PUB and Data-Based Mechanistic Modelling: theImportance of Parsimonious Continuous-Time Models

Peter C. Young and Renata J. RomanowiczCentre for Research on Environmental Systems and Statistics,

Lancaster University, Lancaster, LA1 4YQ, U.K.p.young@lancaster.ac.uk

Abstract: The problem of Prediction in Ungauged Basins (PUB) is intimately linked with the concept ofregionalisation; namely the transfer of information from one catchment that is gauged to another that is not.But such regionalisation exercises can be dangerous and should be attempted only with great care. Thepresent paper addresses what the authors believe to be one essential aspect of regionalisation: namely, theimportance of considering only ‘top-down’ models that are parametrically efficient (parsimonious) and fully‘identifiable’ from the available catchment data. We argue further that many mechanistic model parametersare more naturally defined in the context of continuous-time, differential equation models (normally derivedby the application of natural ‘laws’, such as mass and energy conservation). As a result, there are advantagesif such models are identified and estimated directly in this continuous-time, differential equation form, ratherthan being formulated and estimated as discrete-time models. The arguments presented in the paper areillustrated by an example in which the top-down, Data-Based Mechanistic (DBM) approach to modelling isapplied to a set of precipitation-flow data. This involves the application of an advanced method ofcontinuous-time transfer function identification and estimation; and the interpretation of this estimated modelin physically meaningful terms, as required by the DBM modelling approach.

Keywords: Continuous-time transfer functions; data-based mechanistic; rainfall-flow; regionalisation.

1. INTRODUCTIONFrom a conceptual standpoint, most mathematicalmodels of hydrological systems are formulated onthe basis of natural laws, such as dynamicconservation equations, often expressed in terms ofcontinuous-time (CT), linear or nonlineardifferential equations. Paradoxically, TransferFunction (TF) models, which have been growingin popularity over the last few years because oftheir ability to characterise hydrological data inefficient parametric (or ‘parsimonious’) terms, arealmost always presented in the alternative,discrete-time (DT) terms. One reason for thisparadox is that hydrological data are normallysampled at regular intervals over time, so formingdiscrete time series that are in a most appropriateform for DT modelling. Another is that most of thetechnical literature on the statistical identificationand estimation (calibration) of TF models dealswith these DT models. Closer review of thisliterature, however, reveals apparently less wellknown publications [e.g. Young and Jakeman,1980] dealing with estimation methods that allowfor the direct identification of CT (differentialequation) models from discrete-time, sampled data.

This paper first discusses the formulation,identification and estimation of CT models. It then

considers a practical example in which the Data-Based Mechanistic (DBM) approach to modelling[e.g. Young, 2001 and the prior references therein]is applied to a typical set of effective rainfall-flowdata from the ephemeral Canning River of WesternAustralia.

2. CONTINUOUS-TIME TF MODELS

Let us consider first a conceptual catchmentstorage equation in the form of a continuous-time,linear storage (tank or reservoir) model: see, forexample, the review papers by O’Donnell, Doogeand Young in Kraijenhoff and Moll [1986]; or therecent book by Beven [2001]. Here, the rate ofchange of storage S(t) in the channel is defined interms of water volume Qi (t) entering the reservoir

(e.g. river reach) in unit time, minus the volumeQo (t) leaving in the same time interval, i.e.,

dS(t)dt

= GQi (t ! " ) !Qo (t) (1)

where Qi (t ! " ) represents the input flow ratedelayed by a pure time delay of ! time units toallow for pure advection; and G is a ‘gain’parameter inserted to represent gain (or loss) in the

system. Making the reasonable and fairly commonassumption that the outflow is proportional to thestorage at any time, i.e.,

Qo (t) = !S(t) ,

and substituting into (1), we obtain,

T dQo (t)dt

= GQi (t ! " ) !Qo (t); T = 1#

(2)

This can be written in the following, continuous-time Transfer Function (TF) form,

Qo (t) =G

1+ TsQi (t ! " ) (3a)

where s is the derivative operator, i.e. s = d / dt .Five important, physically interpretable modelparameters are associated with this model. TheSteady State Gain (SSG), denoted by G , isobtained by setting the s operator in the TF tozero (i.e. d / dt = 0 in a steady state). It shows therelationship between the equilibrium output andinput values when a steady input is applied. Forthis reason, if the input and output have similarunits, G is ideal for indicating the physical lossesor gains occurring in the system. In the case of aflow-routing model, it indicates whether water hasbeen added or lost between the upstream anddownstream boundaries and the percentage ofwater lost or gained can be defined by LossEfficiency TE = 100(1!G) , which will benegative if G > 1 . The Residence Time or TimeConstant, T , is the time required for the reservoiroutput to decay to e!1 (~ 37%) of its maximumvalue in response to a unit impulse input. Finallythe pure Advective Time Delay ! indicates thetime it takes for a flow increase upstream to befirst detected downstream: and Tt = T + ! definesthe Travel Time of the system. These fiveparameters typify the equilibrium and dynamiccharacteristics of the TF model and provide aphysical interpretation of the TF model in terms ofits mass transfer and dispersive characteristics.

The first order TF model (3a) is often written inthe form,

Qo (t) =b0

s + a1

Qi (t ! " )

b0 =GT

; a1 =1T

(3b)

because this is the form in which the model isnormally estimated. Typically, a Channel or FlowRouting model for a river catchment will contain anumber of elemental models, such as (3a,b),connected in a manner that relates to the structureof the catchment. For instance, a serial connectionof n such elements constitutes the ‘lag-and-route’model of a single river channel [Meijer, 1941;Dooge, 1986] and, with ! = 0 , it becomes the wellknown ‘Nash cascade’ model [Nash, 1959]. More

complex river systems can be represented by amain channel of this type, with tributariesmodelled in a similar manner. Also, a typical TFmodel between effective rainfall and flow containsa parallel connection of two or more such storageelements [see Young, 1992, 2001]. A practicalexample of this kind is described later in Section 4.Related serial and parallel arrangementscharacterise the mass conservation equation of theAggregated Dead Zone (ADZ) model for thetransport and dispersion of a solute in a riverchannel (e.g. Wallis et al, 1989 and the priorreferences therein). And ADZ-type models canlead to more complex interactive water qualitymodels, including chemical and biologicalinteraction, such as DO-BOD models [e.g. Beckand Young, 1975].

Serial, parallel or even feedback connections ofelemental first order TF models such as (3a,b)lead, in general terms, to the following multi-orderTF model:

x(t) = B(s)A(s)

u(t ! " ) y(t) = x(t) + #(t) (4a)

where A(s) and B(s) are polynomials in s of thefollowing form:

A(s) = sn + a1sn!1 + a2s

n!2 + … + anB(s) = b0s

m + b1sm!1 + b2s

m!2 + … + bmin which n and m can take on any positiveinteger values. Here u(t) and x(t) denote,respectively, the deterministic input and outputsignals of the system at its upstream anddownstream boundaries, respectively; ! is a pureadvective time (transport) delay affecting the inputsignal u(t) ; and y(t) is the observed output,which is assumed to be contaminated by a noise orstochastic disturbance signal !(t) . This noise isassumed to be independent of the input signal u(t)and it represents the aggregate effect, at the downstream boundary, of all the stochastic inputs to thesystem, including distributed unmeasured inputs,measurement errors and modelling error. If !(t)has rational spectral density, then it can bemodelled as an Auto-Regressive (AR) or Auto-Regressive, Moving-Average (ARMA) process butthis restriction is not essential. Also, depending onthe objectives of the modelling study, it may benecessary, in a complete system consisting ofmany sub-elements such as (3a,b), to considernoise inputs within the system, associated withcollections of sub-elements that have distinctphysical meaning: e.g. stochastic lateral inflows.

Multiplying throughout equation (4a) by A(s) andconverting the resultant equation to alternativeordinary differential equation form, we obtain:

dny(t)dt n

+ a1dn!1y(t)dt n!1 +! any(t)

= dmu(t ! " )dtm

+! + amu(t ! " ) +#(t)(4b)

where !(t) = A(s)"(t) . The structure of this model,in either form (4a) or (4b), is defined by the triad[n m ! ] .

To date, the most popular form of TF modellinghas been carried using the discrete-time (DT)equivalents of the models (3a,b) and (4a,b). In thecase of equation (3a,b), this discrete-time TFmodel takes the form:

Qo,k =!0

1+"1z#1 Qi,k#$ (5a)

Here, Qo,k is the flow measured at the kth sampling

instant, that is at time t = k!t , where !t is thesampling interval in time units. Qi,k!" is the input

flow at time t = (k ! " )#t time units previously,where ! is the advective time delay, normallydefined as the nearest integer value of ! / "t (thusincurring a possible approximation error); and z!1

is the backward shift operator, i.e. z!rQo,k = Qo,k!r .

The values of the parameters !1 and !o can berelated to the parameters of the model (4a) invarious ways depending upon how the input flowQi (t) is assumed to change over the samplinginterval (since it is not measured over thisinterval). The simplest and most commonassumption is that this remains constant over thisinterval (the so-called ‘zero-order hold’, ZOH,assumption), in which case the relationships are asfollows:

!1 = " exp("a1#t)

$1 =b0

a1

{1" exp("a1#t)}(5b)

Note that, because these relationships are functionsof the sampling interval !t , for every unique CTmodel such as (3a,b), there are infinitely many DTequivalents (5a,b), depending on the choice of !t ,all with different parameter values. Following fromthe definition of this first order DT model at thechosen !t , the general multi-order equivalent ofthe model (5a,b) is defined as follows:

xk =B(z!1 )A(z!1 )

uk!" yk = xk + #k (6)

where uk!" , xk , yk and !k are the sampled values

of u(t ! " ) , x(t) , y(t) and !(t) in the model(5a,b); and

A(z!1 ) = 1+ a1z!1 + a2z

!2 + … + anz! p

B(z!1 ) = b0 + b1z!1 + b2z

!2 + … + bmz!q

Normally p = n but q may be equal or greaterthan m .

3. CONTINUOUS-TIME TF MODELS:IDENTIFICATION AND ESTIMATION

The statistical estimation of the CT model (4a,b) isstraightforward if completely continuous-time dataare available. However, the hydrologist is normallyconfronted by discrete-time, sampled data and theproblem of modelling continuous-time modelssuch as this, based on discrete-time, sampled dataat sampling interval !t , is not so obvious. Thisproblem can be approached in two main ways.

1. The Direct Approach: here, it is necessary toidentify the most appropriate, identifiable CTmodel structure (4a,b) defined by the triad[n m ! ] ; and then estimate the TF parameters

ai , i = 1, 2, ... n, bj , j = 0,1, ...m, and ! that

characterise this structure. Of course, someapproximation will be incurred in thisestimation procedure because the inter-samplebehaviour of input signal u(t) is not knownand it must be interpolated over this interval insome manner (see above).

2. The Indirect Approach: here, the identificationand estimation steps are first applied to the DTmodel (6). This estimated model is thenconverted to the CT model (4a,b), again usingsome assumption about the nature of the inputsignal u(t) over the sampling interval !t . Inthe first order, ZOH case, this conversion isgiven by the relationships in equation (5b) but,in more general terms, the multi-orderconversion must be carried out in a computer,using a conversion routine such as the D2CMalgorithm in the Matlab Control Toolbox.

Various statistical methods of identification andestimation have been proposed to implement thetwo approaches outlined above and these havebeen formulated in both the time and frequencydomains. However, only estimation in the timedomain will be considered here, and only onedirect estimation method will be utilised: theRefined Instrumental Variable method fo rContinuous-time Systems (RIVC) proposed byYoung and Jakeman [1980: see also Young, 1984].The RIVC algorithm is available in both theCAPTAIN (see http://www.es.lancs.ac.uk/cres/captain/) and CONTSID2 (see http://www.cran.uhp-nancy.fr/) Toolboxes for Matlab. TheRIVC method is the only time domain method thatcan be interpreted in optimal statistical terms, soproviding an estimate of the parametric errorcovariance matrix and, therefore, estimates of theconfidence bounds on the parameter estimates.

RIVC is a close relative of the R e f i n e dInstrumental Variable (RIV) algorithm for theidentification and estimation of discrete-time TFmodels [Young and Jakeman, 1979; Young, 1984].This has been used in a wide variety ofhydrological applications over the past 30 years.These include calibration of rainfall-flow modelssuch as IHACRES Jakeman et al [1990] and itsData-Based Mechanistic (DBM) model equivalent[e.g. Young 2001 and the prior references therein].In the example below, the RIV method is used forindirect identification and estimation of CTmodels. For comparative purposes, however, theindirect estimation is also achieved using the well-known Prediction Error Minimisation (PEM)algorithm in the Matlab System IdentificationToolbox. In both cases, the D2CM algorithm withthe ZOH interpolation assumption (see above) isused to convert the estimated DT algorithms fromdiscrete to continuous-time form.

Finally, model structure identification is based ontwo statistical criteria. First, the Coefficient ofDetermination, RT

2 , based on the error between the

sampled output data yk and the simulated CTmodel output at the same sampling instants (this isnormally equivalent to the well known Nash-Sutcliffe Efficiency). Second, the YIC statistic,which is a measure of model identifiability and isbased on how well the parameter estimates aredefined statistically. These statistics are discussedin more detail in Appendix 3 of Young [2001].

4. PRACTICAL EXAMPLE

Figure 1 shows a portion of effective rainfallu(t) and flow y(t) data from the River Canning,an ephemeral River in Western Australia. A longerset of these data has been analysedcomprehensively in Young et al [1997], usingdiscrete-time DBM modelling. The effectiverainfall series plotted in the lower panel of Fig.1 isgenerated from the effective rainfall nonlinearityidentified in this earlier study, with the effectiverainfall generated by the equation

u(t) = r(t)y(t)! ,

where r(t) is the measured rainfall and ! is apower law parameter. Here, the flow y(t) is actingas a surrogate measure of the catchment storage,rather than attempting to model this storageseparately in some conceptual manner [see e.g.Young 2001].

In the present paper, we employ the smaller dataset shown in Figure 1 in order to obtain linear,continuous-time DBM models between theeffective rainfall u(t) , as defined in the abovemanner, and the flow y(t) .

Figure 1. Daily effective rainfall and flow data forthe River Canning in Western Australia with CTmodel output in upper panel (dotted line).

4.1 Direct CT Identification and EstimationLet us consider first identification and estimationbased on the measured daily data shown in Figure1. The RIVC algorithm, in combination with theRT2 and YIC statistics, identifies a [2 3 0] second

order model with the parameter estimates:a1 = 0.457 (0.032) a2 = 0.0248 (0.0045)b0 = 0.0138 (0.001) b1 = 0.0505 (0.002)b2 = 0.0046 (0.0008)

where, here and elsewhere in the paper, the figuresin parentheses are the estimated standard errorsobtained from the RIVC algorithm. This modeloutput (shown as the dotted line in upper panel ofFig.1) explains the data very well with RT

2 = 0.980(98% of the output flow explained by thesimulated CT model output) and a residualvariance of 0.00723 (standard deviation 0.085cumecs, where the maximum flow is 3.86 cumecs).The model can also be decomposed by standard TFdecomposition [Young, 1992] into a parallelpathway form. The three pathways are: (i) aninstantaneous effect, accounting for 7.4% of theflow, which is a measure of rapid flow that occursin the very short term (within one samplinginterval; here one day); (ii) a ‘quick’ pathway,accounting for 54.1% of the flow, that represents alinear store with a residence time of about 2.53days, probably the result of surface and shallow,sub-surface processes in the catchment; and (iii) a‘slow’ or ‘base-flow’ pathway, accounting for38.5% of the flow, that passes through a muchlonger 15.9 day residence time, linear store,probably representing the effects of deepergroundwater processes and the displacement of‘old water’.

4.2 Indirect CT Identification and EstimationThe first step of indirect CT modelling involvesDT identification using the RIV algorithm (see

earlier). This identifies a similar structure [2 3 0]TF with the following parameter estimates:!1 = "1.6034 (0.008) !2 = 0.6244 (0.007)

#0 = 0.0140 (0.001) #1 = 0.0151 (0.002)#2 = 0.0252 (0.0013)

However, since this is a discrete-time model, it hasto be converted to continuous-time form. TheD2CM algorithm in Matlab (see section 3.)accomplishes this conversion, using a ZOHapproximation (i.e. the input effective rainfall isassumed to be constant over the sampling interval).The resulting CT model parameter estimates are:

a1 = 0.4711 a2 = 0.0264b0 = 0.0140 b1 = 0.0514 b2 = 0.0049Note that there are no standard errors quoted herebecause these are not available after conversionand must be computed separately in some manner,usually by Monte Carlo simulation analysis (seelater). In this case, the model is quite similar to thedirectly estimated CT model in the previoussection 4.1.

4.3 Simulation AnalysisFrom the above results, it is clear that, at the dailysampling interval, direct and indirect methods ofestimation produce quite similar estimation resultsin this example. However, this is not always thecase and the direct approach has a number ofadvantages. First, it provides direct estimates of thedifferential equation model parameters, without theneed for conversion from discrete to continuoustime. Moreover, after TF decomp-osition into firstorder linear storage elements, the derivedparameters have an immediate physicalinterpretation. Second, the RIVC analysis providesa direct estimate of the parametric error covariancematrix, so allowing for the specification of errorbounds on these physically meaningful parameters,as well as the model output or model-basedforecasts. Finally, the estimated covariance matrixprovides the information required for sensitivityand uncertainty analysis based on numerical MonteCarlo Simulation (MCS) analysis. For instance,this can be used to obtain empirical probabilitydistribution functions (normalised histograms)associated with any derived parameters, such asthe parallel flow partition percentages [see e.g.Young, 2001]. Or it can allow for thequantification of uncertainty in model forecastsassociated with the parametric estimation errors.

While these advantages, in themselves, do notmake the direct CT approach irresistible, CTestimation has one other advantage: it providesmore robust estimates of the model parameters atrapid sampling rates, where the indirect DT-basedapproaches encounter difficulties. Thesedifficulties are associated with the fact that, when

the sampling interval is short, the denominatorroots (eigenvalues) of the DT model approach theboundary of the unit circle and the associated TFparameter estimates become much less welldefined. For instance, in the case of the DT model(5a,b), where the root is !a1 , it is clear that as

!t" 0 so a1 ! 1.0 and small changes in theparameter lead to large changes in the residencetime T = 1 / a1 . This can lead to poorer confidencein the parameter estimates and, more importantly, alack of robustness that leads to the DT estimationmethods encountering convergence problems.Also, the DT models at rapid sampling intervalshave much poorer numerical properties when usedfor simulation or control system design.

These questions of what sampling interval is ‘best’for parameter estimation purposes have beenknown for some considerable time but there are nocompletely general analytical results to guide themodeller in this regard. However, experience overmany years has led to a ‘rule-of-thumb’ that a goodrange of sampling intervals for DT models isbetween 10 and 50 times the system bandwidth; orequivalently, between 1/10th and 1/50th of the timeconstant. Since hydrological systems tend to becharacterised by combinations of quick and slowmodes of behaviour, as in the present example, thisrule has to applied carefully and can restrict thechoice of sampling interval. For instance, on thisbasis, the sampling interval for the slow mode(T ! 16 days) should not be greater than 16/50days, or about 8 hours; while in the case of thequick mode (T ! 2.5 days) it should not be greaterthan 2.5/50 days, or about 1 hour.

In this example, let us explore the consequences ofusing different sampling intervals. In order to dothis, the input effective rainfall data can beinterpolated so that it has 288 five minute samplesover each day. Then this rapidly sampled seriescan be used as the input to the RIVC estimatedcontinuous time model, in order to generate asimulated flow series at this sampling interval.Similar simulations can be performed over a wholerange of sampling intervals from 5 minutes to oneday. These simulated outputs can then be used asthe basis for a MCS study at each of thesesampling intervals. Here, for each realisation at theselected sampling interval, the output data is thesum of the ‘noise-free’ simulated output at thissampling interval and an independent white noisesignal with similar variance to the estimatedresidual error of the RIVC model (0.0072). TheMCS results at each sampling interval are based on50 realisations of this type; and the performance ofeach algorithm (RIVC, RIV and PEM) is measuredby the number of times in the 50 realisations thatthe algorithm fails to converge to a reasonablemodel. Here a ‘reasonable’ model is defined as one

in which the estimate a1 of the a1 parameter iswithin 3 standard deviations of the true value(0.457 ± 0.096 ).

The full results from this MCS study are given inYoung [2004]. They show that the direct RIVCalgorithm rarely fails to converge: it has no failuresfor sampling intervals from 5 minutes to one hour;and a maximum of 2 failures in 50 at a samplinginterval of 18 hours (a mean failure rate of only0.3%). On the other hand, the indirect RIV andPEM-based indirect approaches perform poorly,particularly at high sampling rates, with meanfailure rates of 7.4% and 17.4% respectively. Asexpected, the lowest number of failures for theRIV-based method occurs for longer samplingintervals of !t > 1.0 hour, where the failure rate isonly 3%. This is superior to the PEM performance.

5 DISCUSSION AND CONCLUSIONSThis paper has outlined an approach to Data-BasedMechanistic (DBM) modelling of hydrologicalsystems based on the direct identification andestimation of continuous-time (transfer function ordifferential equation) models. It has also arguedthat such models are more appropriate to PUBregionalisation studies than their more widelyknown discrete-time equivalents. The majoradvantages of continuous-time models in thisregard are:

• The model parameters values are unique:unlike discrete-time models, they are not afunction of the data sampling interval.

• The model is in an ordinary differentialequation form that can be related directly tothe formulation of physically meaningfulmodels, such as those derived from mass,energy and momentum conservation.

• Model parameter estimation is superior over awider range of sampling intervals, particular atfast sampling intervals (e.g. 15 min. forrainfall-flow measurements).

These advantages, together with the parsimony thatis a natural consequence of DBM transfer functionmodelling, should mean that any relationshipsbetween the CT model parameters and physicalmeasures of the catchment characteristics shouldbe clearer and better defined statistically, asrequired for PUB applications.

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