The mathematical models that are included in the program describe how a watershed responds to precipitation falling on it or to upstream water flowing into it. While the equations and the solution procedures vary, all the models have the same components in common.

State Variables

These terms in the model's equations represent the state of the hydrologic system at a particular time and location. For example, the deficit and constant-rate loss model that is described in Chapter 5 tracks the mean volume of water in natural storage in the watershed. This volume is represented by a state variable in the deficit and constant-rate loss model's equations. Likewise, in the detention model of Chapter 10, the pond storage at any time is a state variable; the variable describes the state of the engineered storage system.

Parameters

These are numerical measures of the properties of the real-world system. They control the relationship of the system input to system output. An example of this is the curve number that is a constituent of the SCS curve number runoff model described in Chapter 5. This parameter, a single number specified when using the model, represents complex properties of the real-world soil system. If the number increases, the computed runoff volume will increase. If the number decreases, the runoff volume will decrease.
Parameters can be considered the "tuning knobs" of a model. The parameter values are adjusted so that the model accurately predicts the physical system response. For example, the Snyder unit hydrograph model has two parameters, the basin lag, tp, and peaking coefficient, Cp. The values of these parameters can be adjusted to "fit" the model to a particular physical system. Adjusting the values is referred to as calibration. Calibration is discussed in Chapter 9.
Parameters may have obvious physical significance, or they may be purely empirical. For example, the Muskingum-Cunge channel model includes the channel slope, a physically significant, measurable parameter. On the other hand, the Snyder unit hydrograph model has a peaking coefficient, Cp. This parameter has no direct relationship to any physical property; it can only be estimated by calibration.

Boundary Conditions

These are the values of the system input—the forces that act on the hydrologic system and cause it to change. The most common boundary condition in the program is precipitation; applying this boundary condition causes runoff from a watershed. Another example is the upstream (inflow) flow hydrograph to a channel reach; this is the boundary condition for a routing model.

Initial Conditions

All models included in the program are unsteady-flow models; that is, they describe changes in flow over time. They do so by solving, in some form, differential equations that describe a component of the hydrologic system. Solving differential equations that involve time always requires knowledge about the state of the system at the beginning of the simulation.
The solution of any differential equation is a report of how much the output changes with respect to changes in the input, the parameters, and other critical variables in the modeled process. For example, the solution of the routing equations will tell us the value of ∆Q/∆t, the rate of change of flow with respect to time. But in using the models for planning, designing, operating, responding, or regulating, the flow values at various times are needed, not just the rate of change. Given an initial value of flow, Q at some time t, in addition to the rate of change, then the required values are computed using the following equation in a recursive fashion:

Q_t=Q_{t-\Delta t} + (\frac{\Delta Q}{\Delta t})


In this equation, Qt-∆t is the initial condition; the known value with which the computations start.
The initial conditions must be specified to use any of the models included in the program. With the volume-computation models, the initial conditions represent the initial state of soil moisture in the watershed. With the runoff models, the initial conditions represent the runoff at the start of the storm being analyzed. With the routing models, initial conditions represent the flows in the channel at the start of the storm. Moreover, with the models of detention storage, the initial condition is the state of storage at the beginning of the runoff event.