# Shifting the focus of uncertainty analysis from parameters to functions

*Contributed by Shervan Gharari and Hoshin V. Gupta*

To construct a working process-based model of an environmental system, modelers make a great many decisions. The model is fundamentally, therefore, an assemblage of hypotheses regarding how the natural system works. Those hypotheses can be categorized into 4 hierarchical levels:

**Level 1**– *System Diagram and Conservation Law Hypotheses:* in which the boundaries of the system are defined, the ingoing and outgoing fluxes are specified, the fundamental system-level state/latent variables are identified (thereby implicitly characterizing which system-level processes are important), and relevant conservation laws are applied as overarching constraints (Figure-1a).

**Level 2**– *System Architecture Hypothesis: *in which decisions are made about the structure and organization of state-variable components within the system boundaries, thereby defining the requisite level of detail regarding internal components of the system, their interconnections (internal processes and fluxes), and the mappings between inputs, state-variables and outputs (Figure-1b).

**Level 3**– *Process Parameterization Hypothesis: *in which the functional forms of the processes by which physical gradients give rise to internal (state-state) and external (state-output) fluxes are mathematically defined and parameterized. For example, the equations that specify evaporation and transpiration are linked to available soil moisture, plant characteristics, and atmospheric variables (Figure-1c).

**Level 4-** *Parameter Specification Hypothesis: *in which the values of the parameters (in the above-mentioned process parameterization equations) are defined, to enable the model to be applied under specific conditions (e.g., to specific locations). This may involve additional hypotheses that relate the parameters to geometric, material, and other properties of the system (Figure-1d).

At every level, these hypotheses add regularizing information. This information is expressed via changes in uncertainty regarding any aspect of the system; for instance, regarding system state-variables (e.g. soil moisture) or fluxes (e.g., actual evapotranspiration, streamflow, etc.). If the provided information conforms with reality, the uncertain model simulation trajectories should tend to converge towards the observations. However due to the lack of knowledge, the decisions at each level are inherently uncertain, so that developing a working model that conforms closely with the evidence (observations) can be quite challenging. Knowing this, in hydrological modeling, while it is now quite common to talk about parameter uncertainty, this was not always the case, and it has taken years (even decades) for hydrologists to arrive at the stage where parameter uncertainty analysis is now a common practice. However, considerably less attention has been given to the challenge of acknowledging and accounting for model structural (hypothesis) uncertainty. Recently, efforts have been devoted to the development of flexible modeling approaches to address this need (Bulygina & Gupta 2009, 2010, 2011; Nearing and Gupta 2015; Gharari et al. 2021; Kirchner 2009; Koster & Mahanama 2012; Lamb & Beven 1997). However, only a handful of studies have, so far, investigated the uncertainties associated with the process parameterization hypotheses, the functional forms of which are often taken for granted (such as the Richards Equation).

In Gharari et al. (2021), we attempt to address the following questions:

**Question 1-** Can we characterize and quantify the information content introduced by the above-mentioned hypotheses at each level of model development?

To achieve a working model, decisions must be made at all 4 of the above-mentioned levels. To disentangle the conditional information content provided by each level, the constraints imposed by the hypotheses at each sub-levels must progressively be relaxed. For example, instead of treating parameters as fixed (deterministic) values, we can relax their specification by assigning parameter value ranges (or probability distributions). Similarly, to investigate the strength of information provided by structural decisions regarding the functional forms of process relationships, the specifications of the process parameterization equations must be relaxed. Drawing upon the information theoretic idea of formulating maximum entropy hypotheses, this relaxation can be achieved via the notion of “

minimally restrictive process parameterization equations—MR-PPEs”, which impose only physically required restrictions on the forms of the process equations (such as monotonicity and conservation) rather than imposing pre-specified deterministic forms that one cannot be sure are applicable at the given scale and in the given context (location, hydro-geo-climatology, etc.).The answer to this question is rather significant, as it helps to clarify which level of model development (i.e., which hypotheses) need to be further refined and investigated. Instead of directing the quest to a search for the “right” parameter values, the focus should be broadened to include the hypotheses regarding how to parameterize the processes at the given scale, and how to select the “correct” (i.e., more appropriate) system architecture (level and nature of system discretization) given the observation on hand.

**Question 2-** How do the modeling levels interact? Given the interplay between levels, and given the hierarchical and conditional nature of the model development hypotheses, can the same parameter value or process formulation be meaningfully used across various model architectures? How might one need to adapt a so-called “physical” (arguably “empirical” at some level) process formulation, such as the Richards Equations, so that the valuable information encoded in it becomes useful across a variety of different architectural configurations.

This issue is perhaps one of the most ignored topics in hydrological modeling. In practice, theoretically formulated process parameterizations, such as the Richards Equations, are commonly used in combination with any form selected for the model architecture, regardless of modeling scale, resolution, and hydro-geo-climatic context. Considerable effort is then devoted to numerical implementations of these equations. Instead, it may be more productive to note that the appropriate form of process parameterization can be strongly conditional on the model system architecture that has been selected, and can therefore vary significantly from one implementation to another. One way to investigate this issue is through the concept of MR-PPEs. And, given the power enabled by modern machine learning, it becomes possible that the forms of such MR-PPEs can be inferred from data, while remaining consistent with existing physics-based restrictions and specified model architectures, which in turn will be reflected in the confidence that we can have in functional forms implemented within our models.

A cultural shift is needed to prioritize the limits of our knowledge and information in data, rather than a quest for a superior modeling performance that is not logically justified. We suggest this is the beginning of an era in which, combined with increasing computational power, a shift from parameter uncertainty to functional uncertainty in hydrology is both needed and possible. Onward!