Equifinality and Flux Mapping: A New Approach to Model Evaluation and Process Representation Under Uncertainty

Abstract

Uncertainty analysis is an integral part of any scientific modeling, particularly within the domain of hydrological sciences given the various types and sources of uncertainty. At the center of uncertainty rests the concept of equifinality, that is, reaching a given endpoint (finality) through different pathways. The operational definition of equifinality in hydrological modeling is that various model structures and/or parameter sets (i.e., equal pathways) are equally capable of reproducing a similar (not necessarily identical) hydrological outcome (i.e., finality). Here we argue that there is more to model equifinality than model structures/parameters, that is, other model components can give rise to model equifinality and/or could be used to explore equifinality within model space. We identified six facets of model equifinality, namely, model structure, parameters, performance metrics, initial and boundary conditions, inputs, and internal fluxes. Focusing on model internal fluxes, we developed a methodology called flux mapping that has fundamental implications in understanding and evaluating model process representation within the paradigm of multiple working hypotheses. To illustrate this, we examine the equifinality of runoff fluxes of a conceptual rainfall‐runoff model for a number of different Australian catchments. We demonstrate how flux maps can give new insights into the model behavior that cannot be captured by conventional model evaluation methods. We discuss the advantages of flux space, as a subspace of the model space not usually examined, over parameter space. We further discuss the utility of flux mapping in hypothesis generation and testing, extendable to any field of scientific modeling of open complex systems under uncertainty.

Challenges in modeling and predicting floods and droughts: A review

“…Suitable or “best” model structures can be determined using process‐based model evaluation (Clark, Fan, et al, 2015) or within a model‐intercomparison framework, such as the modular assessment of rainfall‐runoff models toolbox MARRMot (Knoben, Freer, Fowler, Peel, & Woods, 2019), the Framework for Understanding Structural Errors (Clark et al, 2008), or the hydrological modeling frameworks SUPERFLEX (Fenicia, Kavetski, & Savenije, 2011), SUMMA (Clark, Nijssen, et al, 2015), and Raven (Craig et al, 2020). Insights into model behavior and process representation can also be gained using flux maps by inspecting different flow components (e.g., baseflow, infiltration excess, and interflow; Khatami, Peel, Peterson, & Western, 2019). Coupled non‐stationary landscape‐flood‐inundation models : The non‐stationarity of the physical landscape is currently less well understood than non‐stationarity of the water cycle (e.g., Slater, Khouakhi, & Wilby, 2019), even though it may be equally important for those seeking to obtain accurate projections of future extremes.…”

Toward Data‐Driven Generation and Evaluation of Model Structure for Integrated Representations of Human Behavior in Water Resources Systems

“…However, we find that for this problem, alternate metrics do not significantly change the rank order of models within each class (see supporting information). In regard to improving regression metrics, the water resources field has thoroughly considered how error metrics for natural process models can incorporate available process knowledge (e.g., Gupta et al., 2009; Khatami et al., 2019; Lamontagne et al., 2020). These approaches are also relevant in scenarios lacking process knowledge but with known statistical relationships in the error signals.…”

“…For data‐driven system identification this is especially challenging given the large space of possible model structures and conflicting performance metrics (Curry & Dagli, 2014). The concept of equifinality has been widely explored in hydrology and water resources (Khatami et al., 2019), as well as in agent‐based models (Williams et al., 2020). However, with the exception of a recent example from the social sciences (Vu et al., 2019), the equifinality problem is rarely approached in integrated studies by global search over model structures that considers both performance and complexity during training.…”

“…First, data‐driven system identification typically results in a tradeoff between model performance and complexity (Bastidas et al., 2006; Hogue et al., 2006; Pande et al., 2009). Second, additional criteria may be required for model evaluation, such as interpretability and agreement with available theory (Khatami et al., 2019; Knüsel et al., 2019). Opportunities remain for data‐driven methods to identify model structures of water resources system components for which theory is still being developed, such as varied human influences.…”

Historical development of rainfall‐runoff modeling

“…There are many other issues relating to rainfall‐runoff modeling that could be discussed within the framework of a primer on RRM, but space precludes their inclusion. For example (a) calibration and evaluation (Bathurst, Ewen, Parkin, O'Connell, & Cooper, 2004; Duan et al, 2006; Ewen & Parkin, 1996; Fowler et al, 2018; Fowler, Peel, Western, Zhang, & Peterson, 2016; Gupta, Kling, Yilmaz, & Martinez, 2009; Klemeš, 1986a, 1986b; Parkin et al, 1996; Saft, Peel, Western, Perraud, & Zhang, 2016; Vaze et al, 2010), (b) equifinality (Beven, 2006; Beven & Freer, 2001; Khatami, Peel, Peterson, & Western, 2019; Savenije, 2001), (c) uncertainty (Beven, 2019a; Kavetski, Kuczera, & Franks, 2006a, 2006b; Nearing et al, 2016; Nearing & Gupta, 2015); (d) consistent modeling across multiple time steps (Ficchi, Perrin, & Andréassian, 2019); (e) modeling framework, methodology and philosophy (Clark et al, 2008, 2011, 2015; Crooks, Kay, Davies, & Bell, 2014; Fenicia et al, 2011; Hrachowitz & Clark, 2017); (f) plausibility and influence of internal fluxes (Ficchi et al, 2019; Guo, Westra, & Maier, 2017; Khatami et al, 2019); and (g) models of everywhere (Beven, 2007; Beven, 2019b; Blair et al, 2019; Wood et al, 2011). The reference list in this primer would be incomplete if reference was not made to “Rainfall‐Runoff Modelling The Primer” in which Beven (2012) deals with the evolution of rainfall‐runoff modeling including the above topics and more.…”