Log-Pearson Type III Distribution

Flood frequencies for streamgages are computed by fitting annual peak-flow data to statistical distributions. A widely used method for computing flood frequencies in the United States involves fitting data to the log-Pearson type III (LP3) distribution. Bulletin 17B describes methods to fit annual peak flows to an LP3 distribution based on three parameters—mean, standard deviation, and a skew coefficient—derived from logarithms (base 10) of annual peak flows to estimate flood magnitudes (IACWD, 1982). The following equation shows the relation between the probability of floods and the three parameters:

The mean, standard deviation, and coefficient of skewness can be directly computed from the logarithms of annual peak-flow data. The coefficient of skewness for an annual peak-flow dataset is needed to determine the frequency factor K_p, which is obtained from appendix 3 of Bulletin 17B (IACWD, 1982).

Skew Coefficient

The skew coefficient describes the asymmetry of a probability distribution about its mean. Computations of the skew coefficient can be affected by sample size, and small sample sizes result in higher uncertainty in the skew. Bulletin 17C recommends the use of a weighted skew derived from at-site data (station skew) and a generalized skew value to compensate for the uncertain estimates of skew related to insufficient samples sizes (England and others, 2018). The weighted skew is computed with the following equation:

The generalized or regional skew coefficient is determined from an analysis of long-term streamgages within a study area. A map of generalized skew coefficients originally published in 1976 for the United States is available in plate 1 of Bulletin 17B; however, this map was based on an analysis of relatively few streamgages with minimal treatment of low outliers and high outliers, no adjustments for historical data, and no detailed evaluation of individual frequency curves (IACWD, 1982; Stedinger and Griffis, 2008). A more recent statewide map of generalized skew coefficients has been developed for drainage basins with relatively low-permeability soil in Nebraska by Soenksen and others (1999). The map was based on analysis of 224 streamgages in Nebraska and surrounding States using the methodology of Bulletin 17B (IACWD, 1982). For streamgages in basins with high-permeability soils (defined as average soil permeability greater than 2.5 inches per hour), an equation is presented for computing a generalized skew value (Soenksen and others, 1999).

The station skew for streamgages on unregulated reaches was weighted with a generalized skew coefficient derived from the statewide skew-isoline map produced by Soenksen and others (1999). The generalized skew was not used to weight the skew at the Willow Creek near Pierce, Nebr., streamgage (NeDWEE station 00232500) because it is affected by regulation; rather, the station skew was used.

Multiple Grubbs-Beck Test

The Grubbs-Beck test (Grubbs and Beck, 1972) is recommended by Bulletin 17B for the detection of low outliers in flood-frequency analysis (IACWD, 1982). In most annual peak-flow datasets, multiple low annual peaks may have excessive influence on the flood-frequency analysis. Cohn and others (2013) describe a generalization of the Grubbs-Beck method known as the “Multiple Grubbs-Beck test,” which allows for the identification of multiple potentially influential low flows (PILFs) using a standard procedure. The PILFs are defined as annual peak flows that meet three criteria: their magnitude is much smaller than the flood quantile of interest, they are present below a statistically significant break in the flood-frequency data plot, and they have excessive influence on the estimated frequency of large floods (Cohn and others, 2013). All values in the annual peak-flow series that are smaller than the flood identified as a PILF are also categorized as PILFs using this procedure. Bulletin 17C recommends the replacement of the Grubbs-Beck test with the more robust Multiple Grubbs-Beck test (England and others, 2018). Thus, the Multiple Grubbs-Beck test was used in the flood-frequency analysis of streamgages in the Elkhorn River Basin.

Expected Moments Algorithm

The Expected Moments Algorithm (EMA) method was used for the flood-frequency analysis of Nebraska streamgages (Cohn and others, 1997). The EMA method uses a similar approach to the recommendations of Bulletin 17B to fit the LP3 distribution to the logarithms of the annual peak flows (IACWD, 1982). The Bulletin 17B and EMA procedures yield identical results when no historic data from before the initiation of the streamgage, missing periods, or low outliers are present (Cohn and others, 1997). The EMA method requires the computation of mean, standard deviation, and skew of the log-transformed annual peak discharges (Cohn and others, 1997). The EMA procedure differs from the Bulletin 17B in the treatment of uncertain data, historical data, low outliers, and missing periods (IACWD, 1982; Cohn and others, 1997). For historical and missing periods, the EMA method uses censoring to deal with unknown peaks above or below thresholds (Cohn and others, 1997). When using the EMA method, the analyst assigns perception thresholds that describe the upper and lower bounds of the unknown or missing peaks to the gaps in the systematic record and periods between historical floods (Veilleux and others, 2014). In many cases, it is assumed that the upper perception threshold in a gap in the historical period is equal to the previous historical flood; that is, the unrecorded peaks were assumed to be lower than prior flooding events, and if the flood was regarded as impactful, it would have prompted an action to document the flooding event. The EMA method also uses a different method than Bulletin 17B for computing uncertainty related to estimates of the magnitudes and frequencies of floods. Cohn and others (2001) describe analytic expression that can be used to compute the confidence intervals for EMA estimates. The use of the EMA method is included in the recommendations in Bulletin 17C (England and others, 2018).

PeakFQ

The Peak flow FreQuency (PeakFQ) analysis software (Veilleux and others, 2014; U.S. Geological Survey, 2022) can implement the procedures of Bulletin 17B or Bulletin 17C including EMA procedures for flood-frequency analysis of streamflow records (Veilleux and others, 2014; England and others, 2018). The Multiple Grubbs-Beck test has also been implemented in PeakFQ and can be used with the standard Bulletin 17B procedures or with the Bulletin 17C EMA analysis (Veilleux and others, 2014).

The PeakFQ software was used to analyze flood frequency on the selected streamgages in the Elkhorn River Basin in Nebraska. The Multiple Grubbs-Beck test was used in PeakFQ to detect low outliers, and historic peaks were used when available. The weighted skew option was used for streamgages with annual peak flows unaffected by regulation. For the Willow Creek near Pierce, Nebr., streamgage (NeDWEE station 00232500), the station skew option was selected because that streamgage is affected by regulation, regulated peaks were used, and the beginning year was changed to the year corresponding to the water year of the first annual peak flow affected by regulation.

Low-Flow Statistics

In addition to examining peak flows, low flows were estimated for selected recurrence intervals at 21 streamgages in the Elkhorn River Basin. In contrast to peak flows for which the maximum effects often transpire upon the occurrence of the peak instantaneous flow, the greatest effects of low-flow extremes usually occur over days or weeks. Consecutive periods of daily flow are averaged over n days to create n-day flow time series for each year of record for any given streamgage. For example, the 7-day flow time series represents an average of daily flows during each 7-day period in a given streamflow record; therefore, the minimum 7-day flow for a given year represents the driest 7-day period during that year.

Time series representing the 1-day, 7-day, and 30-day flows were created for each climate year and streamgage used in this study. The values of n for the n-day flows were selected to include short duration low-flow periods (1-day flows), long duration low-flow periods (30-day flows), and durations important to regulatory standards (7-day flows) (U.S. Environmental Protection Agency, 2023). Because low-flow periods can often span water years that end in September and begin in October, it is common practice to summarize n-day flows by climate year, which is defined as the period from April 1 to March 30 and is named for the calendar year in which it ends (Kiang and others, 2018). The year associated with a computed n-day annual value is the calendar year.

Similar to peak-flow frequency analysis, the n-day time series are fit to the LP3 distribution to determine annual exceedance probability of low-flow magnitudes (Kiang and others, 2018). When used for low flows, the n-day frequency analysis is used to estimate the probability that flow will not exceed different flow levels in any given year. For example, the 7Q10 is the 7-day minimum flow that on average is not exceeded only once every 10 years. Specific low-flow frequency statistics are denoted using the xQy nomenclature, where x is the number of days over which flows are averaged, and y is the recurrence interval in years. Daily streamflow values were obtained from the NWIS database (U.S. Geological Survey, 2023) and the NeDWEE streamflow database (Nebraska Department of Water, Energy, and Environment, 2023), and the Hydrologic Toolbox software was used to calculate the n-day time series and annual exceedance probability flows for the study (Barlow and others, 2022).

Nonstationarity Analysis

Bulletin 17C recommends testing peak-flow datasets for nonstationarity (temporal trends) (England and others, 2018). The PeakFQ computer program performs a Kendall’s Tau nonparametric trend test to evaluate the statistical significance of potential trends for annual peak flows. The Hydrologic Toolbox software was used to evaluate the statistical significance of potential trends in low flows. The PeakFQ and Hydrologic Toolbox software use the Kendall’s Tau non-parametric trend test.

When interpreting the statistical test for trends, a likelihood-based approach as proposed by Hirsch and others (2015) was used instead of reporting significant trends based on arbitrary cutoff values. The trend likelihood values are calculated using the probability values (p-values) from each test with the following formula: trend likelihood=1–(p-value/2). When the trend likelihood value is between 0.85 and 1.00, the trend is categorized as “likely increasing” or “likely decreasing” and indicates there is at least an 85-percent probability of the trend occurring in that direction. Alternatively, if the trend likelihood value is between 0.70 and 0.85, the trend is categorized as “somewhat likely increasing” or “somewhat likely decreasing” and indicates there is at least a 70- to 85-percent probability of the trend occurring in that direction. For likelihood values below 0.70, the trend is classified as “about as likely as not,” meaning there is less than a 70-percent chance of the trend being either upward or downward.

The likelihood method was used because relying exclusively on traditional hypothesis testing might create a misleading impression of no trend because the result of the hypothesis test is interpreted as lacking evidence for a trend, whereas the likelihood method provides further insight in indicating the plausible chance that a trend has occurred.

Tables presenting the nonstationarity results contain the p-value and slope for each test so that a more traditional interpretation of the trends can be made if desired. In previous studies a p-value of less than or equal to 0.05 was used to determine if a trend was significant (Dietsch and others, 2009; Dietsch and Strauch, 2020).

Peak-Flow Frequency

Results of the peak-flow analysis using PeakFQ are shown (app. 1). The period of record, number of peaks used, and peak-flow magnitudes for each streamgage are listed in table 1.1. Generalized skew values and weighted skews are also shown. The 95-percent confidence limits for the EMA estimates also are presented. The estimates of flood magnitudes and frequencies in most cases were similar to those computed by Soenksen and others (1999).

The results of the flood-frequency analysis can be used to estimate the flood magnitudes and frequencies at streamgages in the study. Further analysis is needed at ungaged stations to provide estimates of magnitudes and frequencies of floods. Regression equations developed from basin characteristics derived from a GIS framework such as StreamStats (Ries and others, 2017) would aid in the estimation of magnitudes and frequencies of floods at ungaged stations.

Low-Flow Statistics Results

The recurrence interval and corresponding nonexceedance flow for the 1-day, 7-day, and 30-day low-flows are available for each station in table 1.2. The 95-percent confidence limits for each nonexceedance flow are also available in table 1.2. If input flow values contained “0” or censored flow values, a confidence interval could not be calculated, and that confidence limit is reported as “N/A” (Kiang and others, 2018). Estimates of the low flows ranged from 0 to 637.48 cubic feet per second (ft³/s), 0.016 to 747.6 ft³/s, and 0.212 to 924.47 ft³/s for the 1-, 7-, and 30-day low flows, respectively.

Nonstationarity Analysis Results

Results of the peak-flow nonstationarity analysis are shown in table 2 and figure 2. Of the 23 streamgages analyzed for peak-flow frequency, 4 showed trends that were likely increasing for annual peak flows, whereas 3 indicated trends that were somewhat likely increasing. For 11 streamgages, the trend was categorized as about as likely as not. Additionally, 2 streamgages exhibited trends that were somewhat likely decreasing, and 3 streamgages showed trends that were likely decreasing. Nonstationarity in peak flows may be caused by a single factor (flow regulation, land-use change, modifications to water infrastructure, or climate); however, nonstationarity is more likely the result of a combination of several anthropogenic influences (Vogel and others, 2011).

Results of the low-flow nonstationarity analysis for the 1-, 7-, and 30-day time series for 21 streamgages are shown in table 3 and figure 3. In the 1-day low-flow time series, 13 streamgages exhibited a trend likelihood categorized as likely increasing, 2 were identified as somewhat likely increasing, 5 were deemed about as likely as not, none showed trends that were somewhat likely decreasing, and 1 streamgage demonstrated a likely decreasing trend. In the 7-day low-flow time series, 13 streamgages exhibited a trend likelihood categorized as likely increasing, 4 were identified as somewhat likely increasing, 2 were deemed about as likely as not, 1 showed a somewhat likely decreasing trend, and 1 streamgage demonstrated a likely decreasing trend. Lastly, for the 30-day low-flow time series, 14 streamgages exhibited a trend likelihood categorized as likely increasing, 1 was identified as somewhat likely increasing, 4 were deemed about as likely as not, 1 showed a trend that was somewhat likely decreasing, and 1 streamgage demonstrated a likely decreasing trend. Spatially, the nonstationarity analysis results indicated likely increasing or somewhat increasing trend likelihoods for the 1-day, 7-day, and 30-day low-flows for many of the Elkhorn streamgages downstream from the Elkhorn River at Ewing, Nebr., streamgage (station number 06797500) and on eastern tributaries during the period of record (fig. 3). Although further analysis of the reasons for the increasing trends were outside the scope of the study, Dietsch and Strauch (2020) used streamflow data collected during 1961 through 2011 at Elkhorn River, Salt Creek, and Platte River Basin sites in Nebraska to determine that precipitation was not a significant factor in increasing streamflow trends and trends were more frequent in streamgages with a period of record that began before 1961. The only streamgage with a likelihood of likely decreasing for the 1-day, 7-day, and 30-day low flows was Willow Creek near Pierce, Nebr., streamgage (NeDWEE station 00232500), which is located immediately downstream from the outlet of a reservoir that was completed in 1984.