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                       DESIGN RAINFALL DISTRIBUTIONS
                                 FOR THE
                             STATE OF WYOMING


                 Patrick T. Tyrrell
                 Victor R. Hasfurther         August, 1983







                       Department of Civil Engineering
                           College of Engineering
                           University of Wyoming





                         Research Project Technical
                       Completion Report (A-036-WYO)
                        Agreement No. 14-34-0001-2154


                               Prepared for:
                     U. S. Department of the Interior



             The research on which this report is based was financed
        in part by the U. S. Department of the Interior, as authorized
        by the Water Research and Development Act of 1978 (P.L. 95-467).

             Contents of this publication do not necessarily reflect
        the views and policies of the U.S. Department of the Interior,
        nor does mention of trade names or commercial products
        constitute their endorsement or recommendation for use by the
        U. S. Government.



                        Wyoming Water Research Center
                            University of Wyoming
                              Laramie, Wyoming

INTRODUCTION

The design of hydraulic structures for use in ungaged drainage basins requires some estimate of flood flows and their frequency of occurrence. Because no historical streamflow data exist for these drainages, floods are estimated either by regional frequency analysis or, with the help of digital computers, by parametric rainfall-runoff event simulation.

Computer models dealing with rainfall-runoff event simulation are commonly used today by engineers and hydrologists. These models are used to predict flood hydrographs given an input rainfall volume, distributed over time in some manner, and certain geomorphic basin parameters.

Studies exist in the literature documenting the effects of time distribution of rainfall on runoff hydrographs. The reader is referred to works by Wei and Larson (1971), Yen and Chow (1980), and Shanholtz and Dickerson (1964) as examples. Because this relationship between the time distribution of rainfall and hydrograph characteristics exists, the separate study of storm rainfall is essential for accurate flood prediction regardless of other variables that also influence the runoff process. Additionally, methods of constructing design storms are available and in wide use, but they are general in nature and assume storms occur with the same temporal distribution across much of the country. Because of the drastic climatic differences between the areas encompassed by existing procedures, it was felt their design curves are not likely to be representative of the actual time distribution of storms in semi-arid regions such as Wyoming. It was, therefore, decided to develop a new design storm construction procedure applicable to the State of Wyoming based on observed storm rainfall in Wyoming. This new design storm methodology is the topic addressed herein.

REVIEW OF PREVIOUS WORK

Relatively few precipitation studies made to date deal with the temporal distribution of rainfall as used by hydrologists and engineers in parametric flood prediction.

The Soil Conservation Service (SCS) method (1973) presents two temporal rainfall distribution curves for runoff prediction. For studies in Hawaii, Alaska, and the coastal side of the Sierra Nevada and Cascade mountain ranges, the Type I and IA curves are used. The Type II curve is applied in the remaining part of the United States, Puerto Rico, and the Virgin Islands. These curves are based on generalized rainfall depth- duration curves obtained from published data of the U.S. Weather Bureau (National Oceanic and Atmospheric Administration). All design storms developed with this method, regardless of duration, are based on the 24-hour volume for a given frequency and location.

The Bureau of Reclamation method (1977) is developed in two parts, one for the United States east of the 105° meridian and the other for areas west of the 105° meridian. The procedure requires arranging hourly rainfall increments in a specified sequence depending on the duration and type of storm (thunderstorm or general storm). Maximum 6-hour point rainfall values are used in designing general storms, and maximum 1-hour point rainfall values are used in designing thunderstorms.

The U.S. Weather Bureau procedure (1961) uses depth-duration-frequency (DDF) curves in design storm construction. In this method rainfall intensities are obtained from the DDF curves for a given frequency and duration at a certain locality. These intensities are then rearranged arbitrarily to form a storm pattern.

Kerr, et al., (1974) present a method of hyetograph construction for the State of Pennsylvania. Cumulative dimensionless rainfall versus time graphs used by the method are derived from historical rainfall data. The curves allow the user much flexibility because, rather than define a single storm sequence, they bracket a range of possible storm patterns. Picking the time distribution of a design storm is up to the user, providing he stays within the limits of the bracketing curves and the minimum and maximum intensities given.

Huff (1967) presents a procedure derived from heavy storms observed in Illinois. His distribution patterns are based on the time quartile in which the majority of rain occurs for a given storm. For each quartile storm type, frequency values are given so that the user knows the return period of his design storm.

A method described in Keifer and Chu (1957) uses intensity-duration- frequency curves for hyetograph design at a given location. In general, the proposed storm pattern is fit to exponential growth and decay curves with the most intense part of the storm defined by a parameter termed the "advanceness ratio." This method was developed in Chicago for urban sewer design but can easily be used in other areas of the country where adequate rainfall records are available.

Frederick, et al. (1981) developed annual maximum precipitation events for different durations. The largest precipitation amounts for the selected durations which coincide with a given duration event are selected. The events are stratified according to magnitude and ratios of shorter to longer duration precipitation totals are formed. Accumulated probabilities of this ratio are suggested as a tool to estimate precipitation increments necessary in the synthesis of precipitation mass curves. By analyzing the relative timing of the shorter duration event within the longer duration event, a characteristic time distribution can be developed.

METHODOLOGY

Accumulation of Rainfall Data
The study of time distribution of rainfall requires historic data recorded as nearly continuously as possible. Because continuously recorded rainfall data were not available in the quantities needed for this study, discrete data were used. Hourly measurements from the National Oceanic and Atmospheric Administration (NOAA) publications (1948-1979) provided the data base for the study of general storms while the five-minute incremental precipitation data available in Rankl and Barker (1977) were used in thunderstorm analysis. Table I describes the precipitation stations used from both sources.

The definition of a storm had to be established before usable information could be obtained from the data. In this report, the criteria used for defining a storm are as follows:

          General Storm - preceded and followed by at least two hours of

                               zero rainfall
 
                            TABLE I.

          PRECIPITATION STATIONS PROVIDING DATA FOR STUDY
___________________________________________________________________
Reference   Location Name or       Major Drainage          Recording
  Number        Number                 Basin       Source  Interval
    1      Casper WSO AP           North Platte    NOAA1   Hourly
    2      Cheyenne WSFO AP        North Platte    NOAA    Hourly
    3      Douglas Aviation        North Platte    NOAA    HOURLY
    4      Encampment              North Platte    NOAA    Hourly
    5      Jelm                    North Platte    NOAA    Hourly
    6      Laramie 2 WSW           North Platte    NOAA    Hourly
    7      Medicine Bow            North Platte    NOAA    Hourly
    8      Oregon Trail Crossing   North Platte    NOAA    Hourly
    9      Pathfinder Dam          North Platte    NOAA    Hourly
   10      Phillips                North Platte    NOAA    Hourly
   11      Pine Bluffs             North Platte    NOAA    Hourly
   12      Rawlins FAA AP          North Platte    NOAA    Hourly
   13      Saratoga 4 N            North Platte    NOAA    Hourly
   14      Seminoe Dam             North Platte    NOAA    Hourly
   15      Shirley Basin Station   North Platte    NOAA    Hourly
   16      Torrington 1 S          North Platte    NOAA    Hourly
   17      Wheatland 4 N           North Platte    NOAA    Hourly
   18      Buffalo                 Powder          NOAA    Hourly
   19      Douglas 17 NE           Powder          NOAA    Hourly 
   20      Dull Center             Powder          NOAA    Hourly
   21      Gillette 18 SW          Powder          NOAA    Hourly
   22      Hat Creek 14 N          Powder          NOAA    Hourly
   23      Lance Creek             Powder          NOAA    Hourly
   24      Moorcroft               Powder          NOAA    Hourly
   25      Mule Creek              Powder          NOAA    Hourly
   26      Newcastle               Powder          NOAA    Hourly
   27      Osage                   Powder          NOAA    Hourly
   28      Pine Tree 9 NE          Powder          NOAA    Hourly
   29      Powder River            Powder          NOAA    Hourly
   30      Recluse                 Powder          NOAA    Hourly
   31      Sheridan WSO AP         Powder          NOAA    Hourly
   32      Story                   Powder          NOAA    Hourly
   33      Boysen Dam              Big Horn        NOAA    Hourly
   34      Lander WSO AP           Big Horn        NOAA    Hourly
   35      Meteetse 1 ESE          Big Horn        NOAA    Hourly
   36      Powell Field Station    Big Horn        NOAA    Hourly
   37      Riverton                Big Horn        NOAA    Hourly
   38      Tensleep 4 NE           Big Horn        NOAA    Hourly
   39      Thermopolis             Big Horn        NOAA    Hourly
   40      Thermopolis 25 WNW      Big Horn        NOAA    Hourly
   41      Worland                 Big Horn        NOAA    Hourly
   42      Big Piney               Green           NOAA    Hourly
   43      Mountain View           Green           NOAA    Hourl

                        TABLE I. continued

          PRECIPITATION STATIONS PROVIDING DATA FOR STUDY
___________________________________________________________________
Reference   Location Name or       Major Drainage          Recording
  Number        Number                 Basin       Source  Interval
   44      Mud Springs             Green           NOAA    Hourly
   45      Rock Springs FAA AP     Green           NOAA    Hourly
   46      Lake Yellowstone        Yellowstone     NOAA    Hourly
   47      Jackson                 Snake           NOAA    Hourly
   48      Moran 5 WNW             Snake           NOAA    Hourly
   49      Evanston 1 E            Bear            NOAA    Hourly
   50      06631150                North Platte    USGS2   5-minutely
   51      06634910                North Platte    USGS    5-minutely
   52      06634950                North Platte    USGS    5-minutely
   53      06644840                North Platte    USGS    5-minutely
   54      06648720                North Platte    USGS    5-minutely
   55      06648780                North Platte    USGS    5-minutely
   56      06312910                Powder          USGS    5-minutely
   57      06312920                Powder          USGS    5-minutely
   58      06313050                Powder          USGS    5-minutely
   59      06313180                Powder          USGS    5-minutely
   60      06316480                Powder          USGS    5-minutely
   61      06382200                Powder          USGS    5-minutely
   62      06233360                Big Horn        USGS    5-minutely
   63      06238760                Big Horn        USGS    5-minutely
   64      06238780                Big Horn        USGS    5-minutely
   65      06256670                Big Horn        USGS    5-minutely
   66      06267260                Big Horn        USGS    5-minutely
   67      06267270                Big Horn        USGS    5-minutely
   68      06274190                Big Horn        USGS    5-minutely
______________________________________________________________________
1 NOAA (1948-1979)
2 Rankl and Barker (1977)
=====================================================================

                        - at least four hours in duration
                        - at least one. half (0.5) inch in volume
          Thunderstorm  - preceded and followed by at least one hour of
                          zero rainfall
                        - at least twenty minutes and at most four hours
                          in duration
                        - at least one-half (0.5) inch in volume
These criteria are arbitrary but consistent with similar criteria put forth by Huff (1967), Ward (1973), and Croft and Marston (1950). Minimum duration requirements were used to make sure the time distribution of any storm was described by at least four data points. In all, 531 general storms and 72 thunderstorms were examined.

The period of record represented by the data at most stations covers the years 1969-1979, though the lack of definable storms at some stations required data from as far back as 1948. Because the development of design storms inherently assumes future rainfall events will occur with the same distribution as past events, the use of data from stations with variable periods of record is acceptable.

Description of Study Areas
The State of Wyoming was divided into its major surface water drainage basins for this study. This was done to see if differences in storm rainfall characteristics exist between basins. Figure 1 shows the entire State of Wyoming divided into these major drainages.

Analysis of Storm Parameters
Determining if differences in storm rainfall characteristics exist between basins requires statistical analysis of certain storm parameters.

Definitions of parameters used in describing storm rainfall follow:

     Storm Duration  - the amount of elapsed time, in hours, from the
                          beginning to the end of a storm.
     Storm Volume    - the total amount of rainfall measured during a
                          storm, in inches.
     Storm Intensity - the average rainfall rate during a storm, in inches
                          per hour, calculated by dividing a storm's volume
                          by its duration.
     Percent Time to Peak Intensity - that amount of time, expressed as a
                          percent of total storm duration, from the beginning
                          of a storm to the period of most intense rainfall.
     Pattern Index   - the area beneath a dimensionless cumulative rainfall
                          versus time curve, expressed as a decimal or as a
                          percent.
Pattern Index and Percent Time to Peak Intensity were the parameters used for determining if differences in the time distribution of rainfall exist between basins. This determination was made using a one-way analysis of variance technique for samples of unequal size. The procedure, described in Miller and Freund (1977), tests for differences in the population means for the populations from which the samples were taken. Such tests indicate if significant differences in parameter values exist between all the major drainages. If differences existed, the state would have to be divided accordingly before design storms could be constructed. If no differences existed, the state as a whole could be analyzed with the resulting design storms applicable statewide.

Construction of Design Curves
All the observed dimensionless mass rainfall curves are superimposed on one graph to create a family of "probable" storm patterns. Such an approach to design storm development is described in Kerr, et al. (1974). The method's most attractive feature is its flexibility, allowing the user his choice of three given design hyetographs, as well as the freedom to construct his own hyetograph, within limits. Such flexibility is desirable when, for example, a person is designing a structure based on peak flowrate in one instance and on runoff volume in another. The use of several curves can allow maximization of either peak flowrate or runoff volume for a given storm volume. A single design curve does not have this ability.

Figure 2 is a set of design curves. All of the storms used in the development: of this set of curves are non-dimensionalized and plotted on one graph of percent rainfall versus percent time. The bold vertical lines at each ten percent time increment represent the range of all storm data used. In the center of the plot is the mean curve. The curve is fit through the points representing the average cumulative percent rainfall at each ten percent time increment. It should be noted that the mean curve does not describe the average observed storm, rather it shows average accumulated rainfall with time based on all storms used. Also drawn on the plot: are ten percent and 90 percent limit curves. The ten percent limit curve represents, at a given percentage of storm duration, that value above which ten percent of the storms had accumulated more precipitation. Similarly, ten percent of the storms had each accumulated less than the value described by the 90 percent limit line at a given percentage of storm duration. It is not correct to assume that ten percent of the storms were totally above the ten percent limit line or totally below the 90 percent limit line. The use of ten percent as the cutoff when defining the upper and lower limit lines is arbitrary but reasonable. Using a smaller cutoff percentage resulting in a broader set of enveloping limit curves would be too general to accurately predict probable storm patterns. A larger cutoff value would result in a narrower envelope and a loss in flexibility of the method.

Under the assumption that future rainfall events will have the same time distribution as past events, these limit curves are the boundaries of a region of probable storm sequences. The user of the curves has the freedom to use either limit curve, or the mean curve, when choosing a design storm. In fact, he may pick his own storm sequence as long as he stays between the limit curves at all times and adheres to the maximum and minimum slope guidelines printed at the top of Figure 2. These guidelines are constructed in a manner similar to the limit curves in that for each ten percent time interval they represent intensities exceeded by ten percent of the storms (the steeper line) as well as intensities exceeded in 90 percent of the storms (the less steep line). In using these intensity guidelines, the designer cannot create a storm with an intensity greater than the value defined by the steep line or less than that defined by the shallow line for the appropriate ten percent increment of storm duration. The number accompanying each of these lines at the top of Figure 2 is the slope of that line.

Designing storms in this manner makes the utmost use of historical rainfall patterns while allowing the user flexibility in choosing the time distribution which will provide the critical peak flowrate or runoff volume for his purpose.

Comparison of Storm Design Methods
The creation of new storm patterns for use in a particular region is logically accompanied by a comparison of the results of using the new method with results obtained using established design storm techniques. Such a comparison will prove the need for the new region-specific design curves if the existing general methods do not produce similar runoff characteristics when applied to a given event.

The different storm designs are compared by inputting them to four different rainfall-runoff simulation models and examining the runoff hydrographs produced. Thunderstorm and general storm runoff are simulated with each model. For each model and storm type the infiltration parameters are held constant so that any differences noted in outflow hydrograph characteristics can be attributed to differences in the input hyetographs. The models used are described in Table II. In addition to the design storm construction method presented in this paper, techniques given by the U.S. Soil Conservation Service (1973) and the U.S. Bureau of Reclamation (1977) are used for comparative purposes. These last two methods have already been described in the review of previous work.

DESIGN STORM RESULTS

Statistical Analysis
Examination of the linear regression and analysis of variance (ANOVA) tests performed on the rainfall data leads to the following conclusions:
  1. A difference in the time distribution of thunderstorm rainfall compared to general storm rainfall exists for the entire State of Wyoming.
  2. The time distribution of both thunderstorms and general storms is not dependent upon the drainage basin in which the storms occur.
  3. No relationship exists between time distribution character- istics and duration of general storms or thunderstorms.
                                               TABLE II

                 DESCRIPTION OF DIGITAL COMPUTER MODELS USED IN DESIGN STORM COMPARISONS

   =======================================================================================================
                                              Method of estimating           Method of constructing
     Model                Citation            infiltration                   outflow hydrograph
   _______________________________________________________________________________________________________
   SCS Triagular          U.S. Soil Conser-   Uses a "minimum infiltration   Relates incremental excess
   Hydrograph             vation Service      rate" and runoff curve num-    precipitation to incremental
                          (1972).             ber based on soil type.        runoff with a hydrograph
                                                                             that is triangular in shape.

   HEC-1                  U.S. Army Corps     Uses an exponentially decay-   Derives outflow hydrograph
                          of Engineers        ing function that depends on   from either (1) unitgraph
                          (1973).             rainfall intensity and ante-   input by either, or (2) Clark
                                              cedent losses.                 (1945) synthetic unitgraph.

   HYMO                   Williams and Hann   Similar to SCS method          Uses dimensionless unitgraph
                          (1973). U.S. De-    above; uses curve number and   (described by exponential
                          partment of Agri-   minimum infiltration rate.     expressions relating flowrate
                          culture.                                           to time) and a "dimensionless
                                                                             shape parameter."

   USGS                   Dawdy, David R.,    Uses the Philip (1954) var-    Performs finite difference
                          John C. Shaake, Jr.,iation of the Green-Ampt       solution of kinematic wave
                          and William M.      (1911) equation.  Method in-   equation for each channel and
                          Alley (1978).       cludes soil-moisture account-  overland flow segment in drain-
                          U.S. Geological     ing between storms.            age basin.
                          Survey.
   =======================================================================================================
Inferred by 1 and 2 above is the need for only one set of general storm design curves and one set of thunderstorm design curves for use statewide. Conclusion 3 says that design storms of varying duration, i.e., 1-, 2-, or 3-hour thunderstorms or 6-, 12-, or 24-hour general storms, can all be handled with the same set of design curves. Table III lists the results of selected important linear regression and ANOVA tests used in drawing these conclusions. The rest of the statistical analysis results can be found in Tyrrell (1982).

Probably the most outstanding characteristic of the storms analyzed is their individual diversity. This same finding is corroborated in the paper by Kerr, et al. (1974) for storms in Pennsylvania. It is precisely because of this diversity that the use of an enveloping set of curves is preferred to the use of a single storm pattern when attempting to predict runoff. Presentation and Use of Design Curves

Figures 2 and 3 are the design curves for thunderstorms and general storms, respectively, constructed according to the procedures outlined previously. Figure 2 is to be used when the duration of the design storm of interest is less than four hours. Figure 3 is used for events four hours long or longer.

                                                TABLE III

                  RESULTS OF SELECTED STATISTICAL ANALYSIS OF RAINFALL CHARACTERISTICS
    ======================================================================================================

                                           Linear Regression
    ______________________________________________________________________________________________________
                                                            Correlation
    Dependent Variable       vs   Independent Variable      Coefficient(R)      Conclusion                

    Pattern Index for all         Duration of all storms.          .167     No significant relationship.
    storms.

    *Duration of all general      Percent time to Peak In-         .055     No significant relationship.
     storms-North Platte          tensity-general storms- 
     drainage.                    North Platte drainage.

    *Duration of all thunder-     Percent time to Peak In-         .170     No significant relationship.
     storms-North Platte          tensity-thunderstorms-
     drainage.                    North Platte drainage.
    ______________________________________________________________________________________________________



                                          TABLE III, continued

                  RESULTS OF SELECTED STATISTICAL ANALYSIS OF RAINFALL CHARACTERISTICS
    ======================================================================================================

                                          Analysis of Variance
    ______________________________________________________________________________________________________
  
    Null Hypothesis (Ho)                         F Statistic                   Conclusion
                                             Data   F.05    F.10     _______________________________________
    
    Pattern Index values for general         1.22   2.44   1.99    Do not reject Ho; conclude no difference
    storms are equal for all five                                  in Pattern Index due to drainage basin
    major drainages.                                               location.

    Pattern Index values for thunder-         .79   3.14   2.38    Do not reject Ho; conclude no difference
    storms are equal for three                                     in Pattern Index due to drainage basin
    major drainages.                                               location.

    *Pattern Index values are equal for     24.65   3.91   2.74    Reject Ho; conclude some difference in
    thunderstorms and general storms-                              Pattern Index due to type of storm.
    North Platte River drainage.
    _______________________________________________________________________________________________________
    *Results from the North Platte drainage data analysis are presented as an example.  Results from the
     other basins are similar.

    ======================================================================================================

Following is a list of steps involved in using the design curves:

  1. Select the storm type to be simulated at a certain location; for example, the 10-year, 6-hour event at Buffalo, Wyoming. Consult some source of rainfall frequency data, such as the Rainfall Frequency Atlas by Miller, et al. (1973), to find the volume of rain expected for this event.
  2. Select the appropriate set of design curves. For the example above, the general storm curves (Figure 3) are applicable because the duration is longer than four hours.
  3. Select one curve from the plot, either the ten percent or ninety percent limit curve, the mean curve, or some non-standard curve. When choosing a non-standard curve, the user must remember to stay on or between the limit curves at all times. Also, the steepness (intensity) of a curve in any ten percent time interval is dictated by the "maximum and minimum allowable intensities" shown at the top of the design curves. A non-standard curve must not be more steep than the steeper of these two lines (the maximum intensity line), or less steep than the line with smaller slope (the minimum intensity line) in any given ten percent interval of storm time. Examples of non- standard time distributions are given in following sections of this report.
  4. Using the curve from Step 3, select the percent rainfall values that correspond to the percent time values. A maximum time interval length of one hour is suggested. Table IV recommends percent time increments to be used for storms of varying duration.
  5. Organize the data obtained in Step 4 into the form required by whatever model is being use; i.e., rainfall either as actual depth or a percent of storm value, sequences either cumulative or incremental.
  6. Run the model with infiltration and geomorphic parameters as required.
                                TABLE IV

          RECOMMENDED TIME INTERVALS AND CORRESPONDING PERCENT
              TIME INCREMENTS FOR OBTAINING RAINFALL VERSUS
                      TIME DATA FROM DESIGN CURVES

=====================================================================
                                                      Interval as a
 Storm           Recommended          Number of         Percent of
Duration        Time Interval         Intervals       Storm Duration
30 minute          5 minute               6              16.67%
 1 hour           10 minute               6              16.67%
 2 hour           15 minute               8              12.50%
 3 hour           15 minute              12               8.33%
 6 hour           30 minute              12               8.33%
12 hour            1 hour                12               8.33%
24 hour            1 hour                24               4.17%

=====================================================================
It is recommended that the user run several simulations with different hyetographs to determine the critical runoff volume or peak flowrate. The suite of design curves used probably will include both limit curves, the mean curve, and several curves chosen arbitrarily by the user.

A parameter not included in this study is the areal distribution of rainfall. Therefore, the user of the method presented here is obliged to reduce point rainfall values when working with large drainage basins. Methods of reducing point rainfall with increasing drainage basin area are presented in Design of Small Dams (U.S. Bureau of Reclamation, 1977) and in the Rainfall Frequency Atlas (Miller, et al., 1973). These reductions are necessary because of the tendency of point rainfall values to overestimate actual areal precipitation on large areas.

Because this new design method depicts "probable" events, rather than extreme events (i.e., ultra-high-intensity bursts or long periods of very intense rain), it should not be used when designing for runoff due to "probable maximum" rainfall. Existing methods for probable maximum design (as in Small Dams) should be consulted for those cases.

RESULTS OF DESIGN STORM COMPARISONS

General Information
The purpose of this section is to compare the use of differing design storms in parametric flood prediction. Computer models used are HEC-1, HYMO, HYDRO (SCS Triangular Hydrograph method), and USGS (USGS distributed routing model). The reader is referred back to Table II for descriptions of these models. Design storms recommended by the U.S. Bureau of Reclamation (1977) and the U.S. Soil Conservation Service (1973) are used in the comparison.

The procedure followed in the comparison was to input differing design storms to a model, while leaving all geomorphic and loss parameters unchanged, and examine differences in the simulated outflow hydrograph peak and volume. Variations thus found are attributable to variations in the input hyetograph.

Some problems were encountered in the use of existing design storms. For example, the SCS method, rather than using a rainfall volume based on a certain duration for a given frequency, uses the 24-hour amount for designing storms of all durations. This practice results in slightly different storm volumes than those found in the Miller, et al. (1973), publication for varying durations. Despite this anomaly, the SCS hyetograph was used without a volume correction. Thus, a valid method-by-method comparison is ensured. The Bureau of Reclamation (BUREC) method also involves an odd twist basing its storm volumes on fractions and multiples of the 6-hour value for a given frequency. Modern practice has corrected this deficiency by allowing the use of volumes expected for various durations, not a manipulation of the 6-hour amount, while retaining the recommended time sequence. The BUREC method also typically calls for basing designs on runoff from a 3-hour thunderstorm and an 18-hour general storm. Because there exists no 18-hour duration precipitation data, no storms of this length were used in comparison. Also, a 2-hour thunderstorm was deemed most representative of short duration events (thus, the 3-hour event was not used).

Storms selected for the comparisons were 2, 6 and 24 hours in duration. The 2-hour event is considered a thunderstorm; the other two are general storms. A small drainage in the Powder River Basin provided the geomorphic data for the simulations. Storm volumes (U.S. Weather Bureau, 1961) for the duration's listed above (with a 1O-year return period) at this location are:

                     2-hour  -  1.60"
                     6-hour  -  2.00"
                    24-hour  -  2.75"
                    while the geomorphic parameters for the basin are:
                    Drainage Area         -  0.83 mi2
                    Water Course Length   -  1.38 mi.
                    Elevation Difference  -   125 feet
Model Parameters
Table V lists the loss parameters used with each model. The values of these parameters were not changed at any time. "NA" means the particular model does not use that parameter. It should be emphasized that values of loss parameters for the HYDRO, HYMO, USGS, and HEC-1 models are not calibrated values; they are values presented by Haie (1980) as representative for the Powder River Basin of Wyoming. A requirement of the USGS program, however, forced optimization of PSP. An optimization range of 4.0 - 6.0 was, therefore, used. The resulting small fluctuations in the value of PSP were not felt to harm the objectivity of the testing procedure. Because of the soil moisture accounting capability of the USGS model, antecedent rainfall and evaporation data was needed to "prepare" the soil prior to the occurrence of the storm event. Arbitrary, but consistent, amounts of .03 inches of daily precipitation and .01 inches of daily evaporation were applied for thirty days leading up to the simulated storm.

Because all the results presented herein were obtained using non-calibrated infiltration parameters, they are useful for comparison purposes only.

                                 TABLE V

             LOSS PARAMETERS USED WITH RAINFALL RUNOFF MODELS

                           FOR STORM COMPARISON


==========================================================================
                 Mon. Infil-
          Curve  tration Rate
Model    Number    (in/hr)     STRKR1  DRTKR1  RTIOL1  ERAIN1    TC1    R1

HYDRO       72      .15         NA       NA      NA      NA      NA     NA

HYMO        72      .15         NA       NA      NA      NA      NA     NA

HEC-1       NA       NA        .80      .20     2.75    .70     1.0    5.0



          PSP*    KSAT*    RGI*    BMSN*     EVC*    RR*    DRN/(24˙KSAT)*

USGS      5.0     0.10     10.0     5.0      0.7     0.9         0.5

==========================================================================
*For definition of parameters refer to dawdy, et al. (1978).
1The reader is referred to the HEC-1 users manual (U.S. Army Corps of
 Engineers, 1973) for definitions of these infiltration parameters.
Design Hyetographs
Tables VI, VII, and VIII present the design hyetographs used for each duration given as cumulative rainfall amounts. The "WYO" distribution sequences come from the curves presented in Figures 2 and 3. Those WYO storms designated A, B, C. etc., correspond to non-standard curves arbitrarily picked by the authors. These hyetographs can be graphically constructed by plotting the tabular values on a percent rainfall versus percent time basis, if the reader wishes to compare the
                                          TABLE VI

                   COMPARATIVE HYETOGRAPHS FOR 10 YEAR, 2-HOUR THUNDERSTORM

                                 Cumulative Rainfall (inches)
 =====================================================================================================
  Time,        *SCS            WYO:    10%     90%
 Minutes     Type II   BUREC   Mean   Limit   Limit     A     B     C     D     E     F     G     H
 _____________________________________________________________________________________________________

   0         ----      ----    ----   ----    ----     ----  ----  ----  ----  ----  ----  ----  ----
  15          .06       .14     .35    .75     .06      .30   .67   .75   .35   .35   .30   .35   .35
  30          .15       .36     .66   1.10     .24      .38   .77  1.02   .66   .66   .38   .58   .66
  45          .45       .65     .91   1.30     .50      .50   .83  1.09   .91   .91   .50   .64   .80
  60         1.17      1.26    1.14   1.44     .75      .83   .85  1.12   .98  1.14   .75   .75   .82
  75         1.30      1.39    1.30   1.50    1.01     1.10  1.01  1.15  1.01  1.17  1.01  1.01  1.01
  90         1.37      1.49    1.42   1.55    1.25     1.34  1.25  1.25  1.25  1.25  1.25  1.25  1.25
 105         1.43      1.55    1.52   1.58    1.44     1.57  1.44  1.44  1.44  1.44  1.44  1.44  1.44
 120         1.47      1.60    1.60   1.60    1.60     1.60  1.60  1.60  1.60  1.60  1.60  1.60  1.60
 _____________________________________________________________________________________________________

 *Based on 10 year, 24-hour volume (2.75")


 =====================================================================================================


                                             TABLE VII

                     COMPARATIVE HYETOGRAPHS FOR 10 YEAR, 6-HOUR GENERAL STORM

                                    Cumulative Rainfall (inches)
    =====================================================================================================
    Time,          *SCS                    WYO:          10%           90%
    Minutes       Type II      BUREC       Mean         Limit         Limit          C             G    
    _____________________________________________________________________________________________________

      0            ----        ----        ----         ----          ----          ----          ----
     30             .04                     .18          .34           .04           .04           .34
     60             .10         .14         .36          .68           .10           .10           .68
     90             .17                     .56         1.00           .22           .36           .84
    120             .24         .32         .74         1.24           .34           .68           .88
    150             .41                     .92         1.44           .50          1.00           .94
    180            1.41         .54        1.12         1.60           .68          1.34           .98
    210            1.62                    1.30         1.72           .90          1.68          1.04
    240            1.72        1.50        1.46         1.82          1.12          1.82          1.12
    170            1.80                    1.64         1.88          1.34          1.88          1.34
    300            1.86        1.82        1.76         1.94          1.56          1.94          1.56
    330            1.92                    1.90         1.98          1.78          1.98          1.78
    360            1.96        2.00        2.00         2.00          2.00          2.00          2.00
    _____________________________________________________________________________________________________

    *Based on 10 year, 24-hour volume (2.75")


    =====================================================================================================


                                                  TABLE VIII 

                        COMPARATIVE HYETOGRAPHS FOR 10 YEAR, 24-HOUR GENERAL STORM
                                        Cumulative Rainfall (inches)
  =====================================================================================================
  Time,   SCS            WYO:    10%    90%
  hours  Type II  BUREC  Mean   Limit  Limit    A      B      C      D      E      F        G       H
  _____________________________________________________________________________________________________

    
    0     ---      ---   ---    ---    ---    ----   ----   ----   ----   ----   ----     ----    ----
    1     .03      .05   .11    .22    .03     .03    .03    .03    .03    .22    .22      .22     .22
    2     .06      .14   .25    .47    .06     .06    .06    .06    .06    .47    .47      .47     .47
    3     .09      .22   .36    .72    .08     .08    .08    .08    .08    .72    .72      .72     .61
    4     .13      .33   .50    .94    .14     .14    .14    .14    .14    .94    .94      .94     .63
    5     .17      .44   .66   1.16    .22     .22    .22    .25    .22   1.16   1.16     1.10     .66
    6     .22      .55   .77   1.38    .30     .30    .30    .50    .30   1.38   1.38     1.16     .72
    7     .28      .66   .91   1.54    .39     .39    .39    .72    .39   1.54   1.54     1.18     .74
    8     .34      .80  1.02   1.71    .47     .47    .58    .94    .47   1.71   1.60     1.21     .77
    9     .41      .96  1.16   1.84    .58     .58    .80   1.18    .58   1.84   1.65     1.27     .83
   10     .51     1.71  1.27   1.98    .69     .74   1.02   1.38    .69   1.93   1.68     1.29     .85
   11     .65     1.95  1.40   2.09    .80     .96   1.27   1.62    .80   1.98   1.71     1.32     .88
   12    1.82     2.09  1.54   2.20    .94    1.18   1.49   1.84    .94   2.01   1.73     1.35     .94
   13    2.13     2.15  1.65   2.28   1.07    1.40   1.71   2.06   1.18   2.04   1.76     1.38    1.07
   14    2.26     2.20  1.79   2.37   1.24    1.62   1.93   2.31   1.40   2.06   1.79     1.43    1.24
   15    2.34     2.25  1.90   2.45   1.38    1.84   2.15   2.45   1.62   2.09   1.84     1.49    1.38
   16    2.42     2.31  2.01   2.50   1.54    2.09   2.37   2.50   1.84   2.12   1.87     1.54    1.54





                                            TABLE VIII continued

                       COMPARATIVE HYETOGRAPHS FOR 10 YEAR, 24-HOUR GENERAL STORM

                                        Cumulative Rainfall (inches)
  =====================================================================================================
  Time,  SCS            WYO:    10%     90%
  hours Type II  BUREC  Mean   Limit   Limit    A      B      C      D      E      F        G       H
  _____________________________________________________________________________________________________


   17    2.48     2.37  2.12   2.53    1.68   2.28   2.53   2.53   2.06   2.15   1.90     1.68    1.68
   18    2.54     2.42  2.26   2.59    1.84   2.50   2.59   2.59   2.26   2.17   1.90     1.84    1.84
   19    2.58     2.47  2.34   2.64    2.01   2.64   2.64   2.64   2.48   2.20   2.01     2.01    2.01
   20    2.62     2.53  2.42   2.67    2.15   2.67   2.67   2.67   2.64   2.20   2.15     2.15    2.15
   21    2.66     2.59  2.53   2.70    2.28   2.70   2.70   2.70   2.70   2.28   2.28     2.28    2.28
   22    2.70     2.64  2.61   2.72    2.45   2.72   2.72   2.72   2.72   2.45   2.45     2.45    2.45
   23    2.72     2.69  2.67   2.72    2.59   2.72   2.72   2.72   2.72   2.59   2.59     2.59    2.59
   24    2.75     2.75  2.75   2.75    2.75   2.75   2.75   2.75   2.75   2.75   2.75     2.75    2.75
  _____________________________________________________________________________________________________



  =====================================================================================================

visually with the standard 10%, 90% and mean WYO curves. The reader can see that, due to the discrepancy previously described, the SCS storm volumes do not quite equal the volumes given by the BUREC and WYO storms in Tables VI and VII.

The 6-hour event was the last of the three to be evaluated. Results from the earlier runs for the 2- and 24-hour events were used to indicate which of the lettered (A, B, C, etc.) WYO curves would probably give the largest peak runoff flowrate. As a result, the 6-hour event was run with only the "C" and "G" arbitrary curves used in addition to the mean, ten percent limit, and 90 percent limit curves.

Tables IX, X, and XI present the results of the model runs for the 2-hour, 6-hour and 24-hour events, respectively. Generally, results from HEC-1, HYMO, and HYDRO simulations show that for longer events the WYO curves produce less runoff (Peak and Volume) than the other methods, while for shorter events the WYO curves produce greater runoff. Results from USGS model runs differed from the other models* results by predicting, for all three storm durations, smaller runoff peaks and volumes due to the WYO design curves when compared to established procedures. Because of these results, it is suggested that current methods may lead to consistent over- design of hydraulic structures, at least when long (durations of 6 or more hours) events are stated as part of the design criteria. Also, the ability of any one of the group of WYO curves to produce greater runoff than the others is dependent upon the model used. These results are further detailed in the following section.

                               TABLE IX

        RUNOFF CHARACTERISTICS FOR 10 YEAR 2-HOUR THUNDERSTORM
========================================================================
                                           MODEL:
                             HYDRO       HYMO       HEC-1       USGS____
                          Peak  Vol.   Peak Vol.   Peak Vol.  Peak  Vol.
Design Storm              (cfs) (in.)  (cfs)(in.)  (cfs)(in.) (cfs) (in.)
_________________________________________________________________________
SCS Type II               47.8  .098   11.7 .036   38   .39   41.1  .162
BUREC                     65.3  .137   17.3 .053   36   .38   40.2  .162
WYO-Mean                  61.7  .139   12.9 .040   28   .31   16.0  .094
 10% Limit                61.8  .123   19.9 .061   42   .45   33.2  .146
 90% Limit                76.1  .135   30.7 .100   29   .32   20.6  .107
 -A                       79.6  .134   41.7 .133   31   .34   24.7  .118
 -B                       75.3  .133   30.9 .100   32   .39   23.1  .124
 -C                       62.2  .124   17.2 .064   34   .42   22.2  .138
 -D                       72.6  .132   30.7 .100   29   .35   19.3  .105
 -E                       62.5  .130   21.0 0.80   28   .34   18.0  .102
 -F                       76.1  .135   30.7 .100   27   .31   19.9  .105
 -G                       76.1  .135   30.7 .100   27   .32   19.2  .103
 -H                       76.7  .134   30.7 .100   28   .33   18.9  .103

========================================================================

                               TABLE X

        RUNOFF CHARACTERISTICS FOR 10 YEAR 6-HOUR GENERAL STORM
========================================================================
                                           MODEL:
                             HYDRO       HYMO       HEC-1       USGS____
                          Peak  Vol.   Peak Vol.   Peak Vol.  Peak  Vol.
Design Storm              (cfs) (in.)  (cfs)(in.)  (cfs)(in.) (cfs) (in.)
_________________________________________________________________________
SCS Type II               85.3  .175   42.7 .143   36   .38   47.1  .184
BUREC                     81.6  .251   37.6 .205   20   .23   19.4  .116
WYO-Mean                  52.8  .275   18.9 .094    2   .03    6.7  .065
 10% Limit                50.5  .208   26.9 .103   11   .14    8.5  .075
 90% Limit                83.6  .287   54.8 .261   10   .12   12.4  .085
 -C                       89.1  .221   49.4 .164   18   .22   16.7  .101
 -G                       83.6  .226   55.8 .261   10   .16   10.5  .082
========================================================================


                               TABLE XI

       RUNOFF CHARACTERISTICS FOR 10 YEAR 24-HOUR THUNDERSTORM
========================================================================
                                           MODEL:
                             HYDRO       HYMO       HEC-1       USGS____
                          Peak  Vol.   Peak Vol.   Peak Vol.  Peak  Vol.
Design Storm              (cfs) (in.)  (cfs)(in.)  (cfs)(in.) (cfs) (in.)
=========================================================================
SCS Type II              138.6  .346   57.9 .285   30   .34   43.1  .189
BUREC                     95.5  .268   45.9 .221   14   .16   14.4  .103
WYO-Mean                   0     0      0    0      0    0     1.49 .043
 10% Limit                24.3  .107   14.7 .091    0    0     2.22 .051
 90% Limit                 8.0  .085    6.5 .074    0    0     2.88 .056
 -A                       50.9  .428   35.5 .352    0    0     5.16 .072
 -B                       37.6  .400   29.0 .327    0    0     4.91 .070
 -C                       50.9  .384   36.6 .319    0    0     5.18 .069
 -D                       37.6  .412   27.7 .343    0    0     4.94 .071
 -E                       24.3  .134    1.7 .005    0    0     2.22 .057
 -F                       24.3  .120   14.7 .099    0    0     2.31 .057
 -G                        8.1  .075    6.1 .063    0    0     2.82 .056
 -H                        8.1  .085   6.5  .074    0    0     2.88 .056
=======================================================================

DISCUSSION OF RESULTS

The most significant difference between the WYO design storm methodology and those developed by the Soil Conservation Service and Bureau of Reclamation is the use of totally dimensionless curves. By non- dimensionalizing the time axis, the average intensities of designed storms is decreased as the storm durations are increased. For example, if two general storms of the same volume but differing durations, say 6 hours and 12 hours, were distributed over time according to the mean curve of Figure 3, the 12-hour storm would have half the intensity of the 6-hour event at any point along the curve. This explains why the WYO curves tend to produce smaller runoff peaks than the other methods for long events, and larger peaks for short events. Such a change in intensity with duration may seem inappropriate at first, but analysis of one hundred runoff- producing storms recorded by Ranki and Barker (1977) shows that, while there is not a good linear relationship (R = 53%), the peak intensity of a storm appears to decrease with increasing storm length. Figure 4 suggests this graphically. It, therefore, seems reasonable for the WYU storm design technique to make long storms generally less intense than short storms.

Lower rainfall intensity, as obtained from the WYO curves, is the reason zero runoff is predicted in some instances for the 24-hour event. For example, referring to Table XI, no runoff is produced using the WYO mean curve with the HYDRO and HYMO models. One will notice that, for general storms, the WYO mean curve is almost a 45░ line indicating an almost constant intensity storm. For the 24-hour event, this constant intensity (.11 in/hour) is less than the minimum infiltration loss of .15 in/hour. Thus, no runoff occurs. Similarly, the HEC-1 model produces zero runoff in several instances. Because shorter storms do produce runoff according to HEC-1, the reason for zero predicted runoff in the longer storms obviously also involves low rainfall intensity and associated infiltration losses.

It is interesting to note that choosing a WYO curve for producing peak runoff flowrate or volume depends on the computer model to be used. For instance, referring to Table IX, the WYO 90 percent limit curve produces more runoff (peak and volume) than the ten percent limit curve when HYDRO and HYMO are used. When HEC-1 is used, the ten percent limit curve yields the greatest runoff peak and volume. The user of these curves is, therefore, warned not to assume that a peak-producing hyetograph for one model will perform similarly with a different simulation scheme. Always test several curves for their peak-producing ability when changing models, or when changing storm durations with the same model.

SUMMARY AND CONCLUSIONS

Summary
Parametric flood prediction on ungaged basins in Wyoming requires the use of temporal storm patterns that realistically represent anticipated local rainfall events. Because methods of hyetograph construction currently in use are very general in application, this requirement is not met. Therefore, a design storm methodology based on analysis of time distribution characteristics of 603 observed storms in Wyoming is presented. The "WYO" method of storm design uses not one, but several mass rainfall curves, allowing flexibility of use and maximization of runoff from a given storm volume.

Comparisons were made between the WYO method and design storms recommended by the U.S. Soil Conservation Service and U.S. Bureau of Reclamation using HEC-1, HYMO, HYDRO (Triangular Hydrograph), and USGS Distributed Routing rainfall-runoff models.

Conclusions

  1. The time distribution of both thunderstorms and general storms is not dependent upon the drainage basin in which the storms occur.
  2. The most outstanding characteristic of the storms analyzed is their individual diversity. No relationship exists between time distribution characteristics and duration of general storms or thunderstorms. However, a difference in the time distribution of thunderstorm rainfall, compared to general storm rainfall, exists.
  3. One set of thunderstorm design curves and one set of general storm design curves can be used to create design hyetographs for the entire State of Wyoming.
  4. The "WYO" design storm methodology should not be used to design for "probable maximum" type events because the most intense rainfall values have been neglected by the definition of ten percent and 90 percent limit curves.
  5. Simulation of runoff peak and volume using WYO design curves is sensitive to storm duration and choice of model.
  6. WYO curves typically predict greater runoff peaks than Soil Conservation Service or Bureau of Reclamation synthetic hyetographs for short duration events, and less runoff for long duration events, according to HEC-1, HYMO, and HYDRO model results.
  7. WYO curves consistently produce less runoff than Soil Conservation Service or Bureau of Reclamation synthetic hyetographs when the USGS Distributed Routing model is used.

REFERENCES

Clark, C. 0., 1945, Storage and the Unit Hydrograph, Am. Soc. Civil Engineers Trans., V. 110, pp. 1419-1488.

Croft, A. R., and Richard B. Marston, 1950, Summer Rainfall Characteristics in Northern Utah, Trans. Am. Geophysical Union, Vol. 31, No. 1, pp. 83-95.

Dawdy, David R., John C. Schaake, Jr., and William M. Alley, 1978, Distributed Routing Rainfall-Runoff Model, U. S. Geological Survey, Water-Resources Investigations 78-90, 146 p.

Frederick, Ralph H., John F. Miller, Francis P. Richards and Richard W. W. Schwerdt, 1981, Interduration Precipitation Relations For Storms - Western United States, NOAA Technical Report NWS 27, U. S. Department of commerce, 195 pp.

Green, W. H. and G. A. Ampt, 1911, Studies on Soil Physics; I, Flow of Air and Water Through Soils: Jour. Agr. Research, V. 4, p. 1-24.

Haan, Charles T., and B. J. Barfield, 1978, Hydrology and Sedimentology of Surface Mined Lands, Lexington, Kentucky: University of Kentucky.

Haie, Nairn, 1980, Rainfall-Runoff Models for Ephemeral Streams in the Eastern Powder River Basin of Wyoming, M. S. Thesis, University of Wyoming, 56 pp.

Huff, F. A., 1967, Time Distribution of Rainfall in Heavy Storms, Water Resources Research 3(4):1007-1019.

Keifer, Clint J., and Henry Hsien Chu, 1957, Synthetic Storm Pattern for Drainage Design, Journal of the Hydraulics Division, ASCE, Vol. 83, No. HY4.

Kerr, R. L., T. M. Rachford, B. M. Reich, B. H. Lee, and K. H. Plummer, 1974, Time Distribution of Storm Rainfall in Pennsylvania, Pennsylvania State University, Institute for Research on Land and Water Resources, 34 pp.

Miller, Irwin, and John E. Freund, 1977, Probability and Statistics for Engineers, Second Ed. Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Miller, J. F., R. H. Frederick and R. J. Tracey, Precipitation-Frequency Atlas of the Western United States, Vol. II, Wyoming. NOAA Atlas 2. U. S. Dept. of Commerce, Silver Spring MD, 1973

National Oceanic and Atmospheric Administration (NOAA), 1948-1979, Hourly Precipitation Data for Wyoming, National Climatic Center Asheville, North Carolina.

Philip, J. R., 1954, An Infiltration Equation with Physical Significance: Soil Sci. Soc. Am. Proc., V. 77, p. 153-157.

Ranki, J. G., and D. S. Barker, 1977, Rainfall and Runoff Data from Small Basins in Wyoming, Wyoming Water Placing Program Report No. 17, 195 pp.

Shanholtz, V. 0., and W. H. Dickerson, 1964, Influence of Selected Rainfall Characteristics on Runoff Volume, West Virginia University Agricultural Experiment Station, Bulletin 4971.

Tyrrell, Patrick T., 1982,. Development of Design Rainfall Distribu- tion for the State of Wyoming, M. S. Thesis, University of Wyo- ming, 71 pp.

U. S. Army Corps of Engineers, 1973, HEC-1 Flood Hydrograph Package, Users and Programmers" Manuals, HEC Program 723-X6-L2010.

U. S. Bureau of Reclamation, 1977, Design of Small Dams, U. S. Dept. of the Interior, Washington, D.C.: U. S. Government Printing Office.

U. S. Soil Conservation Service, 1973, A Method for Estimating Volume and Rate of Runoff in Small Watersheds, SC-TP-149, Department of Agriculture.

U. S. Weather Bureau, 1961, Rainfall-Frequency Atlas of the United States for Durations from 30 Minutes to 24 Hours and Return Periods from 1 to 100 Years. Tech. Paper 40, 115 pp.

U. S. Weather Bureau, 1978, Use of DDF Curves in Storm Construction, In Hydrology and Sedimentology of Surface Mined Lands, Haan and Barfield (1978), pp. 49-52.

Ward, Tim, 1973, Quantification of Rainfall Characteristics, Unpublished. CWRR-DRI, University of Nevada System, Reno, Nevada.

Wei, Tseng C., and C. L. Larson, 1971, Effects of Areal and Time Distribution of Rainfall on Small Watershed Runoff Hydrographs, Minnesota Univ., Minneapolis, Water Resources Research Center, Bulletin No. 30, 130 pp.

Williams, J. R., and R. W. Hann, 1973, HYMO: Problem-Oriented Computer Language for Hydrologic Modeling, U. S. Department of Agriculture,Agriculture Research Service.

Yen, Ben Chie, and Ven Te Chow, 1980, Design Hyetographs for Small Drainage Structures, Journal of the Hydraulics Division, ASCE, Vol. 106.,No. HY6.


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