The unit hydrograph is a well-known, commonly-used empirical model of the relationship of direct runoff to excess precipitation. As originally proposed by Sherman in 1932, it is "…the basin outflow resulting from one unit of direct runoff generated uniformly over the drainage area at a uniform rainfall rate during a specified period of rainfall duration." The underlying concept of the unit hydrograph is that the runoff process is linear, so the runoff from greater or less than one unit is simply a multiple of the unit runoff hydrograph.
To compute the direct runoff hydrograph with a UH, the program uses a discrete representation of excess precipitation, in which a "pulse" of excess precipitation is known for each time interval. It then solves the discrete convolution equation for a linear system:

1) Q_{n}=\sum_{m=1}^{n \leq M} P_{m} U_{n-m+1}


where Qn = storm hydrograph ordinate at time n\Delta t; P_m = rainfall excess depth in time interval m\Delta t to (m+1)\Delta t; M = total number of discrete rainfall pulses; and U_{n-m+1} = unit hydrograph ordinate at time (n-m+1)\Delta t. Q_n and P_m are expressed as flow rate and depth respectively, and U_{n-m+1} has dimensions of flow rate per unit depth. Use of this equation requires the implicit assumptions:

  1. The excess precipitation is distributed uniformly spatially and is of constant intensity throughout a time interval \Delta t.
  2. The ordinates of a direct-runoff hydrograph corresponding to excess precipitation of a given duration are directly proportional to the volume of excess. Thus, twice the excess produces a doubling of runoff hydrograph ordinates and half the excess produces a halving. This is the so-called assumption of linearity.
  3. The direct runoff hydrograph resulting from a given increment of excess is independent of the time of occurrence of the excess and of the antecedent precipitation. This is the assumption of time-invariance.
  4. Precipitation excesses of equal duration are assumed to produce hydrographs with equivalent time bases regardless of the intensity of the precipitation.

Parametric vs. Synthetic Unit Hydrographs

The alternative to specifying the entire set of unit hydrograph ordinates is to use a parametric UH. A parametric unit hydrograph defines all pertinent unit hydrograph properties with one or more equations, each of which has one or more parameters. When the parameters are specified, the equations can be solved, yielding the unit hydrograph ordinates.  For example, to approximate the unit hydrograph with a triangle shape, all the ordinates can be described by specifying:

  • Magnitude of the unit hydrograph peak.
  • Time of the unit hydrograph peak.

The volume of the unit hydrograph is known; it is one unit depth multiplied by the watershed drainage area. This knowledge allows us, in turn, to determine the time base of the UH. With the peak, time of peak, and time base, all the ordinates on the rising limb and falling limb of the unit hydrograph can be computed through simple linear interpolation. Other parametric unit hydrograph are more complex, but the concept is the same.

A synthetic unit hydrograph relates the parameters of a parametric unit hydrograph model to watershed characteristics. By using the relationships, it is possible to develop a unit hydrograph for watersheds or conditions other than the watershed and conditions originally used as the source of data to derive the UH. For example, a synthetic unit hydrograph model may relate the unit hydrograph peak of the simple triangular unit hydrograph to the drainage area of the watershed. With the relationship, an estimate of the unit hydrograph peak for any watershed can be made given an estimate of the drainage area. If the time of unit hydrograph peak and total time base of the unit hydrograph is estimated in a similar manner, the unit hydrograph can be defined "synthetically" for any watershed. That is, the unit hydrograph can be defined in the absence of the precipitation and runoff data necessary to derive the UH.  Chow, Maidment, and Mays (1988) suggest that synthetic unit hydrograph fall into three categories:

  1. Those that relate unit hydrograph characteristics (such as unit hydrograph peak and peak time) to watershed characteristics. The Snyder unit hydrograph is such a synthetic UH.
  2. Those that are based upon a dimensionless UH. The SCS unit hydrograph is such a synthetic UH.
  3. Those that are based upon a quasi-conceptual accounting for watershed storage. The Clark unit hydrograph and the ModClark model do so.

All of these synthetic unit hydrograph models are included in the program.