Sensitivity to scale and resolution in process models can be viewed slightly differently from the perspective of lumped parameter models and distributed parameter models. In the latter, the problem focuses on appropriate size of the cells within the grid or triangular mesh. In the former where spatial discretization is often implicit, there is nevertheless some spatial extent that is being modeled, which, for example, in hydrology will be catchment units and yet an area is almost infinitely divisible into catchment units.
The question of catchment size and grid size and their effect on parameterization and uncertainty in simulation results has been a focus of attention in hydrology for some time, particularly to separate out which environmental controls are scale dependent and which are not (for two reviews, see Wood et al., 1990 and Clifford, 2002). This is not to play down the importance of other aspects of spatial variability, such as in rainfall (e.g., Arnaud et al., 2002), but it is now well established that there is a tendency for topographic variability to dominate predicted spatial patterns of storm runoff, particularly for small catchments.
In progressively reducing the size of catchment area by discretizing to smaller subcatchments, the resulting increase in resolution should lead to a reduction in the variance in subcatchment response. This led Wood et al. (1988) to propose the existence of a threshold resolution or representative elementary area (REA) as the fundamental building block for catchment modeling. The REA represents the critical resolution below which it is necessary to account for internal heterogeneity. This was found in their studies to be about 1 km2. For spatial units larger than the REA, only the statistical representations of control variables (e.g., mean values) need to be known. This has implications for distributed parameter models where generally the discretization is less than 1 km2 and thus will require a fuller specification of the variables.
Zhang and Montgomery (1994) studied the use of a high-resolution digital elevation model (DEM) in two small catchments in order to study the effects of changing grid size on parameter estimation and simulation of hydrographs on very small catchments (1.5 km2). They found that the grid size of the DEM significantly affected both. They concluded that a 10 m grid size represented “a reasonable compromise between increasing spatial resolution and data handling requirements” for topographically driven models. Although, in large drainage basins, the hydrograph will be dominated by channel routing, the influence of DEM grid size on the production of predicted runoff should be an important consideration in interpreting simulations. On the other hand, small grid sizes will undoubtedly raise problems concerning the accuracy of the DEM at that resolution with resultant errors in parameter determination propagating to the simulation.
Bruneau et al. (1995) have also carried out a sensitivity analysis of grid size and time step on runoff simulation for a 12 km2 catchment. They found that choices of grid size and time step were not independent and that there is an optimum region of values for model building. In their study, this was a grid size of less than 50 m and a time step of 1 to 2 h. Significantly, larger grid sizes with medium time steps were found to result in some parameter values to be meaningless, thus giving inconsistent runoff simulations. They also found that degraded outputs were more sensitive to larger time steps than to larger grid sizes. Molná and Julien (2002) have tested a distributed parameter model (CASC2D) on two basins of 21 km2 and 560 km2 using grid sizes from 127 to 914 m. This becomes a case of needing less resolution to model larger features, particularly where computation time becomes intractable with smaller grid sizes. For the smaller basin, grid sizes up to 380 m were acceptable provided calibration was carried out to upwardly adjust overland and channel roughness coefficients. However, these grid sizes are approaching the REA (above).
Grid size for the larger basin was found to be critical for shorter rainfall events where equilibrium conditions are unlikely to be met. This takes us straight back to Chapter 4, Figure 4.9: we can’t disentangle space–time and, by the same token, we can’t treat them in a disjointed way in our process models. Modeling large, rapid events is likely to require a different type of model. While the above findings to do with scale have been explored in the context of surface hydrology, the same broad relationships will apply equally to the modeling of other processes (Li, 2007). In Chapters 5 and 6, we looked at coastal oil spill modeling, first at the hydrodynamic modeling that distributed the tidal currents across the study area and then at the trajectory of the floating oil. One constituent of the trajectory model was the spreading action, mostly a random component, but dominated by the tidal and wind-blown currents. Modeled over 3 h on a 200 m grid, an oil spill drifted toward the coast to eventually make landfall on a beach (Figure 9.1(a/f)). Figure 9.1(g/l) is a further simulation using a 400-m grid.
There are noticeable differences, but at the same time broad similarity—the oil still ends up on the same beach. The density values of cells are certainly different. The model has lost its resolving power for the smaller currents; if some oil just moves across a grid boundary, then it moves the full 400 m of the grid. At the same time, it is important not to go for a too fine of a grid. The interpolation of currents from the larger finite element network to the smaller trajectory modeling grid leads to a reduction in the size of the current component at each grid node because of the shorter step in the calculations. If the grid size gets too small, then the random component in the spreading starts to dominate over tide and wind to produce an unacceptable result. The model becomes unstable. Hence, finer resolution is not necessarily more accurate.