The mathematical problem of finding the best parameters for a selected model (or models) is what systems engineers refer to as a constrained optimization problem. That is, the range of feasible, acceptable parameters (which systems engineers would call the decision variables) is limited. For example, a Muskingum x parameter that is less than 0.0 or greater than 0.5 is unacceptable, no matter how good the resulting fit might be. Thus, searching outside that range is not necessary, and any value found outside that range is not be accepted. These limits on x, and others listed in Table 27, are incorporated in the search.
During the search with either the univariant gradient or Nelder and Mead algorithm, the program checks at each iteration to ascertain that the trial values of the parameters are within the feasible range. If they are not, the program increases the trial value to the minimum or decreases it to the maximum before it continues.
In addition to these inviolable constraints, the program will also consider user-specified soft constraints. These constraints define desired limits on the parameters. For example, the default range of feasible values of constant loss rate is 0-300 mm/hr. However, for a watershed with dense clay soils, the rate is likely to be less than 15 mm/hr—a much greater value would be suspect. A desired range, 0-15 mm/hr, could be specified as a soft constraint. Then if the search yields a candidate parameter outside the soft constraint range, the objective function is multiplied by a penalty factor. This penalty factor is defined as:
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Penalty$=2 \prod_{i=1}^{n}\left(x_{i}-c_{i} \mid+1\right)$ |
in which xi = estimate of parameter i; ci = maximum or minimum value for parameter i; and n = number of parameters. This "persuades" the search algorithm to select parameters that are nearer the soft-constraint range. For example, if the search for uniform loss rate leads to a value of 300 mm/hr when a 15 mm/hr soft constraint was specified, the objective function value would be multiplied by 2(300-15+1) = 572. Even if the fit was otherwise quite good, this penalty will cause either of the search algorithms to move away from this value and towards one that is nearer 15 mm/hr.
Table 29.Calibration parameter constraints.
Model | Parameter | Minimum | Maximum |
Initial and constant-rate loss | Initial loss | 0 mm | 500 mm |
| Constant loss rate | 0 mm/hr | 300 mm/hr |
SCS loss | Initial abstraction | 0 mm | 500 mm |
| Curve number | 1 | 100 |
Green and Ampt loss | Moisture deficit | 0 | 1 |
| Hydraulic conductivity | 0 mm/mm | 250 mm/mm |
| Wetting front suction | 0 mm | 1000 mm |
Deficit and constant-rate loss | Initial deficit | 0 mm | 500 mm |
| Maximum deficit | 0 mm | 500 mm |
| Deficit recovery factor | 0.1 | 5 |
Clark's UH | Time of concentration | 0.1 hr | 500 hr |
| Storage coefficient | 0 hr | 150 hr |
Snyder's UH | Lag | 0.1 hr | 500 hr |
| Cp | 0.1 | 1.0 |
Kinematic wave | Lag | 0.1 min | 30000 min |
Baseflow | Manning's n | 0 | 1 |
| Initial baseflow | 0 m3/s | 100000 m3/s |
| Recession factor | 0.000011 | - |
Muskingum routing | K | 0.1 hr | 150 hr |
| X | 0 | 0.5 |
| Number of steps | 1 | 100 |
Kinematic wave routing | N-value factor | 0.01 | 10 |
Lag routing | Lag | 0 min | 30000 min |