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Agricultural and Forest Meteorology 101 (2000) 81–94

A comparison of two statistical methods for spatial interpolation ofCanadian monthly mean climate data

David T. Pricea,∗, Daniel W. McKenneyb,1, Ian A. Nalderc,2,Michael F. Hutchinsond,3, Jennifer L. Kestevend,3

a Canadian Forest Service, Northern Forestry Centre, 5320-122 Street, Edmonton, Alta., Canada T6H 3S5b Canadian Forest Service, Great Lakes Forestry Centre, 1219 Queen Street East, Sault Ste. Marie, Ont., Canada P6A 5M7

c Department of Renewable Resources, Faculty of Agriculture, Forestry and Economics, University of Alberta,Edmonton, Alta., Canada T6G 2E3

d Centre for Resource and Environmental Studies, The Australian National University, Canberra ACT 0200, Australia

Received 11 August 1999; received in revised form 8 December 1999; accepted 9 December 1999

Abstract

Two methods for elevation-dependent spatial interpolation of climatic data from sparse weather station networks werecompared. Thirty-year monthly mean minimum and maximum temperature and precipitation data from regions in western andeastern Canada were interpolated using thin-plate smoothing splines (ANUSPLIN) and a statistical method termed ‘Gradientplus Inverse-Distance-Squared’ (GIDS). Data were withheld from approximately 50 stations in each region and both methodswere then used to predict the monthly mean values for each climatic variable at those locations. The comparison revealedlower root mean square error (RMSE) for ANUSPLIN in 70 out of 72 months (three variables for 12 months for both regions).Higher RMSE for GIDS was caused by more frequent occurrence of extreme errors. This result had important implicationsfor surfaces generated using the two methods. Both interpolators performed best in the eastern (Ontario/Québec) region wheretopographic and climatic gradients are smoother, whereas predicting precipitation in the west (British Columbia/Alberta) wasmost difficult. In the latter case, ANUSPLIN clearly produced better results for most months. GIDS has certain advantagesin being easy to implement and understand, hence providing a useful baseline to compare with more sophisticated methods.The significance of the errors for any method should be considered in light of the planned applications (e.g., in extensive,uniform terrain with low relief, differences may not be important). ©2000 Elsevier Science B.V. All rights reserved.

Keywords:Climate; Temperature; Precipitation; Spatial interpolation; Topographic dependence; Canada; Thin-plate smoothing spline;ANUSPLIN; GIDS

∗ Corresponding author. Tel.:+1-780-435-7249;fax: +1-780-435-7359.E-mail addresses:[email protected] (D.T. Price),[email protected] (D.W. McKenney),[email protected] (I.A. Nalder),[email protected] (M.F. Hutchinson),[email protected] (J.L. Kesteven).

1 Tel.: +1-705-759-5740; fax:+1-705-759-5700.2 Currently resident in Australia. Tel.:+61-2-6254-3322.3 Tel.: +61-2-6249-4783; Fax:+61-2-6249-0757.

1. Introduction

The development of methods to interpolate climaticdata from sparse networks of stations has been a fo-cus of research for much of this century (Thiessen,1911; Shepard, 1968; Hughes, 1982; Hutchinson andBischof, 1983; Phillips et al., 1992; Daly, 1994). Re-cent events, including the IPCC Second Assessment

0168-1923/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved.PII: S0168-1923(99)00169-0

82 D.T. Price et al. / Agricultural and Forest Meteorology 101 (2000) 81–94

Report on climate change (Houghton et al., 1996)and the Kyoto Protocol of late 1997, have sparkedadditional interest in climate data interpolation. Pre-diction of the impacts of a changing climate on thedistribution and functioning of terrestrial ecosystemsrequires as a first step, the development of reliable,spatially-explicit models of current climate. For manyof the areas of greatest concern, such as the borealforest and tundra biomes of central and northernCanada, station coverage is often very sparse, andthe long-term records often incomplete. In addition,many researchers attempt to predict future ecosys-tem responses to climatic change using output fromgeneral circulation models (GCMs) and regionalclimate models (RCMs) (Boer et al., 1992; Cayaet al., 1995). Although there are undoubtedly manyconceptual problems and practical limitations to us-ing coarse-scale climate model output for predictingecosystem responses, any attempt to do this will gen-erally require an unbiased method for interpolation tothe scale of operation of most ecosystem models. Im-proved methods of climate interpolation will enhanceour ability both to quantify effects of climate (and cli-mate variability) on natural and managed ecosystems,including forests, wetlands and agroecosystems, andto forecast the possible impacts of climate change.In most cases, simulation of ecosystem responses toclimate does not require an exact representation ofreality, so interpolation from sparse or incompleterecords is quite acceptable. However, because manynatural vegetation types occur in mountainous re-gions, it is reasonable to suppose that elevation isa key factor influencing the climate experienced bythese ecosystems. Hence it is generally necessary toinclude elevation as an independent variable in theinterpolation method.

In some instances, it may be preferable to usea simple method applied to the region of interestthan to use a more sophisticated approach whichcould be marginally more accurate, but requiresconsiderably more time and money to implement.In this paper, therefore, two elevation-dependentclimate interpolators are compared. One of these,ANUSPLIN (Hutchinson, 1995a, 1999), has beendeveloped and tested over several years and is nowwidely used. The other interpolator, Gradient plusInverse-Distance-Squared (GIDS) (Nalder and Wein,1998), is less well known but attractively simple

and appears to give results adequate for modellingforest ecosystem responses to climate — at least inrelatively flat terrain.

Recently, Price et al. (1998) generated national-scalegridded climate surfaces for Canada using the GIDSweighting method of Nalder and Wein (1998).Nalder and Wein (2000, in press) developed GIDSas a straightforward method for interpolating cli-mate data obtained from a sparse regional net-work of stations in northwestern Canada to thepositions of a large number of survey plots lo-cated in boreal forest stands. Using multiple lin-ear regression to estimate regional gradients intemperature and precipitation with latitude, longi-tude and elevation, GIDS was found to comparevery favourably with several other interpolationmethods, including universal kriging, while havingthe benefit of straightforward implementation andoperation.

McKenney et al. (2000, submitted for publication)have also developed national climatic grids, expand-ing on previous work in Ontario and the Great Lakes(Mackey et al., 1996) using the ANUSPLIN soft-ware of Hutchinson (1999) (see also Hutchinson,1991, 1995a; Hutchinson and Gessler, 1994). ANUS-PLIN is based on smoothing splines as described byWahba (1990), Hutchinson (1984) and Wahba andWendelberger (1980). Additional FORTRAN pro-grams in the ANUSPLIN modeling package can beused to generate interpolated grids and, hence, dig-ital climate maps. In this paper, we compare GIDSwith ANUSPLIN through a data-withholding pro-cess and attempt to identify the ‘better’ approach togenerating digital grids of monthly mean climate forCanada.

Both methods of climate interpolation have alreadybeen used in spatial modelling of forest ecosystems.ANUSPLIN has been used to generate climatolo-gies in Ontario to determine the risk of infection ofpine forest by Sceleroderris disease and to assessclimatic effects on the distributions of breeding birds(Venier et al., 1998a, 1998b). The GIDS methodswas developed and used in a study of productivityand succession in the western boreal forests of cen-tral Canada (Nalder and Wein, 1998; Nalder et al.,2000, submitted for publication). GIDS was also re-cently applied to a study of forest vegetation effectson the climate of North America (E.H. Hogg, per-

D.T. Price et al. / Agricultural and Forest Meteorology 101 (2000) 81–94 83

sonal communication, 1999, Canadian Forest Service,Edmonton).

2. Methods

2.1. GIDS

GIDS relies on multiple linear regression (MLR)analysis of data from a set of nearby stations to esti-

Vp=

[N∑

i=1

Vi + (X0 − Xi)CX + (Y0 − Yi)CY + (Z0 − Zi)CZ

d2i

][

N∑i=1

1

d2i

] (1)

mate the local gradients for each climate variable,treating latitude, longitude and elevation as inde-pendent variables. The climate value for a targetpoint in the neighbourhood is then predicted fromthe same station data, using the MLR coefficients tocorrect for the differences from each station’s posi-tion. The contributions of each station to the finalestimate are inverse-weighted by their squared dis-tances to the target point, i.e., GIDS assumes thatclimate data are spatially autocorrelated. Clearly,there are some important caveats to this assumption,particularly in mountainous regions where topog-raphy tends to reduce spatial correlations (at leastwhen considering two dimensional space) and thusaffects the predictability of both temperature andprecipitation.

Following Nalder and Wein (1998), computerprograms were written to perform MLRs for eachmonthly climate variable at the location of every teststation or grid point, using routines for the matrixsolution of linear equations from Press et al. (1986).The solutions yielded sets of coefficientsCX, CY andCZ, representing the observed gradients of variableVin response to independent variablesX, Y, Z, denot-ing longitude, latitude and elevation asl, respectively.To account for the effect of possible non-linearitiesassociated with convergence of the meridians athigh latitudes, differences in spherical coordinatesbetween the target location and the neighbouringstations were also mapped on to planar distance co-

ordinates using an Azimuthal Equidistant projectionalgorithm given in Snyder (1987). In practice, thistransformation was found to have negligible impactson the averageR2 of the MLRs and was not usedfurther.

The MLR coefficients were then used to predicteach climate variable,Vp, at each target climate sta-tion, from the coordinates and elevation of the neigh-bouring stations, using

whereN is the number of neighbouring stations con-tributing data to the MLR,X0, Y0 andZ0 are longitude,latitude and elevation of the target station andXi , Yi ,and Zi are the corresponding coordinates of theithneighbouring station respectively, whiledi is the greatcircle distance to, andVi the value ofV observed at,stationi. For this study,N was arbitrarily set to 40.

2.2. ANUSPLIN

ANUSPLIN is a suite of FORTRAN programs de-veloped at the Australian National University that cal-culates and optimizes thin plate smoothing splines fit-ted to data sets distributed across an unlimited numberof climate station locations (Hutchinson, 1991, 1999).It has been applied in numerous regions including Aus-tralia, New Zealand, Europe, South America, Africa,China and parts of southeast Asia. A general modelfor a thin plate spline functionf fitted ton data valueszi at positionsxi is given by (Hutchinson, 1995a):

zi = f (xi) + εi (i = 1, . . . , n) (2)

where thexi typically represent longitude, latitude andsuitably scaled elevation. Theεi are zero mean ran-dom errors which account for measurement error aswell as deficiencies in the spline model, such as localeffects below the resolution of the data network. Theεi are assumed to have a covariance matrixVσ 2 whereV is a known positive definiten×n matrix, usually di-agonal, whileσ 2 is usually unknown. The functionf


is estimated by minimizing:

(z − f )T V −1(z − f ) + ρJm(f ) (3)

wherez = (z1 . . . , zn)T , f = (f1, . . . , fn)

T , andTsignifies the matrix transpose, with

fi = f (xi) (4)

andJm(f) is a measure of the roughness of the splinefunctionf defined in terms ofmth order partial deriva-tives off. The positive numberρ is called the smooth-ing parameter. It is determined objectively by mini-mizing the Generalized Cross Validation (GCV) statis-tic, a measure of the predictive error of the surface.The GCV is calculated by implicitly removing eachdata point and summing the square of the difference ofeach omitted data point from a surface fitted to all theremaining data points (see Hutchinson and Gessler,1994). The procedure provides an estimate of the valueof σ 2.

For the interpolation of mean temperatureV wasset to the identity matrix, as local effects are the maincontribution to model error. For the interpolation ofmean precipitationV was set to a diagonal matrix withentriesvii given by

vii = σ 2i

ni

(5)

whereσ i2 is the year-to-year variance of the monthly

totals at locationxi andni is the number of years ofrecord. This is the approximate Method 5.1 describedin Hutchinson (1995a). In the case of precipitation,year-to-year variation is a significant contributor tomodel error when using means for different periods.Further details can be found in the references citedearlier. To properly scale the independent variables,longitude was transformed by the cosine of the centrallatitude. This had only a marginal effect on the precip-itation surface. Elevation units are specified in km toscale this term appropriately (see Hutchinson, 1995a).

2.3. Data and comparison

Monthly climate data for two regions of thecountry were extracted from Environment Canada’sCD-ROM of Canadian 1961–1990 climate normals(Environment Canada, 1994). The variables selectedfor this study were: monthly mean precipitation,P,

and monthly mean daily maximum and minimumtemperatures,Tmax and Tmin, respectively. The twostudy regions (southern British Columbia/Alberta andsouthern Ontario/southwestern Québec) were chosento represent a diverse range of topography and avail-able data. The BC/Alberta region extends across theCanadian Pacific coast, coastal and interior moun-tain ranges including the Rockies, and parts of thewestern boreal forest and prairies. As is typical inmost parts of the world, climate data are particularlysparse in higher elevation areas. The Ontario/Québecregion is comparatively flat but includes some higherelevation locations and areas downwind of the GreatLakes. In combination, these study areas representedthe climatic conditions found in both forested andagricultural regions across much of southern Canada.The locations of the stations providing data for eachvariable in both regions are shown in Fig. 1. For eachof Tmax, Tmin andP, respectively, data were availablefrom 434, 436 and 406 stations in the BC/Albertaregion, and from 407, 405 and 371, stations inOntario/Québec.

For each study region, approximately 50 stationswere selected at random from the data set and with-held from the interpolation calculations. The perfor-mance of the two interpolation methods was then com-pared by using them both to estimateP, Tmax andTmin at the coordinates of the withheld stations. Resid-uals were calculated as the differences between theestimated and observed values for each station foreach month. Following Hulme et al. (1995) and Daly(1994), precipitation residuals were calculated as per-centage differences from the observed data. This ap-proach provides a better relative measure of the dif-ferences as compared to absolute values, particularlywhen the spatial variation in monthlyP is large, as isthe case in BC. The residuals and squared residualswere pooled to determine Mean Errors (ME) and RootMean Square Errors (RMSE), respectively.

The ME was used to detect bias in the two meth-ods. Bias can be very important, both statistically andwhen using the interpolated data for further analysisor modelling. In comparison, RMSE is sensitive to thesize of outliers and was used as an indicator of themagnitude of extreme errors (i.e., lower RMSE indi-cates greater central tendency and generally smallerextreme errors). Box plots were developed to displaythe residuals for both methods for each variable. As


Fig. 1. Locations of climate stations used for comparison of the GIDS and ANUSPLIN spatial interpolation methods, applied to 1961–1990AES normals for study areas in British Columbia/Alberta (left) and Ontario/Quebec (right). (a) monthly mean daily maximum temperature;(b) monthly mean daily minimum temperature; (c) monthly mean total precipitation. Symbols distinguish the stations used for interpolation(o) from those withheld as control stations used for error assessment (x).

a further assessment, the number of months in whichone method gave lower RMSEs than the other werecounted and checked for significance using a simplebinomial test.

2.4. Map generation

To assess the behaviour of each interpolationmethod, maps were generated for a monthly vari-able giving poor agreement between observed andpredicted data (July precipitation in the BC/Albertastudy region was selected for this test). Using a 1 kmDigital Elevation Model (DEM) for this region (ob-tained by resampling the GTOPO30 data set (Verdin

and Jenson, 1996) to a Lambert Conformal Conicprojection), precipitation values were estimated forthe centroids of each DEM pixel. The interpolatedgrids of precipitation data were then imported intoARC/INFO GRIDTMand printed as coloured images.

3. Results

Figs. 2 and 3 present the box plots for BC/Albertaand Ontario/Québec respectively, forP, Tmax andTmin, obtained with ANUSPLIN and GIDS. Notethat much smaller scales were used for plotting theBC/Alberta data because the residuals were gener-


Fig. 2. Box plots for British Columbia/Alberta forP, Tmax and Tmin, obtained with ANUSPLIN and GIDS.


Fig. 3. As Fig. 2, but for the Ontario/Quebec study region. Note that much smaller scales were used for plotting the BC data because theresiduals were generally much larger than those for Ontario.


ally larger than those for Ontario/Québec. The boxplots suggest that the two interpolation methods weregenerally comparable in their predictive accuracy,although careful inspection shows that ANUSPLINperforms rather better with the BC/Alberta data. Inparticular, the magnitudes of the residuals obtainedfor BC/Alberta precipitation are considerably largerusing GIDS. The central tendency of the residualsresulted in generally small ME, confirming that bothinterpolators were substantially unbiased.

For temperature in Ontario/Québec, 50% of thepredictions (the 25–75% percentiles) were generallywithin ±0.5◦C of the recorded values. In most casespredictions fell within±1◦C of the observed data,although estimates of winterTmin were noticeablypoorer. For most months, approximately half of theOntario/Québec precipitation predictions were within±10% of observed values, although the spread wasgenerally wider for some winter months, particularlywhen using GIDS. Results for BC temperature weresimilar except that there were greater numbers of out-liers, and wider spreads for the 25–75% percentiles,with both interpolation methods. This is to be expectedgiven the severe elevation gradients and the sparsecoverage of climate stations sampling higher eleva-tion locations for extended periods. Both methodshad greater difficulty predicting precipitation valuesin BC/Alberta for the withheld stations; i.e., severaloutliers exceeded 100% difference between observedand predicted data. The winter months of December,January and February were generally worst althoughGIDS had some greater residuals in April, Octoberand November. Part of the explanation for this couldbe poorer quality data during winter months (e.g.,difficulties in converting snowfall depth to equivalentwater content), leading to greater variances in surfacefitting and larger validation errors. Overall, the differ-ences in mean errors between the two interpolatorswere generally very small, but GIDS was more proneto extreme errors. This tendency is seen not only inthe boxplots (Figs. 2 and 3), but was also reflected inthe quality of the maps generated for the region (seeFig. 4).

Table 1 compares pooled RMSE values obtainedusing the two interpolators for each of 72 cases (i.e.,12 months for three variables in two regions). Thoseobtained using ANUSPLIN were lower than those forGIDS in 70 out of the 72 possible cases. The bino-

mial test (assuming a null hypothesis of no differencebetween the two interpolation methods) showed thisresult to be highly significant for all three variables inboth regions. For bothTmin andTmax, ANUSPLIN pro-duced lower mean RMSE in all 12 months (p≈0.0002)for both study regions. In the case ofP, mean RMSEwas lower in all 12 months for BC/Alberta, but in only10 months out of 12 (p<0.02) for Ontario/Québec.

Further inspection of Table 1 reveals that althoughthe differences between the two interpolators weregenerally relatively small, the gains obtained fromusing ANUSPLIN were greatest in the most diffi-cult cases — i.e., BC/Alberta winter variables, wherecomplex topography, low station density and poten-tially poorer quality measurements (mentioned ear-lier) were combined. In the particular case of winterprecipitation in BC/Alberta (October–February inclu-sive), mean RMSE values obtained using GIDS wereat least double those obtained from ANUSPLIN. Dif-ferences in mean RMSE for mid-winter temperaturevariables were also generally greater than for othermonths, although this effect is less pronounced forOntario/Québec. Winter temperature data are morevariable, for a number of possible reasons (e.g., forBC/Alberta, the standard deviations of the station ob-servations ofTmax were 5.5 and 2.8◦C for January andJuly, respectively). With the exception of BC/Albertawinter precipitation, differences in RMSE betweenmethods were always much smaller than the RMSEsobtained with either method.

Because GIDS is an exact interpolator, greater vari-ability results in a rougher surface, producing largeRMSEs at locations between stations. In comparison,ANUSPLIN’s smoothing routines result in smaller av-erage errors in the interpolated surface. This tendencyof ANUSPLIN to give lower RMSEs for temperaturewas supported by a visual comparison of maps gener-ated by the two interpolation methods for July meanTmax in the BC/Alberta study region (data not shown).In this case, mean RMSEs were 0.85 and 0.99◦C forANUSPLIN and GIDS, respectively. The only obvi-ous difference on the map was an evident problemon Vancouver Island where GIDS predicted increasingtemperature for mountain locations in the centre of theisland. ANUSPLIN predicted decreasing temperature,which seems more reasonable. The likely explanationfor this is the close proximity of Port Alberni, whichis actually quite far from the coast and therefore sub-


Fig. 4. Distribution of monthly mean total precipitation for July in the British Columbia/Alberta study region as predicted by (a) GIDSand (b) ANUSPLIN. Colour bars indicate rainfall totals in millimetres.


Table 1Summary of pooled Root Mean Square Error (RMSE) values obtained as the differences between observed and interpolated values forclimate stations withheld from the analyses using GIDS and ANUSPLINa

January Febuary March April May June July August September October November December

Ontario/Quebec regionTmin (GIDS) 0.99 1.14 0.93 0.72 0.86 0.66 0.83 0.84 0.83 0.81 0.72 0.86Tmin (ANUSPLIN) 0.88 0.98 0.85 0.58 0.75 0.60 0.77 0.81 0.82 0.74 0.66 0.83Tmax (GIDS) 0.68 0.66 0.58 0.73 0.77 0.63 0.54 0.58 0.56 0.85 0.28 0.45Tmax (ANUSPLIN) 0.61 0.59 0.56 0.66 0.67 0.58 0.48 0.52 0.52 0.72 0.24 0.44%P (GIDS) 18 13 18 10 10 11 15 15 16 13 11 15%P (ANUSPLIN) 17 13 20 9 9 10 14 14 13 12 10 13

British Columbia/Alberta regionTmin (GIDS) 2.79 1.58 1.38 1.02 0.99 1.04 1.21 1.34 1.26 1.17 1.34 1.71Tmin (ANUSPLIN) 2.39 1.42 1.23 0.97 0.95 0.98 1.10 1.23 1.18 1.11 1.26 1.46Tmax (GIDS) 2.14 0.98 1.26 1.01 0.84 0.80 0.99 1.03 1.06 0.66 0.92 1.24Tmax (ANUSPLIN) 1.72 0.86 1.06 0.91 0.70 0.70 0.85 0.87 0.88 0.58 0.90 0.95%P (GIDS) 125 82 46 39 23 18 22 20 22 57 52 71%P (ANUSPLIN) 41 41 26 30 22 17 19 19 17 24 22 31

a In total, 72 cases were compared — three variables for 12 months in two regions. Variables are monthly mean minimum and maximumtemperatures in◦C (Tmin andTmax, respectively) and monthly mean total precipitation (P, with differences expressed as percentages). Boldface figures indicate the smaller RMSE obtained in each comparison.

jected to a more inland climate despite its low eleva-tion. The small change in elevation relative to the coastgenerated an extremely large (inverted) lapse rate thatGIDS applied to the higher elevation locations nearby.

Figs. 2 and 3 show that the greatest differences be-tween interpolated values and observed data (in bothmeans and variances) occurred for precipitation (aswould be expected). The maps generated for July to-tal precipitation in BC/Alberta differ appreciably invisual terms (Fig. 4). As seen from Table 2, althoughthe mean values differ only slightly, GIDS generatesgreater extremes of high and low precipitation, witha significantly larger standard deviation. The probableexplanation for this is that GIDS is markedly moresensitive to local gradients derived from the stationdata (e.g., compare the Fraser Valley mountains at49.5◦N 123.5◦W in Fig. 4a and b). In some cases,these extremes are clearly unrealistic, particularly in

Table 2Summary statistics for comparison of maps of July precipitationin British Columbia/Alberta study region (Fig. 4)a

GIDS ANUSPLIN

Minimum 0.0 15.0Maximum 258.0 136.6Mean 70.4 69.5Standard deviation 23.3 17.8

a Units are in mm.

the coastal mountain regions where zero precipita-tion is predicted for some summits (Fig. 4, 52.5◦N126.0◦W). As with Tmax on Vancouver Island, thelikely explanation is that large changes in precipitationwith small changes in elevation close to sea level re-sult in significant underestimates when the gradient isapplied at higher elevation. GIDS is clearly respond-ing to observations of extreme rainfall, e.g., at two lo-cations on the coast north of Vancouver Island, (Fig.4, 52.5◦N 128.0◦W), where single stations evidentlycause GIDS to estimate high rainfall in their immedi-ate vicinity. Conversely, the minimum curvature sur-faces fitted by ANUSPLIN minimise changes in gra-dients, giving rise to smoother transitions in surfacevalues across the data network. These transitions aregenerally more interpretable and believable than thoseproduced by GIDS. For example, Fig. 4 reveals somevery abrupt boundaries between regions of high andlow rainfall in the GIDS interpolation — particularlynoticeable at the northern end of Vancouver Island(50.0◦N 126.5◦W) and on the mainland coast directlyopposite (51.0◦N 126.0◦W).

4. Discussion

Given previous comparisons showing the generalsuperiority of GIDS over other spatial interpolation


methods, including kriging, in regions with sparse cov-erage of climate stations (Nalder and Wein, 1998), thisstudy provided an important vindication of the advan-tages of using spline interpolation — at least for therange of climates occurring in Canada. Substantial ef-forts are currently being directed at producing highresolution digital climate grids using ANUSPLIN forCanada’s GIS, remote sensing and ecosystem mod-elling communities (McKenney et al., 1996). Never-theless, both interpolation methods performed reason-ably well in providing estimates of all variables inthese two regions of the country.

The box plots revealed larger differences for theBC/Alberta region, particularly in the case of monthlymean precipitation. Pairedt-tests performed on theRMSE values showed that the latter was the only sta-tistically significant case. This result is almost cer-tainly a consequence of the sparser station coverageand shorter records at high elevation sites in BC. Ingeneral, thin-plate smoothing splines such as ANUS-PLIN enable better prediction in such regions becausethey calibrate a spatially varying dependence on el-evation that uses all available data points. AlthoughGIDS does respond to local gradients derived from thenearest data points, it can run into problems in moun-tainous regions where station coverage is sparse.

Perhaps the most serious problem with GIDS isthat when the climate stations closest to the gridpoint are still a large distance away, the regressioncoefficients are relatively sensitive to the inclusion orexclusion of single stations. At these distances, evensmall changes in the coefficients can cause dramaticchanges in the values predicted for the interpolationpoint. In those cases where the station being addedto the set of closest stations contributes markedlydifferent data to those from the station just removed(e.g., because one is at low elevation on the coast andthe other is at higher elevation in a montane valley),abrupt boundaries will result. For example, the sharpboundary seen in Fig. 4a (around 51.0◦N 126◦W),was caused by the station at Holberg, BC, (50.65◦N128.0◦W, elevation 579 m, July precipitation 123 mm)being replaced by a station at Campbell River, BC(50.05◦N 125.32◦W, elevation 31 m, July precipita-tion 46 mm). The large differences in elevation andprecipitation between these two stations caused alarge change in the MLR coefficients, even thoughonly one in 40 stations changed. The obvious conclu-

sion is that GIDS is not as appropriate for use in areaswhere climatic gradients are both steep and variable,e.g., for precipitation in mountainous regions.

When generating digital climate maps such as Fig.4, it is difficult to say which of the two methods is moreaccurate, but the statistics for success in predicting thevalues at ‘withheld’ stations indicate that ANUSPLINis superior. Subjective assessment of the maps shownin Fig. 4 confirms that the GIDS plot is ‘rougher’ —a consequence of it being an exact technique, i.e., thesurface it generates passes through each climate stationdata value, which may result in unrealistically steepgradients at some locations. This is due mainly to theinverse-distance-squared averaging which causes thesurface to form singularities around extreme values. Itshould be noted that some of these results might alsobe due to errors in the data — because large climatedata sets are unlikely to be completely error-free! BothGIDS and ANUSPLIN have been used successfullyto locate possible errors in source data. With ANUS-PLIN, stations with high residuals from the surface aresometimes found to be incorrectly geo-referenced orthe datum has misplaced decimals. With GIDS, whichdoes not smooth the data, errors have been revealed bychecking for large differences between the interpolatedvalues and observations withheld from the analysis.

As with most alternative methods of climate data in-terpolation, both GIDS and ANUSPLIN may producemisleading results in data-sparse regions; e.g., theymay predict negative precipitation, although normallythis would be limited to zero by the interpolation al-gorithm. In the comparison presented here, however,only GIDS generated substantial areas of zero precip-itation at high elevation locations. Fig. 4 shows thatGIDS also generated more extreme values at somemid-elevation locations — a result of extrapolating el-evational gradients observed at low-elevation stationsto higher locations where such gradients probably donot apply. For ANUSPLIN, Hutchinson (1998a, b)has developed a square root transformation to reduceskewness in precipitation data which leads to morestable behaviour in regions of low rainfall (i.e., it re-duces prediction of negative values). It is also worthnoting that the poorest agreement between observedand predicted data occurred for precipitation at a highelevation station in BC, for which only 9 years of datawere available. Hutchinson (1995a) describes regres-sion methods to standardize station means in cases of


shorter climate records that can reduce errors in inter-polated annual mean data by as much as 15%.

The results of our comparisons may be sensitive toedge effects. For these comparisons, climate stationsoutside each of the two defined regions were excluded,creating artificial edges. For a point near the edge ofa region, predictions tend to be less accurate for tworeasons: first, there are often fewer climate stationsclose to that point, and second, the climate stations donot surround the point which leads to a greater likeli-hood of extrapolation rather than interpolation. GIDS,in particular, is sensitive to edge effects because of itsreliance on local gradients, e.g., for BC precipitation,pooled RMSE was 49% when only stations locatedwithin the defined BC/Alberta region were used forinterpolation, but decreased to 26% when all availableclimate stations were used. It is clear that ANUSPLINproduces better results near the coast, where additionalstations beyond the coast boundary are not available.Regardless, when using either interpolator to producea definitive map, it will be desirable to include a gen-erous number of stations beyond the region of interestto minimize edge effects.

Both GIDS and ANUSPLIN have been shown toperform at least as well as universal kriging (Hutchin-son and Gessler, 1994; Nalder and Wein, 1998). Oneof the criticisms levelled at kriging is that it requiresboth questionable assumptions and some subjectivityin interpreting variograms prior to fitting the surface.

Unlike GIDS, ANUSPLIN does require an ele-ment of subjectivity when fitting surfaces for largegeographic regions. In such cases (specifically notincluding the study reported here),‘knots’ (i.e., repre-sentative climate stations) are required. The subjec-tivity comes in deciding how many knot stations touse to obtain an optimal fit of the spline surface to thedata and the most appropriate level of data smooth-ing. Once this number has been set, knot stations areselected automatically by ANUSPLIN’s ‘Selnot’ pro-cedure, after which, interpolation proceeds relativelyquickly. The GCV statistic generated by ANUSPLINassists in objectively identifying the ‘best’ surface,and hence in determining which stations to include asknots. The GCV can be interpreted as a spatially av-eraged standard error, where the calculated standarderrors approximate the GCV over the region coveredby the data network, but increase outside the region(see Hutchinson and Gessler, 1994). In the present

study, the RMSE values for the withheld data were allsimilar to the root GCVs — hence providing a usefulindependent validation of the GCV statistic. In com-parison, GIDS is a ‘brute-force’ method that requiresno optimization — which may be a problem whenusing it to generate large grids. While ANUSPLINcan in principle be used to generate grids over largeareas relatively quickly from a single fitted surface,the potentially limited number of knots may result inlower accuracy — although preliminary tests on Cana-dian data suggest this effect is not significant (D.W.McKenney, personal communication, 1999). If a se-rious reduction in accuracy does occur, the remedy isto break the area of interest into multiple overlappingpatches where each patch has its own set of spline co-efficients, derived from all the available stations. Thiscould result in discontinuities at the boundaries of theoverlapping patches when creating composite maps,but again, this has not been found to be a significantproblem.

The results from both methods could likely be re-fined in actual applications e.g., by adding distancefrom large water bodies or slope aspect as indepen-dent variables. This capacity is already present withinGIDS and ANUSPLIN — the challenge is developingthe required data for both the interpolation componentand the grids. The number of independent variablesused by GIDS would be limited only by the avail-ability of data. A new national DEM created by theCanadian Forest Service and the Canada Centre forTopographic Information (D.W. McKenney, personalcommunication, 1999), will enable some of theseresearch issues to be investigated further.

Interpolated datasets of monthly mean data are use-ful for several purposes. One notable example is theproper parameterization of daily stochastic weathergenerators, now extensively used in the modelling ofregional-scale responses of agricultural and naturalecosystems to climate variability (e.g., Richardson,1981; Racsko et al., 1991; Jones and Thornton, 1997;Wilks, 1999) (see also Hutchinson, 1995b). It shouldbe emphasized however, that neither interpolator isrestricted in its application to monthly mean data:both should also be applicable (with some importantcaveats) to monthly time-series of actual station obser-vations. The latter would be a highly desirable exten-sion of the work reported here, but significantly moreeffort will be required to obtain the best possible re-


sults for interpolating long climate histories over ex-tensive regions. The results of the present study sug-gest that ANUSPLIN should produce better monthlyproducts, but other techniques, including GIDS, couldstill prove useful.

One example of an alternative interpolator is thePRISM model developed by Daly (Daly, 1994; Dalyet al., 1997; Daly and Johnson, 1999), which is beingused to prepare monthly precipitation and tempera-ture maps for southern British Columbia (Eric Taylor,Canadian Atmospheric Environment Service, personalcommunication, 1999). PRISM is a knowledge-basedapproach that combines local regression-based mapoutput with interpretation and evaluation by humanexperts. The knowledge base supporting PRISM fea-tures several station-weighting functions, including:multi-scale topographic facets to simulate rain shad-ows; a two-layer atmosphere to account for temper-ature inversions and mid-slope precipitation maxima;coastal proximity to handle climate gradients alongcoastal strips; and effective terrain height to accountfor the varying effects of topographic barriers on pre-cipitation. These ‘expert’ weighting functions simulateidentifiable climatological processes that have been in-vestigated and validated in regions where such dataare available, and are then assumed to apply to broadlycomparable regions where data are scarce (Christo-pher Daly, Oregon State University, personal commu-nication, 1999). Such an approach may be useful inmountainous regions where high elevation locationsare often under-represented by long-term observingstations. These assumptions should be tested for eachapplication, however, because it is possible that theexpert weightings will instead lead to increased biasesin the interpolated values. Unfortunately, testing canbe problematic since the greatest bias will generallyoccur in the locations where validation data are par-ticularly sparse.

5. Conclusions

In a comparison of two elevation-dependent spa-tial interpolators applied to 30-year monthly climatemeans for selected regions of Canada, the ANUSPLINmethod developed by Hutchinson (1995a) proved gen-erally superior to the GIDS method of Nalder andWein (1998). Both subjective assessment and statis-

tical analyses showed that ANUSPLIN is generallymore accurate in predicting climate variables at thelocations of climate stations withheld at random fromthe source datasets. Additionally, ANUSPLIN is ableto generate smoother and more credible gradients atregional boundaries and at locations where climate sta-tion coverage is poor, notably at higher elevation sites.The generally better accuracy of a thin plate spline oc-curs because it better calibrates a continuous spatiallyvarying dependence on elevation using all availabledata points. This explains why local regression meth-ods (such as GIDS) can have difficulties in areas withnon-uniform gradients.

GIDS is a useful technique nevertheless, becauseit is relatively intuitive, objective and easy to imple-ment, and performs well in comparison to most othercommonly used spatial interpolation methods. It thusprovides a useful standard against which to comparemore sophisticated interpolators such as ANUSPLIN.Both methods have the potential for increased accu-racy by introduction of additional independent vari-ables known to have effects on local climate.

Acknowledgements

Kathy Campbell, Kevin Lawrence, Pia Papadopoland Marty Siltanen of Canadian Forest Service pro-vided technical support in statistical analysis and ingeneration of the figures. We would also like to thankBill Hogg and Anna Deptuch-Stapf of EnvironmentCanada for provision of data and encouragement increating gridded climatologies for Canada.

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