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Intensity-Duration-Frequency Curves Example - Lecture Notes | CIVE 322, Study notes of Hydrology

Colorado State University (CSU)Hydrology
Prof. Jorge A. Ramirez

IDF Procedure Material Type: Notes; Professor: Ramirez; Class: Basic Hydrology; Subject: Civil Engineering; University: Colorado State University; Term: Fall 2018;

Typology: Study notes

2017/2018

Uploaded on 09/30/2018

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Hydrologic Science and Engineering
Civil and Environmental Engineering Department
Fort Collins, CO 80523-1372
(970) 491-7621
CIVE322 BASIC HYDROLOGY
Intensity-Duration-Frequency (IDF) Curves
Example
Intensity-Duration-Frequency (IDF) curves describe the relationship between rainfall intensity,
rainfall duration, and return period (or its inverse, probability of exceedance). IDF curves are
commonly used in the design of hydrologic, hydraulic, and water resource systems. IDF curves
are obtained through frequency analysis of rainfall observations.
Procedure
Data. From rainfall measurements, for every year of record, determine the annual maximum
rainfall intensity for specific durations (or the annual maximum rainfall depth over the specific
durations). Common durations for design applications are: 5-min, 10-min, 15-min, 30-min, 1-hr,
2-hr, 6-hr, 12-hr, and 24-hr (see for example Table 1 below.)
As discussed in class, the development of IDF curves requires that a frequency analysis be
performed for each set of annual maxima, one each associated with each rain duration. The basic
objective of each frequency analysis is to determine the exceedance probability distribution
function of rain intensity for each duration. In class, we discussed two options for this frequency
analysis:
1) Use an empirical plotting position approach to estimate the exceedance probabilities
based on the observations.
2) Fit a theoretical Extreme Value (EV) distribution (e.g., Gumbel Type I) to the
observations and then use the theoretical distribution to estimate the rainfall events
associated with given exceedance probabilities.
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Civil and Environmental Engineering Department

Fort Collins, CO 80523- 1372

CIVE322 BASIC HYDROLOGY Intensity-Duration-Frequency (IDF) Curves Example Intensity-Duration-Frequency (IDF) curves describe the relationship between rainfall intensity, rainfall duration, and return period (or its inverse, probability of exceedance). IDF curves are commonly used in the design of hydrologic, hydraulic, and water resource systems. IDF curves are obtained through frequency analysis of rainfall observations. Procedure Data. From rainfall measurements, for every year of record, determine the annual maximum rainfall intensity for specific durations (or the annual maximum rainfall depth over the specific durations). Common durations for design applications are: 5-min, 10-min, 15-min, 30 - min, 1-hr, 2 - hr, 6-hr, 12-hr, and 24-hr (see for example Table 1 below.) As discussed in class, the development of IDF curves requires that a frequency analysis be performed for each set of annual maxima, one each associated with each rain duration. The basic objective of each frequency analysis is to determine the exceedance probability distribution function of rain intensity for each duration. In class, we discussed two options for this frequency analysis:

  1. Use an empirical plotting position approach to estimate the exceedance probabilities based on the observations.
  2. Fit a theoretical Extreme Value (EV) distribution ( e.g. , Gumbel Type I) to the observations and then use the theoretical distribution to estimate the rainfall events associated with given exceedance probabilities.

Civil and Environmental Engineering Department

Fort Collins, CO 80523- 1372

CIVE322 BASIC HYDROLOGY Empirical Plotting Position Approach To illustrate the first approach, select for example the 30-min duration data from Table 1 and proceed as follows:

  1. Rank the observations in descending order (Table 2, Column 1 )
  2. Compute the exceedance probability associated with each rainfall depth using the following expression (Table 2, Column 4 ): (1) where m is the rank, n is the number of observations, p is the exceedance probability and T is the corresponding return period (Table 2, Column 5).
  3. Transform the depth data into rainfall intensity by dividing depth by the corresponding duration (Table 2, Column 6).
  4. Plot empirical distribution of rainfall intensity (Columns 5 and 6 in Figure 1). As indicated above, this procedure is repeated for each of the desired durations. Table 1. Maximum Annual Rainfall Depth for the Shown Duration Maximum Annual Rainfall Depth for the Shown Duration [mm] 5 min 0.08 hr 10 min 0.17 hr 15 min 0.25 hr 30 min 0.50 hr 1 hr 1 hr 2 hr 2 hr 6 hr 6 hr 12 hr 12 hr 24 hr 24 hr 1985 2.8 5.3 8.1 10.9 13.7 14.4 24.2 28 30. 1986 2.5 3.9 4.4 5.9 8.6 14.6 36.8 56.3 84. 1987 1.5 2.5 3.2 5.5 9.9 17.7 33.8 43.2 65. 1988 2 3.2 4.2 5.3 6.8 11.1 27.7 45 51. 1989 3 4.3 5.2 6.9 9.3 15.2 30 45.6 50. 1990 2.4 2.9 3.5 6.2 10.5 17.7 41.4 52.1 78. 1991 2.6 3.6 4.8 6.4 10.7 17.4 36 66.4 100. 1992 1.7 2 3.1 5.3 9.1 15.3 26.1 43.9 54. 1993 2.8 4 4.5 7.4 10.8 15.8 27.2 38.2 64. 1994 1.8 2.7 3.6 5.8 10.1 15 30.9 40.1 60. 1995 2.5 3.2 4.1 5.9 9.4 14.4 33.7 50.7 82. 1996 4.4 6.9 9.9 15.9 21.2 24 46.7 50.3 60. 1997 3.1 3.6 4.3 6.7 10.5 15.9 38.8 54.8 65. 1998 1.9 2.3 2.9 5.3 8.8 14.4 33.5 44.3 48. 1999 2 2.5 3.5 6 10.8 17.4 35.9 48 59. 2000 2 3.5 4 5.9 8.7 15 30.1 45.2 47. 2001 2.9 4 4.2 5.5 7.8 13.2 23.2 36.2 45. 2002 4.4 4.8 4.8 5.7 9.3 14.5 30 38 64. 2003 2.3 4.2 5.6 8.1 8.7 11.8 29.1 45.5 72. 2004 3.9 6.3 7.6 9.2 10.2 15.2 27.7 33 41 2005 3.2 5 6.5 8.5 9.8 13.9 24.1 34.5 43. Mean 2.65 3.84 4.86 7.06 10.22 15.42 31.76 44.73 60. St. Dev. 0.82 1.28 1.80 2.50 2.87 2.61 6.04 8.79 16. p = 1 T = m n + 1

Civil and Environmental Engineering Department

Fort Collins, CO 80523- 1372

CIVE322 BASIC HYDROLOGY

Theoretical Extreme Value (EV) Distribution Approach

To illustrate the second approach, let us select the Gumbel (Type I) distribution as our EV

distribution. The Gumbel Type I distribution is,

where μ is the location parameter and b is the scale parameter.

It can be shown that the value of the random variable XT associated with a given return period, T ,

may be obtained from the following expression,

where is the mean of the observations ( e.g. , arithmetic average of the observations), and S is

the standard deviation of the observations. The frequency factor associated with return period T ,

KT , is given by

Equations (1), (2) and (3) are applied to each set of annual maxima corresponding to each

duration, as follows:

1. Compute the frequency factors associated with the desired return periods ( e.g. , 2, 5, 10,

25, 50, 100, 1000) using equation (4).

Table 3. Frequency Factors

T 2 5 10 25 50 100 1000

KT - 0.164272 0.7194574 1.3045632 2.0438459 2.5922880 3.1366806 4.

2. For each duration ( e.g. , 5-min, 10-min, …etc.), compute the sample mean and sample

standard deviations of the series of annual maxima, ( x 1 ,……, xm ) (see Table 1).

and

G ( x ; μ, β ) =

e

x − μ

β e − e

x − μ β

XT = X + KT S

X

KT = −

[0.5772 + ln(ln(

T

T − 1

))]

X =

m

xi

i = 1 m

∑ S^ =^

m − 1

( xi − X )^2

i = 1 m

Civil and Environmental Engineering Department

Fort Collins, CO 80523- 1372

CIVE322 BASIC HYDROLOGY

  1. Use equation (3) to compute the precipitation intensity associated with each return period.

Return Period T

Duration 30.213 38.904 44.658 51.928 57.322 62.676 80.

5 min 30.213 38.904 44.658 51.928 57.322 62.676 80.

10 min 21.795 28.585 33.080 38.759 42.973 47.155 60.

15 min 18.248 24.600 28.806 34.121 38.063 41.976 54.

30 min 13.303 17.719 20.642 24.336 27.076 29.797 38.

1 hr 9.753 12.287 13.965 16.085 17.657 19.218 24.

2 hr 7.497 8.651 9.415 10.380 11.096 11.807 14.

6 hr 5.128 6.017 6.605 7.349 7.901 8.449 10.

12 hr 3.607 4.254 4.683 5.225 5.626 6.025 7.

24 hr 2.415 3.029 3.436 3.950 4.331 4.710 5.

  1. Plot the results (Figure 2). Figure 2. 0 10 20 30 40 50 60 70 0 5 10 15 20 Rainfall Intensity [mm/hr] Rainfall Duration [hrs] IDF Curves T = 2 years T = 10 years T = 50 years T = 1000 years