formula de weiss

8/10/2019 Formula de Weiss

1/17

BEE 4730 WatershedEngineering Fall 2014

August 28, 2014

HYDROLOGICAL RISK ANALYSIS

The following provide succinct guidelines for performing lognormal and log-Pearson III frequencyanalyses, the two most commonly used in hydrological engineering, and information for developingstrictly empirical risk estimates. The lognormal probability distribution is usually used for precipitationfrequency analyses and log-Pearson III is typically used for watershed discharge analyses.

A. Lognormal:1. Frequency Factor (KT) Methods

1.1. Chows Lognormal frequency factor (tabulatedKT)1.2. Normal or Gausian frequency factor (Numerical approximation forKT)

2. Approximate Analytical Solution for Lognormal analyses and some example built-infunctions (Excel, MATLAB, R)

B. Pearson III Frequency Factor

C. Empirical Rainfall Frequency-Duration Relationships1. Some Hershfield Maps (USWB TR-40)2. Weiss (1962) Equation3. Worlds Largest Events (PMP)


2/17


A. Lognormal

1. Frequency Factor Methods

A simple frequency analysis requires mean, , and standard deviation, , of a set of data (e.g., maximumvalue series) and knowledge of the probability density function that best describes the distribution of the

data. The valuexfor any given probability,P, or return period, T, is calculated using:

vTT CKx 1 (A.1)

Where Cvis coefficient of variation (/) andKTis called the frequency factor1. Tables of frequency

factors are available for most probability distributions but relatively good approximations are availablefor some of the distributions commonly used in water resource engineering.

1The frequency factor is essentially the z, thestandard normalized variablefor probability distributions. Theadoption of the frequency factor approach essentially streamlines the analytical statistics.


3/17


1.1. Chows Lognormal frequency factor (tabulatedKT)Ven Te Chow (1955) provided the simplest methodology by developing a table of frequency factorsspecifically for the lognormal probability distribution function (pdf). The frequency factors are readdirectly off the table using the Cvof the data to determine the table row and probability (P) to determinethe table column. The following example illustrates how to perform this frequency analysis.

DATA (x)Precipitation

(in)Obs. T2(years)

1.22 9.0 = 0.88 inches

1.2 4.5 = 0.242 inches

1 3.0 Cv= 0.2750.9 2.30.7 1.80.7 1.50.7 1.30.6 1.1

ANALYSISP

(from table 1)T= 1/P (years) K

(from table 1)X

(inches)

0.99 1.0 -1.79 0.450.95 1.1 -1.4 0.540.8 1.3 -0.84 0.670.5 2.0 -0.13 0.850.2 5.0 0.77 1.060.05 20.0 1.82 1.320.01 100.0 2.9 1.58

2The observed T is calculated with the Weibull (1939) relationship: T= (1 +N)/ rank,N= total number of data.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100

Return Period

1-hrPrecipitation

(in)

Return Period (years)

Precipitation

(inches)

DATA

ANALYSIS

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 10 100

Return Period

1-hrPrecipitation

(in)

Return Period (years)

Precipitation

(inches)

DATA

ANALYSIS


4/17



5/17


1.2 Normal or Gausian frequency factor (Numerical approximation forKT)A frequency analysis using the lognormal distribution and can be performed using the normal distributionwith 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):

xy log (A.2)

wherexis a data point andyis the log-transformed data point. (You can use ln() as well)

For the normal distribution, the frequency factor equals a quantity called thestandard normal variable,z,which can be approximated as:

32

2

001308.0189269.0432788.11

010328.0802853.0515517.2

www

wwwzKT

(A.3)

5.0

2

1ln

P

w (A.4)

To carry-out your analysis, log-transform your data, calculate the Cvof these transformed data, choose arange of probabilities, and use Eqs. (A.3) and (A.1) to calculate the associated Yvalues, i.e., theoreticallog-transformed event magnitudes for eachP. Then transform your Ys intoXs, which should have thesame units as your original data:

YX 10 (A.5)

If you used ln() in A.2, you would use exp() instead of 10 in A.5. Note also, most computational toolshave functions to calculate thestandard normal variable,z:

Microsoft Excel 2007 or earlier: =norminv(1-P, 0, 1)Microsoft Excel 2010 or later: =norm.inv(1-P, 0, 1)MATLAB: normcdf(1-P, 0 ,1)R: pnorm(1-P, mean=0, sd= )

Recall,zis the integral of the standard normal probability function (a.k.a., cumulative distribution)betweenand 1-P; the standard normal probability function is the normal probability function with amean =0 and a standard deviation = 1.

ALSOChow (e.g., 1964) also developed a relatively simple approach for determiningKTfor the Extreme Valuetype 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.

1lnln5772.0

6

T

TKT

(A.6)

For extremely small events (e.g., drought conditions), engineers will use log-transformed data inconjunction with Eq. (A.6), which is often called EVIII.


6/17


2. Approximate Analytical Solution for Lognormal analyses3

The following approximation for the Normal or Gaussian cumulative distribution function can be fit toobserved data that are Lognormally distributed.

44

4

3

3

2

2112

1 tbtbtbtbxP for t0 (A.7a)

444

3

3

2

211

2

11

tbtbtbtbxP for t< 0 (A.7b)

WhereP(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, toy(Eq. A.2). Then calculate mean, , and standard

deviation, , of theyvalues.

The value tfor any rainfall amountxis:

xt

log (A.8)

Example: Ithaca, NY 1-hour precipitation (1981-1997): data are shown as symbols, the dashed line is thefrequency analysis using the method above and the solid line is using Chows (1955) frequency factormethod.

3This information is adopted from: Abramowitz, M. and I.A. Stegun. 1972.Handbook of Mathematical Functions.Dover Publications, Inc. New York. 930-933.

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

1 .0 0 1 0 .0 0 1 0 0 .0 0

R eturn Pe riod , T (yrs)

Precipitation(

in)


7/17


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 almostalways apply the Pearson Type III frequency factor approach, using log-transformed data (Eq. A.2). Aswith the lognormal distribution, the frequency factors can be obtained from tables or analytical

approximations (Eq. B.1, B.2)

54322323

116

3

11 kzkkzkzzkzzKT (B.1)

6

sCk (B.2)

Where Csis the coefficient of skew of the log-transformed data andzis thestandard normal variableasdefined in Eq. (A.3).


8/17



9/17


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 contiguousUS (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 inchesx = return period variate from Table 1y = duration variate in Table 2A = 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 1Linearized Rainfall Frequency Variate

ReturnPeriod (yr.)

1 2 5 10 25 50 100

Variate -6.93 0 9.2 16.1 25.3 32.1 39.1

Table 2Linearized Rainfall Duration Variate

Duration(Hrs.)

0.17 0.33 0.5 0.67 1 2 3 6 12 24

Variate -37 -24 -15.6 -9.4 0 17.6 28.8 49.9 73.4 100

3. Worlds Largest Events and U.S. PMPs (see maps and graph included)


10/17



11/17



12/17



13/17



14/17



15/17



16/17



17/17


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:

481-486Chow, 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

estimatesUnited States east of the 105thmeridian, 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. Paper40, May. Washington, DC.

Weiss, L.L. 1962. A general relation between frequency and duration of precipitation.Mon. WeatherRev. 90: 87-88.

formula de weiss

Documents