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BEE 4730 Watershed Engineering Fall 2014 HYDROLOGICAL RISK ANALYSIS The following provide succinct guidelines for performing lognormal and log-Pearson III frequency analyses, the two most commonly used in hydrological engineering, and information for developing strictly empirical risk estimates. The lognormal probability distribution is usually used for precipitation frequency analyses and log-Pearson III is typically used for watershed discharge analyses. A. Lognormal: 1. Frequency Factor (KT) Methods 1.1. Chow’s Lognormal frequency factor (tabulated KT ) 1.2. Normal or Gausian frequency factor (Numerical approximation for KT) 2. Approximate Analytical Solution for Lognormal analyses and some example built-in functions (Excel, MATLAB, R) B. Pearson III Frequency Factor C. Empirical Rainfall Frequency-Duration Relationships 1. Some Hershfield Maps (USWB TR-40) 2. Weiss (1962) Equation 3. World’s Largest Events (PMP) August 28, 2014 BEE 4730 Watershed Engineering Fall 2014 A. Lognormal 1. Frequency Factor Methods A simple frequency analysis requires mean, , and standard deviation, , of a set of data (e.g., maximum value series) and knowledge of the probability density function that best describes the distribution of the data. The value x for any given probability, P, or return period, T, is calculated using: xT 1 K T Cv (A.1) Where Cv is coefficient of variation (/) and KT is called the frequency factor1. Tables of frequency factors are available for most probability distributions but relatively good approximations are available for some of the distributions commonly used in water resource engineering. 1 The frequency factor is essentially the “z,” the standard normalized variable for probability distributions. The adoption of the frequency factor approach essentially streamlines the analytical statistics. Watershed Engineering BEE 4730 Fall 2014 1.1. Chow’s Lognormal frequency factor (tabulated KT) Ven Te Chow (1955) provided the simplest methodology by developing a table of frequency factors specifically for the lognormal probability distribution function (pdf). The frequency factors are read directly off the table using the Cv of the data to determine the table row and probability (P) to determine the table column. The following example illustrates how to perform this frequency analysis. DATA (x) Precipitation (in) 1.22 1.2 1 0.9 0.7 0.7 0.7 0.6 ANALYSIS P (from table 1) 0.99 0.95 0.8 0.5 0.2 0.05 0.01 Obs. T2 (years) 9.0 4.5 3.0 2.3 1.8 1.5 1.3 1.1 T = 1/P (years) 1.0 1.1 1.3 2.0 5.0 20.0 100.0 = 0.88 inches = 0.242 inches Cv = 0.275 K (from table 1) -1.79 -1.4 -0.84 -0.13 0.77 1.82 2.9 X (inches) 0.45 0.54 0.67 0.85 1.06 1.32 1.58 1.8 1.6 (inches) Precipitation (in) 1-hr Precipitation 1.4 DATA 1.2 1.0 0.8 ANALYSIS 0.6 0.4 0.2 0.0 1 10 100 Return Period Return Period (years) 2 The observed T is calculated with the Weibull (1939) relationship: T = (1 + N)/ rank, N = total number of data. BEE 4730 Watershed Engineering Fall 2014 Watershed Engineering BEE 4730 Fall 2014 1.2 Normal or Gausian frequency factor (Numerical approximation for KT) A frequency analysis using the lognormal distribution and can be performed using the normal distribution with log-transformed data; which simply means that you use the logs of your data instead of the raw data (i.e., the logs are normally distributed): y logx (A.2) where x is a data point and y is the log-transformed data point. (You can use ln() as well) For the normal distribution, the frequency factor equals a quantity called the standard normal variable, z, which can be approximated as: KT z w 2.515517 0.802853w 0.010328w 2 1 1.432788w 0.189269w 2 0.001308w3 1 w ln 2 P (A.3) 0.5 (A.4) To carry-out your analysis, log-transform your data, calculate the Cv of these transformed data, choose a range of probabilities, and use Eqs. (A.3) and (A.1) to calculate the associated Y values, i.e., theoretical log-transformed event magnitudes for each P. Then transform your Y’s into X’s, which should have the same units as your original data: X 10Y (A.5) If you used ln() in A.2, you would use exp() instead of “10” in A.5. Note also, most computational tools have functions to calculate the standard normal variable, z: Microsoft Excel 2007 or earlier: Microsoft Excel 2010 or later: MATLAB: R: =norminv(1-P, 0, 1) =norm.inv(1-P, 0, 1) normcdf(1-P, 0 ,1) pnorm(1-P, mean=0, sd= ) Recall, z is the integral of the standard normal probability function (a.k.a., cumulative distribution) between –∞ and 1-P; the standard normal probability function is the normal probability function with a mean =0 and a standard deviation = 1. ALSO Chow (e.g., 1964) also developed a relatively simple approach for determining KT for the Extreme Value type I (EVI) distribution, which is most commonly used in the frequency analyses of large events, although the log-normal analysis often works just as well. KT T 6 0.5772 ln ln T 1 For extremely small events (e.g., drought conditions), engineers will use log-transformed data in conjunction with Eq. (A.6), which is often called EVIII. (A.6) Watershed Engineering BEE 4730 Fall 2014 2. Approximate Analytical Solution for Lognormal analyses 3 The following approximation for the Normal or Gaussian cumulative distribution function can be fit to observed data that are Lognormally distributed. Px Px 1 1 2 3 4 1 b1 t b2 t b3 t b4 t 2 4 for t 0 1 2 3 4 1 b1 t b2 t b3 t b4 t 2 (A.7a) 4 for t < 0 (A.7b) Where P(x) is the probability of exceedence for a rainfall amount = x. The constants, bi, are: b1 = 0.196854; b2 = 0.115194; b3 = 0.000344; b4 = 0.019527 To calculate t, first log transform all your data, x, to y (Eq. A.2). Then calculate mean, , and standard deviation, , of the y values. The value t for any rainfall amount x is: t log x (A.8) Example: Ithaca, NY 1-hour precipitation (1981-1997): data are shown as symbols, the dashed line is the frequency analysis using the method above and the solid line is using Chow’s (1955) frequency factor method. 1 .8 P recip itatio n (in ) 1 .6 1 .4 1 .2 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 1 .0 0 1 0 .0 0 1 0 0 .0 0 R e tu rn P e rio d , T (y rs ) 3 This information is adopted from: Abramowitz, M. and I.A. Stegun. 1972. Handbook of Mathematical Functions. Dover Publications, Inc. New York. 930-933. Watershed Engineering BEE 4730 Fall 2014 B. Pearson III Frequency Factor The Log-Pearson Type III is the most common distribution used in stream discharge frequency analyses. Unfortunately, there are no good analytical approximations for this distribution so practitioners almost always apply the Pearson Type III frequency factor approach, using log-transformed data (Eq. A.2). As with the lognormal distribution, the frequency factors can be obtained from tables or analytical approximations (Eq. B.1, B.2) KT z z 2 1 k 1 3 1 z 6 z k 2 z 2 1 k 3 zk 4 k 5 3 3 k Cs 6 (B.1) (B.2) Where Cs is the coefficient of skew of the log-transformed data and z is the standard normal variable as defined in Eq. (A.3). BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 C. Empirical Rainfall Frequency-Duration Relationships 1. Some Hershfield Maps (USWB TR-40) The US Weather Bureau has created isohyetal maps (isohyet is a line of equal rainfall) for the contiguous US (examples are included in this packet). 2. Weiss Equation (1962) An equation to estimate the rainfall amount from a storm of any frequency (>1 yr.) and any duration (> 10 min.) anywhere in the contiguous U.S. is: I = 0.0256(C-A)x + 0.00256[ (D-C) – (B-A) ]xy + 0.01(B-A)y + A I= rainfall amount in inches x = return period variate from Table 1 y = duration variate in Table 2 A = 2-yr, 1-hr storm amount (from Hershfield map) B = 2-yr, 24-hr storm amount (from Hershfield map) C = 100-yr, 1-hr storm amount (from Hershfield map) D = 100-yr, 24-hr storm amount (from Hershfield map) Table 1 – Linearized Rainfall Frequency Variate Return 1 2 5 10 25 Period (yr.) Variate -6.93 0 9.2 16.1 25.3 Table 2 – Linearized Rainfall Duration Variate Duration 0.17 0.33 0.5 0.67 (Hrs.) Variate -37 -24 -15.6 -9.4 50 100 32.1 39.1 1 2 3 6 12 24 0 17.6 28.8 49.9 73.4 100 3. World’s Largest Events and U.S. PMPs (see maps and graph included) BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 BEE 4730 Watershed Engineering Fall 2014 References: Abramowitz, M. and I.A. Stegun. 1972. Handbook of Mathematical Functions. Dover Publications, Inc. New York. 930-933. Chow, V.T. 1955. On the deterimination of frequency factor in log-probability plotting. Trans. AGU. 36: 481-486 Chow, V.T. 1964. Handbook of applied hydrology. McGraw-Hill. New York. Library of Congress Card No. 63-13931. Hansen, E.M., L.C. Schreiner, J.F. Miller. 1982. Application of probable maximum precipiation estimates – United States east of the 105th meridian, NOAA hydrometerological report no. 52, National Weather Service, Washington, DC/ Hershfield, D.N. 1961. Rainfall Frequency Atlas of the United States. US Weather Bureau Tech. Paper 40, May. Washington, DC. Weiss, L.L. 1962. A general relation between frequency and duration of precipitation. Mon. Weather Rev. 90: 87-88.