Improving Low-Frequency Flood Estimation Using the Partial Duration Series Instead of the Annual Maximum Across the United States

Minimizing the risk of infrastructure damage and fatalities from flooding requires reliable estimates of flood quantiles [1- 6]. Achieving reliable flood estimation remains a challenge in the era of nonstationary climate and anthropogenically manipulated watersheds that have experienced land use/cover change, water diversion, and/or damming [7-9]. In nonstationary conditions, every unit change in climate or anthropogenic disturbances necessarily does not correspond to a unit change in peak flow [10-15]. Hence, considering multiple independent peak flow observations throughout the year, alongside the annual maximum, can improve flood estimation accuracy [16, 17].

Limited studies have incorporated the partial duration series (PDS) alongside the annual maximum flood (AMF) to estimate flood quantiles across the U.S. [18-23]. With wet years expected to become wetter [24, 25], a second or subsequent larger flow in a wet year can easily surpass the AMF in a dry year [26, 16]. Combining several independent flooding events per year with an AMF is likely to enhance flood quantile estimates, both at high and low ends, depending on the degree of watershed disturbance [19, 26-30].

Several studies have distinguished between reference and nonreference gauging stations on flood analysis across the U.S. [31-33, 30, 34]. Reference stations measure streamflow within minimally disturbed watersheds, while nonreference stations are located within watersheds with regulated streams, manmade reservoirs, water withdrawal, and/or changes in land use/cover. Studying peak flow at reference sites provides insight into how climate variation and change may affect flood quantile estimates. Conversely, the peak flow information of nonreference sites helps demonstrate how flood estimates may change under concurrent influences of changing climate and anthropogenic disturbances [35, 36].

The Log Pearson Type III (LP3) has been widely recommended as a suitable statistical distribution for analyzing flood frequency across the U.S. [37, 7, 26, 38, 39]. The LP3 distribution typically utilizes the AMF as an input [40- 42, 29]. However, as mentioned earlier, flood quantile estimates with LP3 fitted to the AMF sometimes fail to capture all observed peak flows. In such cases, using a PDS with the Generalized Extreme Value (GEV) distribution may offer improved flood quantile estimates [19, 23, 16, 27, 30].

In the following study, we aim to accomplish two main objectives: 1) develop a PDS using daily streamflow data for sites with more than 100 years of AMF records, and 2) demonstrate that the GEV distribution combined with the PDS yields better performance than the LP3 distribution with AMF for estimating flood quantiles. The reliability of flood analysis can vary geographically between reference and nonreference sites. We constructed PDSs for 24 reference and 75 nonreference sites across the U.S., each having more than 100 years of AMF observations. LP3 and GEV distributions were fitted to both the AMF and PDS to estimate floods with return periods ranging from two to 400 years. This study emphasize the advantages of incorporating the PDS into flood forecasting alongside the AMF, particularly for reference sites compared to nonreference sites. The findings of this research hold significant potential for enhancing the sustainable design of water infrastructures for regions undergoing both changing climate and anthropogenic disturbances.

LP3 and GEV are extensively utilized statistical distributions for estimating flood quantiles [17, 29]. The distribution parameters can be computed using either product or linear moment methods [43]. Typically, the parameters of the LP3 distribution are approximated using the product moment method, while those of the GEV distribution are estimated using the linear moment method [41]. In this study, we followed the LP3 and GEV formulations outlined by [44], which are briefly summarized in this section. We employed station-based skewness without applying any regional correction and developed an “R” code to compute LP3 and GEV distribution parameters.

Log Pearson Type III (LP3)

The Probability Density Function (PDF) of LP3 distribution can be expressed as follows:

Annual maximum flood (AMF)

Among 9,067 USGS gauging stations, we selected 99 stations that have more than 100 years of AMF information. Using data from the “Hydro-Climatic Data Network” (HCDN or Gages-II) [47], we classified 24 sites as reference sites situated in undisturbed watersheds, while the remaining 75 were categorized as nonreference sites located in disturbed watersheds. Reference sites experience minimal or no anthropogenic disturbances, whereas nonreference sites exhibit varying levels of land use/cover change, river regulation, and/or reservoir construction [48]. The geographical distribution of stream gauges is illustrated on Figure 1, with additional detailed information about the study sites provided in supplementary materials Table S1. Daily streamflow and peak flow data were obtained from the USGS National Water Information System (NWIS) data repository using the “dataRetrieval” [49] and “dplyr” [50] packages developed in “R.”

Partial duration series (PDS)

The PDS comprises several high flow events rather than just the largest peak flow of each year [16].To ensure compliance with the AMF assumption, it is essential that the selected floods are independent [21]. Developing a PDS from daily mean streamflow data is somewhat subjective. However, as long as the selected peaks represent independent floods and the selection process does not impose too many restrictions [51], can serve as a reliable representation of flood peaks in the region.

In this study, we initially selected flows greater than the minimum peak flow reported in the AMF. Subsequently, we identified the absolute and relative maximums. The final step involved assessing the independence of these maximums. We utilized temporal independence criteria proposed by [26], considering events spaced at least five days apart plus the natural logarithm of drainage area in square miles, as independent flooding events. The length of PDS records for each site is detailed in supplementary materials Table S1. On average, the PDS contained at least five floods in addition to the annual maximum.

Performance of LP3 and GEV

The performance of a statistical distribution in estimating flood quantiles is assessed through a goodness-of-fit test [29]. When either an LP3 or GEV distribution was fitted to the AMF or PDS, Pearson cross-correlation indicated how accurately the distribution estimated floods with different recurrence intervals. Figure 2 depicts two flood frequency models and the goodness-offit plots for the Connecticut River at Montague City, Massachusetts (USGS 01170500). While both LP3 (Figure 2a) and GEV (Figure 2b) exhibited reliable performance, as evidenced by high correlation coefficients, both distributions missed a few low-frequency floods. This occurred because the goodness of fit primarily evaluated the relative performance of the LP3 or GEV, assigning less weight to the upper and lower tails of the distributions [52]. Therefore, the goodness-of-fit test was supplemented by the ratio (R) of observed to estimated floods to provide a more comprehensive evaluation of the performance of statistical distributions.

When the ratio fell below one, the distribution performed reliably, indicating that the estimated floods exceeded the corresponding observations. Conversely, a ratio greater than one indicated underestimated flood quantiles, suggesting poor performance of the distribution. We compared the performance of LP3 and GEV with both AMF and PDS at reference and nonreference sites for floods with return periods ranging from two to 400 years. However, in case where either LP3 or GEV performed well with both AMF and PDS, preference should be given to AMF data over PDS, as the development process of a PDS series was subjective.

Two-year flood

LP3 performed well in estimating a two-year flood event with both AMF and PDS information (Figure 3a and Table 1). For regional flood analysis in the Northeast and Pacific Northwest, considering the increase in high-frequency flooding prompted by sea-level rise [53, 54], using GEV with PDS provided better quantile estimates than AMF. However, for the Upper Colorado and northeast of the Great Basin, LP3 outperformed GEV (Figure 3a). Since LP3 demonstrated similar performance with both AMF and PDS, AMF usage was preferred (Table 1). Overall, GEV reliably estimated a two-year flood with PDS information. The distinction between reference and nonreference sites did not affect the performance of LP3, while GEV was sensitive to the conditions of the study sites (Table 1).

Table 1: The ratio (R) of observed peak flows to the corresponding flood quantiles (Obs/Est) estimated by LP3 and GEV distributions for reference and nonreference sites.

Five-year flood

LP3 performed well in estimating a five-year flood with both AMF and PDS (Figure 3b and Table 1). However, the performance of GEV was not spatially consistent. Interestingly, adding more information to a flood series through the use of the PDS alongside AMF did not improve the performance of the GEV and actually worsened it (Table 1). At equal numbers of sites, the GEV with AMF could or could not estimate floods well. Given the uncertainty surrounding the use of GEV for a five-year flood estimation, it appeared that LP3, either with AMF or PDS (with a preference for AMF), was a better choice regardless of site being reference or nonreference (Table 1).

Ten-year flood

Similarly to a five-year flood, the performance of GEV worsened when using the PDS instead of the AMF (Table 1). The LP3 distribution with PDS data should be utilized for both reference and nonreference sites, while the use of GEV distribution should be avoided. At three sites (nonreference: USGS 03284000-Kentucky and 09239500-Colorado; reference: USGS 04254500-New York), neither LP3 nor GEV performed well, even with the use of the PDS instead of the AMF (Figure 3c). Other distributions besides LP3 and GEV may improve flood estimations for these sites.

Twenty-five-year flood

in the mid-Atlantic, neither LP3 nor GEV distributions performed well (Figure 3d). Since in that cluster, the number of reference sites was greater than nonreference sites, it might imply an increase in climate nonstationarity for higher flood quantiles in the Northeast [55], where LP3 and GEV failed to capture the peak flows. A smaller number of nonreference sites compared to reference sites, where neither LP3 nor GEV were fitted to peak flow observations, indicated how climate nonstationarity could be masked by anthropogenic disturbances in the northeastern U.S. [56].

Fifty-year flood

It appeared that GEV could perform well for flood estimations in the Northeast (Figure 3e). Since neither LP3 nor GEV performed well at the following nonreference sites (USGS01076500-New Hampshire, 01434000-New York, 03082500-Pennsylvania, and 08313000-New Mexico), utilizing other distributions besides LP3 and GEV is suggested (Figure 3e). With all those sites being nonreference, it might indicate that, depending on the region of interest, anthropogenic nonstationarity might echo climate nonstationarity [56].

The 200-year floods showed similar patterns to 100-year floods (Figure 3g and Table 1). The performance of LP3 was not significantly improved, even when using the PDS instead of the AMF. However, the PDS enhanced the performance of the GEV for

Four-hundred-year flood

Flood frequency analysis for a return period of 400 years was only feasible for the PDS since it comprised of at least 350 peak flow observations on average (Table S1). Regardless of whether the sites were reference or nonreference, the GEV considerably better estimated a 400-year flood than the LP3 (Figure 3h and Table 1). At a cluster of sites in the mid-Atlantic, neither GEV nor LP3 accurately estimate a 400-year flood, even when using the PDS instead of the AMF. Similar to a 200-year flood, the confounding effects of changing climate and anthropogenic disturbances were evident in the Northeast, suggesting greater uncertainties in high flood quantile forecasting.

We constructed a PDS from daily streamflow data gathered from 24 reference and 75 nonreference sites across the U.S., each site having more than 100 years of AMF records. Since the PDS provided multiple flooding events per year instead of just the maximum observed peak flow as indicated by the AMF, its inclusion in flood frequency analysis suggested potential improvement in flood quantile estimates. We applied LP3 (product moment) and GEV (L-moment) distributions to both the AMF and PDS. By comparing results between reference and nonreference sites, we aimed to differentiate between climate variations with anthropogenic nonstationarity.

We have summarized the results for distributions suitable for estimating two- to 400-year flooding events in Table 2. We observed that for high-frequency floods with recurrence intervals of two to 10 years, LP3 fitting to the AMF data performed well. However, for longer return periods, the PDS aided in improving the performance of LP3. There was no clear recommendation for an appropriate distribution for a 25-year flood. It appears this return period acted as a transition, replacing the LP3, which reliably estimated shorter period floods of two to 10 years, with the GEV for estimating longer period flooding events. For high flood quantiles of 50-400 years, the performance of GEV was significantly enhanced using the PDS.

Table 2: Suggested statistical distribution along with appropriate peak flow information for improving flood quantiles estimates across the U.S.

Sites where neither LP3 nor GEV reliably estimated flood quantiles require special attention. This could be indicative of increases in either climate or anthropogenic nonstationarity in the region. One potential solution would be to examine the performance of other distributions introduced for addressing nonstationarity [57]. Poor performance of LP3 or GEV was mainly observed in the Northeast, where climate nonstationarity has affected high flood quantiles in the region [54]. Increases in climate variability could be stimulated by Atlantic circulation patterns [58-61]. Future study should focus on understanding how variations in flood generating mechanisms can affect high and low flood quantile estimates. Reliable flood quantile estimates, considering nonstationarity, help improve flood forecasting, and reduce property damage and loss of life through enhanced infrastructure design, reservoir management, and floodplain determination.

Authors would like to acknowledge and extent their gratitude for the support provided by Kansas State University for the current research. The work was conducted from 2018 to 2019, during Dr. Berton’s tenure as a postdoctoral fellow and Dr. Rahmani’s tenure as an assistant professor in the Department of Biological and Agricultural Engineering at Kansas State University.

No Conflict of Interest

The views expressed in this publication are solely those of the authors and do not necessarily reflect the views of their either formers or current affiliations.

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Table 3: Description of 24 reference and 75 nonreference study sites with more than 100 years of peak flow information across the contiguous United States.