Rouzbeh Berton
^{1}^{,}*
^{}
Vahid Rahmani
^{2}

Author Information

Stantec Consulting Inc., 410 17th St #1500, Denver, CO 80202, USA

Kansas State University, 1016 Seaton Hall, 920 N. Martin Luther King, Jr. Drive, Manhattan, KS 66506, USA

*

Authors to whom correspondence should be addressed.

Received: 10 June 2024 Accepted: 12 August 2024 Published: 19 August 2024

© 2024 The authors. This is an open access article under the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).

ABSTRACT:
Nonstationarity
due to climate variation and anthropogenic disturbances has altered high flow
regimes. However, the extent of change has not been evaluated for undisturbed
versus disturbed watersheds. This article aimed to determine how partitioning
watersheds into undisturbed and disturbed categories can improve the
performance of probability distributions for flood analysis throughout the
United States. We utilized peak flow information for 26 reference (undisturbed)
and 78 nonreference (disturbed) watersheds with drainage areas ranging from 135
to 42,367 km^{2} and record lengths of 100 to 140 years. Results indicated that flood quantile
estimates of the Log Pearson Type III (LP3) distribution were likely being
overestimated for return periods of 2 to 10 years, while flood estimates of 50
years and higher might be underestimated. In contrast, the Generalized Extreme
Value (GEV) distribution outperformed LP3 in estimating floods with return
periods of 50 years or more. These findings enhance flood frequency analysis
and forecasting under nonstationary conditions.

Keywords:
Nonstationary
Climate; Flood Frequency Analysis; Reference vs. Nonreference Sites; Log
Pearson Type III (LP3); Generalized Extreme Value (GEV); Annual Maximum Flood

Globally, total flood-related financial damages are projected to increase to $52 billion by 2050, up from $6 billion in 2005 [1]. The total property damage and loss of life due to the 2014 flooding events across the United States were estimated to be almost $3 billion, accounting for 36% of the overall average during the 30-year period from 1984 to 2013 [2]. A flood risk assessment study for Los Angeles, California indicated that a projected 100-year flood could potentially affect a median of 425,000 people and result in a median economic loss of $56 billion [3].
Beyond the likelihood of experiencing a flood of a certain magnitude and duration, the extent of financial loss also depends on societal reactions to the event and the levels of societal vulnerability [4]. The 21st century increases in US flood damage may be attributed to rising societal vulnerability in addition to increases in precipitation and streamflow [5,6,7,8,9]. Since flood risk is a product of flood hazard, exposure, and societal vulnerability, measures such as providing more reliable flood quantiles estimate can improve societal resilience and decrease flood risk [4].
Reliable flood forecasting requires acknowledging nonstationarity in climate as well as nonstationarity due to anthropogenic disturbances such as land use/cover change, water withdrawal, river regulation, and reservoir construction [10,11,12,13,14]. To facilitate understanding of the influence of nonstationarity on flood forecasting, it is vital to study both undisturbed and disturbed watersheds [15,16]. Disturbed watersheds have experienced anthropogenic disturbances during the flood record period, while undisturbed watersheds have no or minimum disturbances [17]. Studying reference streamflow gauging stations at undisturbed watersheds along with nonreference stations at disturbed watersheds will improve regional flood frequency analysis and forecasting since the distinction in physiographic characteristics can affect flood quantile estimates [18,19,20,21].
Several studies on flood analysis across the US have used peak flow data, taking into account differences between reference and nonreference streamflow gauges due to factors such as urbanization or damming, but they did not clearly separate gauges into distinct clusters (e.g., [12,22,23,24]).
In this study, we considered two clusters of 26 reference and 78 nonreference sites with peak flow records of at least 100 years and no missing data across the contiguous US. We compared the differences between flood quantile estimates at reference sites with those at nonreference sites. We utilized two statistical distributions, Log Pearson Type III (LP3) and Generalized Extreme Value (GEV), to estimate flood quantiles. These robust, resistant, and efficient distributions have been tested extensively (e.g., [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].
This article aims to: (1) investigate whether the choice of LP3 or GEV distribution is limited by a site being reference or nonreference, and (2) determine the impact of partitioning sites into reference and nonreference on estimating the shape parameters of LP3 and GEV distributions. These objectives were studied throughout the contiguous US using the most recent peak flow data to reassess the design flood information. The results can help engineers and water managers better prepared for changes in flood frequency and magnitude in nonstationary conditions and improve infrastructure design and reservoir management.

```latexf_X\left(X\right)=\left|\beta\right|\left[\beta\left(\ell n\left(X\right)-\xi\right)\right]^{\alpha-1}\frac{exp\left[-\beta\left(\ell n\left(X\right)-\xi\right)\right]}{X \Gamma(\alpha)}```

```latex\hat{\mu}_X=\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i```

```latex\hat{\sigma}_X^2=S^2=\frac{1}{n-1}\sum_{i=1}^n\Bigl(X_i-\bar{X}\Bigr)^2```

```latex\hat{\gamma}_X=G=\frac n{(n-1)(n-2)S^3}\sum_{i=1}^n\left(X_i-\bar{X}\right)^3```

```latex\alpha=\frac4{\hat{\gamma}_X^2}```

```latex\beta=\frac2{\hat{\sigma}_x\hat{\gamma}_x}```

```latex\xi=\hat{\mu}_X-\frac\alpha\beta=\hat{\mu}_X-\frac{2\hat{\sigma}_X}{\hat{\gamma}_X}```

```latex\mu_Q=e^\xi\left(\frac{\beta}{\beta-1}\right)^\alpha```

```latex\sigma_Q^2=e^{2\xi}\left[\left(\frac{\beta}{\beta-2}\right)^\alpha-\left(\frac{\beta}{\beta-1}\right)^{2\alpha}\right]```

```latex\gamma_Q=\frac{E\left[Q^3\right]-3\mu_QE\left[Q^2\right]+2\mu_Q^3}{\sigma_Q^3}\quad\text{where}\quad E\left[Q^r\right]=e^{r\xi}\left(\frac{\beta}{\beta-r}\right)^\alpha ```

```latexX_P=\mu_Q+\sigma_QK_P\left(\gamma_Q\right) \text{where}\, K_P\left(\gamma_Q\right)=\frac{2}{\gamma_Q}\left[1+\frac{\gamma_QZ_P}{6}-\frac{\gamma_Q^2}{36}\right]^3-\frac{2}{\gamma_Q}```

```latexIf \,P>0.5\\Z_{P}=\frac{P^{0.135}-\left(1-P\right)^{0.135}}{0.1975}```

```latexq_i=\frac{i-0.375}{n+0.25}```

```latexP_i=1-q_i```

```latexF_X(X)=exp\left\{-\left[1-\frac{\kappa\left(X-\xi\right)}{\alpha}\right]^{1/\kappa}\right\}```

where

```latex\hat{\beta}_r=b_r=\frac{1}{n}\sum_{j=1}^{n-r}\frac{\binom{n-j}{r}X_j}{\binom{n-1}{r}}=\frac{1}{r+1}\sum_{j=1}^{n-r}\frac{\binom{n-j}{r}X_j}{\binom{n}{r+1}}\quad\text{thus}```

```latex\hat{\beta}_0=b_0=\bar{X}```

```latex\hat{\beta}_1=b_1=\sum_{j=1}^{n-1}\frac{(n-j)X_j}{n(n-1)}```

```latex\hat{\beta}_2=b_2=\sum_{j=1}^{n-2}\frac{\left(n-j\right)\left(n-j-1\right)X_j}{n\left(n-1\right)\left(n-2\right)}```

```latex\hat{\beta}_3=b_3=\sum_{j=1}^{n-3}\frac{\left(n-j\right)\left(n-j-1\right)\left(n-j-2\right)X_j}{n\left(n-1\right)\left(n-2\right)\left(n-3\right)}```

```latex\hat{\lambda}_1=\hat{\beta}_0```

```latex\hat{\lambda}_2=2\hat{\beta}_1-\hat{\beta}_0```

```latex\hat{\lambda}_3=6\hat{\beta}_2-6\hat{\beta}_1+\hat{\beta}_0```

```latex\hat{\lambda}_4=20\hat{\beta}_2-30\hat{\beta}_1+12\hat{\beta}_1-\hat{\beta}_0```

```latexc=\frac{2\hat{\lambda}_2}{\hat{\lambda}_3+3\hat{\lambda}_2}-\frac{\ell n2}{\ell n3}```

```latex\kappa=7.8590c+2.9554c^2```

```latex\alpha=\frac{\kappa\hat{\lambda}_2}{\left(1-2^{-\kappa}\right)\Gamma\left(1+\kappa\right)}```

```latex\xi=\hat{\lambda_1}-\frac{\alpha}{\kappa}\Big[1-\Gamma\big(1+\kappa\big)\Big]```

```latex\hat{X}_P=\xi+\frac{\alpha}{\kappa}\Big\{1-\Big[-\ell nP\Big]^\kappa\Big\}```

```latexq_i=\frac{i-0.4}{n+0.2}```

The results were presented in four sections for both reference and nonreference sites within HUC02 water resources regions of the contiguous US (Figure 1). First, we compared and contrasted the ratio of observed peak flow to estimated flood quantiles between reference and nonreference sites. Second, we identified regions where either GEV outperformed LP3 or vice versa. Third, we analyzed how partitioning sites into reference and nonreference categories influenced the spatial distribution of shape parameters for LP3 and GEV. Finally, we reported the results of goodness-of-fit tests for both distributions, categorized by whether the site was classified as reference or nonreference.
*3.1. The Ratio (R) of Observed to Estimated (Obs/Est) Flood Quantiles for Reference and Nonreference Sites*
The increasing flood frequency and magnitude across the US [61,62], have raised concerns about selecting the appropriate flood quantile estimates [12,16,63]. To investigate how disturbances affect estimated flood quantiles using LP3 or GEV, we calculated the ratio of observed peak flows to the corresponding estimated quantiles at both reference and nonreference sites for the return periods ranging from 2 to 200 years. A ratio of “Obs/Est < 1” indicates a good fit between the distribution and peak flow observations (i.e., overestimating floods). Conversely, when “Obs/Est > 1” (i.e., underestimating floods), it may raise concerns about the LP3 or GEV distributions’ capability to accurately capture floods of a specific return period (Table 1).
**Table 1. **Percentage of reference (out of 26) and nonreference (out of 78) sites for the ratio (R) of observed peak flows to the corresponding flood quantile estimates (Obs/Est). The ratio of Obs/Est < 1 represents the overestimation of flood while flood is underestimated when Obs/Est > 1.
The results showed that for reference sites, the performance of the LP3 distribution in capturing observed peak flows improved as the return period decreased. LP3 performed particularly well in estimating low flood quantiles with return periods of 2 to 25 years. Conversely, for flood quantiles with return periods of 50 to 200 years, GEV outperformed LP3 (Table 1). Both GEV and LP3 underestimated (R > 1) observed peak flows with return periods of 50 to 200 years in 42–54% and 85–100% of the reference sites, respectively (Table 1). Similar patterns were observed for nonreference sites. LP3 performed better than GEV in estimating floods with return periods of 2 to 10 years. However, for return periods of 25 to 200 years, both GEV and LP3 underestimated observed peak flows in 42–53% and 87–99% of the nonreference sites, respectively (Table 1).
*3.2. Regions Where LP3 Outperformed GEV or Vice Versa*
We observed no spatial pattern for floods with return periods of 2, 5, and 200 years at both reference and nonreference sites where LP3 failed to capture the observed peak flow (Figure 2a,c,m). LP3 performed well for floods with a return period of 10 years at reference sites on the West coast and nonreference sites in the Northeast (Figure 2e). However, LP3 estimates for return periods of 50 and 100 years matched the observed quantiles at only 10% of the study sites (Figure 2i,k). Wang et al. (2011) [39] also emphasized the superiority of the LP3 distribution over GEV in non-dam-affected sites in California. Reinders and Munoz (2024) [21] also suggested fitting GEV to peak flows in colder and wetter climates, likely because of established high-flow regimes from year to year, while recommending the use of the log-normal distribution for dry regions, likely due to their flashy peak flow regimes.
The GEV distribution showed poor performance in estimating flood quantiles for reference sites located in the Central US (return period of 10 years, Figure 2f), West US (return period of 25 years, Figure 2h), South and West US (return period of 50 years, Figure 2j), West US (return period of 100 years, Figure 2l), and Northeast US (return period of 200 years, Figure 2n). However, GEV accurately captured observed peak flow in the Northeast and Central US for floods with the return periods of 25, 50 and 100 years (Figure 2h,j,l). Additionally, GEV performed well for lower flood quantiles of 2 years in the Northeast (Figure 2b) and 10 years in the West US (Figure 2f). Pal et al. (2022) [64] also adopted the GEV distribution to estimate events with return periods of 2, 5, 10, 25, and 50 years over northeastern US and reported satisfactory model performance.
**Figure 2. **The spatial performance of LP3 and GEV distributions. R > 1 represents the underestimation of flood while flood was overestimated when R < 1.
For nonreference sites, GEV effectively estimated flood quantiles with a return period of 200 years (Figure 2f), but it struggled to capture peak flows with return periods of 5, 10, and 25 years in the West US (Figure 2d,f,h). However, GEV performed well for smaller floods with return periods of 2 and 10 years in the West coast and Northeast, respectively (Figure 2b,f). Detailed results on the regional performance of LP3 and GEV are summarized in Table 2.
**Table 2. **Regions that LP3 was preferred over GEV or vice versa in capturing the observed peak flow in the contiguous United States. “NA” refers to no specific region identifed.
*3.3. Partitioning Sites into Reference and Nonreference Affected Shape Parameters*
The shape parameter (Equation (5)) plays a crucial role in accurately estimating flood quantiles with the LP3 distribution. The LP3 shape parameter is always positive as it equals the reciprocal of skewness squared. When skewness is close to zero, the shape parameter becomes a very large positive number. In Equation (8), if $$\beta<0$$, then $$\left(\frac{\beta}{\beta-1}\right)<1$$. When $$\left(\frac\beta{\beta-1}\right)$$ is raised to the power of $$\alpha $$ then $$\left(\frac{\beta}{\beta-1}\right)^\alpha<<1$$. Consequently, the LP3 estimated mean value (i.e., $$\mu_{\varrho}$$) becomes lower than the sample mean, and the distribution may fail to capture the observed peak flow accurately. Therefore, studying the variability of the shape parameter across the US will help identify regions where LP3 may not perform successfully.
The reference sites located in the Pacific Northwest and New England regions showed the highest shape parameters for LP3 (Figure 3a). For nonreference sites, the highest values of shape parameters were distributed across water resources regions east of Pacific Northwest, east of the Great Basin, Upper Colorado, and west of Missouri (Figure 3b). When considering both reference and nonreference sites together, the spatial patterns of shape parameters were primarily influenced by nonreference sites (Figure 3c).
Examining the shape parameters of the LP3 distribution without distinguishing between reference and nonreference sites can give a misleading impression of the effectiveness of LP3 in the Pacific Northwest and New England regions. When considering only reference sites (Figure 3a), floods were underestimated in the Pacific Northwest and New England despite the high shape parameters of LP3, likely due to nonstationarity in climate. However, when both reference and nonreference sites were considered together, the LP3 shape parameter decreased in these regions, resulting in flood quantile estimates that were closer to the observed peak flows.
**Figure 3. **The spatial pattern of shape parameter for LP3 distribution throughout the US for (<b>a</b>) reference sites, (<b>b</b>) nonreference sites, and (<b>c</b>) reference and nonreference sites together. The larger the shape parameter was, the lower the performance of LP3 distribution. The solid empty black circles identified regions with large shape parameters. The surface contours were interpolated among study sites via ordinary Krigging with spherical semivariogram model in ArcGIS.
The GEV distribution with a positive shape parameter ($$\kappa$$) has a finite upper bound. When $$\kappa>0$$ in Equation (31), the term $$\frac\alpha\kappa\left\{1-[-\ell nP]^\kappa\right\}$$ will not be large enough so that the summation with $$\xi$$ making it less likely to capture the observed peak flow. Conversely, a negative shape parameter corresponds to a distribution with a thicker right-hand tail [50,65]. A shape parameter of 1/3 implies that moments of order 3 and higher are infinite [24]. When $$\kappa<0$$ in Equation (31), the term $$\frac\alpha\kappa\left\{1-[-\ell nP]^\kappa\right\}$$ becomes a large number, and the summation with $$\xi$$ will more effectively estimate the observed peak flow.
At reference sites, the GEV shape parameter was predominantly positive in the southeast of the Pacific Northwest, northeast of the Great Basin, north of Upper Colorado, west of Missouri, and New England regions (Figure 4a). As mentioned earlier, a positive shape parameter ($$\kappa>0$$) in GEV distributions increases the likelihood of underestimating floods. For nonreference sites, positive shape parameters were observed in the east of the Pacific Northwest, east of the Great Basin, Upper Colorado, Great Lakes, Ohio, and Tennessee water resources regions (Figure 4b). However, without distinguishing between reference and nonreference sites, the spatial pattern of the GEV shape parameter did not precisely align with either category (Figure 4c).
**Figure 4. **The spatial pattern of shape parameter for GEV distribution throughout the US for (<b>a</b>) reference sites, (<b>b</b>) nonreference sites, and (<b>c</b>) reference and nonreference sites together. In regions with negative shape parameter, GEV performed better in capturing the observed peak flow. The solid empty black circles identified regions with large shape parameters. The surface contours were interpolated among study sites via ordinary Krigging with spherical semivariogram model in ArcGIS.
Similar to the LP3 results, combining reference and nonreference sites can lead to misleading conclusions. Regions C and D (Figure 4b) exhibited higher positive shape parameters compared to Region A (Figure 4a), suggesting a greater likelihood of underestimating flood quantiles in those regions. Region G (Figure 4c), which encompassed Regions A, C, and D showed shape parameters with magnitudes similar to Region A, indicating that GEV might estimate floods closer to observed peak flows in Region G. However, when compared to Region C and D (Figure 4b) with higher positive shape parameters and a higher chance of underestimating floods, it becomes clear that the combining results from reference and nonreference sites can potentially lead to misleading interpretations.
*3.4. Goodness-of-Fit*
The goodness-of-fit test evaluates how well a distribution fits the peak flow observations and provides insights into the explanatory power of the distribution [49]. It does not, however, ascertain the true population distribution. Empirical probabilities for LP3 and GEV distributions were estimated using the Blom’s and Cunnane’s empirical probability functions (Equations (14) and (32)). Following the approach suggested by Serago and Vogel (2018) [22], we conducted a goodness-of-fit test. The cross-correlation coefficients between observed and estimated flood quantiles, summarized in Table 3, ranged from 0.8157–0.9986, indicating high correlations for both reference and nonreference sites. It suggests that LP3 and GEV distributions can effectively estimate flood quantiles across the US.
It is important to interpret the goodness-of-fit results with caution, as the test only assesses whether LP3 or GEV distributions adequately fit the observed data and not necessarily the true population distribution. As discussed earlier, there were regions across the US where LP3 and GEV did not perform well in estimating flood quantiles. Particularly, critical flows, such as those with larger return periods, may not be accurately estimated by these distributions. This can potentially lead to incidents and failures during the operational life of hydraulic structures, such as large dams. Examples of the goodness-of-fit tests for LP3 and GEV distributions are presented in Figure 5. These illustrations help visualize how well these distributions match the observed peak flow data.
**Figure 5. **Examples of goodness-of-fit for LP3 and GEV distributions tested at reference (<b>a</b>,<b>b</b>) and nonreference (<b>c</b>,<b>d</b>) sites. Despite the closeness of cross-correlation values to “1”, LP3 or GEV might not be able to capture high or low flows (<b>c</b>,<b>d</b>).

This study aimed to understand how spatial heterogeneity influences the choice of statistical distribution for flood frequency analysis in undisturbed and disturbed watersheds across the US. We used LP3 and GEV distributions to estimate flood quantiles at 26 reference and 78 nonreference sites throughout the contiguous US, each with more than 100 years of record. Comparing results from reference and nonreference sites provided insights into the effects of climate and anthropogenic factors such as land use/cover change, river regulation, irrigation, water withdrawal, and urbanization on flood frequency analysis.
We compared observed peak flows to estimated quantiles for return periods ranging from 2 to 200 years. For design purposes, LP3 distribution provided more reliable flood quantile estimates for return periods of 2 to 10 years, whereas caution is advised when using LP3 estimates for return periods exceeding 50 years. Conversely, GEV performed better than LP3 in estimating floods with return periods of 50 years or more. The reliability of GEV estimates for return periods of 10 years or less varied regionally and might be more effectively substituted with LP3 estimates. Spatial variations in LP3 or GEV performance corroborates the findings of Reinders and Munoz (2024) [21], indicating the importance of considering hydroclimatic properties when selecting an appropriate distribution for flood frequency analysis, rather than solely relying on LP3 as a nationally suggested distribution.
The shape parameter of the LP3 distribution exhibited spatial variability depending on whether the study site was a reference or nonreference site. In regions such as the Pacific Northwest, Great Basin, Upper Colorado, Missouri, and New England, the LP3 flood quantile estimates may not be reliable due to higher values of the shape parameters compared to other regions in the US. Conversely, in regions where LP3 flood quantile estimates were not reliable, the GEV distribution with a negative shape parameter effectively captured observed floods. Our findings emphasize the importance of distinguishing between reference and nonreference sites when reporting the shape parameters of both LP3 and GEV distributions. Accurate estimates of flood quantiles are crucial for safeguarding infrastructure throughout its service life, thereby minimizing potential loss of life, property damage, and infrastructure failures.
The goodness-of-fit tests determined whether LP3 or GEV could represent the true population distribution. However, caution is needed when interpreting high cross-correlation values from these tests, as the performance of LP3 and GEV distributions may differ significantly depending on whether the site is a reference or nonreference site.
Future studies can integrate stations with less than 100 years of record and explore record extension procedures such as MOVE [15,66]. Instead of relying solely on GEV and LP3 distributions, frequency analysis can utilize alternative statistical distributions (e.g., those available in RMC BestFit v1.0 developed by USACE, available at: https://www.rmc.usace.army.mil/Software/RMC-BestFit/). Considering partial duration series to augment data points for upper tercile flows could also be explored at stream gauge stations with less than 30 years of information (e.g., Berton and Rahmani (2024) [67]).

The following supporting information can be found at: https://www.sciepublish.com/article/pii/251, Table S1: The description of 26 reference and 78 nonreference study sites with more than 100 years of peak flow data across the contiguous United States.

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 between 2018–2019, 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.

R.B.: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Data Curation, Writing—Original Draft Preparation, Writing–Review & Editing, Visualization; V.R.: Conceptualization, Validation, Resources, Writing—Review & Editing, Supervision, Project Administration, Funding Acquisition.

Not applicable.

Not applicable.

This research was funded by the Department of Biological and Agricultural Engineering at Kansas State University through the contribution number of “19-102-J” of the Kansas Agricultural Experiment Station.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Addtionally, the views expressed in this paper are those of the authors and do not necessarily reflect the official positions or opinions of their affiliated institutions.

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