Frequency Storm

The objective of the frequency-based hypothetical storm that is included in the program is to define an event for which the precipitation depths for various durations within the storm have a consistent exceedance probability. Nesting the various precipitation depths leads to the notion of a "balanced" storm. For example, consider a synthetic storm with 0.1 annual exceedance probability (AEP.) If the storm is 6 hours long, it will also contain the 3-hour storm, and the 1-hour storm. When actual historical gage records are examined, it is not necessarily true that the 0.1-AEP storm 6 hours long contains the 0.1-AEP storm 1 hour long. However, generating nested storms does produce consistent results that are valuable for design and regulation purposes.

Basic Concepts and Equations

The development of a frequency storm begins with precipitation depths entered by the user. Each depth is associated with a duration. The shortest duration is often called the peak intensity duration and usually should match the simulation time interval. The longest duration may be up to 10 days; the exact length depends on the purpose for developing the frequency storm. A precipitation depth must be entered for each duration, from the selected shortest to the selected longest. All depth-duration precipitation values must be for the same annual exceedance probability.
The depth-duration values entered by the user are first augmented using relationships found in HYDRO-35 (Fredrick, Myers, and Auciello, 1977.) Analysis of high-resolution precipitation gages found that depths for intermediate durations can be reliably estimated as:

The estimated 10-minute and 30-minute depths are inserted into the depth-duration values entered by the user to create an augmented depth-duration relationship.
The augmented depth-duration relationship is next adjusted for storm area. The values entered by the user are so-called point values. Point values represent the precipitation characteristics observed at a point in the watershed. Precipitation at a point (perhaps measured by a rain gage) can be very intense and change rapidly over a short time, but high intensity cannot be sustained simultaneously over a large area. As the area of consideration increases, average intensity decreases. For example, a small thunderstorm may release an intense burst of rainfall over a small area. However, the physical dynamics of thunderstorms do not allow for the intense rainfall to be widespread. Further, if you were to consider a large area around the thunderstorm, the same precipitation volume averaged over the large area would result in a much lower intensity. For a specified frequency and storm duration, the average rainfall depth over an area is less than the depth at a point. To account for this, the U.S. Weather Bureau (1958) used averages of annual series of point and areal values for several dense, recording-raingage networks to develop reduction factors. The factors indicate how much point depths are to be reduced to yield areal-average depths. The factors, expressed as a percentage of point depth, are a function of area and duration, as shown in Figure 18. These factors are used in HEC-HMS to automatically adjust the depth-duration values entered by the user and the augmented values based on the storm area, which is also specified by the user.
In accordance with the recommendation of HEC (USACE, 1982), no adjustment should be made for durations less than 30 minutes. A short duration is appropriate for a watershed with a small time of concentration. A small time of concentration, in turn, is indicative of a relatively small watershed, which, in turn, requires no adjustment. HEC-HMS implements this recommendation.

Figure 18.Reduction of point rainfall depth as storm area increases.

The last adjustment of the depth-duration relationship accounts for the type of input that is used, and the type of output that is desired: annual duration or partial duration. Virtually all precipitation analyses that give depth for a specific duration use partial duration values. This means that the entire precipitation record was scanned and all events greater than a threshold value were included in the statistical analysis. In an analysis of this type, some years may contribute multiple events while other years provide none. However, in some cases the input precipitation data may use annual duration instead. The user must select the type of precipitation data that will be entered, and the type of output that is desired. When the input and output types do not match, the reduction factors shown in Table 11 are used to convert the data as necessary. Note that the conversion only applies to relatively frequent storms. As the annual exceedance probability decreases, the difference between annual and partial duration statistics becomes negligible.

Table 11.Reduction factors for converting partial-duration input to annual-duration output.

Finally, the program is able to use the processed depth-duration relationship to compute a hyetograph. It interpolates to find depths for durations that are integer multiples of the time interval selected for runoff modeling. Linear interpolation is used, after taking logarithms of both the depth and duration data. Performing the interpolation in log-log space improves the quality of intermediate estimates (Herschfield, 1961.) The interpolation yields successive differences in the cumulative depths, thus computing a set of incremental precipitation depths, each of duration equal to the selected computation interval.
The alternating block method (Chow, Maidment, Mays, 1988) is used to develop a hyetograph from the incremental precipitation values (blocks). This method positions the block of maximum incremental depth first at the specified location in the storm. The user may choose from values of 25, 33, 50, 67, or 75% of the time measured from the beginning of the storm to the total storm duration. The remaining blocks are arranged then in descending order, alternating before and after the central block. When the maximum incremental depth is not located at 50%, the arranged blocks will stop alternating as soon as the front (25 and 33%) or back (67 and 75%) of the storm is filled; remaining blocks are placed on the free side of the storm. Figure 19 is an example of this temporal distribution; this shows the rainfall depths for a 24-hour hypothetical storm with a 1-hour computation interval.

Figure 19.Example of distribution of frequency-based hypothetical storm.

Parameter Estimation

In the United States, depths for various durations can be obtained from a variety of sources. Several products are available for the entire country, including TP-40 (Herschfield, 1961) for durations from 30 minutes to 24 hours and TP-49 (Miller, 1964) for durations from 2 to 10 days. The Eastern part of the country has extra data for short durations in HYDRO-35 (Fredrick, Myers, and Auciello, 1977.) Some locations have specialized data developed locally, for example the Midwest has available Bulletin 71 (Huff and Angel, 1992.) The Pacific coast, inter-mountain, and Rocky Mountain states are covered by NOAA Atlas 2 (Miller et al., 1973) with a separate volume for each state. The Ohio River Valley and the Southwest Dessert are covered by NOAA Atlas 14 (Bonnin et al., 2004). These various reports are all similar in that they contain maps with isopluvial lines of constant precipitation depth. Each map is labeled with an annual exceedance probability and storm duration. Knowing the location of the watershed on the map, the depth for each required duration and exceedance probability can be interpolated between the isopluvial lines.
Each of the maps included in the reference sources is developed independently of other maps. That is, the map for the 0.01 probability of exceedance and 1-hour duration is developed separately from the 0.01 probability of exceedance and 6-hour duration. Because of this independence there can be inconsistencies in the values estimated from the maps. If this raw data is input to the program, there can result fluctuations in the computed hyetograph. These fluctuations can be reduced by smoothing the data before entering it into the program. The precipitation depth values for the range of durations, all with the same exceedance probability, should be plotted. A smooth line should be fit through the data. A best-fit line can be used without attempting to fit a line of a particular function. Precipitation values can then be determined from the smooth line, and those values input to the program.
The storm area should be set equal to the drainage area at the evaluation location. The evaluation location will be where the flow estimate is needed, for example at the inflow to a reservoir or at a particular river station where a flood damage reduction measure is being designed. When there are several evaluation locations in a watershed, separate storms must be prepared for each location. Failure to set the storm area equal to the drainage area at the evaluation location leads to incorrect depth-area adjustments and either over or underestimation of the flow for a particular exceedance probability.

As stated earlier, virtually all precipitation maps are given in partial duration form, so input is assumed to be of partial duration type. The selection of the output type as partial or annual duration depends on the intended use of the computed results. Frequently the computed results are used for floodplain regulation or the design of flood damage reduction measures. In these cases, the type of output is determined by the type of damages that are expected. Some damages are generally assumed to happen only once in a year. For example, the time required to rebuild residential housing usually means it can only be damaged once in a year. If two large floods occurred in the same year, the housing would be flooded twice before it could be rebuilt and no additional damage would occur. Annual duration output should be selected. Partial duration output should be used if damages can happen more than once in the same year. This is often the case in agricultural crops, where fields can be plowed and replanted after a flood only to be reflooded. Partial duration output may also be appropriate when ever the recovery time is so short that multiple damaging floods may happen in the same year.