# Spatial Hydrographs of River Flow and Their Analysis for Peak Event Detection in the Context of Satellite Sampling

Arnaud Cerbelaud, Cédric David, Sylvain Biancamaria, Jeffrey Wade, Manu

Tom, Renato Prata de Moraes Frasson, George Allen, Hana Thurman, Denis
Blumstein

► To cite this version:

Arnaud Cerbelaud, Cédric David, Sylvain Biancamaria, Jeffrey Wade, Manu Tom, et al. Spatial Hydrographs of River Flow and Their Analysis for Peak Event Detection in the Context of Satellite Sampling. Water Resources Research, 2025, 61 (4), 10.1029/2024WR038444. http://dx.doi.org/10.1029/2024WR038447

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Distributed under a Creative Commons Attribution 4.0 International License
Water Resources Research®

RESEARCH ARTICLE

10.1029/2024WR038444

**Key Points:**

* Peak flow event spatial coverage is analyzed along river profiles in the Mississippi basin using in situ gages and a hydrography network

* Peak flow events range from around 75 to 1,800 km in length with a median (mean) value of 330 (520) km along the largest rivers

* Sampling intervals must be a factor of 4 finer than median peak flow event lengths to estimate lengths within 84% ± 3% 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., et al. (2025). Spatial hydrographs of river flow and their analysis for peak event detection in the context of satellite sampling. *Water Resources Research*, 61, e2024WR038444. https://doi.org/10.1029/2024WR038444

Received 18 JUL 2024
Accepted 26 MAR 2025

© 2025 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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Spatial Hydrographs of River Flow and Their Analysis for Peak Event Detection in the Context of Satellite Sampling

Arnaud Cerbelaud¹ ID, Cédric H. David¹ ID, Sylvain Biancamaria² ID, Jeffrey Wade¹, Manu Tom¹ ID, Renato Prata de Moraes Frasson¹ ID, George H. Allen³ ID, Hana Thurman³ ID, and Denis Blumstein²,⁴

¹Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA, ²Université de Toulouse, LEGOS (CNES/CNRS/IRD/UT3), Toulouse, France, ³Department of Geosciences, Virginia Tech, Blacksburg, VA, USA, ⁴CNES, Centre National D'Études Spatiales, Toulouse, France

**Abstract** The study of river dynamics has long relied on the analysis of traditional in situ hydrographs. This graphical representation of temporal variability at a given location is so ubiquitous that the mere term “hydrograph” is widely recognized as a time series. While such a “temporal hydrograph” is well suited for in situ data analysis, it fails to represent hydrologic variability across space at a given time; a perspective that characterizes satellite-based hydrologic observations. Here we argue that the concept of “spatial hydrograph” should be the focus of its own dedicated scrutiny. We build “space series” of river discharge and present their analysis in the context of peak flow event detection. We propose the use of peak event spatial coverage, referred to as “length”, as an analog to event duration. Our analysis is performed in the Mississippi basin using a dense in situ network. We reveal that peak flow events range in length from around 75 to 1,800 km with a median (mean) value of 330 (520) km along the basin's largest rivers. Our analysis also suggests that spatial sampling needs to be a factor of 4 (2) finer in resolution than peak flow lengths to detect 81% ± 13% (70% ± 20%) of events and to estimate their length within 84% ± 3% (67% ± 12%) median accuracy. We evaluate the connection between temporal and spatial scales of peak flows and show that events with longer durations also affect larger extents. We finally discuss the implications for the design of satellite missions concerned with capturing floods across space.

**Plain Language Summary** The study of river flooding has traditionally relied on the analysis of temporal variability in streamflow. While such an approach is well suited for gages which observe rivers at a given location through time, it has limits for satellite observations which instead observe rivers at a given time through space. Here we propose a change of perspective in the study of floods by evaluating their spatial extent analogously to their temporal duration. Analyzing a dense network of stream gages in the Mississippi basin over 2010–2022, we find that the longitudinal distance over which rivers show extreme discharge, that is, the “length” of flood events, varies widely, with a median distance of 330 km. To effectively capture the spatial extent of a flood along a stream, we find that monitoring should be a factor of four denser than the typical spatial length of these events. This means if a major flood extends over 330 km of river length, the data should be collected at least every 80 km. We also show that events that last longer also generally affect larger extents. Our findings have implications for flood monitoring projects relying on satellite data, ensuring accurate detection of flood events in space.

1. Introduction

The study of hydrologic variability has relied for decades on the analysis of hydrographs (e.g., Beven, 1977; Beven & Hornberger, 1982; Chow, 1964; Maidment, 1993; McDonnell & Beven, 2014; Sherman, 1932). Hydrographs are typically identified in the hydrological community as a time series of discharge, width, or height at a fixed, well-defined geographical location. These time series are instrumental tools for analyzing core durations in hydrologic variability, including time to peak of the rising limb, residence time, and peak event duration (e.g., Beven, 2020; McMillan, 2021). Time series are also widely summarized in temporally aggregated metrics of comparison between models and observations such as root mean square error, Nash Sutcliffe Efficiency, Kling Gupta Efficiency, among others.

In oceanography and meteorology, both temporal and spatial scales are commonly considered jointly, for example, under the scope of wavenumber-frequency plots or Hovmöller diagrams (representation of a state variable as a function of both time and latitude/longitude; Hovmöller, 1949) in wave analysis. While such analysis
frameworks do not transfer to hydrological sciences per se, it is particularly surprising that the representation and analysis of discharge as a function of 1-dimensional space (i.e., along the channel) has been scarcely explored. Indeed, our search on the Web of Science did not return any relevant reference on the notion of “spatial” hydrographs, with a few exceptions. These exceptions include Hovmöller diagrams of discharge displayed by Biancamaria et al. (2011) and Tourian et al. (2017) and space-based river widths longitudinal profiles measured from optical and SAR imagery to estimate discharge series in the context of flow wave propagation (Alsdorf et al., 2007; Brakenridge et al., 1998; Smith & Pavelsky, 2008). On a broader level, the connection between spatial and temporal scales in hydrological processes has been explored to a greater extent in the past (e.g., the heterogeneity between temporal hydrographs at different locations, or the relationship between catchment area and precipitation variability or flood frequency; Blöschl & Sivapalan, 1997, 1995; Breinl et al., 2020; Skøien et al., 2003). However, the formal analysis of discharge series indexed in space (i.e., as a function of the distance measured along the river's centerline from the source of the river, called chainage or streamwise distance in geomorphology) surprisingly appears to have never been formally introduced.

The representation of discharge data along the chainage, hereafter referred to as a “spatial hydrograph” or “discharge space series”, has likely rarely been investigated by the hydrological community because building and maintaining in situ gaging stations in sequence at multiple locations is costly and burdensome. Consequently, the spatial resolution of discharge ground data along rivers falls short of the fine temporal resolutions available today with modern data loggers (e.g., 5 min). With routing mechanisms through open channels relatively well known for over a century (Barré de Saint Venant, 1871; Manning, 1891), hydrologists have thus traditionally relied on discharge temporal dynamics at a few river stations (often at confluences and the outlet) and on advanced modeling frameworks (e.g., Alfieri et al., 2020; Collins et al., 2024; Hales et al., 2022; Harrigan et al., 2020; Lin et al., 2019; Yang et al., 2021) to gain insights into the overall management of river basins.

Nevertheless, the occasional presence of consecutive gages has been used to approximate average flow wave travel times using lagged cross-correlations (e.g., Allen et al., 2018; David, Maidment, et al., 2011). Assuming flow waves propagate at a celerity between ~0.5 and 1.4 m.s⁻¹ (David, Habets, et al., 2011; David et al., 2013, 2015, 2019; Oki et al., 1999; Smith & Pavelsky, 2008), the distance traveled by a flow wave in just 1 day could range from 43 to 121 km. In addition, because the impact of precipitation and runoff production spatial variability on local streamflow hydrographs (Potdar et al., 2021) is oftentimes diminished by the damping and dispersive effects of routing, the impacts can be undetectable by the time the water reaches a downstream gage (Pokhrel & Gupta, 2011). Having sufficient spatial sampling for observations along rivers would thus improve our knowledge and understanding of fine-scale flood processes.

Yet the measurement density required along a river to precisely identify flood boundaries, as well as the extent to which flow waves propagate over distance, remains uncertain. A change of perspective in the study of rivers from temporal hydrographs to spatial hydrographs can help to determine the required spatial sampling configuration to accurately detect, characterize, and in turn estimate the spatial scales of flow waves. Spatial hydrographs also unveil their significance in the particular context of repetitive satellite missions, that is, satellite missions that follow a pre-determined repeating orbit, providing recurring observations over their intended targets. Indeed, the sampling limitations of satellites lead to incomplete observations of rivers in both space and time (Sikder et al., 2021; Wade et al., 2025). While altimetry missions with sub-weekly (or even daily) revisit at virtual stations (i.e., where the satellite ground track crosses a water body) have recently been praised in hydrology (Hydro-space, 2021), the inherent trade-offs of orbital dynamics necessitate a careful consideration of spatial sampling requirements to ensure comprehensive coverage of rivers while fluvial floods propagate.

In this context, the recent Surface Water and Ocean Topography (SWOT; Fu et al., 2024) mission has introduced a significant innovation through its implementation of a wide swath measurement technique enabled by the mission's main payload, namely the Ka-band Radar Interferometer (KaRIn) instrument (Enjolras & Rodriguez, 2009; Peral et al., 2024). This advancement provides altimetry “images”, that is, spatial snapshots of rivers/basins, a departure from the traditional approach of relying solely on single virtual stations with conventional nadir altimetry measurements. SWOT is thus providing information on spatially distributed phenomena such as the propagation of flood waves along rivers (Biancamaria et al., 2016; Durand et al., 2010, 2016; Paiva et al., 2015; Pavelsky, 2014) and will allow, for the first time ever, the analysis of spatial hydrographs for width, elevation, and discharge at the node (200 m) or reach (~10–20 km) scale (Altenau et al., 2021; Durand et al., 2023; Frasson et al., 2017).
While the goal of this paper is not to exhaustively review the opportunities behind the use of spatial hydrographs, it proposes one way to leverage their utility by studying peak flow events (PFEs), making the concept of discharge space series the focus of its own dedicated scrutiny. PFEs in rivers refer to periods of significantly increased water flow rate and are critical in hydrology because they often indicate the river's response to specific natural or anthropogenic conditions like heavy rainfall, snowmelt, or dam releases. Better monitoring of these events is critical for flood forecasting and mitigation. In this study, we consider these events to occur at a particular location whenever discharge exceeds a local flood peak threshold. In previous work, PFEs above the 90th discharge percentile were revealed to last between 2 days upstream to 30 days downstream of the largest rivers of the Mississippi basin (Cerbelaud et al., 2024b). The detection of PFEs in spatial hydrographs allows PFE lengths, that is, the cumulative distance between consecutive locations where discharge exceeds its threshold, to be estimated analogously across the basin at any given time. Here we aim to answer the following research questions: (a) How can we detect peak flow events in space using spatial hydrographs? (b) How "long" are flow waves along rivers of the Mississippi River basin at a given time? (c) What are the required spatial sampling strategies for capturing PFEs across river networks using spaceborne instruments? and (d) How can the connections between temporal and spatial scales of peak events inform global orbital trade-offs? Hence, instead of studying when PFEs occur and how long they last at a given location as was done previously (Cerbelaud et al., 2024b), we focus on estimating where they occur and how spread out they are at a given time. To document the variability in the lengths of PFEs over time, we rely on a dense network of in situ gages in one of the world's largest and most highly anthropized river basin. We then evaluate PFE lengths under different spatial sampling scenarios to discuss the implications for the design of satellite missions concerned with the capture of peak river discharge in space, particularly in the context of the SMall Altimetry Satellites for Hydrology (SMASH, Blumstein et al., 2019) constellation that is currently under study by the scientific community with the support of the French Space Agency CNES. We also take advantage of the spatial hydrograph concept to investigate the connection between temporal and spatial scales of flooding in the Mississippi basin to further inform mission observation trade space.

## 2. Methods

Our methodology consists of three key steps:

1. Data preparation: reference discharge observations from in situ gages are selected and matched to corresponding river reaches within a digital hydrography network (Section 2.1).

2. Spatial hydrograph construction: spatial hydrographs are built through spatial indexing using river network topology. Peak flow events are identified in space using the newly defined spatial hydrographs, and their spatial coverages (i.e., lengths) are defined and measured (Section 2.2).

3. Spatial sampling strategy: key variables, such as the sampling interval and ratio, are introduced to guide the observation of peak flow events in space using suitable spatial sampling strategies, particularly in the case of remote sensing (Section 2.3).

This methodology is eventually directly applied to various SMASH-like orbital configurations to evaluate their relevance for observing the full extent of peak flow events in space.

### 2.1. Reference In Situ Discharge Observations

We use instantaneous streamflow observations on the Mississippi River Basin retrieved from the USGS Water Services Test Tool (2024) over the 2010–2022 period (Figure 1). We use the same data set of gages as Cerbelaud et al. (2024b), using the Global River Width From Landsat (GRWL; Allen & Pavelsky, 2018a) database and the MERIT Hydro Basins river network data set (Lin et al., 2019; Yamazaki et al., 2019; Yang et al., 2021) to select satellite-observable reaches. The data set contains 518 stations with at least 10 years of daily observations over the 2010–2022 period. We match (or “snap”) all 518 stations in our study to the nearest reach in the MERIT Hydro Basins hydrography network by geometric distance with a 0.005° (~500 m) buffer. The allocation to the digital river network is successful for all stations, with a unique reach allocated to each gage, and a matching distance of approximately 20 m (25th percentile) to 100 m (75th percentile). Upstream area for all 518 gaged reaches is greater than 1,000 km² and average, 5th, 25th, 50th, 75th, and 95th percentiles of upstream area are respectively 93,700, 1,800, 4,800, 13,300, 38,000, and 237,000 km². Based on the estimated length of river reaches documented in MERIT Hydro Basins, the median distance between gages is 75 km (see Section 2.2 for the matching of gages to digital hydrography). The mean and 75th quantile distance values are both equal to 138 km. The full
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Map of USGS gages used in this study. Cumulative distribution function of the distance between consecutive stations (i.e., irregular spatial resolution Δx of our data set) as computed from the MERIT Hydro Basins vector hydrography is shown in the top right corner.</figcaption></figure>

distribution of our data set's irregular spatial resolution Δx is displayed in the top right corner of Figure 1. The interested reader is referred to Cerbelaud et al. (2024b) for more information about how this gage data set was established and to the Data Availability Statement to access the actual data set.

## 2.2. Spatial Hydrographs and PFE Characterization in Space

### 2.2.1. Building Spatial Hydrographs

To build space series and detect PFEs along longitudinal river profiles at any given date, a spatial index representing how gages are positioned relative to one another is needed. We associate each station with its next downstream station using the MERIT Basins river network topology iteratively and compute the longitudinal length (i.e., the curvilinear distance, as opposed to a direct length that ignores river sinuosity) between both gages. We then determine that 133 stations do not have an upstream station and are thus the starting point of our space series. The 133 series of consecutive stations, hereafter referred to as UTD (Upstream To Downstream) lists, are thus established as individual space series (see Figure 2 for an example of a UTD list, the Yellowstone-Missouri-Mississippi river continuum). We define the variable *x* both to designate a gage/station or equivalently its distance from the most upstream gage in a UTD list (starting point of the *x*-axis in the spatial hydrograph) measured along the river centerline, called chainage. At each date *t* and for every UTD list, the series of discharge records $Q(x, t)$ composed of all the gages *x* between the most upstream gage to the last common outlet gage ("Mississippi river at Belle Chasse, LA") are called "reference" discharge space series. Note that although they are all unique, a significant number of UTD lists share gages, particularly toward the downstream gages. The effect of duplicates in event detection is discussed at the end of Section 2.2.2.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Spatial hydrographs of river discharge and their analysis for PFE detection. (a) Discharge values on 7 May 2017, in the Mississippi River Basin, depending on whether the value exceeds the station-specific 90th quantile (red) or not (cyan). (b) Space-time representation of streamflow along the Yellowstone-Missouri-Mississippi river continuum (1 of 133 UTD lists) at three dates and illustration of the calculation of PFE lengths *L*<sub>90,t</sub>(*t*) and resulting median length *L*<sub>90</sub>(*t*) on reference data. (c) Illustration on 200-km sampled data. Note: On 14 May 2017, in the reference gage data space series (purple), *N*<sub>90</sub> = 2 events are detected, *S*<sub>90</sub> = 12 stations are involved in the events, and *L*<sub>90</sub> ~ 1110 km.</figcaption></figure>
### 2.2.2. PFE Definition and Detection in Spatial Hydrographs

We define PFEs to occur at a particular location $x$ of the constructed spatial hydrographs whenever discharge $Q(x, t)$ exceeds the long-term 90th discharge percentile of the station $Q_{90}(x)$ (Figure 2). This peak threshold has already been used in recent studies associated with the detection of river flow variability from space (Cerbelaud et al., 2024b; Sikder et al., 2021), although other thresholds or peak detection technique based on recurrence intervals could equally be used (England et al., 2018). Rather than recording when discharge lies above the threshold at a particular gage to estimate the event duration (Cerbelaud et al., 2024b), we aim to determine the event spatial extent at a particular moment in time. Using the constructed spatial hydrographs, we designate PFEs in space as, at any given time $t$, lists of two or more consecutive stations (from upstream to downstream) in a UTD list where streamflow $Q(x, t)$ continuously exceeds $Q_{90}(x)$ (see red bubbles in Figure 2). Then, we estimate PFE lengths as the cumulative distance between the most upstream station $x_{up}$ and most downstream station $x_{down}$ where discharge continuously exceeds its threshold.

We document for each day of the study period (4,748 days) the number of PFEs across all the UTD lists of the basin, their median length, and the number of stations involved in all the events of the day. This last variable allows for a consistency check between our PFE spatial detection methodology and the classic temporal approach used in Cerbelaud et al. (2024b). We thus define three variables $N_{90}(t)$, $L_{90}(t)$ and $S_{90}(t)$, corresponding respectively to the number of events, their median length, and the number of stations involved in all the events where $Q(x, t) \ge Q_{90}(x)$, for all UTD lists at time $t$. The median length $L_{90}(t)$ is derived from the list of all PFE lengths recorded over all the UTD lists at time $t$. The median is preferred over the mean to better control for outlier effects:

$$L_{90}(t) = \text{median}[L_{90,i}(t) = x_{i,\text{down}} - x_{i,\text{up}}, i = 1, \dots, N_{90}(t)] \quad (1)$$

Given the distance between consecutive gages (Figure 1), multiple small PFEs can occasionally be recorded as one using our approach if the distance between two consecutive gages is larger than the length of the PFEs. We assume here that, considering the temporal analysis of Cerbelaud et al. (2024b) and the estimates of average flow wave celerity (~1 m.s⁻¹; Smith & Pavelsky, 2008), the shortest PFEs corresponding to 1–2 days durations should cover around 85–170 km in space and hence mostly be correctly recorded as one event over two consecutive gages in our data set. However, we acknowledge that our gage-based spatial hydrograph methodology is not well suited to investigate short local flow waves. In addition, when scanning through UTD lists sharing a significant number of downstream gages, some events (or part of them) are recorded multiple times. To alleviate for such potential double counting, we removed any PFE that is strictly included in a larger one. When two PFEs partly share the same downstream stations, both are kept for the calculation of $L_{90}(t)$ and $N_{90}(t)$, but duplicate stations are only counted once for the calculation of $S_{90}(t)$. Overall, a single unique PFE can only be counted once at a given date $t$ and can only be recorded in a single UTD list.

Note that we adopt in this work a “snapshot” perspective, focusing on spatial events observed each day, as they could/would be seen by satellites at the time of overpass. We do not seek to track events over time by recording prior upstream occurrences of the same events. Consequently, events persisting over multiple days are counted each day they remain in the network and the sum of the number of events $N_{90}(t)$ over a given period cannot be interpreted as a number of basin-scale PFEs over that period. While this does not undermine the purpose of our analysis, we additionally note that the use of daily data (instead of, e.g., hourly), increases the relative probability of capturing long events compared to intra-daily events.

## 2.3. Gage-Based Spatial Sampling Strategy

### 2.3.1. Spatial Sampling Variables

Once spatial hydrographs are constructed and the PFE detection methodology is established on the reference discharge space series, the reference series are spatially sub-sampled to analyze the implications for observational scenarios in river-monitoring satellite missions. We define two variables to refer to the sampling scenarios. The first one is the sampling interval $\Delta L$, which is set as a fixed distance. We implement 25- to 250-km sampling intervals, $\Delta L$, with a 25 km increment, composing 10 sampling scenarios, over each of the 133 UTD space series. A 25 km increment was chosen as a trade-off between oversampling and undersampling given that only 10% of stations are less than 25 km apart and half of stations are 75 km apart or less. The second variable is the sampling
ratio ΔL/L90(t), defined as the sampling interval as a percentage of a date's reference PFE median length L90(t). Hence, for ΔL/L90(t) = 30%, each streamflow space series is sampled every 30% of the reference L90(t) of the studied date t. A single value of ΔL in a spatial sampling experiment thus leads to various values of ΔL/L90(t) temporally. This ratio between ΔL and L90(t) effectively helps defining how to capture PFEs in space and estimate their lengths with a desired level of accuracy and consistently at different times of the year. Note that L90(t) is a representative length of all events at time t (the median)—some might be larger, some smaller—that we use as indication to set the spatial sampling ratio.

For a single sampling interval of ΔL km, sampling can start anywhere between x = 0 (most upstream gage) and x = ΔL - Δx(x) km, with Δx the (irregular) discharge space interval in the series (i.e., the distance between consecutive stations, or spatial resolution). This introduces an uncertainty range in the sampling performance for a single value of ΔL, as some events can be caught or missed based on where the sampling starts. Like in our previous study (Cerbelaud et al., 2024b), 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 interval but shifted in space may not adequately observe them. Hence, we implement 5 sub-scenarios per ΔL values to account for this effect as much as possible.

Spatial sampling is then performed as follows for each value of ΔL (i.e., 10 sampling scenarios). For each of the 133 UTD lists, we start upstream at the first station x = 0. We first select as initial station the one closest to chainage x = h/5 * ΔL km, with h = 0...4, for all 5 sub-scenarios. Depending on how many stations are in the [0, 4/5 * ΔL] space interval, we eventually obtain between 1 and 5 sub-sampled UTD lists with one initial gage. We then scan through all the remaining downstream stations of the UTD list and only select the closest stations every ΔL km from that initial station. We obtain up to 5*133 sub-sampled UTD lists for each of the 10 sampling scenarios. Note that because the distance Δx between two consecutive stations is irregular, the sub-sampled UTD lists do not always precisely reflect the sampling interval ΔL. This limitation comes from the use of in situ gages for this work and is discussed in Section 4.

### 2.3.2. PFE Detection in Sampled Spatial Hydrographs

When applying event detection on these “sampled” data, PFE lengths are artificially inflated or deflated at the start and end of events due to the computation on larger, still irregular, space intervals (Figure 2c). A linear interpolation at the start and end of spatial events is thus implemented to better approximate the lengths L90,i(t), especially on sampled data. With Δx the discharge space interval in the series, reference or sampled, and x the first gage where Q(x, t) ≥ Q90(x), a simple cross-multiplication is applied to calculate the starting length ΔL90,i(t):

$$ \Delta L_{90,i}(t) = \Delta x(x) \frac{Q(x, t) - Q_{90}(x)}{Q(x, t) - Q(x - \Delta x(x), t)} \quad (2) $$

A similar calculation is performed at the end of events.

We apply the event detection procedure of Section 2.2.2 on the sub-sampled UTD lists to compute the PFE number Ñ90(t), median length L̃90(t) and number of stations involved S̃90(t) from the sampled space series at each date t. Sampled variables will be hereafter referred to with a ~. The absolute value of the relative error for all reference variables X = N90, L90, S90 per date associated with a given sampling period ΔL or sampling ratio ΔL/L90(t) is:

$$ \varepsilon(X(t)) = \left| \frac{\tilde{X}(t)}{X(t)} - 1 \right| \quad (3) $$

To account for the previously mentioned serendipity effect, each sampling performance (i.e., for each value of ΔL or ΔL/L90(t)) is portrayed by the ensemble of sub-scenarios available with its associated average and best/worst error values. Results reflecting multiple dates at once are shown using the median of all dates errors rather than the mean to limit the influence of outliers.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Cumulative distribution functions of $N_{90}(t)$, $L_{90}(t)$ and $S_{90}(t)$ over the 4,748 days examined in this study (2010–2022). The curves depict the temporal statistics of the detection of PFEs in the 133 UTD space series of the Mississippi River Basin.</figcaption></figure>

# 3. Results

## 3.1. PFEs Across Space in the Mississippi River Basin

Multiple PFEs across the basin can be observed on 7 May 2017, as an illustration (Figure 2a). Each unique list of consecutive “red” stations is recorded as a PFE (and only once, even if it is present in multiple UTD lists). Between 30 April, 7 May and 14 May 2017, the PFE highlighted in the dotted frame in the Yellowstone-Missouri-Mississippi UTD list can be seen shifting downstream of several hundreds of km, with upstream stations receding and downstream stations flooding progressively (Figure 2b).

The temporal median (mean) of the number of events $N_{90}(t)$ over the whole basin is 6 (12) spatial events per day (Figure 3). No events are recorded on ~16% of the dates studied. The temporal median (mean) of the number of stations involved in the PFEs $S_{90}(t)$ is 19 (39) stations per day. Hence only 7.5% of stations are involved in PFEs each day on average. $S_{90}(t)$ peaks at around 200 stations for the more widespread events, representing almost 40% of all stations at high flow on the same day (see Figure 3 for the distribution of each variable).

Over the 4,748 days of discharge space series analyzed (2010–2022), a total of ~185,000 occurrence of PFEs are recorded (~39*4,748, one occurrence = 1 day at one station), which corresponds to ~357 days of PFEs per station on average. According to our past temporal investigation of PFEs (Cerbelaud et al., 2024b) and from our own reanalysis of these results, ~62 temporal PFEs were recorded over the same period on average per station, with a mean duration of 5.6 days. This corresponds to a total of ~347 days of PFEs recorded per station and over the whole period of study. Both the temporal and spatial methods are thus consistent (357 vs. 347 days of PFEs per station), with slight differences likely due to the downgraded temporal resolution of the data used in this study (daily) versus the finer hourly in our previous work (Cerbelaud et al., 2024b).

Peak flow events approximately range from around 75 to 1,800 km in length across the Mississippi River Basin. The temporal median (mean) of the length of events $L_{90}(t)$ is 330 (520) km, with 20% of days featuring a median event length smaller than 200 km or larger than 800 km (Figure 3). We emphasize again that any PFE strictly included within a larger one is removed from this analysis, and each PFE can only be recorded in one UTD list. Therefore, potential partial duplicate information is limited and can only originate from PFEs featuring distinct upstream locations but identical downstream locations.

Finally, integrating into our spatial framework the exactly analogous analysis developed by Cerbelaud et al. (2024b) on the duration of PFEs detected from temporal hydrographs, the connection between the temporal and spatial scales of flooding can be revisited and discussed. It emerges that events with longer durations also affect larger maximum extents during their occurrence (250 km median length for 0–3 days long events vs. 520 km for 9–12 days events, Figure 4). This highlights the need for both high temporal and high spatial sampling to monitor PFEs in small flashy sub-basins, which can only be achieved from space using constellations of satellites (see Section 4). In addition, with PFE median (mean) lengths of ~330 (520) km estimated in this work and median (mean) durations of ~3.9 (5.6) days reported in Cerbelaud et al. (2024b), we can assume that flow
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Maximum length $L_{90}(t)$ measured during PFEs depending on their duration (days).</figcaption></figure>

waves travel at a celerity of approximately 0.98–1.07 m.s⁻¹ across the Mississippi basin, which is remarkably consistent with previous space-based flow wave propagation studies (Allen et al., 2018; Smith & Pavelsky, 2008).

## 3.2. Sampling of PFE Number, Length and Number of Stations Involved

Following the investigation of PFEs using spatial hydrographs at their “native” resolution, we seek to study the required spatial sampling strategies for satellite missions concerned with the capture of PFEs in space. An example of PFE detection with a 200-km sampling interval is illustrated in Figure 2b. An error on the median length of events $\epsilon(L_{90}) \sim 23\%$ is found for date A ($L_{90} = 505$ km and $\tilde{L}_{90} = 620$ km) as opposed to only around $\sim 11\%–14\%$ for dates B and C (respectively $L_{90} = 1,020$ and $1,110$ km, $\tilde{L}_{90} = 1,160$ and $990$ km; Figure 2b). As expected, smaller sampling intervals are needed to accurately capture PFEs when smaller lengths, that is, closer, or fewer stations, are involved (such as for date A).

The distribution function of the number of events $N_{90}(t)$ is not much affected by a coarser spatial sampling, at least up to $\Delta L = 250$ km (Figure 5a). The mean number of spatial events recorded per day decreases from 12 (reference) to 9 in the coarsest scenario ($\Delta L = 250$ km). This represents a 25% decrease in the number of recorded events that could impact anthropized areas without flood monitoring. The average number of stations involved in these events $S_{90}(t)$ is much more impacted and decreases from 39 (reference) to 16 ($\Delta L = 250$ km). This means that although the number of stations involved in the events decreases significantly with larger sampling intervals, a significant number of events are still observed. The impact on the distribution of the length of events $L_{90}(t)$ is also visible in Figure 5a. The larger the sampling interval $\Delta L$, the fewer small events are recorded (rising edge of the $L_{90}(t)$ CDF). In particular, $L_{90} = 400$ km corresponds to the 2nd decile for $\Delta L = 250$ km sampling interval as opposed to the 6th decile for the reference data. The mean $L_{90}$ per day increases from 520 km (reference) to 805 km for $\Delta L = 250$ km.

The estimation of event lengths (number of events and stations involved) from spatially sampled series suffers indeed from two main challenges leading to upward (downward) biases. First, small events are increasingly missed with higher $\Delta L$ due to the increased number of stations removed, so the longest events are more likely to be captured. Second, distinct but spatially consecutive events can be counted as one single/longer event when stations that are below their quantile threshold and located in-between the events are removed following the sampling process.

## 3.3. Guiding Principle for Spatial Sampling of PFEs

The accuracy in the capture of PFEs for a given sampling interval $\Delta L$ depends on their actual length $L_{90,i}(t)$. To help defining how to capture PFEs in space and estimate their lengths with a desired level of accuracy and
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Effects of spatial sampling on PFE number, length and number of stations involved. (a) Cumulative distribution functions of $N_{90}(t)$, $L_{90}(t)$ and $S_{90}(t)$ over the 4,748 days examined in this study (2010–2022) as a function of the sampling interval $\Delta L$. (b) Median absolute relative errors $\epsilon(X)$ on the PFE number $N_{90}(t)$ (red), median length $L_{90}(t)$ (blue) and number of stations involved $S_{90}(t)$ (green) from sampling as a function of the sampling ratio $\Delta L/L_{90}(t)$ (in %, maximum sampling interval $\Delta L = 250$ km) for all acquisition dates, with their best/worst performance interval. Date A from Figure 2b is placed when sampled at $\Delta L = 200$ km for illustration purposes.</figcaption></figure>

consistently at different times of the year, we explore a parallel with digital signal processing (similarly to Cerbelaud et al. (2024b) with time series). Perhaps the best analogy to the question of optimal sampling for capturing PFEs in a discharge series lies in the Nyquist-Shannon sampling 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 Nyquist-Shannon theorem does not apply to spatial hydrographs *per se*, we can draw inspiration from it to establish a general guiding principle regarding the impact of PFE lengths on spatial sampling requirements to observe floods across space. Hence, we quantify the range in error $\epsilon$ for $N_{90}(t)$, $L_{90}(t)$ and $S_{90}(t)$ as a function of the sampling ratio $\Delta L/L_{90}(t)$ on each day, and for all dates combined between 2010 and 2022 (Figure 5b).

Median $\epsilon(N_{90})$, $\epsilon(L_{90})$ and $\epsilon(S_{90})$ over all dates from Figure 5b are given in Table 1. The large error intervals emphasize the erratic dimension in the estimation of the length $L_{90}(t)$ and even more so the number of PFEs $N_{90}(t)$ and the number of stations involved $S_{90}(t)$ when using spatially sampled data, solely due to serendipity (i.e., different sampling patterns with the same sampling interval). This variability is albeit exacerbated by the nature of our data set (gage-based discharge series with irregular space interval $\Delta x$).
<table><caption>**Table 1**</caption><caption>Range in Median Error ε for N<sub>90</sub>(t), L<sub>90</sub>(t) and S<sub>90</sub>(t) as a Function of the Sampling Ratio ΔL/L<sub>90</sub>(t) Over All Dates (± Refers to the Serendipity Effect)</caption><thead><tr><th>Sampling ratio ΔL/L<sub>90</sub>(t)</th><th>1/4</th><th>1/2</th></tr></thead><tbody><tr><td>Error on the number of events ε(N<sub>90</sub>)</td><td>19% ± 13%</td><td>30% ± 20%</td></tr><tr><td>Error on the median length of events ε(L<sub>90</sub>)</td><td>16% ± 3%</td><td>33% ± 12%</td></tr><tr><td>Error on the number of stations involved in events ε(S<sub>90</sub>)</td><td>28% ± 15%</td><td>44% ± 18%</td></tr></tbody></table>

The error range $\epsilon(L_{90})$ for date A, centered around 21%, can also be read on Figure 5b with a sampling interval $\Delta L = 200$ km as an example ($\Delta L/L_{90}(t = A) \sim 0.4$). Another representation of $\epsilon(L_{90})$ as a function of $L_{90}(t)$ for several values of $\Delta L$ is given in Figure S1.

Echoing the Nyquist-Shannon theorem, a potential guiding principle for river spatial sampling requirements to capture PFEs and their lengths in rivers is thus: a sampling interval at least 4x (2x) finer in resolution than the median PFE length is needed to estimate, on a median basis, the number of events and their median length within a 19% ± 13% (30% ± 20%) and 16% ± 3% (33% ± 12%) median relative error range, respectively.

### 3.4. Implications for Satellite Sampling

Following the definition of a guiding principle for spatial sampling of river flow to detect PFEs in space, we evaluate our findings in the context of a hypothetical concept for a nadir altimetry satellite constellation with sub-weekly revisits (SMASH, Blumstein et al., 2019) using three possible repetitive orbit strategies ($a = 1$ day revisit period, 250 km intertrack distance at the equator; $b = 2$ days, 125 km; and $c = 4$ days, 62.5 km). Our previous temporal study demonstrated that 1-day, 2-day and 4-day revisit periods allow capturing PFE durations accurately at 93%, 46% and 15% of gages, respectively (Cerbelaud et al., 2024b). However, the spatiotemporal trade-off imposed by orbital configurations implies that shorter revisit periods can only be obtained at the cost of coarser spatial sampling. The following investigation aims to offer a spatial analysis component for evaluating trade-offs in spaceborne river observation.

To be consistent with our PFE length estimation methodology and definition of the sampling interval $\Delta L$, we sample our rivers in space following the exact same methodology described in Section 2.3 (i.e., following the river network), rather than using the simulated orbit tracks. The limitations implied by this choice are discussed in Section 4. At the latitude of the Mississippi River Basin, the ground track spacing is around 200, 100, and 50 km respectively for the three scenarios. In the following paragraph, acquisition dates (AD) refer to the dates that would be sampled in the three scenarios (every 1, 2 or 4 days). To illustrate “satisfactory” ADs and linking them to our guiding principle, we choose a 35% error limit on $\epsilon(L_{90})$ corresponding approximately to when $L_{90}(t) \gtrsim 2 \Delta L$ (Figure 5b and Table 1).

Among all spatial events occurring over 2010–2022 in the reference data, 20% of them are never observed with $\Delta L = 200$ km, that is, in scenario (a). Among all dates featuring a $L_{90}(t)$ strictly positive (i.e., when at least one spatial event is recorded, which corresponds to 84% of dates), 41% show $\epsilon(L_{90})$ errors below 35% in that same scenario (a) (Figure 6a, 1,644 “satisfactory” acquisition dates out of 3,984). However, only 23% of event dates with median length <330 km are “satisfactorily” captured with such coarse spatial sampling. Still, the 1-day temporal revisit associated with the 200-km spatial sampling would enable a great capture of event durations across the basin, and would be invaluable for small, low-order streams with high-frequency dynamics (Cerbelaud et al., 2024b). Overall, scenario (a) is by far the best one for capturing and monitoring in time the biggest spatial events spanning over 330 km (median $L_{90}(t)$) and more in length, while providing accurate temporal information at most virtual stations.

With finer sampling intervals, among all spatial events occurring over 2010–2022 in the reference data, only 8%/4% of them are never observed with $\Delta L = 100$ km/50 km, that is, scenarios (b)/(c). Scenario (b) features an overall 64% “satisfactory” coverage of overall event lengths (Figure 6b). In scenario (c), with the best spatial sampling capabilities (4-day & 50 km), 82% of acquisition dates result in “satisfactory” estimation of spatial event lengths (Figure 6c, see also Figure 5a). In addition, PFE median lengths <330 km are “satisfactorily” captured in 49%/72% of acquisition dates, with 477/350 dates out of 981/484 featuring $\epsilon(L_{90})$ errors lower than 35%, in
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Median length of events *L*<sub>90</sub>(*t*) (bar heights) as a function of time for three conceptual mission scenarios with “satisfactory” (green) and “bad” (red) acquisition dates (AD) depending on the implemented spatial sampling. Here, “satisfactory” acquisition dates refer to when PFE median length error ε(*L*<sub>90</sub>) is below 35% with the spatially sampled data.</figcaption></figure>
scenarios (b)/(c) (Figures 6b and 6c). Overall, the 100-km & 2-day sampling scenario (b) provides the best coverage of spatial events with median lengths <330 km of all three scenarios (477 “satisfactory” acquisition dates). However, scenarios (b) and (c) only provide 2- to 4-day revisit periods, thus limiting their benefits for temporal analysis to large rivers with lower frequency temporal dynamics. Furthermore, a 4-day temporal sampling would significantly deviate from our spatial hydrograph design which requires that all river reaches in the basin are observed within an interval of no longer than 1 day (see Section 4 for more on this topic).

## 4. Discussion

In this study, we reveal an estimation of PFE lengths in the Mississippi basin based on USGS gage data, along with a recommended spatial sampling interval between consecutive observations along rivers. With a reference PFE median length $L_{90} \sim 330$ km and the need for a sampling interval at least 4x (2x) finer in resolution to accurately estimate PFE coverage, the 80 km (160 km) mark provides a minimal spatial sampling benchmark to achieve. These findings allow us to present implications for potential satellite constellation configurations, factoring in the spatiotemporal orbital trade-off. However, a few caveats need to be highlighted. First, limitations on the transferability of our derived guiding principle (Figure 5b) along with its practical implications (Figure 6) to other catchments must be acknowledged, as discharge dynamics vary globally depending on rainfall patterns (Mandapaka et al., 2009), environmental conditions (tropics, temperate areas) and geomorphic properties of river basins (Rigon et al., 2011). Second, because our PFE length estimation and sampling methodologies rely on curvilinear distance following the MERIT Hydro Basin hydrograph, we cannot directly transpose our findings to actual requirements for ground track spacing of nadir altimetry missions. For a given orbit strategy, ground tracks indeed can cross rivers at multiple angles and the resulting river spatial sampling is irregular. We acknowledge that there are thus important limitations to generalize our results to actual satellite acquisition strategies.

However, to contextualize our study within the framework of past and current nadir altimetry missions used to observe river elevation, this study reveals spatial sampling requirements aligning with ENVISAT, SARAL and ERS-1/2 (80 km) or Sentinel 3A/B combined ground track spacing (52 km). Nevertheless, the intertrack distance reported for satellite missions results from a complete cycle (e.g., 35 days for ENVISAT or SARAL) and final neighboring tracks are not sensed during consecutive orbits. Hence, data across a basin coming from a single satellite with sole nadir capabilities are not entirely acquired on the same day at the reported ground track spacing. The spatial analysis of discharge on a given river would thus involve a bundle of acquisition dates irregularly spanned and interpreting the spatial hydrograph would be nonsensical. Our assumption of daily snapshots of river profiles can only be approximately valid if one considers that all the data is acquired by a constellation of satellites with a global (a) 1 or (b) 2 days revisit period and an intertrack spacing (a) $D \sim 250$ km or (b) $D \sim 125$ km over a full cycle, such that any river of length up to $\sim 2,500$ km can be sensed with a spatial sampling of $D$ km in a maximum of a little over a day (cf. the SMASH constellation concept comprising 10 small satellites; Blumstein et al., 2019).

Another shortcoming for application to satellite sampling arises from our framework completely disregarding uncertainties in altimetry measurements and water surface elevation retrieval (e.g., range window tracking system, waveform retracking algorithms; Biancamaria et al., 2018, 2017; Cretaux et al., 2017; Zakharova et al., 2020) and discharge algorithms (Durand et al., 2016, 2023; Garambois et al., 2017; Kouraev et al., 2004; Papa et al., 2012; Paris et al., 2016; Tarpanelli et al., 2013; Tourian et al., 2013, 2017). Indeed, we intrinsically assume in this work that any water surface elevation data retrieved could be accurately transformed into discharge. Yet, consistently estimating global discharge from altimetry (or from other types of satellite data) is still considered an important challenge (e.g., Elmi et al., 2024).

We underline that while analyzing time series data at a single station (i.e., the common hydrograph representation) involves a one-dimensional approach with regular time steps (e.g., hourly, daily), our spatial framework features as many dimensions as UTD lists (hence 133) for each date and is based on space series with irregular spatial steps (see Figure 1 or Figure 2). This introduces a bias in both the estimation of event lengths $L_{90,i}(t)$ and the sampling process, for instance in parts of the Missouri which are not frequently gaged. Although model simulations could provide seamless and near-regular $\sim 10$ km resolution daily model outputs (Lin et al., 2019; Yang et al., 2021, 2023) for a more consistent spatial analysis (as was done in Sikder et al., 2021), their ability to reliably capture the variability of discharge series remains limited due to parameter uncertainties, calibration challenges, and the inability to fully capture localized hydrological processes. In comparison, gage data show superior reliability and
accuracy in representing observed variability, which is critical to this study's specific objectives. In particular, the presence of hydraulic structures, reservoirs and lakes which can dampen or block the propagation of flow waves and impact the lengths of PFEs along rivers is naturally accounted for in our analysis. We also note that the selection of $Q_{90}(x)$ as the PFE threshold directly impacts the number and length of events we measure, with smaller and more events resulting from higher quantile thresholds. Nonetheless, our spatial sampling guiding principle was designed around the sampling ratio $\Delta L/L_{90}(t)$ so that it could be transposable to event lengths computed with other thresholds. In future studies, the selection and analysis of peak events could be achieved through more elaborate techniques involving precipitation intensity, patterns, and spatial extent, in an event-wise manner. Such investigations would allow not only measuring the lengths of PFEs, but also understanding the mechanisms underlying the observed spatial scales of events.

Finally, as previously described, this study cannot apply to traditional single-satellite exact repeat nadir altimetry missions. Nevertheless, data from geodetic altimetry missions such as CryoSat-2 with its 7.5 km intertrack distance at the equator and drifting daily measurements of water surface elevation during part of its 30-day sub-cycle share characteristics with our gage-based spatial series. Indeed, CryoSat-2 has shown potential for water propagation monitoring during PFEs (Boergens et al., 2019; Schneider et al., 2017; Villadsen et al., 2015). In addition, contrary to nadir altimeters, swath-based sensors like optical and SAR imagers or the KaRIn instrument aboard the SWOT satellite observe rivers with continuous spatial sampling (but at different resolutions), albeit neighboring swaths are also not acquired during consecutive orbits. This study hence emphasizes the need to consider PFE spatial scales not only for spatial sampling requirements in high revisit frequency nadir altimetry constellations and geodetic missions, but also for field of view (FOV) requirements for swath-based instruments. Finally, the guideline established in this study also suggests that models utilizing digital hydrography networks with reach lengths of approximately 10 km (i.e., spatial resolution; e.g., MERIT Basins) can effectively capture PFEs with a minimum length of 40 km.

## 5. Conclusions and Perspectives

Time series of discharge have been highly instrumental in hydrology for decades. Here we propose the concept of spatial hydrographs, or river discharge space series, as a valuable counterpart, and make it the focus of its own dedicated scrutiny. Using in situ gages and a hydrography data set, our work enriches existing literature by quantifying the spatial occurrence of PFEs and their lengths along river chainage as an analog to PFE duration, effectively applying a space-for-time substitution. PFEs are among the most difficult river traits to capture in time, making them an interesting analog in space. We carry the analysis each day across a 13-year period on the Mississippi River Basin. The robustness of our approach lies in employing a spatially dense network of in situ observations within one of the world's largest and most heavily anthropized basins.

Our study reveals reference PFE median (mean) spatial extent in the form of a length, $L_{90} \sim 330$ (520) km based on the 90th quantile peak threshold over the 13 years of data, with around 6 events and 19 stations (out of 518) involved in PFEs each day on a median basis, and no events recorded on ~16% of the dates studied. We assess the accuracy in the detection of PFEs and the estimation of their lengths under different spatial sampling configurations using the ratio between the sampling interval and the reference PFE median length as a guide, allowing us to establish a rule of thumb applicable at multiple dates and potentially over other basins. We show that a sampling interval 4-times (2-times) finer in resolution than the median flood length should allow capturing ~81% (70%) of PFEs on a given day (median accuracy) across the Mississippi River Basin and estimate their length with ~84% (67%) median accuracy. We also take advantage of the spatial hydrograph concept to investigate the connection between temporal and spatial scales of flooding and find that PFEs with longer durations affect larger maximum extents during their occurrence.

The framework established in this study anticipates the integration of SWOT 2D altimetry “images” and the analysis of flow waves across both temporal and spatial dimensions. Beyond exact repeat missions, the potential of geodetic-type orbits (long repeat, e.g., CryoSat-2) for mapping river longitudinal profiles and constructing time series of river levels calls for more investigation in the context of spatial hydrographs (Jiang et al., 2024). Further analysis on the factors governing the spatial scales of flow waves (e.g., precipitation, water surface slope, river morphology, hydraulic structures) will also constitute next steps for future research. Our work complements other existing studies dealing with temporal sampling requirements for capturing PFEs in time and estimating their durations, with recent works advocating a revisit frequency of approximately 2–3 days (Cerbelaud et al., 2024b;
Munasinghe et al., 2023) or at least no more than 5 days (Blaen et al., 2016; Gehring et al., 2023; Sikder et al., 2021; Stroud et al., 2024). While transitioning altimetry missions to sub-weekly revisits has been encouraged by the community of remote sensing scientists in the field of hydrology for some years (Hydro-space, 2021), this study underscores the importance of preserving adequate spatial sampling of rivers (every ~80–160 km) within the constraints posed by orbital trade-off limitations.

## Data Availability Statement

The global river network used in this study is available in MERIT Hydro Basins (2019) and the GRWL data set in Allen and Pavelsky (2018b). 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. (2024a). We follow the guidelines of NASA's Transform to Open Science (TOPS) mission for our open science practices. © 2025 California Institute of Technology. Government sponsorship acknowledged.

## 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), Earth Science U.S. Participating Investigator (NNH20ZDA001N-EUSPI), and Surface Water and Ocean Topography Science Team (NNH23ZDA001N-SWOTST) Programs.

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