User's Manual for Program PeakFQ, Annual Flood-Frequency Analysis Using Bulletin 17B Guidelines

Tables

Table 1. Peak-flow codes used by program PeakFQ.
Table B.1.1. Specification file output keywords that apply to the entire run.
Table B.1.2. Specification file keywords that apply to a specific station.
Table B.2. WATSTORE station header record formats.
Table B.3. WATSTORE station option record formats.
Table B.4. WATSTORE peak-flow record formats.
Table B.5. WATSTORE basin characteristics record formats.
Table C.1. Attributes associated with annual peak-flow data sets.
Table C.2. Sources of attributes associated with peak-flow data sets.

SYMBOLS

Symbol, explanation:

a,	a constant characteristic of a particular plotting position
Ḡ,	generalized skew coefficient.
,	historically-adjusted station skew coefficient.
,	skew coefficient of frequency curve passing through *0.50. *0.10, and *0.01.
|G|,	absolute value of the station-skew coefficient.
G,	station skew coefficient.
GW,	weighted-skew coefficient estimate used in final log-Pearson Type III frequency curve.
g,	desired skew coefficient.
H,	historical period length.
K,	confidence coefficient.
KN,	10-percent significance-level critical value for outlier test statistic for sample of size N from the normal distribution.
KH,	10-percent significance-level critical value for the outlier test statistic for sample of size H, where H is the length of the historic record period.
k,p,	ordinates for skew and exceedance probability (p) for curve passing through *0.50. *0.10, and *0.01.
kg,p,	Pearson Type III standardized ordinates for desired skew (g) and exceedance probability (p).
kp,	standard normal frequency factor for probability p.
k´p,	frequency factor for expected probability frequency curve.
k(1-α),	standard normal deviate with exceedance probability 1-alpha.
kγ,p,	the Pearson Type III frequency factor.
,	historically-weighted logarithmic mean.
,	Bulletin 17B mean.
',	mean of frequency curve passing through *0.50. *0.10, and *0.01.
,	historically-weighted rank of the mth largest observed peak.
m,	rank of the mth largest observed peak.
MSE,	mean-square error (standard error of estimate squared).
MSEḠ,	mean-square error of generalized skew coefficient.
MSEG,	mean-square error of station skew coefficient.
Ñ,	effective number of peaks above flood base, QO.
NBB,	number of peaks below the flood base, including any zeros and low outliers.
NHO,	number of high outliers.
NHP,	number of historic peaks.
NS,	number of systematic peaks.
NX,	number of peaks between QO and QH.
N,	record length in years.
n,	sample size from normal population of flood logarithms.
m,	historically-weighted probability plotting position of the mth ranked observed peak.
O,	estimated probability of a flood exceeding the flood base.
p,	exceedence probability.
p',	normal exceedance probability corresponding to k'p.
,	conditional frequency curve describing only those peaks above the flood base.
*,	intermediate unconditional frequency curve.
*p,	ordinates of the unconditional curve.
p,	final Bulletin 17B-estimated frequency curve.
QH,	historical threshold streamflow.
QO,	flood base streamflow.
s,p,	systematic frequency curve ordinate at exceedance probability p.
RMSE,	root mean square error.
S,	sample logarithmic standard deviation.
,	Bulletin 17B standard deviation.
,	historically-weighted logarithmic standard deviation.
',	standard deviation of frequency curve passing through *0.50. *0.10, and *0.01.
tn-1,	Student’s t random variate with n-1 degrees of freedom.
tn-1,p,	Student’s t quantile with n-1 degrees of freedom and exceedance probability p.
W,	weight given to systematic peaks below historical threshold.
X´,	logarithmic magnitudes of historic peaks and high outliers.
,	sample logarithmic mean.
X,	logarithmic magnitudes of systematic peaks between QO and QH.
XH,	logarithmic high-outlier test threshold.
XL,	logarithmic low-outlier test threshold.
α,	confidence level.
γ,	population skew coefficient.
μ,	population mean.
σ,	population standard deviation.
<,	less than.
>,	greater than.
≥,	greater than or equal to.

Note: Most symbols and explanations from Interagency Advisory Committee on Water Data (1982) and Lepkin and others (1979).

Abstract

Estimates of flood flows having given recurrence intervals or probabilities of exceedance are needed for design of hydraulic structures and floodplain management. Program PeakFQ provides estimates of instantaneous annual-maximum peak flows having recurrence intervals of 2, 5, 10, 25, 50, 100, 200, and 500 years (annual-exceedance probabilities of 0.50, 0.20, 0.10, 0.04, 0.02, 0.01, 0.005, and 0.002, respectively). As implemented in program PeakFQ, the Pearson Type III frequency distribution is fit to the logarithms of instantaneous annual peak flows following Bulletin 17B guidelines of the Interagency Advisory Committee on Water Data. The parameters of the Pearson Type III frequency curve are estimated by the logarithmic sample moments (mean, standard deviation, and coefficient of skewness), with adjustments for low outliers, high outliers, historic peaks, and generalized skew. This documentation provides an overview of the computational procedures in program PeakFQ, provides a description of the program menus, and provides an example of the output from the program.

Introduction

Program PeakFQ performs statistical flood-frequency analyses of annual-maximum peak flows (annual peaks) following procedures recommended in Bulletin 17B of the Interagency Advisory Committee on Water Data (1982), referred to hereinafter as Bulletin 17B. The following sections document the implementation of the Bulletin 17B guidelines in program PeakFQ. This information is intended to assist the user with selection of program options and interpretation of the program output. Program users should refer to Bulletin 17B for the complete and definitive description of the recommended procedures.

The Bulletin 17B procedures treat the occurrence of flooding at a site as a sequence of annual random events or trials. The magnitudes of the annual events are assumed to be independent random variables following a log-Pearson Type III probability distribution; that is, the logarithms of the annual peak flows are assumed to follow a Pearson Type III distribution. This distribution defines the probability that any single annual peak will exceed a specified streamflow. Given this annual exceedance probability, other probabilities, such as the probability that a future design period will be free of exceedances, can be calculated by standard methods, as described in Appendix 10 of Bulletin 17B. Program PeakFQ estimates the parameters of the log-Pearson Type III frequency distribution from the logarithmic sample moments (mean, standard deviation, and coefficient of skewness) of the record of annual flows, with adjustments for low outliers, high outliers, historic peaks, and generalized peak skew. The parameter values are used to calculate the percentage points (or quantiles) of the log-Pearson Type III distribution for selected exceedance probabilities.

Peak Streamflow Records

The U.S. Geological Survey maintains a peak-flow file in the National Water Information System (NWIS) data base. The contents and format of data retrieved from the peak-flow file are described in Appendices B.2 (Station Header Records) and B.4 (Peak-Flow Records). Program PeakFQ uses the station identification number and name to label the printed output and may use the station latitude and longitude to look up the generalized skew. The Bulletin 17B statistical computations use only the annual-peak discharge and discharge-qualification codes from the peak-flow records; the gage-height information and all information about partial-duration or secondary peaks is ignored.

The annual peak-flow data fall into two classes: systematic and historic. The systematic record includes all annual peaks observed in the course of one or more systematic gaging programs at the site. In a systematic gaging program, the annual peak is observed (or estimated) for each year of the program. Several systematic records at one site can be combined, provided that the hydrologic conditions during the periods of record are comparable. The gaps between distinct, systematic-record periods can be ignored, provided that the lack of record in the interim was unrelated to the hydrologic conditions. Thus, if a flood record was interrupted for lack of funds for data collection, the interruption could be ignored and the available data could be used as if no interruption had occurred. On the other hand, if the record was interrupted because of prolonged drought or excessive flooding, the interruption should not be ignored but rather should be used, if possible, as evidence for adding one or more estimated peaks to the systematic record. Thus, the systematic record is intended to constitute an unbiased and representative sample of the population of all possible annual peaks at the site.

In addition to the systematic record, some stations have a historic record consisting of generally isolated high-magnitude peaks that occurred outside the period of systematic data collection. In contrast to the systematic record, the historic record consists of annual peaks that would not have been observed except for a recognition that an unusually large peak had occurred. Flood information acquired from old newspaper articles, letters, personal recollections, and other historical sources almost invariably refers to floods of noteworthy, and hence extraordinary, size. Similarly, paleoflood information, determined by analysis of geological or botanical evidence, is considered historic information and almost always refers to extraordinary floods. Thus, historic records, by the conditions of their collection, form a biased and unrepresentative sample of flood experience. Despite this bias, however, the historic record can be used to supplement the systematic record provided that all historic peaks above some historic threshold have been recorded.

The systematic record also may contain one or more large-magnitude peaks for which historic information is available or which exceed some historic peaks. Such peaks are called high outliers if they are greater than the high outlier threshold. They are not considered historic peaks because they are part of the systematic record. In particular, the peak of record is not considered a historic peak if it occurred during a period of systematic data collection. Although high outliers are part of the systematic record, they also are treated in the same way as historic peaks in the historic-record adjustment procedure.

Qualification codes are assigned to some peaks to identify (1) basin or environmental conditions that may have affected the magnitude of the streamflow, (2) measurement conditions that may have affected the accuracy of the recorded value and (3) historic peaks. These codes are described in Appendix B.4 and also in the NWIS-Web Help System at URL -- http://nwis.waterdata.usgs.gov/nwis/help?codes_help#flow_qual_cd. Note that an individual peak flow can have more than one qualification code associated with it. Program PeakFQ recognizes several of these codes and uses them to control the statistical computation. For example, discharge code 4 (discharge less than indicated value, reported as the mnemonic letter code L in the PeakFQ output file) automatically triggers the zero-flow and conditional-probability adjustments. Table 1 identifies the peak-flow file qualification codes used by program PeakFQ, explains how these codes are interpreted by the program, and briefly describes how the PeakFQ program handles the associated peaks.

Table 1. Peak-flow codes used by program PeakFQ.

[NWIS--National Water Information System]

Principles of Computations

The Bulletin 17B computational analysis is illustrated in figure 1. The following sections provide an overview of the major computational steps.

Figure 1. General flow chart for flood-frequency computations (modified from Interagency Advisory Committee on Water Data, 1982).

Systematic-Record Analysis

The systematic-record analysis involves the computation of the mean, standard deviation, and coefficient of skewness (, S, and G, respectively) of the common logarithms of the annual peak flows in the systematic record. At some sites, annual peaks of magnitude zero can occur; more generally, the annual peak may occasionally fall below or be equal to some lower limit of measurement called the gage base (which usually is zero but may be greater than zero). To account for this possibility, the number of peaks below the gage base is computed in addition to the mean, standard deviation, and skewness of the logarithms of the above-base systematic peaks. The statistics of the above-base systematic peaks and the number of peaks below the gage base are used to compute the systematic-record frequency curve. If there are no zeroes or below-base peaks, the systematic-record frequency curve is computed as follows:

,

(1)

where s,p	=	systematic frequency-curve ordinate at exceedance probability p, and
kg,p	=	the Person Type III standardized ordinates for station skew g and exceedance probability p.

If there are zeroes or below-base peaks, the statistics of the above-base systematic peaks are used to define a conditional above-base systematic-record frequency curve, which is then adjusted by the conditional-probability adjustment, as described in a subsequent section and in Bulletin 17B, Appendix 5.

The systematic-record frequency curve is an initial estimate of the Bulletin 17B frequency curve. This initial estimate is adjusted to account for historic data, high and low outliers, and regional (generalized) skew information.

Outlier Tests

Peaks that depart substantially from the trend of the remaining peaks are outliers. The first adjustments of the initial frequency curve involve detecting and accounting for high and low outliers. The sequence of these tests and adjustments depends on the station skew coefficient, G, computed in the first step. Because a relatively large skew coefficient of either sign (G > +0.4 or G < -0.4) is likely to be caused by an outlier on the corresponding end of the sample, this possibility is checked first and any necessary adjustment is applied before checking for outliers on the other end. If the skew coefficient is of moderate size (-0.4 ≤ G ≤ +0.4), the existence of both high and low outliers can be checked before applying any adjustments.

Program PeakFQ tests for high outliers using the following equation:

,

(2)

Program PeakFQ does not automatically use the high-outlier test threshold in the analysis. Flood peaks considered high outliers should be evaluated in the context of the flood record at the site and at nearby sites. If the records indicate a high outlier(s) is the maximum in an extended period of time, the outlier(s) should be treated as historic data. For this case, the program requires the user to specify the high-outlier threshold and length of historic period in order for the high-outlier and historic-peak adjustment to be applied. The computations for performing the adjustment are described in the next section. If the record does not contain sufficient information to adjust for high outliers, they should be retained as part of the systematic record. For this case, no values are specified for the high-outlier threshold and length of historic period.

Program PeakFQ tests for low outliers using the following equation:

,

(3)

If an adjustment for historic data has previously been made, then the following equation is used to detect low outliers:

,

(4)

where	=	historically-weighted logarithmic mean,
KH	=	10-percent significance-level critical value for outlier test statistic for sample of size H, where H, is the length of the historic record period, and
	=	historically-weighted logarithmic standard deviation.

The computation of and is described in the next section.

The frequency curve is automatically adjusted for the effect of low outliers using the conditional probability adjustment described later.

Historic-Record Adjustment

Recalculation of the statistics of the above-base peaks is required after the detection of outliers or historic information, as specified in Appendix 6 of Bulletin 17B. The logical basis for the calculation is the following:

Historic-adjustment criterion: It is assumed that every annual peak that exceeded some historic threshold streamflow (Q_H) during the historic period (H) has been recorded as either a historic peak or a systematic peak (high outlier). In other words, the record is complete for peaks above Q_H during the time period H.

The historic period H includes the systematic-record period plus one or more years that have no systematic record. This criterion implies that the unrecorded portion of the historic period contains only peaks below the threshold Q_H. Figure 2 presents a definition sketch showing the time periods and streamflows used in the historic-record adjustment.

Figure 2. Definition sketch showing time periods and discharges used in historic recordd adjustment.

The Bulletin 17B historic adjustment, in effect, fills in the ungaged portion of the historic period with an appropriate number of replications of the below-QH portion of the systematic record. The adjustment is accomplished by weighting the below-threshold systematic peaks in proportion to the number of the below-threshold years in the historic period, as illustrated in figure 2, as follows:

,

(5)

where W is the weight to be applied to the below-threshold systematic peaks and N_S, N_HP, and N_HO are the numbers of systematic peaks, historic peaks, and high outliers, respectively. Then the effective number of peaks, Ñ, above the flood base (Q_O) is

,

(6)

where N_BB is the number of peaks below the flood base, including any zeros and low outliers.

The corresponding estimated probability of a flood exceeding the flood base is

,

(7)

Applying the historic weight W to those systematic peaks below the historic threshold Q_H (and above the flood base Q_O) yields the following formulas for the historically-weighted mean (), standard deviation (), and skewness ():

,

(8)

,

(9)

, and

(10)

in which X´ denotes logarithmic magnitudes of historic peaks and high outliers and X denotes logarithmic magnitudes of systematic peaks between the flood base Q_O and the historic threshold Q_H. These formulas are equivalent to those given in Appendix 6 of Bulletin 17B.

These formulas remain correct even if there is no historic information (in which case H = N_S), no high or low outliers, and no below-gage base peaks. Thus, these formulas are used in PeakFQ to calculate the Bulletin 17B statistics in all cases, including the calculation of the unadjusted systematic record statistics.

Conditional-Probability Adjustment

After the peak-streamflow frequency curve parameters have been determined, the historically weighted frequency curve can be tabulated. If no low outliers, zero flows, or below-gage base peaks are present, this process is simply a matter of looking up the Pearson Type III standardized ordinates, k_g,p, for the desired skew coefficient (g) and probability (p) and computing the logarithmic frequency curve ordinates by the formula:

.

(11)

When peaks below the flood base are present, however, the above calculation determines a conditional frequency curve describing only those peaks above the base. To account for the fraction of the population below the flood base, the following argument is used: the probability that an annual peak will exceed a streamflow x (above the flood base) is the probability that the peak will exceed the base at all, multiplied by the conditional probability that the peak will exceed x, given that the peak exceeds the base. The first of these factors is just the probability, _O, calculated in equation (7); the second factor is the probability on the conditional frequency curve at streamflow x. Thus the unconditional curve, *, assigns a probability [_O]p to the streamflow having exceedance probability p on the above-base curve. Conversely, an exceedance probability p on the unconditional curve * corresponds to the probability p/_O on the original above-base curve . Thus, the ordinates of the unconditional curve can be computed directly by the formula:

,

(12)

in which , , and are the logarithmic mean, standard deviation, and skew coefficient, respectively, of the above-base distribution.

Because this distribution does not have the Pearson Type III shape, it is used only as an intermediate step in constructing an equivalent Pearson Type III curve. First, the three points *0.50, *0.10, and *0.01 are computed using the above formula. Then a logarithmic-Pearson Type III curve is passed through these three points; the curve mean, standard deviation, and skew coefficient, ´, ´, and ´, respectively, are found by solving the three simultaneous equations:

,

(13)

An exact solution requires a laborious interpolation in the Pearson Type III tables; the Bulletin 17B guidelines present a direct formula based on a linear approximation. Note that ´, ´, and ´ represent the contributions of all the observed peaks, those below the base as well as those above, whereas , , and do not. The resulting unconditional frequency curve, when floods below the base have been detected, then is:

.

(14)

This defines only the part of the distribution above the flood base; the part below the flood base is not defined, and is of no practical importance.

These conditional-probability adjustments are used not only to construct the final Bulletin 17B frequency curve, but also to construct a systematic-record frequency curve that takes into account any zero flows or below-gage base peaks (but not low outliers).

Computation of Weighted Skew Coefficient

The station- (or sample-) skew coefficient, which reflects the average of the cubed deviations from the sample mean, is highly sensitive to the observations in both the upper and lower tails of the sample. As a result, the estimated station-skew coefficient and extreme-flood quantiles may be strongly affected by idiosyncrasies of the sample, and may be unrepresentative of long-term flood characteristics. To help counter this problem, Bulletin 17B uses a generalized skew, which is a skew coefficient representative of neighboring stations, as explained in a subsequent section.

The station skew and generalized skew are combined in a weighted average that is expected to be more accurate than either of its constituents. Guidelines for estimating generalized skew are given in Bulletin 17B and are summarized in a subsequent section of this manual. Program PeakFQ does not perform generalized skew estimation. Instead, program PeakFQ either looks up the generalized skew from a digitized copy of the map in Bulletin 17B or reads it from user-supplied input (see preceding section). The following paragraphs explain the weighted skew computation.

The station skew, the generalized skew, and the weighted skew are quantities that are estimated from flood records at and near the station under study. As such, they are subject to estimation errors. The error in each of the skew statistics is characterized by two properties, the expected value (bias) and the standard deviation (standard error), representing systematic errors and random-sampling variability, respectively. The random and systematic errors are combined in a single composite property called mean-square error (MSE), which is the expected value of the difference between the estimated and true values of the statistic (station, generalized, or weighted skew). The MSE is the sum of the squares of the bias and standard error. The MSE often is reported in terms of its square root, the Root Mean Square Error (RMSE), which is directly comparable to the quantity being estimated (rather than to its square) and can be expressed as a percentage. If the bias is small relative to the standard error, the RMSE is approximately equal to the standard error. Because of its wide availability and usefulness, the RMSE is used as input to program PeakFQ; the program squares the input RMSE to obtain the MSE values used in equation 15.

The station- and generalized-skew coefficients are combined in a weighted average to form a better estimate of the skew coefficient for a given watershed. Under the assumption that the generalized-skew coefficient is unbiased and independent of the station-skew coefficient, the MSE of the weighted-skew estimate is minimized by weighting the station- and generalized- skew coefficients in inverse proportion to their individual MSE's. This concept is expressed in equation 15, adapted from Tasker (1978), which is used in computing the weighted-skew coefficient:

,

(15)

The MSE (or RMSE) of the generalized skew is estimated in conjunction with the development of the generalized-skew value. In program PeakFQ, if the user does not specify a value for the generalized-skew coefficient, the value is obtained from a digitized version of Plate 1 of Bulletin 17B, and the corresponding value of MSE_Ḡ= 0.302 is used in equation 15. (The corresponding RMSE value is 0.55.) Otherwise, the user must supply the RMSE of the generalized skew as input data along with the user-supplied generalized-skew value.

The MSE of the station skew for log-Pearson Type III random variables is obtained from the results of Monte Carlo experiments by Wallis and others (1974). Their results show that the MSE of the logarithmic station skew is a function of record length and population skew. This function (MSE_G) is approximated with sufficient accuracy for use in calculating the weighted skew by the equation:

,

(16)

in which |G| is the absolute value of the station-skew coefficient (used as an estimate of population-skew coefficient) and N is the record length in years. If the historic adjustment (Bulletin 17B, Appendix 6) has been applied, the historically-adjusted skew coefficient, , and historic period, H, are used for G and N, respectively, in equation 16.

Bulletin 17B indicates that equations 15 and 16 may underestimate the weight given to the station skew if the station skew is large and the record is long, or if the magnitude differs from the generalized skew by more than 0.5. In these cases, Bulletin 17B suggests that the peak-flow data and the flood-producing characteristics of the basin be examined to determine whether greater weight should be given to the station skew.

Expected-Probability Adjustment

The final steps in the Bulletin 17B analysis, as implemented in program PeakFQ, are to compute the expected-probability frequency curve and a set of upper and lower confidence limits. These computations are optional and are intended primarily as an aid to the interpretation of the principal Bulletin 17B-estimated frequency curve given by p above.

The expected probability concept deals with the following problem. A sample of size n will be drawn from a normal population (of flood logarithms), and the flood having a specified exceedance probability p will be estimated by the quantity + k_pS, in which and S are the ordinary sample mean and standard deviation, respectively, and k_p is the standard normal frequency factor for probability p. Because it is computed from a random sample, the estimate + k_pS is a random variable, that usually will differ from the true p-probability flood. Thus, one is led to ask how the probability of another flood exceeding the estimate + k_pS compares with the specified (nominal) probability p. For a normal population, one has:

(17)

where t_n-1 is Student’s t random variate with n-1 degrees of freedom. This probability has come to be known as the “expected probability” (Beard, 1960; Bulletin 17B, Appendix 11). For nominal exceedance probabilities less than 0.50 (floods above the median), the expected probability exceeds the nominal probability. The expected probability can be made equal to the nominal probability by replacing k_p by the frequency factor:

,

(18)

in which t_n-1,p is the Student‘s t value with n-1 degrees of freedom and exceedance probability p. The visible effect of this adjustment is to increase the slope of the estimated frequency curve in proportion to the statistical variability of the sample statistics.

This normal-population result is applied to the Bulletin 17B-estimated Pearson Type III distribution with mean, standard deviation, and skew coefficient, , , and G_W, by first looking up the normal exceedance probability p´ corresponding to k´_p and, second, applying the Pearson Type III frequency factor, k_G,p´ having skew coefficient and probability, to the sample mean and standard deviation, as follows: + (k_GW,p´). Even this estimate, however, when evaluated for any particular sample, normally will misrepresent the true p-probability flood. With respect to a large number of flood records, however, the fraction of floods actually exceeding the estimated p-probability floods will be correct. Nonetheless, the Bulletin 17B guidelines specify that the basic flood-frequency curve (without expected probability) is the curve to be used for estimating flood risk and forming weighted averages of independent flood-frequency estimates.

Confidence Limits

Finally, one-sided confidence limits for the p-probability flood are computed. A one-sided confidence limit is a sample statistic—hence a random variable—having a specified probability (confidence level) of exceeding (or not exceeding) a specified population characteristic. In the Bulletin 17B analysis, these statistics are of the form + KS, where and S are the sample mean and standard deviation, respectively, after all Bulletin 17B tests and adjustments and K is a confidence coefficient chosen to satisfy the following equation:

(19)

in which α is the confidence limit, μ, σ, and γ are the population mean, standard deviation, and skew coefficient, respectively, k_γ,p is the Pearson Type III frequency factor, and the right-hand side of the inequality is the population p-probability flood. The population parameters are unknown, but constant. The idea is to find a K-value such that + KS, which can be computed from the sample, and is a random variable, will have a high probability of being an upper (or lower) bound on the unknown population p-probability flood. In any particular sample the computed value + KS may fail to bound the population characteristic, but, over a number of samples, the specified fraction, -α (or 1-α), will yield correct bounds. A value of close to unity yields upper confidence limits and a value close to zero yields lower limits. In particular, the upper 95-percent confidence limit has α = 0.95; the lower 95-percent limit has α = 0.05. The value of K is found by rearranging the probability statement as follows:

(20)

in which n is the sample size. If the underlying population were normally distributed (γ = 0), and if and S were the ordinary sample mean and standard deviation, respectively, then the random variable on the left-hand side of the inequality would have the noncentral t distribution with n-1 degrees of freedom and noncentrality parameter (k_γ,p). If the underlying population had a small skew, if the sample were large, and if the population skew coefficient, γ, could be replaced by the Bulletin 17B estimated skew coefficient, G_W, then one could assume that the variate would have approximately the noncentral t distribution. Building upon this foundation, one obtains:

,

(21)

which is the noncentral t value with exceedance probability 1-α. A standard large-sample approximation for the noncentral t distribution then yields the result:

,

(22)

in which k_(1-α) is the standard normal deviate with exceedance probability 1-α and G_W is the Bulletin 17B weighted-skew coefficient. As stated above, an α-value near unity yields upper confidence limits whereas a value near zero yields lower limits. This result is equivalent to that in the Bulletin 17B guidelines.

Probability-Plotting Positions

Probability plotting positions are estimates of the exceedance probabilities of observed peak flows. They are computed by the formula p = (m-a)/(N-2a+1) (equation 10 in Bulletin 17B), in which m is the rank of an observed peak (m = 1 for highest peak), N is the sample size, and a is a constant characteristic of a particular plotting-position formula. Bulletin 17B and Program PeakFQ use the Weibull plotting-position formula (a = 0) by default, although other a-values can be specified. The probability-plotting positions are not used in the Bulletin 17B computations, but are used in graphic displays of the observed data in relation to the fitted frequency curve.

If there is historical information, the probability-plotting positions are adjusted using the same logic that underlies the calculation of the historically-weighted mean, standard deviation, and skew coefficient. The actual sample of size N is augmented by (W-1) “virtual” copies of the observed peaks below the historic threshold to fill out the entire historic period (H). In the ranked record, each below-threshold observed peak is preceded by (W-1)/2 of its virtual copies and followed by the remaining (W-1)/2 copies. In the augmented ranked series, if there are Z peaks above the historic threshold, then the rank of the first below-threshold observed peak is Z + (W-1)/2 + 1. The rank of the second below-threshold observed peak is Z + W + (W-1)/2 + 1. In general, the historically adjusted rank of the m^th ranked observed peak is:

(23)

The historically-weighted plotting positions _m then are:

(24)

These equations are equivalent to equations 6-6 through 6-8 in Appendix 6 of Bulletin 17B. As indicated above, program PeakFQ uses the Weibull plotting-position formula (a = 0) by default, although other a-values can be specified.

Estimation of Generalized Skew

The skew of a frequency distribution has a great effect on the shape and thus the values of the distribution, particularly in the extreme tail, which is of most concern in flood-risk estimation. The skew coefficient of the station record (station skew coefficient, G) is sensitive to extreme events; thus it is difficult to obtain an accurate estimate of the skew coefficient from a small sample. The accuracy of the estimated skew coefficient can be improved by weighting the station-skew coefficient with a generalized-skew coefficient estimated by pooling information from nearby sites.

Program PEAKFQ does not perform generalized-skew estimation. The program either looks up the generalized skew in a digitized version of the map (Plate I) in Bulletin 17B or reads the generalized skew and its associated RMSE from user-supplied input. The estimation of the generalized skew is performed by the flood-frequency analyst.

The discussion in this section concerns Bulletin-17B guidelines for development of appropriate generalized-skew coefficients for flood-frequency analysis. The following discussion is modified from Bulletin 17B (p. 10-14).

Bulletin 17B includes a map (Plate I) showing generalized-skew values that may be used in the absence of detailed generalized-skew studies. This map and its corresponding MSE of 0.302 (RMSE = 0.550) were developed when Bulletin 17 was first introduced in 1976 and have not been updated.

Additional peak-flow records have become available since that time. Also, the procedures used to develop the map do not conform in all respects to Bulletin 17B. Generalized-skew estimates developed in accordance with Bulletin 17B procedures should preferably be used if available. Nonetheless, Plate I is still considered an alternative for use with Bulletin 17B for those who prefer not to develop their own generalized-skew estimates. Program PeakFQ contains a digitized version of this map, which is used if the user does not specify a generalized skew and RMS error.

The Bulletin-17B recommended procedure for developing generalized-skew coefficients requires the use of at least 40 stations, or all stations within a 100-mile radius. The stations used should have 25 or more years of record. It is recognized that in some locations, a relaxation of these criteria may be necessary. The actual procedure includes analysis by three methods: (1) skew isolines drawn on a map; (2) skew prediction equation; and (3) the mean skew coefficient from selected stations. Each of the methods is discussed separately.

To develop the isoline map, each station-skew coefficient is plotted at the centroid of its drainage basin and the plotted data are examined for any geographic or topographic trends. If a pattern is evident, then isolines are drawn and the average of the squared differences between observed and isoline values, MSE, is computed. The square root of the MSE (RMSE or RMS error) should be computed to permit a better appraisal of the expected magnitude of the discrepancy between the generalized and station skews relative to the absolute magnitude of the skews. The MSE or RMSE will be used in appraising the accuracy of the isoline map. If no pattern is evident, then an isoline map cannot be drawn and is, not considered further.

A prediction equation should be developed that would relate either the station-skew coefficients or the differences from the isoline map to predictor variables that affect the skew coefficient of the station record. These would include watershed and climatologic variables such as drainage area, channel slope, and precipitation characteristics. The prediction equation should be used preferably for estimating the skew coefficient at stations with variables that are within the range of data used to calibrate the equation. The MSE (or RMSE) should be computed as the average (or square root of the average) of the residuals between the observed station skews and the fitted relation. If the relation is fitted by linear regression, then the standard error of regression can be taken as equivalent to the RMSE. The MSE (or RMSE) will be used to evaluate the accuracy of the prediction equation.

Determine the arithmetic mean and variance (or standard deviation) of the skew coefficients for all stations. In some cases, the variability of the runoff regime may be so large as to preclude obtaining 40 stations with reasonably homogeneous hydrology. In these situations, the arithmetic mean and variance of about 20 stations may be used to estimate the generalized-skew coefficient. The drainage areas and meteorologic, topographic, and geologic characteristics should be representative of the region around the station of interest. The variance (or standard deviation) is taken as comparable to the MSE (or RMSE) and is used to appraise the accuracy of the mean value as a prediction of the skew.

Select the method that provides the most accurate estimate of the skew coefficient. Compare the MSE from the isoline map to the MSE for the prediction equation. The smaller MSE should then be compared to the variance of the data. If the MSE is significantly smaller than the variance, the method with the smaller MSE should be used and that MSE used in equation 15 to predict the weighted skew coefficient. If the smaller MSE is not significantly smaller than the variance, neither the isoline map nor the prediction equation provides a significantly more accurate estimate of the skew coefficient than the mean value. In this case, the mean skew coefficient should be used because it provides as accurate an estimate as the more complicated alternatives; the variance should be used in equation 15 for the MSE of the generalized skew MSE_Ḡ.

The accuracy of a regional generalized skew relations is generally not comparable to the accuracy of Plate I in Bulletin 17B. Whereas the average accuracy of Plate I is given, the accuracies of subregions within the United States are not given. A comparison should be made only between relations that cover approximately the same geographical area.

Computer Program

The following sections describe the computer program PeakFQ for performing the Bulletin 17B flood-frequency analysis. There are two methods for running PeakFQ: a Windows version (called PKFQWin) and a batch version (called PKFQBat).

Windows Version (PKFQWin)

The program PKFQWin provides a user interface to the PeakFQ batch model. The opening screen of the program is shown below in figure 3.

Figure 3. Example of opening screen of program PKFQWin showing the station specifications tab before an input file has been opened.

When first opened, most of the interface is disabled. The interface is designed to follow a logical procession toward running PeakFQ: Use the File:Open menu item to open a PeakFQ data file. View/edit the Station Specifications that are populated by the data. View/edit the Output Options. Click the Run PEAKFQ button and View Results. Click the Save Specs button to store a desired set of specifications for future use.

File Open

The File:Open menu item is used to open any of the file types that can be used by PeakFQ. These include:

• Standard WATSTORE format Peak Flow data, as described in Appendices B.2, B.3, and B.4. Data in this format can be found on the NWIS-Web at http://nwis.waterdata.usgs.gov/usa/nwis/peak.
• Watershed Data Management (WDM) files, as described in Appendix C.
• PeakFQ Specification (PSF) files, as described in Appendix B.1.

Selecting the File:Open menu item opens the Open PeakFQ File dialogue. As shown in figure 4, this dialogue can open any of the three file types discussed above. After opening a file, the Station Specifications tab will be populated based on the contents of the file. Initial station specification values are derived from different sources for the three file formats: WATSTORE - station header, station option, and peak-flows records; WDM - data and attributes from each station’s data set; PSF - data file (WATSTORE or WDM) plus specifications for overriding initial values.

Figure 4. Example of the File Open window in program PKFQWin, obtained by selecting Open from the File Menu.

Station Specifications

Once the selected data file has been opened and read, the Station Specifications tab is populated.

The default values on the Station Specifications tab are determined by the contents of the input file, including any WATSTORE “I” records or WDM attributes that may be present. The default values may be further updated if a PSF file was opened that contains station specifications for overriding defaults. The example in figure 5 shows many of the different options for the various fields. In particular, note how the same station may be run multiple times with different options between the runs (see Station IDs 03606500, 06600500, and 11274500 in figure 5).

Figure 5. Example of the Station Specification Tab of program PKFQWin after a file containing 8 sets of data has been opened.

By default, all stations found on the data file will be included in the analysis. If a station is not to be analyzed, the Include in Analysis? field may be changed to no by either typing “No” or double-clicking to activate the pull-down menu.

By default, the Beginning Year and Ending Year fields contain the water years of the first and last peaks, respectively, in the input file for the station. If a WATSTORE “I” record or WDM attribute is present, positive values in the Beginning Year and (or) Ending Year fields will be provided as the defaults. This will determine the period of record to be used in the calculations. These fields may be updated interactively by clicking in the desired station row and typing the new value.

The Historic Period field displays the value of any user-specified historic period that may have been present in a WATSTORE “I” record or WDM attribute. If no user-specified value has been given, then a value of zero (0) is displayed and no historic adjustment is applied during computation, the historic peaks are ignored, and any high outliers are treated as normal systematic peaks. If positive, the historic period contains the systematic record as a subset and historic adjustment will be applied during the computation. This field may be updated interactively by clicking in the desired station row and typing the new value.

The Skew Option is, by default, “Weighted” between “Station” and “Generalized” skews (WTD, Bulletin 17B weighted skew.) If a WATSTORE “I” record or WDM attribute is present with a station option code of S or G, the default skew option code will be “Station” or “Generalized.” The three options are available for selection in a pull-down menu in the Skew Option field.

The Generalized Skew is, by default, based on the station latitude and longitude using the generalized-skew map accompanying Bulletin 17B. If a WATSTORE “I” record or WDM attribtute is present and the generalized skew field is non-blank, that value will be provided as the default. This field may be updated interactively by clicking in the desired station row and typing the new value.

The Gen Skew Std Error field is, by default, 0.55, corresponding to the standard error of the generalized-skew map accompanying Bulletin 17B. If a WATSTORE “I” record or WDM attribute is present and the standard error of generalized-skew field is non-blank, that value will be provided as the default. This field may be updated interactively by clicking in the desired station row and typing the new value.

The Low-Outlier-Threshold field displays the value of any user-specified low-outlier threshold that may have been present in a WATSTORE “I” record or WDM attribute. If no user-specified value has been given, then a value of 0.0 is displayed, and the low-outlier threshold is computed by PeakFQ using the Bulletin 17-B low outlier test. Any input peaks less than the low-outlier threshold are accounted for by the conditional-probability adjustment. Occasionally, it may be necessary or appropriate to override the Bulletin-17-B low-outlier test if, for example, the test criterion is very close to one of the input peaks or if there are several very low peaks. The Low-Outlier-Threshold field may be updated interactively by clicking in the desired station row and typing the new value.

The Hi-Outlier Threshold field displays the value of any user-specified historic/high-outlier threshold that may have been present in a WATSTORE “I” record or WDM attribute. If no user-specified value has been given, then a value of 0.0 is displayed. This field is used only if the Bulletin-17-B historic-record adjustment has been specified by the user in the Historic-Period field. If a value greater than zero is displayed in the Hi-Outlier-Threshold field, that value will be used as the historic/high-outlier threshold in the Bulletin-17-B historic-record adjustment. Otherwise, the lowest historic-coded input peak will be used as the historic/high-outlier threshold. If there are no historic-coded peaks and a historic adjustment for a high outlier is needed, the user must specify the required high-outlier threshold. The Hi-Outlier-Threshold field may be updated interactively by clicking in the desired station row and typing the new value.

The Gage Base Discharge represents the lower limit of measureable flood peak at a station; this is zero (0) by default. If a WATSTORE “I” record or WDM attribute is present, a positive value in the field will be provided as the default. A negative or zero value will be ignored by the program. If positive, this gage-base discharge will supersede the gage base inferred from any “less than” NWIS qualification code (4) in the peak record. Note that this gage-base discharge is not the same as the partial-duration base discharge that may be in the station header “Y” record. This field may be updated interactively by clicking in the desired station row and typing the new value.

By default, Urban and (or) Regulated Peaks are not (“No”) included in the computations. These peaks are indicated by a “6” or “C” in the NWIS qualification code field. If a WATSTORE “I” record or WDM attribute is present, this field will default to “Yes” if the Station option field contains a “K.”

The Latitude and Longitude fields contain, by default, the values from the WATSTORE station header “H” record or the WDM attributes. They are used to compute the generalized skew if it is not entered. These fields may be updated interactively by clicking in the desired station row and typing the new value.

The Mean Square Error, Lowest Historic Peak, and Highest Systematic Peak fields are informational and cannot be modified. These values are determined from the peak record for the station.

Output Options

The Output Options tab is used to modify options for output that will be used for all of the stations processed. These include the output file name, saving additional output to other files, including additional information in the output file, plot types and styles, and confidence limits.

The Output File panel in figure 6 contains the name of the file that will be used for all regular output from the program. This includes a summary of the input data, computed results in tabular format, and any warning or error messages that may be issued. By default, the output file name will use the prefix of the input file name and have the .prt suffix. If a .psf file is used and the O FILE record is included, that file name will be used. A different name may be specified for output by choosing the Select button and entering the name for the file. See appendix D.3 for an example output file. See appendix A for information on the error and warning messages that may be written to this file.

Figure 6. Example of the Output Options tab of program PKFQWin after an input file has been opened.

Three additional types of information may be included in the regular output file by selecting the appropriate check boxes. Selecting Output Intermediate Results will result in additional messages and tabulated information that may be useful for debugging purposes. Selecting this check box is equivalent to specifying O DEBUG YES in the .psf file. Selecting Print Plotting Positions will result in the Empirical Frequency Curves table being included in the regular output; this table contains the points used to generate the annual exceedance probability plot. Selecting this check box is equivalent to specifying O PLOT PRINTPOS YES in the .psf file. Selecting Line Printer Plots results in a plot rendered using keyboard characters; this option is included for consistency with older versions of the program and is equivalent to specifying PRINTER or BOTH for O PLOT STYLE in the .psf file.

The

Additional Output

Within PKFQWin, a variety of Graphic Plot Formats are available for the annual exceedance probability plot. These include:

• Computer Graphics Metafile (CGM)
• PostScript (PS)
• Raster Bitmap (BMP)
• Windows Metafile (WMF)

By default, NONE are produced. Click on the appropriate radio button for the desired format. There will be one file for each station analysis. If the radio button for CGM, PS, or WMF is selected, temporary BMP files are generated to be used for viewing from within PKFQWin; these files are deleted at the end of the session. If the BMP format is selected, the files are retained at the end of the session.

By default, the Plotting Position used is 0.0, this is the Weibull plotting position. Other named plotting positions include Median/Beard (0.3), Bolm (0.375), Cunnane (0.4), and Gringorten (0.44). The plotting position is entered as a numeric value and is not restricted to the named values. See the description of O PLOT POSITION in appendix B.1 for a description of how the plotting position is computed.

Upper and lower Confidence Limits for the Bulletin 17B estimates are drawn on the graph and also tabulated in the output file. By default, the 95-percent confidence limits are used (0.95).

Results

The Results tab shown in figure 7 allows viewing of the various forms of PeakFQ results. These include the main output file, the additional output file (if in use), and graphic plots.

Figure 7. Example of the Results tab of program PKFQWin after the Run PEAKFQ button has been selected.

The View buttons in the Output File and Additional Output frames are used to open those files for viewing. The files will be viewed with the system’s default viewer of Text files.

The Graphs list displays the available plots from the stations that were analyzed. This list is populated only if the user selects something other than None for the Graphic Plot Format on the Output Options tab. The default base file names are the station IDs. If a station is analyzed more than once, an index is attached to the station ID to make its graph name unique. The View button under the list of graphs will cause the selected graphs to be displayed, each in a separate window.

The graphs viewed from the PKFQWin interface are in Bitmap (BMP) format. If, on the Output Options tab, the user selected another graphic format (for example, CGM, PS, WMF), the graphs will also be stored in that selected format for use outside of PeakFQWin. The Bitmap files will not be saved for later use unless that was the selected graphic format.

Save Specs

The Save Specs feature shown in figure 8 (menu option or command button) allows the user to save a set of specifications for future use. The specifications from the last PeakFQ run will be written to a PSF file. The PSF file will contain only specifications that are not the default values for the run.

Figure 8. Example of the Save Specifications File window in program PKFQWin, obtained by selecting the Save Specs button at the bottom of the PEAKFQ window.

Batch Version

The PeakFQ batch program is run from a command prompt by typing the executable file name followed by an input specification file. It may be desirable to pipe the output to a file to capture any messages. For example:

        PEAKFQ TEST2.PSF>TEST2.RUN

Paths to any of these files may also be included.

The batch program is given instructions for the run through the PeakFQ specification file (*.psf). The .psf extension is not required, but is useful for file organization. The only required elements of the specification file are the input data file and the output file. Thus, the simplest example of a specification file might look like this:

I ASCI Test2.inp
O FILE Test2.OUT

Defaults for all output and station specifications are defined in the code. These specifications may then be updated by the input data file either through Watstore “I” cards or WDM attributes. Finally, specification updates may be made through the PeakFQ specification file.

Details of all specification file elements are found in Appendix B.1.

REFERENCES CITED

Beard, L.R., 1960, Probability estimates based on small normal-distribution samples: Journal of Geophysical Research, v. 65, no. 7, p. 2143-2148.

Flynn, K.M., Hummel, P.R., Lumb, A.M., and Kittle, J.L., Jr., 1995, Users manual for ANNIE, version 2, a computer program for hydrologic data management: U.S. Geological Survey Water-Resources Investigations Report 95-4085, 211 p., http://water.usgs.gov/software/annie.html.

Interagency Advisory Committee on Water Data, 1982, Guidelines for determining flood-flow frequency: Bulletin 17B of the Hydrology Subcommittee, Office of Water Data Coordination, U.S. Geological Survey, Reston, Va., 183 p., http://water.usgs.gov/osw/bulletin17b/bulletin_17B.html.

Kirby, W.H., 1981, Annual flood frequency analysis using U.S. Water Resources Council guidelines (program J407): U.S. Geological Survey Open-File Report 79-1336-I, WATSTORE User’s Guide, v. 4, chap. I, sec. C, 56 p.

Lepkin, W.D., DeLapp, M.M., Kirby, W.H., and Wilson, T.A., 1979, National Water Data Storage and Retrieval System WATSTORE User’s Guide: U.S. Geological Survey Open-File Report 79-1336-I, v. 4, ch. I, secs. A, B, and C.

Tasker, G.D., 1978, Flood Frequency Analysis with a Generalized Skew Coefficient: Water Resources Research, v. 14, no. 2, p. 373-376.

Wallis, J.R., Matalas, N.C., and Slack, J.R., 1974, Just a Moment: Water Resources Research, v. 10, no. 2, p. 211-219.

APPENDIX A. PeakFQ Diagnostic Messages

Diagnostic messages are produced when real or potential errors are detected. The diagnostic messages included in the PeakFQ output file are substantially the same as those produced by the original J407 procedure in WATSTORE (Kirby, 1981). These messages are listed and briefly explained below.

Most of the messages have the following general format:

***iiinnns - text data

APPENDIX B. PeakFQ Data Formats

Appendix B contains detailed documentation of text files read by PeakFQ. Appendix B.1 describes the PeakFQ specification file used to run the batch program. Appendices B.2 - B.4 describe the WATSTORE standard formats used by PeakFQ.

• Appendix B.2 describes the station header records
• Appendix B.3 describes the station option records
• Appendix B.4 describes the peak-flows records

Appendix B.1. Batch Specifications

This appendix gives detailed descriptions of the PeakFQ specification (PSF) file. Running the batch version of PeakFQ, whether stand-alone or from the PKFQWin interface, requires a specification file as a command line argument. There are only two required records in PeakFQ specification files. These are the input data file and the main output file. The input data file record must start with “I”, followed by either the “ASCI” (for Watstore text files) or “WDM” (for WDM files) keyword, followed by the name of the input data file. Here are examples of each:

I ASCI Test2.inp
I WDM Test.wdm

The main output file record must start with “O”, followed by the “FILE” keyword, followed by the output file name. For example:

O FILE Test2.out

Other output specification records (also starting with “O”), are used to define output options that apply to the entire run. These specifications are described in table B.1.1.

Table B.1.1. Specification file output keywords that apply to the entire run.

Note: Each keyword is preceded by the letter O and a space.

The remaining specifications are made for each station being analyzed in the run. If specifications are to be made for a station, the first record must indicate the station to which the specifications apply:

STATION <staid>

where <staid> is either the alphanumeric Station ID from the WATSTORE file or the data-set number from the WDM file.

Table B.1.2 describes the available station specifications. This sequence of a STATION record followed by any desired specifications is then repeated for each station to be analyzed in the run.

Two additional keywords may be found in PSF files, particularly those generated and used by PKFQWin. These are VERBOSE and UPDATE. The VERBOSE keyword will only be found at the start of a PSF file and indicates that all possible specifications are written out in the file, even if they are the default value. The UPDATE keyword will only be found at the end of a PSF file and indicates that as PeakFQ performs the run, it should write out the PSF file in VERBOSE mode.

The following sample PSF File is written in VERBOSE mode and contains just the first station from the Test2.inp sample data file (included in program distribution.)

I ASCI TEST2.INP
O File TEST2.OUT
O Plot Style None
O Plot PrintPos Yes
O Plot Position     0.00000
O Additional None
O Debug No
O EMA No
O Confidence    0.950000
Station 03606500
   SkewOpt Weighted
   GenSkew   -0.500000
   SkewSE    0.550000
   BegYear         1897
   EndYear         1973
   HistPeriod     0.00000
   Urb/Reg No
   LoThresh     0.00000
   HiThresh     0.00000
   GageBase     0.00000
   Latitude     36.0386
   Longitude     88.2283

Table B.1.2. Specification file keywords that apply to a specific station.

Appendix B.2. Station Header Records (WATSTORE standard format)

The optional station header records are described in table B.2. These records contain some fields not read by PeakFQ; for completeness, these fields are included in the description. If latitude and longitude are not provided on an H record, either latitude and longitude or generalized skew must be input elsewhere.

If included in the input file, the H, N, and Y records must contain the station identification number. The Record identifier is required for all records in the input file. Only fields described as required or optional are read by the PeakFQ program. Example:

columns             1         2         3         4         5         6         7         8
           ----+----0----+----0----+----0----+----0----+----0----+----0----+----0----+----0
           Z                               USGS
           H 03606500      3602190881342004747017SW 6040005 205.00         380.58
records    N 03606500      BIG SANDY RIVER AT BRUCETON, TENN
           Y 03606500      2000.00

Table B.2. WATSTORE station header record formats.