Hydrological And Shallow Flow Modeling: Solutions Converge

Hydrological modeling is at the core of many environmental, agricultural, and engineering applications, research, and practices. Decision-making on managing our water resources has become increasingly model-based and with this, we need to be able to assess errors and uncertainties using hydrological models. These models have been developed to understand and predict how precipitation in a catchment is partitioned into different conceptual hydrological components (surface, subsurface, water storage in canopies, ponds, etc).

Early efforts of model development focussed on physical-based descriptions of interacting processes ranging from evaporation, surface water flow, and porous media flow, solving partial differential equations that conserve mass, energy, and momentum. In recent decades, researchers have advanced mathematical and numerical methods which are capable of predicting the behavior of many such processes based on parameters which attempt to characterize in simple ways the complexity of hydrological catchments in a spatially-distributed fashion, so that the hydrological and hydrodynamic states can be predicted everywhere in the catchment. However, the interactions of hydrological and biological processes have increased the complexity of hydrological predictions, as feedback mechanisms between water, soil, and vegetation are difficult to identify and to quantitatively describe.

Exploring such complex interactions in space and time has led to the development of simple approaches in an attempt to capture the essence of complex interactions neglecting details of processes assumed to be of little significance. This simple approach is based on describing interactions between discrete cells in space. Rules how the state of a cell is affected by its neighbours are assigned based either on heuristic knowledge of small scale interactions or rules of nature. This approach dates back to the 1940s, when computational power was low or not available, a simple mathematical approaches were developed and termed Cellular Automata (CA).

More recently this methods have received a lot of attention as it allows to generate patterns observed in nature and to simulate self-organized system beahviour using simple rules at the cell scale. More recently, simple cellular automata have been proposed to describe a deterministic physical system such as surface runoff, erosion, and landscape evolution. In this particular case, rules of how cells are interacting and communicating are based on flux laws derived from mass, energy or momentum conservation defined for discrete time steps. In a way, it represents a “bottom up” approach in terms of spatial and temporal scales to modelling larger scale spatial and temporal patterns. In contrast, “conventional” hydrological modelling is based on an continuum approach, which uses numerical methods to discretisize the spatial land temporal domain of interests representing a kind “top down” approach.

A recent review paper (Caviedes-Voullième et al., 2018) shows that both approaches converge to the same set of equations being solved. Many applications have been proposed since for this type of computational solver. One such application has been in Computational Hydrology. In this context, the space defined by a catchment can be discretised into cells of a chosen resolution. Such cells simply represent small parts of the catchment, on which hydrologically-relevant properties, such as terrain elevation, vegetation cover, soil type and so on are represented.

In the last 30 years, a wide range of CA models in the context of hydrology, particularly in surface flow modeling, have been proposed, tested and applied for research. The base component of these CA models as applied to surface flow (or hydrological modelling in general) is the rule on which water is allowed to move from one cell to the next. This water volume exchange rule is what differentiates the different approaches. Some CA models simply assume that water follows the steepest path (regardless of anything else). Others, arguably more physical in their formulation, enforce that water flows following the slope of the water surface and that the rate of flow is somwhow proportional to such slope. In such way, it is possible to write a computational rule which allows to compute the change in water volume in a cell and how it transfers to or receives water from its neighbors.

Different variations of these ideas, as well as other simple rules, have been proposed and tested, each with their own strengths and drawbacks and different levels of accuracy in predicting surface water flow. From a different perspective, surface water flow has been described mathematically by the so-called Saint-Venant equations (first derived in the 19th century), nowadays called the shallow-water equations. This systems of equations enforces basic phsyical principles such as conservation of mass and momentum to describe how water flows over a surface.

Solving this set of equations to predict the movement of the water surface in time is mathematically complex and computationally intensive, although possible. Historically though, simplified forms of the shallow-water equations have been used in practice. One such simplification is the Zero-Inertia (ZI) approximation (also often termed diffusive-wave approximation). This simplification implies that inertia effects can be assumed to be neglected, and that water basically moves following the slope of the water surface, and is hindered by friction effects. To solve this equation, traditional numerical approximations are often used. The Finite Volumes (FV) method -one of the methods of choice for solving Fluid Dynamics problems- basically divides space into a number of cells, and solves the equation at hand between each pair of cells by computeing a ”flux”, that is, the water volume exchange. Interestingly, given appropriate and meaninful choices, the CA computational rule can be shown to be identical to the FV approximation of the zero-inertia equation.

The relevance of this insight has multiple facets. Firstly, it evidences the methodological convergence from researchers in different fields and with different approaches. Second, it shows that the underlying mathematical structure of many of the available models is the same, but different assumptions are made which allow to formulate different fluxes. Moreovore, be acknowledging the identity between the methods, it becomes clear that resarchers, modellers and practitioners can benefit from the knowledge, insight and experience from both the application-oriented CA community and the numerical community solving the ZI equation with formal solvers for partial-differential equations.

These insights are wide and span everything from formal mathematical properties of the system such as the stability of its numerical solution, to techniques which allow to keep te solution accurate and efficient, and to practical implementation and usability issues. It is a consequence of this that many of the existing models, may they be called CA or FV-ZI can be groped into the same category, with similar properties. Many CA models which do not have the same assumptions and indeed a different formulation must be critically assessed ad-hoc in terms of their properties and applicability, as they do not follow the same formal reasoning.

These findings are described in the article entitled Cellular Automata and Finite Volume solvers converge for 2D shallow flow modelling for hydrological modelling, recently published in the Journal of Hydrology. This work was conducted by Daniel Caviedes-Voullième and Christoph Hinz from the Brandenburg University of Technology, and Javier Fernández-Pato from CSIC-Universidad Zaragoza.

Reference:

1. Caviedes-Voullième, D.; Fernández-Pato, J. & Hinz, C. Cellular Automata and Finite Volume solvers converge for 2D shallow flow modelling for hydrological modelling Journal of Hydrology, 2018, 563, 411-417