Estimating River Depth From Remote Sensing Swath Interferometry

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 SwathInterferometry Measurements of River

Height, Slope, and WidthMichael 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 wouldprovide new measurements of inland water surface elevation(WSE) for rivers, lakes, wetlands, and reservoirs. SWOT WSEestimates would provide a source of information for character-izing streamflow globally and would complement existing in situgage networks. In this paper, we evaluate the accuracy of riverdischarge estimates that would be obtained from SWOT measure-ments over the Ohio river and eleven of its major tributaries withinthe context of a virtual mission (VM). SWOT VM measurementsare obtained by using an instrument measurement model coupledto simulated WSE from the hydrodynamic model LISFLOOD-FP,using USGS streamflow gages as boundary conditions and vali-dation data. Most model pixels were estimated two or three timesper 22-day orbit period. These measurements are then input intoan algorithm to obtain estimates of river depth and discharge. Thealgorithm is based on Manning’s equation, in which river widthand slope are obtained from SWOT, and roughness is estimated apriori. SWOT discharge estimates are compared to the dischargesimulated by LISFLOOD-FP. Instantaneous discharge estimatesover the one-year evaluation period had median normalized rootmean square error of 10.9%, and 86% of all instantaneous errorsare less than 25%.

Index Terms—Hydrology, interferometry, radar, remote sensing.

I. INTRODUCTION

R IVER discharge denotes the volume flowrate of watermoving through a fluvial channel. River discharge has tra-

ditionally been estimated in situ by relating water surface ele-

Manuscript received March 20, 2009; revised August 25, 2009.First pub-lished November 10, 2009; current version published February 24, 2010. Thiswork was supported in part by the NASA Terrestrial Hydrology Program; in partby the NASA Physical Oceanography program; in part by the Jet PropulsionLaboratory, California Institute of Technology, Pasadena, CA, under a contractwith NASA; in part by the Climate, Water, and Carbon Program of The OhioState 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 In-stitute of Technology, Pasadena CA 91109 USA (e-mail: [email protected]).

D. E. Alsdorf is with the Byrd Polar Research Center, School of Earth Sci-ences, 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 onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTARS.2009.2033453

vation (WSE) measurements taken nearly continuously to pe-riodic measurements of flow velocity and channel cross-sec-tional area, from which instantaneous river discharge is derived.Stream gages provide the WSE or stage and are located sparselyat individual points along channels. From these instantaneousmeasurements of discharge and concurrent stage measurements,a “rating curve” is developed that relates discharge to WSE. Therating curve, or stage-discharge relationship, is then applied tothe continuous stage measurements to produce discharge as acontinuous derived measurement.

As discussed by Alsdorf et al. [1], there have been a varietyof attempts to characterize river discharge via remote sensingmeasurements. One approach is to use airborne [2], [3] or space-borne [4] measurements to estimate the fluvial surface velocity.Another approach relates river width or inundated area to riverdischarge, either at gaging stations [5], [6] or combined with es-timates of shoreline elevations [7]–[9]. A third approach is usedin studies that relate water elevations measured from radar al-timeters to discharge, e.g., [10]. A fourth approach is to deriveestimates of river discharge from spaceborne measurements ofgravity fluctuations from the Gravity Recover And Climate Ex-periment (GRACE) measurements [11]–[13].

The Surface Water and Ocean Topography (SWOT) missionis a swath mapping radar interferometer that would providemeasurements of inland WSE for rivers, lakes, wetlands andreservoirs. SWOT has been recommended by the National Re-search Council (NRC) Decadal Survey [14] to measure oceantopography as well as WSE over land; the proposed launch datetimeframe recommended by the NRC is between 2013–2016.In contrast with traditional radar altimeters, SWOT will directlymeasure fluvial inundated area as well water elevation, withspatial pixels on the order of 10 s of meters. Average revisittimes will depend upon latitude, with two to four revisits atlow to mid latitudes and up to ten revisits at high latitudes per

22-day orbit repeat period. Although SWOT WSE estimateswill provide a source of information for characterizing stream-flow globally, SWOT is not designed to replace stream gages.Stream gauges can supply timely measurements (e.g., dailyor even real-time) and measure discharge in small tributariesdraining headwater catchments; thus, a close tie to rainfall gen-erated streamflow. In contrast, the SWOT orbit will not supplydaily data nor will the mission operate in real-time mode. Theinstrument’s spatial resolution limits the ability of SWOT toestimate discharge in rivers having a small width. Streamflowestimates derived from SWOT and gages will be complemen-tary. 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 21

point, SWOT measurements will span all rivers, and measureelevations between gages, but will only provide a measurementweekly, not daily, for most locations.

Methods for estimating discharge from SWOT are stillbeing developed. Andreadis et al. [15] used the ensembleKalman filter (EnKF) to update discharge within the contextof 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: syntheti-cally generated SWOT measurements were assimilated into thevariable infiltration capacity (VIC) hydrologic model [16] andthe LISFLOOD-FP hydrodynamic model [17] for a reach ofthe Ohio river. River depth was estimated as a state variable,assuming that channel bathymetry and roughness were knowna priori.. Another approach, also conducted in the context ofthe SWOT VM, used a data assimilation system to estimate theslope of the channel bed elevation from seasonal measurementsof inundated area [18]. Once the bathymetry was estimated,the river depth and discharge could be estimated. While dataassimilation algorithms have the advantage of bringing as muchancillary data to bear on the problem of estimating discharge,they typically rely on ensemble-based estimates of these inputsand thus require significant computational resources. Thishigh computational expense may have implications for globalapplication of assimilation algorithms. A simpler approach thatis less computationally expensive has been outlined based onusing SRTM estimates of water elevation and slope in conjunc-tion with Manning’s equation by [19], assuming that the riverdepth is known a priori.

In this paper, our goal is to present a method of estimatingriver depth directly from SWOT measurements, since a priori.depth estimates will not be available globally. We will performthe further step of evaluating the sensitivity of this method-ology to differences in SWOT orbits. At this time, it is expectedthat SWOT will have two operational phases. During the “fastphase”, the SWOT orbit will repeat every three days, but spatialcoverage will be limited; the duration of the fast phase will bebetween three and six months. During the “nominal phase”, theSWOT orbit will repeat every twenty-two days, but spatial cov-erage will be global; the nominal phase will constitute the rest ofthe mission lifetime. Note that because of the swath sampling,revisits to any given location will occur at least twice during arepeat cycle.

Our overarching goal in this study is to characterize how ac-curately depth and discharge will be estimated. We test this ap-proach as part of the SWOT hydrology VM. Our methodologyproceeds as follows: 1) a model of the “true” discharge, waterelevations, widths, and slopes is constructed using a hydrody-namic model; 2) synthetic SWOT observations are generatedwith realistic (to first order) error characteristics; 3) these ob-servations are used to estimate river depth and to calculate dis-charge; and 4) the SWOT measurements of discharge are com-pared with the true discharge to characterize SWOT dischargeaccuracy.

II. MODELS, METHODS, AND DATA

A. LISFLOOD-FP Model

The LISFLOOD-FP model [17], [20] uses a 1-D finite differ-ence hydrodynamic scheme to solve for water depth, velocity,and discharge in channel flow. Although LISFLOOD-FP alsoincludes functionality for floodplain inundation, in this study weonly utilize the channel solver in order to focus on developinga retrieval algorithm for in-channel flow. In order to achieve aparsimonious implementation, a rectangular river cross sectionis assumed. The model uses the diffusion wave approximation tothe full Saint Venant equations of flow. If this is written in termsof discharge, where Manning’s equation is used to describe ve-locity for a rectangular cross section and large width-to-depthratio, we have

(1)

where is the roughness coefficient, is the river width, isthe flow depth, is the bed slope, and is the distance alongthe channel. Thus, the final term in (1) represents the slope ofthe water surface. By taking this term into account, the diffusionwave approximation can be used to model the effects of changesin flow downstream on the flow conditions upstream, includingbackwater effects. Such changes are manifested in changes inwater surface slope, which will be of crucial importance to ourestimates of channel depth.

B. Study Area, Model Setup and Inputs

Our study area is the Ohio river basin, with a total drainagearea of approximately 529,000 km [21]. We chose a total ofeleven 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 areaof these eleven tributaries comprises a total of 401,012 km , or76% of the drainage area of the mainstem Ohio. The remaining24% of the drainage area is comprised by smaller rivers observ-able by SWOT, and by streams and lateral inflow that SWOTwill not be able to measure. We have chosen to work with theseeleven rivers in order to achieve a parsimonious model setup,and to demonstrate proof-of-concept. Follow-on studies will in-clude all rivers, and examine which will be able to be charac-terized by SWOT measurements. In order to model these eleventributaries in LISFLOOD-FP, estimates of the river centerlines,channel bed elevation along the centerline, and channel widthare needed.

1) LISFLOOD-FP Channel Inputs: The Hydro1K datasetwas used to provide estimates of the river centerline and channelbed elevation. Hydro1K was derived from the GTOPO30 dig-ital elevation model (DEM) at 30 arc-second resolution [22].Stream centerlines are derived from the DEM at approximately1-km spatial resolution as described in [22], and represented asa series of sequential location points (i.e., latitude and longi-tude). Some of these points have additional data describing theriver cross section geometry: width, bed elevation and rough-ness. The DEM elevation will be used in this study to representthe channel bed elevation. The Hydro1K data for the Ohio river



TABLE IDRAINAGE 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 modelare shown in blue, while excluded tributaries are shown in grey. Streamlines arefrom the Hydro1K dataset. River widths are shown by the relative thickness ofthe blue lines. USGS gages used for boundary conditions are shown as circles.

basin is shown in Fig. 1. The LISFLOOD-FP model interpo-lates the latitude and longitude data from Hydro1K to a regulargrid; in this case, we use a spatial resolution of 1 km. Thus, theblue streamlines in Fig. 1 show the extent of the study area thatis simulated in LISFLOOD-FP. The channel roughness (Man-ning’s ) was assumed to be 0.03 for each channel.

The LISFLOOD-FP model requires an estimate of river widthfor each channel segment. For this study, we use the NationalLand Cover Dataset (NLCD) 2001 [23], which was derived from

Landsat5 and Landsat7 imagery [24]. The NLCD classifica-tion was used along with the algorithm developed Smith andPavelsky [25] to extract the river width for all of the rivers inthe Ohio basin, resulting in a raster image of the river width.This raster image was intersected with streamlines from theHydro1K: all river width estimates from the raster image thatfell near each segment in a streamline were averaged to obtainan estimate of the width for that segment. The resulting widthsare shown in Fig. 1.

2) LISFLOOD-FP Boundary Conditions: The diffusionwave implementation of LISFLOOD-FP requires upstreamdischarge boundary conditions for each tributary, as well asa downstream depth boundary condition on the mainstem.Note that the mainstem itself provides the downstream depthboundary condition for the tributaries. In this study, we useUnited States Geologic Survey (USGS) gages to provide thedepth and discharge boundary conditions from 1 June 1991–31May 1992. The location of each gage used as a boundarycondition is shown in Fig. 1. The USGS gage informationand mean annual discharge are shown in Table I. Gages werechosen as far downstream as possible, in order to representthe total flow as completely as possible for each tributary; seeFig. 1. From Table I, the gages represent between 67% and 99%of the drainage area of each tributary. As a whole, the gagesrepresent a total of 342,857.4 km , which is approximately65.2% of the total Ohio river basin drainage area of 525,767.6km from Table I. The downstream boundary condition thusrepresents hydrologic processes operating on the entire basin,of which only 65.2% is represented by our boundary condi-tions. In order to deal with this issue, we calculate a reduced



downstream water depth boundary condition from theoriginal boundary condition

(2)

where is the sum of the mean annual discharge at the elevengages used as upstream boundary conditions (Table I) andis the mean annual discharge from downstream boundary con-ditions. Equation (2) follows from assuming that Manning’sequation holds, and calculating the reduction in depth from agiven reduction in discharge, given that width and roughnessare constant. Presumably, a significant fraction of the 35.8% ofthe drainage area that is not included in this model would pro-duce runoff in channels large enough to be measured by SWOT,though they are not included in this simplified model, as notedin Section II-B.

C. Depth Estimation Algorithm

Our goal in this paper is to explore a method of estimatingriver depth from SWOT measurements. The LISFLOOD-FPmodel assumes a rectangular cross-section, which implies thatwidth is time-invariant. We proceed by making two assump-tions: 1) most of the time, for most rivers, the discharge atone point along the channel will not likely be significantlydifferent than the discharge at another point along the channela small distance (i.e., several kilometers) away, assuming nomajor changes in contributing area (i.e., no major tributariesbetween the two points); 2) most of the time, for most rivers,the effects of downstream changes in flow will not have asignificant effect on the flow upstream or downstream; in otherwords, the kinematic wave approximation holds. We will referto these two assumptions as “the continuity assumption”, andthe “the kinematic assumption,” hereafter; moreover, we willinvestigate how well they hold for our model setup, below.Note that the continuity assumption is subject to errors due tolateral inflows entering a river through channels that are toonarrow to be accurately characterized by SWOT, as discussedin Section II-B2. Note also that we do not invoke the continuityassumption if a tributary joins the river that is wide enough tobe accurately characterized by SWOT.

Given the continuity assumption, we can equate the dischargeat pixel and pixel at every time

(3)

Given the kinematic assumption, we have

(4)where is the depth at some initial measurement time, and

is the change in water depth at time . Width, slope, androughness are defined as for (1), and roughness is assumed notto vary in time. Note that , , and are all SWOT ob-servables, and and are unknown. The depth at any time

is given from and . Since our objective in this study isto estimate depth, we will make the assumption that roughnesscan be adequately estimated from ancillary data; we will discuss

roughness estimation at the end of this section. Equation (4) canbe recast such that it is linear in and by recognizing that ,

, and are known and defining

(5)

resulting in

(6)

This can be rearranged to yield

(7)

where the vector contains the unknown initial depths

(8)

the matrix contains combinations of the observed , with anumber of rows corresponding to the total number of measure-ment times,

......

(9)

and the vector contains combinations of and

...(10)

Assuming that there are more than two independent measure-ment times, (7) represents an overconstrained set of linear equa-tions, which can be readily solved by finding the value ofthat minimizes the least-squares differences for equations. Inorder for (7) to be solvable by this method, there must be morethan two linearly independent rows in (9); otherwise, will besingular, and will not be solvable. As roughness and widthare time-invariant, all temporal variability in will be due totemporal variability in slope. Slope time series variability in themodel will be discussed in our results, below. It should be notedthat roughness could also be solved for using this approach, al-though the minimization of residuals required to solve (7) wouldthen be over a set of nonlinear equations, which would be morecomplex.

The depth estimation analysis in (7) will be applied onlybetween pixels if slope time series variability as measured bythe coefficient of variation is greater than some arbitrarythreshold , and if the matrix is nonsingular. The lattercondition will be evaluated by the matrix condition numberin the norm; is defined as the ratio of the largest singularvalue of to the smallest singular value [27] as determined bya singular value decomposition of . The analysis will only beperformed if is less than some arbitrary threshold.



The depth estimation analysis laid out in this section proceedsas follows. For each pixel in the model domain, we first testwhether or not the slope time series coefficient of variation thatpixel exceeds . If not, we will estimate depth usingthe interpolation methods presented in Section II-D. Otherwise,we search the 5-km neighborhood of pixel and define pixelswithin the neighborhood that satisfy the condition that ex-ceeds . (The “5-km neighborhood” is defined as five pixelsupstream and five pixels downstream of pixel ). If no pixels arefound that satisfy this condition, we will estimate depth usingthe interpolation methods presented in Section II-D. If one ormore pixels are found that satisfy the condition, then we imple-ment (6) by choosing pixel from the 5-km neighborhood ofpixel . If one pixel is found that satisfies the condition, thenpixel is chosen to be that pixel. If multiple pixels are foundthat satisfy the condition, then pixel is chosen to be the pixelwith the greatest value of . Thus

(11)

where is the number of pixels within the 5 km neighborhoodof pixel that meet the required slope time series condition.Note that in order to avoid violation of the continuity assump-tion, depth estimation will only be performed for two pixelsand along the mainstem if there is not a tributary joining inbetween the two pixels.

D. Interpolation of Depth Estimates

There are two issues that could pose a difficulty to the methodlaid out above. For the purpose of clarifying discussion, we de-fine a river “section” as a part of a river without tributaries en-tering. Within the context of this study, each of the eleven trib-utaries constitute a single section. In contrast, the mainstem iscomposed of twelve sections, where each section runs from theinflow of one tributary to the next. First, it is to be expected thatsome—but not all—pixels of a given river section will meet theslope timeseries variability condition described above. How willthe depth be estimated for pixels that do not meet this condition?The second issue is that for some sections of the mainstem Ohioriver, there may not be any pixels that meet the slope timeseriesvariability condition. How will depth be estimated within theseriver sections?

To deal with the first issue, we perform an interpolation overall pixels for a given section of the river where depth estimatesfor some pixels were obtained from the algorithm describedin Section II-C. First, the discharge at the initial time is esti-mated for all locations where measurements exist from Man-ning’s equation. Second, the discharge estimates obtained inthe first step are averaged together. Third, it is assumed thatthe discharge for all locations where no depth estimate is avail-able is equal to the average discharge obtained in the secondstep. Fourth, the initial depth for locations where no depth es-timate is available is calculated from the discharge assumed inthe third step. To deal with the second issue, we have availablethe depth and discharge estimates at the initial time for sectionsof the river where depth was successfully estimated. We firstconstruct a power law between discharge at the initial time and

Fig. 2. Ohio basin rivers (streamlines) as represented in the LISFLOOD-FPmodel are shown, along with representative SWOT measurement swaths fromthe 22-day 78-degree orbit for ascending (a) and descending (b) passes. Thelabels near the bottom of each swath indicate the day on which the measurementoccurred. For simplicity, only measurements on the first seven days of the orbitare shown.

drainage area for all rivers and river sections that have uniquevalues of drainage area. We then use this power law to predictthe discharge in the sections of the river where no depth esti-mates could be obtained using the depth algorithm described inthe previous section. The depth at the initial time is then calcu-lated from the discharge estimate obtained from the power law.

E. Obtaining SWOT Observations

SWOT observations are obtained by overlaying SWOTswaths on the LISFLOOD-FP pixels. This is done by firstgenerating a SWOT ground track based on orbital elements; theground track is represented as a series of latitude, longitude, andspacecraft heading as a function of time, as measured from thebeginning of the orbit period. From this ground track, a swathof SWOT ground coverage polygons is generated from geomet-rical considerations, and the specified swath width of 140 km.As an example, Fig. 2 shows the SWOT coverage of the Ohioriver basin model as represented by the LISFLOOD-FP modeldescribed above. After overlaying the swaths, a spatial inter-section of each LISFLOOD-FP pixel and each of the groundcoverage polygons is performed to determine the precise timeat which each pixel is measured.

After the measurement times are determined, syntheticSWOT measurements of river slope, river width, and riverheight are generated at each model pixel by using the LIS-FLOOD-FP output as the “true” model states and corruptingthe true states with measurement error. In this study, we includeonly measurement error of river height, as the spatial resolution



of the LISFLOOD-FP model at 1 km resolutionprecludes accurate representation of SWOT errors of slope andwidth. In reality, SWOT pixels will have a spatial resolution inthe cross-track direction that ranges from 10 m in the far swathto 60 m in the near swath. Spatial resolution in the along-trackdirection in the best-case scenario will be 2 m due to syntheticaperture processing. Along-track spatial resolution may havethe potential to degrade slightly due to temporal decorrelationof the scene [26], however; studies to investigate these effectsare ongoing. In this study, we make the very conservativeassumption that SWOT spatial resolution in both along-trackand cross-track will be approximately 50 m, and thateach of these 50 50 m pixels will be characterized by zeromean Gaussian random errors with a standard deviation of0.5 m. SWOT errors for river height are simulated bynoting that the dominant errors in river height measurementsderive from thermal noise, are additive in nature, and cantherefore be modeled as

(12)

where is the number of SWOT pixels that would be con-tained within a LISFLOOD-FP model pixel; is calculatedindividually for each pixel from the river width , ,and

(13)

In order to solve for the initial depth as described above, es-timates of are obtained between successive SWOT mea-surement times from the sum of true water heights and randomlygenerated based on (12).

F. Error Metrics and Evaluation

Depth errors will be evaluated by comparing the estimatedand true values of at each pixel on each river. Two types ofdischarge errors are considered here: 1) the difference betweenestimated and true values at the time of a SWOT measurementwill be referred to as instantaneous discharge errors, hereafter;2) the difference between the monthly average of all instanta-neous discharge estimates and the monthly average of the truedischarge from LISFLOOD-FP will be referred to as monthlydischarge errors, hereafter. For instance, the normalized rootmean square error (NRMSE) for instantaneous discharge error

will be evaluated for pixel as

(14)

where is the total number of SWOT observations, and isthe total number of days simulated.

III. RESULTS AND DISCUSSION

We first present results from the evaluation of the LIS-FLOOD-FP model (3.1) and show examples of the SWOT

observations derived from model results (3.2). We then evaluatethe extent to which our assumptions hold (3.3). Finally, wepresent results from the depth estimation (3.4), the resultingdischarge errors, and the sensitivity of the latter to various orbitconfigurations (3.5).

A. LISFLOOD-FP Model Evaluation

Because our primary goal in this study was to evaluate amethodology for estimating river depth from SWOT observa-tions, we did not perform any model calibration. In order toverify that the LISFLOOD-FP model set up is producing reason-able results, we compare the discharge predicted at the down-stream model outlet and the discharge observed at the mostdownstream USGS gage available. Note that the drainage-area-adjusted water depth data from this gage are used as the down-stream boundary condition for LISFLOOD-FP. The gages usedfor the upstream boundary conditions represent only 65.2% ofthe total Ohio river drainage area, and the USGS gage obvi-ously represents all of the drainage area. Thus, in order to assessthe timing of streamflow predicted by LISFLOOD-FP, we havescaled the USGS discharge time series by the ratio of the sum ofthe mean annual discharge for each upstream gage to the meanannual discharge for the downstream gage. The modeled andmeasured discharge time series are shown in Fig. 3; the mod-eled discharge clearly matches the observed discharge to firstorder. Discharge for both the model and gage ranges from 2,000m s to 4,000 m s between June and December. DuringDecember there is a significant increase in discharge to approx-imately 17,000 m s for the gage; this peak is somewhat un-derestimated by the model. Both model and gage decrease toapproximately 4,000 m s in mid-February, before showingtwo peaks in March and April. From this, we conclude that themodel is adequately representative of reality to investigate ourmethods of estimating depth and discharge.

B. SWOT Observations

Examples of SWOT observations derived from the LIS-FLOOD-FP model results are shown in Fig. 4. The simulatedelevation of the water surface for the Wabash river is shownalong with SWOT measurements derived from these measure-ments as described in Section II-E. Synthetic SWOT measure-ments are shown on two different days, on 11 November and on24 November. Water elevations along a flow distance of 228 and212 km are measured on the two days, respectively, indicatingthat the 140-km swath is at an angle less than perpendicularfor the flow direction at the point where the river was crossed.We would expect that the monthly sampling error ultimatelyderived from these measurements would be closely tied to thenumber of times each pixel in the domain is sampled in eachmeasurement cycle or in each month. Fig. 5 shows a histogramof the number of times each pixel was measured within theSWOT measurement cycle. Most pixels are measured eithertwo or three times in the 22-day cycle, while 818 of the 5,860pixels (14.1%) were measured four times. Over the one-yearperiod, the average number of measurements per month is3.75, indicating approximately weekly sampling. Note thatsampling regimes are latitude-dependent, with (on average)higher latitudes being sampled more frequently.



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 pointdownstream 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 meandischarge observed at the upstream boundary condition gages to the mean discharge observed at this downstream gage.

Fig. 4. Examples of the synthetic SWOT height observations (circles) gener-ated from the true water elevation simulated by LISFLOOD-FP (lines) used forthe 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 de-pendent upon the assumption that the kinematic approximationcan be used to represent the flow processes, and that continuityholds between pixels separated by small spatial distances. Weevaluated the kinematic wave assumption by assuming that thechange in water depth with distance is zero in (1), and usingthe true model simulated values of , , , and to estimatedischarge (referred to as “kinematic discharge,” hereafter),then comparing the kinematic discharge with the true modeldischarge. For both absolute discharge RMSE, and dischargeNRMSE calculated using (14), most pixels have very low levelsof 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 thedischarge at each pixel with the discharge at a lag of five pixels.For both absolute discharge RMSE and discharge NRMSE

Fig. 5. Number of times each of the 5,860 model pixels is measured in a 22-daycycle with 78-degree inclination angle is shown.

calculated using (14), most pixels have very small error dueto the continuity assumption. For 94.8% of all pixels from alltributaries the NRMSE is less than 5%. As a further test, wecalculated the errors due to both assumptions separately duringthe first and second half of the year (i.e., during both high-and low-flow conditions), and obtained very similar results.Errors associated with the continuity assumption are greaterthan those for the kinematic assumption; this may be due to thefact that where tributaries join the mainstem, two pixels mayhave significantly different contributing areas; this is dealt withas described in Section II-C. Note that we have assumed thatstreamflow does not increase due to runoff processes in for agiven river section between tributaries. Future studies of thismethodology should examine the sensitivity of the continuityassumption to these runoff processes. Within the context of thisstudy, based on the extent to which the continuity and kine-matic assumptions hold, it is expected that the depth estimationalgorithm described above will be accurate.

Another potential limitation of our method is that it relies onslope time series variability to obtain accurate estimates of riverdepth. In Fig. 6(a), the values of the slope time series coefficientof variation is shown for the Tennessee river. The low valuesof for chainage 0–200 km and from 400–1000 km imply that



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 pixelsis shown, for the 96% of model pixels with � less than 0.75 (b).

it will be difficult to estimate depth for those pixels. The largervalues of between 200–400 km are likely caused by changesin the channel slope and channel width at these parts of the river;the large values of near the confluence of the mainstem after1000 km chainage are likely due to the effects of the mainstemon the Tennessee tributary. Based on the fact that some pixelsat least have significant slope time series variability, it is ex-pected that the depth estimation algorithm described above willachieve accuracy consistent with the underlying assumptions. Ahistogram of values is shown in Fig. 6, for all model pixelswith less than 0.75 (95.8% of the pixels). The mean value of

is 0.1586, and 1293 pixels (22%) have a value of greaterthan 0.25.

D. Estimating Depth

There was adequate slope time series variability to estimateinitial depth for a total of 227 pixels. The depth estimate for theCumberland river is shown in Fig. 7(a). Depth estimates werederived from SWOT observables using the algorithm describedin Section II-C for a number of pixels near the downstream por-tion of the river, where there was adequate temporal variabilityin the slope coefficient of variation. For five sections on themainstem Ohio, and for the Licking river and Kentucky river,there were no initial depth estimates; the power law approachwas used to estimate depth for these pixels (see Section II-D).Relative depth error over all 5,860 pixels from all rivers is shownin Fig. 7(b). The depth errors show a very slight positive bias,with a mean relative error of 4.1%. The standard deviation of thedepth error is 11.2%. There are several outliers, with maximumand minimum errors of 89% and 56%, respectively.

As noted above, the expected mission lifetime will be greaterthan three years. The method we have presented used one yearof data to achieve the results described above. It is potentiallyof great value to understanding the availability of data productsto know how many measurements are required before the depthwill be able to be estimated. In Fig. 8, we show the standarddeviation of the depth error as a function of how many days ofdata were used to calculate the depth. As the time series be-comes longer, the accuracy of the depth estimate improves. Thestandard deviation of the error after 12 months is approximately

Fig. 7. Initial depth profile for the Cumberland river is shown (a). The truedepth at the initial time is taken from model output (solid line), the depth esti-mates are derived from SWOT observables as described in Section II-C (circles),and interpolated to the pixels where SWOT observations are not available as de-scribed in Section II-D (dashed line). The relative depth error for depth at theinitial time for all pixels is shown in (b).



Fig. 8. Sensitivity of the standard deviation of the relative error in the initialdepth is shown as a function of the number of simulation days used to obtainthe depth estimate.

one half the error after 11 months. The depth algorithm was per-formed for only 204 pixels for the 11-month case, and for 227pixels for the 12-month case. For the Cumberland river, the al-gorithm was performed for 16 pixels for the 12-month case (seeFig. 7), but was not performed for any pixels for the 11-monthcase. This is due to highly variable river slopes during May1992: for the Cumberland river during May 1992 was greaterthan averaged over the other eleven months by a factor 5.02.Thus, for the 11-month case, the 648 pixels for the Cumber-land river were estimated via the interpolation algorithm de-scribed in Section II-D. The mean of the relative depth errorfor 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 Cum-berland river depth estimates is the reason for the change in theoverall error from the 11-month to the 12-month case shown inFig. 8. These results indicate that the method accuracy for esti-mating depth from the slope time series is generally better thanthat using the interpolation algorithm if no pixel in a given riveris measured. Moreover, the model results for the Cumberlandriver, and the sensitivity of the depth estimates to slope vari-ability, indicates a need for future studies to explore the possibleseasonal dependence of .

E. Estimating Discharge

Example discharge results for the Kanawha river are shownin Fig. 9. The true discharge shows a large flood wave propa-gating through the river channel over the course of four days:285, 286, 287, 288. On day 285, discharge is increasing from0–200 km, but is constant from 200–400 km. On day 286, dis-charge at the upstream boundary has peaked, and discharge isdecreasing along the remainder of the river. On day 287, theflood wave peak is around 375 km, and on day 288, discharge isincreasing along the entire course of the Kanawha river. SWOTmeasurements occurred on day 285 and 288, and different partsof the river were sampled. On day 285, the low flow conditionwas measured, and on day 288, the increasing discharge pro-file was measured. Thus, a partial picture of the discharge dy-

Fig. 9. Spatial profiles of discharge along the Kanawha river for four days areshown. 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 and288 are shown as points.

namics were obtained from the synthetic SWOT measurements.The large amount of scatter in the discharge estimates is due tothe random error added to the SWOT height observations. Theseerrors 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 5,860pixels, resulting in many spatial and temporal series of dis-charge to examine. We summarize these errors by calculatingthe NRMSE of the discharge time series at each pixel using(14). Both instantaneous discharge errors and monthly dis-charge errors (as described in Section II-F) are shown inFig. 10. Instantaneous discharge errors compare only estimateddischarge to true discharge only during the measurement times,with the initial depth estimated using the algorithm presentedin Sections II-C and II-D. Monthly discharge errors use truedischarge at the SWOT measurement times to calculate amonthly discharge estimate, and compare with the true monthlydischarge; thus, no depth error is included in the monthlydischarge errors. The median instantaneous discharge error is10.9%, with 86% of all instantaneous errors less than 25%.Similarly, the median monthly discharge error is 14.7%, with87% of all monthly errors less than 25%. As a final analysis,we combined both error due to temporal sampling and deptherror, and found that the median error from both error sourcescombined was 22%.

As noted above, the mission will consist of a fast phase (3 dayperiod), and a nominal phase (22 day period). During the fastphase, spatial coverage is not global; only 2,264 model pixels(39%) were sampled. During the fast phase, however, pixels aremeasured at least once every three days, or approximately tentimes per month. As noted above, pixels are measured on av-erage 3.75 times per month in the nominal phase, which is farless frequently. The monthly discharge errors reflect this; themedian NRMSE from Table II is 3% for the fast phase, and14.7% for the nominal phase, with a 78 degree inclination angle.We also tested whether or not the temporal sampling was sen-sitive to the inclination angle of the orbit. The 74 degree incli-nation angle is the minimum required to capture the outlets ofthe major Arctic rivers. The 78 degree inclination angle would



Fig. 10. Histograms of instantaneous NRMSE discharge error (top) andmonthly NRMSE discharge error (bottom) are shown.

TABLE IIMONTHLY 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 circula-tion, and is the maximum allowable inclination angle, due toother considerations. We calculated monthly discharge errorsfor the nominal phase and for the fast phase for a 74 and a 78degree inclination. During the fast phase, the monthly dischargeerror was identical for both inclination angles. During the nom-inal phase, the median NRMSE was 14.7% and 15.8% for the78 degree and 74 degree orbits, respectively; the error associ-ated with the 74 degree orbit was thus 7.4% greater. Thus, inthe context of this study, the monthly sampling error was notsensitive to the inclination angle of the orbit.

The instantaneous discharge results of Fig. 10 (top panel) areessentially an experiment in examining how the depth errors cal-culated from SWOT observables propagate into discharge esti-

mates. We can examine this more rigorously by using a first-order Taylor series expansion to approximate the sensitivity ofdischarge to depth, which yields an estimate of (instantaneous)discharge errors due to depth errors

(15)

Normalizing this expression by discharge leads to a relative sen-sitivity of discharge to depth error

(16)

The implication of this equation is that any errors in depth aremultiplied by a factor of 1.67; thus, to attain discharge accuracyof 10%, depth errors must be limited to 6%. This sensitivity isillustrated as the red line in Fig. 11. However, estimates of dis-charge anomaly will be much less sensitive than absolute dis-charge to errors in the initial depth. SWOT will yield highly ac-curate measurements of water height anomaly and (thus) depthanomaly . Defining discharge anomaly as the differ-ence between discharge at time and time 1, we have

(17)

The sensitivity of this expression to error in the initial depth isgiven by

(18)

Normalizing this by and rearranging gives

(19)

Note that we have removed the dependence of the error metricon slope in order to clearly show the differences between the ex-pressions for absolute and relative discharge error due to depth:(16) and (19) are identical except for the term in square bracketsin (19). The term in square brackets is a function only of the ratiobetween the change in depth at time and the initial depth; giventhis ratio, the error in discharge anomaly due to depth is a linearfunction of the error in depth, as shown in Fig. 11. Dischargeanomaly is most sensitive to depth error for less than zero;this is intuitive, since if the depth at time is much larger thanthe initial depth, the effect of the initial condition will be mini-mized. From Fig. 11, discharge anomaly is much less sensitiveto depth error than absolute discharge. For instance, to achieve adischarge anomaly accuracy of 10%, depth errors must be lim-ited to 10.7%, for a relative increase of depth of 25% over theinitial time.

IV. SUMMARY AND CONCLUSIONS

An algorithm for estimating river depth from SWOT mea-surements was presented and tested using LISFLOOD-FPmodel output. The algorithm uses the time series of SWOTmeasurements to obtain an estimate of river depth at an arbitraryinitial time. River depth at other times can then be estimated



Fig. 11. Sensitivity of absolute discharge (dashed line) and discharge anomalyare shown in the top panel; the circles, squares, diamonds, and triangles referto�� values of�0.25, 0.25, 0.75, and 1.25, respectively (a). Absolute dis-charge (b) and discharge anomaly (c) are shown for a pixel near the mainstemOhio downstream boundary condition, where the circles are discharge estimatesand the line is the true discharge.

from the time variability of the SWOT height measurementsand the depth at the initial time. The LISFLOOD-FP modelwas integrated for one year over the eleven largest tributaries of

the Ohio river basin. River depth at the initial simulation timewas successfully estimated for the 5,860 model pixels with amean (standard deviation) relative error of 4.1% (11.2%). Fromthese depth estimates and SWOT observables, discharge wasestimated, assuming that roughness was known. Instantaneousdischarge estimates over the one-year evaluation period hadmedian NRMSE of 10.9%, and 86% of all instantaneous errorswere less than 25%. As a separate experiment, we sampled thetrue discharge time series at the SWOT measurement times, andused only the sampled estimates to calculate monthly discharge.The median monthly discharge error is 14.7%, and 87% of allmonthly errors are less than 25%. Combining both error due totemporal sampling and depth error, the median error was 22%.

From this preliminary analysis, we conclude that the depthalgorithm presented here has potential for use in developing anestimate of river depth from SWOT measurements. In contrastto depth estimation approaches based on data assimilation pre-sented by [15] and [18], no ensemble hydrodynamic simulationsare required, which significantly reduces the computational ex-pense and may make this approach more feasible for global ap-plication. A method to interpolate the SWOT estimates of dis-charge such as an Optimal Interpolation (OI) scheme [28] hasthe potential to improve averaged estimates of discharge fromthe instantaneous estimates, such as the monthly discharge er-rors analyzed here. Future work will investigate methods to es-timate the roughness coefficient and depth simultaneously, willexplore the spatiotemporal characteristics of slope timeseriesvariability, and will explore the role of slope and width errorsin discharge estimates.

ACKNOWLEDGMENT

The authors would like to thank the two anonymous reviewerswho provided helpful comments and improved the quality of thepaper.

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Michael Durand received the B.S. degree inmechanical engineering and biological systemsengineering from Virginia Polytechnic Institute,Blacksburg, in 2002, and the M.S. and Ph.D. degreesin civil engineering from the University of Cali-fornia, Los Angeles, in 2004 and 2007, respectively.

He is currently a Postdoctoral Researcher with theByrd Polar Research Center, The Ohio State Univer-sity, Columbus.

Ernesto Rodríguez received the Ph.D. degree inphysics from the Georgia Institute of Technology,Atlanta, in 1984.

Since 1985, he has been with the Radar Scienceand Engineering Section, Jet Propulsion Laboratory,California Institute of Technology, Pasadena. Hisresearch interests include radar interferometry,altimetry, sounding, terrain classification, and EMscattering theory.

Douglas E. Alsdorf received the M.Sc. degreein geophysics from The Ohio State University,Columbus, in 1991, and the Ph.D. degree in geo-physics from Cornell University, Ithaca, NY, in1996.

He is currently an Associate Professor with theSchool of Earth Sciences at The Ohio State Univer-sity and Director of the Climate, Water, and CarbonProgram and the Interim Directory of the Institutefor Energy and the Environment at The Ohio StateUniversity.

Mark Trigg received the B.Eng. degree in mechan-ical engineering from the University of Surrey, U.K.,in 1991, and the M.Sc. degree in soil and water engi-neering in 1997 from Cranfield University, U.K. Heis currently pursuing the Ph.D. degree in geographywith the Hydrology Research Group at the Univer-sity of Bristol, U.K. The topic of his current researchis Amazon flood wave hydraulics and floodplain dy-namics.


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Estimating River Depth From Remote Sensing Swath Interferometry