20
IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 3, NO. 1, MARCH 2010
Estimating River Depth From Remote Sensing Swath
Interferometry Measurements of River
Height, Slope, and Width
Michael Durand, Ernesto Rodríguez, Douglas E. Alsdorf, and Mark Trigg
Abstract—The Surface Water and Ocean Topography (SWOT)
mission is a swath mapping radar interferometer that would
provide new measurements of inland water surface elevation
(WSE) for rivers, lakes, wetlands, and reservoirs. SWOT WSE
estimates would provide a source of information for characterizing streamflow globally and would complement existing in situ
gage networks. In this paper, we evaluate the accuracy of river
discharge estimates that would be obtained from SWOT measurements over the Ohio river and eleven of its major tributaries within
the context of a virtual mission (VM). SWOT VM measurements
are obtained by using an instrument measurement model coupled
to simulated WSE from the hydrodynamic model LISFLOOD-FP,
using USGS streamflow gages as boundary conditions and validation data. Most model pixels were estimated two or three times
per 22-day orbit period. These measurements are then input into
an algorithm to obtain estimates of river depth and discharge. The
algorithm is based on Manning’s equation, in which river width
and slope are obtained from SWOT, and roughness is estimated a
priori. SWOT discharge estimates are compared to the discharge
simulated by LISFLOOD-FP. Instantaneous discharge estimates
over the one-year evaluation period had median normalized root
mean square error of 10.9%, and 86% of all instantaneous errors
are less than 25%.
Index Terms—Hydrology, interferometry, radar, remote sensing.
I. INTRODUCTION
IVER discharge denotes the volume flowrate of water
moving through a fluvial channel. River discharge has traditionally been estimated in situ by relating water surface ele-
R
Manuscript received March 20, 2009; revised August 25, 2009.First published November 10, 2009; current version published February 24, 2010. This
work was supported in part by the NASA Terrestrial Hydrology Program; in part
by the NASA Physical Oceanography program; in part by the Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, CA, under a contract
with NASA; in part by the Climate, Water, and Carbon Program of The Ohio
State University; and in part by the U.K. Natural Environment Research Council
(Grant NER/S/A/2006/14062).
M. Durand is with the Byrd Polar Research Center, The Ohio State University,
Columbus, OH 43210 USA (e-mail: [email protected]).
E. Rodríguez is with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109 USA (e-mail: [email protected]).
D. E. Alsdorf is with the Byrd Polar Research Center, School of Earth Sciences, and the Climate, Water, and Carbon Program, The Ohio State University,
Columbus, OH 43210 USA (e-mail: [email protected]).
M. Trigg is with the School of Geographical Sciences, University of Bristol,
Bristol, BS8 1SS, U.K. (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSTARS.2009.2033453
vation (WSE) measurements taken nearly continuously to periodic measurements of flow velocity and channel cross-sectional area, from which instantaneous river discharge is derived.
Stream gages provide the WSE or stage and are located sparsely
at individual points along channels. From these instantaneous
measurements of discharge and concurrent stage measurements,
a “rating curve” is developed that relates discharge to WSE. The
rating curve, or stage-discharge relationship, is then applied to
the continuous stage measurements to produce discharge as a
continuous derived measurement.
As discussed by Alsdorf et al. [1], there have been a variety
of attempts to characterize river discharge via remote sensing
measurements. One approach is to use airborne [2], [3] or spaceborne [4] measurements to estimate the fluvial surface velocity.
Another approach relates river width or inundated area to river
discharge, either at gaging stations [5], [6] or combined with estimates of shoreline elevations [7]–[9]. A third approach is used
in studies that relate water elevations measured from radar altimeters to discharge, e.g., [10]. A fourth approach is to derive
estimates of river discharge from spaceborne measurements of
gravity fluctuations from the Gravity Recover And Climate Experiment (GRACE) measurements [11]–[13].
The Surface Water and Ocean Topography (SWOT) mission
is a swath mapping radar interferometer that would provide
measurements of inland WSE for rivers, lakes, wetlands and
reservoirs. SWOT has been recommended by the National Research Council (NRC) Decadal Survey [14] to measure ocean
topography as well as WSE over land; the proposed launch date
timeframe recommended by the NRC is between 2013–2016.
In contrast with traditional radar altimeters, SWOT will directly
measure fluvial inundated area as well water elevation, with
spatial pixels on the order of 10 s of meters. Average revisit
times will depend upon latitude, with two to four revisits at
low to mid latitudes and up to ten revisits at high latitudes per
22-day orbit repeat period. Although SWOT WSE estimates
will provide a source of information for characterizing streamflow globally, SWOT is not designed to replace stream gages.
Stream gauges can supply timely measurements (e.g., daily
or even real-time) and measure discharge in small tributaries
draining headwater catchments; thus, a close tie to rainfall generated streamflow. In contrast, the SWOT orbit will not supply
daily data nor will the mission operate in real-time mode. The
instrument’s spatial resolution limits the ability of SWOT to
estimate discharge in rivers having a small width. Streamflow
estimates derived from SWOT and gages will be complementary. Whereas in situ gages have real-time capability at a single
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DURAND et al.: ESTIMATING RIVER DEPTH FROM REMOTE SENSING SWATH INTERFEROMETRY MEASUREMENTS
point, SWOT measurements will span all rivers, and measure
elevations between gages, but will only provide a measurement
weekly, not daily, for most locations.
Methods for estimating discharge from SWOT are still
being developed. Andreadis et al. [15] used the ensemble
Kalman filter (EnKF) to update discharge within the context
of a data assimilation scheme. The Andreadis et al. approach
(conducted in the context of what has been termed the SWOT
“virtual mission” or VM) was executed as follows: synthetically generated SWOT measurements were assimilated into the
variable infiltration capacity (VIC) hydrologic model [16] and
the LISFLOOD-FP hydrodynamic model [17] for a reach of
the Ohio river. River depth was estimated as a state variable,
assuming that channel bathymetry and roughness were known
a priori.. Another approach, also conducted in the context of
the SWOT VM, used a data assimilation system to estimate the
slope of the channel bed elevation from seasonal measurements
of inundated area [18]. Once the bathymetry was estimated,
the river depth and discharge could be estimated. While data
assimilation algorithms have the advantage of bringing as much
ancillary data to bear on the problem of estimating discharge,
they typically rely on ensemble-based estimates of these inputs
and thus require significant computational resources. This
high computational expense may have implications for global
application of assimilation algorithms. A simpler approach that
is less computationally expensive has been outlined based on
using SRTM estimates of water elevation and slope in conjunction with Manning’s equation by [19], assuming that the river
depth is known a priori.
In this paper, our goal is to present a method of estimating
river depth directly from SWOT measurements, since a priori.
depth estimates will not be available globally. We will perform
the further step of evaluating the sensitivity of this methodology to differences in SWOT orbits. At this time, it is expected
that SWOT will have two operational phases. During the “fast
phase”, the SWOT orbit will repeat every three days, but spatial
coverage will be limited; the duration of the fast phase will be
between three and six months. During the “nominal phase”, the
SWOT orbit will repeat every twenty-two days, but spatial coverage will be global; the nominal phase will constitute the rest of
the mission lifetime. Note that because of the swath sampling,
revisits to any given location will occur at least twice during a
repeat cycle.
Our overarching goal in this study is to characterize how accurately depth and discharge will be estimated. We test this approach as part of the SWOT hydrology VM. Our methodology
proceeds as follows: 1) a model of the “true” discharge, water
elevations, widths, and slopes is constructed using a hydrodynamic model; 2) synthetic SWOT observations are generated
with realistic (to first order) error characteristics; 3) these observations are used to estimate river depth and to calculate discharge; and 4) the SWOT measurements of discharge are compared with the true discharge to characterize SWOT discharge
accuracy.
21
II. MODELS, METHODS, AND DATA
A. LISFLOOD-FP Model
The LISFLOOD-FP model [17], [20] uses a 1-D finite difference hydrodynamic scheme to solve for water depth, velocity,
and discharge in channel flow. Although LISFLOOD-FP also
includes functionality for floodplain inundation, in this study we
only utilize the channel solver in order to focus on developing
a retrieval algorithm for in-channel flow. In order to achieve a
parsimonious implementation, a rectangular river cross section
is assumed. The model uses the diffusion wave approximation to
the full Saint Venant equations of flow. If this is written in terms
of discharge, where Manning’s equation is used to describe velocity for a rectangular cross section and large width-to-depth
ratio, we have
(1)
where is the roughness coefficient, is the river width, is
the flow depth,
is the bed slope, and is the distance along
the channel. Thus, the final term in (1) represents the slope of
the water surface. By taking this term into account, the diffusion
wave approximation can be used to model the effects of changes
in flow downstream on the flow conditions upstream, including
backwater effects. Such changes are manifested in changes in
water surface slope, which will be of crucial importance to our
estimates of channel depth.
B. Study Area, Model Setup and Inputs
Our study area is the Ohio river basin, with a total drainage
area of approximately 529,000 km [21]. We chose a total of
eleven of the major Ohio river tributaries to include in the model;
the eleven tributaries and their respective drainage areas from
[21] are listed in Table I. From Table I, the contributing area
of these eleven tributaries comprises a total of 401,012 km , or
76% of the drainage area of the mainstem Ohio. The remaining
24% of the drainage area is comprised by smaller rivers observable by SWOT, and by streams and lateral inflow that SWOT
will not be able to measure. We have chosen to work with these
eleven rivers in order to achieve a parsimonious model setup,
and to demonstrate proof-of-concept. Follow-on studies will include all rivers, and examine which will be able to be characterized by SWOT measurements. In order to model these eleven
tributaries in LISFLOOD-FP, estimates of the river centerlines,
channel bed elevation along the centerline, and channel width
are needed.
1) LISFLOOD-FP Channel Inputs: The Hydro1K dataset
was used to provide estimates of the river centerline and channel
bed elevation. Hydro1K was derived from the GTOPO30 digital elevation model (DEM) at 30 arc-second resolution [22].
Stream centerlines are derived from the DEM at approximately
1-km spatial resolution as described in [22], and represented as
a series of sequential location points (i.e., latitude and longitude). Some of these points have additional data describing the
river cross section geometry: width, bed elevation and roughness. The DEM elevation will be used in this study to represent
the channel bed elevation. The Hydro1K data for the Ohio river
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IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 3, NO. 1, MARCH 2010
TABLE I
DRAINAGE AREA OF EACH OF THE ELEVEN TRIBUTARIES INCLUDED IN THE MODEL AS ESTIMATED BY [21], AS WELL AS INFORMATION FROM THE USGS GAGE
USED AS A BOUNDARY CONDITION FOR EACH TRIBUTARY: GAGE ID, CONTRIBUTING AREA, AND DISCHARGE AVERAGED OVER THE STUDY PERIOD
Fig. 1. Map of the Ohio river basin is shown; tributaries included in the model
are shown in blue, while excluded tributaries are shown in grey. Streamlines are
from the Hydro1K dataset. River widths are shown by the relative thickness of
the blue lines. USGS gages used for boundary conditions are shown as circles.
basin is shown in Fig. 1. The LISFLOOD-FP model interpolates the latitude and longitude data from Hydro1K to a regular
grid; in this case, we use a spatial resolution of 1 km. Thus, the
blue streamlines in Fig. 1 show the extent of the study area that
is simulated in LISFLOOD-FP. The channel roughness (Manning’s ) was assumed to be 0.03 for each channel.
The LISFLOOD-FP model requires an estimate of river width
for each channel segment. For this study, we use the National
Land Cover Dataset (NLCD) 2001 [23], which was derived from
Landsat5 and Landsat7 imagery [24]. The NLCD classification was used along with the algorithm developed Smith and
Pavelsky [25] to extract the river width for all of the rivers in
the Ohio basin, resulting in a raster image of the river width.
This raster image was intersected with streamlines from the
Hydro1K: all river width estimates from the raster image that
fell near each segment in a streamline were averaged to obtain
an estimate of the width for that segment. The resulting widths
are shown in Fig. 1.
2) LISFLOOD-FP Boundary Conditions: The diffusion
wave implementation of LISFLOOD-FP requires upstream
discharge boundary conditions for each tributary, as well as
a downstream depth boundary condition on the mainstem.
Note that the mainstem itself provides the downstream depth
boundary condition for the tributaries. In this study, we use
United States Geologic Survey (USGS) gages to provide the
depth and discharge boundary conditions from 1 June 1991–31
May 1992. The location of each gage used as a boundary
condition is shown in Fig. 1. The USGS gage information
and mean annual discharge are shown in Table I. Gages were
chosen as far downstream as possible, in order to represent
the total flow as completely as possible for each tributary; see
Fig. 1. From Table I, the gages represent between 67% and 99%
of the drainage area of each tributary. As a whole, the gages
represent a total of 342,857.4 km , which is approximately
65.2% of the total Ohio river basin drainage area of 525,767.6
km from Table I. The downstream boundary condition thus
represents hydrologic processes operating on the entire basin,
of which only 65.2% is represented by our boundary conditions. In order to deal with this issue, we calculate a reduced
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DURAND et al.: ESTIMATING RIVER DEPTH FROM REMOTE SENSING SWATH INTERFEROMETRY MEASUREMENTS
downstream water depth boundary condition
original boundary condition
from the
23
roughness estimation at the end of this section. Equation (4) can
be recast such that it is linear in and by recognizing that ,
, and are known and defining
(2)
(5)
where
is the sum of the mean annual discharge at the eleven
gages used as upstream boundary conditions (Table I) and
is the mean annual discharge from downstream boundary conditions. Equation (2) follows from assuming that Manning’s
equation holds, and calculating the reduction in depth from a
given reduction in discharge, given that width and roughness
are constant. Presumably, a significant fraction of the 35.8% of
the drainage area that is not included in this model would produce runoff in channels large enough to be measured by SWOT,
though they are not included in this simplified model, as noted
in Section II-B.
resulting in
(6)
This can be rearranged to yield
(7)
where the vector
contains the unknown initial depths
(8)
C. Depth Estimation Algorithm
Our goal in this paper is to explore a method of estimating
river depth from SWOT measurements. The LISFLOOD-FP
model assumes a rectangular cross-section, which implies that
width is time-invariant. We proceed by making two assumptions: 1) most of the time, for most rivers, the discharge at
one point along the channel will not likely be significantly
different than the discharge at another point along the channel
a small distance (i.e., several kilometers) away, assuming no
major changes in contributing area (i.e., no major tributaries
between the two points); 2) most of the time, for most rivers,
the effects of downstream changes in flow will not have a
significant effect on the flow upstream or downstream; in other
words, the kinematic wave approximation holds. We will refer
to these two assumptions as “the continuity assumption”, and
the “the kinematic assumption,” hereafter; moreover, we will
investigate how well they hold for our model setup, below.
Note that the continuity assumption is subject to errors due to
lateral inflows entering a river through channels that are too
narrow to be accurately characterized by SWOT, as discussed
in Section II-B2. Note also that we do not invoke the continuity
assumption if a tributary joins the river that is wide enough to
be accurately characterized by SWOT.
Given the continuity assumption, we can equate the discharge
at pixel and pixel at every time
(3)
Given the kinematic assumption, we have
(4)
is the depth at some initial measurement time, and
where
is the change in water depth at time . Width, slope, and
roughness are defined as for (1), and roughness is assumed not
, and
are all SWOT obto vary in time. Note that ,
are unknown. The depth at any time
servables, and and
is given from and
. Since our objective in this study is
to estimate depth, we will make the assumption that roughness
can be adequately estimated from ancillary data; we will discuss
the matrix contains combinations of the observed , with a
number of rows corresponding to the total number of measurement times,
..
.
(9)
..
.
and the vector contains combinations of
and
..
.
(10)
Assuming that there are more than two independent measurement times, (7) represents an overconstrained set of linear equations, which can be readily solved by finding the value of
that minimizes the least-squares differences for equations. In
order for (7) to be solvable by this method, there must be more
than two linearly independent rows in (9); otherwise, will be
singular, and will not be solvable. As roughness and width
are time-invariant, all temporal variability in will be due to
temporal variability in slope. Slope time series variability in the
model will be discussed in our results, below. It should be noted
that roughness could also be solved for using this approach, although the minimization of residuals required to solve (7) would
then be over a set of nonlinear equations, which would be more
complex.
The depth estimation analysis in (7) will be applied only
between pixels if slope time series variability as measured by
is greater than some arbitrary
the coefficient of variation
, and if the matrix is nonsingular. The latter
threshold
condition will be evaluated by the matrix condition number
in the
norm; is defined as the ratio of the largest singular
value of to the smallest singular value [27] as determined by
a singular value decomposition of . The analysis will only be
performed if is less than some arbitrary threshold.
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IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 3, NO. 1, MARCH 2010
The depth estimation analysis laid out in this section proceeds
as follows. For each pixel in the model domain, we first test
whether or not the slope time series coefficient of variation that
exceeds
. If not, we will estimate depth using
pixel
the interpolation methods presented in Section II-D. Otherwise,
we search the 5-km neighborhood of pixel and define pixels
within the neighborhood that satisfy the condition that
ex. (The “5-km neighborhood” is defined as five pixels
ceeds
upstream and five pixels downstream of pixel ). If no pixels are
found that satisfy this condition, we will estimate depth using
the interpolation methods presented in Section II-D. If one or
more pixels are found that satisfy the condition, then we implement (6) by choosing pixel from the 5-km neighborhood of
pixel . If one pixel is found that satisfies the condition, then
pixel is chosen to be that pixel. If multiple pixels are found
that satisfy the condition, then pixel is chosen to be the pixel
with the greatest value of . Thus
(11)
is the number of pixels within the 5 km neighborhood
where
of pixel that meet the required slope time series condition.
Note that in order to avoid violation of the continuity assumption, depth estimation will only be performed for two pixels
and along the mainstem if there is not a tributary joining in
between the two pixels.
D. Interpolation of Depth Estimates
There are two issues that could pose a difficulty to the method
laid out above. For the purpose of clarifying discussion, we define a river “section” as a part of a river without tributaries entering. Within the context of this study, each of the eleven tributaries constitute a single section. In contrast, the mainstem is
composed of twelve sections, where each section runs from the
inflow of one tributary to the next. First, it is to be expected that
some—but not all—pixels of a given river section will meet the
slope timeseries variability condition described above. How will
the depth be estimated for pixels that do not meet this condition?
The second issue is that for some sections of the mainstem Ohio
river, there may not be any pixels that meet the slope timeseries
variability condition. How will depth be estimated within these
river sections?
To deal with the first issue, we perform an interpolation over
all pixels for a given section of the river where depth estimates
for some pixels were obtained from the algorithm described
in Section II-C. First, the discharge at the initial time is estimated for all locations where measurements exist from Manning’s equation. Second, the discharge estimates obtained in
the first step are averaged together. Third, it is assumed that
the discharge for all locations where no depth estimate is available is equal to the average discharge obtained in the second
step. Fourth, the initial depth for locations where no depth estimate is available is calculated from the discharge assumed in
the third step. To deal with the second issue, we have available
the depth and discharge estimates at the initial time for sections
of the river where depth was successfully estimated. We first
construct a power law between discharge at the initial time and
Fig. 2. Ohio basin rivers (streamlines) as represented in the LISFLOOD-FP
model are shown, along with representative SWOT measurement swaths from
the 22-day 78-degree orbit for ascending (a) and descending (b) passes. The
labels near the bottom of each swath indicate the day on which the measurement
occurred. For simplicity, only measurements on the first seven days of the orbit
are shown.
drainage area for all rivers and river sections that have unique
values of drainage area. We then use this power law to predict
the discharge in the sections of the river where no depth estimates could be obtained using the depth algorithm described in
the previous section. The depth at the initial time is then calculated from the discharge estimate obtained from the power law.
E. Obtaining SWOT Observations
SWOT observations are obtained by overlaying SWOT
swaths on the LISFLOOD-FP pixels. This is done by first
generating a SWOT ground track based on orbital elements; the
ground track is represented as a series of latitude, longitude, and
spacecraft heading as a function of time, as measured from the
beginning of the orbit period. From this ground track, a swath
of SWOT ground coverage polygons is generated from geometrical considerations, and the specified swath width of 140 km.
As an example, Fig. 2 shows the SWOT coverage of the Ohio
river basin model as represented by the LISFLOOD-FP model
described above. After overlaying the swaths, a spatial intersection of each LISFLOOD-FP pixel and each of the ground
coverage polygons is performed to determine the precise time
at which each pixel is measured.
After the measurement times are determined, synthetic
SWOT measurements of river slope, river width, and river
height are generated at each model pixel by using the LISFLOOD-FP output as the “true” model states and corrupting
the true states with measurement error. In this study, we include
only measurement error of river height, as the spatial resolution
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DURAND et al.: ESTIMATING RIVER DEPTH FROM REMOTE SENSING SWATH INTERFEROMETRY MEASUREMENTS
of the LISFLOOD-FP model at 1 km resolution
precludes accurate representation of SWOT errors of slope and
width. In reality, SWOT pixels will have a spatial resolution in
the cross-track direction that ranges from 10 m in the far swath
to 60 m in the near swath. Spatial resolution in the along-track
direction in the best-case scenario will be 2 m due to synthetic
aperture processing. Along-track spatial resolution may have
the potential to degrade slightly due to temporal decorrelation
of the scene [26], however; studies to investigate these effects
are ongoing. In this study, we make the very conservative
assumption that SWOT spatial resolution in both along-track
will be approximately 50 m, and that
and cross-track
each of these 50 50 m pixels will be characterized by zero
of
mean Gaussian random errors with a standard deviation
are simulated by
0.5 m. SWOT errors for river height
noting that the dominant errors in river height measurements
derive from thermal noise, are additive in nature, and can
therefore be modeled as
(12)
where
is the number of SWOT pixels that would be conis calculated
tained within a LISFLOOD-FP model pixel;
individually for each pixel from the river width ,
,
and
(13)
In order to solve for the initial depth as described above, estimates of
are obtained between successive SWOT measurement times from the sum of true water heights and randomly
based on (12).
generated
F. Error Metrics and Evaluation
Depth errors will be evaluated by comparing the estimated
and true values of
at each pixel on each river. Two types of
discharge errors are considered here: 1) the difference between
estimated and true values at the time of a SWOT measurement
will be referred to as instantaneous discharge errors, hereafter;
2) the difference between the monthly average of all instantaneous discharge estimates and the monthly average of the true
discharge from LISFLOOD-FP will be referred to as monthly
discharge errors, hereafter. For instance, the normalized root
mean square error (NRMSE) for instantaneous discharge error
will be evaluated for pixel as
(14)
is the total number of SWOT observations, and
where
the total number of days simulated.
is
III. RESULTS AND DISCUSSION
We first present results from the evaluation of the LISFLOOD-FP model (3.1) and show examples of the SWOT
25
observations derived from model results (3.2). We then evaluate
the extent to which our assumptions hold (3.3). Finally, we
present results from the depth estimation (3.4), the resulting
discharge errors, and the sensitivity of the latter to various orbit
configurations (3.5).
A. LISFLOOD-FP Model Evaluation
Because our primary goal in this study was to evaluate a
methodology for estimating river depth from SWOT observations, we did not perform any model calibration. In order to
verify that the LISFLOOD-FP model set up is producing reasonable results, we compare the discharge predicted at the downstream model outlet and the discharge observed at the most
downstream USGS gage available. Note that the drainage-areaadjusted water depth data from this gage are used as the downstream boundary condition for LISFLOOD-FP. The gages used
for the upstream boundary conditions represent only 65.2% of
the total Ohio river drainage area, and the USGS gage obviously represents all of the drainage area. Thus, in order to assess
the timing of streamflow predicted by LISFLOOD-FP, we have
scaled the USGS discharge time series by the ratio of the sum of
the mean annual discharge for each upstream gage to the mean
annual discharge for the downstream gage. The modeled and
measured discharge time series are shown in Fig. 3; the modeled discharge clearly matches the observed discharge to first
order. Discharge for both the model and gage ranges from 2,000
m s to 4,000 m s between June and December. During
December there is a significant increase in discharge to approximately 17,000 m s for the gage; this peak is somewhat underestimated by the model. Both model and gage decrease to
approximately 4,000 m s in mid-February, before showing
two peaks in March and April. From this, we conclude that the
model is adequately representative of reality to investigate our
methods of estimating depth and discharge.
B. SWOT Observations
Examples of SWOT observations derived from the LISFLOOD-FP model results are shown in Fig. 4. The simulated
elevation of the water surface for the Wabash river is shown
along with SWOT measurements derived from these measurements as described in Section II-E. Synthetic SWOT measurements are shown on two different days, on 11 November and on
24 November. Water elevations along a flow distance of 228 and
212 km are measured on the two days, respectively, indicating
that the 140-km swath is at an angle less than perpendicular
for the flow direction at the point where the river was crossed.
We would expect that the monthly sampling error ultimately
derived from these measurements would be closely tied to the
number of times each pixel in the domain is sampled in each
measurement cycle or in each month. Fig. 5 shows a histogram
of the number of times each pixel was measured within the
SWOT measurement cycle. Most pixels are measured either
two or three times in the 22-day cycle, while 818 of the 5,860
pixels (14.1%) were measured four times. Over the one-year
period, the average number of measurements per month is
3.75, indicating approximately weekly sampling. Note that
sampling regimes are latitude-dependent, with (on average)
higher latitudes being sampled more frequently.
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IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 3, NO. 1, MARCH 2010
Fig. 3. LISFLOOD-FP modeled discharge at the downstream model outlet is shown (solid line) as well as the discharge from the USGS gage at the farthest point
downstream on the Ohio river (dashed line). Note that the discharge from the USGS gage has been adjusted by multiplying by the ratio of the sum of the mean
discharge observed at the upstream boundary condition gages to the mean discharge observed at this downstream gage.
Fig. 5. Number of times each of the 5,860 model pixels is measured in a 22-day
cycle with 78-degree inclination angle is shown.
Fig. 4. Examples of the synthetic SWOT height observations (circles) generated from the true water elevation simulated by LISFLOOD-FP (lines) used for
the Cumberland river on day 163 (a) and day 175 (b) of the simulation period.
C. Evaluation of Assumptions
As described above, our algorithm to estimate depth is dependent upon the assumption that the kinematic approximation
can be used to represent the flow processes, and that continuity
holds between pixels separated by small spatial distances. We
evaluated the kinematic wave assumption by assuming that the
change in water depth with distance is zero in (1), and using
the true model simulated values of , , , and to estimate
discharge (referred to as “kinematic discharge,” hereafter),
then comparing the kinematic discharge with the true model
discharge. For both absolute discharge RMSE, and discharge
NRMSE calculated using (14), most pixels have very low levels
of error due to the kinematic assumption. Indeed, for 98.9%
of all pixels from all tributaries, the NRMSE is less than 5%.
We evaluated the continuity assumption by comparing the
discharge at each pixel with the discharge at a lag of five pixels.
For both absolute discharge RMSE and discharge NRMSE
calculated using (14), most pixels have very small error due
to the continuity assumption. For 94.8% of all pixels from all
tributaries the NRMSE is less than 5%. As a further test, we
calculated the errors due to both assumptions separately during
the first and second half of the year (i.e., during both highand low-flow conditions), and obtained very similar results.
Errors associated with the continuity assumption are greater
than those for the kinematic assumption; this may be due to the
fact that where tributaries join the mainstem, two pixels may
have significantly different contributing areas; this is dealt with
as described in Section II-C. Note that we have assumed that
streamflow does not increase due to runoff processes in for a
given river section between tributaries. Future studies of this
methodology should examine the sensitivity of the continuity
assumption to these runoff processes. Within the context of this
study, based on the extent to which the continuity and kinematic assumptions hold, it is expected that the depth estimation
algorithm described above will be accurate.
Another potential limitation of our method is that it relies on
slope time series variability to obtain accurate estimates of river
depth. In Fig. 6(a), the values of the slope time series coefficient
is shown for the Tennessee river. The low values
of variation
for chainage 0–200 km and from 400–1000 km imply that
of
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DURAND et al.: ESTIMATING RIVER DEPTH FROM REMOTE SENSING SWATH INTERFEROMETRY MEASUREMENTS
27
Fig. 6. Coefficient of variation of the slope time series at various points along the Tennessee river is shown (a). The histogram of across all model pixels
is shown, for the 96% of model pixels with less than 0.75 (b).
it will be difficult to estimate depth for those pixels. The larger
between 200–400 km are likely caused by changes
values of
in the channel slope and channel width at these parts of the river;
near the confluence of the mainstem after
the large values of
1000 km chainage are likely due to the effects of the mainstem
on the Tennessee tributary. Based on the fact that some pixels
at least have significant slope time series variability, it is expected that the depth estimation algorithm described above will
achieve accuracy consistent with the underlying assumptions. A
values is shown in Fig. 6, for all model pixels
histogram of
less than 0.75 (95.8% of the pixels). The mean value of
with
is 0.1586, and 1293 pixels (22%) have a value of
greater
than 0.25.
D. Estimating Depth
There was adequate slope time series variability to estimate
initial depth for a total of 227 pixels. The depth estimate for the
Cumberland river is shown in Fig. 7(a). Depth estimates were
derived from SWOT observables using the algorithm described
in Section II-C for a number of pixels near the downstream portion of the river, where there was adequate temporal variability
in the slope coefficient of variation. For five sections on the
mainstem Ohio, and for the Licking river and Kentucky river,
there were no initial depth estimates; the power law approach
was used to estimate depth for these pixels (see Section II-D).
Relative depth error over all 5,860 pixels from all rivers is shown
in Fig. 7(b). The depth errors show a very slight positive bias,
with a mean relative error of 4.1%. The standard deviation of the
depth error is 11.2%. There are several outliers, with maximum
and minimum errors of 89% and 56%, respectively.
As noted above, the expected mission lifetime will be greater
than three years. The method we have presented used one year
of data to achieve the results described above. It is potentially
of great value to understanding the availability of data products
to know how many measurements are required before the depth
will be able to be estimated. In Fig. 8, we show the standard
deviation of the depth error as a function of how many days of
data were used to calculate the depth. As the time series becomes longer, the accuracy of the depth estimate improves. The
standard deviation of the error after 12 months is approximately
Fig. 7. Initial depth profile for the Cumberland river is shown (a). The true
depth at the initial time is taken from model output (solid line), the depth estimates are derived from SWOT observables as described in Section II-C (circles),
and interpolated to the pixels where SWOT observations are not available as described in Section II-D (dashed line). The relative depth error for depth at the
initial time for all pixels is shown in (b).
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28
IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 3, NO. 1, MARCH 2010
Fig. 9. Spatial profiles of discharge along the Kanawha river for four days are
shown. The true discharge profiles are shown as lines, where the solid, dashed,
dotted, and dash-dot lines refer to days 285, 286, 287, and 288, respectively.
The discharge estimates derived from SWOT measurements on days 285 and
288 are shown as points.
Fig. 8. Sensitivity of the standard deviation of the relative error in the initial
depth is shown as a function of the number of simulation days used to obtain
the depth estimate.
one half the error after 11 months. The depth algorithm was performed for only 204 pixels for the 11-month case, and for 227
pixels for the 12-month case. For the Cumberland river, the algorithm was performed for 16 pixels for the 12-month case (see
Fig. 7), but was not performed for any pixels for the 11-month
case. This is due to highly variable river slopes during May
1992: for the Cumberland river during May 1992 was greater
than averaged over the other eleven months by a factor 5.02.
Thus, for the 11-month case, the 648 pixels for the Cumberland river were estimated via the interpolation algorithm described in Section II-D. The mean of the relative depth error
for the Cumberland river for the 12-month case was 2.23 cm,
but was 62.7 cm for the 11-month case. This bias in the Cumberland river depth estimates is the reason for the change in the
overall error from the 11-month to the 12-month case shown in
Fig. 8. These results indicate that the method accuracy for estimating depth from the slope time series is generally better than
that using the interpolation algorithm if no pixel in a given river
is measured. Moreover, the model results for the Cumberland
river, and the sensitivity of the depth estimates to slope variability, indicates a need for future studies to explore the possible
seasonal dependence of .
E. Estimating Discharge
Example discharge results for the Kanawha river are shown
in Fig. 9. The true discharge shows a large flood wave propagating through the river channel over the course of four days:
285, 286, 287, 288. On day 285, discharge is increasing from
0–200 km, but is constant from 200–400 km. On day 286, discharge at the upstream boundary has peaked, and discharge is
decreasing along the remainder of the river. On day 287, the
flood wave peak is around 375 km, and on day 288, discharge is
increasing along the entire course of the Kanawha river. SWOT
measurements occurred on day 285 and 288, and different parts
of the river were sampled. On day 285, the low flow condition
was measured, and on day 288, the increasing discharge profile was measured. Thus, a partial picture of the discharge dy-
namics were obtained from the synthetic SWOT measurements.
The large amount of scatter in the discharge estimates is due to
the random error added to the SWOT height observations. These
errors could be partially mitigated by utilizing a low-pass filter;
e.g., a polynomial could be fitted to the height measurements,
following the approach of [19].
There are 366 days of simulation time and a total of 5,860
pixels, resulting in many spatial and temporal series of discharge to examine. We summarize these errors by calculating
the NRMSE of the discharge time series at each pixel using
(14). Both instantaneous discharge errors and monthly discharge errors (as described in Section II-F) are shown in
Fig. 10. Instantaneous discharge errors compare only estimated
discharge to true discharge only during the measurement times,
with the initial depth estimated using the algorithm presented
in Sections II-C and II-D. Monthly discharge errors use true
discharge at the SWOT measurement times to calculate a
monthly discharge estimate, and compare with the true monthly
discharge; thus, no depth error is included in the monthly
discharge errors. The median instantaneous discharge error is
10.9%, with 86% of all instantaneous errors less than 25%.
Similarly, the median monthly discharge error is 14.7%, with
87% of all monthly errors less than 25%. As a final analysis,
we combined both error due to temporal sampling and depth
error, and found that the median error from both error sources
combined was 22%.
As noted above, the mission will consist of a fast phase (3 day
period), and a nominal phase (22 day period). During the fast
phase, spatial coverage is not global; only 2,264 model pixels
(39%) were sampled. During the fast phase, however, pixels are
measured at least once every three days, or approximately ten
times per month. As noted above, pixels are measured on average 3.75 times per month in the nominal phase, which is far
less frequently. The monthly discharge errors reflect this; the
median NRMSE from Table II is 3% for the fast phase, and
14.7% for the nominal phase, with a 78 degree inclination angle.
We also tested whether or not the temporal sampling was sensitive to the inclination angle of the orbit. The 74 degree inclination angle is the minimum required to capture the outlets of
the major Arctic rivers. The 78 degree inclination angle would
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DURAND et al.: ESTIMATING RIVER DEPTH FROM REMOTE SENSING SWATH INTERFEROMETRY MEASUREMENTS
29
mates. We can examine this more rigorously by using a firstorder Taylor series expansion to approximate the sensitivity of
discharge to depth, which yields an estimate of (instantaneous)
due to depth errors
discharge errors
(15)
Normalizing this expression by discharge leads to a relative sensitivity of discharge to depth error
(16)
The implication of this equation is that any errors in depth are
multiplied by a factor of 1.67; thus, to attain discharge accuracy
of 10%, depth errors must be limited to 6%. This sensitivity is
illustrated as the red line in Fig. 11. However, estimates of discharge anomaly will be much less sensitive than absolute discharge to errors in the initial depth. SWOT will yield highly accurate measurements of water height anomaly and (thus) depth
. Defining discharge anomaly
as the differanomaly
ence between discharge at time and time 1, we have
(17)
Fig. 10. Histograms of instantaneous NRMSE discharge error (top) and
monthly NRMSE discharge error (bottom) are shown.
The sensitivity of this expression to error in the initial depth is
given by
(18)
TABLE II
MONTHLY SAMPLING ERROR FOR BOTH THE NOMINAL PHASE (22-DAY
PERIOD) AND THE FAST PHASE (THREE-DAY PERIOD) OF THE MISSION, AND
FOR TWO INCLINATION ANGLES, ASSUMING PERFECT DISCHARGE ESTIMATES
permit further oceanographic study of the Arctic Ocean circulation, and is the maximum allowable inclination angle, due to
other considerations. We calculated monthly discharge errors
for the nominal phase and for the fast phase for a 74 and a 78
degree inclination. During the fast phase, the monthly discharge
error was identical for both inclination angles. During the nominal phase, the median NRMSE was 14.7% and 15.8% for the
78 degree and 74 degree orbits, respectively; the error associated with the 74 degree orbit was thus 7.4% greater. Thus, in
the context of this study, the monthly sampling error was not
sensitive to the inclination angle of the orbit.
The instantaneous discharge results of Fig. 10 (top panel) are
essentially an experiment in examining how the depth errors calculated from SWOT observables propagate into discharge esti-
Normalizing this by
and rearranging gives
(19)
Note that we have removed the dependence of the error metric
on slope in order to clearly show the differences between the expressions for absolute and relative discharge error due to depth:
(16) and (19) are identical except for the term in square brackets
in (19). The term in square brackets is a function only of the ratio
between the change in depth at time and the initial depth; given
this ratio, the error in discharge anomaly due to depth is a linear
function of the error in depth, as shown in Fig. 11. Discharge
less than zero;
anomaly is most sensitive to depth error for
this is intuitive, since if the depth at time is much larger than
the initial depth, the effect of the initial condition will be minimized. From Fig. 11, discharge anomaly is much less sensitive
to depth error than absolute discharge. For instance, to achieve a
discharge anomaly accuracy of 10%, depth errors must be limited to 10.7%, for a relative increase of depth of 25% over the
initial time.
IV. SUMMARY AND CONCLUSIONS
An algorithm for estimating river depth from SWOT measurements was presented and tested using LISFLOOD-FP
model output. The algorithm uses the time series of SWOT
measurements to obtain an estimate of river depth at an arbitrary
initial time. River depth at other times can then be estimated
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30
IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 3, NO. 1, MARCH 2010
the Ohio river basin. River depth at the initial simulation time
was successfully estimated for the 5,860 model pixels with a
mean (standard deviation) relative error of 4.1% (11.2%). From
these depth estimates and SWOT observables, discharge was
estimated, assuming that roughness was known. Instantaneous
discharge estimates over the one-year evaluation period had
median NRMSE of 10.9%, and 86% of all instantaneous errors
were less than 25%. As a separate experiment, we sampled the
true discharge time series at the SWOT measurement times, and
used only the sampled estimates to calculate monthly discharge.
The median monthly discharge error is 14.7%, and 87% of all
monthly errors are less than 25%. Combining both error due to
temporal sampling and depth error, the median error was 22%.
From this preliminary analysis, we conclude that the depth
algorithm presented here has potential for use in developing an
estimate of river depth from SWOT measurements. In contrast
to depth estimation approaches based on data assimilation presented by [15] and [18], no ensemble hydrodynamic simulations
are required, which significantly reduces the computational expense and may make this approach more feasible for global application. A method to interpolate the SWOT estimates of discharge such as an Optimal Interpolation (OI) scheme [28] has
the potential to improve averaged estimates of discharge from
the instantaneous estimates, such as the monthly discharge errors analyzed here. Future work will investigate methods to estimate the roughness coefficient and depth simultaneously, will
explore the spatiotemporal characteristics of slope timeseries
variability, and will explore the role of slope and width errors
in discharge estimates.
ACKNOWLEDGMENT
The authors would like to thank the two anonymous reviewers
who provided helpful comments and improved the quality of the
paper.
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Fig. 11. Sensitivity of absolute discharge (dashed line) and discharge anomaly
are shown in the top panel; the circles, squares, diamonds, and triangles refer
to zz values of 0.25, 0.25, 0.75, and 1.25, respectively (a). Absolute discharge (b) and discharge anomaly (c) are shown for a pixel near the mainstem
Ohio downstream boundary condition, where the circles are discharge estimates
and the line is the true discharge.
1
0
from the time variability of the SWOT height measurements
and the depth at the initial time. The LISFLOOD-FP model
was integrated for one year over the eleven largest tributaries of
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Michael Durand received the B.S. degree in
mechanical engineering and biological systems
engineering from Virginia Polytechnic Institute,
Blacksburg, in 2002, and the M.S. and Ph.D. degrees
in civil engineering from the University of California, Los Angeles, in 2004 and 2007, respectively.
He is currently a Postdoctoral Researcher with the
Byrd Polar Research Center, The Ohio State University, Columbus.
Ernesto Rodríguez received the Ph.D. degree in
physics from the Georgia Institute of Technology,
Atlanta, in 1984.
Since 1985, he has been with the Radar Science
and Engineering Section, Jet Propulsion Laboratory,
California Institute of Technology, Pasadena. His
research interests include radar interferometry,
altimetry, sounding, terrain classification, and EM
scattering theory.
Douglas E. Alsdorf received the M.Sc. degree
in geophysics from The Ohio State University,
Columbus, in 1991, and the Ph.D. degree in geophysics from Cornell University, Ithaca, NY, in
1996.
He is currently an Associate Professor with the
School of Earth Sciences at The Ohio State University and Director of the Climate, Water, and Carbon
Program and the Interim Directory of the Institute
for Energy and the Environment at The Ohio State
University.
Mark Trigg received the B.Eng. degree in mechanical engineering from the University of Surrey, U.K.,
in 1991, and the M.Sc. degree in soil and water engineering in 1997 from Cranfield University, U.K. He
is currently pursuing the Ph.D. degree in geography
with the Hydrology Research Group at the University of Bristol, U.K. The topic of his current research
is Amazon flood wave hydraulics and floodplain dynamics.
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