but exhibit a much smaller relative nugget effect than the wind speed semi-variogram may improve prediction using cokriging, particularly if, as we found, the nugget effect of the cross semi-variogram between wind speed and elevation is small (Figure 5(c)). The cokriging estimates exhibited a greater range, with a lower minimum and a higher maximum and provided the best representation of the original data range. Cokriging appeared to have less speckling or bulls eye effects in mountainous areas than kriging and the standard deviations were somewhat smaller. The technique therefore appears more precise overall and performed well for daily mean wind speed data (Figure 8(b)) Statistical assessment RMSE and ME (bias) were used to evaluate the performance of the interpolation method, together with the minimum and maximum interpolated values of the output wind surface (Table III). Overall, the surfaces under predicted the wind speed range. The TSA interpolated wind speed range was typically the narrowest compared to the original data and did not produce representative results. Ordinary kriging and cokriging adhered to the original wind speed range most consistently. The ME for all the methods were generally small (Table III). In terms of RMSE, cokriging, which had the lowest value (1.47), showed clear superiority over the other methods. Overall, the four different assessment measures consistently identified cokriging as the best method for interpolating surfaces, followed by ordinary kriging and universal kriging. To examine the quality of the cokriging predictions, a scatter plot between the cross-validated estimation and the observation was plotted (Figure 9). A best linear line fitted to the estimated points had an R 2 value of Table III. Summary statistics of the seven interpolation methods. Method Min estimate Max estimate ME RMSE TSA IDW Local polynomial TPS Ordinary kriging Universal kriging Cokriging Figure 8. The daily mean (a) and standard deviation (b) wind speed surfaces for cokriging, 27 Mar 2001.

10 956 W. LUO ET AL. Estimates :1 line best linear fit estimate R 2 = (a) Normalised absolute ME TSA IDW LP TPS OK UK Cokriging Observed values Data set Figure 9. Scatter plot of the cokriging estimates against the observed values for daily mean wind speed on 27 Mar but shows that wind speeds below 6 were overestimated Reliability of results In the absence of more extensive testing from the trial data, it is not possible to state that one method was superior to another. Therefore, a reliability analysis of the interpolation techniques was carried out using daily mean wind speed data from 10 days selected randomly from the period 1998 to 2002 (Table IV). A total of 70 wind speed surfaces were generated from the 10 days and seven interpolation techniques. Summary statistics rather than analysis of variance (ANOVA) were used to evaluate the performance of the interpolation method with particular importance given to ME and RMSE (Figure 10). The reason for this choice was that hypothesis testing using ANOVA assumes that the means compared are drawn from populations with an equal variance. As this analysis provides a comparison across different dates, the assumption of equal variances was not valid. The ME and RMSE, normalized by the mean of the estimators, were used to evaluate the performance Table IV. Summary statistics for ten randomly selected days. Datasets Selected date Number of stations Mean WS WS range WS variance ( ) Feb Feb Apr Jun Oct Feb Jan Mar Sep May (b) Normalised RMSE TSA IDW LP TPS OK UK Cokriging Data set Figure 10. Comparative statistics on normalized absolute ME (a) and RMSE (b) for ten randomly selected days during the period of each method on different days. The striking feature of Figure 10 is that some methods appear to be consistently poor interpolators while the performance of all methods is strongly related to the dataset. As the temporal scale (daily in this project) is short, preliminary data analyses are especially important to determine the suitability of a particular interpolation technique. For all cases with similar mean wind speed, when data ranges and variances are large, the performance of all interpolation techniques suffered, e.g. comparing dataset 3 (21 Apr 2001) and dataset 5 (29 Oct 1999) (Table IV). There was also evidence that decreasing the number of input data impairs interpolator performance negatively as indicated by higher RMSE values, e.g. comparing dataset 6 (09 Feb 1999) and dataset 9 (12 Sep 2001). Of all methods studied, LP, TSA, kriging and cokriging techniques generally had the smallest ME, followed closely by TPS, with IDW resulting in the largest. In particular, the ME for kriging and cokriging is relatively low for all 10 days, confirming these geostatistical interpolators were substantially unbiased. For normalized RMSE,

11 WIND SPEED INTERPOLATION FROM IRREGULAR NETWORKS 957 cokriging was ranked the lowest for all 10 days tested while TSA and TPS was comparatively high. And the normalized RMSE were approximately the same for all other interpolation techniques. A binomial test (assuming a null hypothesis of no difference between cokriging and other interpolators) showed that a single interpolation method, cokriging, had clear superiority over other methods for all 10 days tested (p 0.001). Further inspection of the binomial test result reveals that at the 95% significance interval (α = 0.05), on at least 76% of days (from 1998 to 2002) cokriging performed better than the other interpolators. This indicated that cokriging is generally the best method for wind speed interpolation with good temporal consistency. 4. Discussion Range, variance and input data values are important attributes to consider when selecting a spatial interpolation method. Collins and Bolstad (1996) found that several variable characteristics (range, variance, correlation with other variables) can influence the choice of a spatial interpolation technique. These results from this study agree with MacEachren and Davidson (1987) who concluded that data measurement accuracy, data density, data distribution and spatial variability have the greatest influence on the accuracy of interpolation. As with most alternative methods of smoothing, both TSA and LP produced misleading results in station-sparse areas, e.g. they may predict negative wind speed values, which were normally truncated to zero to avoid impossible results. Generated wind speed values were low and implausible for the mountainous area of northern England (Figure 6(a) and (c)). Another serious problem with these smoothing algorithms is that they are always unrepresentative of the original data range, which is caused by the limitation of polynomial regression (Myers, 1990). Although the global and LP methods are more flexible and result in a smaller RMSE, based on the interpolated surfaces, they both cannot be preferred as a meaningful method for estimating wind speed. However, they could be useful for the removal of broad trends prior to further spatial analysis and geostatistical methods (Davis, 1986; Burrough and McDonnell, 1998). IDW is an exact interpolator so high variability in the input data consequently produces a rougher surface (Figure 6(b)). The surface generated passes through the value at each weather station which does capture the local gradients, but may lead to unrealistically steep gradients in areas with poor station coverage, e.g. the Southwest and Wales. IDW interpolation is easily affected by uneven distributions of observational data points since similar weight will be assigned to each of the points even if it is in a cluster and this probably explains why the predictions tend to be less accurate for the boundary areas. Differences in ME between IDW and other interpolators were significant (Figure 10), and IDW was more prone to produce biased estimates. TPS produces a natural looking surface, but this may just be a reflection of intuitive preference; in fact the surface is always much smoother than the underlying reality. TPS is particularly poor for datasets that exhibit abrupt changes over small distances and this disadvantage is clearly revealed (Figure 6(d)) where predicted wind speeds are decreasing for mountainous locations near the coast (e.g. in coastal mountain regions of north England). TPS is generally not recommended for interpolation for irregularly spaced data (Collins and Bolstad, 1996). Kriging is a somewhat more sophisticated interpolation technique which does explicitly account for spatial variance and, in contrast to TSA and TPS, there was a strong tendency for kriging to give lower RMSE values. Tabios and Salas (1985) found kriging to be superior to other conventional interpolation techniques such as LP and IDW, which is in accordance with the observations of this study. Overall, the differences of RMSE between ordinary kriging and universal kriging were generally very small, but universal kriging tended to have higher errors. One of the advantages of kriging is that it provides estimation errors to measure the interpolation uncertainty. This error information reflects the density and distribution of control points and the degree of spatial correlation within the surface, and therefore is very useful in analysing the reliability of each feature in the kriged map.theerrormapcanalsobeusedtodeterminewhere more information is needed so that future sampling can be planned if necessary. The benefit of using elevation as a covariate (Figure 10) corresponds with the results from Hevesi et al., (1992) who showed cokriging was capable of producing estimates of greater precision than kriging. In addition, cokriging appears to more closely reflect the wind features in mountainous areas. Cokriging is the most time-consuming interpolation technique as it requires fitting two semi-variograms and one semi-crossvariogram for each dataset, but to meet the high accuracy required in this study it was the preferred technique. Both surface assessment and statistical comparisons showed that cokriging was most likely to produce the best estimation of a continuous surface for wind speed temporal consistency. However, there is the issue of bias in the resulting surface (Figure 9) which overestimates wind speeds slower than 6 and underestimates the fastest wind speeds. The relatively poorer performance of the interpolation at higher altitude, faster wind speed, locations is of less importance for the intended use of these surfaces in predicting risk from pest invasion. Cropping areas in the UK are mostly restricted to lower lying ground across the south and up the eastern side of England where the wind speed range is more often in the range where the interpolations work best. The research presented illustrates that even the best interpolation method, cokriging, did not adequately address the wind speed variability in southwest England and Wales. In addition there was a tendency to underestimate larger wind speed values due to the lack of high altitude meteorological stations. Only elevation was considered as a covariate when interpolating wind data but other factors would affect wind conditions. From the

12 958 W. LUO ET AL. same digital elevation model used to determine average altitude around the meteorological stations, slope and aspect information could be obtained and used as additional covariates to possibly improve wind speed estimation (Burrough and McDonnell, 1998). As a result, it is critical that additional data should be incorporated into any further spatial interpolations if a more realistic representation of wind speed across the more problematic parts of the study area is to be achieved. Further geostatistical techniques are another possibility to improve the performance of the interpolation, such as regressionkriging (Hengl et al., 2004), but these are beyond the scope of the present paper. 5. Conclusions When compared with the other interpolation methods evaluated, the results of this study indicate that cokriging was most likely to produce the best estimation of a continuous surface for wind speed and that the result had temporal consistency. The cokriging maps show more details than the kriging maps, due to the inclusion of elevation as a covariable in estimation. In this case study, the additional complexity of cokriging was worthwhile because smaller prediction errors were produced compared to kriging. However, due to a general lack of high altitude meteorological stations across England and Wales, interpolated surfaces were prone to underestimate larger wind speed values in upland areas. Further research is recommended to test whether other environmental factors, such as aspect, might allow explanation of a larger proportion of the spatial variability displayed by wind speed. As there are some additional geostatistical methods that were not evaluated in this study, it may be necessary to explore these methods to determine if they could generate a better representation of wind speed across the study area. 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