## 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].