Precipitation Analysis Areal Precipitation Estimation Depth - Area Analysis

1. Precipitation Analysis • Areal Precipitation Estimation • Depth - Area Analysis • Depth - Duration Analysis • Intensity - Duration Analysis • Intensity-Duration-Frequency Analysis

2. Quantitative Description of Rainfall • A rainfall event, or storm, describes a period of time having measurable and significant rainfall, • preceded and followed by periods with no measurable rainfall. • The time elapsed from start to end of the rainfall event is the rainfall duration. For small catchment rainfall duration is measured in minutes, while for very large catchments it may be measured in days. Rainfall durations of 1, 2, 3, 6, 12 and 14 h are common in hydrologic analysis and design. Rainfall depth is measured in mm, cm, or in and considered to be uniformly distributed over the catchment area. For instance, a 60-mm, 6-h rainfall event produces 60 mm of depth over a 6-h period.

3. Rainfall depth and duration tend to vary widely, depending on geographic location, climate, and time of the year. • Other things being equal, larger rainfall depths tend to occur more infrequently than smaller rainfall depths. Average rainfall intensity is the ratio of rainfall depth to rainfall duration. For example, a rainfall event producing 60 mm in 6 h represents an average rainfall intensity of 10 mm/h. Rainfall intensity varies widely in space and time. Typically, rainfall intensities are in the range 0.1 - 30.0 mm/h, but can be as large as 150 to 350 mm/h in extreme cases.

4. Rainfall Frequency Rainfall Frequency refers to the average time elapsed between occurrences of two rainfall events of same depth and duration. The actual elapsed time varies widely and can therefore be interpreted only in a statistical sense. For instance, if at a certain location a 100-mm rainfall event lasting 6-h occurs on the average once every 50 y, the 100-mm, 6-h rainfall frequency for this location would be 1 in 50 years, 1/50, or 0.02. The reciprocal of rainfall frequency is referred to as return period or recurrence interval. In the case of the previous example, the return period corresponding to a frequency of 0.02 is 50 y.

5. Rainfall Frequency • Generally, larger rainfall depths tend to be associated with longer return period. • The longer the return period, the longer the historical record needed to find out the statistical properties of the distribution of annual maximum rainfall. Due to the shortage of long rainfall records, extrapolations are usually necessary to estimate rainfall depths associated with long return periods. The first step in designing a water-control works is to determine the probable recurrence of rainfall of different intensity and duration so that an economical size of structure can be provided. Where human life is endangered, however, the design should handle runoff from storms even grater than have been recorded.

6. Temporal Rainfall Distribution Hyetograph Depth (%) depth or intensity Time (%) Time The temporal rainfall distribution represents the variation of rainfall depth within a storm duration. It can be expressed in either discrete or continuous form. The discrete form is referred to as hyetograph, a histogram of rainfall depth (or rainfall intensity) with time increments as abscissas and rainfall depth (or rainfall intensity) as ordinates. The continuous form in the temporal rainfall distribution, a function describing the rate of rainfall accumulation with time. Rainfall duration (abscissas) and rainfall depth (ordinates) can be expressed in percentage of total value, the result is dimensionless temporal rainfall distribution.

7. Spatial Rainfall Distribution 3 4 5 4 Isohyets Rainfall varies spatially, i.e., the same amount of rain does not fall uniformly over the entire catchment. Isohyets are used to depict the spatial variation of rainfall. An isohyte is a contour line showing the loci of equal rainfall depth.

8. Depth-Area Duration Analysis • In designing hydraulic structures for controlling river flow, an engineer needs to know the areal rainfall of the area draining to the control point. • The techniques of relating areal rainfall depths to area by analyzing several storms gives depth-area relationships for different specific durations. • Generally, the greater the catchment area, the smaller the averaged storm depth i.e. P  1/A(Example on Page 46)

9. Depth – Area- Duration Analysis • The purpose: To determine the maximum precipitation amounts over various area sizes during the passage of storms of various durations for the computation of probable maximum precipitation (PMP) estimates. • The y-axis is the area size and X-axis Maximum average areal precipitation. • Example: During a 36-hour storm event, the heaviest 12-hour 100 sq-mi precipitation was about 11 inches.

10. Depth - Duration Analysis Log-Log Lin-Lin Depth Depth Time Time Storm depth (h) and duration (t) are directly related, i.e., storm depth increases with duration. An equation relating storm depth and duration is h = c tn(1) where c is a coefficient and n is an exponent (a positive real number less than 1). Typically n varies between 0.2 and 0.5, indicating that storm depth increases at a lesser rate than storm duration. The applicability of such an equation, however, is limited to the regional or local conditions for which it is derived.

11. Intensity - Duration Analysis Storm intensity (i) and duration are inversely related. Differentiating Equation 1 with respect to duration (time), we have; (2) where a = cn, and m = 1-n. Since n is less than 1, it follows that m is also less than 1. Equation 2 is used for duration’s exceeding two hours. Another intensity-duration model (for shorter duration less than 2 hours) is the following: A. N Talbot’s formula (3) Where a and b are constants to be determined by regression analysis. Simple Linear Model Y = c + dt

12. Intensity - Duration Analysis Intensity Time A general intensity-duration model combining the features of Equations 2 and 3 is (4) Taking log Log(i) = log (a) - m log (t+b) Therefore various values of b may be assumed and graphs plotted till a straight-line is obtained.

13. Example Determine the equation relating rainfall intensity and duration for the following 10-y frequency rainfall data. Duration (min) 5 10 15 30 60 120 180 Intensity (cm/h) 8.0 5.0 4.0 2.5 1.5 1.0 0.8 Solution The data suggest that the relation is of hyperbolic type, with greater intensities associated with shorter durations. Therefore, an equation of the following type is applicable: A. N Talbot’s formula This equation can be linearalized in the following way: y = 1/i c = b/a d = 1/a

14. Calculations Model t i (min) (cm/hr) 5 8.0 10 5.0 15 4.0 30 2.5 60 1.5 120 1.0 180 0.8 Total = X Y X2 XY t 1/i t2 t (1/i) 5 0.1250 25 0.625 10 0.2000 100 2 15 0.2500 225 3.75 30 0.4000 900 12 60 0.6667 3600 40 120 1.0000 14400 120 180 1.2500 32400 225 420 4 51650 403 i = a/(t+b) (cm/hr) 4.9330 4.2584 3.7462 2.7527 1.7987 1.0624 0.7538 c = 0.006422495 d = 0.170602665 a = 1/d = 155.70 b = a c = 26.56

16. Calculations Model

17. Intensity-Duration and Frequency of Rainfall In hydrologic design projects, the use of intensity-duration-frequency relationships is recommended. Intensity refers to rainfall intensity, duration refers to rainfall duration and frequency refers to Return Period, which is expected value of the recurrence interval (time between occurrence). “It is defined as the average period of time (years) within which the depth of rainfall for a given duration will be equalled or exceeded once on the average.” The IDF relationships are also referred to as IDF curves. A general expression for IDF can be obtained by assuming that the constant a in Equation 4 [i = a / (t+b)m] is related to return period as: a = K Tx Where k is a coefficient, T is return period and x is an exponent. Therefore Where t in min, T in years and I in cm/hour. The values of k, b, m, and x are evaluated from measured data. Sherman Equation

18. IDF Curves of Maximum rainfall in Chicago, USA (from Chow, Maidment and Mays)