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\begin{document}

\title{Lake topography and active storage from satellite observations of flood
frequency}



\author[1]{Alice César Fassoni-Andrade}%
\author[1]{Rodrigo Cauduro Dias De Paiva}%
\author[1]{Ayan Santos Fleischmann}%
\affil[1]{Institute of Hydraulic Research, Federal University of Rio Grande do Sul}%


\vspace{-1em}



  
  \date{June 17, 2024}


\begingroup
\let\center\flushleft
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\selectlanguage{english}
\begin{abstract}
Topography is critical information for water resources management in
lakes, and remote sensing provides a unique opportunity to estimate it
in ungauged regions. We introduce here a new method which estimates near
shore topography of water bodies based on a flood frequency map and time
series of water levels by assuming the equivalence between flood
frequency and water level exceedance probability at a given area. Test
cases are performed for two lakes and 12 hydropower reservoirs in Brazil
using the proposed Flood2Topo app. This new application generates the
bottom level pixel by pixel and a level-area-active storage
relationships directly from the topography map, without the need to fit
functions. Flood extent estimates from the Landsat based JRC Global
Surface Water (GSW) dataset, current state-of-the-art, were used to run
Flood2Topo, together with water levels from satellite altimetry and
in-situ gauges. Results show bottom level root mean square deviation
(RMSD) values of 18.5 cm and 146 cm for Lake Poop\selectlanguage{ngerman}ó (Bolivia) and Lake
Curuai (Amazon basin), respectively. For reservoir active storage, RMSD
normalized values ranged from 2\% to 11.09\% for 11 reservoirs (average
NRMSD of 6.39 \%). The method can be applied to any area seasonally
flooded, for instance, it is applicable in 35.8 \% (86\%) of the global
water surface area mapped by occurrence map from GSW dataset, when
considering the number of pixels with occurrence between 0 and 95\%
(99\%) over 35 years. This is a promising tool for obtaining data for
hydrodynamic simulations and monitoring of ungauged water bodies.%
\end{abstract}\selectlanguage{ngerman}%



\sloppy


\textbf{Lake topography andactive storage from satellite observations of
flood frequency}

Alice C\selectlanguage{ngerman}ésar Fassoni-Andrade\textsuperscript{1}*, Rodrigo Cauduro Dias de
Paiva\textsuperscript{1}, Ayan Santos Fleischmann\textsuperscript{1}

\textsuperscript{1} Institute of Hydraulic Research, Federal University
of Rio Grande do Sul (UFRGS), Porto Alegre, RS, Brazil. Caixa Postal
15029. Av. Bento Gonçalves, 9500. CEP: 91501-970.

*Corresponding author: Alice Fassoni-Andrade (alice.fassoni@ufrgs.br)

Key Points:

\begin{itemize}
\tightlist
\item
  New method for estimation of near shore topography and active storage
  for inland water bodies
\item
  Flood2Topo app converts water levels and Landsat based flood frequency
  into water body topography
\item
  Accurate representation of spatial patterns, 0.18 to 1.4 m errors on
  lake bottom level and \textless{}11\% error for reservoir active
  storage
\end{itemize}

Abstract

Topography is critical information for water resources management in
lakes, and remote sensing provides a unique opportunity to estimate it
in ungauged regions. We introduce here a new method which estimates near
shore topography of water bodies based on a flood frequency map and time
series of water levels by assuming the equivalence between flood
frequency and water level exceedance probability at a given area. Test
cases are performed for two lakes and 12 hydropower reservoirs in Brazil
using the proposed Flood2Topo app. This new application generates the
bottom level pixel by pixel and a level-area-active storage
relationships directly from the topography map, without the need to fit
functions. Flood extent estimates from the Landsat based JRC Global
Surface Water (GSW) dataset, current state-of-the-art, were used to run
Flood2Topo, together with water levels from satellite altimetry and
in-situ gauges. Results show bottom level root mean square deviation
(RMSD) values of 18.5 cm and 146 cm for Lake Poopó (Bolivia) and Lake
Curuai (Amazon basin), respectively. For reservoir active storage, RMSD
normalized values ranged from 2\% to 11.09\% for 11 reservoirs (average
NRMSD of 6.39 \%). The method can be applied to any area seasonally
flooded, for instance, it is applicable in 35.8 \% (86\%) of the global
water surface area mapped by occurrence map from GSW dataset, when
considering the number of pixels with occurrence between 0 and 95\%
(99\%) over 35 years. This is a promising tool for obtaining data for
hydrodynamic simulations and monitoring of ungauged water bodies.

1 Introduction

The bathymetry of natural and artificial lakes is critical information
for navigation, hydrological, ecological and geomorphological studies
(Dörnhöfer \& Oppelt, 2016; J. Gao, 2009). At local scales, it has been
estimated through methods such as echo sounding, multi-beam SONAR and
LiDAR (Abdallah et al., 2013; J. Gao, 2009), while at larger scales
recently released global databases (Downing et al., 2006; Bernhard
Lehner et al., 2011; Bernhard Lehner \& Döll, 2004; Messager et al.,
2016; Verpoorter et al., 2014) have improved the understanding of lakes
and reservoirs and their impacts on global hydrology and biogeochemical
cycles (Bastviken et al., 2004; Döll et al., 2009; Grill et al., 2019;
Hanasaki et al., 2006; Shin et al., 2019; Zajac et al., 2017; Zhou et
al., 2016). These global databases, however, lack information on
topography, which is typically described in terms of level-area-volume
relationships or 3D maps of bottom level. Level-area-volume
relationships are used to monitor the variations in water storage and
are usually necessary to estimate reservoirs inflows using water balance
equations (Deng et al., 2015). In hydrodynamic mathematical modeling,
bathymetric data are required to estimate storages and flow propagation
across lakes. While the level-area-volume relationships are used in
lumped, simplified level-pool routing methods, the availability of
bathymetric data allows a more complete description of the hydrodynamics
of water bodies (Dargahi \& Setegn, 2011; Fread, 1993; Li et al., 2014;
Lopes et al., 2018; Munar et al., 2018; Zhang et al., 2017). In river
modeling systems, channel bathymetry is also required, and remote
sensing-based methods have been proposed (Brêda et al., 2019;
Domeneghetti, 2016; Durand et al., 2008; Moramarco et al., 2019).

Several studies have estimated level-area-volume relationships from
remote sensing data as water level and flood extent, for applications in
hydrological modeling, water balance calculations, lake volume
monitoring, and others (Avisse et al., 2017; Busker et al., 2019; Cai et
al., 2016; Collischonn \& Clarke, 2016; Crétaux et al., 2015, 2016; Duan
\& Bastiaanssen, 2013; Gao et al., 2012; Hoek et al., 2019; Ma et al.,
2019; Tong et al., 2016; Zhang et al., 2014). In general, this
estimation involves fitting a function to the lake area and water level
data and then integrating the resulting curve to obtain a level-storage
relationship (Busker et al., 2019; Huilin Gao et al., 2012; S. Zhang et
al., 2014). This approach usually requires that the date of the water
level corresponds to the date of the image used to estimate the water
surface area. Although, the fitted function is subject to error, the
level-area-volume relationship is easier to obtain than 3D topographic
mapping because noise and errors are compensated in the calculation of
total volume and total area.

In turn, 3D topographic mapping by bathymetric surveys remains a time-
and cost-intensive method, requiring extensive fieldwork and data
analyses (Messager et al., 2016). Furthermore, the direct retrieval of
bathymetry from satellite imagery is limited for situations in clear and
shallow water bodies where the remotely detected signal is dominated by
bottom reflected radiation (Carbonneau et al., 2006; Dilbone et al.,
2018; J. Gao, 2009; C. J. Legleiter \& Overstreet, 2012; Carl J.
Legleiter et al., 2009; Sandidge \& Holyer, 1998). Alternatively, the
topography can be estimated between the smallest and largest water
surface areas using water level and flood extent data, such as the
approaches presented by Arsen et al. (2013), Feng et al. (2011),
Getirana et al. (2018) and Tseng et al. (2016). The methods, however,
still have limitations. The approaches presented by Feng et al. (2011),
Arsen et al. (2013), Tseng et al. (2016), and Getirana et al. (2018),
for instance, require delineating the shorelines of the water body in
each flood extent image. Therefore, in regions with many lakes, such as
in the Amazon floodplain and in delta areas, such methods have high
processing costs, in addition to inherent difficulties in determining
the shorelines due to complex flow patterns and connections among lakes
(Alsdorf et al., 2007; Park \& Latrubesse, 2017; Trigg et al., 2012).
Another caveat of these approaches is the need to obtain water level
observations and images on the same date. The method proposed by Tseng
et al. (2017) only partly resolves these limitations by estimating the
bottom levels on a pixel by pixel basis from a flood frequency map
computed from multi-temporal satellite images, but this approach does
not account for the temporal variation of water surface elevation.

This study presents a flood frequency-based method to estimate 3D
topography above the lowest observed water surface elevation and active
storage in regions subject to varying flooded areas, using long-term
remote sensing-based flood extent maps and water level observations
obtained in-situ and/or via satellite altimetry. In addition to
obtaining the bottom level for each pixel, the proposed method does not
require simultaneous water level data and flood extent image. The method
can be applied to any water body (lakes, reservoirs, and intermittent
rivers) subject to flooded area variation.

This paper is organized as follows: section 2 introduces the proposed
method, available (current and future) datasets to be used with the
method, and the developed Flood2Topo app. Section 3 shows the
experimental design for validation of the method. Results and discussion
are divided into validation for natural lakes and reservoirs (section
4), followed by a discussion on other proposed methods, as well as
future developments and major caveats (section 5).

2 Flood frequency-based method

2.1 Conceptual formulation

A straightforward method to estimate topography above the lowest
observed water surface elevation and active storage of lakes and
reservoirs is proposed here based on the following steps and
assumptions: 1) the water level duration curve is known for at least one
site within the study area; and 2) a flood frequency map is available
for the region. The water body bottom level map is then obtained through
the combination of the water level duration curve and the flood
frequency map, with the same spatial resolution as the flood map and the
same vertical reference as the water level data (e.g., the Earth
Gravitational Model geoid 2008 - EGM08, sea level, or an arbitrary
reference).

The proposed approach considers the flooding process in a given area
where variations in water surface elevation produce variations in flood
extent. Changes in water level along the area can be represented by the
water level duration curve, i.e. the water level cumulative probability
in decreasing order (the probability that the water level in the area
exceeds a certain value in a particular period of time), and the
variation in flood extent can be synthesized by the flood frequency at
each point (or pixel) in the area, as shown in Figure 1.

Thus, given the flood probability map \(F(x,y\)), and the water
level duration curve \(P\left(h\right)=Prob\{h^{\prime}\geq h\}\), the bottom level at a pixel
\(Z(x,y\)) is defined to be the water level \(h\)
whose probability of exceedance is equal to the flood frequency,
i.e.,\(Z(x,y)\ =P^{-1}(\ F\left(x,y\right)\ )\). In other words, the relationships between (1)
water level and its exceedance probability, and (2) flood frequency and
the associated bottom level, are the same.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\par\null

\textbf{Figure 1.} Flow chart of the flood frequency-based method to
estimate topography.

For any pixel with \(F(x,y\)) equal to 100\%,
\(Z(x,y\)) is equal to the lowest water level observed over the
period of time, so that the method will be inaccurate to estimate bottom
levels for any permanently flooded pixel. In addition, because the
calculation is performed pixel by pixel, the water level duration curve
and the flood frequency must be known for each pixel
\((x,y\)). If only one water level time series is available
for the water body, a uniform water level (i.e., a horizontal surface)
will be assumed for the whole area (\(P\left(h\right)\ \)is defined at a
station). For areas with variable water levels, a distributed
information of water level gauges (in-situ or virtual stations) are
preferable (\(P\left(h\right)\) is defined at-many-stations). In this
case, discrete information of water level can be interpolated throughout
the extent of the water body to represent the water surface slope.

The method is exemplified in Figure 2 by the representation of a digital
elevation model (DEM) composed of nine pixels with defined elevations
(\(z_{1}\) to \(z_{4}\) ) in which the horizontal
water surface increases with each time step (t\textsubscript{1} to
t\textsubscript{4}). During this period, pixels with bottom
level\(Z\left(x,y\right)=z_{3}\) have a flood frequency\(F\left(x,y\right)=25\%\) (Figure
2). Similarly, the water level in the DEM remains above elevation
z\textsubscript{3} during 25\% of the time (Figure 2), i.e., the
probability of the water level being greater than \(z_{3}\) is
25\% (\(\text{Prob}\left\{h^{{}^{\prime}}\geq z_{3}\right\}=25\%\) or\(P\left(h\right)=25\%\)). This means that the
flood probability is equal to the probability that the water level is
higher than the bottom level (\(F\left(x,y\right)=P\left(h\right)\)).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure 2.} Illustration of the flood frequency-based method to
estimate topography. Blue pixels are the flooded ones.

2.2 Method uncertainties and caveats

The method can be applied in any area that is not permanently flooded,
and the size of the estimated area depends on the water level and flood
frequency data spatial resolution. These observations should ideally
cover the same time period and be representative of the entire range of
water levels (i.e., from minimum to maximum water levels). In other
words, more extensive and complete time series data lead to more
accurate topography map. In addition, the data temporal resolution must
be greater than the water body dynamics, i.e. it must be enough to
represent the filling and draining process. Furthermore, the
non-stationarity of data must be considered as several processes promote
changes in water level and topography over time, such as dams, erosion,
deforestation, sediment deposition and climate change.

The method uncertainties depend on the water level and flood frequency
data that are used. The water level time series can be derived from
in-situ gauge stations or satellite altimeters (i.e., virtual stations).
Altimetry levels usually represent the surface averages from multiple
tracks and/or footprints, and several altimetry databases also provide
level uncertainties. The level accuracy depends on several factors such
as the operation and maintenance of in-situ gauge stations and the
accuracy of altimeters. However, if the water level error is not
systematic, the water level duration curve will not change because the
error will cancel itself with many observations.

The accuracy of the flood frequency map depends on the errors of the
images used to map the water coverage and the classification accuracy.
The images can be derived from optical or radar sensors. Active sensors,
such as synthetic aperture radars (SAR), have advantages over optical
sensors because they can observe the surface at any weather condition
and at any time (day and night), and can map below-canopy inundation (G.
J.-P. Schumann \& Moller, 2015). However, long term radar image
collections are still scarce, limiting flood frequency mapping. On the
other hand, optical images collections have high data availability, such
as Landsat missions with data since 1972. Errors in water cover
classification using optical images include a possible underestimation
of the water coverage extent in canopy inundation areas, such as in
lakes that have floating aquatic vegetation or submerged flooded
vegetation. Since these areas are classified as flooded only when the
water level is equal to or greater than the vegetation height, the flood
frequency map represents the flood above the vegetation, and therefore
the estimated topography has a vegetation height bias. In addition, the
seasonal bias in the flood frequency map can occur due to temporal
distribution of the valid observations, that is, cloudy seasons may have
fewer valid images than dry seasons. Furthermore, the scan line
corrector (SLC) failure in some satellites, such as Landsat-7,
introduces a slatted appearance in the images, carrying these artifacts
into the flood frequency map (Pekel et al., 2016). In section 2.3, the
state-of-the-art of altimeters and flood extension databases is
presented.

The number of water level stations used to represent the water surface
elevation also affects the accuracy of the bottom level estimation. For
instance, distributed information of water level within the water body
allows the representation of the water surface slope. However, if only
one information is considered (i.e., assumption of a horizontal water
surface), the maximum error of the estimate (Max.\selectlanguage{ngerman} Error) can be
calculated considering the slope (Slope) and the maximum
length (Max. length) of the water body by Equation 1.

\(Max.\ Error\ =\ Slope\ \times\ Max.\ length\) (1)

Slope in natural and artificial lakes depends on factors as wind fetch,
volume of the inflow hydrograph and length and depth of the lake (Fread,
1993). In floodplain shallow lakes, additional uncertainties arise due
to the terrain heterogeneity and the connectivity with the main river
and other floodplain units.

Finally, the method is not able to estimate bottom levels below the
lowest observed water surface elevation, i.e. inactive volumes. However,
this information can be approximated from 3D topography extrapolations
and interpolations considering the edge slope of the water body, as
presented by Getirana et al. (2018) and Tseng et al. (2016). Absolute
volume can also be estimated by extrapolating the stage-area curve, as
presented by Collischonn and Clarke (2016) and Busker et al. (2019).

2.3 Data sources

In this section, the current and future datasets available for deriving
water levels and flood frequency are reviewed, with a focus on remote
sensing methods.

2.3.1 Water level

The water level duration curve can be calculated from water level time
series from in-situ gauge stations or satellite altimeters, such as
ICESat-1 and 2 (https://openaltimetry.org), Jason-1, 2 and 3, ENVISAT,
and TOPEX/Poseidon. The accuracy of these altimeters ranges from 0.10 m
(ICESat-1, Urban et al., 2008) to 1.07 m (Jason-1, Jarihani et al.,
2013), with a 10 to 91 days repeat cycle. O'Loughlin et al. (2016)
provides an analysis for the most popular altimetry missions. In
addition, multi-mission datasets with time series are available for
rivers, lakes, and reservoirs. 1) Hydroweb-THEIA (Crétaux et al., 2011);
2) Global Reservoir and Lake Elevation Database (G-REALM; Birkett et
al., 2011); 3) Global Land Surface Altimetry Data (DLAS/ICESat GLA14;
Zwally et al., (2014); 4) Database for Hydrological Time Series over
Inland Waters (DAHITI; Schwatke et al., 2015); and 5) Global River Radar
Altimetry Time Series (GRRATS; Data are available at DOI
10.5067/PSGRA-SA2V1; Coss et al., 2019). Duan and Bastiaanssen (2013)
describe the satellites used in some of these databases, the available
data period, and the vertical reference system. In addition to
databases, altimetry data can be processed by users with web
applications, such as Altimetry Explorer (AltEx) for JASON-2 and JASON-3
data (Markert et al., 2019), or specific software such as MAPS, and
Satellite Water Gauging (SWG).

2.3.2 Flood frequency

The flood frequency at each pixel in the region of interest can be
obtained from temporal maps of surface water cover, e.g., through
classification of optical imagery using water indexes and methods as
reviewed by Huang et al. (2018). Examples of spaceborne sensors are:1)
Landsat (since 1984) with 16-day timescales and spatial resolution of 30
m (Du et al., 2014; Feyisa et al., 2014; Fisher et al., 2016; Jiang et
al., 2014; Sheng et al., 2016; Yamazaki et al., 2015); 2) MODerate
Resolution Imaging Spectroradiometer (MODIS, since 2000) with daily
timescales and spatial resolution of 250--500 m in visible and near-
infrared bands (Huilin Gao et al., 2012; Khandelwal et al., 2017; Klein
et al., 2017; Sun et al., 2011); 3) Sentinel 2A and 2B (since 2015) with
10 m spatial resolution in visible and near- infrared bands at 5--10 day
intervals (W. Huang et al., 2018; Kordelas et al., 2018; Christian
Schwatke et al., 2019); 4) RapidEye constellation (5 satellites, since
2008) with 5 m resolution and temporal resolution of 1-5.5 days (Cooley
et al., 2017, 2019; Garousi\selectlanguage{english}-Nejad et al., 2019); and 5) PlanetScope
constellation (120CubeSats, since 2014) with daily revisit time globally
at 3-5 m spatial resolution (Cooley et al., 2017, 2019).

Based on the decadal-scale monitoring of Landsat missions, a few global
datasets of surface water extent have been developed, such as the JRC
Global Surface Water dataset (GSW; Pekel et al., 2016) available in
Google Earth Engine platform (Gorelick et al., 2017), and the recent
Landsat Level 3 Dynamic Surface Water Extent dataset (DSWE; Jones,
2019), with data available for the US. GSW dataset represents a monthly
record of water presence on a global scale between March 1984 and
October 2015, corresponding to images from the Landsat program
classified as water, non-water, and no-data with a spatial resolution of
30 m, and the classification accuracy showed omission errors less than
5\% and commission errors less than 1\% (Pekel et al., 2016).

Time series of water surface area of specific lakes and reservoirs that
have been corrected for contamination from clouds, cloud shadows, and/or
terrain shadows are also available: 1) Surface area time series of the
6,817 global reservoirs (Zhao and Gao, 2018;
https://ceprofs.civil.tamu.edu/hgao/); 2) Database for Hydrological Time
Series of Inland Waters (DAHITI; Schwatke et al., 2019); and 3) Global
lake/reservoir area time series (GLATS; Yao et al., 2019).

Examples of SAR satellites used for water covered mapping are: 1)
Sentinel 1 (Cazals et al., 2016; W. Huang et al., 2018); RADARSAT
(Bolanos et al., 2016); 2) Envisat Advanced Synthetic Aperture Radar
(ASAR, Santoro et al., 2015); 3) JERS-1 (Hess et al., 2003, 2015); and
4) ALOS/PALSAR (Arnesen et al., 2013; Chapman et al., 2015). New
opportunities will emerge with new satellites, such as NASA-ISRO
Synthetic Aperture Radar and SWOT (Surface Water and Ocean Topography),
which will be launched in 2021. SWOT will allow estimates of both flood
extent and distributed information of water levels of inland water
bodies (Biancamaria et al., 2016). Bonnema and Hossain (2019) presented
a synthetic experiment to investigate the capability of SWOT to estimate
reservoir characteristics as the stage-area relationship.

2.4 Flood2Topo application

Flood frequency to Topography (Flood2Topo) is a MATLAB application
developed to convert a flood frequency map into a topography map (bottom
levels), computing also the stage-area-active storage curves. The
algorithm takes as input a flood frequency map (GeoTIFF format) and a
single water level time series of the water body (txt format), both
provided by the user.

Besides flood frequency, water level information is required for each
pixel to obtain the bottom level, as described in section 2.1. In the
Flood2Topo app, the estimation was simplified by choosing a single water
level time series for all pixels and assuming that the water surface is
horizontal along the region of interest. To use water level time series
data for each pixel in the case of a non-horizontal surface, the script
must be modified. The estimated topography is used to infer a stage-area
relationship which is then numerically integrated to yield a
stage-active storage relationship. The stage-area curve is elaborated
considering, for each bottom level value \selectlanguage{english}\selectlanguage{english}in the topographic map
(\emph{x} ), the number of pixels with values \selectlanguage{english}\selectlanguage{english}less than or equal to
\emph{x} multiplied by the spatial resolution, resulting in the area
related to each stage (\emph{y} ). Therefore, it is not necessary to
adjust functions from level versus area points, like traditional
approaches. This processing is done by considering all pixels in the
topography map: if the map contains more than one water body or noise,
the area of interest must be delimited before using the Flood2Topo app
to yield individual stage-storage curves. Furthermore, since the method
does not recover the topography below the lowest observed water surface,
the active volume represents the volume above the lowest observed water
level. The application with documentation and an example is available at
XX.

Flood frequency maps can be obtained from different sources, as
described in section 2.3.2. As an option for the user, an Earth Engine
script is provided (Text 1, Supporting Information or XX) to calculate
the flood frequency map for a defined period from the GSW - JRC Monthly
Water History v1.0 dataset (Pekel et al., 2016), available in Google
Earth Engine platform (Gorelick et al., 2017). In the script, the open
water cover frequency for each pixel was calculated as the ratio between
the number of observations classified as water and the number of valid
observations, i.e. with information during the selected period (user
defined).

3 Validation experiments

In order to evaluate the proposed method, Flood2Topo was used to
estimate the topography of two natural lakes, Lake Poopo in the Bolivian
Altiplano and Lago Grande de Curuai in the Central Amazon floodplains,
and 12 hydropower reservoirs in Brazil (Figure 3). The lakes were chosen
due to data availability, and the reservoirs were selected to cover a
wide variety of reservoir shapes, volumes, and areas, given the limited
30 m spatial resolution of Landsat images. To understand the global
representativeness of the assessed lakes, Figure 3l and 3m present the
distributions of reservoir areas and volumes from global databases on
lakes and reservoirs (GRanD and HydroLAKES), and show that the 12
selected reservoirs are representative of the largest 42.6\% of dams in
the GRanD database in terms of area (smallest value = 5.5 km2 for Quebra
Queixo dam; \textasciitilde{} 61 pixels), and the largest 34.8\% in
terms of volume (smallest value=136.6 hm3 for Quebra Queixo dam).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 3.} a) -- k) Location, maximum area, and volume of
natural lakes and reservoirs used for validation of the method. Terrain
elevation from SRTM DEM (Farr et al., 2007; spatial resolution of 90m).
l) and m) Cumulative frequency of area and volume of 1.43 million
natural and artificial lakes and 6,862 reservoirs from the HydroLAKES
(Messager et al., 2016) and GRanD (B Lehner et al., 2011) datasets, as
well the values related to the 12 analyzed reservoirs.

As described in Section 2.4, Flood2Topo requires two information: a
water level time series and a flood frequency map. The map was
elaborated for each lake from the developed Earth Engine script (Text 1,
Supporting Information or code), and the assessed period is the same for
both GSW Monthly and water level datasets for each lake. The water level
was derived from in-situ and virtual stations, as described below.

The topography of Lake Poop\selectlanguage{ngerman}ó was estimated using 726 water level
estimates from a satellite altimetry virtual station (available at the
Theia website) for the period from 17-Feb-2000 to 01-Oct-2015, while for
the Lago Grande de Curuai 11,628 observations from Vila Curuai in-situ
gauge station for the period from 1-Mar-1984 to 31-Dec-2015 were used
(see Figure 3 for location of the stations). Altimetry data are
referenced in relation to geoid EGM08 (Theia dataset), and the Vila
Curuai gauge station was referenced in relation to geoid EGM08 according
to values recommended by Moreira (2016). Therefore, estimated
topographies are also referenced in relation to geoid EGM08. The
reservoirs topography was estimated using in-situ water level time
series from ONS (Brazilian \emph{Operador Nacional do Sistema Elétrico}
; available at \textless{}www.ons.org.br\textgreater{}). The assessed
period of water level for each reservoir are shown in Table S1
(Supporting Information).

Lakes' topographies were validated with the following datasets. Arsen et
al. (2013; called hereafter Arsen approach) estimated the Poopó
topography interpolating isolines (isobaths) created from Landsat images
and altimetry data, which was compared with Flood2Topo estimates. The
Curuai bathymetry was obtained in previous studies with an extensive
echo-sounding survey (Barbosa, 2005; Barbosa et al., 2006; Rudorff et
al., 2014a). Since reservoir topographies were not available, the method
was validated in terms of the stage-active storage relationship i.e., by
removing the dead volume from the total reservoir volume.

4 Results

4.1 Natural lake

Lake Poopó is a shallow lake that can be completely dry at certain years
and is situated in the Bolivian Altiplano (Figure 3). The lake bottom
level was well represented relatively to the Arsen approach (Figure 4),
with a bias of 5.68 cm and root mean square deviation (RMSD) of 18.50 cm
(n = 2,524,605). Topographical variation was also well represented, with
a Pearson correlation coefficient of 0.97, an RMSD (bias-corrected) of
17.61 cm, and a coefficient of determination (R\textsuperscript{2},
bias-corrected) of 0.93. To disregard the bias, RMSD and
R\textsuperscript{2} values were determined by considering the anomaly
of values, i.e., by computing the difference between values and their
means.

As the lake was dry in some periods (Max flood frequency \textless{}
100\%), it was possible to estimate the bottom level in the whole area
of Lake Poopó. The satisfactory results also suggest that for this
estimation the assumption of a horizontal water surface was sufficient.
In addition, terrain heterogeneities were better estimated with the
proposed method than with the Arsen approach, revealing a more detailed
topography (see detail in Figure 4). This is because Flood2Topo
estimation is performed pixel by pixel, avoiding the loss of detail in
the isobath interpolation of the Arsen approach.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 4.} a) Locally derived (Arsen et al., 2013) and b)
Flood2Topo based topographies for the Lake Poop\selectlanguage{ngerman}ó. c) Water level (m)
time series at the satellite altimetry virtual station.

4.2 Floodplain lake

The other evaluated lake is the Lago Grande de Curuai, which is located
in the middle-lower reach of the Amazon River and has great variation in
flooded areas and water levels associated with flooding from the Amazon
mainstem. The floodplain is composed of several lakes that are very
shallow in the low water period. The observed (Rudorff et al., 2014a)
and estimated topographies are presented in Figure 5. Figure 5b and d
also exemplify the slatted appearance in the estimated topography and
flood frequency map due to the Landsat SLC-off (Pekel et al., 2016).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 5.} Observed (a) and estimated (b) topographies in the
Lago Grande de Curuai floodplain; c) Flood frequency map and d) water
level (m) time series at Vila Curuai station.

The bottom level was overestimated with a bias of 0.60 m and an RMSD of
1.46 m (n = 158,387). The estimated minimum value was 4 m, the smallest
observed value in the water level time series. Thus, bottom level was
overestimated in regions with flood frequencies equal to 100 \% (e.g.,
the channels highlighted in Figure 5c). In addition, the lower bottom
levels were poorly represented for two reasons. First, while the whole
floodplain is subject to the Amazon River flood pulse (Rudorff et al.,
2014b), the water level measurements were obtained at a single location
in the floodplain located on the south edge of the lake (Figure 5c); and
second, the minimum level in the Vila Curuai station (Figure 5d) is
affected by the impoundment of water in low-water periods, meaning that
the water level at the station may be higher than the river water level
and in the lakes that remain connected with the river. Therefore, the
water level in Vila Curuai is not fully representative of the water
level variation across the floodplain. Despite the underestimation, the
topographical variation was reasonably well- represented. Estimations
yielded a Pearson correlation coefficient of 0.68, an RMSD
(bias-corrected) of 1.33 m, and an R\textsuperscript{2}(bias-corrected)
of 0.36.

4.3 Reservoirs

The observed and estimated stage-active storage relationships of 12
Brazilian reservoirs (location in Figure 3), the volume RMSD
(hm\textsuperscript{3}), and the normalized RMSD (NRMSD = RMSD/observed
V\textsubscript{100}) are presented in Figure 6. Table S1 (Supporting
Information) includes, for each reservoir, the percentage of the
reservoir area that has a flood frequency equal to 100\%
(\(\frac{\text{FF}_{100}}{\text{Area}}\)) during the assessed period, and the lowest and
highest level (L\textsubscript{0} and L\textsubscript{100})
corresponding respectively to the minimum (V\textsubscript{0} = 0) and
maximum active storage (V\textsubscript{100}) for both observation and
estimations.

In general, there is a satisfactory agreement between observed and
estimated curves, with NRMSD values ranging from 2\% to 11.09 \% (with
the exception of the Barra Bonita reservoir with NRMSD = 27.05\%), and
average NRMSD of 6.39 \%. Other authors found average NRMSD values of
7.4\% (n = 18; Busker et al., 2019), 7.2\% (n = 5; Gao et al., 2012),
and 17.54\% (n = 5; Zhang et al., 2014) using optical images and
satellite altimetry for reservoirs.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure 6.} Water level versus active storage observed and
estimated for 12 artificial reservoirs in Brazil.

Although the method only recovers topography in regions with flooded
area variation, it presented satisfactory results to estimate the
stage-volume relationship in small lakes with small variation in the
flooded area, such as the Quebra Queixo reservoir, where only 14\% of
the reservoir area (0.77 km\textsuperscript{2}) has a variable flooding
area. In fact, Figure 7a shows that there is no clear relation between
the reservoirs size and the NRMSD values. The percentage of the
reservoir area that is permanently flooded (\(\frac{\text{FF}_{100}}{\text{Area}}\)) is also
not related to NRMSD, as observed in Figure 7b. Both reservoirs with
areas where flood frequency varies greatly (e.g., Itumbiara
with\(\frac{\text{FF}_{100}}{\text{Area}}\) = 45.96\%) and reservoirs with large permanently
flooded areas (e.g., Tucuru\selectlanguage{ngerman}í with\(\frac{\text{FF}_{100}}{\text{Area}}\) = 85.86\%) presented
low NRMSD. Busker et al. (2019) estimated stage-area relationships using
Landsat images and satellites altimetry for 135 global lakes and pointed
out that low R\textsuperscript{2}values were related, among others, to a
nearly constant lake area and small lake sizes. Although we did not use
satellite altimetry, we did not observe a similar relationship. On the
other hand, although the amplitude of level variation (observed
L\textsubscript{100} - L\textsubscript{0}) has no clear relationship
with NRMSD (Figure 7c), two out of the three reservoirs with the highest
NRMSD have the smallest variations in level amplitude (Boa Esperança
with L\textsubscript{100} - L\textsubscript{0} = 6 m and Quebra Queixo
with L\textsubscript{100} - L\textsubscript{0} = 5 m). Therefore, the
proposed method is satisfactory for small and large reservoirs, and the
NRMSD can be related to the amplitude of water level variation (within
the range of values tested in this work, with the lower limit tested
being 5 m).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure 7.} NRMSD versus area, the fraction of the reservoir area
that has a flood frequency equal to 100\% (\(\frac{\text{FF}_{100}}{\text{Area}}\)), and
amplitude of level variation (observed L\textsubscript{100} -
L\textsubscript{0}).

5 Discussion

The presented method can be applied to any lake worldwide. Since we use
30 m spatial resolution Landsat images, our ability to resolve small
lakes is limited (e.g., the smallest assessed lake is 5.5
km\textsuperscript{2}, equivalent to \textasciitilde{} 61 pixels). The
assessed sample of lakes is not representative of Earth small lakes,
since 2.2\% and 0.8 \% of lakes (n = 1.43 million; HydroLAKES dataset;
Messager et al., 2016), or 42.6\% and 34.8 \% of reservoirs (n = 6,862;
GRanD dataset; B Lehner et al., 2011) are larger than 5.5
km\textsuperscript{2} and 136,6 hm\textsuperscript{3}, respectively
(Figure 3). Therefore, the method needs to be better evaluated in small
lakes using better spatial resolution images. This is possible, for
instance, using Sentinel images (5 m) and the PlanetScope constellation
(3-5 m). The images from this constellation can be used to depict
high-frequency dynamics of flood extent on a daily basis.

Although the analyzed lakes are not representative of global small
lakes, the global applicability of the method is promising. Considering
the GSW dataset, the most complete dataset currently available, the
seasonal water cover in the contemporary timeframe represents 22.5 \%
(0.81 million km\textsuperscript{2}) of the water covered mapped from
October 2014 to October 2015 (Pekel et al., 2016). However, as the water
coverage is variable in time and space, and the flood frequency map
depends on the assessed period, usually longer than one year, the method
applicability can be greater. For example, assuming the water level time
series data is available, the method is applicable in 35.8 \% (86\%) of
the global water surface area mapped by Pekel et al. (2016; Occurrence
map), when considering the number of pixels with occurrence between 0
and 95\% (99\%) over 35 years. This is an approximate estimate as the
coast regions (ocean) are considered in the map.

The method accuracy is related to the input data. Regarding the water
level, satellite altimetry virtual station was only used for Lake Poop\selectlanguage{ngerman}ó
with expected accuracy of less than 40 cm, and in situ stations were
used in other lakes. The water cover images from GSW dataset has high
accuracy in the classification with low errors of omission
(\textless{}5\%) and commission (\textless{}1\%). However, the estimated
topography can have a bias related to the seasonality of valid data
and/or the presence of flooded vegetation, as described in section 2.2.
To further understand these uncertainties, a detailed analysis is
performed for the Curuai floodplain since this region of the Amazon
basin has high cloud cover in the austral summer (Martins et al., 2018),
presence of aquatic macrophytes in lakes (Sartori et al., 2011), and
availability of an in situ surveyed bathymetry.

Figure 8 shows the flood frequency versus errors in the bottom level
estimation. Note that the vegetation height bias in the Curuai
floodplain occurs mainly in regions of lower flood frequency or in
higher elevations. This bias could be removed using altimetry data such
as ICESat-1 and ICESat-2. For example, by fitting a relationship between
flood frequency and the topography error in relation to the altimetry
level.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure 8.} Flood frequency versus error in the estimation of the
bottom level of the Lago Grande de Curuai floodplain.

The occurrence map developed by Pekel et al. (2016) was used to evaluate
the bias related to seasonal variation in the number of valid
observations across the year once this map represents the average of all
monthly flood frequency, i.e., this map normalizes the occurrences.
However, the topography estimate using occurrence map does not improve
by comparing the flood frequency computed over the full period: bias
increased from 0.60 to 0.72; RMSD increased from 1.46 to 1.57; Pearson's
correlation coefficient decreased from 0.68 to 0.63 and
R\textsuperscript{2} reduced from 0.36 to 0.3. This difference can be
related to the high cloud cover in the summer, since the average of all
monthly flood frequency is affected if the flood frequency of summer
months is very poorly represented. Therefore, normalize the flood
frequency did not improve the estimate.

The maximum error due to not considering water surface slope was
estimated using Equation 1, and considering the Amazon River water
surface slope in this area (1.25 cm/km; average value observed by
Birkett et al., 2002 at \selectlanguage{ngerman}Óbidos station) and the floodplain longitudinal
extent (\textasciitilde{} 90 km). It resulted in a maximum error of 1.12
m, which is relatively small if compared to the amplitude of the water
surface elevation at Vila Curuai station (5 m; Figure 5).

Estimating bathymetry is a major challenge in the remote sensing
community due to the time required to obtain and process data. Methods
using echo sounders, multi-beam SONAR and LiDAR require time-consuming
fieldwork as well as data processing. Empirical approaches to optical
imaging are limited to certain special water conditions (e.g., low
sediment load) and still require field data (J. Gao, 2009; Carl J.
Legleiter et al., 2009). Methods using flood extent require water
surface classification and shoreline delineation for each image in a
time series (Arsen et al., 2013; Feng et al., 2011; Getirana et al.,
2018). Besides, none of these studies benefited from using the
pre-processed GSW dataset for topography estimation. More recently,
Busker et al. (2019) used the GSW database to estimate lake volume,
which reduced processing time and storage demands, but they still had to
fit functions for the stage-area relationship as in the other,
previously developed methods.

The method proposed in this study distinguishes from others as it uses
temporal information (i.e. water level curve duration and flood
frequency map), not requiring that data need to be acquired at the same
time, and it is performed pixel by pixel. Therefore, it allows
topography retrieval for areas connecting large lakes. Similarly, an
estimate of the level-active storage relationship can be directly
obtained from the topography map, without the need to fit functions or
make subjective decisions. The proposed technique is also a
straightforward method that can be used with currently available remote
sensing datasets.

In addition to lakes, the method can be extended to estimate the
bathymetry of intermittent rivers, since they are associated to flood
frequencies smaller than 100\%. Such rivers are found on all continents,
may represent up to 50\% of the global river network, and are usually
associated with data scarcity (Datry et al., 2014; Abdollah A. Jarihani
et al., 2015). In this estimation, the bed's slope can be used to
consider the inclination of the water surface.

Estimation of wetlands topography is also possible, as in the complex
floodplains of the Amazon Basin with multiple lakes (Latrubesse, 2012;
Melack \& Hess, 2010; Trigg et al., 2012). Due to the complexity of
these environments, which might include large areas of flooded forest,
the flood frequency mapping should be performed using radar or
considering only the open water areas using optical images. In addition,
the approximation of a horizontal water surface might be inaccurate in
floodplains (Jung et al., 2010). To overcome this problem, the method
must be adjusted by accounting for distributed information across the
floodplain using different water level gauges (or virtual stations) or
estimating the floodplain water level using the mainstem surface water
elevation, by interpolation, for example. Currently, the spatial
resolution of space altimeters is limited to a few points within the
water body, but this could be improved by adopting distributed
information with new swath remote sensing products such as SWOT
(Biancamaria et al., 2016).

For the hydrodynamic simulation of flood dynamics in complex
river-floodplain systems, the proposed method is also very promising,
since current modeling approaches typically neglect the topography or
dynamic storage of floodplain lakes due to data unavailability (Neal et
al., 2012; G. J. P. Schumann et al., 2016; Wilson et al., 2007).
Similarly, analyses of the temporal variation of topography could be
performed, for example to study lakes with changing patterns, such as
the Lake Chad (H. Gao et al., 2011), Lake Eyre (Leon \& Cohen, 2012),
and Lake Poyang (Feng et al., 2011) due to dredging or more complex
morphodynamics.

\par\null

6 Conclusions

A simple flood frequency-based method was proposed for estimating the
topography and active volume of water bodies. The approach requires that
the analyzed area is not permanently flooded, in addition to
availability of water level data (at least at one station), and a flood
frequency information for each pixel, which can be obtained from
currently available datasets (e.g., in-situ or satellite altimetry, and
GSW dataset). The developed Flood2Topo app provides as outputs the
topography map and level-area-active storage relationships.

We validated 3D topography in two natural lakes and 12 artificial
reservoirs in Brazil. The estimated bottom level in lakes presented BIAS
of 5.68 cm and 60 cm; RMSD of 18.5 cm and 146 cm; and
R\textsuperscript{2} of 0.93 and 0.36. Test cases in artificial
reservoirs showed that the active volume of reservoirs can be estimated
from 3D topography and the stage-area-volume relationship with NRMSD
values ranging from 2\% to 11.09\% for 11 reservoirs (average NRMSD of
6.39 \%).

The growing number of databases of satellite altimetry and surface water
mapping (e.g., 46\% of the databases cited in section 2.2 are from
2019), in addition to improved image processing capability (e.g., Google
Earth Engine platform), and the perspectives with forthcoming missions
(e.g., SWOT to be released in 2021), improve the feasibility of
topography estimation by the proposed method. This sets up great
opportunities to quickly and simply monitor the volumes of ungauged
water bodies (natural lakes, reservoirs, intermittent rivers, and
wetlands), as well as using topography for hydrological applications,
such as hydrodynamic simulation of flood dynamics. On the other hand,
there is still potential for improvements of the method, as well as open
major challenges, such as: 1) application in flooded vegetation areas
with radar images; 2) estimation in small lakes using high resolution
images; 3) use of distributed water level across the studied region
(i.e., considering non-horizontal water surfaces); and 4) estimation of
the bottom levels below the lowest observed water surface.

Acknowledgments

The Flood2bathy application developed in this study and intermediate
data sets are freely available for noncommercial use only from the
developer's webpage (XXX). The authors would like to thank CNPq
(Conselho Nacional de Desenvolvimento Científico e Tecnológico) for
partially funding this research under the grant numbers 140352/2016-3
(Alice César Fassoni-Andrade) and 141161/2017-5 (Ayan Santos
Fleischmann), and CAPES (Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior) - Finance Code 001. They are grateful to Conrado
Rudorff, Evlyn Novo, Claudio Barbosa, Walter Collischonn and Rafael
Kayser for their insights. The authors would also like to thank Adalbert
Arsen and Claudio Barbosa for making available the topography data of
Lake Poopó and Lago Grande de Curuai, respectively. This study was
carried out within the SWOT-MOD project
(https://swot.jpl.nasa.gov/st\_projects.htm).

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