Math Geosci (2011) 43: 265–292DOI 10.1007/s11004-011-9331-z

Estimating Freshwater Acidification Critical LoadExceedance Data for Great Britain UsingSpace-Varying Relationship Models

Paul Harris · Steve Juggins

Received: 17 June 2010 / Accepted: 31 October 2010 / Published online: 19 March 2011© International Association for Mathematical Geosciences 2011

Abstract In this study, two distinct sets of analyses are conducted on a freshwateracidification critical load dataset, with the objective of assessing the quality of variousmodels in estimating critical load exceedance data. Relationships between contextualcatchment and critical load data are known to vary across space; as such, we caterfor this in our model choice. Firstly, ordinary kriging (OK), multiple linear regres-sion (MLR), geographically weighted regression (GWR), simple kriging with GWR-derived local means (SKlm-GWR), and kriging with an external drift (KED) are usedto predict critical loads (and exceedances). Here, models that cater for space-varyingrelationships (GWR; SKlm-GWR; KED using local neighbourhoods) make more ac-curate predictions than those that do not (MLR; KED using a global neighbourhood),as well as in comparison to OK. Secondly, as the chosen predictors are not suited toproviding useable estimates of critical load exceedance risk, they are replaced withindicator kriging (IK) models. Here, an IK model that is newly adapted to cater forspace-varying relationships performs better than those that are not adapted in thisway. However, when site misclassification rates are found using either exceedancepredictions or estimates of exceedance risk, rates are intolerably high, reflecting muchunderlying noise in the data.

Keywords Acidified surface-waters · Catchment characteristics · Geographicallyweighted regression · Indicator kriging · Environmental risk · Nonparametric

P. Harris (�)National Centre for Geocomputation, National University of Ireland, Maynooth, Co. Kildare, Irelande-mail: Paul.Harris@nuim.ie

S. JugginsSchool of Geography, Politics and Sociology, University of Newcastle, Newcastle upon Tyne, UK

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1 Introduction

Resulting from the deposition of sulfur dioxide and nitrogen oxides, the acidificationof freshwaters is an environmental threat to freshwaters across large areas of up-land Britain (Mason 1993). While these acidifying compounds have natural sourcesin volcanic eruptions and emissions from the oceans, over 90% derive from anthro-pogenic activities, such as the combustion of fossil fuels at power stations and in-dustrial plants, vehicle exhausts, and agriculture. Acidified freshwaters are hostileto aquatic life; as such, continuous assessment and informed management strategiesare fundamental for their protection. One approach to protecting freshwaters involvesthe calculation of acid deposition critical loads. Critical loads are calculated in sucha way as to indicate a freshwater site’s capacity to buffer the input of strong acidanions (Nilsson and Grennfelt 1988). Critical loads are thresholds and can be com-pared directly to current and future deposition data. For sites where the depositionexceeds the critical load, acidification and associated environmental damage are ex-pected. The above approach reflects spatial variability in critical loads consideredjointly with spatial variability in deposition and allows for selecting exceeded sitesfor preferential management. The susceptibility of freshwaters to acidification tendsto vary according to the geology and land use of the catchment in question. For GreatBritain, the slow weathering geologies of upland Scotland and Wales are particularlyvulnerable (Hornung et al. 1995).

The calculation of a site’s critical load requires surface water chemistry data.Given that it is costly to collect this type of data, research has looked at ways ofpredicting critical loads at sites where water chemistry data are unavailable. In thisrespect, research has endeavoured to link critical load variation with inexpensivecatchment data for Great Britain as well as at similar spatial scales (Kernan et al.1998, 2001). These studies found moderate relationships between critical load andcatchment characteristics. In addition, they noted that the strength and nature of theserelationships could vary according to sample scale in both attribute space and geo-graphic space. However, these early studies applied only basic regression techniques,such that any local modelling, using arbitrary aspatial or spatial partitions, was rudi-mentary in design. With such results in mind, the critical load dataset of this studywas explored using geographically weighted summary statistics (GWSSs) (Harrisand Brunsdon 2010) and geographically weighted regression (GWR) (Harris et al.2010a). These studies yielded results that agreed with those of the earlier research,which demonstrates that relationships between critical load and catchment data couldvary across space. However, the use of geographically weighted models was found tobe much more flexible and informative.

As our third study of the critical load data, the present research extends such in-vestigations by proposing that GWR now be used as a predictor of critical load (andcritical load exceedance) in the expectation that it will perform well in comparisonto geostatistical predictors. In particular, the prediction accuracy using GWR andthe prediction accuracy using simple kriging (SK) with GWR-derived local means(SKlm-GWR) are compared to those of ordinary kriging (OK), multiple linear re-gression (MLR) and kriging with an external drift (KED). All multivariate modelsare informed by the catchment data, but only GWR, SKlm-GWR, and KED (usinglocal neighbourhoods) are able to cater for relationships that vary across space.

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From the outset, these predictors are designed in such a way that locally-relevantestimates of critical load prediction uncertainty and, in turn, locally-relevant estimatesof critical load exceedance risk, are rendered unlikely (Journel 1986; Goovaerts 2001;Heuvelink and Pebesma 2002). As such, these models are replaced with correspond-ing indicator kriging (IK) models. An IK model directly predicts the conditional cu-mulative distribution function (ccdf) at a target location and is thus specifically de-signed to measure local uncertainty. For this study, IK is calibrated in simple IK (SIK)and simple IK with local means (SIKlm) forms. The multivariate SKlm adaptation ofIK results in a soft indicator coding, where SIKlm-MLR is a standard model (MLRas a trend component), whilst SIKlm-GWR is a novel adaptation (GWR as a trendcomponent). Although the results of our analyses should further our understandingof the critical load process, this study places a clear methodological emphasis on themodelling techniques themselves.

2 Data

In this study, critical load values from the steady-state water chemistry (SSWC)model (Henriksen et al. 1992) for total acidity (sulphur and nitrogen) in GreatBritain are modelled. Steady-state approaches calculate values in such a way thatexceedances (critical load minus deposition) reflect potential future damage oncesteady-state is achieved. The critical load data stems from a water chemistry samplingprogramme for Great Britain that is in turn part of the UK Department of Transportand Regions critical loads mapping programme (CLAG Freshwaters 1995). Waterchemistry samples were taken during the autumn or early spring from the years 1992through 1994. In order to allow for the minimum critical load to be calculated andmapped, sites were chosen to represent the most sensitive water body within eithera 10 km grid square (for medium to high sensitive areas) or within a 20 km gridsquare (for low or non-sensitive areas). This unbalanced (or preferential) samplingcampaign creates a need for declustered data, which can ensure unbiased estimates ofany (global) moment or model parameter. Two data subsets were found: (i) a spatiallyrepresentative (declustered) dataset of 497 sites for model calibration, and (ii) a spa-tially representative (declustered set-aside) dataset of 189 sites for model validation(Fig. 1(a)). This data post-processing ensured that the distribution summary statisticsfor the calibration and validation datasets are almost identical. Units for critical load(and deposition) data are given in keq H+ ha−1 year−1.

Data that characterise a freshwater site’s catchment are used in order to shed lighton critical load variation. Three continuous and one class catchment covariates arespecified here. The continuous covariates are termed weighted geological sensitiv-ity (Wt.GSP), weighted soil buffering capacity (Wt.SBCP) and weighted soil criti-cal load (Wt.SCLP). The nominal nine-class covariate is termed dominant land cover(LC9D). Table 1 summarizes the range of values that the weighted covariates can takeaccording to an expected acid buffering capacity. Low critical load values would beexpected to correspond to low Wt.GSP/Wt.SBCP/Wt.SCLP values (and vice-versa).The land cover covariate is described in Table 2. The origins of the catchment data

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Fig. 1 (a) Location of model calibration and validation sites shown with an OK surface to all criticalload data for context (note that the Orkney and Shetland Islands in the far NE of Great Britain have nosampled sites). (b) Critical load histogram (calibration data). Local correlation surfaces for critical loadrelationships with (c) Wt.GSP and (d) Wt.SCLP

can be found in Kernan et al. (1998, 2001), while details of the particular data for-mulations used in this study can be found in Harris and Brunsdon (2010) as well asHarris et al. (2010a).

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Table 1 Continuous weightedcatchment covariates Buffering capacity Wt.GSP Wt.SBCP Wt.SCLP Acid sensitivity

Low 1.0 10.0 0.1 High

↓ ↓ ↓ ↓ ↓High 4.0 80.0 4.0 Low

Table 2 Nine-class land cover covariate

LC9D class Description LC9D class Description

1 Water and built/bare ground 6 Coniferous woodland

2 Mown/grazed turf 7 Lowland semi-natural grass/moor

3 Meadow/verge/seminatural 8 Upland semi-natural grass/bog moor

4 Tilled land 9 Upland semi-natural shrub moor

5 Deciduous woodland

3 Modelling Techniques

3.1 Models for Accurate Prediction

All models can yield prediction standard errors; however, for reasons given above,such local uncertainty measures are not (directly) used in this study. Prediction stan-dard errors from MLR and GWR fits are however needed in their respective SIKlmformulations (see Sect. 3.2.2).

3.1.1 MLR and GWR

For the dependent variable z and the independent variables y1, y2, . . . , yK , the MLRmodel has the form zα = χ0 + ∑K

k=1 χkyαk + rα , with sample data being denotedby α = 1, . . . , n. With ordinary least squares being used to find the estimator χ , aprediction from MLR at a target location x is found from

zMLR(x) = χ0 +K∑

k=1

χkyk(x). (1)

The GWR model can be defined as zα = χ0(xα) + ∑Kk=1 χk(xα)yαk + rα , where

χk(xα) is a realization of the continuous function χk(x) at sample location α (Bruns-don et al. 1996). In particular, a local MLR is calibrated at any location x, where ob-servations close to x are spatially weighted according to some kernel function. Thus,GWR parameters are estimated using a weighted least squares (WLS) approach, withweights changing according to location. Given that GWR allows a local regression tobe calibrated at any location as well as at sample locations α, its parameters can bemapped and explored. Similarly, GWR can be used as spatial predictor, with

zGWR(x) = χ0(x) +K∑

k=1

χk(x)yk(x). (2)

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For this study, the weighting matrix in GWR is specified using an (isotropic) expo-nential kernel, chosen from a preliminary experiment of GWR fits using a suite ofkernel-types. For this kernel, weights at location x accord to w(x) = exp(−dα/b),where the bandwidth is the distance b and dα is the distance between x and a sam-ple location α. An optimal bandwidth is found in an adaptive form using leave-one-out cross-validation, where the root mean squared prediction error (RMSPE) (seeSect. 3.1.3) is calculated for a range of bandwidths and the bandwidth that gives theminimum RMSPE is considered optimal. In this case, the bandwidth is a non-linearparameter, reflecting a fixed local sample size that exerts the greatest influence oneach local regression fit. Note that MLR and GWR both assume spatially indepen-dent residuals with rα ∼ N(0, σ 2I ); this is not the case for kriging.

3.1.2 OK, KED and SKlm-GWR

From a classical geostatistics viewpoint, such as that presented in Goovaerts (1997),predictions from kriging are always variations on the linear predictor z(x) − m(x) =∑n

α=1 λα(x)[z(xα) − m(xα)], where λα(x) is the kriging weight assigned to z(xα),which is taken as a realisation of the random variable Z(xα). The expected values ofthe random variables Z(x) and Z(xα) are denoted by m(x) and m(xα), respectively.The prediction error is also defined as a random variable Z(x) − Z(x), where thevariance of this prediction error σ 2

E(x) = VAR{Z(x) − Z(x)} is minimized under theconstraint that E{Z(x)−Z(x)} = 0. The weights λα(x) are found by solving a systemof linear equations calibrated with parameters of the covariogram, which is a modelof the data’s spatial dependence.

Hence, Z(x) is decomposed into a mean m(x) plus a residual R(x) component.R(x) is modelled as a stationary random function with E{R(x)} = 0 and the covari-ogram COV{R(x),R(x + h)} = E{R(x) · R(x + h)} = CR(h), where h is the separa-tion distance vector h = xα −xβ . For KED, m(x) is modelled as an unknown throughthe linear function m(x) = ∑K

k=0 χkyk(x) (i.e., an MLR fit with y0(x) = 1); this isfiltered from the linear predictor by means of constraints which give

zKED(x) =n∑

α=1

λKEDα (x)z(xα) (3)

as the KED predictor at x. Here, the weights λKEDα (x) are found from the system

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

n∑

β=1

λKEDβ (x)CR(xα − xβ) +

K∑

k=0

μKEDk (x)yk(xα) = CR(xα − x),

n∑

β=1

λKEDβ (x)yk(xα) = yk(x),

where CR(xα − xβ) is the spatial covariance of R(x) between sample locations xα

and xβ;CR(xα − x) is the spatial covariance of R(x) between sample locations xα

and target location x; and the use of constraints entails that Lagrange parameters

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μKEDk (x) are needed. If K = 0, the KED system reverts to the simpler OK system in

which m(x) is modelled as an unknown constant, and the OK predictor at x is

zOK(x) =n∑

α=1

λOKα (x)z(xα). (4)

For SK, m(x) is taken as a known constant m and the SK predictor at x is zSK(x) =m + ∑n

α=1 λSKα (x)[z(xα) − m]. The weights λSK

α (x) are found from the system∑n

β=1 λSKβ (x)CR(xα − xβ) = CR(xα − x). The SKlm-GWR model replaces m with

known local means found from a GWR fit (Harris et al. 2010b; Lloyd 2010). Hence,the SKlm-GWR predictor at a target location x is the sum of the GWR predictionfrom (2) and the SK prediction of the residual, namely

zSKlm-GWR(x) = zGWR(x) + rSK(x). (5)

The SKlm-GWR predictor is an explicit model in which the mean and residualprocesses are dealt with separately in a two-stage procedure. The OK/KED predictordeals with the mean and residual processes in an implicit fashion, in which all modelequations are solved at once. A SKlm-MLR predictor is not pursued since KED witha global neighbourhood (KED-GN) is a theoretically equivalent model (Hengl et al.2007). All systems are defined with the residual covariogram CR(h), but when themean is taken as some constant (SK and OK) only the raw data covariogram C(h) isneeded.

All forms of kriging can be applied using local neighbourhoods. This approxima-tion can account for local fluctuations in the mean and can ease computational bur-dens. In this study, only OK and KED are applied using local neighbourhoods whoseoptimal size is found using the same leave-one-out cross-validation procedure as thatused in GWR for finding the bandwidth. Crucially, the use of local neighbourhoodsin KED (KED-LN) caters for relationships that can vary across space (as the MLRcomponent fit is also local) (Wackernagel 2003). In this respect, KED-LN acts as adirect geostatistical alternative to GWR in cases where a non-stationary relationshipprocess is being modelled.

As is common practice, we find variograms instead of covariograms (whereγ (h) = C(0) − C(h) is used to relate the two). Here, we use restricted maximumlikelihood (REML) to find relatively unbiased variogram parameters for our OK/KEDmodels (Schabenberger and Gotway 2005). Optimal parameterisation via REML isnot suited to SKlm-GWR; as such, the residual variogram γR(h) from the GWR fit isfirst estimated with the usual classical estimator (Schabenberger and Gotway 2005)and then modelled using the WLS fitting approach proposed by Zhang et al. (1995).After some initial experimentation, only (isotropic) exponential variogram model-types are considered, namely γ (h) = c0 + c1(1 − exp(−h/a)). This model catersfor a (small-scale) nugget variance c0, a (large-scale) structural variance c1 (wherec0 + c1 = σ 2), and a correlation range a.

3.1.3 Model Validation

The prediction accuracy of each model is measured by the following: (a) meanprediction error, MPE = (1/N)

∑Nv=1{z(xv) − z(xv)}; (b) RMSPE =

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√(1/N)

∑Nv=1{z(xv) − z(xv)}2; and (c) mean absolute prediction error, MAPE =

(1/N)∑N

v=1 |z(xv) − z(xv)|. Here, z(xv) is the actual data, z(xv) is the predicteddata, and N = 189 is the size of our validation dataset. Note that for an optimal band-width (or neighbourhood), RMSPE follows a leave-one-out approach; as such, itsdefinition is slightly different from that given.

3.2 Models for Accurate and Precise Prediction Uncertainty Estimates

In the classical geostatistics framework, uncertainty at a target location x can be ac-counted for in three distinct ways: (i) a conditional distribution defined by the krigingprediction and its variance from (any form of) kriging; (ii) a conditional distribu-tion from a specific nonlinear method such as multi-Gaussian kriging (MK) (Emery2005), disjunctive kriging (DK) (Emery 2006b) or IK (i.e., methods that all strive tobuild the ccdf F(x; z|(Info)) = Pr{Z(x) ≤ z|(Info)} where z is some threshold and|(Info) refers to the local conditioning information); and (iii) some conditional simu-lation approach (Journel 1989).

For a number of reasons, a method from (i) or (iii) may be overlooked in favour ofone from (ii). Firstly, a basic linear method from (i) is not recommended because itssmoothing property makes it ill-suited to applications where conditional distributionsare needed (Emery 2005). Secondly, unlike methods from (ii), a method from (i) onlyprovides a prediction and an estimate of its error variance (i.e., the kriging variance)at a target location; it does not provide the shape of the error distribution (Emery2005). Thirdly, a method from (ii) is useful at locations where predictions from amethod of (i) are close to a given threshold, since it is at these locations that it is mostdifficult to correctly diagnose an exceedance (Webster 1999). Lastly, for point (notarea) prediction, a method from (iii) is not recommended, as it will simply produceresults that are broadly similar to a much simpler method of (ii), on which it is usuallybased (Goovaerts 2001).

Methods from all three approaches (i, ii, and iii) are in someway compromised byvariance and/or variogram stationarity decisions. These model decisions often entailthat estimates of local uncertainty have little or no relationship to the actual localvariability of the sample data. To counter this, methods can be adapted to allow thevariance and/or variogram to vary across space. Haas (1990), Cattle et al. (2002)and Lyall and Deutsch (2002) present adapted methods from approaches (i), (ii), and(iii), respectively. However, it is routinely observed that the IK method of (ii) cansimilarly account for the sample data in uncertainty modelling, even without anylocal variance/variogram adaptation (Goovaerts et al. 2005).

This valuable property of IK stems from its data-driven make-up (in contrast toDK and MK, which are both model-driven), and can be taken as our first reason forchoosing an IK methodology in this study. We also prefer IK (over DK or MK) be-cause it is recommended for datasets that have: (a) extreme observations, (b) a strongskew, and (c) a high degree of connectivity (Goovaerts 2009). All of these features arepresent in the critical load data (Sect. 4.1). Empirical comparisons between IK, DKand MK have often preferred (Goovaerts et al. 2005), or at least found little seriousfault (Lark and Ferguson 2004; Emery 2006a) with the IK approach. Outputs fromIK will allow us to estimate the risk of a critical load exceedance at any freshwater

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site. In this study, our exceedance thresholds will be site-specific, and will correspondto the site’s deposition rate at a given time point. Specifically, we will compare theperformance of: (a) a univariate, simple IK (SIK) model; (b) a multivariate SKlmadaptation of IK (SIKlm), where MLR is used as a trend component (SIKlm-MLR);and (c) SIKlm, where GWR is used as a trend component (SIKlm-GWR).

3.2.1 Steps of a Basic IK Model

Following the descriptions of IK presented in Goovaerts (1997) and Deutsch andJournel (1998), the steps of a basic IK model can be summarized as follows:

1. Transform the data z(xα) into proportions below and above a set of chosen thresh-olds zq with q = 1, . . . ,Q, to give a set of binary variables. Define this indicatortransform as

i(xα; zq) ={

1 if z(xα) ≤ zq,

0 otherwise.

2. Estimate and model the Q indicator variograms.3. Perform Q separate kriging runs (each using an indicator dataset and the param-

eters of its indicator variogram model) to find a set of Q estimated conditionalprobabilities (i.e., construct a discrete ccdf) at each target location.

4. Correct any order relation problems (as each conditional probability is estimatedindividually) and constrain the estimated probabilities to lie within [0, 1] at eachtarget location.

5. Interpolate/extrapolate each discrete local ccdf to find its continuous form andthen use this ccdf to assess the risk of exceedance for some critical threshold.

6. If required, calculate the mean and standard deviation of each local ccdf (i.e., theIK prediction and its standard error). Other moments can also be found.

Problems with IK largely revolve around (a) a tendency for the indicator variogramfor the important tail of the sample distribution to depict nugget variation (Step 2);(b) the method used to correct any order relationship problems (Step 4); and (c) themethod used to extrapolate the tails of the discrete ccdf (Step 5). A simplification ofIK is possible where the median indicator variogram is used for all thresholds, whichalleviates problem (a). Further, this simplification can be adapted with locally-definedthresholds, which can reduce order relationship problems (b) and can also provide abetter resolution of the discrete ccdf in areas of predominantly low- or predominantlyhigh-valued data (address problem (c)).

3.2.2 SIK, SIKlm-MLR and SIKlm-GWR

For the particular SK-based IK models of this study, we use the AUTO-IK FORTRANprogram presented by Goovaerts (2009). This program permits both SIK and ordinaryIK, and is fully automatic. We adapt AUTO-IK to incorporate both the SIKlm-MLRmodel outlined by Goovaerts et al. (2005) as well as our new SIKlm-GWR model.We choose an automatic IK program because it provides a convenient way to per-formance test our study models in an objective manner. As a result, we can formally

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define these models in the following manner, bearing in mind that in all cases, Q = 19thresholds are taken as sufficient (i.e., we use the five-percentiles of the sample cu-mulative distribution, with no additional thresholds in the upper tail); isotropic spatialcorrelation structures are assumed; and a global kriging neighbourhood is specified.

For SIK, ccdf values are estimated by applying SK to indicator transforms of thedata, yielding the estimator

FSIK(x; zq |(Info)

) =n∑

α=1

λαq(x)i(xα; zq), (6)

where the SK system∑n

β=1 λβq(x)CI (xα −xβ; zq) = CI (xα −x; zq), calibrated withthe model parameters of the indicator covariogram CI (h; zq), is solved to find theweights λαq(x) for each threshold zq . As usual, covariograms are derived from var-iograms, with each indicator variogram being estimated by the classic estimator andthen modelled using a WLS variogram fit (i.e., the same procedure as that used withthe SKlm-GWR model in Sect. 3.1.2). For each estimated variogram, exponential andspherical model-types are fitted, with the best-fitting model being retained. Followingthe definition of an exponential model given in Sect. 3.1.2, a spherical model can bedefined as γ (h) = c0 + c1((3h/2a) − 0.5(h/a)3) if h ≤ a; and as γ (h) = c0 + c1 ifh > a.

Once the nineteen ccdf values are estimated at each target location, these proba-bilities are constrained to lie within [0, 1], while all order relationship anomalies arecorrected. Continuous ccdfs are then found via the computation of one hundred per-centiles from each discrete ccdf. This step requires an interpolation between the setof nineteen probabilities and an extrapolation beyond the smallest and largest prob-abilities. Here, only linear interpolation is used, where difficult extrapolations at thetails are avoided through an interpolation between tabulated bounds of the samplehistogram. The mean and the standard deviation of each ccdf are estimated using

zSIK(x) = 1

100

100∑

l=1

zp(x), (7)

σSIK(x) =√√√√ 1

100

100∑

l=1

[zp(x) − zSIK(x)

]2, (8)

with p = 0.01 × (l − 0.5).

For the two SIKlm models, the ccdf values are estimated at location x using

FSIKlm(x; zq |(Info)

) = j (x; zq) +n∑

α=1

λ∗αq(x)

[i(xα; zq) − j (xα; zq)

], (9)

where the probabilities j (xα; zq) are known as soft indicators as they lie within [0,1],whilst the probabilities i(xα; zq) are known as hard indicators, given that they canonly be zero or one. The soft probabilities are found using

j (x; zq) = Pr{Z(x) ≤ zq |(Info)

} = G[(

Φ(zq) − zREG(x))/

σREG(x)], (10)

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where G[·] is the standard Gaussian cumulative distribution function and wherezREG(x) and σREG(x) are (i) the MLR prediction and its standard error at locationx for the SIKlm-MLR model, and (ii) the GWR prediction and its standard error atlocation x for the SIKlm-GWR model. Predictions from MLR and GWR are definedin (1) and (2), respectively. Definitions for the prediction standard errors from MLRand GWR can be found in Harris et al. (2010b). Equation (10) means that the distribu-tion of the chosen regression estimator (MLR or GWR) is used as a prior probabilitydistribution for z(x), where the soft probabilities are found directly for the normalscore transforms of the thresholds zq . The weights λ∗

αq(x) in (9) are found by solvinga SK system with the residual variable R(x; zq) = I (x; zq) − j (x; zq). The residualindicator variograms for a SIKlm model are estimated and modelled using the sameprocedure as that defined for SIK. Similarly, once the nineteen ccdf values are esti-mated at each target location, all other aspects of a SIKlm model mirror those of theSIK model.

3.2.3 Model Validation

The value of an IK model’s prediction uncertainty estimates (at the validation sites) ismeasured in (i) a global-sense, by the mean square standardised residual (MSSR), and(ii) a local-sense, by prediction confidence interval (PCI) diagnostics. In this study,MSSR = (1/N)

∑Nv=1[{z(xv) − zIK(xv)}/σIK(xv)]2 where an MSSR value less than

1 implies that the IK standard errors tend to overestimate the actual IK errors (andvice versa). The accuracy of an IK model’s PCIs can be assessed using coverageprobabilities. For example, if symmetric 95% PCIs were calculated at each valida-tion site, then a correct modelling of local uncertainty would entail there is a 0.95chance that the actual value z(xα) falls within the interval. If a coverage probabilityis found for a range of symmetric PCIs (e.g., from a 5% to a 95% PCI in incrementsof 5%) and the results plotted against the probability interval p, then an accuracyplot is found. Accuracy plots can be summarized by the G-statistic, which is definedas G = 1 − (1/Q∗)

∑Q∗q=1 wq |p∗

q − pq | with 0 ≤ G ≤ 1 and where a value of 1 issought. In this case p∗

q and pq are the observed and expected coverage probabilities,respectively, wq is equal to 1 if p∗

q > pq and is equal to 2 otherwise, and Q∗ repre-sents the discretization level used in the computation. The weights wq give favour tothe accurate case (i.e., where the fraction of true values falling in the PCI is largerthan expected). For cases where two IK models provide similar accuracy plots, onemodel can be preferred if its PCI widths containing the actual value are narrower (i.e.,more precise). The average width of these local PCIs should also be smaller than theglobal PCI inferred from the sample histogram. In this respect, the average width ofthe PCIs that include the actual value are calculated for each p and plotted againstthe corresponding PCI inferred from the global histogram. This PCI width plot canbe summarized with the standardized width (SW) diagnostic, which is the ratio ofthe local PCI width versus the global PCI width, averaged over a series of p. Thisdiagnostic should be as small as possible.

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4 Analysis

4.1 Exploration of the Critical Load Data: Results of Previous Studies

Following the GWSSs investigations conducted in Harris and Brunsdon (2010), thereis evidence for: (a) both global and local spatial trends in critical load variability;(b) skew nonstationarity (globally, the critical load distribution is positively skewedwith its median value 42% lower than its mean); and (c) influential spatial outliers innorth-western England. Furthermore, a multi-modal histogram suggests evidence oftwo or three critical load populations (see Fig. 1(b)). The correlation between criticalload and Wt.GSP/Wt.SBCP/Wt.SCLP is moderate with p = 0.58/0.65/0.54; andsimilar critical load explanatory powers are likely with the nominal LC9D. In general,change in the relationship between these data only occurs at a fairly large spatialscale. In this study, Wt.GSP has the strongest non-stationary relationship to criticalload, whilst Wt.SBCP has the strongest stationary relationship. Correlation surfacesdepicting the two strongest non-stationary relationships are given in Figs. 1(c) and (d)(Wt.GSP and Wt.SCLP with critical load). Surfaces are specified with an exponentialkernel using an adaptive bandwidth of 3.62% (which matches that of an AIC-definedGWR model in Harris et al. 2010a) and are shown with randomisation test results(at a 95% level of confidence) where places of unusual correlations are highlighted(Fotheringham et al. 2002; Harris and Brunsdon 2010).

The AIC-led investigations described in Harris et al. (2010a), indicate that a MLRmodel with the form CriticalLoad = f (Wt.GSP,Wt.SBCP,LC9D2,LC9D3,LC9D4)

provides the most parsimonious global regression fit, whereas a GWR modelof form CriticalLoad = f (Wt.GSP,Wt.SBCP,Wt.SCLP,LC9D2,LC9D3,LC9D4)

provides the most parsimonious local (and overall) fit (i.e., Wt.SCLP is only impor-tant locally). More specific investigations with the same GWR model indicated thatWt.GSP and Wt.SCLP have strong non-stationary relationships with critical load,whilst all other critical load relationships (i.e., those with Wt.SBCP and the threeland cover classes) could be viewed as stationary. Furthermore, the perceived natureand strength of critical load relationships can sometimes depend on a few (outlyingbut valid) observations.

4.2 Critical Load Prediction with MLR, GWR and Kriging

4.2.1 Covariate Subset Selection

For this study, we first conduct prediction-only investigations, in order to evaluatewhether

(i) CriticalLoad = f (Wt.GSP,Wt.SBCP,Wt.SCLP,LC9D2,LC9D3,LC9D4),(ii) CriticalLoad = f (Wt.GSP,Wt.SBCP,LC9D2,LC9D3,LC9D4), or

(iii) CriticalLoad = f (Wt.GSP,Wt.SCLP)

should be taken as the underlying trend function in the following five multivariatemodels: MLR, GWR, KED-GN, KED-LN and SKlm-GWR. All three options followfrom the results of Sect. 4.1, with the third option being the one most suited to a non-stationary model. Other covariate subset permutations were not considered. Thus,

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Table 3 Prediction accuracy at validation sites: covariate subset selection

Model MPE RMSPE MAPE

Meana −0.04 5.61 4.59

OK −0.05 4.46 3.14

1. Covariates: Wt.GSP, Wt.SBCP, Wt.SCLP, LC9D2, LC9D3, and LC9D4

MLR-1 −0.12 4.21 2.96

GWR-1 −0.16 4.15 2.84

KED-GN-1 −0.11 4.18 2.89

KED-LN-1b −0.24 4.24 2.92

SKlm-GWR-1 −0.17 4.13 2.82

2. Covariates: Wt.GSP, Wt.SBCP, LC9D2, LC9D3, and LC9D4

MLR-2 −0.16 4.23 2.98

GWR-2 −0.29 4.20 2.94

KED-GN-2 −0.15 4.22 2.91

KED-LN-2b −0.19 4.22 2.92

SKlm-GWR-2 −0.17 4.06 2.76

3. Covariates: Wt.GSP and Wt.SCLP

MLR-3 0.13 4.47 3.22

GWR-3 0.09 4.05 2.82

KED-GN-3 0.10 4.09 2.87

KED-LN-3b 0.03 3.91 2.72

SKlm-GWR-3 0.03 3.99 2.77

aMean is where the calibration data mean is used as the critical load predictor

bFor neighbourhood selection, catchment data with a small random error addition (between 0.0001 and0.001) was necessary to avoid matrix instability. Sensitivity analyses indicated that this had a negligibleeffect on model performance. Once a neighbourhood was defined, this random error addition was notneeded for KED-LN-3, our chosen study model

fifteen multivariate models are assessed for prediction accuracy at the validation sitesand the results are given in Table 3. The prediction results for the univariate OKmodel are also presented at this stage.

As expected, the multivariate models almost always out-perform OK, thereby pro-viding some value to the catchment data. Here, OK is only of value, relative to a MLRmodel calibrated with the third (smallest) covariate subset. Generally, little is gainedby specifying any spatial model with respect to the aspatial MLR model in caseswhere the land cover and Wt.SBCP covariates are included. The GWR, KED, andSKlm-GWR models are of much greater value (relative to a MLR model) when thesecovariates are excluded (i.e., with the third covariate subset). This fact is related tomodel misspecification, since it entails that important spatial effects can be (partially)attributable to missing covariates.

Results suggest that we should continue our analysis using only models speci-fied with the third trend function, where both critical load relationships are stronglynon-stationary. This decision should not be dismissed as though it were a fabrication

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designed to promote our (non-stationary focused) model evaluation; indeed, it is infact common for some catchment covariates to be unavailable at some sites (Kernanet al. 1998, 2001). In any case, the most accurate critical load predictions are foundwith spatial models using this reduced trend function. This result follows from thefact that we allow spatial information to have a more important role in the predictorover covariate information. We thus assume that, for whatever reason, Wt.GSP andWt.SCLP are the only covariates available; and as such, in the remainder of the studywe only interpret models that use this covariate information.

For our chosen covariate subset, KED-LN is the most accurate predictor andMLR is the least accurate. The difference between their RMSPE values is 0.56keq H+ ha−1 year−1; that is, KED-LN offers a prediction improvement of 12.5%. Be-hind KED-LN, SKlm-GWR performs reasonably well. It can be observed that KED-GN performs almost as well as GWR, even though it models stationary relationships.This is not surprising, given that any deficiency in KED-GN’s trend component pre-dictions can be re-addressed in its residual predictions. The strong performance ofKED-GN also reflects its status as a best linear unbiased predictor. In summary, mod-els that account for relationships that vary across space are the most accurate criticalload predictors.

4.2.2 Model Calibration Details

The bandwidth function for our chosen GWR model is well-behaved, with a clearminimum reached at 2% of local sample data (see Fig. 2(a)). The optimal krigingneighbourhoods for OK and for our chosen KED-LN model are taken at 20% and16% of local sample data, respectively. Variography for the kriging models is givenin Figs. 2(b) to 2(d), where all estimated variograms are specified with a lag inter-val of 20 km. For OK, the variogram parameters are estimated at c0 = 18.74, c1 =33.63, and a = 450 km; for KED, they are estimated at c0 = 17.15, c1 = 3.76, anda = 114 km; and for SKlm-GWR, they are estimated at c0 = 12.16, c1 = 2.40, anda = 303 km. The nugget variances are estimated relatively high, reflecting an under-lying noise in the critical load data (Sect. 5). This noise explains why none of ourmodels can be considered a particularly accurate predictor of critical load (althoughreassuringly, all models predict better than the calibration data mean). Observe alsothat the spatial dependence in the residual process (for KED and SKlm-GWR) is notparticularly strong.

4.2.3 Actual Versus Predicted Relationships

To provide a local assessment of model prediction accuracy, scatterplots and localcorrelation surfaces of z(xv) and z(xv) are presented in Figs. 3, 4, 5 for the five mul-tivariate models (using our chosen covariate subset) and the OK model. Correlationsurfaces are specified with a bi-square kernel using an adaptive bandwidth of 25%and are shown with randomisation test results (at a 95% level of confidence) whereplaces of unusual correlations are highlighted. All correlations were checked for anysevere scaling issues; the correlation surfaces can only reflect a rough guide to re-gional differences in model performance, as the data are on point support.

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Fig. 2 (a) Bandwidth function for GWR (shown with selective filtering of bandwidth size). REML vari-ogram fits for (b) OK and (c) KED (both fits shown with corresponding classic variogram estimators forcontext only). (d) WLS variogram fit for SKlm-GWR (note that the WLS fit is to a judged truncation ofthe estimator). Observe that each variogram plot is given with a different semivariance scale

From the scatterplots, all models suffer from an under-prediction of the largestcritical load values. From the surfaces, OK is the least accurate in southern Scotlandand northern England (an area of high critical load variability) and in central to south-east England (an area of negative skew, dominated by high critical load values). Forthe multivariate models, the use of catchment information clearly improves predic-tion accuracy in central to southeastern England (and to a lesser extent, in southernScotland and northern England). However, prediction accuracy for MLR and KED-GN is the weakest of all models in the far northwestern corner of Scotland, whichsuggests that the latter is a region most at odds with an assumption of stationary re-lationships. All multivariate models are relatively accurate in Wales and southeastern

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Fig. 3 Actual versus predicted for (a) OK and (b) MLR: scatterplots and local correlation surfaces

England, and the non-stationary relationship models (GWR, KED-LN, and SKlm-GWR) can provide similar levels of accuracy in parts of eastern England and easternScotland. All models have weak prediction accuracy in northwestern England (andsouthwestern Scotland), an area of known critical load outliers.

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Fig. 4 Actual versus predicted for (a) GWR and (b) KED-GN: scatterplots and local correlation surfaces

4.3 Critical Load Prediction Uncertainty with IK

4.3.1 Model Calibration

A representative subset of the nineteen indicator variograms for SIK, SIKlm-MLR,and SIKlm-GWR is presented in Figs. 6(a) to (c). Variograms for the 5% (the lowest),25%, 50%, 75%, and 95% (the highest) thresholds are given. For SIK, variogramsat the lower thresholds can display a large nugget effect (defined as c0/(c0 + c1)),

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Fig. 5 Actual versus predicted for (a) KED-LN and (b) SKlm-GWR: scatterplots and local correlationsurfaces

together with a long correlation range. The SIK variogram at the lowest thresholdsuffers from an unusual behaviour at the origin, which is likely due to the influenceof outlying data. The SIK variogram at the highest threshold displays a relativelysmall nugget effect but with a short correlation range, reflecting the existence of afew small clusters of high critical loads.

A consequence of adopting a detrended IK model is most clearly seen in the be-haviour of the middle threshold variograms. Here, for SIK, variograms at the 50%

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Fig. 6 Estimated and modelled indicator variograms at the 5%, 25%, 50%, 75%, and 95% thresholdsfor (a) SIK, (b) SIKlm-MLR, and (c) SIKlm-GWR. All variograms are rescaled by the variance of theindicator variable. Variograms for SIK reflect raw information, whilst variograms for SIKlm-MLR andSIKlm-GWR reflect residual information

and 75% thresholds depict a clear drift in the spatial process. However, this be-haviour largely disappears in the corresponding residual variograms for SIKlm-MLRand SIKlm-GWR, as catchment information now accounts for this trend. This in turncreates a fairly weak spatial dependence across most thresholds for both detrended

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Table 4 Prediction and prediction uncertainty accuracy at validation sites: IK models

Model MPE RMSPE MAPE MSSR G SW EC

OK −0.05 4.46 3.14 0.913 0.922a 0.738a 0.03KED-LN 0.03 3.91 2.72 0.796 0.904a 0.686a 0.08SIK −0.08 4.47 3.15 0.944 0.961 0.844 0.37SIKlm-MLR 0.09 4.12 2.84 1.011 0.971 0.722 0.41SIKlm-GWR 0.06 4.03 2.78 1.004 0.965 0.641 0.47

aEstimated in a manner that enables direct comparison with the IK models

Fig. 7 Actual versus predicted for SIKlm-GWR: scatterplots and local correlation surfaces

IK models. Weak spatial dependence of this sort is most prevalent in SIKlm-GWR.In summary, the variography for all three IK models appears to be reasonable.

4.3.2 Assessment of IK Models: Accuracy and Precision

The IK models are first assessed for critical load prediction accuracy, with the re-sults given in Table 4. In order to provide a context, these results are given with theprevious results of OK and KED-LN. The prediction accuracy of the IK models isrelatively good where SIK is almost as accurate as OK and where SIKlm-MLR andSIKlm-GWR both perform reasonably well in comparison to KED-LN. Given thatSIKlm-GWR is the most accurate IK model, its scatterplot, as well as its local corre-lation surface of z(xv) and z(xv), is presented in Fig. 7 (using the same specificationsas described in Sect. 4.2.3). In this case, SIKlm-GWR performs in a similar mannerto the other non-stationary relationship models, which is a promising result for ournew model.

Next, each IK model’s estimates of critical load prediction uncertainty are evalu-ated and the corresponding results for the OK and KED-LN models are also given.

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Fig. 8 Accuracy plots (a) KED-LN, (b) SIK, (c) SIKlm-MLR, and (d) SIKlm-GWR. An output abovethe 45° line is favoured for a more accurate uncertainty model

For the latter, parametric models, an assumption of multivariate normality is needed,which entails that z(x) and its standard error σK(x) can be taken as the two definingmoments of a normal distribution at x. Definitions for OK and KED-LN standarderrors can be found in Goovaerts (1997). Performance results are given in Table 4,together with the associated plots in Figs. 8 and 9 (for the IK models and KED-LNonly). Models that perform well in a global sense (via MSSR) are, firstly, SIKlm-GWR, secondly, SIKlm-MLR, and, thirdly, SIK. In a local sense and via the G-statistic, SIKlm-MLR performs the best, yielding marginally more accurate resultsthan those derived through SIKlm-GWR. Here, both models suffer from dips belowthe 45o line in their accuracy plots (i.e., their PCIs contain a lower than expectedproportion of true values in such places). However, because SIKlm-GWR has a muchlower SW diagnostic than SIKlm-MLR, it is viewed as the best performing modelin this local sense (i.e., SIKlm-GWR provides the most accurate and precise PCIs).All IK models perform better than their standard kriging counterpart (OK and KED-LN), which is not surprising given that IK is specifically designed to measure localuncertainty.

Further locally-orientated assessments can be made by relating the absolute pre-diction errors |z(xv) − z(xv)| to the (estimated) prediction (or ccdf) standard errors

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Fig. 9 PCI width plots (a) KED-LN, (b) SIK, (c) SIKlm-MLR, and (d) SIKlm-GWR. An output abovethe 45° line is favoured for a more precise uncertainty model

σK(xv). These data should have a regression fit with a slope of one that passes throughthe origin. As expected, OK and KED-LN perform weakly in this respect, whereasthe IK models perform relatively well. Scatterplots of this data for KED-LN and theIK models are presented in Fig. 10. In this case, all IK models are able to providevariability in their ccdf standard error distribution, and in doing so, these estimatescan correlate with the absolute prediction errors. These correlations (termed EC) aregiven in Table 4, where, again, the SIKlm-GWR model performs the best of all.

It can also be useful to investigate the spatial pattern of a model’s estimates oflocal uncertainty. To this end, prediction (or ccdf) standard error surfaces are givenin Figs. 11(a) and (b) for KED-LN and SIKlm-GWR, respectively. Clearly, KED-LN standard errors exhibit little spatial variation, whilst the ccdf standard errors forSIKlm-GWR do. It is well known that kriging standard errors from any standardmodel depend solely on data geometry (e.g., large values tend to be where samplingis sparse). Relating KED-LN standard errors to calibration data density allows us tosee this effect (see the kernel density estimation (KDE) surface given in Fig. 11(c)).Conversely, SIKlm-GWR standard errors can reflect the actual local variation foundin the critical load data (compare Fig. 11(b) with the standard deviation surface givenin Fig. 11(d)). For example, a region of low ccdf standard errors can be found innorthern Scotland, reflecting an area of low critical load variation (and low critical

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Fig. 10 Scatterplots of absolute (actual) prediction errors versus (estimated) standard deviation of localprobability distribution: (a) KED-LN, (b) SIK, (c) SIKlm-MLR, and (d) SIKlm-GWR

loads). The SIKlm-GWR standard errors can also reflect data geometry (Goovaerts2009) as those from KED-LN do, but this is not immediately apparent. For Fig. 11,the KDE surface is specified with a quartic kernel using an adaptive bandwidth of5%, and all other surfaces are specified with a bi-square kernel using an adaptivebandwidth of 25%.

4.4 Site Misclassification with Critical Load Exceedance Data

The mapping of exceedances is fundamental to the critical load concept. Maps areused for policy-making and for setting target deposition loads. For exceeded sites,two avenues are likely: (a) reduction of (near-by) deposition rates, or (b) physical

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Fig. 11 (a) Prediction standard error surface for KED-LN (smoothing uses the validation site estimates);(b) ccdf standard error surface for SIKlm-GWR (smoothing uses the validation site estimates); (c) KDEsurface to assess the spatial configuration of the calibration data; and (d) local standard deviation (SD)surface for the critical load calibration data

neutralization of freshwater acidity (e.g., by the addition of some alkali compound).We now investigate two aspects of the application of the critical load concept asper the use of a critical load prediction model. Firstly, critical load predictions areused simply to provide critical load exceedance predictions. Secondly, critical loadprediction uncertainty estimates are used to estimate the risks of a critical load ex-

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Table 5 Percentages of falsenegatives, false positives, andfalse classifications

Values above the criticalprobability threshold (CPT) areclassified as critical loadexceeded (observe that CPTchanges for each model)

Model CPT (%) False negative False positive False

Via critical load prediction accuracy:

OK – 90.7 0.0 25.9

SIK – 88.9 0.0 25.4

KED-LN – 75.9 3.7 24.3

SIKlm-GWR – 72.2 4.4 23.8

Via critical load prediction uncertainty accuracy:

OK 37.9 51.8 23.0 31.2

SIK 44.1 50.0 22.2 30.1

KED-LN 41.0 42.5 19.3 25.9

SIKlm-GWR 42.9 38.9 17.8 23.8

ceedance. In both cases, the results are compared with actual exceedance data. Inthis study, a deposition dataset for the years 1992 to 1994 is used, and this data is(naively) assumed to be free of error (Sect. 5). At the validation sites, depositionranges from 0.56 to 4.90 keq H+ ha−1 year−1 and actual critical loads range from 0 to26.40 keq H+ ha−1 year−1. Hence, with a simple subtraction, the actual exceedancedata can be seen to range from −24.88 to 4.73 keq H+ ha−1 year−1. This results in29% of the sites having true or positive exceedances. It should be noted that the ex-ceedance performance of a given critical load prediction model will depend on thespatial distribution of our chosen deposition dataset.

We only assess the exceedance performance of OK, SIK, KED-LN, and SIKlm-GWR, because this selection provides us with a useful univariate to multivariate com-parison while also including our best performing models (i.e., KED-LN for criticalload prediction and SIKlm-GWR for critical load prediction uncertainty). For eachmodel (and for both exceedance assessments), the percentage of exceeded sites andthe percentage of not-exceeded sites that are incorrectly identified are classified usingrates of false negatives and false positives. Thus, a validation site that is wrongly de-clared as safe from acidification is given a false negative, while a site that is wronglydeclared as acidified is given a false positive. We also calculate the overall misclassi-fication rate.

Table 5 summarizes the exceedance prediction results. All models have weak ex-ceedance prediction accuracy, which produces a high rate of false negatives com-bined with a low rate of false positives. Such results are not good enough within thecontext of practical freshwater management. Surprisingly, the most accurate modelis SIKlm-GWR, with an overall misclassification rate of 23.8%. We calculate ex-ceedance risk by relating a validation site’s estimated critical load conditional distri-bution to a threshold, which is defined by the site’s deposition value. In this case, weuse critical probability thresholds (CPTs) so that a validation site can be classified (asexceeded or not exceeded) according to some specified level of risk. For this study,a CPT is chosen in such a way that the estimated percentage of exceedances exactlymatches the actual percentage of exceedances found in the calibration data, whichis 30% (Goovaerts and Journel 1995). Accordingly, the resultant misclassificationrates are given in Table 5. Again, all models produce higher rates of false negatives

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than of false positives. The SIKlm-GWR model is the best performing model with anoverall—but again, intolerable—misclassification rate of 23.8%.

5 Discussion and Conclusions

Our results demonstrate the clear value of accounting for relationships that varyacross space, when specifying a critical load prediction model. For the particularstudy data as well as for the particular set of models compared, the non-stationarymodels out-perform their stationary counterparts in all aspects. Firstly, it is foundthat kriging with an external drift (using local neighbourhoods) is the most accu-rate critical load predictor. This predictor is marginally more accurate than krigingwith geographically weighted regression (GWR) as its trend component. These re-sults concur with the simulated data study of Harris et al. (2010b) for the same modelforms. Secondly, it is found that a novel indicator kriging (IK) model (where GWR isagain used as a trend component) provides the most accurate and precise critical loadprediction uncertainty estimates. The same IK model also predicts critical loads rel-atively well. This is a very promising result and IK models of this form are likely tobe of value in other applications where non-stationary relationships are evident. Fur-thermore, this GWR-detrended IK model could be embedded within a conditionalsimulation algorithm in cases where issues of spatial uncertainty are important.

A caveat to these results is that model calibration and output depend strongly onreliable data. As already discussed in Harris et al. (2010a), strong misgivings can behad concerning the reliability of both the critical load and catchment data. Uncer-tainty in the critical load data was borne out by the high nugget variances estimatedin this study’s variography. Further research could gauge how sensitive the results ofthis study are to such data uncertainties. Ultimately, however, it is likely that data reli-ability should be improved, namely through a re-examination of the critical load andcatchment data sampling and measurement programme. For the moment, a compro-mise may be arrived at by predicting into critical load exceedance bands. This type ofprocedure would discriminate between sites that have a clear exceedance and thosewhose exceedances are small. It would also provide a useful practical tool, since itwould allow for those sites clearly exceeding to be managed as such, and for thosesites whose deposition is close to its critical load to be re-sampled for water chemistryso as to confirm their status.

This study, together with the related (non-stationary variance/variogram) study byHarris et al. (2010c) has now evaluated numerous predictors using the same dataset.Most of these predictors adopt a classical stance to uncertainty. Given that this view-point can be rather restrictive, we are currently researching Bayesian space-varyingrelationship models such as that presented by Gelfand et al. (2003) with a particularfocus on how they can be adapted to provide locally-relevant measures of predictionuncertainty. An application of such models to the critical load data may help accountfor the data uncertainties discussed above.

Finally, we assumed that the deposition corresponding to each freshwater site’scritical load is free from error. This is clearly a naïve assumption, as the depositiondata are model predictions from a relatively sparse network of monitoring sites and

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have substantial uncertainty. Future research should not only predict critical loads andtheir error (i.e., site-specific threshold prediction), but also predict deposition and itserror (i.e., contaminant prediction). This will enable the results of the two sets ofpredictors to interact, with models for each process providing their own assessmentsof prediction uncertainty. Van Meirvenne and Goovaerts (2001) present a noteworthyexample of such interacting prediction models, in which IK is used to investigate soilcontamination in the vicinity of a zinc-ore smelter.

Acknowledgements Research presented in this paper was funded by a Strategic Research Cluster grant(07/SRC/I1168) from the Science Foundation Ireland under the National Development Plan. The authorsgratefully acknowledge this support. Thanks are also due to the first author’s PhD studentship at NewcastleUniversity and to Martin Kernan at University College London who provided the study data. Critical loaddata similar to those used here can be found at http://critloads.ceh.ac.uk/index.htm.

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