THE SENSITIVITY OF TWO DISTRIBUTED NON-POINT SOURCE POLLUTION MODELS TO THE SPATIAL ARRANGEMENT OF...

THE SENSITIVITY OF TWO DISTRIBUTED NON-POINT SOURCEPOLLUTION MODELS TO THE SPATIAL ARRANGEMENT

OF THE LANDSCAPE*

PETER FISHER,1 ROBERT J. ABRAHART2 AND WERNER HERBINGER3

1Department of Geography, University of Leicester, Leicester, LE1 7RH, UK2School of Geography, University of Leeds, Leeds, LS2 9JT, UK

3Institut fuer Wasserbau und Kulturtechnik, Universitaet Karlsruhe, 76128 Karlsruhe, Germany

ABSTRACT

Distributed hydrological models are becoming increasingly complex with respect to spatial phenomena, and withthe widespread availability of spatial data from GIS, this trend is likely to increase. In all such models the spatialarrangement of phenomena, such as soil properties and land-use categories is fundamental, and so the arrangementshould have an in¯uence on the model output. Testing for this in¯uence we term spatial sensitivity analysis. Here, wereport on the spatial sensitivity of two widely used models, AgNPS (agricultural non-point source pollution model) andANSWERS (areal nonpoint source watershed environment response simulation). The input spatial data were subjectedto spatially random mixing to varying degrees, such that the organized landscape became disorganized. The chemicaldischarge from AgNPS, and the sediment and water discharge from ANSWERS, are examined. In both cases mostoutputs exhibited little or no sensitivity to the spatial distribution of most input data. Only in®ltration-related inputsproduced large variations, but these changes were not in the sense that might have been predicted. Although theanalytical methods used require further re®nement, there must now be some doubt as to the validity of the models, andwhether they repay their computational complexity. Furthermore, it is felt that spatial sensitivity analysis should becomea fundamental part of the veri®cation of all such models. # 1997 by John Wiley & Sons, Ltd.

Hydrological Processes, vol. 11, 241±252 (1997)

(No. of Figures: 1 No. of Tables: 7 No. of Refs: 30)

KEY WORDS AgNPS; ANSWERS; distributed models; randomization; spatial sensitivity analysis

INTRODUCTION

One widely used type of hydrological model today is the deterministic distributed process model (Beasleyet al., 1980; Rogers et al., 1985; Abbott et al., 1986a, b; Young et al. 1989a, b). Typically, these modelsrequire as input a gridded dataset giving a complete model of the spatial distribution of each inputparameter across the catchment. The model then simulates the processes that operate on the input stormevent as it is a�ected by the variables in each cell, cascading the surface runo�, subsurface ¯ow andentrainment of particulate and soluble pollutants. The transport, as the discharge, is modelled from cell tocell, and typically either a distributed and/or a summary solution is found.

The process of developing and testing such models usually employs some sensitivity analysis. McCuen(1973) identi®es the bene®ts of sensitivity analysis in all three phases of the modelling process: formulation,calibration and veri®cation. The methods used to date have included sensitivity to the magnitude of eachattribute and to the resolution of the spatial and temporal data. In the present work a new form of sensitivityanalysis is proposed, called spatial sensitivity analysis.

This paper starts with a review of past studies of sensitivity analysis. Next a theoretical basis forspatial sensitivity analysis is presented, and that method is then applied to the outcome of two non-point

CCC 0885±6087/97/030241±12 Received 17 July 1995# 1997 by John Wiley & Sons, Ltd. Accepted 17 November 1995

HYDROLOGICAL PROCESSES, VOL. 11, 241±252 (1997)

*This paper is based on one presented at the ASA±CSSA±SSSA Bouyoucos Conference, on Applications of GIS to the Modeling ofNon-Point Source Pollutants in the Vadose Zone, Riverside, CA, 1±3 May 1995

source pollution models, in order to assess the sensitivity of these models. The conclusion reviews theresults of this preliminary work for the two models analysed and its implications for distributed modelsin general.

SENSITIVITY ANALYSIS

To date, two types of sensitivity analysis are generally recognized. The ®rst, attribute sensitivity analysis, iswidely used in model validation. The second, resolution sensitivity analysis, is appropriate to any distributedmodel where parameters are being sampled over space or time.

Attribute Sensitivity Analysis

Attribute sensitivity analysis is commonly called, simply, sensitivity analysis. It is used to examine changesin the model output with respect to changes in variable and parameter input. Mar (1974) states thatsensitivity analysis may be performed on models constructed without data, and draws attention to thosecomponents where further research and the introduction of new variables would enhance model perform-ance. It also shows the relative importance of the parameters to the model, and shows where most care mustbe taken in collecting data to produce reliable conclusions (Beven, 1989).

The simplest and most widely used representation of sensitivity analysis is to show the percentage changein the variable and the output graphically (McCuen, 1973). This is achieved by systematically changing onevariable, and observing the output. It is, however, limited, since it only demonstrates sensitivity to variationsin one variable at a time. More sophisticated methods do exist. McCuen (1973) showed how all inputs maybe varied simultaneously using a partial di�erence equation. The latin hypercube and the associated stepwiseregression is another method for sensitizing all parameters simultaneously (McWilliams, 1987).Problems with sensitivity analysis exist. Beven (1985) showed that di�erent combinations of sensitized

variables may yield equally acceptable results, but, in calibrating the SHE model, Bathurst (1986) demon-strated that the scope for achieving acceptable calibrations based on di�erent combinations of parameters islimited when several di�erent hydrographs are considered.As well as systematic variations of variables, Monte Carlo simulation can also be used. Jakeman et al.

(1990) employed input values obtained from probability distributions, and those inputs that were importantto the model were then identi®ed from a statistical analysis of the results.

Resolution Sensitivity Analysis

The sensitivity of a model to the spatial and temporal resolution of its input data is assessed by examiningthe e�ect of varying the sampling interval of the input parameters in geographical space, and has beenreceiving increasing attention as a crucial aspect of the parameterization of a model. Bathurst (1986)examined the e�ect of both spatial and temporal resolution on the outcome of the SHE model. He showedthat the model was sensitive to these parameters, and recommended that the spacings should be smallcompared with the spatial and temporal variability being modelled. Similar investigations have shown thesensitivity of TOPMODEL to spatial and temporal resolution (Zhang and Montgomery, 1994; Bruneauet al., 1995). They show the model to be more sensitive to increases in time intervals than spatial resolution,with the model becoming inconsistent with observed data and producing meaningless results for largespatial grid sizes, and medium temporal steps.

Brown et al. (1993) have also reported on the sensitivity of the ANSWERS (areal nonpoint sourcewatershed environment response simulation) model to the spatial resolution of the inputs. They used bothsemivariograms and fractal dimensions to examine landscape diversity, and showed that the degree ofgeneralization is a signi®cant factor in determining the outcome. They showed the level of generalization atwhich the model produced repeatable results, and also the characteristics of the landscape that caused it toproduce unreliable output. Similar ®ndings were reported by Farajalla and Vieux (1995) who examined yetanother model and used entropy as a statistical parameter to assist the identi®cation of an acceptable gridresolution.

242 P. FISHER ET AL.

SPATIAL SENSITIVITY ANALYSIS

Theory

As has been stated earlier, deterministic distributed process models in hydrology use the output from alluphill grid cells as part of the input to a downhill element. In that lower element, the reaction with thephysical variables of the landscape (land use, management practice, soil type, slope, etc.) is determined,inputs from other uphill elements added, a new component of atmospheric input included and output to thenext element downhill modelled. This basic process is repeated for every element in the catchment until thesummary values have been determined for the output from the catchment. This being the case, the orderingor spatial pattern of the physical variables in the landscape can be expected to have a profound e�ect on themagnitude of the output values from the catchment. Indeed, this is one principle of soil conservation, andruno� control. For example, strip cropping along the contour alternates a fodder crop with a cereal (perhapsalfalfa and corn) to minimize soil erosion. This e�ect also encourages in®ltration, and so reduces waterdischarge. Chemical pollutants in solution can also be expected to be reduced in a manner similar to theintroduction of nitrate bu�er zones between farmland and river channels (Haycock and Burt, 1993).

A further method of sensitivity analysis for a model can therefore be suggested. The spatial pattern in thelandscape of the variables input to a model should be expected to control, to some extent, the outputvariables. This approach is termed spatial sensitivity analysis here, and can actually be applied to all spatialmodels from both the physical and built environments.

Lodwick et al. (1990) have used the term geographical sensitivity analysis, but it seems to mean varyingthe magnitude of attributes in a GIS coverage used as input to a spatial model, with no concern for thelocation of the attribute varied. On the other hand, Stoms et al. (1992) show the sensitivity of a habitatmodel for the California condor to the positional accuracy of condors sightings.

Application

Spatial sensitivity analysis has been applied to a number of areas, and a number of methods exist foraltering spatial patterns. Working with categorical data, Fisher (1991a) has shown sensitivity in valuationbased on soils for agricultural land taxation. He used the US national soil mapping standards as a baselineand perturbed soil map units according to that and information in the relevant soil survey. Goodchild et al.(1992) presented a method for simulating required levels of spatial autocorrelation in the categorical ®eldsby working on each category, as a binary occurrence in turn. Fisher (1991b) has applied an error term toperturb a surface model (a digital elevation model), and explored the sensitivity of the view-shed calculatedfrom that model.

What distinguishes the activities of Fisher (1991a, 1991b) and Goodchild et al. (1992), among manyothers, from that reported here is that they have been examining the sensitivity of models to errors inmapping. They have not been working on the sensitivity of models.

De Roo et al. (1992), on the other hand, have examined the e�ect of varying the spatial pattern in oneparameter used in the ANSWERS model. They observed that ANSWERS is very sensitive to in®ltrationrates, and that in their study area in®ltration had very low levels of spatial autocorrelation (nugget variance).They showed that simulating spatial autocorrelation for the in®ltration rate (as a continuous ®eld) reducedestimates of peak and total runo�, and of sediment yield by a few per cent each.

Method

Both process models analysed here (AgNPS and ANSWERS) use a gridded database of the geographicalvariables, and for each grid cell a set of attributes is required. These can be divided broadly into continuous(e.g. slope) and categorical data (e.g. land use). Whatever the nature of the data, the same method ofperturbation is used here:

1. The user speci®es the level of change required from within the limits of 0 to 100% of the area of thedatabase.

2. Two grid cells are selected at random and their attributes are changed.

243NON-POINT SOURCE POLLUTION MODELS

3. If the user-speci®ed level of change has been reached then the process terminates, otherwise step 2is repeated.

The randomization process used assumes that all grid cells have an equal chance of being selected, and thatsampling is without replacement, i.e. once changed the grid cell will not be selected in a subsequent call tostep 2.

Figure 1 shows the progressive e�ect of increasing percentages of change on a categorical input such asland use. It can be seen that the overall e�ect of this operation is to disorganize the landscape. The existingpattern with large contiguous and homogenous areas is changed to one with a heterogeneous distribution.This e�ectively reduces the level of organization or spatial autocorrelation in the landscape; placingareas of woodland within arable ®elds, and small patches of low in®ltration in areas dominated by highin®ltration.

As documented here, the process deliberately includes, as valid changes, those when both grid cells startwith the same attribute; these could have been disquali®ed.

Many alternative methods might have been used, including sequential indicator simulation (Beirkens andBurrough, 1993a, b). The method implemented, however, not only has the beauty of simplicity, but alsodoes simulate one of the classic approaches to pollution and erosion reduction, namely the diversi®cation ofland use.

In the research reported here, each variable studied was submitted to 25, 50, 75 and 100% swapping(Figure 1), and the model was rerun. Ten separate swappings to each proportion were determined and themodel rerun. Thus, a total of 40 model runs were performed for each variable considered.


Figure 1. Mixing the spatial pattern of the land covers of an area. (A) the original arrangement with (B) 25%, (C) 50%, (D) 75%, and(E) 100% of grid cells in the image area mixed. Note: In (E) it is apparent that some grid cells appear to retain the cover type from theoriginal (A). They have actually been changed such that they now have the cover type of a di�erent grid cell in the original data, it justhappens that the cell they have been swapped with had the same cover type. The same e�ect is present in all other realizations, and

means that actually somewhat less than the target area has been changed

SPATIAL SENSITIVITY ANALYSIS OF AgNPS

The Model

AgNPS is an event-based distributed process model developed by the USDA for the Minnesota PollutionControl Agency (Young et al., 1989a). It is intended to provide estimates of runo� water quality bymodelling single storm events in basins over 20 000 hectares. The basic outputs of the model are sedimenttransport and water quality including nitrogen, phosphorus and oxygen demand. The model incorporatesboth point source and non-point source pollution.

The model requires ®ve catchment-wide input variables and 13 distributed variables for each cell,including locational references. Some of the primary distributed variables are SCS curve number, averageslope, slope shape, average ®eld slope length, soil erodibility (K ), cropping factor (C ), management practice(P ) and aspect.

Young et al. (1989b) examined the sensitivity of the model to absolute values of the attributes. They foundthat in order of importance of the e�ect on output the model is most sensitive to land slope, soil erodibilityfactor, storm energy intensity, SCS curve number and cropping factor.

Setting Up

The Eagle Basin was selected for spatial sensitivity analysis of the AgNPS model. Data for this catchmentis supplied with the model for demonstration and training purposes. It is therefore fully populated with data,and easily available. The catchment is presumed to be in Minnesota, although this is undocumented. It is12 920 acres, divided into 323 cells of 40 acres each. The model was run with a characteristic stormprecipitation of 4.46 inches and a storm energy intensity value of 63 (the value with the test data) being theproduct of storm total kinetic energy and maximum 30-minute intensity (Young et al., 1989b). Althoughrelatively high, these are values recommended in the tutorial materials with the model, and may berepresentative of the continental Minnesota climate. Only chemical outputs from the catchment werestudied, not sediment output.

Sensitivity to Slope

The results of spatial sensitivity to land surface slope (reported as the most sensitive parameter forattribute sensitivity) are shown in Table I. We can see that, although total nitrogen and total phosphorus insediment show changes with increasing swapping, the absolute change of the mean of the sensitized data isnot very di�erent from the original, and the standard deviation around the mean estimate is negligible; thecoe�cient of variation is never more than 7% in either parameter. No other variable shows any variation atall with change in the pattern of the land slope in the catchment.


Table I. Sensitivity of AgNPS to the spatial arrangement of slope

Mixing (%)

0 25 50 75 100

Total nitrogen in sediment (lbs/acre)Mean 2.30 2.35 2.46 2.54 2.59S.D. Ð 0.05 0.12 0.13 0.16

Total soluble nitrogen in runo� (lbs/acre) 2.22 No changeSoluble nitrogen in runo� (ppm) 3.82 No changeTotal phosphorus in sediment (lbs/acre)

Mean 1.15 1.17 1.23 1.27 1.32S.D. Ð 0.02 0.06 0.07 0.09

Total soluble phosphorus in runo� (lbs/acre) 0.47 No changeSoluble phosphorus in runo� (ppm) 0.80 No changeTotal soluble chemical oxygen demand (lbs/acre) 52.79 No changeSoluble chemical oxygen demand in runo� (ppm) 91 No change

Sensitivity to Soil Erodibility

Mixing the pattern of soil erodibility has a very similar e�ect (or rather lack of e�ect) to land slope, and isshown in Table II. Again total nitrogen and total phosphorus in sediment show changes, but here thecoe�cient of variation is even smaller, never more than 2%. All other outputs are una�ected by change inthe pattern.

Sensitivity to Cropping Factor

The pattern of cropping factor has the same e�ect as land slope and soil erodibility, except that thecoe�cient of variation ranges to just over 4% (Table III).

Sensitivity to SCS Curve Number

Unlike patterns in the other three input parameters, the spatial pattern of the SCS curve number(a measure of in®ltration) causes changes in all the outputs studied (Table IV). These can be grouped intotwo types of change.

1. Total nitrogen and total phosphorus in sediment and both oxygen demand output variables show meanestimates with mixing, which are little changed from the original estimate and small coe�cients ofvariation from the mean (never as much as 2% in all four output variables).


Table II. Sensitivity of AgNPS to the spatial arrangement of soil erodibility

Mixing (%)

0 25 50 75 100



Mean 1.15 1.12 1.09 1.05 1.02S.D. Ð 0.02 0.02 0.02 0.01


Table III. Sensitivity of AgNPS to the spatial arrangement of cropping factor

Mixing (%)

0 25 50 75 100



Mean 1.15 1.15 1.15 1.14 1.12S.D. Ð 0.04 0.05 0.04 0.04


2. By contrast, the remaining variables all show increases in the means of the estimated values with mixing,while the standard deviations stay relatively constant, and so the coe�cient of variation actually falls.The magnitude of change is quite large. Thus, estimates of soluble nitrogen in the sediment increasefrom 2 to 5 lbs/acre, and of soluble phosphorus in sediment from 0.8 to 2.6. Approximately, a threefoldincrease seems typical.

The changes observed in the output parameters are contrary to the expected result with increased mixing.Owing to the mixing of cell values input to the model and so to the creation of neighbouring cells with verydi�erent values of curve number (in®ltration), it was expected that pollutants would decrease in amount, butthey are seen to increase, and the mean of the estimates are not even approximately equal to the originalmodel output in more than half of the outputs.

Summary

Most outputs are insensitive to the spatial pattern of the input data, to the extent that, of the inputparameters tested, the vast majority of output variables showed absolutely no change as a result of themixing. When change occurred it was contrary to the predicted nature of the change. The same basic result,which is not reported here, was found for a second catchment, which comes with the model for demon-stration and training.

SPATIAL SENSITIVITY ANALYSIS OF ANSWERS

The Model

ANSWERS was developed in the 1970s (Beasley et al. 1980), and, like AgNPS, it is an event-orienteddistributed model which predicts watershed behaviour during and immediately following a rainfall event


Table IV. Sensitivity of AgNPS to the spatial arrangement of SCS curve number

Mixing (%)

0 25 50 75 100


Total soluble nitrogen in runo� (lbs/acre)Mean 2.22 3.22 3.89 5.07 5.95S.D. Ð 0.57 0.69 0.64 0.46

Soluble nitrogen in runo� (ppm)Mean 3.82 5.53 6.85 8.53 10.21S.D. Ð 0.98 0.93 1.10 0.79

Total phosphorus in sediment (lbs/acre)Mean 1.15 1.15 1.16 1.16 1.16S.D. Ð 0.005 0.01 0.003 0.005

Total soluble phosphorus in runo� (lbs/acre)Mean 0.47 0.74 1.67 1.24 1.52S.D. Ð 0.15 0.24 0.16 0.15

Soluble phosphorus in runo� (ppm)Mean 0.80 1.28 1.67 2.13 2.61S.D. Ð 0.26 0.24 0.27 0.27

Total soluble chemical oxygen demand (lbs/acre)Mean 52.79 53.78 54.43 55.35 56.39S.D. Ð 0.66 0.65 0.66 0.49

Soluble chemical oxygen demand in runo� (ppm)Mean 91 92.4 93.5 95.1 96.8S.D. Ð 1.35 1.27 1.17 0.92

(primarily single storm). The primary outcomes of the model are runo� and erosion predictions. In the studypresented here the version of ANSWERS distributed with GRASS v 4.1 was used in all analyses.

The model uses four primary geographical parameters: soil, land use, elevation-based slope and aspect,and channel information, as well as details of the storm event. Thus, for each soil type eight variablesare required: total porosity, ®eld capacity, steady-state in®ltration, di�erence between steady-state andmaximum in®ltration, the rate of decrease in in®ltration with increasing soil moisture, in®ltration controlzone depth, antecedent soil moisture and erodibility. Land use types have six variables recorded.

Since it has been available for a number of years ANSWERS has been the subject of considerablesensitivity analysis. Thus, Rewerts and Engel (1991) showed that it was sensitive to the algorithm forcalculating slope from a DEM, while Wu et al. (1993) showed that ANSWERS provides a more consistentprediction of conditions than AgNPS or CREAMS. Brown et al. (1993) have examined the e�ect of spatialresolution of the database, and De Roo et al. (1989, 1992) have shown that the model is extremely sensitiveto in®ltration, both in terms of its absolute amount and its spatial pattern.

The ANSWERS model provides temporally discrete results so that for any simulation run of a particularstorm the hydrograph can be displayed, along with the sediment discharge. To make the most of theseoutputs, the Euclidean deviation (ED) was determined between the original estimate and the mean estimatewith the various levels of mixing, being the sum of all deviations at the discrete steps.

Setting Up

The tests presented here were conducted on the Weiherbach catchment in Germany, which is situatedbetween the cities of Heidelberg, Heilbronn and Karlsruhe. The catchment is dominated by loess soils(pararendzinas and colluvial soils). Various forms of arable agriculture dominate the land use, but grass(21% of the area) and woods (13%) both occur. Slopes range up to 20% showing an approximately normaldistribution with a median value of 9%. The area is highly prone to soil erosion. The catchment is the focusfor the development of a new soil erosion model (Buck and Plate, 1992). All tests reported here used a 90minute extreme rainfall event which was recorded on 25 April 1994.

Sensitivity to Slope

The results of sensitivity analysis of ANSWERS to the arrangement of slopes is shown in Table V. Thepattern reported here is not as stark as the results for AgNPS; there are changes in all output parameters.


Table V. Sensitivity of ANSWERS to the spatial arrangement of slope

Mixing (%)

0 25 50 75 100

Peak runo� (mm/h)Mean 23.11 23.18 23.05 23.08 23.03S.D. Ð 0.29 0.27 0.26 0.26

HydrographE.D. Ð 2.08 2.95 5.70 7.44

Total sediment discharge (kg)Mean 27643.0 27751.4 29 389.5 30266.5 30926.8S.D. Ð 493.98 495.18 573.60 705.61

Cumulative sediment lossE.D. Ð 7253.9 11282.0 17697.7 22480.4

Maximum sediment concentration (mg/l)Mean 15847.0 18952.1 17253.1 18113.2 18703.8S.D. Ð 1284.18 518.49 886.42 990.26

Total soil loss from all cells (kg)Mean 528542.0 544913.4 555969.7 567869.6 576975.8S.D. Ð 8996.5 8267.3 8870.1 10679.8

Peak runo�, however, changes very little, all values being approximately 23mm/h with a very small standarddeviation in the estimate. Maximum sediment concentration with simulated patterns shows a variablechange with increasing mixing, all being greater than the estimate with no mixing of slope. There is,however, no consistent rise with increasing mixing, although again the coe�cient of variation is alwayssmall.

Total sediment discharge from the catchment shows a progressive increase with increasing mixing, againwith a relatively small standard deviation (giving a coe�cient of variation of less than 3%). The totalamount of soil removed from all cells (which may be stored in the next cell downslope) shows a similarincrease with increasing mixing of slope to the total sediment discharged from the catchment.

Sensitivity to Land Use

The in¯uence of land use is reported in Table VI. The amount of variation in estimates is again smallcompared with the mean estimates (the coe�cient of variation is never more than 4%).

Peak runo� progressively decreases with mixing, from 23 to 21mm/h, and the hydrograph showsincreasing deviation from the original as measured by the Euclidean distance. On the other hand,total sediment discharge hardly changes at all and the maximum sediment concentration thereforeincreases. Total sediment loss from all grid cells shows an increase with increasing mixing, although it is notconsistent.

Sensitivity to Soil Types

Table VII reports changes related to variations in the pattern of soil types, which cause changes in thepattern of the eight related variables, including soil in®ltration, which others have shown to be the mostimportant variable in attribute sensitivity modelling.

Peak runo� again shows e�ectively no change, as does the hydrograph. Sediment-related variables,however, do change but usually by small amounts compared with the standard deviations. Thus, totalsediment discharge can be seen to increase by less than 2000 kg between the unmixed simulation and thatwith 100% mixing, and the Euclidean distance from the curves of cumulative sediment loss changes getlarger with respect to the original. Maximum sediment concentration increases as does loss of sediment fromall cells.


Table VI. Sensitivity of ANSWERS to the spatial arrangement of land use

Mixing (%)

0 25 50 75 100


HydrographE.D. Ð 1.36 2.80 3.93 5.12

Total sediment discharge (kg)Mean 27643.0 28082.8 27854.9 27740.7 27739.0S.D. Ð 497.25 554.14 472.16 813.72




Summary

As with AgNPS, the results of spatial sensitivity analysis of the ANSWERS model show many outputs tobe all but insensitive to changes in the spatial pattern of variables input to the model. Furthermore, althoughthe expectation was that the changes would generally cause a decrease in discharges and yields withmixing of the spatial pattern resulting from the creation of adjacency of erosion-favourable and erosion-unfavourable grid cells, this is not observed. Increases are much more common.

CONCLUSION

Limited testing of the e�ect of spatial pattern on the predictions of two non-point source pollution modelshas been reported. The results show those models (1) to be frequently insensitive to the spatial pattern ofsingle input variables; and (2) to produce changes with increasing mixing that are contrary to the expectedchanges; in no instance are changes reported that match the investigator's prior expectation that thereshould be reductions in all yields.

There could be a number of reasons for these ®ndings.

1. A lack of variability of important parameters within the catchments studied;

2. Key model components which have not been taken into account in the investigations; and/or

3. Variables that are not subject to spatial mixing in any run swamp the e�ect of that mixing.

The implications of the work presented here, however, are signi®cant. That the models tested seem to beinsensitive to spatial pattern in the data, suggests that they are failing to model e�ectively the distributedprocesses they are designed to address. That being the case it is reasonable to question whether the com-putational e�ort being expended on these distributed models is purposeful, and whether the lumped modelsare not more appropriate. Indeed, it is possible to ask whether our understanding of processes modelled bythese distributed models is actually good enough to be formulated in a computer model. Calibration againstmeasured catchments and attribute sensitivity analysis may be successful due to the basic similarity of manyof the formulae in the models to those in lumped models.


Table VII. Sensitivity of ANSWERS to the spatial arrangement of soil types

Mixing (%)

0 25 50 75 100


HydrographE.D. Ð 0.14 0.30 0.46 0.60

Total sediment discharge (kg)Mean 27643.0 28037.3 28391.7 28726.7 29035.6S.D. Ð 146.28 181.42 181.90 232.62




Another, more encouraging, interpretation of the results does, however, present itself. Since there isgenerally little change in the models as a result of spatial mixing, the models may actually prove to be veryrobust in their estimation, such that rearranging the distribution of single input parameters has little e�ecton outputs. That in®ltration-related inputs to the models do have an e�ect shows the importance of thoseparameters and the need for precise calibration of them. On the other hand, it is still disturbing that withthese key parameters, the sense of change in the output is not that which was expected.

Perhaps the results suggest that some variables are better incorporated within a model as distributed andsome as lumped variables. Such a mixed-parameter model would have numerous advantanges for modelusers knowing which variables require extensive parameterization and which more global values, for aparticular catchment.

The results reported do seem to recommend further research along the lines followed in this paper. Spatialsensitivity analysis, as de®ned here, would appear to provide a valid method for assessing distributedmodels. The fact that the two models addressed here fail to show consistent results under testing highlightspossible shortcomings in those models. The approach would appear to be appropriate to the nature of themodels, however, and the failure a matter of concern for those developing and using the models.

ACKNOWLEDGEMENTS

The assistance and comments of a number of individuals are gratefully acknowledged. The work reportedwas performed by R.J.A. and W.H. as MSc dissertations while studying at the University of Leicester.R.J.A. was funded by the UK ESRC, and W.H. by the German DAAD; the contributions of bothorganizations are acknowledged. Ad DeRoo of the University of Utrecht, David Unwin of BirkbeckCollege, and AndrewMillington and Jo Wood at Leicester all provided useful assistance and insights, whichare gratefully acknowledged.

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