U.S. Department of the Interior

U.S. Geological Survey

Fact Sheet 2012–3038

March 2012

Calculating Weighted Estimates of Peak Streamflow Statistics

Printed on recycled paper

Data Sources

The at-site estimate of the logarithm of the peak 

streamflow statistic, Qas, typically is calculated from the 

record of annual peak flows at the streamgage, according 

to methods described in Bulletin 17B (Interagency Committee on Water Data, 1981). A corresponding estimate, 

Qre, is obtained from regression equations derived from 

a multivariate analysis of at-site values and explanatory 

basin characteristics from multiple streamgages in a 

region (Stedinger and Tasker, 1986).

If the mean square errors (MSE) of the two independent estimates are known, then an optimal weighted 

average is inversely proportional to the uncertainty of 

the individual estimates and can be calculated by the 

following:

MSE[log(Qas)] and MSE[log(Qre)] represent the uncertainty corresponding to each estimate. When the MSE 

corresponding to log(Qas) is large, the uncertainty is also 

large, and so the optimal weight assigned to log(Qas) is 

relatively small. Conversely, when the MSE is small, the 

variance is also small, and so the optimal weight is correspondingly large.

Application of this weighting technique is easy if 

estimates of the quantiles, Qas and Qre, and the associated 

uncertainties, MSE[log(Qas)] and MSE[log(Qre)] are 

available. Because the MSE[log(Qas)] is not specifically 

reported at present (2012) from PEAKFQ, the Weighted 

Independent Estimates (WIE) program was developed by 

the U.S. Geological Survey Office of Surface Water to 

compute this statistic and automate the process described 

above. It is anticipated that these statistics will be reported 

in future versions of PEAKFQ (Flynn and others, 2006) 

and WREG (Eng and others, 2009).

The WIE Program

The WIE program calculates the following:

A. The variance of the at-site estimate using the 

Expected Moments Algorithm (Cohn and others, 

2001, eq. 65). The log-Pearson Type III (LP3) 

distribution parameters that can be obtained from 

the PEAKFQ-generated output file must be supplied by the user. These parameters include the 

number of years of systematic record, skew type, 

MSE of the regional skew used, and the mean, 

standard deviation, and skew in log units for the 

streamgage.

B. The variance of the regression estimate (Stedinger 

and Tasker, 1986) from either

1. The variance at a specific streamgage, or

2. The average variance of the regional 

 regression model.

In either case, results from the regression model 

generated by WREG must be supplied by the 

user. These results include the standard error of 

the model, the number of explanatory variables, 

and the covariance matrix [(U) or T 1 (X X) − Λ ] of 

the explanatory variables used in the regression 

equation.

According to the Federal guidelines for flood-frequency estimation, the uncertainty of peak streamflow statistics, 

such as the 1-percent annual exceedance probability (AEP) flow at a streamgage, can be reduced by combining 

the at-site estimate with the regional regression estimate to obtain a weighted estimate of the flow statistic. 

The procedure assumes the estimates are independent, which is reasonable in most practical situations. The 

purpose of this publication is to describe and make available a method for calculating a weighted estimate from 

the uncertainty or variance of the two independent estimates.

MSE[log( )] log( ) MSE[log( )] log( ) ˆ log( ) MSE[log( )] MSE[log( )]

re as as re

as re

QQ QQ Q Q Q⋅+ ⋅ = +
C. The optimal weighted estimate, 

 

log(Q ˆ ), and the 

corresponding MSE.

The WIE computer program is available as a package 

that includes an executable program file, detailed instructions for use, and other supporting files. The package 

containing the compressed files for the WIE program is 

available for download at http://water.usgs.gov/usgs/osw/

swstats/freq.html.

The executable program wie.exe calculates the variances from the two independent estimates, the weighted 

average peak streamflow, and the associated variance. 

The program must be executed in a disk operating system 

(DOS) or Microsoft Windows system. To execute the 

WIE program after downloading, enter the program’s 

name (wie.exe) in a DOS window or double-click the left 

mouse button. The user will be prompted to input data 

from the LP3 analysis of the at-site streamgage (results 

from PEAKFQ) and from a regional regression equation 

(results from WREG). Program results are displayed on 

the screen. All program results and inputs also are saved 

in a single output file named wie_out.txt and filed in the 

same directory or folder as the location of the executable 

program file.

Example Sessions for the WIE Program

In the subsequent paragraphs, two example sessions are given for the WIE program. The first example 

addresses the variance at a specific streamgage (Task B.1), 

and the second example addresses the average variance of 

the regional regression model (Task B.2). Data from the 

same streamgage are used in both examples.

A computer session for calculating the estimate using 

the variance at a specific streamgage is shown in figure 1; 

the numbers shown in red are entries specified by the user 

and are required to execute the program. The program 

begins by collecting data needed to calculate the variance 

from the PEAKFQ analysis. Next, WIE calculates and 

displays the peak streamflow estimate and the variance on 

the basis of the at-site streamgage data and the statistics 

that were entered manually. Finally, the program requests 

information needed from the regional regression equation to calculate the variance at the given streamgage 

(Task B.1). For this task, the user is asked to enter (1) the 

peak flow estimate at the streamgage from the regression 

model, (2) the average standard error of the regression 

model, (3) the number of explanatory variables in the 

regression model, (4) the covariance matrix T 1 (X X) − Λ of 

the regression model, row by row, and (5) the values of 

the explanatory variables in log space. After these data are 

entered, the variance at the given streamgage is calculated 

as well as the weighted average peak streamflow estimate 

and the associated variance (weighted average). The 

upper and lower 95-percent confidence intervals also are 

calculated and displayed.

The second example is similar to the first except that 

the average variance of the regional regression model 

(Task B.2) is used. Figure 2 displays a computer session 

for the second example. Data from the PEAKFQ analysis 

in example 1 were also used in the second example; 

however, data that reflect calculations using the average 

variance of the regional regression model were entered. 

For the second example, the user is asked to enter the peak 

streamflow estimate from the regional regression equation 

and the standard error of prediction. The program then 

calculates the variance of the regional regression model 

and the weighted average peak streamflow estimate and 

the associated variance. Confidence intervals for the upper 

and lower 95-percent also are calculated and displayed.

References

Cohn, T.A., Lane, W. L., and Stedinger, J.R., 2001, 

Confidence intervals for Expected Moments Algorithm 

flood quantile estimates: Water Resources Research, 

v. 37, no. 6, p. 1695–1706.

Eng, Ken, Chen, Yin-Yu, and Kiang, J.E., 2009, User’s 

guide to the Weighted-Multiple-Linear Regression 

Program (WREG version 1.0): U.S. Geological Survey 

Techniques and Methods, book 4, chap. A8, 40 p.

Flynn, K.M., Kirby, W.H., and Hummel, P.R., 2006, 

User’s manual for program PeakFQ, annual floodfrequency analysis using Bulletin 17B guidelines: 

U.S. Geological Survey Techniques and Methods, 

book 4, chap. B4, 42 p.

Interagency Advisory Committee on Water Data, 1981, 

Guidelines for determining flood flow frequency: 

Bulletin 17B of the Hydrology Subcommittee, Office 

of Water Data Coordination, U.S. Geological Survey, 

Reston, Va., revised March 1982, 183 p.

Stedinger, J.R., and Tasker, G.D., 1986, Regional hydrologic analysis, 2. Model-error estimators, estimation 

of sigma and log-Pearson Type III distributions: Water 

Resources Research, v. 22, no. 10, p. 1487–1499.

For questions about calculating weighted estimates, please contact

h2osoft@usgs.gov

By Timothy A. Cohn, Charles Berenbrock, 

Julie E. Kiang, and Robert R. Mason, Jr.
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 

 Weighting of Independent Estimates from 

 PeakFQ and Regional Regression Model 

 

 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 ~~~~~~~~~~~~~~~~~~~~~~~

 Enter PeakFQ information

 -----------------------

 Enter station number

13186000 

 Enter number of observations

 53

 Enter fitted parameters of M, S, G (Log units)

 3.628 0.1789 -0.361 

 Enter skew type (W=Weighted, S=Station, G=Generalized)

 Please enter a W, S, or G

W

 Enter mse of generalized skew

0.150

 Enter non-exceedance probability to be estimated

 (for example, 0.01 for the 1-percent chance flood)

 0.010

 estimate = 9923. (cfs)

 log of estimate = 3.9967 

 variance of estimate (log) = 0.22620E-02

 ---------------------------------------

 Enter Regional Regression Equation info

 ---------------------------------------

 Enter regression estimate (cfs)

 11451.

 Depending on availability of RRE data, select the

 appropriate method to use to calculate the

 variance of the RRE

 (1) Station (At-Site) predictors

 Need: U matrix, Xo matrix, and SEM

 (2) Basin wide average predictors

 Need: SEP

 Enter 1 or 2:

 1

 Enter the average standard error of model (SEM):

 0.1588 

 What is the type and/or units of SEM?

 1--log units (average)

 2--percent (average)

 3--percent (upper or lower)

 1

 Calculated variance of model (SEM) = 0.25217E-01

 How many explanatory variables are in the RRE?

 3

 Input all U matrix values for row 1

 0.33276 -0.10035E-01 -0.18763 -0.20337E-01

 Input all U matrix values for row 2

 -0.10035E-01 0.21217E-02 0.53802E-02 -0.22203E-02

 Input all U matrix values for row 3

 -0.18763 0.53802E-02 0.11659 -0.38839E-03

 Input all U matrix values for row 4

 -0.20337E-01 -0.22203E-02 -0.38839E-03 0.20716E-01

 Input the explanatory values for the site in log units

 Input explanatory value 1 (log units)

 2.8073 

 Input explanatory value 2 (log units)

 1.5406 

 Input explanatory value 3 (log units)

 1.3522 

 Calculated variance of sampling error (log) = 0.26747E-02

 Total variance of prediction of RRE (log) = 0.27892E-01

 ----------------------

 Results from weighting

 ----------------------

 Weighted estimate (cfs) = 10030.

 log(weighted estimate) = 4.001 

 upper 95% confidence interval (cfs) 12330.

 lower 95% confidence interval (cfs) 8160.

 variance of estimate (log) = 0.20923E-02

 Press "Enter" to end program

Figure 1. A WIE screen session for calculating the weighted estimate of peak streamflow-frequency statistics 

using the variance at a specific streamgage. (Input data are in red print.)
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 

 Weighting of Independent Estimates from 

 PeakFQ and Regional Regression Model 

 

 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 ~~~~~~~~~~~~~~~~~~~~~~~

 Enter PeakFQ information

 -----------------------

 Enter station number

13186000 

 Enter number of observations

 53

 Enter fitted parameters of M, S, G (Log units)

 3.628 0.1789 -0.361 

 Enter skew type (W=Weighted, S=Station, G=Generalized)

 Please enter a W, S, or G

 W

 Enter mse of generalized skew

 0.150

 Enter non-exceedance probability to be estimated

 (for example, 0.01 for the 1-percent chance flood)

 0.010

 estimate = 9923. (cfs)

 log of estimate = 3.9967 

 variance of estimate (log) = 0.22620E-02

 ---------------------------------------

 Enter Regional Regression Equation info

 ---------------------------------------

 Enter regression estimate (cfs)

 11451.

 Depending on availability of RRE data, select the

 appropriate method to use to calculate the

 variance of the RRE

 (1) Station (At-Site) predictors

 Need: U matrix, Xo matrix, and SEM

 (2) Basin wide average predictors

 Need: SEP

 Enter 1 or 2:

 2

 Enter the average standard error of prediction (SEP):

 48.4 

 What is the type and/or units of SEP?

 1--log units (average)

 2--percent (average)

 3--percent (upper or lower)

 3

 Calculated variance of prediction (log) = 0.29390E-01

 ----------------------

 Results from weighting

 ----------------------

 Weighted estimate (cfs) = 10025.

 log(weighted estimate) = 4.001 

 upper 95% confidence interval (cfs) 12329.

 lower 95% confidence interval (cfs) 8152.

 variance of estimate (log) = 0.21003E-02

 Press "Enter" to end program

Figure 2. A WIE screen session for calculating the weighted estimate of peak streamflow-frequency statistics using 

the average variance of the regional regression model. (Input data are in red print.)