Parameter Adjustment Procedure

The parameter adjustment procedure is a directed trial-and-error process by which the parameters are iteratively adjusted to move the model behavior closer to the observed data. The choice of procedure is related to the measure of closeness selected (see above). If the calibration is performed by an expert hydrologist having a great deal of familiarity with the nuances of the model, the method of manual parameter adjustment guided by visual comparison can be extremely effective. However, manual calibration has several drawbacks. First, the procedure requires a great deal of subtlety in evaluating the visual goodness of fit, something that takes time and training to develop. Even to the trained eye, there may appear to exist numerous equally "good" parameter sets that are difficult to distinguish (Beven, 1992; Freer, 1996). Different good parameter sets will appear to match the data well in different ways, and moving from one set to another will trade-off an improvement in matching some parts of the data against deterioration in matching other parts of the data (Gupta, 1998). In practice, the calibration expert can also support the qualitative visual comparison with one or more quantitative scalar measures (e.g., see Table 1). However, this evaluation process still tends to be greatly complicated by the large number of model parameters to be adjusted and their tendency to have interacting and compensating effects on the output. Furthermore, the process can be very time intensive, particularly when the model contains numerous subcomponents and a large number of parameters (e.g., manual calibration of the SAC-SMA model can take several person-days of dedicated effort). These difficulties tend to limit widespread utility of the more complex and sophisticated models.

An alternative to manual parameter adjustment is to use the speed and power of a computer to automatically search the feasible parameter space for "better" solutions. In this approach, the measure of closeness is typically one of the scalar measures of closeness described earlier. A great deal of research has gone into the development of an automatic parameter adjustment procedure that gives satisfactory results while being reliable (effective) and efficient. A satisfactory result is one that gives model simulations similar to those obtained by an expert manual calibration, while resulting in parameter estimates that are conceptually realistic; a reliable procedure is one

TABLE 2 Summary of Five Major Characteristics Complicating the Optimization Problem in CRR Model Calibration

1. Regions of attraction

More than one main convergence region

2. Minor local optima

Many small "pits" in each region

3. Roughness

Rough response surface with discontinuous derivatives

4. Sensitivity

Poor and varying sensitivity of response surface in region

of optimum, and nonlinear parameter interaction

5. Shape

Nonconvex response surface with long curved ridges

that consistently provides satisfactory results; and an efficient procedure is one that requires only small amounts of computer time.

The earliest attempts at automatic calibration drew on a class of function optimization techniques called "local search" procedures; examples include the pattern search method (Hooke, 1961), the rotating directions method (Rosenbrock, 1960), the downhill simplex method (Nelder, 1965), and various versions of the Gauss-Newton quadratic approximation method (Luenberger, 1984). It quickly became apparent that such methods were highly unreliable; independent trials of the algorithm initiated from different initial parameter estimates would converge to widely differing solutions. A study by Duan (1993) demonstrated conclusively the reasons for this poor performance; the response surface of the scalar measure being optimized typically has several characteristic properties (see Table 2) that local search

Figure 3 Function response surface showing multiple regions of attraction.

Figure 3 Function response surface showing multiple regions of attraction.

algorithms are not able to handle well. The most important of these are the existence of more than one primary region of attraction (see Fig. 3) as well as large numbers of local optima throughout the feasible space (see Fig. 4). The focus therefore shifted to trying the existing "global search" methods including adaptive random search (Brazil, 1987), the genetic algorithm (Wang, 1991; Tanakamaru, 1995), and the multistart simplex (Duan, 1992; Gan, 1996). The most successful method to date has been the shuffled complex evolution (SCE-UA) method recently developed at the University of Arizona (Duan, 1992, 1994; Sorooshian, 1993), which has proved to be both reliable and relatively efficient (see Fig. 5).

It should be noted that "manual" and "automatic" parameter adjustment approaches have mutually complementary strengths and weaknesses, which suggests the implementation of a hybrid approach that draws on the strengths of each (while minimizing their weaknesses). The strength of the manual approach is its ability, when successful, to provide very satisfying model calibrations because visual comparisons draw on the human ability to perceive patterns that are not easy to detect using numerical techniques. The strength of the automatic approach is that it can very quickly and rapidly find the region(s) of the parameter space that give relatively close matching of the simulated flows and observed data, while manipulating large (even bewildering) numbers of mutually compensating and interacting parameters. The hybrid procedure therefore involves two steps. In the first step, the automatic procedure is used to quickly find several solutions that seem to have similar ability to match the data when measured using one or more of the scalar

Figure 4 Locations of local optima in three-dimensional parameter subspace; each dot represents optima for a local region.
Figure 5 Convergence of model parameter for 10 different trials of the SCE-UA method; dotted line represents the true paramenter value.

numerical measures of closeness described earlier. These then become the starting point for a manual procedure of refinement in which the expertise of the hydrologist can be used to further improve the solution. The computer does what it does better than a human, which is to search through large numbers of options very quickly and reject the unacceptable ones. The human does what humans do better than a computer, which is to use perceptual discrimination to make qualitative distinctions that are difficult to describe mathematically.

Developments of this hybrid approach through the use of multi-objective procedures can be found in the work of Gupta (1998) and Yapo (1998). The multiobjective global optimization strategy, MOCOM (multi-objective complex evolution), is used to identify the set of solutions that provide a "trade-off" in simultaneously minimizing several criteria that measure the goodness of fit of the model to the calibra-

Figure 6 Identification of trade-off solutions using multiobjective optimization strategy.

tion data (Fig. 6). The hydrologist can then use visual means to identify the most perceptually appealing solution(s).

Evaluation Procedure

Once a model has been calibrated by one of the methods outlined above, it is useful to evaluate the result by testing its performance using data not employed for model calibration. For flood forecast models, one might (if possible) select a period of data of comparable length to the calibration period containing several significant storm events. Visual comparison of the observed and simulated outputs for this evaluation period and a check of the goodness-of-fit statistics can reveal any obvious divergence of the model performance from reality. This can also give a reasonable estimate of the approximate forecasting accuracy that can be expected when the model is used for real-time forecasting. Particular attention should be given to any tendencies for the model simulations to be biased at different streamflow levels. Admittedly, the process of model evaluation is somewhat subjective; however, if the simulation performance over the evaluation period is essentially similar to that over the calibration period, the model can then be used for flood forecasting with some understanding of its expected level of performance.

4 FORECASTING AND STATE UPDATING

The calibrated model can be implemented for real-time flood forecasting. The main issue here is that of the "lead time" (duration between time of making the forecast and time of actual occurrence). Clearly, the benefit of a flood forecast lies both in its accuracy and in its being available as early as possible before the actual event occurs. With this in mind, the model time step At must necessarily be shorter than the time of concentration of the watershed so that the precipitation data available up to time t are used to compute the model-simulated streamflow at time t + At; this is called a one-step-ahead forecast. For small watersheds, this time step At may be on the order of only a few hours, while for larger watersheds the time step may be one day or more. To maximize the forecast lead time, it is desirable that the precipitation measurement be either phoned or radioed in to the forecast center within minutes of its occurrence, or even telemetered in by automatic recording gauges and processed immediately through the model. If a forecast with longer lead time is required, it becomes necessary to obtain independently generated precipitation forecasts to feed to the model in place of precipitation measurements. The U.S. National Weather Service uses a "quantitative precipitation forecast" system to enable several time-step-ahead forecasts to be made for many watersheds (Funk, 1991).

A second issue is that of model state updating (also called data assimilation or filtering). Because the accuracy of each forecast can be evaluated as soon as the observed flow for that time step becomes available, this information should, in principle, be useful for adjusting the internal model states to maximize the accuracy of the next forecast. For example, underprediction of the observed flow may indicate that the model storages that represent the wetness of the various watershed compo-

nents are too dry and should be adjusted accordingly. Kitanidis (1980a, 1980b) rewrote the SAC-SMA model in a state-space form and implemented an extended Kalman filter to enable the model to correct for data errors. Because of the mathematical complexity of reworking a model into a state-space form, state updating has not become widely popular for use with flood forecast models. As the use of the simpler "hybrid" models becomes popular, we can expect to see more exploitation of systems-theoretic methods such as state updating to improve the performance of watershed models.

Finally, it is important to consider the forecasting uncertainty associated with the uncertainty in the model structure and parameter estimates. Some interesting (and somewhat similar) Monte Carlo approaches for representing forecast uncertainty include the generalized likelihood uncertainty estimation (GLUE) method [see, e.g., Beven (1992) and Freer (1996)], the Monte Carlo set membership (MCSM) procedure [see, e.g., Keesman (1990) and van Straten (1991)], and the prediction uncertainty (PU) method [see, e.g., Klepper (1991)]. For example, the GLUE procedure estimates the range of forecast uncertainty by estimating the likelihood associated with the individual forecasts given by different "equifinal" parameter sets in the feasible space.

5 EMERGING DIRECTIONS

It is the thesis of this chapter that the trend in hydrologie modeling for runoff forecasting will be toward a successful marriage of hydrologie science and systems theory, implemented through the coupling of hybrid watershed models with automated procedures for calibration and data assimilation for state updating. We can expect to see clear and rapid progress in all three of these components. Experiments with data from numerous watersheds will help in establishing general guidelines about the level of conceptual detail required to model the dominant watershed responses that are observable in the input-output data. The development of multi-objective calibration procedures (Gupta, 1998; Yapo, 1998) has already begun to merge the strengths of the manual and automated calibration procedures into an effective hybrid calibration method. The simplicity of the hybrid model structures will enable approximate Kalman filtering methods (or other uncertainty estimation methods) to be implemented for improving online forecasts. In addition, radar-based precipitation estimates are already replacing gage-based data and will encourage the development of "distributed" structures but parsimoniously parameterized hybrid watershed models. Finally, because the hybrid modeling approach provides us with a simple functional representation of the watershed, we can also expect progress in understanding how to apply watershed models to ungaged basins.

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