# Peak Flow Event Durations in the Mississippi River Basin and Implications for Temporal Sampling of Rivers

Arnaud Cerbelaud, Cedric H. David, Sylvain Biancamaria, Jeffrey Wade, Manu Tom, Renato Prata de Moraes Frasson, Denis Blumstein

► To cite this version:

Arnaud Cerbelaud, Cedric H. David, Sylvain Biancamaria, Jeffrey Wade, Manu Tom, et al. Peak Flow Event Durations in the Mississippi River Basin and Implications for Temporal Sampling of Rivers. Geophysical Research Letters, 2024, 10.1029/2024GL109220. http://hal-04720400

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# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL109220

**Key Points:**

* Peak flow events in the Mississippi basin range from 2 days upstream to 30 days downstream, hence exhibiting acute spatial variability

* Decadal mean, standard deviation and quantiles of hourly discharge can all be computed within 2.5% error up to an 8-day sampling period

* Sampling periods must be a factor of 4 finer than median peak flow event durations to estimate durations within 90 ± 5% median accuracy

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

A. Cerbelaud,
arnaud.cerbelaud@jpl.nasa.gov

**Citation:**

Cerbelaud, A., David, C. H., Biancamaria, S., Wade, J., Tom, M., Prata de Moraes Frasson, R., & Blumstein, D. (2024). Peak flow event durations in the Mississippi River basin and implications for temporal sampling of rivers. *Geophysical Research Letters*, *51*, e2024GL109220. https://doi.org/10.1029/2024GL109220

Received 15 MAR 2024

Accepted 12 MAY 2024

**Author Contributions:**

**Conceptualization:** Arnaud Cerbelaud, Cédric H. David, Sylvain Biancamaria, Denis Blumstein

**Data curation:** Arnaud Cerbelaud

**Formal analysis:** Arnaud Cerbelaud, Cédric H. David, Sylvain Biancamaria, Jeffrey Wade, Manu Tom, Denis Blumstein

**Funding acquisition:** Cédric H. David, Sylvain Biancamaria, Denis Blumstein

**Investigation:** Arnaud Cerbelaud, Cédric H. David, Jeffrey Wade, Manu Tom, Renato Prata de Moraes Frasson

© 2024 Jet Propulsion Laboratory, California Institute of Technology and The Author(s). Government sponsorship acknowledged.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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## Peak Flow Event Durations in the Mississippi River Basin and Implications for Temporal Sampling of Rivers

Arnaud Cerbelaud¹, Cédric H. David¹, Sylvain Biancamaria², Jeffrey Wade¹, Manu Tom¹, Renato Prata de Moraes Frasson¹, and Denis Blumstein²,³

¹Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA, ²Université de Toulouse, LEGOS (CNES/CNRS/IRD/UT3), Toulouse, France, ³CNES, Centre National D'Études Spatiales, Toulouse, France

**Abstract** The impact of an episodic river flood is intimately linked to its duration. Yet it is still unclear how often should a river be observed to accurately determine the occurrence and duration of extreme events. Here we assess flow statistics along with peak flow event detection and duration as a function of the discharge sampling period for large tributaries of the Mississippi basin using hourly gages over 2010–2022. Median event durations above high quantiles spatially vary from around 2 days upstream to 30 days downstream. Discharge mean, standard deviation, and quantiles can all be estimated within 2.5% error for sampling periods up to 8 days. A minimum temporal sampling 4× (2×) finer than peak flow event median duration is required to detect 95 ± 3% (85 ± 5%) of events and to estimate their duration within 90 ± 5% (75 ± 10%) median accuracy. Our findings have direct implications for future satellite missions concerned with capturing flood events.

**Plain Language Summary** High flow events and their duration in rivers are key aspects of flood studies. Yet, it is still unclear how often a river should be observed to detect floods and determine their durations. Analyzing hourly stream gage data in the Mississippi River basin from 2010 to 2022, we calculate that flood events vary widely in duration from upstream (2 days) to downstream (30 days). We show that observing large rivers every few days is sufficient to compute common long-term river flow statistics within 2.5% error. However, capturing most floods and accurately assessing their duration (within 10% error) requires monitoring at least four times more frequently than the typical duration of these events. This means that if a major flood typically lasts 10 days, the river data should be collected at least every 2–3 days. Our findings have implications for the design of monitoring systems concerned with the accurate capture of floods, including those using Earth orbiting satellites.

## 1. Introduction

With floods being consistently and increasingly among the world's costliest natural disasters (David & Frasson, 2023; Tellman et al., 2021; UN Report, 2020), the exploration of extreme streamflow events using gage measurements is a central topic in hydrology. A wide range of methodologies have been employed to study how fluvial floods propagate and the temporal and spatial scales over which peak discharge events occur (Mishra et al., 2022). Global endeavors have relied on trustworthy news/science report information to estimate median flood durations of around 10 days worldwide (Dartmouth Flood Observatory (DFO); Brakenridge, 2024; Najibi & Devineni, 2018). Alternatively, streamflow fluctuations can be quantified through the implementation of peak flow events (PFEs) detection techniques. Multiple methods have been designed to detect PFEs (e.g., Greimel et al., 2016). Hydrologists commonly carry out frequency analysis to assess return periods or annual exceedance probabilities for categorizing floods (England et al., 2018; Van Campenhout et al., 2020). While peak magnitudes are often the key variable of interest, the simple definition of flood thresholds allows detecting high flow events and estimating their duration. In our analysis of data produced by Sikder et al. (2021), PFEs above the 90th discharge quantile were found to last 10 days on average based on global simulations, consistent with the reports from the DFO. Harmonic analysis was also used to determine the time span across which floods most commonly occur (5 days for US rivers; Stroud et al., 2024). Flow wave travel time, which governs the downstream propagation of flood events, was estimated at around 6 days globally (median value to reach basin terminus) using empirical relations or lagged cross-correlations in combination with routing models (Allen et al., 2018).

The space-time resolution of discharge data is key to the detection of PFEs. While the study of local hydrological processes typically requires the use of multiple sub-daily gages in small-sized basins (Ficchì et al., 2016; Giani et al., 2022), a few stations strategically placed along major rivers providing daily to monthly data may be
**Methodology:** Arnaud Cerbelaud, Cédric H. David

**Project administration:**

Arnaud Cerbelaud, Cédric H. David

**Resources:** Arnaud Cerbelaud, Cédric H. David, Sylvain Biancamaria

**Software:** Arnaud Cerbelaud

**Supervision:** Cédric H. David

**Validation:** Arnaud Cerbelaud

**Visualization:** Arnaud Cerbelaud

**Writing – original draft:**

Arnaud Cerbelaud, Cédric H. David, Sylvain Biancamaria, Jeffrey Wade, Manu Tom, Renato Prata de Moraes Frasson

**Writing – review & editing:**

Arnaud Cerbelaud, Cédric H. David, Renato Prata de Moraes Frasson

sufficient for larger-scale analysis (Allen et al., 2018; David et al., 2011, 2013; Gudmundsson et al., 2021; Oki et al., 1999). Unfortunately, the spatiotemporal availability of discharge observations is becoming increasingly limited due to an ongoing worldwide decline in gage data availability (Group et al., 2001; Hannah et al., 2011; Ruhi et al., 2018), making it a challenge to determine PFE characteristics globally. Additionally, most gages today are disproportionally located in wide, perennial rivers draining largely urbanized watersheds (Krabbenhoft et al., 2022). Most basins in remote areas and developing countries remain ungaged and thus require other sources of data or modeling endeavors for flood monitoring (Hrachowitz et al., 2013). Remote sensing, and in particular radar altimetry, has emerged as an increasingly common way of measuring water surface elevation and approximating stream discharge from space for Earth's largest rivers (Kouraev et al., 2004; Papa et al., 2012; Paris et al., 2016; Tarpanelli et al., 2013; Tourian et al., 2017; Zakharova et al., 2020). These estimates of discharge are made at virtual stations (VS), located where the satellite ground track crosses a water body.

Past altimetry mission revisit periods have extended from 10 days (TOPEX-Poseidon, Jason-1/2/3) to 35 days (ENVISAT, SARAL, ERS-1/2), while more recent missions still lie in-between (27 days for Sentinel 3A/B; 10 days for Sentinel-6; 21 days for SWOT, reduced to 11 days on average thanks to wide swath measurement overlaps). From a theoretical standpoint, annual discharge estimates resulting from a 10-day sampling rate are comparable to what daily rates provide (±5%) at least for the largest rivers (>100 m³.s⁻¹; Fekete et al., 2012). SWOT irregular temporal sampling should have very limited impact on the frequency distribution of estimated discharge compared to daily observations (Nickles et al., 2019). However, only 22% of SWOT measurements are expected to fall within 1 day of event peak discharges (Nickles et al., 2019) and around half of floods occurring in SWOT lifetime are likely to be missed (Frasson, Schumann, et al., 2019). Based on global model simulations, sampling periods of at least 5 days were advised to adequately characterize the global distribution of the number and duration of PFEs (Sikder et al., 2021).

Yet it is still unclear how often rivers should be observed so that the magnitude and duration of their PFEs can be determined with a given accuracy. This fundamental question has likely rarely been addressed before because temporal sampling specifications are comparatively less relevant in the case of in situ measurements. Gage data loggers have been able to support hourly resolutions at least for decades. However, the nature of Earth-orbiting repetitive satellite missions (e.g., orbital mechanics) imposes tight constraints on the spatiotemporal sampling possibilities. These constraints generate a trade space such that one generally must choose between (a) frequently revisiting a few places or (b) infrequently revisiting many places. Previous work has noted the space/time resolution of radar altimetry missions as one of its primary limitations for hydrology (Falck et al., 2021; Fekete et al., 2012; Hydrospace, 2021; Sikder et al., 2021; Tourian et al., 2016).

Perhaps the best analogy to the question of optimal sampling for capturing discharge peaks lies in the Nyquist-Shannon sampling theorem (NS theorem; Shannon, 1949). The theorem states that when sampling an integrable continuous function of finite bandwidth, the sample rate must be at least twice the bandwidth of the signal to avoid aliasing distortion. Even though the NS theorem does not technically apply to hydrographs since discharge oscillations originate from multiple random inputs, it is unknown whether a sampling period at least twice as frequent as PFE timescales is needed to adequately capture streamflow variability.

The objective of this study is to answer the following research questions: (a) What is the duration of PFEs in one of the world's largest and most highly anthropized basins? (b) How spatially variable are PFEs across a river basin (small/large rivers, upstream/downstream)? And (c) What are the implications for satellite sampling to capture PFE occurrence and duration? To document the variability in the duration of PFEs and their impact on requirements for sub-weekly sampling of Earth's rivers, we here rely on in situ observations at the continental scale rather than on global model simulations (Sikder et al., 2021). The use of gage data allows for the most accurate capture of variability in streamflow time series. We analyze 518 USGS gages in the Mississippi river basin from 2010 to 2022. Improvements from previous studies include high-resolution sampling with 95 sub-weekly scenarios (from 2 to 192 hr/8 days sampling periods) based on hourly in situ observations. We outline a potential guiding principle for hydrograph sampling based on the ratio between the sampling period and the PFE median duration at any location as an analogy to the NS theorem. We eventually discuss the implications for the design of sub-weekly satellite missions concerned with the capture of peak discharge in rivers.
## 2. Methods

### 2.1. Reference Hourly Discharge Observations

We use 15-, 30-, and 60-min temporal resolution streamflow observations on the Mississippi river basin (MRB) retrieved from the USGS Water Services over the 2010–2022 period. In situ discharge data are then averaged at an hourly resolution. These time series are called “reference” discharge. All selected gages are located along river reaches featured in the Global River Width From Landsat (GRWL; Allen & Pavelsky, 2018a) database (river width ≥30 m). We implement a second more restrictive condition so that upstream area is greater than 1000 km² (ensuring river width ≥ 90 m according to Frasson, Pavelsky, et al., 2019) using information derived from the MERIT river network data set (Lin et al., 2019; Yamazaki et al., 2019a). These requirements are set to guarantee that our gaged river reaches are likely observable using current radar altimetry technology. At the end of our selection process, only 518 stations with at least 10 years of hourly observations over the 2010–2022 period are kept. Average and 5th/25th/50th/75th/95th percentiles of upstream area at all 518 gaged reaches are respectively 93,700 km² and 1,800/4,800/13,300/38,000/237,000 km². The resulting network of MRB gages covers about 40% (3,200,000 km²) of the contiguous United States, including all or parts of 32 states (tributary Canadian provinces are not included). Instrumented rivers flow through various landscapes and with varying degrees of human interventions, experience different climates, and hence feature heterogeneous concentration times.

### 2.2. Discharge Variability and PFE Characterization

To evaluate the sampling requirements for observing river temporal dynamics, we first compute several standard streamflow statistics on each station reference time series: long-term mean ($Q_{mean}$), standard deviation ($Q_{std}$), and 75th, 90th, and 95th percentiles ($Q_{75}$, $Q_{90}$, $Q_{95}$). We further characterize hydrograph variability through the estimation of the number of PFEs and their median duration. PFEs are here designated as, at any given station, consecutive periods of time when streamflow exceeds a predefined threshold. We thus define two variables, $N_p$ and $T_p$, corresponding respectively to the number of events per year and the median duration of events when $Q(t) \ge Q_p$, with $p$ equal to the 75th, 90th or 95th percentile (Figure 1). We acknowledge that the $Q_p$ exceedance thresholds chosen as a focus here are useful to provide various estimates of flood durations but differ from what is commonly used in flood planning and may have limited practical utility concerning infrastructures subject to flooding.

For a given station, the median duration $T_p$ is derived from the list of all event durations $T_p^i$ over the available period. The median is preferred over the mean to better control for the influence of $T_p^i$ outliers that can originate from occasional hydrograph oscillations around a given quantile (Figure 1). To further mitigate the impact of potential sub-daily variations of $Q(t)$ around $Q_p$, two features are added to the PFE detection workflow: (a) hydrographs undergo a smoothing process through the application of a ±6-hr centered rolling average window; (b) any instance when $Q(t) \ge Q_p$ lasting less than 24 hr is excluded from the analysis. This second condition eliminates a great quantity of excessive sub-daily variations that can occur around the quantile threshold intermittently, yet we acknowledge that we are likely missing some flash flooding events in the smallest rivers. However, we believe that the selected rivers should not be so prone to flash-flooding considering their width.

The smoothing process underwent preliminary testing with rolling windows ranging from 2 to 24 hr. Improvements in the removal of $T_p^i$ outliers were minimal past the 6-hr mark. Consequently, a 6-hr smoothing window is retained to maintain proximity to the original time series (Figure 1). Anomalies from sampling are also seldom observed due to missing data windows over weeklong periods, especially during high flow events (instruments may be damaged during floods). Missing data windows cannot be considered for the calculation of the PFE duration. Yet, because these windows can be overlooked when applying sampling periods greater than the actual window duration, upward biases in event durations can be observed from sampling. We implement a “NoData” filling with backward propagation to avoid such sampling anomalies, with a gap limit of 7 days (Figure 1). We acknowledge that this procedure may on the other hand lead to upward biases in the reference PFE durations when high flow events are filled.

Then, the reference discharge time series are subsampled (Section 2.3). When applying event detection on these “sampled” data, event durations are artificially inflated or deflated at the start and end of events due to the computation on larger time steps. A linear interpolation at the start and end of events is thus implemented to better approximate the durations $T_p^i$ on sampled data. With $\Delta T$ the discharge time step in the series (reference or
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Conceptual space-time representation of streamflow illustrating the upstream-downstream contrast in PFE timing and duration $T_{90}$. Rolling average smoothing and “NoData” filling is depicted.</figcaption></figure>

sampled) and t the first date when $Q(t) \ge Q_p$, a simple cross-multiplication is applied to calculate the starting duration $\Delta T_p^i$:

$$ \Delta T_p^i = \Delta T \frac{Q(t) - Q_p}{Q(t) - Q(t - \Delta T)} $$

A similar calculation is performed at the end of events.

## 2.3. High-Resolution Temporal Sampling Strategy

We define two variables to refer to the sampling scenarios. The first one is the sampling period $\Delta T$. The second one is the sampling ratio $\Delta T/T_p$, defined as the sampling period as a % of a station's reference PFE median duration $T_p$. Hence, when $\Delta T/T_p = 30\%$, each streamflow time series is sampled every 30% of its reference $T_p$. A single value of $\Delta T$ thus leads to various values of $\Delta T/T_p$ spatially. This second variable is defined in analogy to the NS theorem to indicate the required ratio between $\Delta T$ and $T_p$ to capture PFEs and their durations with a desired level of accuracy and consistently across the whole catchment. We then define the sampled discharge metrics $\tilde{X}$ as
the streamflow mean ($\tilde{Q}_{\text{mean}}$), standard deviation ($\tilde{Q}_{\text{std}}$), high quantiles ($\tilde{Q}_{p}$), PFE number of events per year ($\tilde{N}_{p}$) and median duration of events ($\tilde{T}_{p}$) ($p = 75,90,95$) from the sampled time series. The absolute value of the relative error for all aforementioned reference streamflow characteristics $X$ per station associated with a given sampling period (ratio) $\Delta T$ ($\Delta T/T_{p}$) is:

$$ \varepsilon(X) = \left| \frac{\tilde{X}}{\tilde{X}} - 1 \right| $$

A 2- to 192-hr (8 days) sampling period $\Delta T$ is implemented with a 2h increment over each hourly station time series, composing 95 sampling scenarios. Note that for a single sampling period of $k$ hours, sampling can start anytime between $t = 0$ and $t = k - 1$ hours. Hence, $k$ sub-scenarios are possible and must be accounted for, for each $\Delta T$. This introduces an uncertainty interval in the sampling performance for a single value of $\Delta T$ or $\Delta T/T_{p}$, as some events can be caught or missed based on when the sampling starts. We refer to this as a “serendipity” effect, where some PFEs may be observed accurately with one sampling pattern, while another pattern with the same sampling frequency but shifted in time may not observe them. Hence, in this study, each sampling performance is portrayed by an ensemble of sub-scenarios with its associated average and best/worst error values. Results reflecting multiple stations at once are shown using the median of all stations errors rather than the mean to limit the influence of outlier hydrographs.

# 3. Results

## 3.1. PFEs in the Mississippi River Basin

We select $Q_{90}$ as the primary PFE threshold as it was shown in previous literature to lead to a consistent 10 days average event duration worldwide (Najibi & Devineni, 2018; recomputed data from Sikder et al., 2021). We discuss this choice in Section 4. Our analysis of the MRB gages suggests that $Q_{90}$ values range mostly from around 10 to 10,000 m$^3$.s$^{-1}$ (up to 38,000 m$^3$.s$^{-1}$ at the Mississippi river at Vicksburg), with a spatial median (mean) of 150 (775) m$^3$.s$^{-1}$ and a high spatial standard deviation of 3,165 m$^3$.s$^{-1}$. To relate our PFE threshold to more traditional measures of peak flows, we report an average and standard deviation of the ratio between $Q_{90}$ and average annual maximum flows at all stations of 44% and 30%, respectively.

Heterogeneous values of $T_{90}$ are found across the basin, from around 2 days to more than 30 days, with a spatial median (mean) of 3.9 (5.6) days and a spatial standard deviation of 5.2 days. As expected, a clear upstream-downstream contrast is observed (Figure 2a). Headwater streams and comparatively small tributaries feature shorter PFE durations compared to downstream, larger drainage locations. However, some $T_{90}$ values vary unexpectedly along rivers. Longer $T_{90}$ values are measured in the headwaters of the Arkansas, Platte, Missouri, and Mississippi rivers. These might be due to the presence of snowpack at upper elevations leading to high flow events for 2–4 weeks typically during spring runoff. Tides and diversions affect results downstream of the Mississippi with shorter $T_{90}$ at the last gage (Figure 2a). Values of $N_{90}$ range from around 1 event per year to around 14, with a spatial median (mean) of 5.5 (6.2) events per year and a spatial standard deviation of 3.4 events per year. Once again, an upstream-downstream contrast is present with fewer events detected at downstream locations, which is consistent with their longer durations.

The disparity in hydrograph variability between upstream and downstream locations is also visible in Figure 2b. Station A upstream of the Ohio river features high frequency oscillations, with $N_{90} = 90$ events over 13 years of data and $T_{90} = 2.5$ days. Only about 40 events are detected at stations B and C downstream in contrast, with $T_{90} = 6.2$ and 9.6 days, respectively.

## 3.2. Sampling of Discharge Magnitude Statistics

The distributions of long-term mean and high quantiles of streamflow are mostly unaffected by sub-weekly sampling. Errors lie nearly entirely below $\pm 1\%$ even for the longest 8-day sampling period investigated in this work (Figure S1 in Supporting Information S1). In particular, the errors on the sampled discharge quantiles $\tilde{Q}_{p}$ are all small enough to confidently use $\tilde{Q}_{p}$ as thresholds for detecting PFEs in all sampling scenarios. PFE detection accuracy from sampling is described in the next sections. The quality of sampled discharge thresholds for in situ gages is consistent with the model simulations of Sikder et al. (2021). The error on the standard deviation of
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) PFE median duration *T*<sub>90</sub> (days) at each station of the study area. (b) Space-time representation of streamflow along the Ohio River at three stations and illustration of the calculation of PFE median duration *T*<sub>90</sub> on reference hourly data (top) and 2-day sampled data (bottom).</figcaption></figure>

discharge from sampling is slightly larger, which demonstrates higher impact of sampling to capture data dispersion, that is, in our case, temporal variability. However, error ranges still do not exceed ±2.5% for the vast majority of stations (Figure S1 in Supporting Information S1).

### 3.3. Sampling of PFE Number and Duration

The estimation of event durations from sampled series suffers three main challenges leading to upward biases. First, with longer $\Delta T$, short events are increasingly missed, and long events are more likely to be captured. Second, multiple consecutive events can be counted as one single/longer event as the sampling fails to capture streamflow receding below $Q_{90}$ between consecutive events. Third, since the edges of discharge rising and receding limbs are mostly convex, the starting and ending points of events are often estimated to be longer with linear interpolation.

An example of a 2-day sampling period is illustrated in Figure 2b. An error $\epsilon$ ($T_{90}$)~50% is found for station A ($\tilde{T}_{90}$ = 3.7 days) as opposed to only around ~5% for stations B and C (respectively $\tilde{T}_{90}$ = 6.5 and 10.2 days; Figure 2b). As expected, shorter sampling periods are needed to capture PFEs in flashier river reaches such as where station A is located.

### 3.4. Guiding Principle for Temporal Sampling of PFEs

To establish a general guiding principle regarding the impact of PFE durations on sampling requirements, we quantify the range in error $\epsilon(N_{90})$ and $\epsilon(T_{90})$ as a function of the sampling ratio $\Delta T/T_{90}$ at every station, and for all stations combined (Figure 3a).

Over all stations, median $\epsilon(N_{90})$ lies between 2% and 8% for a sampling ratio $\Delta T/T_{90} = 1/4$, and between 9% and 20% for $\Delta T/T_{90} = 1/2$. Median $\epsilon(T_{90})$ ranges from 5% to 15% for a sampling ratio $\Delta T/T_{90} = 1/4$, and 13%–36% for $\Delta T/T_{90} = 1/2$ (Figure 3a). Even though one would assume that the effective timing of sampling should not significantly affect the results over a dozen years of data at more than 500 stations, these error intervals reflect the meaningful volatility in the estimation of the number and even more so the duration of PFEs solely due to serendipity. The error ranges $\epsilon(T_{90})$ for stations A, B and C on the Ohio river can also be read on Figure 3a with a sampling period $\Delta T = 2$ days as an example. Similarly to the aggregated Figure 3a,

station-specific error plots are available in Figure S2 in Supporting Information S1. Another representation of $\epsilon(T_{90})$ as a function of $T_{90}$ for several values of $\Delta T$ is given in Figure S3 in Supporting Information S1.

Echoing the NS theorem, a potential guiding principle for temporal river sampling requirements to capture PFEs and their durations in rivers is: a sampling period at least 4× (2×) faster than the median PFE duration is needed to estimate, on a median basis, the number of events and their median duration within a 5 ± 3% (15 ± 5%) and 10 ± 5% (25 ± 10%) relative error range, respectively.

### 3.5. Implications for Satellite Sampling

We evaluate our findings in the context of a hypothetical concept for a space mission with sub-weekly revisits (Blumstein et al., 2019) using three possible repetitive orbit strategies ($\Delta T$ = 1 day, 250 km intertrack distance at the equator; 2 days, 125 km; 4 days, 62.5 km). To illustrate “good” virtual stations corresponding approximately to where $T_{90} \ge 2\Delta T$, we choose a 25% error limit on $\epsilon(T_{90})$. Most reaches featuring a $T_{90} \ge 2$ days, hence most of the observed river reaches, are “well” captured in the 1-day sampling scenario (Figure 3b, 1477 “good” VS). However, the spatial sampling is relatively sparse (~200 km at this latitude) which limits the number of overall VS (1595). With longer sampling periods, PFE durations are accurately captured only in the major river systems and/or the downstream locations, with 1290 (768) reaches out of 2823 (5269) featuring errors lower than 25% in
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Median absolute relative errors $\epsilon(X)$ on the PFE number $N_{90}$ and median duration $T_{90}$ from sampling as a function of the sampling ratio $\Delta T/T_{90}$ (%), maximum sampling period $\Delta T = 192$ hr) for all stations, with their best/worst performance interval. Stations A, B, and C from Figure 2 are placed when sampled at $\Delta T = 2$ days for illustration purposes. (b) Conceptual mission scenarios with constant capabilities with “good” (green) and “bad” (red) observed reaches (i.e., virtual stations). Here, “good” reaches refer to locations where the closest station features PFE median duration error $\epsilon(T_{90})$ below 25%. The MERIT and GRWL data sets are used for the river network.</figcaption></figure>

the 2-day (4-day) sampling scenarios (Figure 3bii/iii). However, such scenarios allow visiting more locations along the major streams, thus revealing the spatial patterns in the river network. Overall, the 2-day sampling scenario provides slightly better spatial sampling on the biggest rivers (348 “good” reaches out of 594 VS where upstream area >10⁵ km²) while the 1-day scenario offers more accurate observation of the smaller headwater streams (1153 “good” reaches out of 1227 VS where upstream area <10⁵ km²; Figure 3b inset tables). Note that the notion of “good” stations represented in Figure 3b assumes a significant constraint on what is considered an acceptable estimation of river flow dynamics through a 25% limit on $\epsilon(T_{90})$. Indeed, $\epsilon(Q_i)$, $i = \text{mean, std, 75th, 90th and 95th, are mostly below 2.5\% even up to an 8-day sampling for most stations (Figure S1 in Supporting Information S1). Such precision already yields significant information on discharge temporal dynamics and average magnitude of PFEs, without going into the detailed monitoring of PFEs. The 4-day sampling scenario
would for instance provide more than 3 times more VS than the 1-day scenario with >97.5% accuracy on discharge statistics (5259 vs. 1595), allowing a much better investigation into spatial variability.

4. Discussion and Conclusions

In a sampling analysis of a synthetic global river discharge data set, Sikder et al. (2021) estimated that temporally sampled data will not be able to accurately capture event duration unless events are much longer than the sampling interval. Our work sheds additional light on this matter by specifically evaluating PFE occurrence and duration across space and time through the perspective of high-resolution sampling. PFEs are indeed among the most difficult river traits to capture, making them crucial in establishing precise sampling requirements. In contrast to earlier research based on model simulations, the robustness of our study lies in employing a dense network of in situ observations within one of the world's largest and most heavily anthropized basins. Thanks to high resolution gage data, we quantify the expected accuracy in discharge quantile estimation and the detection of PFEs as a function of the ratio between the sampling period and the reference PFE median duration, allowing our guiding principle to be applicable at multiple locations. Our results reveal sub-weekly temporal sampling requirements aligning with previous research conducted using global-scale simulations (minimum 5-day sampling interval; Sikder et al., 2021) which, while geographically widespread, lack a guarantee of accuracy on real discharge conditions. Discharge statistics, including high quantiles associated with PFE magnitude, are substantiated to be very well captured with sub-weekly measurements (error ≤2.5%). The detection of PFEs and the estimation of their duration are confirmed to pose a more difficult challenge in terms of sampling accuracy. Our study reveals reference PFE median number *N*<sub>90</sub> ~ 5.5 events per year and duration *T*<sub>90</sub> ~ 4 days across the MRB, while Sikder et al. (2021) estimated a median number/duration of events of around 5 per year/25 days globally (based on a 75th quantile threshold, or, in our analysis of their results, 4 per year/10 days based on the 90th). At the global scale, the DFO also indicates a median flood duration of ~10 days in 2015 (Najibi & Devineni, 2018). Should our guiding principle be applicable globally, a ~2–3 days revisit frequency (4 times faster than the median flood duration) would allow capturing ~95% of PFEs and estimate their duration with ~90% accuracy for half of Earth's biggest rivers. This is in line with Munasinghe et al. (2023) who found using satellite imagery that a sampling period of ~2.4 days would be sufficient to sample ~90% of the floods recorded by DFO. Given SWOT's revisit frequency of ~11 days, the detection of PFEs will primarily rely on opportunistic observations. Precise estimation of their duration will only be feasible for events spanning at least 22 days (i.e., for the largest rivers on the planet).

However, limitations on the transferability of our derived guiding principle along with its practical implications to other catchments must be acknowledged, as PFE patterns vary among the world's basins depending on rainfall triggering events (spatial distribution, intensity, magnitude and duration; see Mandapaka et al., 2009), environmental conditions and geomorphic properties of river basins (Rigon et al., 2011). Ideally, the same workflow should be applied to a worldwide network of hourly gages, though such network does not presently exist. Another shortcoming for application to satellite sampling arises from our framework completely disregarding numerous operational constraints, which represent entire optimization challenges for space missions (Gorr et al., 2023), as well as uncertainties in altimetry measurements (e.g., waveform retracking algorithm, range window tracking system; Biancamaria et al., 2018, 2017; Zakharova et al., 2020) and discharge algorithms (Durand et al., 2016, 2023). Furthermore, we note that the selection of *Q*<sub>90</sub> as the PFE threshold directly impacts the durations we measured, with shorter durations resulting from higher quantile thresholds. Nonetheless, our guiding principle should remain mostly unaffected by this choice as it is specifically tailored around the sampling ratio Δ*T*/T<sub>p</sub>.

The sampling period and the detection accuracy for PFEs are intimately linked. Frequent observations generally benefit quality estimations for flashier upstream rivers but may be excessive further downstream. This trade space must together be accounted for when designing an observing system fulfilling one's preferred accuracy needs with implications on event capture across basins.

Our findings have direct implications on the design of a future surface water satellite mission aimed at unveiling the high-frequency dynamics of Earth's freshwater fluxes in rivers. In particular, the high-resolution temporal information delivered by such a mission (e.g., the SMASH—SMall Altimetry Satellites for Hydrology—constellation concept from the French Space Agency CNES; Blumstein et al., 2019) along with the unprecedented spatial coverage provided by the current SWOT mission could work in synergy to address fundamental questions related to the water cycle.
Acknowledgments

This research was carried out at the Jet
Propulsion Laboratory, California Institute
of Technology, under a contract with the
National Aeronautics and Space
Administration (NASA) including grants
from the Terrestrial Hydrology
(NNH17ZDA001N-THP) and Earth
Science U.S. Participating Investigator
(NNH20ZDA001N-EUSPI) Programs.

Data Availability Statement

The MERIT global river network is available in Yamazaki et al. (2019b) and the GRWL data set in Allen and Pavelsky (2018b). The satellite altimeter orbit file used in this study was computed at CNES and is available in Cerbelaud et al. (2024). All in situ USGS discharge data sets are available in the USGS “Water Services Test Tool (2024)” under Instantaneous Values Service. The software and data set allowing for the computation of sampled discharge data and hydrological analysis related to this study are made publicly available for research purposes in Cerbelaud et al. (2024). © 2024 California Institute of Technology. Government sponsorship acknowledged.

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