Dynamical grouping and representative computation: a new approach to reduce computational efforts in distributed, physically based modeling on the lower mesoscale

Abstract:

Lower mesoscale catchments as systems of organized complexity and related modeling approaches

Though hydrological modelling is a story of great and ongoing success, we still struggle to provide predictions for systems that exhibit organized complexity at the intermediate scale of a few to a few hundred km². Such catchments can often be characterized as heterogeneous systems that display "some degree" of organization (Dooge 1986), where hydrological response is strongly controlled by multivariate statistical and topological characteristics of key patterns in the landscape - topography, vegetation, soil properties, preferential pathways (Schulz et al. 2006, Uhlenbrook 2006, Zehe and Sivapalan 2009) - which dynamically interact with the space time patterns of boundary conditions as for instance precipitation. Both major hydrological modeling approaches, conceptual and physically based, have limitations for application to such systems: Conceptual models such as Topmodel (Beven and Kirkby 1979) or HBV (Lindström et al. 1997) implicitly conceptualize the process patterns and redistribution processes of water inside the catchment and the underlying controls by means of effective states, effective parameters and effective fluxes, which makes them subject to considerable equifinality and predictive uncertainty (Uhlenbrook et al. 1999, Neumann et al. 2010). This is in part due to the fact that available observations, which could be used to constrain model parameters, often cannot be straightforwardly related to them (Binley et al. 1989, Güntner et al. 1999, Hundecha and Bárdossy 2004, Merz and Blöschl 2005, Kirchner 2006).

Reductionist, physically based modeling approaches such as MIKE SHE (Refsgaard and Storm 1995, Christiansen et al. 2004), HYDRUS (Šimunek et al. 1999, Šimunek et al. 2005, Gärdenäs et al. 2006) or CATFLOW (Zehe et al. 2001) on the other hand can be parameterized and validated by distributed observations, and hence are much more likely to provide 'the right answers' (catchment response) for 'the right reasons' (distributed internal dynamics). Their major drawbacks for application on the lower mesoscale however are i) their large demand of high-resolution data for model setup and operation, and ii) numerical issues and CPU demand that rapidly grow with catchment size.

There are several approaches to address the challenge of hydrological modeling on the lower mesoscale: The hillslope storage Boussinesq model (HSB) proposed by Troch et al. (2004) which is tailored for hilly landscapes with shallow, permeable, weakly heterogeneous soils. The REW approach (Reggiani et al. 1998, Reggiani and Rientjes 2005) which is based on a joint, zero dimensional and thermodynamically consistent treatment of the momentum, energy and mass balance within representative elementary watersheds as least model entities. The mesoscale Hydrologic Model mHM (Samaniego et al., 2010) relies largely on standard conceptual process representations, except the soil moisture accounting which is based on the Brooks and Corey formulation. The main asset of the mHM is a multiscale approach for parameter regionalization, which assures conservation of the simulated fluxes when stepping across scales. This approach has been successfully applied to a broad range of catchment sizes at spatial resolutions between 1 km and 100 km (Samaniego et al., 2010) and has proven to yield spatially transferable parameter sets. Another high-resolution mesoscale model approach, ROGeR, was recently proposed by Steinbrich and Weiler (2011). It makes explicit use of knowledge about local runoff formation processes and has been successfully applied in several mesoscale watersheds with different geology, soils, topography and land-use.

Reducing CPU demands by exploiting spatio-temporal similarity of states and dynamics

In this research, we propose an alternative approach to maintain distributed and physically based modeling up to the lower mesoscale by exploiting spatio-temporal similarity of states and dynamics in the landscape and related model elements. It is built on the fundamental hypothesis that the landscape is composed of many similarly structured entities (in this text we refer to them as Elementary Functional Units or EFUs) which, due to long-term co-evolution, belong to a few, typical landscape EFU classes. EFUs of the same class in a similar state and exposed to similar forcing should produce similar integral responses based on similar internal dynamics. If a model is also built from such EFUs, then such sets of (temporarily) functionally similar model EFUs could be grouped to Dynamical Functional Units (DFUs), and their dynamics and responses could with sufficient precision be described by computing the dynamics of just one or a few group representatives and by assigning the results back to all group members.

We will present results of this approach from studies in two catchments: The 3.5 km² Weiherbach catchment in South-West Germany and selected watersheds in the 288 km² Attert catchment in Luxembourg.

For the Weiherbach, forcing data, detailed knowledge on land use, soil hydraulic properties and a distributed physically based model were available from previous studies (Zehe et al. 2001). The CATFLOW model discretization consists of 169 hill slopes (see Fig. 1, panel a), which are all similar with respect to soils, catena, macropore distribution and land use. We could hence assume structural similarity among the hillslopes, except for their size. From the analysis of a 5 year rainfall time series, a relation between rainfall duration, depth and maximum intensity for relevant events (those producing runoff) could be established, i.e. the single parameter maximum rainfall intensity of an event Imax was sufficient to characterize the forcing. For the same period, we analyzed soil moisture observed at different locations along the catena and at different depths and found that the mean soil moisture of the top 30 cm (ISM) was a robust indicator of the slopes initial states prior to a rainfall event. Hillslope response to a rainfall event was characterized by the event runoff coefficient Cr. As can be seen from Fig. 1 panel b, the hillslope response could be characterized by 3 distinct 'dynamic modes', namely 'no runoff' for small rainfall events, 'infiltration excess' for low initial soil moisture and high rainfall, 'saturation excess' for high initial soil moisture and high rainfall. We used these simple descriptors of structure, state, forcing and function as basis for our grouping rules: All slopes of similar structure, initial state and forcing were assigned to one DFU-group. As in the Weiherbach the hillslopes are structurally similar and as rainfall could be assumed homogeneous throughout the catchment due to its small size, a reasonable start was to assign all 169 slopes into one group. Then we selected, one after another, a single hillslope from the group, computed its dynamics and runoff production in full resolution, and assigned the results, scaled only by hillslope size, back to the remaining 168 slopes. For two large rainfall-runoff events, the resulting simulations are shown in Fig. 1 panels c and d. For the 150 mm event (panel c), we identified the 15 best performing representatives (restriction to 15 for better visibility in the plot). In comparison to the fully distributed simulation, the very simple one-group representative computation showed good agreement. However, these 15 best-performing representatives did not perform quite as good when applied to a different event (panel d). This suggests that more discriminative grouping rules and descriptors of structure, state and function are needed.

For the Attert basin, a newly developed hydrological model ('CAOS' = 'Catchments As Organized Systems') was set up for two hillslopes in the northern, forested schist areas of the basin. It is physically based and explicitly treats vertical and lateral preferential flow, but with the restriction that fluxes are only considered 1-dimensional (either vertical or lateral). The model consists of hierarchical objects closely resembling real physical landscape elements, with the catchment object on top, followed by hillslope and riparian zone objects which are further divided into elementary functional Units (EFUs), which are assumed to be laterally homogeneous (ca. 1.000 m²), but further subdivided in layers of a few tens of centimeters. The preferential flow paths macropores, rills, subsurface hillslope lateral flow paths and the river network are explicitly represented. For both sites, excellent data from geophysical exploration, interdisciplinary sprinkling experiments, ecological exploration, several sensor clusters and soil hydraulic tests are available and have been used for model setup. Based on these realistic and behavioral hillslope models, we will present results from representative computation tests across a wide range of model states and forcing. With this we can evaluate the feasibility of grouping and representative computation among structurally identical hillslopes which are in different states and are exposed to different forcing. Once the CAOS model is set up for the entire Attert basin, we can additionally explore the 'groupability' of hillslopes or EFUs with similar, but non-identical structure. This is planned for future studies.

Figure 1:

a) Left panel: Spatial discretization of the CATFLOW model in the Weiherbach catchment by 169 hillslopes.

b) Runoff coefficient Cr for varying initial conditions: initial soil moisture (ISM) and maximum intensity of rainfall forcing Imax.

c) and d) Comparison of the fully distributed simulation (bold black line) with the 15 best representative simulations for 2 storm events in the Weiherbach. Panel c: Imax=150mm/h; panel d) Imax=80mm/h. Note that the selection of the best 15 representatives was done for the 150 mm/h event, and held constant for the 80 mm/h event

Altogether, the concepts of dynamical grouping and representative computation have the potential to avoid many redundant computations, while preserving state and process resolution. Apart from CPU considerations, looking at model dynamics in terms of similarities yields new and exciting insights in the dynamical (self-) organization of states and dynamics of the landscape.

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