Modeling phosphorus in the environment - Chapter 6 pot

It is important to recognize from the outset, however, that this will be difficult:1 to evaluate model structures as working hypotheses about the functioning ofcatchment systems independ

in Phosphorus Models

Keith Beven

Lancaster University, Lancaster, United Kingdom

Trevor Page

Lancaster University, Lancaster, United Kingdom

Malcolm McGechan

Scottish Agricultural College, Bush Estate, Penicuik,

United Kingdom

CONTENTS

6.1 Sources of Uncertainty in Modeling P Transport to Stream Channels 132

6.2 Sources of Uncertainty 133

6.3 Uncertainty Is Not Only Statistics 134

6.4 Uncertainty Estimation: Formal Bayes Methods 135

6.5 Uncertainty Estimation Based on the Equifinality Concept and Formal Rejectionist Methods 137

6.6 Uncertainty as Part of a Learning Process 140

6.7 An Example Application 142

6.7.1 The MACRO Model 142

6.7.2 Study Site and Data 143

6.7.2.1 Drainage Discharge and Phosphorous Concentrations 143

6.7.2.2 Slurry Applications 144

6.7.3 MACRO Implementation within a Model Rejection Framework 144

6.7.4 Results and Discussion 146

6.7.4.1 Using Initial Rejection Criteria 146

6.7.4.2 Using Relaxed Rejection Criteria 148

6.7.4.3 Simulations for the Period from 1994 to 1995 148

6.7.4.4 Simulations and Parameterizations for the Period 1995 to 96 150

6.8 Learning from Rejection: What If All the Models Tried Are Nonbehavioral? 153

6.9 What Are the Implications for P Models? 155

Acknowledgments 157

References 157

6.1 SOURCES OF UNCERTAINTY IN MODELING

P TRANSPORT TO STREAM CHANNELS

The starting point for this contribution is the extensive review of Beck (1987) marizing his arguments at the end of the review, he posed the following questions:Are the basic problems of model identification ones primarily of inadequatemethod or of inadequate forms of data?

Sum-What opportunities are there for the development of improved, novel methods

of model structure identification, particularly regarding exposing the failure

of inadequate, constituent model hypotheses?

How can an archive of prior hypotheses be appropriately engaged in inferringthe form of an improved model structure from diagnosis of the failure of

an inadequate model structure? Moreover, in what form should the edge of the archive be most usefully represented?

knowl-What does a lack of identifiability imply for the distortion of a model ture, and what are the consequences of a distorted model structure in terms

of the environmental modeler has been model calibration If a model has at leastapproximately the right sort of functionality, then there are generally sufficientdegrees of freedom to be able to adjust effective values of the parameters to get anacceptable fit to the data and to declare some sort of success in reporting results inscientific articles and reports to decision makers This obviously does not mean thatwhat is being reported is good science if the calibration allows compensation forerrors in model structure as a representation of the processes actually controllingwater quality variables, including phosphorus (P) concentrations in different forms.Perhaps we are now reaching a stage where it might be possible to take account ofsome of the sources of uncertainty in predicting water quality more explicitly, Pbeing a particularly interesting and practically relevant example

It is important to recognize from the outset, however, that this will be difficult:(1) to evaluate model structures as working hypotheses about the functioning ofcatchment systems independently of errors in the input data used to drive themodel and the calibration of effective parameter values; and (2) to estimate effec-

tive values of physical and geochemical parameter values a priori by measurement.

The struggle to improve water-quality modeling remains as much a struggle againstthe limitations of current measurement techniques as against the limitations ofcurrent model structures

6.2 SOURCES OF UNCERTAINTY

The sources of uncertainty in the modeling process are manifold, and, generallyspeaking, good methodologies have not been developed for assessing the nature andmagnitude of uncertainties from different sources They are thus frequentlyneglected For example, some uncertainty exists in the input and boundary conditiondata used to drive a model Such uncertainties include measurement errors in assess-ing the inputs at the measurement scale, together with interpolation errors in spaceand time to provide the values required at the lumped or distributed element scale

of the model The interpolation error may be made worse by a lack of resolution inthe measurements in space and time and by nonstationarity in the processes con-trolling the inputs Rainfall is a good example There are issues about all themeasurement techniques available to estimate rainfalls, both at a point using gauges

or over an area using radar or microwave techniques Point ground-level ments may be sparse in space, whereas the spatial and temporal variability of rainfallintensities may vary markedly between events Rainfall may show fractal character-istics in space and time, but analyses suggest that there may be nonstationarity inthe fractal scaling between events Thus, interpolation of the measurements to pro-vide the inputs — and an estimate of their uncertainty — at the space and timescales of the model may be difficult What is clear is that a point measurement ofrainfall is, under many circumstances, not a good estimate of the rainfall inputsrequired by the model The two variables may, because of time and space variability,actually be related but different variables — they are incommensurate Yet rainfalldata are essential to drive models that will predict the fluxes in hydrological pathwaysthat will control the transport of P However, the number of nonhypothetical hydro-logical modeling studies that have attempted to include a treatment of rainfallestimation error is very small indeed

measure-The problem is compounded by other uncertainties Most particularly for based simulations, errors in the estimation of model initial conditions may beimportant Errors may be associated with the model structures used due to theincorrect representation of some processes or the neglect of processes (e.g., prefer-ential flow pathways) that are important in the real system There may be errors inestimating or calibrating the effective values of parameters in the model that maycontrol the predictions of P mobilization and transport in different pathways Finally,there may be errors in the observations used to evaluate the model predictions or tocalibrate the model parameters

event-Unfortunately, the possibility of assessing all these different sources of error islimited In general, only the total model error produced can be assessed by comparing

an observation, which is not error-free, with a model-predicted variable produced

by a model, which is subject to structural and input errors Unless some very strong

— and usually difficult to justify — assumptions are made about the nature of thesources of error, disaggregating the total model error into its component parts will

be impossible It is an ill-posed problem The result will be an inevitable ambiguity

in model calibrations and error assessment — an ambiguity that also brings with itdifficulty in transferring information gained in one application to applications atother sites or different hydrological conditions

6.3 UNCERTAINTY IS NOT ONLY STATISTICS

The aim of science, however, is a single true description of reality The ambiguityarising from uncertainties from these different sources means that this aim is difficult

to achieve in applications to places that are all unique in their characteristics and

uncertainties It follows that many descriptions may be compatible with current

understanding and available observations, called the equifinality thesis (Beven 1993,

2006a) One way of viewing these multiple descriptions is as different workinghypotheses of how a system functions The concept of the single description mayremain a philosophical axiom or theoretical aim but will generally be impossible toachieve in practice in applications to real systems (Beven 2002a, 2002b)

This view is actually fundamentally different to a statistical approach to modelidentification In both frequentist and Bayesian approaches to statistics, the uncer-tainty associated with a model prediction is often assumed to be adequately treated

as a single lumped additive variable in the form

O(X, t) = M(Θ, εθ, I, εI, X, t) +ε(X, t) (6.1)

where O(X, t) is a measured output variable, such as discharge, at point X and time t; M(Θ, εθ, I, εI, X, t) is the prediction of that variable from the model with parameter

set Θ with errors εθand driven by the input vector I with errors εI; and ε(X, t) is the

model error at that point in space and time Transformations of the variables ofEquation 6.1 can also be used where appropriate to constrain the modeling problem

to this form A logarithmic transformation, for example, can be used for an errorthat is multiplicative —that is, increasing with the magnitude of the model prediction

— as a simple way of allowing for heteroscedascticity in the errors with nonconstantvariance Other transformations can also be used to try to stabilize the statisticalcharacteristics of the error series (Box and Cox 1964) Normal statistical inferencethen aims to identify the parameter set Θ that will be in some sense optimal, normally

by minimizing the residual error variance of a model of the model error, whichmight include its own parameters for bias and autocorrelation terms with the aim

of making the residual error independent and identically distributed, even thoughthere may be good physical reasons why errors that have constant statistical char-acteristics in hydrological and water quality modeling should not be expected (see,e.g., Freer et al 1996)

The additive form of Equation 6.1 allows the full range of statistical estimationtechniques, including Bayesian updating, to be used in model calibration Theapproach has been widely used in hydrological and water resources applications,including flood forecasting involving data assimilation (e.g., Krzysztofowicz 2002;Young 2001, 2002 and references therein), groundwater modeling, including Bayesianaveraging of model structures (e.g., Ye et al 2004), and rainfall-runoff modeling(e.g Kavetski et al 2002; Vrugt et al 2002, 2003)

In principle, the additive error assumption that underlies this form of uncertainty

is particularly valuable for two reasons: (1) it allows checking of whether the actualerrors conform to the assumptions made about the structural model of the errors;and (2) if this is so, then a true probability of predicting an observation, conditional

on the model, can be predicted as the likelihood L(O(X, t) | M(θ, I, X, t)) These

advantages, however, may be difficult to justify in many real applications where

poorly known input errors are processed through a nonlinear model subject to

structural error and equifinality (see Hall 2003; Klir 1994 for reviews of more

generalized mathematizations of uncertainty, including discussion of fuzzy set

meth-ods and the Dempster-Shafer theory of evidence) One implication of the limitations

of the additive error model is that it may actually be quite difficult to estimate the

true probability of predicting an observation, given one or more models, except in

ideal cases because the model structural error has a complex and nonlinear effect

structured in both time and space on the total model error, ε(X, t).

This implies that a philosophically different approach to the statistical approach

might be worth investigating In the statistical approach, the error model is generally

evaluated as conditioned on finding the best maximum likelihood model In

evalu-ating models as multiple working hypotheses, it is often more interesting to estimate

the likelihood of a model conditioned on some vector of observations such as L(M(θ,

εθ, I, εI, X, t) | O(X, t)) and, in particular, to reject those models as unacceptable

hypotheses that should have a zero likelihood

This is the basis for the Generalized Likelihood Uncertainty Estimation (GLUE)

methodology, first proposed by Beven and Binley (1992) It can be argued that the

formal statistical approaches are a special case of the GLUE methodology within

which the formal assumptions of a defined error model can be accepted such that

the formal likelihood function can be used to weight model predictions It can also

be argued that the GLUE methodology is a special case of formal statistical inference,

in which informal likelihood measures replace a formal likelihood function with its

rigorous assumptions about the nature of the error model GLUE can indeed make

use of formal likelihood measures if the associated assumptions can be justified It

is perhaps better, however, to consider the two approaches as based on different

philosophical frameworks to the uncertainty estimation problem

6.4 UNCERTAINTY ESTIMATION: FORMAL

BAYES METHODS

The traditional approach to model calibration in hydrological modeling has been to

simplify Equation 6.1 to the form

O(X, t) = M(θ, I, X, t) +ε(X, t) (6.2)with the aim of minimizing the total error in some way This assumes that the effect

of all sources of error can be subsumed into the total error series as if the model

was correct and that the input and boundary condition data and observations were

known precisely

Furthermore, if the total error ε(X, t) can be assumed to have a relatively simple

form — or can be suitably transformed to a simple form — then a formal statistical

likelihood function can be defined, dependent on the assumed error structure Thus,

for an evaluation made for observations at a single site for total model errors that

can be assumed to have zero mean, constant variance, independence in time, and aGaussian distribution, the likelihood function takes the form

(6.3)

where εt= O(X, t) − M(θ, I, X, t) at time t, T is the total number of time steps, and

σ2 is the residual error variance For total model errors that can be assumed to have

a constant bias, constant variance, autcorrelation in time, and a Gaussian distribution,the likelihood function takes the form

(6.4)where µ is the mean residual error (bias) and α is the lag 1 correlation coefficient

of the total model residuals in time More complex error structure assumptions willlead to more complex likelihood functions, with more parameters to be estimated

A significant advantage of this formal statistical approach is that when the

assumptions are satisfied, the theory allows the estimation of the probability with

which an observation will be predicted, conditional on the model and parametervalues, and the probability density functions of the parameter estimates, which underthese assumptions will be multivariate normal As more data are made available, theuse of these likelihood functions will also lead to reduced uncertainty in the estimatedparameter values, even if the total error variance is not reduced O’Hagan (2004)

suggested that this is the only satisfactory way of addressing the issue of model

uncertainty; without proper probability estimate statements about modeling, tainty will have no meaning

uncer-There is an issue, however, about when probability estimates based on additive,

or transformed, error structures are meaningful From a purely empirical point ofview, a test of the actual model residuals ε(X, t) for validity relative to the assumptions

made in formulating the likelihood function might be considered sufficient to justifyprobability statements of uncertainty From a theoretical point of view, however,there has to be some concern about treating the full sources of error in Equation 6.2

in this type of aggregated form Model structural errors will, in the general case, benonlinear, nonstationary, and nonadditive Input and boundary condition errors, aswell as any parameter errors, will also be processed through the model structure innonlinear, nonstationary, and nonadditive ways

Kennedy and O’Hagan (2001) attempted to address this problem by showingthat all sources of error might be represented within a hierarchical Bayesian frame-work In particular, where any model structural error is simple in form, it might bepossible to estimate this as what they called a “model inadequacy function,” or, more

recently, “model discrepancy function” (O’Hagan 2004) In principle, this could takeany nonlinear form, although the most complex in the cases they considered was aconstant bias, which can, in any case, be included as a parameter in Equation 6.4.The aim is to extract as much structural information from the total error series aspossible, ideally leaving a Gaussian independent and identically distributed residualerror term The model discrepancy function can then also be used in prediction,under the assumption that the nature of the structural errors in calibration will besimilar in prediction.

It should be noted, however, that the model discrepancy function is not a directrepresentation of model structural error It is a compensatory term for all the unknownsources of error in Equation 6.1, conditional on any particular realization of themodel, including specified parameter values and input data These sources of errorcould, in principle, be considered explicitly in the Bayesian hierarchy if goodinformation were available as to their nature This will rarely be the case in hydro-logical modeling applications, where, for example, rainfall inputs to the system may

be poorly known for all events in some catchments and where even the mostfundamental equation — the water balance — cannot be closed by measurement(Beven 2001, 2002b) Thus, disaggregation of the different error components will

be necessarily poorly posed, and ignoring potential sources of error, including modelstructural error, may result in an overestimation of the information content of addi-tional data and may lead to an unjustified overconfidence in estimated parametervalues (see discussion in Beven and Young 2003) In representing the modelingprocess by the simplified form of Equation 6.2, the error model is required tocompensate for all sources of deficiency

6.5 UNCERTAINTY ESTIMATION BASED

ON THE EQUIFINALITY CONCEPT

AND FORMAL REJECTIONIST METHODS

The equifinality thesis is the central concept of the GLUE methodology (Beven andBinley 1992; Beven and Freer 2001) The GLUE methodology does not purport to

estimate the probability of predicting an observation given the model but rather attempts to evaluate the predicted distribution of a variable that is always conditional

on the model or models considered, the ranges of parameter values considered, the

evaluation measures used, and the input and output data available to the applicationfor model evaluation The prediction distributions do not consider the residual errorassociated with a particular model run explicitly There is instead an assumption thatthe error series associated with a model run in calibration will have similar charac-teristics in prediction — note the similar assumption about model structural error

in the formal likelihood approach just described Thus, in weighting the predictions

of multiple models to form the predictive distribution for a variable, there is animplicit weighting of the error series associated with those models, without the need

to consider different sources of error explicitly; explicit error models can be handled

in this framework by treating them as additional model components (see, e.g.,Romanowicz et al 1998)

One of the most interesting features of the GLUE methodology is the mentarity of model equifinality and model rejection Equifinality accepts that mul-tiple models may be useful in prediction and that any attempt to identify an optimalmodel might be illusory But if multiple models are to be considered acceptable or

comple-behavioral, it is evident that models can also be rejected (given a likelihood of zero)

where they can be shown to be nonbehavioral (given unacceptable simulations ofthe available observables) Thus, there is always a possibility that all the modelstried will be rejected — unlike the statistical approach where it is possible tocompensate for model deficiencies by some error structure

However, at this point the limitations of implicit handling of error series in the GLUEmethodology become apparent since it is possible that some hypothetical perfect modelcould be rejected if driven by poor input and boundary condition data or if comparedwith poor observation data Thus, there is a need for a more explicit consideration ofsources of error in this framework while retaining the possibility of model rejection

A potential methodology has been proposed by Beven (2005, 2006a) Equation6.1 can be rewritten to reflect more sources of error as

O(X, t) +εO(X, t) +εC(∆x, ∆t, X, t) = M(θ, εθ, I, εI, X, t) +εM(θ, εθ, I, εI, X, t) +εr (6.5)The error terms on the left-hand side of Equation 6.5 represent the measurementerror, εO(X, t), and the commensurability error between observed and predicted vari-

ables, εC(∆x, ∆t, X, t) The model term, M(θ, εθ, I, εI, X, t), will reflect error in input

and boundary conditions, model parameters, and model structure The error term, εM(θ,

εθ, I, εI, X, t), can now be interpreted as a compensatory error term for model

defi-ciencies, analogous to the discrepancy function in the Bayesian statistical approach ofO’Hagan (2004) but that must also reflect error in input and boundary conditions,model parameters, and model structure Finally, there may be a random error term, εr.Equation 6.5 has been written in this form to both highlight the importance

of observation measurement errors and the commensurability error issue and toreflect the real difficulty of separating input and boundary condition errors, param-eter errors, and model structural error in nonlinear cases There is no generaltheory available for doing this in nonlinear dynamic cases One simplification can

be made in Equation 6.5: If applied on a model-by-model basis, model parameter

error has no real meaning It is the model structure and set of effective parameter

values together that process the nonerror-free input data and determine total modelerror in space and time Thus, Equation 6.5 could be rewritten, for any modelstructure, as

O(X, t) +εO(X, t) +εC(∆x, ∆t, X, t) = M(θ, I, εI, X, t) +εM(θ, I, εI, X, t) +εr (6.6)and εM(θ, I, εI, X, t) is a model specific error term

The question that then arises within this framework is whether εM(θ, I, εI, X, t)

is acceptable in relation to the terms εO(X, t) +εC(∆x, ∆t, X, t) This is equivalent

to asking if the following inequality holds:

Omin(X, t) < M(θ, I, εI, X, t) < Omax(X, t) for all O(X, t) (6.7)

where Omin(X, t) and Omax(X, t) are acceptable limits for the prediction of the output

variables given εO(X, t) and εC(∆x, ∆t, X, t), which together might be termed an effective observation error The effective observation error takes account of both

real measurement errors and commensurability errors between observed and dicted variables When defined in this way, the effective observation error needsneither zero mean or constant variance nor to be Gaussian or stationary in the form

pre-of its distribution in space or time, particularly where there may be physical straints on the nature of that error Note that the commensurability error might beexpected to be model implementation dependent in that the difference betweenobserved and predicted variables may depend on model time and space discretisa-tions and measurement scales in relation to expected time and space heterogeneities

con-of the observable quantities However, it should really be possible to develop amethodology for making prior estimates of both measurement and commensurabilityerrors, since they should be independent of individual model runs An objectiveevaluation of each model run using Equation 6.7 should then be possible If a model

does not provide predictions within the specified range, for any O(X, t), then it should

The approach can also be relativist in taking account of the performance ofdifferent models within the set of behavioral models (Beven 2004b, 2005) Within

the behavioral range, for all O(X, t), a positive weight could be assigned to the model predictions, M(θ, I, εI, X, t), according to the level of past performance The simplest

possible weighting scheme that need not be symmetric around the observed value,

given an observation O(X, t) and the acceptable range [Omin(X, t), Omax(X, t)], is the

triangular relative weighting scheme, but other bounded weighting schemes could

be used — including truncated Gaussian forms A core range of observationalambiguity, or equal weighting, could be added if required (Beven 2006a)

This methodology gives rise to some interesting possibilities Within this work there is no possibility of a representation of model error being allowed tocompensate for poor model performance, even for the “optimal model,” unless theacceptability limits are made artificially wide to avoid rejecting all of the models

frame-— but this might not generally be considered to be good practice If no model proves

to be behavioral, then it is an indication that there are conceptual, structural, or dataerrors, though it may still be difficult to decide which is the most important There

is perhaps then more possibility of learning from the modeling process on occasionswhen it proves necessary to reject all the models tried

However, this type of evaluation requires that consideration also be given to

input and boundary condition errors, since, as noted before, even the perfect model

might not provide behavioral predictions if it is driven with poor input data error.Thus, the combination of input and boundary data realization — within reasonable

bounds — and model structure and parameter set in producing M(θ, I, εI, X, t) should

be evaluated against the effective observational error The result will hopefully still

be a set of behavioral models, each associated with some likelihood weight Anycompensation effect between an input realization — and initial and boundary con-ditions — and model parameter set in achieving success in the calibration periodwill then be implicitly included in the set of behavioral models

There is also the possibility that the behavioral models defined in this way

do not provide predictions that span the complete range of the acceptable erroraround an observation The behavioral models might, for example, provide sim-

ulations of an observed variable O(X, t) that all lie in the range O(X, t) to Omax(X, t) or even in just a small part of it They are all still acceptable but are apparently

biased This provides real information about the performance of the model orother sources of error that can be investigated and allowed for specifically at thatsite in prediction rather than being lost in a statistical representation of modelerror

6.6 UNCERTAINTY AS PART OF A LEARNING PROCESS

Both Bayesian and equifinality (rejectionist set-theoretic) concepts allow the eling process to be set up within a learning framework, using data assimilation toupdate the model each time new data become available This can be for short-termforecasting with the aim of minimizing forecast uncertainty as conditioned on thenew data or in a simulation context with the aim of refining the model representation

mod-of the system mod-of interest as new information is received to update the Bayes likelihoodfunction or the weights associated with the set of behavioral models using the Bayesequation, originally proposed by Thomas Bayes in 1724 (see Bernado and Smith1994; Howson and Urbach 1993)

In formal Bayes theory, the posterior likelihood is intended to represent the

probability of predicting an observation, given the true model, L(Y |θ) where Y is the

observation vector and θ is the parameter vector

Lp (O|θ ) ∝ Lo(θ) L(θ|Y) (6.8)

where Lp (O|θ) is the posterior probability of predicting observations O given a

model with parameter set θ, Lo(θ) is the prior likelihood of parameter set θ, and

L(θ|Y) is the likelihood given data Y However, Bayes’s equation was originally stated in the more general conditioning form for hypotheses, H, given evidence, E, as

where Lp(Hk|E) is the posterior likelihood for hypothesis Hk given the evidence E;

Lo(Hk) is a prior likelihood for Hk; and L(E|Hk) is the likelihood of predicting the

evidence E given Hk Here, the hypotheses of interest are each model of the system,including its parameter values and any other ancillary hypotheses

When this type of Bayes conditioning is first applied, the prior likelihoods can

be chosen subjectively based on any available evidence about each model as esis In later evaluations, which may be as each new piece or set of data becomesavailable, the posterior for the last updating step becomes the prior for the nextupdating step It is therefore important that the errors at each updating step remainconsistent with the assumptions that underlie the definition of the likelihood function.The equifinality approach as previously formulated in the extended GLUE meth-odology is also effectively Bayesian in nature in that each time a new modelevaluation is made using Equation 6.10, the distribution of behavioral models andthe weights associated with them can be reevaluated as a combination of the priorlikelihood weights and the new evaluation for each model The approach is lessformal in that more choices can be made about how to weight different behavioralmodels and how to combine the weights in successive evaluations It is easy toinclude model evaluations based on multiple criteria within the GLUE methodology

hypoth-in this way Bayes’s equation implies a multiplicative combhypoth-ination but other types

of combinations, such as a weighted average of multiple evaluation measures, can

be used to provide a likelihood weight for each behavioral model These choicesmust, however, be made explicit so that they can be reproduced by others if necessary.The most important part of the learning process is the successive application ofEquation 6.7 This defines the set of behavioral models and the rejection of non-behavioral models In the case of multicriteria evaluation, this may mean that amodel that is successful on one criterion may be rejected on another

This type of learning process will increasingly represent the way models areimplemented The possibility of the routine application of data assimilation within

a learning framework raises some interesting questions about the nature of modelingand model evaluation Effectively, repeated updating and correction of model pre-dictions will allow the data assimilation process to compensate for errors in modelinputs and model structure Model evaluation therefore becomes more difficult Inreal-time forecasting, this may not be such a problem In fact, the desire is for thedata assimilation process to compensate for errors in model inputs and modelstructures if this results in improved forecasts with maximized accuracy and mini-mized uncertainty

This may not be the case, however, in simulation where such compensation maynot be desirable if it leads to a model structure being accepted when it should berejected There is, of course, a fundamental difficulty in deconstructing the causes

of model error and isolating the model structural error alone (see discussions inBeven 2005, 2006a; Beven and Young 2003; Kavetski et al 2002) Though we donot want to accept a model structure because of the compensation allowed by dataassimilation, equally we would not want to make the error of rejecting a perfectlygood model because of errors in the input data Differentiating between these sources

of error may be very difficult, if only because the nature of both input errors andmodel structural errors may be nonstationary in time

Thus, there is a question as to whether the use of data assimilation can, in asimulation context, reveal deficiencies in either model structures or input data.

There is an analogy here with the State Dependent Parameter (SDP) estimation

methodology used by Young (2003) and Romanowicz et al (2004) In the SDPapproach, an initial estimate of a linear transfer function model is used within arecursive data assimilation framework to examine how the best estimates of theparameters change over time or with respect to some other variable This can lead

to the identification of the dominant nonlinear modes of behavior of the systembased directly on the observations rather than prior conceptual assumptions aboutthe system response

6.7 AN EXAMPLE APPLICATION

The following application of the methodology will illustrate some of the issues thatarise in thinking about different potential sources of error in the modeling processesand the relationship between observed and predicted variables It examines the use

of the MACRO-P model in predicting observations of P concentrations in thedrainage from slurry application experiments on instrumented grassland plots inScotland

The MACRO model of water flow in structured soils was developed by Jarvis andcolleagues (Jarvis 1994) It was adapted for colloid-facilitated transport of contam-inants (Jarvis et al 1999; Villholth et al 2000), and subsequently the version usedhere was adapted for P transport (McGechan 2002; McGechan et al 2001) Only abrief description of the representation of the more important model processes will

be given here as background to the model parameters investigated in the study MACRO is a soil profile model that is divided into a number of vertical layersand is partitioned into micropore (soil matrix) and macropore domains The boundarybetween the two domains is described in the model by the air–entry soil–watertension in the Brooks-Corey equation The two domains function separately and areassociated with their own degree of saturation, hydraulic conductivity, and flux Flow

in the micropores is calculated using the Richards (1931) equation Flow in themacropores is gravity driven at a rate determined by degree of macropore saturation Simulation of soluble P transport requires a single run of the model where theconvection-dispersion equation is solved for each layer at each time step to calculatethe P concentration For simulation of colloid-facilitated P transport, two consecutivemodel runs are required In the first run the concentration of colloids is simulated

in place of the solute, and in the second run the concentration of P sorbed to sites

on the colloids is calculated This requires the concentrations of colloidal particlesand inorganic P, respectively, to be specified For this application, all colloidalparticles were those derived from the slurry applied on specific dates An alternativeoption in the model that was not used — to reduce the number of parameters to beconsidered — was colloid generation by rainfall impact

Sorption of P to the soil matrix is described using the Freundlich isothermequation

where s is the sorbed phase concentration in either micropores or macropores, K d is

the Freundlich sorption coefficient, c is the solution concentration, and n is the

Freundlich sorption exponent For sorption to colloids a similar formulation is

employed using a sorption coefficient for colloids, Kc,and the Freundlich sorptionexponent set to unity

Filtration, which is the physical trapping of particles, in macropores andmicropores leading to the irreversible trapping of colloids, is included using thefollowing equations For macropores,

(6.12)

where F is the filtration sink term, fref is a reference filter coefficient, nf is an empirical exponent, vref is the pore water velocity at which fref is measured, and θ is thevolumetric soil water content for the domain

For micropores,

where fc is the micropore filter coefficient

6.7.2 S TUDY S ITE AND D ATA

The study site is located at Crighton Royal Farm, Dumfries, Scotland The annualaverage precipitation at the site is 1054 mm (Hooda et al 1999), and the precipitationfor the simulation periods chosen for this study are 755 mm (October 1, 1994 toMarch 31, 1995) and 624 mm (October 1, 1995 to April 30, 1996) The two 0.5 hagrass field plots are grassland with a silty clay loam soil that showed significantvertical macroporous flow channels (Hooda et al 1999) The plots were isolatedfrom each other and external areas by a drainage system giving isolation, apart fromdeep percolation, which is thought to be insignificant on these soils (Hooda et al.1999; McGechan et al 1997)

6.7.2.1 Drainage Discharge and Phosphorous Concentrations

From each plot the discharge from a main drain was recorded and summed to aweekly total Flow-proportional sampling was used to obtain an effective P con-centration for the period since the previous sample Different forms of P wereanalyzed, but only total inorganic P (molybdate-reactive phosphorus, MRP) wasused for this study

F= f v nf v−nf c

6.7.2.2 Slurry Applications

Slurry applications were made at a rate of approximately 50 m3 ha−1 the followingtimes: February 14, 1994, 23 kg ha−1 total phosphorus (TP), 5000 gm−3 colloidconcentration; May 25, 1994, 8.4 kg ha−1 TP, 1825 gm−3 colloid concentration;November 21, 1994, 17 kg ha−1 TP, 3700 gm−3 colloid concentration; May 31, 1995,12.7 kg ha−1 TP, 2760 gm−3 colloid concentration; July 7, 1995, 4.1 kg ha−1 TP, 900

gm−3 colloid concentration; and January 31, 1996, 23.9 kg ha−1 TP, 5200 gm−3 colloidconcentration For assumptions made in estimating the P and colloid concentrations

in slurry, see McGechan (2002)

MACRO has a large number of parameter values that must be specified before a run

is made Many of the parameters are effective parameters with values that are difficult

to specify at the scale of application because of scale and heterogeneity effects Herethe most important parameters based on previous experience of calibrating MACROhave been varied across ranges defined to allow for uncertainty in these effectivevalues (Table 6.1) Lacking knowledge about the nature of the prior distributionsand the possible covariation of the parameters, uniform independent distributionshave been assumed, noting that the evaluation of the simulations from each parameter

set will result in posterior likelihood weights that reflect any interactions among the

effective parameter values in achieving an acceptable simulation

For the two periods of interest, as specified previously (1994 to 1995 and 1995

to 1996), 10,000 and 11,500 model simulations were carried out, respectively Ateach weekly time step, MACRO’s estimates of discharge and P concentration werecompared to the observed data described previously Use of weekly discharge totalsand weekly flow-weighted average concentrations in the model evaluation does add

a degree of incommensurability to the model evaluations This is particularly thecase for concentrations that can vary rapidly within hydrological events, particularly

at this relatively small scale, meaning that flow-weighted average concentrationsmay systematically differ from those predicted by the model For discharge theproblem is likely to be less pronounced, as the simulation of weekly discharge willdepend less on correct model dynamics and more on longer-term mass balances;this does, however, mean that the evaluations will not provide a strong constraint

on the model’s description of the flow pathways

To allow for these problems, effective observation errors were defined in a fuzzymanner by allowing model simulations to be considered acceptable within somerange of effective observation error in the GLUE framework as described previously.The errors were taken as ± 40% for flow-weighted P concentrations, −30% and +30

to +300% for discharge The range +30 to +300% was used for the upper limit ofdischarge evaluations as very low flows were known to be underestimated by themeasurement equipment For this reason positive errors were estimated on a linearsliding scale of +30% for flows of ≥ 80 mm wk−1 to +300% for flows ≤ 1 mm wk−1.The ranges were implemented as triangular fuzzy distributions (see Figure 6.1)

54.8 (1) 52.4 (2) 48.5 (3-9)

% (volume)

2.71 (2) 0.972 (3-9)

2.59 (1) 3.31 (2) 1.19 (3-9)

% (volume)

determines when macropores flow

moisture tension (CTEN)

0.5 (1) 0.36 (2) 0.075 (3-8)

1 × 10−26 (9)

6.0(1) 1.92 (2) 0.4 (3-8)

1 × 10−26 (9)

mm hr−1

35.6 (2-3) 29.0 (4) 32.2 (5) 32.6 (6) 32.9 (7) 32.9 (8) 36.3 (9)

55.0 (1) 53.5 (2-3) 43.6 (4) 48.3 (5) 48.9 (6) 48.9 (7) 49.3 (8) 54.4 (9)

Note: Values in brackets represent the numbering of the soil layers in the model (layer 1 at the surface).

a Parameters varied within ranges by a multiplicative factor applied to all layers simultaneously.