Unit Hydrograph - theory

The unit hydrograph is the unit pulse response function of a linear hydrologic system. The unit hydropraph (originally named unit-graph) of a watershed is defined as a direct runoff hydrograph (DRH) resulting from 1 inch (usually taken as 1 cm is SI units) of excess rainfall generated uniformly over the drainage area at a constant rate for an effective 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.

It is a simple linear model that can be used to derive the hydrograph resulting from any amount of excess rainfall. The following assumptions are inherent in this model:

1) The excess rainfall has a constant intensity within the effective duration;

2) The excess rainfall is uniformly distributed throughout the whole drainage area.

3) The base time of the DRH (the duration of direct runoff) resulting from an excess rainfall of given duration is constant.

4) The ordinates of all DRH's of a common base time are directly proportional to the total amount of direct runoff represented by each hydrograph.

5) For a given watershed, the hydrograph resulting from a given excess rainfall reflects the unchanging characteristics of the watershed.

Under natural conditions, the above assumptions cannot be perfectly satisfied. However, when the hydrologic data to be used are carefully  selected so that they come close to meeting the above assumptions, the results obtained by the unit hydrograph model are generally acceptable for practical purposes (Heerde￾gen, 1974). Although the model was originally devised for large watersheds, it has been found applicable to small watersheds from less than 0.5 hectares to 25 km2 (about 1 acre to 10 mi2). Some cases do not support the use of the model because one or more of the assumptions are not well satisfied. For such reasons, the model is considered inapplicable to runoff originating from snow or ice.
Concerning assumption (1), the storms selected for analysis should be of short duration, since these will most likely produce an intense and nearly constant excess rainfall rate, yielding a well-defined single-peaked hydrograph of short time base. Concerning assumption (2), the unit hydrograph may become inapplicable when the drainage area is too large to be covered by a nearly uniform distribution of rainfall. In such cases, the area has to be divided and each subarea analyzed for storms covering the whole subarea.
Concerning assumption (3), the base time of the direct runoff hydrograph (DRH) is generally uncertain but depends on the method of baseflow separation. The base time is usually short if the direct runoff is considered to include the surface runoff only; it is long if the direct runoff also includes subsurface runoff.

Synthetic Unit Hydrograph

The unit hydrograph, derived from rainfall and streamflow data of a watershed, is specific to that watershed and the point on the stream where the flow was measured. To develop unit hydrographs for other locations within the same watershed or for nearby watersheds with similar characteristics, synthetic unit hydrograph methods are used. There are three main types of synthetic unit hydrographs:

1. Those that relate hydrograph characteristics (peak flow rate, duration, etc.) to watershed characteristics (Snyder, 1938; Gray, 1961). It is the Snyder Synthetic Unit Hydrograph.
2. Those based on a dimensionless unit hydrograph (Soil Conservation Service, 1972). It is the SCS Synthetic Unit Hydrograph.
3. Those based on watershed storage models (Clark, 1943). It is the Clark Synthetic Unit Hydrograph.

1) Snyder Synthetic Unit Hydrograph

The Snyder unit hydrograph method uses various relationships to estimate key points of a unit hydrograph, helping to route excess precipitation to the subbasin outlet. Snyder identified three critical characteristics:

1. Basin lag (tp​),
2. Peak discharge per unit area (qp​),
3. Total time base of the hydrograph.

For Snyder, a "standard unit hydrograph" is defined based on the relationship between rainfall duration (tr​) and basin lag (tp​): ​

If the duration of the desired unit hydrograph for the watershed of interest is significantly different than Equation 1 (that is: tpR is quite different from 5.5 tR), the following relationship can be used:

where t_R = lag of the desired unit hydrograph [hours].  For the standard case, Snyder found that q_p can be computed as:

where C_p = a coefficient derived from gaged data in the same region.

For the non-standard case, the peak discharge per unit area of a desired unit hydrograph, q_pR, can be computed as:

Required parameters (Snyder Synthetic UH)

To apply the "Standard" Snyder method in HEC-HMS, the required parameters include:

• Basin lag time (tp​) in hours
• Peak flow coefficient (Cp​)

These parameters are typically estimated using multi-linear regression analyses based on watershed physical characteristics.

Alternative Methods and Additional Parameters:

1. Ft Worth District Method requires:

   • Total stream length to the most hydraulically remote point (mi/km)
   • Stream length to the subbasin centroid (mi/km)
   • Weighted slope between points at 10% and 85% of the total stream length (ft/mi or m/km)
   • Percent urbanization
   • Percent sand content

2. Tulsa District Method requires:

   • Total stream length to the most hydraulically remote point (mi/km)
   • Stream length to the subbasin centroid (mi/km)
   • Weighted slope between points at 10% and 85% of the total stream length (ft/mi or m/km)
   • Percent channelization

These values are typically estimated using GIS analysis.

Estimating parameters (Snyder Synthetic UH)

For the unit hydrograph lag:

where:

• Ct​ = Basin coefficient
• L = Length of the main stream from the outlet to the watershed divide
• Lc​ = Length along the main stream from the outlet to a point nearest the watershed centroid
• C = Conversion constant (0.75 for SI units, 1.00 for the foot-pound system).

Calibration of Ct​ and Cp​

The parameters CtC_tCt​ (from Equation 34) and CpC_pCp​ (from Equation 32) are not physically based and are best determined through calibration.

According to Bedient and Huber (1992):

• CtC_tCt​ typically ranges from 1.8 to 2.2, but can vary from 0.4 in mountainous areas to 8.0 along the Gulf of Mexico.
• CpC_pCp​ ranges from 0.4 to 0.8, where larger values of CpC_pCp​ correspond to smaller values of CtC_tCt​.

2) SCS Synthetic Unit Hydrograph

The SCS dimensionless hydrograph is a synthetic unit hydrograph where:

• Discharge is expressed as the ratio of actual discharge (q) to peak discharge (qp​).
• Time is expressed as the ratio of actual time (t) to the time of rise (Tp​) of the unit hydrograph.

By knowing the peak discharge (qp​) and lag time for the given excess rainfall duration, the unit hydrograph for a watershed can be derived from the dimensionless hydrograph.

Figure 1.1A(a) illustrates a dimensionless hydrograph, created from unit hydrographs of different watersheds. The parameters qp​ and Tp​ can be estimated using a triangular unit hydrograph model (as shown in Figure 7.7.4(b)), where time is in hours and discharge is in m³/s-cm (or cfs/in). 

where C = 2.08 (483.4 in the English system) and A is the drainage area in square kilometers (square miles).

The time of rise (Tp​) of the unit hydrograph can be expressed in terms of lag time (tp​) and the duration of effective rainfall (tr​) using the following relationship:

where:

• Lag time (tp​): The time from the centroid of excess rainfall to the peak discharge.
• Effective rainfall duration (tr​): The duration of the excess rainfall event.
• The rise time (Tp​) represents the time from the beginning of excess rainfall to the peak of the unit hydrograph.
• Since rainfall distribution is assumed uniform over the duration tr​, half of tr​ is added to the lag time tp​.

This equation helps estimate Tp​ when only tp​ and tr​ are known, which is useful in deriving the SCS dimensionless unit hydrograph.