Effect of Streamflow Measurement Error on Flood Frequency Estimation

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Krajewski This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3837694/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 25 Mar, 2024 Read the published version in Stochastic Environmental Research and Risk Assessment → Version 1 posted 8 You are reading this latest preprint version Abstract Significant errors often arise when measuring streamflow during high flows and flood events. Such errors conflated by short records of observations may induce bias in the flood frequency estimates, leading to costly engineering design mistakes. This work illustrates how observational (measurement) errors affect the uncertainty of flood frequency estimation. The study used the Bulletin 17C (US standard) method to estimate flood frequencies of historical peak flows modified to represent the measurement limitations. To perform the modifications, the authors explored, via Monte Carlo simulation, four hypothetical scenarios that mimic measurement errors, sample size limitations, and their combination. They used a multiplicative noise from a log-normal distribution to simulate the measurement errors. They implemented a bootstrap approach to represent the sampling error. They randomly selected M samples from the total N records of the observed peak flows of four gauging stations in Iowa in central USA. The observed data record ranges between 76 and 119 years for watersheds with drainage areas between 500 and 16,000 km 2 . According to the results, measurement errors lead to more significant differences than sampling limitations. The scenarios exhibited differences with median magnitudes of up to 50%, with some cases reaching differences up to 100% for return periods above 50 years. The results raise a red flag regarding flood frequency estimation that warrants looking for further research around observational errors. Flood frequencies Uncertainty Sampling Errors Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1 Introduction Recent investments in updating the world's infrastructure against flood hazards bring to focus uncertainty in flood frequency estimation (FFE), which provides crucial information for managing flood risk and improved engineering design (e.g., Ryberg et al. 2020 ; Zhou et al. 2021 ). The significant uncertainty surrounds estimates of low probability of exceedance quantiles (e.g., Beven 2010 ; Horner et al. 2018 ). According to the literature, sampling limitations (e.g. Apel et al. 2004 ; Kjeldsen et al. 2014 ), stage-discharge errors (e.g. Di Baldassarre et al. 2012 ; Di Baldassarre and Montanari 2009 ; Kuczera 1996 ), and rating curves extrapolations (e.g. Hosking and Wallis 1986 ; Lang et al. 2010 ; Moges et al. 2021 ) are some of the significant uncertainty sources. Considering its relevance, several authors have recognized the importance of addressing uncertainty in flood frequency estimation, proposing different approaches to quantify it (Dixon et al. 2017 ; Hu et al. 2020 ; Vieira et al. 2022 ; Garcia et al. 2020; Gaume 2018 ; Kuczera 1999; Shang et al. 2021). At-site FFE is obtained by fitting historical peak flow records to an extreme value distribution. Rating curves that model the stage-discharge relationship or direct measurements collected during flood events are used to get peak flows. However, these relationships are subject to uncertainty (Horner et al. 2018 ; Vieira et al. 2022 ), leading to biases in FFE (Di Baldassarre et al. 2012 ; Neppel et al. 2010 ). Sensor overtopping during high-flow events (Fig. 1 a) (Dixon et al. 2017 ), channel cross-section changes (Fig. 1 b) (Coxon et al. 2015 ; Guerrero et al. 2012 ; Jalbert et al. 2011 ), hysteresis "loops" (Fig. 1 c) (Coz et al. 2012 ; Muste et al. 2022 ), and extrapolating rating curves (Fig. 1 d) (Moges et al. 2021 ) are some of the most common uncertainty sources sometimes happening simultaneously (Coxon et al. 2015 ). On the other hand, we also have sampling limitations. Most gauges worldwide have between 50 and 100 years of records (Bomers et al. 2019 ) and usually fewer years in undeveloped regions. The community has explored the sample size limitations using analytical and numerical approaches (e.g. Hu et al. 2020 ; Keast and Ellison 2013 ; Rahman et al. 2013 ). However, the sample size is still a dominant source of uncertainty for FFE although, according to some authors, it can only improve over time (e.g., Gaume 2018 ; Payrastre et al. 2011 ). The USGS Bulletin 17C is the official methodology in the United States for FFE analysis. Bulletin 17C uses the Expected Moments Algorithm (EMA) (Cohn et al. 2001 ; 1997 ) with regional skewness correction and stabilization (Beard 1974 ; Griffis and Stedinger 2007 ; Veilleux et al. 2011 ) to estimate the Log-Pearson type III (LP-III) parameters while accounting for local uncertainties. This work explores how measurement errors bias FFE performed following the Bulletin 17C methodology. Our experiment, a mere illustration of a larger problem, used observed annual peak flow data of four USGS streamgauges in Iowa to create scenarios representing potential measurement errors. We simulate the errors using two simple models' assuming independence and dependence on the discharge magnitude. To benchmark the relevance of the errors, we analyzed the scenarios using only 50 years of records, including one case without errors (only sampling). Finally, we computed the differences between the FFE using the Synthetic annual peak flows and the observed ones. Despite the limitation of our study to only four watersheds, the results highlight FFE differences that suggest a need to pay more attention to how measurement error propagates into our risk assessment plans and designs. The structure of this paper is as follows: Section 2 describes the methodology used to generate the scenarios that mimic sampling and measurement errors. Section 3 compares the observed and synthetic FFE and a comparison of all the synthetic scenarios. Section 4 presents the conclusions of our work and provides insights into future work. 2 Methodology We implemented our experiment using observed data at four USGS gauges in Iowa monitoring watersheds with drainage areas between 517 and 16,860 \(k{m}^{2}\) (see Fig. 2 ) and length records between 77 and 119 years. 2.1 Hypothetical uncertainty scenarios We represented the described uncertainty sources using two simple measurement error models: magnitude independent (EI) and dependent (ED). Additionally, we explore the sampling (S) limitation effect. We edited the peak flow observations in the following scenarios: EI, ED, S, and S + EI (SEI). In the EI scenario, we represented measurement errors using the formulation of Potter and Walker ( 1981 ), where the observed peaks ( \({\stackrel{\sim}{Q}}_{p}\) ), assumed in error, are related with the true peaks ( \({Q}_{p}\) ) by a multiplicative factor ( \(s\) ) that follows a log-normal distribution with \(E\left[s\right]=1\) and \(V\left[s\right]={\sigma }^{2}\) where \({\sigma }^{2}\approx 0.16\) (Riggs 1976 ). In the ED scenario, \({\sigma }^{2}\) increases with the magnitude of the observed peak flow relative to the mean peak flow ( \({\stackrel{-}{Q}}_{p}\) ) as follows: $${\sigma }^{2}=\frac{{\stackrel{\sim}{Q}}_{p}}{{\stackrel{-}{Q}}_{p}}\cdot \epsilon$$ 1 where \(\epsilon\) is the observational error (assumed as 10%) that linearly increases with the peak flow magnitude. We implemented the EI and ED scenarios for the total length of the records (TLR). Conversely, in the S scenario, we represented the shortage of records in a peak flow time series by randomly selecting \(50\) years (50Y case) out of the total \(N\) samples of the series. We implemented EI, ED, S, and S + EI (SEI) in this case. In all the combinations, we ran 1,000 realizations to obtain a robust estimation of the FFE uncertainty. 2.2 FFE comparison After generating the hypothetical peak flows ( \(Q{h}_{p,i}\) ) for each scenario and watershed, we used PeakFQ (Flynn et al. 2006 ) software to estimate the FFE of the observed and synthetic records ( \(FF{E}_{o}\) and \(FF{E}_{s}\) , respectively). Then, we compared them using the relative difference ( \(RD\) ) as follows: $$RD= \frac{F{FE}_{s,i}\left(Tr\right)-FF{E}_{o}\left(Tr\right)}{FF{E}_{o}\left(Tr\right)}$$ 4 where \({T}_{r}\) corresponds to the return period and \({i}_{th}\) the realization of a hypothetical scenario. Comparing these different realizations to the observations gives us some ideas of the potential differences. Nevertheless, the observed values themselves are still in error. Therefore, we conducted a more robust analysis that treats each realization as an observation computing \(RD\) for all the \(FF{A}_{s}\) of each scenario. 3 Results and discussion For each watershed, we generated 1,000 realizations using the total length of the records (TLR) and a random 50-year sample case (50Y). We implemented the EI and ED scenarios for the TLR case, and all added the S and SEI in the 50Y case. In the following, we describe and discuss our results. 3.1 FFE differences The error model and the sample size influence the FFE uncertainty, as illustrated in Fig. 2 for the North Raccoon River. In this case, the 1,000 realizations exhibit lower uncertainties in the EI (Figs. 2 a and c) scenario than in ED (Figs. 2 b and d) for the exceedance probabilities above 10%. We attribute the uncertainty increase to the ED's dependence on \({\stackrel{\sim}{Q}}_{p}\) magnitude. Moreover, there is also less uncertainty in the TLR case (Fig. 3 a and b) than in the 50Y case (Figs. 3 c and d). The latter is an expected result previously reported by Hu et al. ( 2020 ). In both cases, most \(FF{E}_{s}\) tend to overestimate the \(FF{E}_{o}\) at low exceedance probabilities over the high end indicating likelihood of FFE underestimation due to observational errors. The results presented in Fig. 3 correspond to one case with varying sample sizes and for different scenarios. According to the results, the sample size and the measurement errors play a crucial role in the FFE uncertainty. In both cases, we obtained \(FF{E}_{s}\) outside of the \(FF{E}_{o}\) confidence interval highlighting the methodology sensitivity to changes in the data. Nevertheless, Fig. 3 's results correspond to one case, limiting our conclusions. Therefore, we expanded the analysis to the four selected watersheds, exploring FFE differences using the RD (see Figs. 4 and 5 ). Positive RD values correspond to FFE overestimations, and negative values correspond to underestimations. We arbitrarily selected 10% in this analysis as an acceptable error range (blue band in Figs. 4 and 5 ). The distribution of the RD values (Fig. 4 ) agrees with the results we presented for the North Raccoon River case, obtaining lower relative differences in the TLR case than in the 50Y one. With 119 years, the Cedar River exhibits lower relative differences with most RD values inside the 10% error line. The Turkey and North Raccoon Rivers, with 102 and 79 years, respectively, developed considerably larger RD values (around 0.5 or 50%) for the return periods ( \({T}_{r}\) ) of 100 and 500 years. Conversely, in Salt Creek (76 years), we obtained relatively low RD median values but large dispersions. Additionally, the 50Y case significantly increases the differences over the extreme quantiles, with most scenarios tending towards overestimation. The median RD values for the EI and ED scenarios remained similar. However, we observe a significant increase in their dispersion. In contrast, the S scenario presents lower RD values with some tendency to underestimation, possibly because its values are limited to observed records. Finally, the SEI and EI scenarios exhibit similar behavior. Excluding the S scenario, we identified several cases with RD median values above 10%, with many instances close to 50%, indicating expected underestimation in FFE when we exclude possible observational errors. Additionally, we compute RD for the confidence intervals of the EI and ED scenarios in the TLR case (Fig. 5 ). The figure shows that differences are more significant for the 95% confidence interval than the 5%. Also, we noticed that both EI and ED exhibit similar RD distributions. Like Fig. 4 's results, we observe the most considerable differences between the Turkey River and the North Raccoon and the largest dispersion over Salt Creek. In contrast with Fig. 4 , the 95% confidence interval RD reaches values larger than 1 (or 100%), indicating significant overestimations on the high-end due to observational errors. Our results indicate possible underestimations of the FFE and its 95% confidence interval for the return periods above 50 years. Moreover, the four scenarios exhibit a similar behavior except for S, where we appreciate lower RD values, possibly due to the exclusion of errors in its observations. Regarding the error scenarios (EI, ED, and SEI), we observed that neglecting existing observational errors may lead to underestimations above 50% on the FFE and above 100% on the 95% confidence interval. 3.2 Comparing scenarios The results of the previous section correspond to comparisons with the available observations and, therefore, are subject to biases. In the following section, we addressed this limitation by assuming that each setup realization is a possible observed scenario. Under this assumption, we conducted pairwise comparisons among the \(F{FE}_{s}\) realizations for the different scenarios, considering the TLR and 50Y cases. According to Fig. 6 , the sample size influences the flood frequency estimation uncertainty, as Gaume ( 2018 ) and Hu et al. ( 2020 ) previously suggested. In the TLR case (first row), the probability of obtaining an RD value lower than 0.1 (10%) is significantly higher than in the 50Y case (second row). Nevertheless, the chances of getting RD values below 10% for high return periods (> 100 years) are relatively low in both cases, indicating significant limitations due to measurement errors. We identify some similarities and differences among the four scenarios. In all the cases, the probability of low RD values decreases with the return period, reaching its minimum value for \({T}_{r}=500\) . The ED scenario developed a slightly lower P(RD < 0.1) than the EI for the TLR case (first row of Fig. 6 ), likely due to the \({\sigma }^{2}\) dependence on the peak magnitude. In the 50Y case, the sampling scenario (S) develops a higher P(RD < 0.1), highlighting the relevance of considering observational errors. Nevertheless, S P(RD < 0.1) also falls below 50% for most 100- and 500-year return period cases. Conversely, the EI, ED, and SEI scenarios exhibit similar probabilities with P(RD < 0.1) below 50% for the return periods of 10 and up. In this last case, ED also shows a more evident lower P(RD < 0.1) for North Raccoon and Salt Creek. Additionally, we observe a P(RD < 0.1) decrease with the upstream area. Examining the 500-year return period in the 50Y case, the probabilities at Cedar Rapids (16,860 \(k{m}^{2}\) ) are above 25%. The same value decreases to around 25% for the Turkey River (4,000 \(k{m}^{2}\) ) and North Raccoon (2,000 \(k{m}^{2}\) ) and below 25% for Salt Creek (517 \(k{m}^{2}\) ). We appreciate a similar behavior for the 10- and 100-year return periods in both the TLR and 50Y cases. Ayalew and Krajewski (2017) suggested a possible existing link between the watershed network topology and the FFE. It is possible that the watershed network and its shape influence the annual peak flow variability and, therefore, the probabilities of obtaining more uncertain FFE. Exploring this connection is a relevant avenue of work. However, it is a question that falls outside this work's scope. The differences between the S and the error scenarios (EI and ED) indicate that flow measurement errors play a crucial role in FFE uncertainty. We observed these differences in the second row of Fig. 6 and, more specifically, in the North Raccoon case. On the other hand, the ED scenario obtained a slighter smaller P(RD < 0.1) in most cases, indicating that a magnitude-dependent error may increase the FFE uncertainty. In this case, we compared two simple error models. Nevertheless, the results highlight the relevance of including streamflow uncertainty when performing FFE. 4 Conclusions We explored the FFE uncertainty due to measurement errors in four watersheds using the Bulletin 17C guidelines. To illustrate the effect of the errors, we used two multiplicative models: one independent and one dependent of the magnitude, following the log-normal distribution proposed by Potter and Walker ( 1981 ). We evaluated the error models using the length of the records and 50-year by bootstrapping observations. We used the 50-year scenario to contrast the effect of the errors. We generated 1,000 realizations for each scenario from where we computed the hypothetical \(FFA\) realizations ( \(FF{A}_{s}\) ) using PeakFQ and obtained differences comparing with the observed \(FFA\) . We obtained differences higher than 50% in multiple cases when comparing the \(FF{A}_{s}\) with the observations ( \(FF{A}_{o}\) ). Besides, we found differences around 100% for the 100- and 500-year return periods. We explored these differences further by comparing the realizations of the hypothetical scenarios, finding significant differences for return periods above 50 years. In both cases, the error scenarios developed higher uncertainties than the bootstrapping ones, highlighting the relevance of measurement error. From the analysis of four watersheds with areas oscillating between 517 and 16,860 \(k{m}^{2}\) we observed that measurement errors increase the uncertainty in relatively small watersheds. However, our results are limited by using a purely statistical error model and the assumption of a stationary behavior. We are aware that the error structure may be more intricate. It can be discontinuous (Potter and Walker 1981 ), relying on the hydraulic and hysteretic conditions (Moges et al. 2021 ; Muste et al. 2020 ), and it is likely non-stationary. Nevertheless, our work illustrates the \(FFE\) uncertainty magnitude due to measurement errors. FFE accuracy is highly relevant as it conditions our infrastructure designs. The environmental crisis and landscape changes will likely bring more frequent floods in the coming years. These changes will also come with increased uncertainties. We can explore further how to address those uncertainties by examining our measurements. Future work should explore the development of physical-based models to represent measurement errors. In the meantime, we should opt for a conservative approach to flood frequency estimation. Declarations Funding: the Iowa Flood Center at the University of Iowa, with grants from the Iowa Department of Transportation (Grant TR-699) and the Mid-American Transportation Center (MATC). Competing interests: The authors have no relevant financial or non-financial interests to disclose. Author contribution: N.V: Numerical analysis, figures, writing, conceptual idea, W.K: Conceptual idea, writing, editing, funding. Acknowledgments: To be edited after acceptance. References Apel H, Thieken AH, Merz B, Blöschl G (2004) Natural Hazards and Earth System Sciences Flood risk assessment and associated uncertainty. Nat Hazards Earth Syst Sci 4:295–308 Beard LR (1974) Technical Report: Flood Flow Frequency Techniques. The University of Texas, Center of Research in Water Resources, Austin Beven K (2010) Environmental modelling: an uncertain future?, 1st edn. 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Cite Share Download PDF Status: Published Journal Publication published 25 Mar, 2024 Read the published version in Stochastic Environmental Research and Risk Assessment → Version 1 posted Editorial decision: Revision requested 28 Feb, 2024 Reviews received at journal 11 Feb, 2024 Reviewers agreed at journal 22 Jan, 2024 Reviewers agreed at journal 19 Jan, 2024 Reviewers invited by journal 19 Jan, 2024 Editor assigned by journal 10 Jan, 2024 Submission checks completed at journal 08 Jan, 2024 First submitted to journal 05 Jan, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3837694","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Short Report","associatedPublications":[],"authors":[{"id":265870120,"identity":"2963d420-2dfb-4864-af59-9b9c8064f76e","order_by":0,"name":"Nicolás Velasquez","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVRIiWNgGAWjYDACZiB+AGMwVDDwQITZCGhJgGs5Q4wWBpgWEGBsg7HwaDFnZ374IKGmjkHenffh48J5h2X4JZIfMHwoO4xTi2Uzm7FBwrHDDIaH2Y2NZ247zCM5I82AccY53FoMDvOwSSSwHajf2MzGJs0L1GJwO8GAmbeNkJZ/dQyGYC1zQFrSPzD/JaQlsY2ZQZ4ZpKUBpCXHgJkRrxagXxL7DjMYMLMxG/McS+eRnP+m4GDPuXTcWs4ffvjgwzdgiPUfY3zMU2Ntz89zfOODH2XWOLUg9B5A4hzAoQgVyDcQpWwUjIJRMApGIgAAf3dK8j2s32gAAAAASUVORK5CYII=","orcid":"","institution":"University of Iowa","correspondingAuthor":true,"prefix":"","firstName":"Nicolás","middleName":"","lastName":"Velasquez","suffix":""},{"id":265870121,"identity":"e7ef9e44-904a-4f46-91ce-f870634312be","order_by":1,"name":"Witold F. Krajewski","email":"","orcid":"","institution":"University of Iowa","correspondingAuthor":false,"prefix":"","firstName":"Witold","middleName":"F.","lastName":"Krajewski","suffix":""}],"badges":[],"createdAt":"2024-01-05 16:14:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3837694/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3837694/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00477-024-02707-1","type":"published","date":"2024-03-25T15:01:02+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":49439502,"identity":"0a3c0376-cec1-4c51-aa5d-f7cba3250a93","added_by":"auto","created_at":"2024-01-10 21:46:37","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":105992,"visible":true,"origin":"","legend":"\u003cp\u003eSources of uncertainties in the flood frequency estimation due to measurement errors.\u003c/p\u003e","description":"","filename":"Fig1uncertaintyerrors.png","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/17b68d560848fec3baa4f644.png"},{"id":49439506,"identity":"1b188439-8a69-48b8-8910-0d7e62c45238","added_by":"auto","created_at":"2024-01-10 21:46:38","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":5209746,"visible":true,"origin":"","legend":"\u003cp\u003eWatersheds analyzed in this study. The colors red, yellow, and blue correspond to the landforms covered by the watersheds. Black polygons represent the watershed's boundaries.\u003c/p\u003e","description":"","filename":"Fig2iowawatersheds.png","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/f437b63b50ae3fbe9aa21294.png"},{"id":49439503,"identity":"2f222046-5bdd-4b8c-b9ff-33830da9a730","added_by":"auto","created_at":"2024-01-10 21:46:38","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":867134,"visible":true,"origin":"","legend":"\u003cp\u003eNorth Raccoon Watershed 𝐹𝐹𝐸𝑜 (black line) and 𝐹𝐹𝐸𝑠(blue lines) for the TLR case (a and b) and the 50Y case (c and d). The red dashed lines correspond to the 𝐹𝐹𝐸𝑜 confidence intervals of 5 and 95% and the orange lines to the 𝐹𝐹𝐸𝑠 ones.\u003c/p\u003e","description":"","filename":"Fig3FFEExamples.png","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/553c0b78903c3e2919324a2b.png"},{"id":49439505,"identity":"18d708d3-7b30-464b-b8bc-70ba2e0da2eb","added_by":"auto","created_at":"2024-01-10 21:46:38","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":436362,"visible":true,"origin":"","legend":"\u003cp\u003eRelative difference (RD) distribution for each scenario (colors), the watershed (columns), and the number of years used out of the total number of years (rows). The violin lines correspond to the extreme values and the mean values.\u003c/p\u003e","description":"","filename":"Fig4Relativedifferences.png","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/f472c2300e5bb5a64c46a8f7.png"},{"id":49440538,"identity":"43228e74-9e1d-4803-a6dd-40ac71c9fa75","added_by":"auto","created_at":"2024-01-10 21:54:38","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":342276,"visible":true,"origin":"","legend":"\u003cp\u003eConfidence intervals of 95 (first row) and 5% (second row) relative difference for the EI and ED cases using the total length of the records.\u003c/p\u003e","description":"","filename":"Fig5Relativedifferences595percentconfidence.png","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/d4622de40807310c794aef1d.png"},{"id":49439507,"identity":"a7c6b192-620d-4812-8a81-110972732a26","added_by":"auto","created_at":"2024-01-10 21:46:38","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":588654,"visible":true,"origin":"","legend":"\u003cp\u003eProbability of obtaining an RD value lower than 0.1 for the TLR case (first row) and the 50Y case (second row). Each column corresponds to a watershed and the colors to the different scenarios.\u003c/p\u003e","description":"","filename":"Fig6allvsallProbofrelerroabove10.png","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/bd9ca72769c1924c15398fe1.png"},{"id":53869554,"identity":"deaa0b19-a086-4d7d-b7ba-1cb2b18dc145","added_by":"auto","created_at":"2024-04-01 15:10:07","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2283738,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3837694/v1/28ff8a84-1191-4e4a-8b87-017a74373691.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Effect of Streamflow Measurement Error on Flood Frequency Estimation","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eRecent investments in updating the world's infrastructure against flood hazards bring to focus uncertainty in flood frequency estimation (FFE), which provides crucial information for managing flood risk and improved engineering design (e.g., Ryberg et al. \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Zhou et al. \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). The significant uncertainty surrounds estimates of low probability of exceedance quantiles (e.g., Beven \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Horner et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). According to the literature, sampling limitations (e.g. Apel et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Kjeldsen et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2014\u003c/span\u003e), stage-discharge errors (e.g. Di Baldassarre et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Di Baldassarre and Montanari \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Kuczera \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1996\u003c/span\u003e), and rating curves extrapolations (e.g. Hosking and Wallis \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; Lang et al. \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Moges et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) are some of the significant uncertainty sources. Considering its relevance, several authors have recognized the importance of addressing uncertainty in flood frequency estimation, proposing different approaches to quantify it (Dixon et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Hu et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Vieira et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Garcia et al. 2020; Gaume \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Kuczera 1999; Shang et al. 2021).\u003c/p\u003e \u003cp\u003eAt-site FFE is obtained by fitting historical peak flow records to an extreme value distribution. Rating curves that model the stage-discharge relationship or direct measurements collected during flood events are used to get peak flows. However, these relationships are subject to uncertainty (Horner et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Vieira et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), leading to biases in FFE (Di Baldassarre et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Neppel et al. \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). Sensor overtopping during high-flow events (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea) (Dixon et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), channel cross-section changes (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) (Coxon et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Guerrero et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Jalbert et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2011\u003c/span\u003e), hysteresis \"loops\" (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ec) (Coz et al. \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Muste et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), and extrapolating rating curves (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ed) (Moges et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) are some of the most common uncertainty sources sometimes happening simultaneously (Coxon et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). On the other hand, we also have sampling limitations. Most gauges worldwide have between 50 and 100 years of records (Bomers et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) and usually fewer years in undeveloped regions. The community has explored the sample size limitations using analytical and numerical approaches (e.g. Hu et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Keast and Ellison \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Rahman et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). However, the sample size is still a dominant source of uncertainty for FFE although, according to some authors, it can only improve over time (e.g., Gaume \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Payrastre et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2011\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe USGS Bulletin 17C is the official methodology in the United States for FFE analysis. Bulletin 17C uses the Expected Moments Algorithm (EMA) (Cohn et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1997\u003c/span\u003e) with regional skewness correction and stabilization (Beard \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1974\u003c/span\u003e; Griffis and Stedinger \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Veilleux et al. \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) to estimate the Log-Pearson type III (LP-III) parameters while accounting for local uncertainties. This work explores how measurement errors bias FFE performed following the Bulletin 17C methodology. Our experiment, a mere illustration of a larger problem, used observed annual peak flow data of four USGS streamgauges in Iowa to create scenarios representing potential measurement errors. We simulate the errors using two simple models' assuming independence and dependence on the discharge magnitude. To benchmark the relevance of the errors, we analyzed the scenarios using only 50 years of records, including one case without errors (only sampling). Finally, we computed the differences between the FFE using the Synthetic annual peak flows and the observed ones. Despite the limitation of our study to only four watersheds, the results highlight FFE differences that suggest a need to pay more attention to how measurement error propagates into our risk assessment plans and designs.\u003c/p\u003e \u003cp\u003eThe structure of this paper is as follows: Section 2 describes the methodology used to generate the scenarios that mimic sampling and measurement errors. Section 3 compares the observed and synthetic FFE and a comparison of all the synthetic scenarios. Section 4 presents the conclusions of our work and provides insights into future work.\u003c/p\u003e"},{"header":"2 Methodology","content":"\u003cp\u003eWe implemented our experiment using observed data at four USGS gauges in Iowa monitoring watersheds with drainage areas between 517 and 16,860 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e (see Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and length records between 77 and 119 years.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Hypothetical uncertainty scenarios\u003c/h2\u003e \u003cp\u003eWe represented the described uncertainty sources using two simple measurement error models: magnitude independent (EI) and dependent (ED). Additionally, we explore the sampling (S) limitation effect. We edited the peak flow observations in the following scenarios: EI, ED, S, and S\u0026thinsp;+\u0026thinsp;EI (SEI). In the EI scenario, we represented measurement errors using the formulation of Potter and Walker (\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1981\u003c/span\u003e), where the observed peaks (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\stackrel{\\sim}{Q}}_{p}\\)\u003c/span\u003e\u003c/span\u003e), assumed in error, are related with the true peaks (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Q}_{p}\\)\u003c/span\u003e\u003c/span\u003e) by a multiplicative factor (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(s\\)\u003c/span\u003e\u003c/span\u003e) that follows a log-normal distribution with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(E\\left[s\\right]=1\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V\\left[s\\right]={\\sigma }^{2}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }^{2}\\approx 0.16\\)\u003c/span\u003e\u003c/span\u003e (Riggs \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e1976\u003c/span\u003e). In the ED scenario, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }^{2}\\)\u003c/span\u003e\u003c/span\u003e increases with the magnitude of the observed peak flow relative to the mean peak flow (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\stackrel{-}{Q}}_{p}\\)\u003c/span\u003e\u003c/span\u003e) as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${\\sigma }^{2}=\\frac{{\\stackrel{\\sim}{Q}}_{p}}{{\\stackrel{-}{Q}}_{p}}\\cdot \\epsilon$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e is the observational error (assumed as 10%) that linearly increases with the peak flow magnitude. We implemented the EI and ED scenarios for the total length of the records (TLR). Conversely, in the S scenario, we represented the shortage of records in a peak flow time series by randomly selecting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(50\\)\u003c/span\u003e\u003c/span\u003e years (50Y case) out of the total \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(N\\)\u003c/span\u003e\u003c/span\u003e samples of the series. We implemented EI, ED, S, and S\u0026thinsp;+\u0026thinsp;EI (SEI) in this case. In all the combinations, we ran 1,000 realizations to obtain a robust estimation of the FFE uncertainty.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 FFE comparison\u003c/h2\u003e \u003cp\u003eAfter generating the hypothetical peak flows (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Q{h}_{p,i}\\)\u003c/span\u003e\u003c/span\u003e) for each scenario and watershed, we used PeakFQ (Flynn et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2006\u003c/span\u003e) software to estimate the FFE of the observed and synthetic records (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{E}_{o}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{E}_{s}\\)\u003c/span\u003e\u003c/span\u003e, respectively). Then, we compared them using the relative difference (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(RD\\)\u003c/span\u003e\u003c/span\u003e) as follows:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$RD= \\frac{F{FE}_{s,i}\\left(Tr\\right)-FF{E}_{o}\\left(Tr\\right)}{FF{E}_{o}\\left(Tr\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{r}\\)\u003c/span\u003e\u003c/span\u003e corresponds to the return period and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({i}_{th}\\)\u003c/span\u003e\u003c/span\u003e the realization of a hypothetical scenario.\u003c/p\u003e \u003cp\u003eComparing these different realizations to the observations gives us some ideas of the potential differences. Nevertheless, the observed values themselves are still in error. Therefore, we conducted a more robust analysis that treats each realization as an observation computing \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(RD\\)\u003c/span\u003e\u003c/span\u003e for all the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{A}_{s}\\)\u003c/span\u003e\u003c/span\u003e of each scenario.\u003c/p\u003e \u003c/div\u003e"},{"header":"3 Results and discussion","content":"\u003cp\u003eFor each watershed, we generated 1,000 realizations using the total length of the records (TLR) and a random 50-year sample case (50Y). We implemented the EI and ED scenarios for the TLR case, and all added the S and SEI in the 50Y case. In the following, we describe and discuss our results.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.1 FFE differences\u003c/h2\u003e \u003cp\u003eThe error model and the sample size influence the FFE uncertainty, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e for the North Raccoon River. In this case, the 1,000 realizations exhibit lower uncertainties in the EI (Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea and c) scenario than in ED (Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb and d) for the exceedance probabilities above 10%. We attribute the uncertainty increase to the ED's dependence on \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\stackrel{\\sim}{Q}}_{p}\\)\u003c/span\u003e\u003c/span\u003e magnitude. Moreover, there is also less uncertainty in the TLR case (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea and b) than in the 50Y case (Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec and d). The latter is an expected result previously reported by Hu et al. (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In both cases, most \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{E}_{s}\\)\u003c/span\u003e\u003c/span\u003e tend to overestimate the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{E}_{o}\\)\u003c/span\u003e\u003c/span\u003e at low exceedance probabilities over the high end indicating likelihood of FFE underestimation due to observational errors.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe results presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e correspond to one case with varying sample sizes and for different scenarios. According to the results, the sample size and the measurement errors play a crucial role in the FFE uncertainty. In both cases, we obtained \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{E}_{s}\\)\u003c/span\u003e\u003c/span\u003e outside of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{E}_{o}\\)\u003c/span\u003e\u003c/span\u003econfidence interval highlighting the methodology sensitivity to changes in the data. Nevertheless, Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e's results correspond to one case, limiting our conclusions. Therefore, we expanded the analysis to the four selected watersheds, exploring FFE differences using the RD (see Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). Positive RD values correspond to FFE overestimations, and negative values correspond to underestimations. We arbitrarily selected 10% in this analysis as an acceptable error range (blue band in Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe distribution of the RD values (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) agrees with the results we presented for the North Raccoon River case, obtaining lower relative differences in the TLR case than in the 50Y one. With 119 years, the Cedar River exhibits lower relative differences with most RD values inside the 10% error line. The Turkey and North Raccoon Rivers, with 102 and 79 years, respectively, developed considerably larger RD values (around 0.5 or 50%) for the return periods (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{r}\\)\u003c/span\u003e\u003c/span\u003e) of 100 and 500 years. Conversely, in Salt Creek (76 years), we obtained relatively low RD median values but large dispersions. Additionally, the 50Y case significantly increases the differences over the extreme quantiles, with most scenarios tending towards overestimation. The median RD values for the EI and ED scenarios remained similar. However, we observe a significant increase in their dispersion.\u003c/p\u003e \u003cp\u003eIn contrast, the S scenario presents lower RD values with some tendency to underestimation, possibly because its values are limited to observed records. Finally, the SEI and EI scenarios exhibit similar behavior. Excluding the S scenario, we identified several cases with RD median values above 10%, with many instances close to 50%, indicating expected underestimation in FFE when we exclude possible observational errors.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAdditionally, we compute RD for the confidence intervals of the EI and ED scenarios in the TLR case (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). The figure shows that differences are more significant for the 95% confidence interval than the 5%. Also, we noticed that both EI and ED exhibit similar RD distributions. Like Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e's results, we observe the most considerable differences between the Turkey River and the North Raccoon and the largest dispersion over Salt Creek. In contrast with Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the 95% confidence interval RD reaches values larger than 1 (or 100%), indicating significant overestimations on the high-end due to observational errors.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eOur results indicate possible underestimations of the FFE and its 95% confidence interval for the return periods above 50 years. Moreover, the four scenarios exhibit a similar behavior except for S, where we appreciate lower RD values, possibly due to the exclusion of errors in its observations. Regarding the error scenarios (EI, ED, and SEI), we observed that neglecting existing observational errors may lead to underestimations above 50% on the FFE and above 100% on the 95% confidence interval.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Comparing scenarios\u003c/h2\u003e \u003cp\u003eThe results of the previous section correspond to comparisons with the available observations and, therefore, are subject to biases. In the following section, we addressed this limitation by assuming that each setup realization is a possible observed scenario. Under this assumption, we conducted pairwise comparisons among the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F{FE}_{s}\\)\u003c/span\u003e\u003c/span\u003e realizations for the different scenarios, considering the TLR and 50Y cases. According to Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the sample size influences the flood frequency estimation uncertainty, as Gaume (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) and Hu et al. (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) previously suggested. In the TLR case (first row), the probability of obtaining an RD value lower than 0.1 (10%) is significantly higher than in the 50Y case (second row). Nevertheless, the chances of getting RD values below 10% for high return periods (\u0026gt;\u0026thinsp;100 years) are relatively low in both cases, indicating significant limitations due to measurement errors.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe identify some similarities and differences among the four scenarios. In all the cases, the probability of low RD values decreases with the return period, reaching its minimum value for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{r}=500\\)\u003c/span\u003e\u003c/span\u003e. The ED scenario developed a slightly lower P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1) than the EI for the TLR case (first row of Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), likely due to the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }^{2}\\)\u003c/span\u003e\u003c/span\u003e dependence on the peak magnitude. In the 50Y case, the sampling scenario (S) develops a higher P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1), highlighting the relevance of considering observational errors. Nevertheless, S P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1) also falls below 50% for most 100- and 500-year return period cases. Conversely, the EI, ED, and SEI scenarios exhibit similar probabilities with P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1) below 50% for the return periods of 10 and up. In this last case, ED also shows a more evident lower P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1) for North Raccoon and Salt Creek.\u003c/p\u003e \u003cp\u003eAdditionally, we observe a P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1) decrease with the upstream area. Examining the 500-year return period in the 50Y case, the probabilities at Cedar Rapids (16,860 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e) are above 25%. The same value decreases to around 25% for the Turkey River (4,000 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e) and North Raccoon (2,000 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e) and below 25% for Salt Creek (517 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e). We appreciate a similar behavior for the 10- and 100-year return periods in both the TLR and 50Y cases. Ayalew and Krajewski (2017) suggested a possible existing link between the watershed network topology and the FFE. It is possible that the watershed network and its shape influence the annual peak flow variability and, therefore, the probabilities of obtaining more uncertain FFE. Exploring this connection is a relevant avenue of work. However, it is a question that falls outside this work's scope.\u003c/p\u003e \u003cp\u003eThe differences between the S and the error scenarios (EI and ED) indicate that flow measurement errors play a crucial role in FFE uncertainty. We observed these differences in the second row of Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e and, more specifically, in the North Raccoon case. On the other hand, the ED scenario obtained a slighter smaller P(RD\u0026thinsp;\u0026lt;\u0026thinsp;0.1) in most cases, indicating that a magnitude-dependent error may increase the FFE uncertainty. In this case, we compared two simple error models. Nevertheless, the results highlight the relevance of including streamflow uncertainty when performing FFE.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eWe explored the FFE uncertainty due to measurement errors in four watersheds using the Bulletin 17C guidelines. To illustrate the effect of the errors, we used two multiplicative models: one independent and one dependent of the magnitude, following the log-normal distribution proposed by Potter and Walker (\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1981\u003c/span\u003e). We evaluated the error models using the length of the records and 50-year by bootstrapping observations. We used the 50-year scenario to contrast the effect of the errors. We generated 1,000 realizations for each scenario from where we computed the hypothetical \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FFA\\)\u003c/span\u003e\u003c/span\u003e realizations (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{A}_{s}\\)\u003c/span\u003e\u003c/span\u003e) using PeakFQ and obtained differences comparing with the observed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FFA\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eWe obtained differences higher than 50% in multiple cases when comparing the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{A}_{s}\\)\u003c/span\u003e\u003c/span\u003e with the observations (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FF{A}_{o}\\)\u003c/span\u003e\u003c/span\u003e). Besides, we found differences around 100% for the 100- and 500-year return periods. We explored these differences further by comparing the realizations of the hypothetical scenarios, finding significant differences for return periods above 50 years. In both cases, the error scenarios developed higher uncertainties than the bootstrapping ones, highlighting the relevance of measurement error.\u003c/p\u003e \u003cp\u003eFrom the analysis of four watersheds with areas oscillating between 517 and 16,860 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k{m}^{2}\\)\u003c/span\u003e\u003c/span\u003e we observed that measurement errors increase the uncertainty in relatively small watersheds. However, our results are limited by using a purely statistical error model and the assumption of a stationary behavior. We are aware that the error structure may be more intricate. It can be discontinuous (Potter and Walker \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1981\u003c/span\u003e), relying on the hydraulic and hysteretic conditions (Moges et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Muste et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), and it is likely non-stationary. Nevertheless, our work illustrates the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(FFE\\)\u003c/span\u003e\u003c/span\u003e uncertainty magnitude due to measurement errors.\u003c/p\u003e \u003cp\u003eFFE accuracy is highly relevant as it conditions our infrastructure designs. The environmental crisis and landscape changes will likely bring more frequent floods in the coming years. These changes will also come with increased uncertainties. We can explore further how to address those uncertainties by examining our measurements. Future work should explore the development of physical-based models to represent measurement errors. In the meantime, we should opt for a conservative approach to flood frequency estimation.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding:\u0026nbsp;\u003c/strong\u003ethe Iowa Flood Center at the University of Iowa, with grants from the Iowa Department of Transportation (Grant TR-699) and the Mid-American Transportation Center (MATC).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests:\u0026nbsp;\u003c/strong\u003eThe authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contribution:\u003c/strong\u003e\u0026nbsp; N.V: Numerical analysis, figures, writing, conceptual idea, W.K: Conceptual idea, writing, editing, funding.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments:\u0026nbsp;\u003c/strong\u003eTo be edited after acceptance.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eApel H, Thieken AH, Merz B, Bl\u0026ouml;schl G (2004) Natural Hazards and Earth System Sciences Flood risk assessment and associated uncertainty. 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Nat Hazards Earth Syst Sci 21:1071\u0026ndash;1085. https://doi.org/10.5194/nhess-21-1071-2021\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"stochastic-environmental-research-and-risk-assessment","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"serr","sideBox":"Learn more about [Stochastic Environmental Research and Risk Assessment](https://www.springer.com/journal/477)","snPcode":"477","submissionUrl":"https://submission.nature.com/new-submission/477/3","title":"Stochastic Environmental Research and Risk Assessment","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Flood frequencies, Uncertainty, Sampling, Errors","lastPublishedDoi":"10.21203/rs.3.rs-3837694/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3837694/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSignificant errors often arise when measuring streamflow during high flows and flood events. Such errors conflated by short records of observations may induce bias in the flood frequency estimates, leading to costly engineering design mistakes. This work illustrates how observational (measurement) errors affect the uncertainty of flood frequency estimation. The study used the Bulletin 17C (US standard) method to estimate flood frequencies of historical peak flows modified to represent the measurement limitations. To perform the modifications, the authors explored, via Monte Carlo simulation, four hypothetical scenarios that mimic measurement errors, sample size limitations, and their combination. They used a multiplicative noise from a log-normal distribution to simulate the measurement errors. They implemented a bootstrap approach to represent the sampling error. They randomly selected M samples from the total N records of the observed peak flows of four gauging stations in Iowa in central USA. The observed data record ranges between 76 and 119 years for watersheds with drainage areas between 500 and 16,000 km\u003csup\u003e2\u003c/sup\u003e. According to the results, measurement errors lead to more significant differences than sampling limitations. The scenarios exhibited differences with median magnitudes of up to 50%, with some cases reaching differences up to 100% for return periods above 50 years. The results raise a red flag regarding flood frequency estimation that warrants looking for further research around observational errors.\u003c/p\u003e","manuscriptTitle":"Effect of Streamflow Measurement Error on Flood Frequency Estimation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-10 21:46:33","doi":"10.21203/rs.3.rs-3837694/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-02-28T12:40:02+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-02-11T22:57:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"26024749-fea1-4a05-90a6-50ab64d2c33d","date":"2024-01-22T18:10:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"83e7810b-65e1-4ad6-ba25-19aa754899fc","date":"2024-01-20T01:02:20+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-01-19T19:49:28+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-01-10T19:36:38+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-01-08T11:40:49+00:00","index":"","fulltext":""},{"type":"submitted","content":"Stochastic Environmental Research and Risk Assessment","date":"2024-01-05T16:12:50+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"stochastic-environmental-research-and-risk-assessment","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"serr","sideBox":"Learn more about [Stochastic Environmental Research and Risk Assessment](https://www.springer.com/journal/477)","snPcode":"477","submissionUrl":"https://submission.nature.com/new-submission/477/3","title":"Stochastic Environmental Research and Risk Assessment","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"af763a0b-7135-42ea-89c0-ed152eb35b15","owner":[],"postedDate":"January 10th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-04-01T15:04:13+00:00","versionOfRecord":{"articleIdentity":"rs-3837694","link":"https://doi.org/10.1007/s00477-024-02707-1","journal":{"identity":"stochastic-environmental-research-and-risk-assessment","isVorOnly":false,"title":"Stochastic Environmental Research and Risk Assessment"},"publishedOn":"2024-03-25 15:01:02","publishedOnDateReadable":"March 25th, 2024"},"versionCreatedAt":"2024-01-10 21:46:33","video":"","vorDoi":"10.1007/s00477-024-02707-1","vorDoiUrl":"https://doi.org/10.1007/s00477-024-02707-1","workflowStages":[]},"version":"v1","identity":"rs-3837694","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3837694","identity":"rs-3837694","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}