On the identification of model structure in hydrological and environmental systems

Authors


Abstract

[1] The paper presents a new recursive estimation algorithm designed expressly for the purpose of model structure identification (not for state estimation or primarily for parameter estimation) and discusses two applications thereof, one to a motivating, hypothetical example and one to data from whole-pond manipulations designed to explore sediment-nutrient-phytoplankton dynamics. The algorithm is the current culmination of a long-term technical development from state estimation using a Kalman filter, through state parameter estimation using an extended Kalman filter, through a recursive prediction error (RPE) algorithm for parameter estimation cast in the state space and recently modified for estimating time-varying model parameters, to an RPE algorithm for estimating time-varying parameters but cast in a parameter space formulation. It is concluded that the algorithm performs well, in the sense of being robust and indeed in revealing specifically where (but less so exactly how) a prior candidate model's structure may be in error.

1. Introduction

[2] The structure of a model is defined by the input, state, and output variables chosen to characterize the behavior of the modeled system, the logic of the interconnections amongst all these variables, and the particular mathematical forms of the various assumed interactions. To say that a model suffers from structural error/uncertainty, or conceptual error, is to indicate error or uncertainty in any one of these facets: most simply, that except for an incorrect mathematical form for the interaction between two variables, all else about the model's structure is correct; more profoundly, that significant and manifest attributes of the system's behavior appear to attach to (unknown) variables entirely omitted from the model. As a problem in hydrology and environmental systems analysis, structural error/uncertainty in a model is a source of uncertainty attracting increasing attention [Beven, 2005; Refsgaard et al., 2006]. Ideally, it should be identified, quantified, accounted for, reduced in its extent, and, if at all possible, rectified by invoking alternative, constituent hypotheses to be incorporated into a model with therefore a necessarily revised structure. In general, as the biogeochemical and ecological content of the model increases, one could argue that the difficulty of dealing with structural error/uncertainty is progressively exacerbated.

[3] In the present paper, our concern lies in the particular problem of taking a prior structure for the model and, by reference to a set of field data, proceeding to an improved posterior model structure. We refer to this as model structure identification. Having studied approaches to solving this problem over many years [see, e.g., Beck, 1987], it has been helpful to separate the overall problem into two significantly distinct subproblems: (1) demonstrating failure in the individual components of the model structure, i.e., failure of some or all of its constituent hypotheses when tested against the field data (as opposed to demonstrating merely aggregate failure of the model as a whole) and (2) probing the model's structure with a view to generating speculations about why it is flawed (in parts) and how those flaws might be eliminated. The significance of this distinction arises from both a broadly Popperian philosophical underpinning to the particular algorithmic approach of recursive estimation to model structure identification, and from the conceptual analog of testing physical engineering structures to the point of failure, in their constituent members [Beck, 1987].

[4] In the following we focus primarily on improving procedures for solving subproblem (1), more recent examination of which [Beck et al., 2002] reveals a further important binary distinction, with significant implications central to the present paper for the design of improved algorithms for model structure identification. When a model is constructed, certain pieces of the science base are presumed known and included in explicit mathematical form, to which we shall subsequently refer as the {presumed known}. This implies a complement, of that which is acknowledged as not known, the {acknowledged unknown}, and therefore not included in the model's structure, except typically under the lumped, and largely conceptual, stochastic processes customarily referred to as the system and/or observation noises. In the light of this distinction, the foregoing reference to structural “error/uncertainty” is not a matter of being pedantic. For there are important differences between discovering that the {presumed known} is in fact in error and discovering that something of significance, not arising from pure chance, resides in the uncertainty of the {acknowledged unknown}. This distinction will prove to be especially important herein.

[5] The paper presents an approach to solving some of the challenges of model structure identification cast essentially within the framework of recursive estimation of the model's parameters (coefficients). Similar kinds of investigations can be supported by other algorithmic frameworks, however, for example, that of Regionalized Sensitivity Analysis and related methods [Chen and Beck, 2002; Beven, 2002; Osidele and Beck, 2001], as most notably and completely realized in Wagener et al. [2003]. Perhaps more important than the algorithmic framework of recursive estimation therefore is the role of assuming that the model's parameters may be described not as time-invariant random variables but as stochastic processes, i.e., capable of exhibiting variations with time. After outlining the development of a novel extension of Ljung's recursive prediction error (RPE) algorithm [Ljung, 1979; Stigter and Beck, 1994, 2004], designed to focus on estimating time-varying model parameters expressly in the context of model structure identification, the performance of this new algorithm is first evaluated against a hypothetical case study and then extended to a full study of the biogeochemical dynamics of a manipulated pond system [see also Lin, 2003].

2. Derivation of the RPE Algorithm

[6] Key to solving the problem of model structure identification in this paper is the idea that the parameters in a model may vary with time (and space).

2.1. Pivotal Perspective: Time-Varying Parameters

[7] Conceiving of parameters in a model as entities changing with time, the notion that they might not actually be “constants,” and applying this outlook within the context of model structure identification, date back at least to the late 1960s [Young, 1978; Beck, 2002], if not earlier [Young, 1984]. The logic of why structural error/uncertainty in a model, which is axiomatic, implies the need to conceive of model parameters as capable in principle of variations in time, is an argument of rather more recent origin [Beck, 2002, 2005], and will not be rehearsed here. Ultimately, progress in acquiring knowledge of any system's behavior is gauged by the search to achieve the goal of a model populated by parameters that are indeed demonstrably constants. Suffice it to say that being able to estimate values for a model's parameters that change with time is therefore indicative of that goal not having been attained (strictly speaking, the goal is essentially ever-receding). This is informative evidence of (1) the fact that the model's structure contains flawed constituent hypotheses or suffers from significant omissions and (2) of the manner in which those flaws might be rectified and omissions filled, as a part of the search for invariance in the model's parameters (and therefore provisional “stability” in the bits of the science base encoded in the model).

[8] While the availability of mathematical filtering theory and recursive algorithms for state parameter estimation may well have prompted conception and characterization of the problem of model structure identification [Beck and Young, 1976], these frameworks are neither the only means of approaching solutions to the (self-styled) problem, nor the only motivation for an interest in the notion of time-varying parameters.

[9] Faced with the recalcitrant problem of a lack of identifiability in calibrating hydrological models, Wheater et al. [1986] sought yet another route to its obviation. Specific segments, blocks, or windows in the empirical hydrological record are especially informative (information-rich) with respect to identifying the values of particular model parameters. Thus, instead of seeking to choose uniquely best, invariant, singular values for all of the parameters across the entirety of the empirical record, i.e., for all (observed) time, it could be more beneficial to search for uniquely best, invariant, singular values for some of the parameters for some segments of the record, i.e., for some of the time. Thus was opened up the possibility, not exploited at the time, of the parameters desirably having different values at different times. Likewise faced with an inevitable lack of model identifiability, in their benchmark paper Gupta et al. [1998] came to the view that further progress in model calibration would only be achieved through radical changes of perspective. They proposed that algorithms of parameter estimation should henceforth be assigned the task of seeking to minimize structural error in the model at all (discrete) points in time. Further, under a Pareto perspective on the attaching optimality, if this meant different “best” values for the model's parameters at different instants in time, so be it. They too had thereby opened up the prospect of entertaining parameters desirably having different values at different times [Gupta et al., 1998].

[10] Motivated by the ubiquitous lack of model identifiability, and conjoining their respective methodological backgrounds, these two schools of thought [Wheater et al., 1986; Gupta et al., 1998] subsequently gave rise to a more complete means of exploring the temporal variability in the identifiability of specific model parameters [Wagener et al., 2003] (which has its counterpart in the framework of recursive estimation [Beck et al., 1990; Stigter and Beck, 2004]). Two further, more recent steps inter alia have more fully illuminated the role of such identifiability analysis in model structure identification [Smith et al., 2005] and expressly exploited temporally varying parameter estimates in an online hydrological forecasting context (not our present purpose), echoing some of the earlier approaches to this problem of the 1970s [Moradkhani et al., 2005a, 2005b].

2.2. Estimating Time-Varying Parameters: Innovations Representation and Structural Error/Uncertainty

[11] It is clear from the work of Wagener et al. [2003] and Chen and Beck [2002] that the broad framework of recursive estimation, with its origins in mathematical filtering theory [Jazwinski, 1970], is not a prerequisite for estimating temporal variations in the estimates of model parameters. This is likewise clear from the work of Beven [2002], who explores the issue of whether a change in model structure (parameter values) can be detected from before to after a significant hydrological event (fire, drought), all the uncertainty notwithstanding. Recursive estimation, however, will be the framework employed henceforth herein.

[12] While current developments have their origins in Kalman filtering [Beck, 1987], with its more contemporary algorithmic realizations in the forms of ensemble filtering [Moradkhani et al., 2005b; Drécourt et al., 2006a, 2006b] and the related Bayesian approach of particle filtering [Moradkhani et al., 2005a] (and the precursor work of Thiemann et al. [2001]), they derive from a seminal paper of Ljung [1979]. His recursive prediction error (RPE) algorithm arose from the need to overcome some of the notorious difficulties of working with the extended Kalman filter (EKF) as a parameter estimator. Rather than viewing the algorithm as a derivative of filtering theory, however, it might more accurately and usefully be understood as a recursive formulation of the scheme for minimizing a model fitting function composed of the sum of squared one-step-ahead prediction errors, which are very closely similar to the innovations errors of filtering theory [Beck et al., 2002]. Indeed, derivation of the RPE algorithm (below) begins with the so-called innovations representation of a system's dynamic behavior.

[13] To place our current version of the RPE algorithm in historical context, having first mechanized the basic algorithm and applied it to the problem of model structure identification, presuming all model parameters to be time-invariant [Stigter and Beck, 1994], it was subsequently modified to cater for the estimation of time-varying parameters using the rather simple, crude expedient of exponential weighting of past data [Stigter and Beck, 2004]. These earlier developments formulated the RPE algorithm in the state space of the model. In this paper, two significant, novel modifications are made: (1) the model's parameters are presumed to vary according to generalized random walk (GRW) processes, with therefore specific noise variances attaching to the presumed degree and type of temporal variability in particular, individual model parameters and (2) the RPE is formulated in the parameter space of the model.

[14] Many models of the behavior of environmental systems can be defined according to the following continuous-discrete innovations format [Ljung, 1979, 1987], i.e.,

equation image
equation image

where t and tk are continuous and discrete time, respectively. u and y are the system's input and output vectors with dimensions r and m, respectively. x is an n-dimensional state vector; and θ is a p-dimensional vector of model parameters. f and h are vectors of nonlinear functions. The structure of the model is most succinctly conveyed in terms of [f, h], which denote the logical interconnections among u, x, y, while θ signifies parameterization of the particular mathematical expressions of all the hypothetical mechanisms believed to underpin these interactions. ν(tk) represents the observation noise and is estimated using the innovation ɛ(tk), the mismatch between the predicted and observed values of the output at the next sampling instant in discrete time, i.e., ɛ(tk) = y(tk) − equation image(tk). The gain matrix K is a weighting matrix and can be thought of as a device for distributing the impacts of the innovations among the constituent representations of the various state variable dynamics, i.e., the representations fi(·) for each state xi. Kν(t) is customarily considered to represent the system noise, composed primarily of unobserved, incoming, input disturbances of the system and, secondarily, of the errors of observation associated with u. Crucially herein, however, provided K can be estimated or enumerated in some way, Kν(t) in equation (1) is a notional representation of a computable quantity, (tk), reflecting those attributes of the system's behavior that are either omitted or not to be included in the model in more specific mathematical forms. Estimating the elements of K can be thought of as akin to the earlier approaches of adaptive estimation, in which elements of the variance-covariance matrices of the system and observation noises are estimated in a noninnovations format formulation of conventional filtering algorithms.

[15] As discussed by Beck [2005; see also Beck et al., 2002; Stigter and Beck, 2004], the above innovations formulation has many advantages with respect to allowing us to probe the model's structure using numerical methods. In fact, structural error/uncertainty enters into equations (1) and (2) through θ, Kν(t), and ν(tk), although these points of entry differ in their interpretation and significance. The principal distinction is between θ, embedded within the choices for [x, θ, f, h], which we presume to know of the system's behavior, to connote what is the {presumed known}, and [Kν(t), ν(tk)], which fall outside the scope of what we believe we know, thus to connote the {acknowledged unknown}. The former can be wrongly presumed in the event, i.e., be found to be in error (a “structural error”), while the latter is more intuitively what we would label “structural uncertainty”. Like θ(t), the other elements of which the structural error/uncertainty is composed, i.e., [Kν(t), ν(tk)], must also in principle be capable of changing with time, self-evidently so for ν(tk), but also for K(t). Furthermore, because of their mutually exclusive relationships with the {presumed known} and {acknowledged unknown} subspaces, the elements of the parameter vector θ and the matrix K can respectively be used as tags for the two subspaces. In other words, any significant variations in the reconstructed estimates of the elements of θ specifically reflect changes or inadequacies of the model's constituent hypotheses in the {presumed known}. Excursions of the reconstructed estimates of the elements of the matrix K significantly from their prior, presumed values of 0.0 indicate a change, or incompleteness, in the model as a whole, as subsumed under the {acknowledged unknown}. Such estimation of θ and K will be simultaneously carried out in the RPE algorithm, as modified (in section 2.3) for the purpose of model structure identification.

2.3. Formulation of the RPE Algorithm for Time-Varying Parameter Estimation

[16] Inspired by Ljung's work [Ljung, 1979, 1987] and departing from the above-defined innovations model (equations (1) and (2)), Stigter [1997] developed a continuous-discrete RPE algorithm and applied it to several case studies for the purposes of state/parameter estimation and model structure identification. It is well documented by Beck et al. [2002] that as a parameter estimator the RPE algorithm has many advantages over the conventional EKF [see also Stigter and Beck, 1994, 2004]. However, demonstration of the failure of the prior model structure, i.e., unequivocal demonstration of the inadequacies in the {presumed known}, is the outcome of merely the first step in the procedure of model structure identification. In addition, according to the essential nature of the propagation of the variance-covariance matrices of the RPE algorithm, once failure of the defective model first occurs, the resulting deformation of the model's structure becomes “crystallized,” so that any further adaptation is suppressed. Any subsequent change or failure of model structure that might have been revealed is thereby masked; in effect, it is subsumed under the {acknowledged unknown}. Consequently, the form of an improved candidate model structure cannot be inferred from the patterns of the estimated parametric trajectories from the RPE algorithm. To support this particular kind of inference it would be necessary to presume the presence of time-varying parameters (TVPs) in the model's {presumed known} subspace, whose reconstructed changing pattern could provide us with hints about the nature of the system's structural inadequacies, thus to make inferences about an improved candidate model structure. The RPE algorithm must be modified therefore to accommodate the simultaneous estimation of both time-invariant and time-varying parameters, such that the structural errors originating in inadequate (or wrong) constituent hypotheses in the {presumed known} can be distinguished from something of a systematic, nonrandom character arising from factors encompassed by the {acknowledged unknown}.

[17] Temporal evolution in each TVP is assumed to be described by a generalized random walk (GRW) process defined as follows [see also Young, 1984, 1999, 2001]:

equation image

where

equation image

(p + n × m) is the number of the parameters in equations (1) and (2), including the elements of the matrix K; ηi(tk) = [η1i(tk), η2i(tk)]T, is a 2 × 1, zero mean, white noise vector that allows for stochastic variability in the parameters and is assumed to be characterized by a (normally diagonal) variance-covariance matrix Qηi. Hereafter we assume that (1) the model function (equation (1)) is written in continuous time, (2) the observation equation (equation (2)) is linear in its argument, i.e.,

equation image

and (3) the observation noise ν(tk) with variance-covariance matrix Λ is Gaussian and independent of all the ηi(tk) defined above.

[18] In addition, a discrete time “observation” equation (equation (6)) that is linear in the parameters can be formulated to describe the locally linearized relationship between the observations y(tk) and the augmented parameter vector α(tk) (including both model parameters θ and elements of the K matrix) as follows:

equation image

[19] The astute reader should not be confused by the similar appearances of H(x, u; tk) and H(θ; tk) in equation (5). Normally, the latter is an m × n matrix with alternate unity and zero elements, which relates the observations vector y(tk) to the state-variable vector x(tk), whereas the former is an m × (p + n × m) time-variable matrix relating the observations vector y(tk) to the augmented parameter vector α(tk). This latter can be obtained through the differentiation of the model output vector equation image(tk) with respect to the augmented parameter vector α(tk). Fortunately, the sensitivity coefficient matrix ψT(tk) calculated in the RPE algorithm (see Stigter [1997] for details) satisfies the property required by H(x, u; tk). The “observation” equation (6) can therefore be rewritten as:

equation image

[20] Straightforwardly, an overall state space model (equations (8) and (9)) can then be constructed by the aggregation of the subsystem matrices defined in equations (3) and (4), with the observation equation defined by equation (7), to estimate the parameters in the augmented parameter vector α(tk), which are in fact treated as “state” variables in this state space model.“State” equations

equation image

“Observation” equations

equation image

[21] If l = 2 × (p + n × m), then F is an l×l block diagonal matrix with blocks defined by the Fi matrices in equation (4); G is an l×l matrix constructed by the concatenation of the corresponding subsystem matrices Gi in equation (4) as well; and η(tk) is an l-dimensional vector containing, in appropriate locations, the white noise input vectors ηi(tk) to each of the GRW models in equation (3). These white noise inputs, which provide the stochastic stimulus for parametric change in the model, have a covariance matrix Q formed from the combination of the individual covariance matrices image To summarize, the modified RPE algorithm for estimating the time-varying parameters and the corresponding variance-covariance matrix has the following form:Prediction step [Young, 1984, 1999]

equation image
equation image

Correction step [Stigter, 1997; Stigter and Beck, 2004]

equation image
equation image

where ɛ(tk) = y(tk) − H(θ; tk)x(tktk−1); and L(tk) is the gain matrix that translates information on the mismatch between the predicted and observed outputs into corrections of the parameter estimates, just as the K matrix fulfils this function for the state estimates in conventional filtering theory (see Stigter [1997] for details). Note that the correction step (equations (12) and (13)) is only an excerpt from the complete continuous discrete RPE algorithm derived by Stigter [1997].

[22] To summarize, being cast in the parameter space, the algorithm of equations (10)(13) differs significantly from all previous versions of the RPE algorithm, which were cast essentially in the state space. It articulates the key principles of Young's [1999] algorithms for estimating time-varying parameters in input-output, “externally descriptive” models into the domain of models that are internally descriptive, i.e., seek to account for the conceptual mechanisms governing the manner in which input stimuli are transcribed into output responses. Indeed, given the similarity of algorithmic approach spanning the two forms of models (“data-based” and “theory-based,” respectively), it contributes significantly to implementation of a recently proposed dual approach to identifying models of environmental systems [Lin and Beck, 2006, 2007]. Above all, it realizes in specific, computational form many of the conceptual features proposed previously as desirable for the design of an algorithm for model structure identification [Beck et al., 2002].

3. Hypothetical Case Study

[23] The RPE algorithm (equations (10)(13)) modified for estimating time-varying parameters has been tested on a hypothetical nonlinear system of biomass-substrate interaction, as for example in the biogeochemical dynamics of a surface water or groundwater system, taking place in a single (idealized) continuously stirred tank reactor (CSTR), or conceptual hydrological store. The true system is defined by the following set of ordinary differential equations:

equation image
equation image

where x1(t) and x2(t) are the concentrations of biomass and substrate in the store (CSTR) at time t, M L−3; μmax is the maximum specific growth rate of the biomass, T−1; KS is the half-saturation constant of the substrate, M L−3; Y is the yield coefficient of biomass on substrate, M M−1; u(t) is the inlet concentration of substrate, M L−3; and q0 is the (constant) dilution rate of the CSTR, T−1. Given the specific parameter values presented in Table 1, the system is simulated using the SIMULINK® software and a typical set of input and output signals is shown in Figure 1. y1(t) and y2(t) are the observed time series of x1(t) and x2(t), respectively. They are corrupted with white Gaussian measurement noise with variance-covariance matrix

equation image

Clearly, the assumption that the biomass observation can be obtained from an online sensor of some kind is rather strong, but acceptable in the context of this hypothetical example.

Figure 1.

Typical set of simulated data for the hypothetical system using parameter values in Table 1.

Table 1. Parameter Values in the Model of the Hypothetical System
ParameterValue
μmax0.3
KS3.0
Y0.6
q00.1
[x1(0), x2(0)]T[5.0, 2.0]T

3.1. TVP Estimation Using the RPE Algorithm

[24] Turning thus to the problem of model structure identification, suppose now that our prior conceptual model of the system's observed behavior is of a simpler structure, being linear in its assumptions about the kinetic interactions between biomass and substrate. That is to say, we have, in effect, the following approximation

equation image

which, when substituted into equations (14) and (15), yields our prior candidate model structure as

equation image
equation image

where θ(t) is a time-varying parameter reflecting the possibility of a changing structure in the dynamics of the state variables (x1(t) and x2(t)), in line with equation (16).

[25] At this point, let us recall our basic concern in this paper, in the overall context of model structure identification, that is: to be able to estimate time-varying model parameters, as they appear in differential equation forms of state space models, in a more systematic manner than previously. We now proceed therefore to embedding this candidate linear model (equations (17) and (18)), of known incorrect structure, into the previously developed RPE algorithm (equations (10) through (13)), for that structure thereby to be reconciled with the observed data set of Figure 1. Self-evidently, this entails estimating the time-varying parameter θ(t) in the linear model, for which purpose the dynamics of θ(t) during the observed period are treated as a random walk (RW) process. Although this is our primary goal in this demonstration example, the algorithm also estimates the time-invariant parameter Y and the elements in the K matrix at the same time. The specifications of the initial conditions of the state variables, the initial values of the parameters, and the leading diagonal elements of the variance-covariance matrices P(t0), Q and Λ are all given in Table 2. It should be noted that the initial values for the two model parameters (θ(t) and Y) are deliberately set to be different from their true values (shown in the parentheses) in order to reflect the customary situation where there is no access to the true initial values of the parameters. Robustness in the performance of the RPE algorithm, with respect to the choices of these initial values and the amplitudes of the observation noise variances, is therefore assessed below.

Table 2. Initial Values and Leading Diagonal Elements for Vectors and Matrices of the RPE Algorithm
State/ParameterInitial ValueP(t0)QΛa
  • a

    Estimated using equation (19); numbers in parenthesis are the true values.

x1(t)5.00.509 (0.5)
x2(t)2.00.205 (0.2)
θ(t)−0.2 (−0.5)1040.005
Y0.8 (0.6)10−30
K elements010−30

[26] The initial values of the main diagonal elements of the variance-covariance matrices for the parameters (including the elements of K) will influence the initial adaptations of the corresponding parameters; that is, the higher the values of P(t0), the more the parameters will change in the transient estimation period. Young [1984] suggests that each diagonal element in P(t0) should take on a large value, in order to obtain a recursive parameter estimate equivalent to the corresponding en bloc estimate and to reflect our a priori unknown knowledge about the parameters. However, in practice, too large values of P(t0) may allow the corresponding parameters to assume “unreasonable” values during the transient stage, such that we cannot obtain stable numerical solutions of the model (which is especially a problem for models cast in differential equation form, as herein, as opposed to discrete time algebraic form). A compromise value for pY(t0) was accordingly chosen: small enough to guarantee stable solutions; yet large enough to allow this parameter's estimate to converge to its true value (or its equivalent en bloc estimate, if the true value is not attainable). The variances attaching to the elements of the matrix K, i.e., the set of variance elements pk(t0), are assigned the same value as pY(t0), since the elements of K are also treated as time-invariant parameters. Since θ(t) is to be estimated as a time-varying parameter, the corresponding pequation image(t0) is given a larger value, hence to reflect our a priori lack of knowledge about this parameter. While the exact values of the variances of the observation noise, i.e., the diagonal elements of Λ, are known in the present hypothetical example, this will not usually be the case. Thus the method suggested in the work of Wadsworth [1998] is adopted for estimation of the variance of an observed time series (see also Pastres et al. [2003] or Drécourt et al. [2006b] for an alternative), where for a time series y(tk), k = 1, 2, …, N,

equation image

[27] Figure 2 presents the results, then, of reconciling the (erroneous) prior linear model structure embedded within the RPE with the (error-corrupted) data of the true system's behavior. It is apparent that the performance of the algorithm is indeed successful, in the sense that comparison with the known true situation is demonstrable. The estimates of the time-invariant parameter Y converge to its true value very quickly, while the estimates of the time-varying parameter θ(t) track its true pattern of variation successfully, notwithstanding the fact that the algorithm commenced processing the data with wrong initial estimates of this parameter.

Figure 2.

RPE algorithm estimation results. (a and b) Parameter estimates (estimated value, cyan solid line; standard error, magenta dotted line; true value, black dashed line), (c and d) biomass reconstruction and associated innovation residuals, and (e and f) substrate reconstruction and associated innovation residuals.

[28] In Figure 3 the recursive estimates for the elements of the K matrix are presented, where it is apparent that all four elements deviate from their initial values of zero, although not significantly for three of the four elements, according to the estimated standard error bounds. At the outset, the behavior of the recursive estimates of these elements may have been influenced by the errors associated with the incorrect initial assumptions about the model's parameter estimates. It is helpful to note here that the two diagonal elements, k1,1(tk) and k2,2(tk), reflect the extent to which uncertainty arising from the mismatch between the predicted and observed outputs, i.e., the estimated innovations ɛ1(tk) and ɛ2(tk) (as in Figures 2d and 2f), is passed through to the state-variable dynamics for x1(t) and x2(t) respectively. The off-diagonal elements can thus be thought of, for this particular example, as the extent to which uncertainty in predicting observed biomass, for instance, is distributed back to the unfolding dynamics of the substrate concentration (and vice versa). Numerically, comparison of the trajectories for the two diagonal elements of K, suggests that relatively more of the perceived mismatch in predicting observed biomass, i.e., ɛ1(tk), is fed back into the dynamics of the biomass state (through k1,1(tk)) than is the case for the substrate state (through k2,2(tk)), since the magnitude of the former is generally greater than that of the latter, as well as being estimated to be significantly different from zero (according to the standard error bounds). When the algorithm uses the correct initial values for the model's parameter estimates, the four elements of the K matrix show no significant migration from their initial values of zero (Figure 4), hence our foregoing assertion regarding the consequences of employing incorrect model parameter estimates a priori, as would normally be the case in practice.

Figure 3.

Trajectories of K matrix elements, recursively estimated by the RPE algorithm using wrong initial values for parameter estimates.

Figure 4.

Parameter estimates and trajectories of K matrix elements, recursively estimated by the RPE algorithm using correct initial values for parameter estimates.

3.2. Testing Robustness in the Performance of the RPE Algorithm

[29] In order to test the effect of the observation noise amplitudes and the prior model parameter estimates on the performance of the modified RPE algorithm, three levels of the observation noise were chosen to corrupt the system output observations (y1(t) and y2(t)) and 100 pairs of initial values were randomly selected for the prior parameter estimates (Y and θ). The low noise level was set at

equation image

the medium noise level at

equation image

and the high noise level at

equation image

The feasible parameter space is [0, 1) for Y and (−3.0, −0.2) for θ in this specific case (given Y = 0.6). In order to dramatize the effect of the initial parameter values on the execution of the RPE algorithm, however, we enlarge the value-selecting interval for θ from (−3.0, −0.2) to [−10.0, 10.0]. The 100 pairs of initial values for Y and θ are randomly selected using the Latin Hypercube sampling method (see Figure 5a).

Figure 5.

(a) Latin hypercube sampling pairs of Y and θ. (b–d) Distributions of the coefficients of determination (RT2) between true values and estimates of the time-varying parameter at three different observation noise levels. Note that the transient stage was not included in calculation of RT2.

[30] The success of the RPE algorithm in estimating the time-varying parameter under these different test scenarios, combinations of different noise levels and different prior parameter estimates, was gauged by the coefficient of determination (RT2) between the true and estimated variations of θ. The coefficient of determination is defined as RT2 = 1 − equation image, where σe2 is the variance of the residuals between true and estimated variations of θ; σ2 is the variance of the true variation of θ. The coefficient of determination takes values from −∞ to 1; the larger value of RT2 means the estimated equation image(t) is closer to the true θ(t). Unity means a perfect match. The distributions of the coefficients of determination at the three different observation noise levels are given in Figures 5b–5d. It is apparent that with higher noise amplitudes present in the system's output observations, the less successfully the modified RPE tracks the true trajectory of the time-varying parameter. It is worth noting that when the noise level is high the majority of the coefficients of determination are less than 0.85. In this case, one should be very careful in choosing initial values for the model parameters.

[31] Figure 6 displays the best and worst estimates of the time-variant and time-invariant parameters at the three different noise levels, along with their true values. Two different pairs of initial parameter values are responsible for the best performances at all three noise levels while another two different pairs of initial parameter values are responsible for the worst performances, as identified in Figure 5a; the best pairs being identified with circles in the top left corner, while the worst pairs are identified with triangles in the bottom left corner. Both the latter occurred when the initial values of both parameters (Y and θ) were far away from their true initial values. For the best pairs, however, the initial value for θ was also far from its true initial value (∼−0.2), while the initial value for Y is close to and slightly larger than its true value (0.6). This means that the RPE is more sensitive to the selection for the initial value of Y (time-invariant parameter) than that of θ (time-variant parameter) in this hypothetical example.

Figure 6.

Best (magenta solid line) and worst (cyan dashed line) parameter estimates for three different noise levels present in the observed system outputs. (a and b) Low noise level, (c and d) medium noise level, and (e and f) high noise level.

3.3. Conclusions From Hypothetical Example

[32] Taking stock of these results, our first interim conclusion is that the modified RPE algorithm performs well in being able to track the trajectory of a time-varying parameter while converging on the true value of an invariant model parameter. Moreover, the former signals in some way the structural error of the candidate model. However, this success is contingent upon prior knowledge of which of the model's parameters is expected to vary with time and which not, since this governs the choice of values assigned to the variance-covariance matrices required for implementing the algorithm (see Table 2). Such prior knowledge might be available in a given case study, as a result of having previously tested the model's structure with the express purpose of exposing its inadequacies, as already discussed by Stigter and Beck [1994] for a model of river water quality.

[33] At a technical level, we observe that the estimated trajectory of the time-varying parameter is insensitive to the values assigned to the elements of Q, which determine, in effect, the propensity of the corresponding parameters to vary with time, if these values are nonzero. Tests (not shown) reveal that the algorithm still generates reasonable estimates for θ(t) across a range of values for the corresponding element of Q from 0.0005 to 0.05. Alternatively, the values of those elements of Q corresponding to the parameters to be estimated as time-varying can be optimized by the maximum likelihood optimization method, using prediction error decomposition, as suggested by Young [1999]. This adds measurably to making implementation of the RPE algorithm more systematic, and much less arbitrary, than the previous experience with the EKF, when applied to the problem of model structure identification [Beck, 1994; Stigter and Beck, 1994].

[34] At a more strategic level, recalling our overall purpose in being able to address the problem of model structure identification more systematically, while bearing in mind the previous work with exponential weighting of past data [Stigter, 1997], we also conclude that the current RPE algorithm performs well in respect of reconstructing the trajectories of time-varying parameters. The ideas of Young [1999; see also Young, 2000, 2001] regarding the role of time-varying parameters in what he calls state-dependent parameter modeling have, we argue, been successfully transferred from recursive estimation algorithms designed primarily for discrete time, algebraic models to the internally descriptive, differential equation models of this paper. Thus, for example, it is apparent from the reconstructed trajectory for θ(tk) (in Figure 2) that this parameter must be correlated in some way with the oscillations in the state variables (Figure 1), as clearly expected through equation (16), possibly more so with those of the substrate concentration (x2). Close inspection of the confidence bounds attaching to the trajectory of θ(tk) reveal that its rising limbs are less well estimated, when the substrate concentration is close to zero and biomass concentration is declining.

[35] Somewhat less satisfactory, however, is performance in respect of being able to employ the estimated trajectories of the model's parameters (θ) and elements of the matrix K to distinguish between the detection of errors in what has been {presumed known} and features of significance in the uncertainty of the {acknowledged unknown} (as originally conjectured by Beck et al. [2002]). The structure of the prior candidate, linear model in this instance has not so much overlooked (or omitted) something of significance, as included an incorrect specification of the nature of the kinetic interaction between substrate and biomass. Yet the results would suggest, through the nonzero estimate of k1,1(tk), something of significance not included in that model's structure, although this may be an artifact of treating the elements of K as invariant quantities.

[36] This reservation apart, the general success of our hypothetical example provides a sound basis on which to proceed to the much more challenging issues of a real case study.

4. Case Study: Dissolved Oxygen and Biomass-Nutrient Dynamics in a Pond System

4.1. Study Site, Experiment, and Data

[37] As part of a wider study [Parker, 2004] of sediment-nutrient associations in the streams and impoundments of the Piedmont province of Georgia, United States, full-scale experimental manipulations of a small aquaculture pond were conducted in the summer and fall of 2000. They followed similar prototype manipulations, and subsequent modeling studies, carried out in 1998 [Zeng et al., 2006]. The pond, located on the Whitehall Estate of the Warnell School of Forestry and Natural Resources of the University of Georgia, is normally used to culture fish for the purposes of teaching and research, but in 1998 and 2000 was lying “dormant”. The volume of the pond is approximately 2000 m3; its surface area is approximately 7000 m2; and its maximum depth (adjacent to its dam) is less than 2 m. As part of a 5-month monitoring campaign, the pond was fertilized on three occasions and the system's responses observed more or less in real time by the University of Georgia's Environmental Process Control Laboratory and ancillary instruments. Typical sampling intervals for variables were of the order of minutes. For present purposes, a subset of this large volume of data, for a period of 40 days from 27 June 2000 through 5 August 2000, is examined in detail, with a view to developing an understanding of the dynamics of dissolved oxygen in association with biomass-nutrient interactions in the pond. We note that missing data have herein been interpolated using the CAPTAIN framework for time series analysis (http://www.es.lancs.ac.uk/cres/captain). The smoothed and interpolated time series are shown in Figures 7 and 8for those variables of water quality relevant to the case study.

Figure 7.

Smoothed and interpolated nutrient observations during the pond experiment. Note that the two spikes of the NH3-N and PO4-P concentration time series clearly indicate the fertilizations applied on 5 and 27 July, respectively.

Figure 8.

Smoothed and interpolated physical and biochemical observations during the pond experiment. Note that PAR observations were measured at 0.5 m below water surface and the vertical solid line marked in the graph of chlorophyll a concentration indicates the timing of the first fertilization.

4.2. Prior Model Structure and Implementation of RPE Algorithm

[38] Much is known about the behavior of biomass-nutrient interactions in an impoundment or lake. In fact, this might be more accurately put as: much is “believed to be known” about such systems. For when one consolidates what is believed to be known into a relatively high-order biogeochemical (state space) model of this particular pond system, there are significant challenges in reconciling such prior concepts with the high-volume, high-quality (HVHQ) data collected previously in 1998 [Zeng et al., 2006]. What is more, it is particularly difficult, when confronting high-order models with HVHQ data, to utilize effectively this procedure to identify those components (constituent hypotheses) of the model that are working successfully and those that are not and, then, to speculate on how the defective components might be replaced with more successful representations. Such difficulties are self-evident in the results of Zeng et al. [2006] and they in turn are indicative of our wider experience in developing models, given now the kinds of comprehensive data sets recoverable through the EPCL and other systems of sensors for real-time monitoring of water quality [Kirchner et al., 2004]. Paradoxically, these difficulties become “self-evident” precisely because of the HVHQ data. For significant discrepancies between the model and the customarily sparse, highly uncertain data of previous exercises would not have been readily apparent and, if they were, could have been largely dismissed as a consequence of the poor data.

[39] Thus, while acknowledging the work of Zeng et al. [2006] as a consolidated archive of the multitude of constituent hypotheses (about biomass-nutrient interactions, in general, and the behavior of the Whitehall pond system, in particular) our prior candidate model structure (Table 3) departs from a considerably simpler perspective on the concepts to be included in mathematical/computational form. Its design is, in effect, a prudent combination of a concise representation of the {presumed known} with scope for detecting matters of possible significance in the {acknowledge unknown}, thereby articulating in practice many of the principles set out by Beck et al. [2002], of how to proceed with developing and using models in the presence of significant structural error/uncertainty. To summarize, our prior model has just two state variables, for the concentrations of algal biomass (x1) and dissolved oxygen (DO; x2) in a pond assumed to behave as a single CSTR. It expressly recognizes the constituent mechanisms of algal growth, mortality, predation (grazing by zooplankton), and loss through settling, as well as photosynthetic production of DO, respiratory consumption of DO, re-aeration at the air-water interface of the pond, withdrawal of DO through degradation of organic matter, as well as (rather vaguely) “other sources” of DO in the pond (see Table 3). All of these can be gathered under the {presumed known}, although some of them begin to touch upon parts of what, of possible significance, might be within the vastness of the {acknowledged unknown}.

Table 3. Formulations, State Variables, and Parameters for the Algae-DO Model
ParameterDescriptionUnits/Value
  • a

    Algal biomass concentration is assumed to be 50 times the chlorophyll a concentration in the water column.

Model Formulations
dx1(t)/dt= r1f(N, P; t)f(I; t)f1(T; t)x1(t) − r2f2(T; t)x1(t) − r3(i)
dx2(t)/dt= cr1f(N, P; t)f(I; t)f1(T; t)x1(t) − cr2f2(T; t)x1(t) + r6f3(T; t) [CS(T; t) − x2(t)] − r4f4(T; t) C(t) + r5(t)(ii)
f(N, P; t)= min(N(t)/[KN + N (t)], P (t)/[KP + P (t)]) 
f(I; t)= [I(t)/IS] exp [1 − I(t)/IS] 
fi(T; t)= θT(t)−20; i = 1, 2, 3, 4 
CS(T; t)= 14.589 − 0.4T(t) + 0.008T(t)2 − 0.0000661T (t)3 
 
State Variables and Environmental (Forcing) Functions
x1(t),ax2(t)algal biomass concentration and dissolved oxygen concentrationmg L−1
C(t), N(t), P(t)partial TOC, NH3-N + NOx-N, and PO4-P concentrationsmg L−1
I(t), T(t)photosynthetically active radiation, water temperatureKJ m−2 h−1, °C
 
Fixed Parameters
cRedfield stoichiometric coefficient3.47
ISsaturation constant for solar radiation70.0 KJ m−2 h−1
KN, KPhalf-saturation constant for ammonium and phosphate concentrations0.2, 0.075 mg L−1
θ1,2,3,4coefficients for Arrhenius equations1.02
r2kinetic rate of loss of algae due to respiration and non-predatory death0.05 d−1
r6global transfer coefficient between air and water at 20°C0.6 d−1
 
Time-Invariant and Time-Varying Parameters (to Be Estimated)
r1specific growth rate of algae at 20°Cd−1
r3loss of algae due to grazing and settling processes, etc.mg L−1 d−1
r4total organic matter decay rated−1
r5(t)sources of DO other than those included in the model explicitlymg L−1 d−1

[40] To ensure identifiability of the model's structure, an essential prerequisite for any such study as this, many of the model's parameters were assumed to be known with certainty. That is to say, they were assigned invariant values and not estimated. Only four parameters, r1, r3, r4, and r5(t), were estimated using the RPE algorithm, of which r1, r3, and r4 are assumed to be time-invariant over the given period studied, while just one parameter, r5(t), is allowed to change with time, so that confounding, correlated variations among the TVPs are eliminated. The variation of this latter is again modeled as a Random Walk stochastic process (as in the hypothetical example). The four elements of the matrix K are also (again) considered as time-invariant, their values being set initially as zero, thus implying rather boldly that the candidate prior model structure describes the behavior of the real system perfectly, i.e., the state variable dynamics are not subject to any stochastic disturbances. Any subsequent, statistically significant excursions of the recursive estimates of the elements of K away from zero would accordingly contradict this presumption and, according to our arguments, be indicative of features of significance within the {acknowledged unknown} subspace.

[41] Initial values of the states and the four parameters to be estimated, together with the leading diagonal elements of the variance-covariance matrices P(t0), Q, and Λ are specified in Table 4 (following the same principles as already discussed for the hypothetical example). The initial values for the states are based on the observations from the pond. The initial values for the kinetic parameters (r1 and r4) are taken from the corresponding parameter ranges in the literature [Bowie et al., 1985] and our previous study of the pond [Zeng et al., 2006].

Table 4. Initial Values and Leading Diagonal Elements for the Algorithm-Relevant Matrices
State/ParameterInitial ValueP (t0)QΛ
x1(t), mg L−12.170.0144
x2(t), mg L−16.890.1180
r1, d−11.510−30
r3, mg L−1 d−10.021040
r4, d−10.1510−30
r5(t), mg L−1d−101040.00005
K elements010−30

4.3. Prior Model: Results and Discussion

[42] Results from reconciling the prior model, embedded within the RPE algorithm, against the field data (Figures 7 and 8), are presented in Figures 911. It is in the nature of a recursive algorithm such as the RPE that the one-step-ahead predictions track rather closely the observed outputs, as in Figure 9.

Figure 9.

RPE algorithm state estimation results, i.e., one-step-ahead predictions (solid line) compared with observations (crosses). (a and b) Algal biomass and associated innovation residuals and (c and d) dissolved oxygen and associated innovation residuals. The vertical solid lines indicate the timing of the first fertilization.

Figure 10.

RPE algorithm parameter estimation results for the a priori model: parameter estimates (solid line) and standard errors (dashed line). The vertical solid lines indicate the timing of the first fertilization.

Figure 11.

RPE algorithm K matrix estimation results: estimates (solid line) and standard errors (dashed line). The vertical solid lines indicate the timing of the first fertilization.

[43] What is of special interest to us, in respect of identifying the successes and failures of the various components of the model's structure, is bound up with the recursively estimated trajectories of the model's parameters and the elements of the matrix K. To begin assessment of these, we first inspect the trajectories of the three time-invariant parameters (r1, r3, and r4), all of which converged to invariant values, and close to the initial estimate chosen from the literature in the case of r1 (Figure 10). All too are not greatly affected by the substantially “exciting” disturbance of the fertilization, which, in principle, should provoke subsequently observed information-rich responses in the system's behavior, especially in respect of the algal growth rate parameter (r1). We might be tempted to conclude therefore that this is evidence of confirmation of the constituent model hypothesis relating to algal growth. Yet again it is important to keep in mind the fact that an algorithm such as the RPE is primed (by the choices of the matrices in Table 4) to translate significant mismatches between overall observed and conjectured behavior preferentially into changes and fluctuations in those quantities deemed to be time-varying. A vital difference between the RPE and EKF algorithms, however, is that in the former such translation is directed not at the model's state variables, but its parameters and the elements of matrix K; and it achieves this with fewer, arbitrary assumptions about the accompanying variance-covariance matrices, here P(t0), Q, and Λ [Stigter and Beck, 1994; Beck et al., 2002].

[44] The estimated trajectory of r5(t), the lumped sources of DO other than those included in the model, i.e., re-aeration and algal photosynthesis, is predominantly positive. Speculating on why this should be so depends, like so many aspects of environmental modeling, on having access to rather intimate experience of the field situation and the conduct of the experiment. During this field campaign, a population of the duckweed Lemma, a species of free-floating macrophytes, was observed to propagate rapidly after the fertilization, starting from the rim of the pond and eventually covering almost its entire surface. One small detail in Figure 10 suggests that a period of three to four days elapsed (after the fertilization) before the duckweed began to have a noticeable impact on the behavior of DO in the pond. This is consistent with field experience, where the biomass of the duckweed was not observed as significant for several days after the fertilization. We note, with some significance, that considerations of duckweed are entirely absent from the prior knowledge base of Zeng et al. [2006], possibly because duckweed was not present on the pond during the 1998 campaign. It is in any case rarely a factor mentioned in the extensive literature on modeling the behavior of eutrophic systems, wherein some models incorporate the behavior of submerged, but not floating macrophytes, which is what the duckweed is.

[45] Figure 11 displays the recursive estimates of the K matrix elements, all of which are perturbed by the observed responses to the act of fertilization of the pond, and more so in for k1,1(tk) and k1,2(tk) than for k2,1(tk) and k2,2(tk). In contrast to the results of the hypothetical example (Figures 3 and 4), all too progress toward values that are estimated to be significantly nonzero. Crudely speaking, it would appear much of significance in accounting for the pond's behavior, relative to this candidate prior model structure, must be present in the {acknowledged unknown}. Technically, element k1,1(tk) indicates that overestimates of the algal biomass response, reflected in negatively valued innovations for that first state variable, i.e., ɛ1(tk), will be fed back to the evolving dynamics of that state as negatively valued disturbances (tending thus to slow the speed of change in the state variable and, if anything, to depress its level). The same is true of the diagonal element k2,2(tk) relating to the DO state variable, as well as to the off-diagonal element k2,1(tk), which distributes the impacts of overestimation of the DO state back to the evolving dynamics of the algal biomass as a negatively valued “force” acting on this first state variable. According to the results of Figure 11, however, off-diagonal element k1,2(tk) acts in the reverse sense, i.e., an overestimate of the algal biomass response is fed back as a positively valued disturbance acting so at to accelerate the speed of the DO response and, in due course, raise its level (if anything). However, what more incisively might be discerned from these results for the elements of matrix K, about the possible nature of the important features omitted from the model and cast under the domain of the {acknowledged unknown}, is not readily forthcoming. They do not point clearly in the direction of the influence of the duckweed, the primary suspected missing feature in the prior candidate model structure. Also, perhaps this is inevitable, given that one could argue parts of the {acknowledged unknown} have been included under the grossly lumped time-varying parameter r5(t) present in the {presumed known} and the fact that the elements of K are assumed to be invariant with time.

4.4. Candidate Posterior Model Structure

[46] Prompted by the insights distilled out of our foregoing attempt at reconciling the two-state prior model structure with the observed data, and drawing as appropriate upon the wider biogeochemical model-building experience of Zeng et al. [2006], it is possible to construct a seven-state posterior model for the Whitehall aquaculture pond (Figure 12). The key innovation in this second model, of course, is the inclusion of an account of the state dynamics of the duckweed population into the {presumed known}.

Figure 12.

Water quality simulation model of the Whitehall Pond. State variables are algal biomass concentration; duckweed biomass concentration; dissolved oxygen (DO) concentration; total inorganic carbon (CT) molar concentration, collective term of [H2CO3*] ([H2CO3] + [CO2(q)]), [HCO3] and [CO32−]; alkalinity (Alk) molar concentration, collective term of [HCO3], 2[CO32−], [OH] and negative [H+]; organic phosphorus (OP) concentration; and Orthophosphate (PO4−P) concentration. Modeling processes: 1, reaeration of dissolved oxygen; 2, carbon dioxide (CO2) exchange with atmosphere; 3, algal photosynthesis; 4, algal respiration; 5, duckweed photosynthesis; 6, duckweed respiration; 7, carbonaceous matter aerobic degradation; 8, algal nonpredatory mortality death and excretion; 9, algal predatory mortality death; 10, algal settling to sediment; 11, duckweed mortality death; 12, duckweed photorespiration (excretion); 13, organic phosphorus mineralization; 14, organic phosphorus settling; 15, ammonium-nitrogen nitrification; 16, nitrate-nitrogen denitrification.

[47] In order to illustrate a first test of this candidate posterior model structure's promise, vis à vis the simpler, prior structure, both structures were subjected to a more conventional exercise in calibration, comprising two steps: adjustment of the parameter estimates by informal trial and error; followed by embedding the model within the Shuffled Complex Evolution–University of Arizona (SCE-UA) method of Duan et al. [1992] in order to refine the estimates of up to six of the model parameters judged to be the more important, such as the maximum specific growth rates of biomasses, and the DO consumption kinetic parameters. Comparisons with observations of the resulting deterministically generated variations in the concentrations of algal biomass and DO in the pond are given in Figures 13 and 14, for the prior and posterior structures respectively. The Nash-Sutcliffe coefficient (E) [Legates and McCabe, 1999] was used to evaluate the closeness of fit between modeled and observed time series. E is defined as follows:

equation image

where the Pi are model-simulated data and the Oi are observed data; equation image is the mean of observations; and N is the total number of observations.

Figure 13.

Comparison of simulation results for the prior model structure with observations: (a) algal biomass and (b) dissolved oxygen concentration.

Figure 14.

Comparison of simulation results for the posterior model with observations: (a) algal and duckweed biomasses and (b) dissolved oxygen concentration. No observations are available for the duckweed biomass.

[48] Both models describe reasonably well the dynamics of the algal bloom triggered by the fertilization, although both tend to overestimate persistently the observed algal biomass (see Figures 13a and 14a). The principal difference between the two models is in their simulation of the DO behavior, with performance of the posterior structure being clearly superior (see Figures 13b and 14b). The simulated trajectory of the duckweed biomass state of the posterior model (Figure 14a) is broadly similar to the pattern of the time-varying parameter r5(t) in the prior structure, albeit with a shift of phase suggestive of a duckweed biomass that blooms and crashes more swiftly in practice than in the (posterior) model.

[49] Beyond these investigations, however, a submodel of the candidate posterior model is now embedded into the modified RPE algorithm, to examine whether inclusion of the duckweed dynamics does indeed eliminate the temporal variation exhibited in r5(t) (in the prior model structure). Choosing just a submodel instead of the full structure of the posterior model, to be embedded into the RPE algorithm, is here motivated by a desire to ensure identifiability of the submodel's parameters. In this submodel, only the dynamics of the interactions of the algal biomass, duckweed biomass, and dissolved oxygen concentrations are included. Except for the four parameters (r1, r3, r4, and r5(t)), which have the same meanings as in the candidate prior model structure, all the other model parameters were presumed to be time invariant. Initial values of the algal biomass and DO concentrations and the four parameters, together with the leading diagonal elements of the variance-covariance matrices P(t0), Q, and Λ, were assigned the same values as previously (Table 4). For variations across the entire period of assessment the duckweed biomass concentrations were taken to be those simulated by the full posterior model (see Figure 14a).

[50] The estimates of the four parameters, from this test of the posterior submodel, are shown in Figure 15. Compared with Figure 10, all three time-invariant parameters (r1, r3, and r4) converged to the same values as those estimated through the prior model. The upward trend apparent in the time-varying parameter (r5(t)) of the prior model structure is no longer apparent in these results from the posterior model, although the sinusoidal oscillation pattern is still present. Such a pattern might follow from the influence of a varying zooplankton population in the pond, amongst other speculative causes that might have been pursued (had observations been available), and would motivate inclusion of additional state variables.

Figure 15.

RPE algorithm parameter estimation results for a submodel of the a posteriori model: parameter estimates (solid line) and standard errors (dashed line). The vertical solid lines indicate the timing of the first fertilization.

[51] Since a practical model, by definition, is only an approximation of the real system, identification of the model's structure in the manner outlined would, in principle, be something of an unending quest, punctuated by provisional candidate structures having the highly desirable property of parameters whose estimates do not vary significantly with time over any observed record. An important element of this case study, however, is the manner in which our analyses enable fruitful learning to take place as a consequence of being able to probe the strengths and weaknesses of the prior model structure in ways rarely employed in the more conventional approaches to model calibration illustrated in the companion study of Zeng et al. [2006]. Especially notable is the fact that characterization of the behavior of a part of the system not normally included in such biogeochemical models, the dynamics of a duckweed population, has been incorporated in this paper into the posterior model structure.

5. Conclusions

[52] We have introduced herein a conceptual framework of what constitutes model structure error/uncertainty, in order to appreciate better how systematic estimation of model parameters that vary with time relates to identifying and rectifying inadequacies of model structure. Three points are crucial in this: (1) recognition of the fact that structural error/uncertainty implies some, if not all, of the model's parameters should be conceived of as stochastic processes (as opposed to random variables), (2) the capacity to distinguish formally between the {presumed known} and {acknowledged unknown} components of the model's structure, and (3) the capacity to parameterize these two components in a manner whereby reconstruction of the associated time-varying parameter trajectories is indicative of inadequacies in the model's structure and of how to begin to rectify them.

[53] Our further extension of the recursive prediction error (RPE) algorithm is formulated in the model's parameter space, thereby both realizing the ideas of Young [1999] regarding state-dependent parameter modeling (but here in continuous time, ordinary differential equation models) and making progress in the incremental process of reorienting the design of recursive estimation algorithms away from state estimation (as in the extended Kalman filter) toward parameter estimation and, in particular, parameter estimation specifically in the service of model structure identification. Relative to our foregoing studies with earlier versions of the RPE algorithm, the current algorithm demonstrates superior performance, in respect of both a hypothetical example and a real case study. Our conclusion is that the reconstructed trajectories of the time-varying parameters, as well as the recursive estimates of invariant model parameters, are now being generated algorithmically on a significantly more reliable basis than hitherto.

[54] As always, there are limitations on our progress. These are manifest first in respect of interpreting the trajectories of the estimates for the elements of the K matrix in the algorithm, which expressly parameterizes the content of the {acknowledged unknown} in the model's structure. However, it has to be said that this paper conveys the results of our first experiences in the use of a novel algorithm in much of a novel problem setting. Second, our results illuminate some of the more strategic challenges in choosing what is to be estimated in the structure of the model (and what not), and what is to be estimated as time varying (and what not). These in turn illuminate the challenges of being logical and consistent in setting up the model structure, and in implementing the RPE algorithm (notably assumptions about its variance-covariance matrices), for testing that structure within this new framework, such as, for example, in maximizing the subsequent possibilities for clearly distinguishing between an error in the {presumed known} and something of nonrandom significance in the {acknowledged unknown}.

[55] Future algorithmic studies will focus on three areas: (1) acquiring further experience with the performance and logic of implementing this version of the RPE algorithm, using the case study material of the present paper and other similar comprehensive sets of high-volume high-quality (HVHQ) data, (2) exploring the scope for improving the performance of further adaptations of the RPE algorithm, specifically in respect of estimating time-varying parameters, for example, through incorporating a Fixed Interval Smoothing (FIS) algorithm [Norton, 1975], and (3) undertaking computational studies on how to account for the consequences of any residual structural error/uncertainty when a model is used for the purposes of forecasting and decision support.

[56] Taking stock of these results with the RPE algorithm in the wider context of system identification, we note that by no means are we recommending reliance on this one algorithmic framework alone for unraveling (and resolving) the issues of inadequate model structures. There is great value in a variety of approach, as formally proposed in our companion papers [Lin and Beck, 2006, 2007] and as often advocated more broadly elsewhere (as by Beck [2002], for example). When one has available HVHQ data on the behavior of aquatic environmental systems, reconciling models with those data, as this paper clearly demonstrates, is a painstaking, forensic science, even for a relatively low-order model. For a higher-order model, or a very high order model (VHOM), the effort of accumulating knowledge in this manner (by engaging the HVHQ data with the VHOM, whose structure will not be a flawless representation of observed behavior) seems a massive undertaking, not least because an algorithm such as the RPE estimator generates so much diagnostic material. Better methods of scientific visualization will surely assist in processing this material (as suggested by Beck [2002, pp. 89–90]), but it seems unlikely that progress will be swift.

Acknowledgments

[57] This paper is a contribution arising from the Network on Environmental Applications (NEA) of the Dynamic Systems Program (DYN) of the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria. The authors want to thank three anonymous reviewers and the Editors for their invaluable comments and suggestions in making this paper more balanced and accessible to readers. M. B. Beck is currently Visiting Professor and Senior Research Associate in the Department of Civil and Environmental Engineering, Imperial College London.

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