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Remote Sensing of Environment 113 (2009) 1025–1045

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Remote Sensing of Environment

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A forward image model for passive optical remote sensing of river bathymetry☆

Carl J. Legleiter a,b,⁎, Dar A. Roberts c

a Department of Geography, University of Wyoming, Laramie, WY 82071, USAb Yellowstone Ecological Research Center, Bozeman, MT 59718, USAc Department of Geography, University of California, Santa Barbara, CA 93106, USA

☆ Revised manuscript submitted to Remote Sensing of⁎ Corresponding author. USGS Geomorphology and S

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E-mail address: [email protected] (C.J. Legleiter).

0034-4257/$ – see front matter © 2009 Elsevier Inc. Adoi:10.1016/j.rse.2009.01.018

a b s t r a c t
a r t i c l e i n f o
Article history:
By facilitating measurement Received 29 July 2008Received in revised form 30 January 2009Accepted 31 January 2009
Keywords:River bathymetryRemote sensingDepthFluvial geomorphologyRadiative transfer

of river channel morphology, remote sensing techniques could enable significantadvances in our understanding of fluvial systems. To realize this potential, researchers must first gainconfidence in image-derived river information, as well as an appreciation of its inherent limitations. Thispaper describes a forward image model (FIM) for examining the capabilities and constraints associated withpassive optical remote sensing of river bathymetry. Image data are simulated “from the streambed up” byfirst using information on depth and bottom reflectance to parameterize models of radiative transfer withinthe water column and atmosphere and then incorporating sensor technical specifications. This physics-basedframework provides a means of assessing the potential for spectrally-based depth retrieval from a particularriver of interest, given a sensor configuration. Forward image modeling of both a hypothetical meander bendand an actual gravel-bed river indicated that bathymetric accuracy and precision vary spatially as a functionof channel morphology, with less reliable depth estimates in pools. A simpler, more computationally efficientanalytical model highlighted additional controls on bathymetric uncertainty: optical depth and the ratio ofthe smallest detectable change in radiance to the bottom-reflected radiance. Application of the FIM to acomplex, natural channel illustrated how the model can be used to quantify the effects of various sensorcharacteristics. Bathymetric accuracy was determined primarily by spatial resolution, due to mixed pixelsalong the banks and sub-pixel scale variations in depth, whereas depth retrieval precision depended on thesensor's ability to resolve subtle changes in radiance. This flexible forward modeling approach thus allowsthe utility of image-derived river information to be evaluated in the context of specific investigations, leadingto more efficient, more informed use of remote sensing methods across a range of fluvial environments.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Remote sensing typically involves solution of an inverse problem,with some parameter of interest inferred from image data. To deriveuseful information on spatial and temporal variations of theparameter, these variations must be related to the magnitude andspectral shape of the measured upwelling spectral radiance. Empiricalrelationships are widely used for this purpose and have formed thebasis of most previous efforts to estimate river depth from image data(e.g., Lejot et al., 2007; Marcus et al., 2003; Winterbottom & Gilvear,1997). While these studies have demonstrated the potential toremotely map channel bathymetry, a physics-based approach wouldprovide greater flexibility and more general insight. For example, byconsidering the radiative transfer processes that govern the interac-tion of solar radiationwith the atmosphere, air–water interface, water

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column, and a shallow, reflective substrate, Mobley et al. (2005)developed a spectrum-matching methodology that does not rely onscene-specific empirical relations. Instead, lookup tables were used todirectly link calibrated hyperspectral image data to the environmentalattributes of interest: bathymetry, water column optical properties,and bottom type (Lesser & Mobley, 2007).

In principle, algorithms of this kind could be extended to fluvialenvironments, but remote sensing of rivers remains in an early stageof development. At present, a physics-based approach is more usefulfor addressing not the inverse but rather the forward problem— giveninformation on channel morphology, imaging conditions, and sensorcharacteristics, simulate image data for a particular river of interest.Although the value of remote sensing ultimately lies in its applicationas an inverse method, an initial consideration of the forward problemallows the capabilities and limitations of the technique to beexamined systematically a priori. Insight generated in this mannercan thus lead to more efficient, more informed use of remote sensingin stream studies.

In the context of river research, remote sensing is increasinglyviewed as a way to expand the scale of inquiry from short, isolatedreaches to entire watersheds (Marcus & Fonstad, 2008). To realize this

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1026 C.J. Legleiter, D.A. Roberts / Remote Sensing of Environment 113 (2009) 1025–1045

potential, scientists must first gain confidence in image-derived riverinformation, as well as an appreciation of its inherent limitations.Effective application of remote sensing techniques to fluvial systemsthus requires ameans of quantifying the extent to which various typesof image data can support specific investigations conducted in a rangeof different settings. Motivated by this objective, we outline herein aforward image modeling framework for remote mapping of riverchannel morphology via depth retrieval from passive optical imagedata. The following sections: 1) describe the algorithms comprisingthe forward image model; 2) illustrate how simulated image data canbe used to quantify depth retrieval accuracy and precision; 3) consideran efficient analytical alternative to computationally intensivenumerical simulation that highlights important controls on bathy-metric precision; and 4) present a case study in which the forwardimage model is applied to a gravel-bed river to examine the effects ofsensor spatial and radiometric resolution.

2. Forward image modeling framework

In essence, the forward image model (FIM) generates images“from the streambed up,” producing simulated data that can be usedto quantify the accuracy and precision of depth retrieval underknown conditions. The model thus enables careful evaluation of theextent to which data collected by a given sensor from a particularriver of interest can satisfy the information requirements of a specificapplication. To achieve this goal, the FIM utilizes information ondepth and bottom reflectance, combines radiative transfer models ofthe water column and atmosphere, and incorporates sensor

Fig. 1. Forward image model structure. The information on imaging conditions and sensorperform the image simulation procedures in the central portion of the figure. The resulting sithe lower right corner. Inputs to the model are noted, processes represented by the modelparallelograms.

characteristics. Fig. 1 summarizes model structure, and the followingsections describe individual components. Briefly, the forward imagemodel:

1) uses the MODTRAN atmospheric radiative transfer model (Berket al., 1989) to simulate the direct and diffuse components of thedownwelling irradiance incident upon the river;

2) generates reflectance spectra for a range of depths and bottomtypes using the Hydrolight radiative transfer model (Mobley,1994;Mobley & Sundman, 2001);

3) propagates the resulting spectra through the atmosphere to createa library of at-sensor radiance values, again using MODTRAN;

4) produces a simulated radiance field by assigning the appropriatespectrum to each location within the channel based on a high-resolution input depth map;

5) adds a buffer of terrestrial radiance to account for mixed pixelsalong the channel banks;

6) spatially aggregates the base radiance field to form image pixels;and

7) performs convolution with the sensor spectral response tocalculate band-integrated radiances.

The modular nature of the FIM allows different scenarios to beexamined by refining or replacing individual model components. Here,we illustrate each phase of the modeling process using an idealizedmeander bend. The atmospheric parameters used to initialize MOD-TRAN and the bottom reflectance spectrum required by Hydrolight aredrawn from field observations along Soda Butte Creek, a gravel-bedriver to which the FIM was applied as a case study (Section 4).

characteristics on the left is combined with the data on river attributes on the right tomulated image data are then used to assess depth retrieval performance, as indicated inare enclosed in rectangles, and intermediate and final model outputs are enclosed in

Table 1Parameters for simulating downwelling irradiance spectra with MODTRAN.

Parameter Value(s) Notes

Albedo 0.04 Irradiance mode for total Ed (λ)Albedo 1.0 Radiance mode for direct

fraction fd (λ)Atmospheric water vapor 1.2 g/cm2 Little effect in 400–800 nm rangeAtmospheric ozone – Default climatological valueAtmospheric model Mid-latitude summer Rural extinctionVisibility 50 kmGround altitude zg 2000 m Soda Butte Creek (SBC) field siteSensor altitude zs 2001 m Radiance mode run for

determining fd (λ)Julian day of year 196 July 15Solar zenith angle 30° SBC* (44.91°N, 110.11°W)

at 17:55 GMTSolar azimuth 145° Measured clockwise from north

* Indicates that SBC is an abbreviation for Soda Butte Creek, the field site examined inthis study, conducted in Yellowstone National Park, USA.

1027C.J. Legleiter, D.A. Roberts / Remote Sensing of Environment 113 (2009) 1025–1045

2.1. Image modeling procedure

2.1.1. Simulation of downwelling spectral irradianceReflectance from the water surface and transmission across the

air–water interface depend on how the downwelling spectralirradiance Ed (λ) is partitioned into direct and diffuse components,associated with the solar beam and scattered skylight, respectively.The FIM thus specifies Ed (λ) via two separate runs of the widely usedMODTRAN atmospheric radiative transfer model (Berk et al., 1989).First, a total Ed (λ) spectrum is obtained using MODTRAN's irradiancemode with an albedo of 0.04, a typical value for the in-air irradiancereflectance R (λ) of water; additional parameters are summarized inTable 1. The direct fraction fd (λ) associated with the solar beam isdetermined by performing a second run in radiance mode with analbedo of 100% and a ‘sensor’ altitude 1 m above ground level. Thisparametrization ensures that all incident radiation is reflected andthat the total at-sensor radiance is not modified along the 1 m paththrough the atmosphere. MODTRAN distinguishes between total anddirect ground-reflected radiance, and these data are used to computefd (λ) and partition Ed (λ) into direct and diffuse components.

2.1.2. Simulation of river channel upwelling radiance spectraGiven this irradiance input, the Hydrolight radiative transfer model

(Mobley, 1994; Mobley & Sundman, 2001) is used to simulateirradiance reflectance spectra R (λ) in air just above the water surface.This model is commonly used in studies of coastal bathymetry andbenthic habitat (e.g. Dierssen et al., 2003; Louchard et al., 2003;Mobley et al., 2005), and has been applied to fluvial environments aswell (Legleiter & Roberts, 2005; Legleiter et al., 2004). Because itsnumerical algorithms account for multiple scattering and providesolutions to the radiative transfer equation accurate to approximatelyone percent for a given set of inputs (Lesser & Mobley, 2007),Hydrolight has become a widely accepted tool for simulating aquaticspectra. The model is based upon a simplified, one-dimensionalgeometry: the in-water light field varies with depth, but not withhorizontal location. For situations where depth and/or bottom albedovary spatially (i.e., rivers), the bottom boundary condition can nolonger be assumed horizontally homogeneous, and the true, three-dimensional light field is more appropriately described using compu-tationally intensive Monte Carlo techniques. Because these probabil-istic methods account for the local spatial pattern of depth and bottomreflectance, they must be parameterized separately for each uniquebottom configuration; for complex river channels, such an approachwould be intractable. Mobley and Sundman (2003) demonstrated,however, that light fields predicted by Hydrolight agree with MonteCarlo simulations to within 10% for both variable substrate types andbottom slopes up to 20°. These results imply that radiance fields aboveheterogeneous bottoms are accurately portrayed by efficient one-

dimensional models that, unlike Monte Carlo methods, provide theflexibility required by a spatially explicit forward image model.Moreover, close agreement between numerical simulations and fieldspectra measured along Soda Butte Creek (Legleiter et al., in press)lends someempirical support to our use ofHydrolight as the core of theFIM.

In addition to direct and diffuse components of Ed (λ), importantinputs to Hydrolight included:

1. Water depth d: varied from 0.02–2 m in 0.01 m increments. Neithershallower nor more closely spaced depths could be simulatedbecause Hydrolight requires that adjacent numerical ‘layers’ of thewater column be separated by at least 0.01 m.

2. Bottom reflectance Rb (λ): a spectrum from the periphyton-coatedstreambed of Soda Butte Creek was used for all model runs.Chlorophyll absorption features were prominent throughout alarge sample (n=199) of field spectra that indicated little variationin Rb (λ) within this gravel-bed channel (Legleiter et al., in press).Using a constant bottom reflectance also served to isolate the effectof depth, the variable of primary interest in this study. Thisassumption could be relaxed, and heterogeneous substratesincorporated into the FIM, if reflectance data for multiple bottomtypes and information on their spatial distribution were available.

3. Suspended sediment concentration Cs: held constant at 4 g m−3, atypical to slightly above average value for late-summer, base flowconditions in our study area. This concentration was multiplied bythe brown earth optical cross-section included in Hydrolight tospecify absorption and scattering coefficients. Water and sus-pended sediment were assumed to be the only optically significantcomponents, but chlorophyll and colored dissolved organic mattercould also be considered, given data on their concentrations andspectral characteristics.

4. Wind speed U: held constant at 2 m/s. Hydrolight accounts for theeffects of an irregularwater surfaceon reflectanceand transmittance atthe air–water interface using stochastic surface realizations parame-terized in terms of U. This approach provides a surrogate for flow-related surface turbulence in rivers, andU=2m s−1 corresponds to awater surface elevation standard deviation of 0.032 m. Again, the FIMcouldaccommodatevariable surface roughness, but, givenour focusondepth retrieval, assuming a constant U simplified the analysis.

We used Hydrolight to generate R (λ) spectra as a series of single-wavelength runs, spaced every 4 nm from 400–800 nm. Before thesespectra were input to MODTRAN, non-Lambertian behavior wasaccounted for using the Q factor, the ratio of upwelling irradianceEu (λ) to upwelling radiance Lu (λ). These directional effects wereincorporated bymultiplying each R (λ) spectrumby π sr, the value ofQfor a Lambertian surface, and then dividing by the Q factor calculatedfor that spectrum. Using dimensionless albedo values defined as R (λ)[π/Q(λ)] produced good agreement between Lu (λ) computed byHydrolight and the total ground-reflected radiance from MODTRAN.

2.1.3. Simulation of at-sensor radiance spectraRadiance upwelling from a river channel is modified during

transmission to the sensor, which also records path radiance LP (λ)scattered into the field of view by the atmosphere. To account forthese effects, we propagated the Hydrolight-generated reflectancespectra through the atmosphere to the sensor usingMODO (Schlapfer,2001), an ENVI/IDL interface for MODTRAN that enables efficientsimulation of total at-sensor radiance LT (λ) values for a series ofreflectance spectra (Schlapfer & Schaepman, 2002). Atmosphericparameters and viewing geometry (Table 1) were specified in a baseinput file and a separate MODTRAN run performed for each spectrumin the series, comprised of Hydrolight simulations for different depths.The sensor altitude zs is determined from the ground altitude zg andthe flying height required for the sensor of interest to achieve thedesired ground sampling distance. In anticipation of our application of

Table 2Input parameters for the MD-SWMS channel builder.

Parameter Value(s) Notes

Channel length 400 m Measured along centerlineBankfull width 40 mNumber of meanderwavelengths

0.5 Corresponds to a single bend

Channel bed slope 0.0025Cross-sectional shape Parabolic Maximum bankfull depth of 2 mCrossing angle 45° Determines tightness of bendBar amplitude 0.75 Dimensionless measure of pool-bar reliefPhase difference 30° Defines location of bar relative to bend apexDischarge 10 m3/s Sub-bankfull flowMean wetted width 23 mMaximum depth 1.67 m


the FIM to Soda Butte Creek (zg=2000 m), we performed simulationsfor flying heights of 955, 1910, and 2865 m above terrain,corresponding to 1, 2, and 3 m pixels, respectively, for the AirborneImaging Spectrometer for Applications (AISA), a hyperspectralinstrument used to acquire images of our study area (Legleiter et al.,in press). Although MODTRAN output has a very high spectralresolution, we resampled the MODTRAN LT (λ) spectra to their native,Hydrolight resolution via convolution with a spectral responsefunction consisting of 100 equally spaced, Gaussian-shaped bandsspanning the 400–800 nm range with a constant full width halfmaximum of 4 nm.

2.1.4. Constructing a base radiance field from input bathymetryThe resulting spectra were compiled in a library of at-sensor

radiance values for each depth, denoted by LT (λ; d). This databasecould easily be expanded to include a range of bottom types,suspended sediment concentrations, water surface states, atmo-spheric conditions, and illumination and viewing geometries; theforward image modeling framework allows considerable flexibilityin this regard. Spectra drawn from the library were then used toconstruct a plausible radiance field based on a high spatialresolution depth map provided as input to the FIM. In this study,we considered both an idealized meander bend and field data froma more complex, natural gravel-bed river in Yellowstone NationalPark, described in Section 4. For the hypothetical channel, a simple,known bed configuration was created using the USGS' Multi-Dimensional Surface Water Modeling System (MD-SWMS; Nelsonet al., 2003). Input parameters for the MD-SWMS “channel builder”module are summarized in Table 2, and depths were determined byrunning the hydrodynamic model for steady, uniform flow condi-tions. The resulting depth map and an example cross-section areshown in Fig. 2a and b. Depth data were extracted on a regular 0.5 mgrid, which thus represented the base resolution of the simulatedradiance field.

For each location s within the channel, the local depth d (s) isrounded to the nearest 0.01mand LT (λ; d) is retrieved from the libraryand assigned to that location. Because Hydrolight cannot simulatedepths shallower than dm=0.02 m, the shallowest spectrum in thedatabase is temporarily assigned to all locations for which d (s)bdm;these shallow channel margins are addressed in the next stage of theFIM process. Although Hydrolight assumes a laterally infinite watercolumn of uniform depth, Mobley and Sundman (2003) demonstratedthat, by adjusting the radiance calculated for a level bottom to accountfor differences in illumination, this one-dimensional model can

Fig. 2. Forward image modeling of a hypothetical meander bend. Input to the FIM consists of(b) indicate the locations fromwhich example spectra were extracted. The base radiance fiespectra shown in (d)). A buffer of terrestrial radiancewas added to account for the influence othe base radiance field to form image pixels (g) produced the spectral mixtures shown in (hyielding band-

^integrated radiance spectra (j).

approximate the radiance field for sloping beds. We used Eqs. (9)and (10) of Mobley and Sundman (2003) to calculate a correctionfactor based on local bed slope and aspect, solar azimuth, and the solarzenith angle inwater. Multiplying the level-bottom LT (λ) by this factorreduces the radiance for areas facing away from the sun (spectrum1 inFig. 2d) and increases the radiance where the bed slopes toward thesun (spectrum 5).

2.1.5. Addition of stream bank radianceAlong the margins of the river, radiance from thewetted channel is

mixed with radiance reflected from adjacent terrestrial features. Toaccount for such contamination of the aquatic signal, we first assigneda cover type to the banks: riparian vegetation, gravel, or sand. At-sensor radiance values for each class, denoted by LL (λ), were definedby propagating field spectra measured along Soda Butte Creekthrough the atmosphere using MODTRAN. An LL (λ) spectrum wasthen assigned to each location within a buffer around the wettedchannel based on a simple configuration typical of meandering rivers,with riparian vegetation on the outer bank and a point bar finingdownstream from gravel to sand (Fig. 2e; Bridge, 2003).

This buffer of terrestrial radiance has the same spatial resolution asthe input depth map and does not influence spectra from the wettedchannel until the base radiance field is aggregated to form imagepixels (Fig. 2f). Shallow cells in the input depth map for which d (s)bdm are treated as spectral mixtures at this stage, however. The waterfraction fw for each cell is calculated as d (s)/dm and the compositeradiance defined as fwLT (λ; dm)+(1− fw) LL (a), where LL (λ) is theat-sensor radiance for the adjacent bank type. These calculationsprovide a plausible representation of the light environment at theland–water interface.

2.1.6. Spatial aggregation of the base radiance field to form image pixelsThe base radiance field, comprised of Hydrolight/MODTRAN-

generated spectra within the wetted channel (corrected for variableillumination), mixed aquatic/terrestrial spectra along the edges of thewater, and a buffer of terrestrial radiance (Fig. 2e), has a resolutiondefined by the cell size of the input depth map. To create image pixelswith a specified edge dimension, this radiance field is spatiallyaggregatedusing aGaussianpoint spread function (Collins&Woodcock,1999, Eq. (21)) implemented as a moving window operation. Thisprocedure creates mixed pixels along the channel margins whereterrestrial radiance is encompassed within the sensor's instantaneousfield of view. For example, spectrum 1 in Fig. 2h has high NIR radiancedue to the presence of vegetationwithin the pixel. The pixel-scale meandepth d ̄ is calculated by equally weighting all cells within the movingwindow, including any zero-depth values along the banks.

2.1.7. Convolution to sensor spectral bandsThe resulting pixel-scale LT (λ) spectra have the same 4 nm

spectral sampling interval as the original Hydrolight simulations, butmost remote sensing systems measure radiance integrated overbroader bands. The FIM thus convolves the Hydrolight/MODTRAN-generated LT (λ) with the spectral response function for the sensorunder consideration. In this paper, we obtained simulated AISAimage data by convolving each spectrum in the aggregated, pixel-scale radiance field with a Gaussian response function defined by thecenter wavelength and full width half maximum of each band (Fig. 2iand j). Whereas Hydrolight/MODTRAN simulations yield spectralradiance values with units of W m−2 sr −1 nm−1, convolutionproduces band-integrated radiance values with units of Wm−2 sr−1;

a high-^resolution depth map (a). The numbered points on the channel cross-

^section in

ld in (c) was constructed using the Hydrolight/MODTRAN spectral library (e.g., the fivef the banks (e), but the spectra in (f) were unaffected at this stage. Spatial aggregation of). Finally, the pixel-

^scale radiance field was convolved to the sensor's spectral bands (i),

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Fig. 3. (a) Image-derived depth estimates computed using the log-transformed band ratio expression (1). (b) Matrix of R2 values and regression equation obtained via the optimalband ratio analysis (OBRA) procedure described in the text.


this is an important distinction for determining whether an imagingsystem is capable of detecting an incremental change in radiance,and therefore depth.

2.2. Depth retrieval performance assessment

These algorithms enable forward modeling of an image from theriver of interest, acquired by a particular sensor. The resultingsimulated data can then be used to identify spectral bands useful forbathymetric mapping and to quantify the accuracy and precision ofimage-derived depth estimates. The FIM thus provides a means ofevaluating the utility of remotely sensed data for specific fluvialapplications. We caution, however, that our approach involves anumber of simplifying assumptions and neglects several factors thatcan degrade the quality of actual data. For the example shown inFig. 2, bottom reflectance, suspended sediment concentration, andwater surface state were held constant, and only depth was allowedto vary within the channel. In practice, heterogeneous substrates,variable water column optical properties, and flow-related patternsof surface turbulence can influence upwelling radiance from theriver, potentially in ways not easily represented by our first-ordermodel. Atmospheric heterogeneity can also modify transmittance to

Fig. 4. Depth retrieval residuals (a) mapped for the hypothetical meander bend, (b) compsummary statistics. Predicted and observed depths at six validation cross-sections are comp

the sensor and introduce variable amounts of path radiance.Because the FIM is essentially one-dimensional, neither in-air norin-water adjacency effects are considered explicitly, though theycould be significant for low-albedo aquatic targets. Similarly, weassume linear mixing while aggregating the base radiance field toform image pixels, both along the banks and within the water,where spectra for different depths are combined. Shadows, sensornoise, and image geometric correction and re-sampling are notaccounted for, either, but these factors could also reduce the fidelityof real image data relative to the idealized simulations generated bythe FIM.

Analyses of simulated image data thus yield upper bounds on theaccuracy and precision theoretically achievable via passive opticalremote sensing of river bathymetry; FIM results are best interpretedas a first-order assessment of the feasibility and potential perfor-mance of spectrally-based depth retrieval. The model does notrepresent a definitive accounting of bathymetric uncertainty and thisimportant caveat must be borne in mind. Nevertheless, we believethat because the framework described here explicitly incorporatesradiative transfer processes in the water column and atmosphere,our physics-based approach provides a useful tool for examining thecapabilities and limitations of remote sensing of rivers. More

ared to the observed depths for cross-section 8, and (c) plotted as a histogram withared in (d).


importantly, the model's flexible, modular nature allows individualcomponents to be refined, removed, or added to enable morerigorous studies.

2.2.1. Optimal band ratio analysisRemote detectors measure radiance reflected from a number of

different sources, including the water surface, water column,streambed, and atmosphere. Of these components, only the bottom-reflected radiance LB (λ) is directly related to depth. Inferringbathymetry from remotely sensed data is complicated by thedependence of LB (λ) not only on d but also Rb (λ), but a simple,ratio-based method effectively distinguishes changes in depth fromvariations in bottom albedo (e.g., Dierssen et al., 2003; Mishra et al.,2007). Using a combination of Hydrolight simulations, field spectra,and hyperspectral image data, Legleiter et al. (in press) demonstratedthat the image-derived quantity X, defined as

X = lnLT λ1ð ÞLT λ2ð Þ

� �ð1Þ

where λ1 and λ2 denote two spectral bands, is strongly linearlyrelated to depth and thus useful for mapping river bathymetry. Thekey to such ratio-based depth retrieval is to select pairs of wave-lengths that are responsive to changes in d but insensitive to otherfactors influencing LT (λ). Optimal band ratio analysis, or OBRA, is asimple technique for determining which band combination yields thestrongest linear relationship with depth. Taking as input pairedobservations of d and LT (λ) measured in n spectral bands, thealgorithm considers each pair of wavelengths in turn, calculates Xvalues via Eq. (1), and performs regressions of d on X. The results arethen used to populate an n×n matrix of R2 values that summarizesspectral variations in the strength of the X vs. d relation; the optimalband ratio is that yielding the highest R2 (Legleiter et al., in press).

In the context of the FIM, input to OBRA consists of band-integrated radiance values and pixel-scale mean depths. These datawere extracted from the simulated AISA image along 12 cross-sectionsspaced equally throughout the bend (Fig. 3a), a typical sampling forfield-based calibration of image-derived depth estimates. Half of thesetransects were provided as input to OBRA and half were retained foruse as validation data. For the homogeneous substrate, opticalproperties, and water surface state specified for this hypotheticalchannel, the R2 matrix displayed in Fig. 3b indicates that red and near-infrared denominator bands yield strong correlations with depth. TheR2 value of 0.952 and standard error of 0.089 m for the optimal bandratio suggest that remote bathymetric mapping can be highlyaccurate, and the spatial pattern of image-derived estimates(Fig. 3a) closely matches the input depth map (Fig. 2a).

2.2.2. Quantifying depth retrieval accuracyApplying the X vs. d relation from OBRA throughout the image

yielded, for each location s, a spectrally-based depth estimate d ̂ (s). Tocharacterize bathymetric accuracy, each prediction was compared tothe corresponding pixel-scale mean depth d ̄ (s) and a residual ε (s)computed as

e sð Þ = P

d sð Þ− d̂ sð Þ: ð2Þ

Positive residuals thus indicate under-predictions of depth [ε (s)N0⇒ d

_(s)Nd(̂s)] and negative residuals correspond to over-predictions

[ε (s)b0 ⇒ d_(s)bd (̂s)]. For the simulated meander bend, where d

_(s)

was known throughout the channel, ε (s) values were used to producea map of depth retrieval errors (Fig. 4a), which indicated that depthswere under-predicted through the apex of the bend. Comparison ofpredicted and observed depths for a validation cross-sectionsuggested that image-derived estimates saturated in deeper water(Fig. 4b), where changes in depth produce only relatively small

changes in radiance. The negative depth predicted on the right edge ofthe channel implied that depth retrieval from very shallow water alsomight be unreliable due to the influence of the banks on the pixel-scale LT (λ). The overall impact of these errors was assessed byexamining the distribution of residuals in Fig. 4c. On average, d ̂ (s)was an unbiased estimator of d

_(s), but the modal ε (s) is negative,

implying that depths were over-predicted by ~0.05 m for much of thechannel (yellow tones in Fig. 4a). Over-predictions greater than0.06 m were uncommon, but a few large negative residuals, on theorder of −0.25 m, occurred along the outer bank. The distinctsecondarymodes at ε (s)=0.2m and 0.3m corresponded to relativelylarge under-predictions in the pool and along the sandy, lower portionof the point bar, respectively.

This analysis not only summarized overall accuracy but alsohighlighted systematic depth retrieval errors. Though evident withina forward modeling context when the true bathymetry was knownexhaustively, these biases could easily go unnoticed in practice, whenonly much smaller samples are available for evaluating image-deriveddepth estimates. For example, a simple plot of predicted vs. observeddepths for six validation cross-sections, typical of the data practicablyobtained via traditional field methods, indicated strong agreement atfirst glance (Fig. 4d). Closer inspection revealed some subtle discre-pancies between d(̂s) and d

_(s), however: d

_(s) plotted above the

regression line for greater depths and the scatter about this relationincreased in shallower water. Interpreting these discrepancies on thebasis of sparse field data would have been difficult, but forward imagemodeling revealed the spatial pattern of depth retrieval errors in thecontext of channel morphology and bank cover type. A distinctadvantage of the FIM is the ability to identify consistent biases inimage-derived depth estimates, explain why these errors occur, andassess the extent towhich such issues could affect a specific application.

2.2.3. Quantifying depth retrieval precisionFor many studies, the utility of image data depends on the ability to

resolve subtle variations in depth across a broad range of depths, andthe precisionwithwhich bathymetry can be remotelymapped becomesan important consideration. We define depth retrieval precision as thesmallest change in depth for which the associated change in LT (λ) isdetectable by the imaging system. Whether an incremental change indepth Δd can be resolved at an initial depth d0 depends on themagnitude of the corresponding change in at-sensor radiance

ΔLT λ;d0 + Δdð Þ = LT λ; d0 + Δdð Þ− LT λ; d0ð Þ ð3Þ

relative to the sensor's noise-equivalent delta radiance ΔLN (λ). Variousdefinitions ofΔLN (λ) have been proposed (e.g., Brando & Dekker, 2003;Schott, 1997), but in this paper we simply equate ΔLN (λ) with thechange in radiance equivalent to one digital number. This quantity isdetermined by the sensor's radiometric resolution, and the smallestchange in radiance an imaging system can resolve is the inverse of thenumber of possible discrete values. For example, for a 12-bit instrument,ΔLN=1/212=0.00024, assumed to be a constant, band-integratedvalue with units of Wm−2 sr−1. Calculations based on this definition ofΔLN (λ) thus yield the theoretical maximum depth retrieval precisionattainable by a sensor. In practice, a more inclusive specification of ΔLN(λ) that considers both scene-specific imaging conditions and sensorsignal-to-noise characteristics should beused (e.g., Giardino et al., 2007;Wettle et al., 2004), and the bathymetric precision predicted by the FIMwill be reduced due to the larger, more realistic ΔLN (λ).

In any case, a change in depth is detectable only if the followinginequality holds:

jΔLT λ;d0 + Δdð Þ j N ΔLN λð Þ: ð4Þ

To evaluate whether a given change in depth satisfies Eq. (4), arange of Δd values are added to the original depth map, the FIM used


to generate an image corresponding to each depth increment in turn,and ΔLT (λ; d0+Δd) calculated by subtracting the at-sensor, band-integrated radiances for each depth-incremented image from thecorresponding values in the image produced from the original depthmap. The resulting ΔLT (λ; d0+Δd) values are then compared to thesensor's ΔLN (λ) to determine where and in which spectral bands agiven Δd is detectable by the imaging system.

To provide a metric of bathymetric precision, we next define theminimum detectable increase in depth, denoted by Δd+ (λ), as thesmallest ΔdN0 for which the condition specified by Eq. (4) is satisfied.Similarly, the minimum detectable depth decrease Δd− (λ) is the leastnegative Δdb0 for which Eq. (4) holds. Note that in general Δd+ (λ)≠ |Δd− (λ) | because an increase in depth does not necessarily produce achange in radiance equivalent in absolute value to the change in radianceproduced by a decrease in depth of the samemagnitude. For example, inabsorption-dominated red and near-infrared wavelengths, a 0.05 mdecrease in depthwill produce a larger positiveΔLT (λ; d0+Δd) than thenegative ΔLT (λ; d0+Δd) associated with a 0.05 m increase in depth,implying that Δd+ (λ)b | Δd− (λ) |.

Conceptually, this algorithm amounts to shifting the bed up ordown by Δd, updating the radiance field, re-simulating the image, andevaluating whether the resulting ΔLT (λ; d0+Δd) is detectable by a

Fig. 5.Metrics of depth retrieval precision for band 21 (693 nm) of the simulatedAISA image, shd) minimum detectable increase in depth Δd+; and (e and f) total bathymetric contour interv

sensor with the specified ΔLN (λ). Although the depth is incrementeduniformly over the entire channel, Δd+ (λ) and Δd− (λ) vary spatiallybecause Eq. (4) is a function not only of Δd but also the original depthd0. As a result, relatively small changes in depth can be resolved insome parts of the channel whereas only a larger Δd would bedetectable in other areas. For the simulated image of a meander bend,Fig. 5a indicates that Δd− (693 nm)b0.07 m for most of the river foran assumed ΔLN (λ) of 1/212=0.00024 W m−2 sr−1, suggesting thatdepth decreases on the order of a few centimeters could be resolvedwith a 12-bit imaging system (Fig. 5b). In the deep, central portion ofthe channel near the bend apex, decimeter-scale changes in depthwere required to satisfy Eq. (4), however. For certain areas, a depthdecrease of 0.15 m, the largest considered, did not produce asufficiently large change in radiance to be detectable. Fig. 5c illustratesa similar pattern for Δd+ (693 nm), with depth increases less than0.1 m detectable for all but the greatest depths, where no Δdb0.15 mcould be resolved for some locations (Fig. 5d).

Our use of the phrase ‘change in depth’ does not imply an increaseor decrease in bed elevation or flow stage over time but rather refersto spatial variations in bathymetry. The ability of an imaging system toresolve these variations can be conceptualized in terms of contourintervals. The fundamental detection limits dictated by the range of

own asmaps and histograms. (a and b)Minimumdetectable decrease in depthΔd−; (c andal width Δdc. See text for details.


depths, bottom albedos, and water column optical properties presentin the channel of interest, as well as sensor characteristics, imply thattruly continuous depth maps cannot be derived from digital imagedata. Spectrally-based depth estimates are instead subject to aninherent uncertainty defined by the smallest discernible increase anddecrease in depth. The total width of a bathymetric contour intervalcan thus be defined as

Δdc λð Þ = Δd + λð Þ + jΔd− λð Þ j : ð5Þ

This quantity represents the fundamental limit of precision fordepth retrieval from remote measurements of LT (λ). The map andhistogram of Δdc (693 nm) shown in Fig. 5e and f indicate thatbathymetric precision was strongly dependent on depth. Throughoutmost of the bend, Δdc (693 nm) was on the order of 0.1 m and tendedto increase from the shallow margins toward the deeper center of thechannel, where a given Δd produced only a small ΔLT (λ; d0+Δd)below the detection threshold Eq. (4) of all but the most sensitiveinstruments. In this case, the blank areas in Fig. 5e indicate that formuch of the deepest part of the bend, depth retrieval precision for a12-bit imaging system is no better than 0.3 m, the largest range ofdepth increments modeled.

The series of depth-incremented images can also be used toexamine spectral variations in depth retrieval precision. For a givenΔd, displaying ΔLT (λ; d0+Δd) values determined via Eq. (3)illustrates the spatial pattern of radiance differences. For example,Fig. 6a indicates that for Δd=0.05 m, ΔLT (693 nm; d+0.05 m) was arelatively large negative value for the shallow margins of the channel

Fig. 6. Change in radiance for a fixed change in depth of Δd=0.05 m, plotted as (a) an imagethe legend, of the distribution of(b) ΔLT (λ; d+0.05 m) and (c) | ΔLT (λ; d+0.05 m) |. The h(λ) and the vertical lines indicate the position of the spectral band featured in (a).

but approached zero in deeper water. Averaged over the reach, ΔLT(693 nm; d0+0.05 m)=−0.000251 W m−2 sr−1, which barelyexceeded ΔLN (λ) in absolute value, implying that a Δd of 0.05 m wasmarginally detectable for this band. Plotting percentiles of the ΔLT (λ;d0+Δd) distribution as spectra can help identify wavelengths usefulfor resolving relatively small changes in depth. Fig. 6b highlights thetransition from scattering-dominated wavelengths in the blue andgreen, for which ΔdN0 ⇒ ΔLT (λ; d0+Δd)N0, and longer absorption-dominated wavelengths, where ΔdN0 ⇒ ΔLT (λ; d0+Δd)b0. In thistransition region, changes in depth corresponded to small changes inradiance that were below the detection threshold indicated by thethin horizontal lines in Fig. 6b; only in bands experiencing strongerscattering or absorption would Δd=0.05 m have been discernible. Ifthe kth percentile of the distribution of absolute changes in radiance,denoted by | ΔLT (λ; d0+Δd) |k, does not exceed the sensor's ΔLN (λ),then Δd will be undetectable over at least k percent of the channel.Conversely, the imaging system is theoretically capable of resolving adepth difference of Δd over at most 100− k% of the channel. Forexample, Fig. 6c indicates that 25% of | ΔLT (693 nm; d0+0.05 m) |values are less than the specified ΔLN (λ), and a Δd of 0.05 m will bedetectable over at most 75% of the channel. This type of analysiscomplements OBRA by identifying those wavelengths most sensitiveto subtle bathymetric variations.

3. Analytical approach to characterizing bathymetric precision

The forwardmodeling framework outlined above fully accounts forthe interaction of light and water in shallow stream channels, to the

for a single AISA band and as spectra for different percentiles, denoted by subscripts inorizontal lines in (b) and (c) represent the sensor's noise-equivalent delta radiance ΔLN


extent that the radiative transfer processes involved are adequatelyrepresented by the one-dimensional Hydrolight model. Becauseimportant, spatially explicit effects are also treated explicitly, theFIM yields a thorough description of the accuracy and precision withwhich the bathymetry of a particular river can be remotely mapped bya given sensor. This type of direct numerical simulation is computa-tionally intensive, however, and the data on bed topography andbottom reflectance required by the FIM can be difficult to obtain. Anexisting, analytical model, developed by Philpot (1989), provides analternative approach that is much more efficient and, in somerespects, more general. To assess the extent to which such theoreticalcalculations could complement, or even obviate, computationaltechniques, we compared FIM results to predictions of bathymetricprecision derived via Philpot's formulation.

A simple and widely used model for the total at-sensor radiancefrom optically shallow water has the following general form (e.g.,Lyzenga, 1978; Maritorena et al., 1994; Philpot, 1989)

LT λð Þ = LB λð Þexp − K λð Þdf g + LW λð Þ: ð6Þ

The first term represents radiance that has interacted with thestreambed, K (λ) is an ‘effective’ attenuation coefficient used to

Fig. 7.Minimum detectable changes in depth Δd plotted as spectra for a fixed initial depth d0suspended sediment concentrations. Solid lines indicate Δd values obtained via direct numerthin vertical line marks the crossover from scattering- to absorption-dominated radiative tplotted, corresponding to decreases in depth in the scattering-dominated region to the leftplotted indicate that no depth increment included in our Hydrolight database was detectab

summarize the optical properties of the water column, and LW (λ)denotes radiance observed over optically deep water (i.e., K (λ)d→ ∞). The latter term thus accounts for radiance scatteredwithin thewater column, reflected from the water surface, and added along theatmospheric path. From this starting point, Philpot (1989, Eq. (20))derived an expression for the smallest depth difference detectable byan imaging system

Δd λð Þ = −1K λð Þ ln 1−ΔLN λð Þ

LB λð Þ exp K λð Þd0f g� �

: ð7Þ

This relation provides an analytical approximation toΔd (λ) valuescomputed via the FIM.

Calculating Δd (λ) with Eq. (7) requires information on K (λ),which is essentially an index of the optical properties of water column.Determining a single, effective attenuation coefficient is complicatedby the need to account for upwelling flux from both the water columnand a reflective substrate (e.g., Dierssen et al., 2003; Maritorena et al.,1994). Because these fluxes are difficult to measure in coastal settings,let alone rivers, K (λ) is typically approximated as 2Kd (λ) (Maritorenaet al., 1994; Philpot, 1989), where Kd (λ) denotes the diffuse

of 0.5 m, ΔLN (λ) values for noise-free 12-, 10-, and 8-bit imaging systems, and a range ofical simulation, dashed lines indicate theoretical predictions of Δd from Eq. (7), and theransfer. For clarity, only those Δd (λ) values associated with decreases in radiance areof the crossover and increases in depth to the right. Wavelengths for which no line isle and/or that Eq. (7) was undefined.


attenuation coefficient for downwelling irradiance. In this study, Kd

(λ) was determined using a relation from Gordon (1989):

Kd λð Þ≈ a λð Þ + bb λð Þcos θswð Þ ð8Þ

where a (λ) and bb (λ) are absorption and back-scattering coefficients,respectively, and θsw is the solar zenith angle in water. Thisformulation has two significant advantages: 1) a (λ) and bb (λ) areinherent optical properties that do not depend on the ambient lightfield; and 2) existing data on the optical properties of pure water andsuspended sediment can be used to express these quantities, andhence Kd (λ), in terms of Cs (Bukata et al., 1995).

Eq. (7) indicates that Δd (λ) also depends on the bottom-reflectedradiance LB (λ), which is a function of the bottom contrast Rb (λ)− R∞(λ) between the streambed and optically deep water. Plausibleestimates of LB (λ) were obtained via the following procedure: 1)parameterizeHydrolight as described in Section 2 to specify the total at-sensor radiance LT (λ); 2) perform an additional radiative transfersimulation for a hypothetical, infinitely deep water body with the sameoptical properties (i.e., suspended sediment concentration) to define LW(λ); and3) substitute 2Kd (λ) fromEq. (8) forK (λ) and rearrange Eq. (6)to calculate LB (λ).

3.1. Comparison of theoretical prediction and direct numericalsimulation of depth retrieval precision

Given estimates of the effective attenuation coefficient K (λ) andbottom-reflected radiance LB (λ), Eq. (7) can be used to determine thesmallest change in depth Δd (λ) detectable at initial depth d0 by animaging systemwith a noise-equivalent delta radiance of ΔLN (λ). Thevalidity of this analytical approach was assessed by comparing thesetheoretical predictions to the minimum detectable depth differencesdetermined via direct numerical simulation. The computationalapproach involved retrieving the LT (λ) spectrum for initial depth d0from the Hydrolight-generated spectral library and then querying thelibrary to identify the smallest depth increment for which Eq. (4) wassatisfied. Whereas the full FIM incorporated spatial information,sensor spectral response, and atmospheric effects, only the originalHydrolight spectra were used in this analysis so that numerical resultscould be directly compared to analytical approximations that did notaccount for these effects. We considered ΔLN (λ) values of 2−12, 2−10,and 2−8 W m−2 sr−1 nm−1, the fundamental minima achievable by12-, 10-, and 8-bit imaging systems, respectively. To examine thespectral variation and depth dependence of depth retrieval precisionacross a range of conditions, we calculated Δd (λ) values forwavelengths from 400–800 nm, depths from 0.05–2 m, and Cs valuesof 0, 4, 8, and 40 g m−3. The latter range encompassed pure water, adoubling, and an order of magnitude increase relative to the 4 g m−3

concentration used in the FIM example above.For the limiting case of sediment-free water, agreement between

theoretically- and numerically-derived Δd (λ) values is superb, andFig. 7a clearly illustrates the importance of sensor radiometricresolution. Whereas a 12-bit imaging system can resolve depthvariations of 0.05 m or less throughout the red and NIR, Δd (λ) for an8-bit system varies from 0.4 m to 0.9 m in this region. The spectralpattern of the 8-bit Δd (λ) curve is dictated by that of K (λ), which isdetermined by the absorption coefficient of pure water in this case,and LB (λ), which is a function of the periphyton Rb (λ) spectrum usedto parameterize Hydrolight.

For the base case Cs of 4 g m−3, agreement between theoretical andcomputational Δd (λ) values remains good for strongly absorbingwavelengths in the vicinity of 700 nm, even for the largest ΔLN (λ)considered (Fig. 7b). Theoretical andnumerical predictionsdivergewhereattenuation isweaker,with Eq. (7) tending to indicate a largerΔd (λ) thancomputed via direct numerical simulation for 600 nmbλb685 nm.

Conversely, to the left of the scattering–absorption crossover, the theoryappears to be overly optimistic — Hydrolight model results indicate thatsatisfying the detection threshold Eq. (4) requires larger decreases indepth than predicted by Eq. (7). The discrepancy between theory andcomputation is greatest near the crossover, where the analytical resultsare most sensitive to water column optical properties.

This sensitivity becomesmore evidentwhenCs is doubled to 8gm−3,which causes the transition from scattering- to absorption-dominatedradiative transfer to shift toward longer wavelengths (Legleiter et al.,inpress). Just beyondthis transition, for 640bλb680nm, Eq. (7) predictsthat only depth decreases are detectable, whereas direct numericalsimulation indicates that only fairly large increases in depth can beinferred from decreases in radiance (Fig. 7c). This difference most likelyresults from an incomplete description of water column optical proper-ties in the analytical model. For both shorter and longer wavelengthsoutside this region, agreement between the analytically- and numeri-cally-derived Δd (λ) values is better than in the Cs=4 g m−3 case, andboth approaches indicate that depth increases can no longer be detectedby an 8-bit sensor for d0=0.5 m. At 710 nm, depth increases of 0.03 mand 0.12 m can be resolved by 12- and 10-bit imaging systems,respectively, butΔd (λ) increases rapidly to either sideof thiswavelengthdue to the sharp increase inK (λ) in the NIR. For 700 nmbλb710 nm,Δd(λ) increases because attenuation by the water column is weaker,producing a smaller change in radiance for a given depth increment. ForλN710 nm, absorption by pure water is so strong that small increases indepth produce relatively large decreases in radiance, which causes thesignal to saturate in deeperwater— the 10-bit sensor in Fig. 7c illustratesthis effect. To the left of the scattering–absorption crossover, decreases indepthon theorderof 0.03mand0.13mcanbedetectedby12- and10-bitinstruments, respectively. Whereas Eq. (7) predicts that an 8-bit systemshould be capable of resolving 0.35–0.40 m depth decreases, directnumerical simulation indicates that a Δd of this magnitude will notproduce a decrease in LT (λ) measurable by such a sensor.

Fig. 7d indicates that even for a Cs value an order of magnitudegreater than that examined with the FIM, a limited amount ofbathymetric information can be inferred from measurements of LT (λ);for d0=0.5 m, only decreases in depth are detectable. Under thesehighly-scattering conditions, a decrease in depth produces a decrease inradiance because the relatively bright water column is truncated by adarker substrate — that is, Rb (λ)bR∞ (λ). At λ=600 nm, boththeoretical predictions and direct numerical simulation indicate that adecrease in depth of 0.03 m produces a sufficiently large decrease in LT(λ) to be detected by a 12-bit system. Agreement between theory andcomputation deteriorates as ΔLN (λ) increases, however, and Eq. (7)over-predicts the change in depth required to produce a detectablechange in radiance. This discrepancy results from an underestimation ofK (λ) by Eq. (8)) for this scattering-dominated water, which translatesinto an underestimate of the change in radiance associatedwith a givenΔd and hence an overly conservative estimate of the smallest change indepth that can be resolved. In general, our results suggest that analyticaland computational predictions of bathymetric precision diverge as thebottom contrast diminishes, as scattering predominates over absorp-tion, and as ΔLN (λ) increases.

The dependence on depth of bathymetric precision is illustrated inFig. 8. For both scattering- and absorption-dominated wavelengthsand two different sediment concentrations, Δd (λ) increases inabsolute value as d0 increases — depth estimates are more uncertainin deeper water. For a given depth, the magnitude of the smallestdetectable change in depth is strongly dependent on sensor radio-metric resolution, and Δd (λ) increases with ΔLN (λ). Moreover, forλ=710 nm, the rate at which Δd (λ) increases with d0 is greater forless sensitive instrumentation, limiting the dynamic range over whichbathymetry can be remotely mapped. For example, Fig. 8a indicatesthat the maximum (d0, Δd) combination determined via directnumerical simulation decreases from (1.6 m, 0.36 m) to (1.1 m,0.52 m) to (0.6 m,1.1 m) as radiometric resolution is reduced from 12-

Fig. 8. Smallest detectable depth difference Δd plotted against the initial depth d0 for an absorption- and a scattering-dominated wavelength (510 and 710 nm, respectively) forsuspended sediment concentrations Cs of 4 g m−3 and 8 g m−3. Three values of the noise-equivalent delta radiance ΔLN (λ) are considered. Solid lines indicate computationally-derived Δd values and terminate where the radiance for the initial depth d0 is not sufficiently different from the radiance for any other depth in the Hydrolight database to bedetected. The dashed lines represent theoretical predictions and end where Eq. (7) is undefined. The shaded areas represent (d0, Δd) combinations not included in the Hydrolightdatabase (a and b), or for which the smallest detectable decrease in depth is larger than the depth (c and d).


to 10- to 8-bits. For the higher Cs conditions depicted in Fig. 8b, theseranges are even further restricted. For this strongly absorbingwavelength, agreement between theory and computation is quitegood, with Eq. (7) yielding slightly larger estimates of Δd (λ) thandirect numerical simulation. The discrepancy between the twoapproaches is greater for Cs=8 g m−3 because Eq. (8) underpredictsK (λ) when scattering is more significant, which translates into alarger Δd (λ) and thus a relatively conservative assessment ofbathymetric precision.

For the scattering-dominated 510 nm wavelength (Fig. 8c and d),decreases in depth correspond to decreases in radiance due to reducedvolume reflectance from a thinner water column. As for the absorption-dominated band, only larger changes in depth can be resolved bysensors with greater ΔLN (λ) detection thresholds, and | Δd (λ) |generally increases with d0. The minor inflection in the (d0, Δd) curvesimplies that in shallower water, slightly larger decreases in depth mustoccur to be detected. These curves terminate before reaching thesmallest depths because further decreasing the thickness of an alreadythin water column does not produce a sufficient decrease in radiance.As Cs increases, a given sensor can resolve smaller depth decreases inshallowerwater due to the greater volume reflectance associatedwith ahigher Cs. For example, for the 8-bit imaging system, direct numerical

simulation indicates that for Cs=8 g m−3, a reduction in depth from0.6 m to 0.04 m can be resolved, but if Cs is halved to 4 g m−3, adecrease in depth from 1.3 m to 0.09 m is the smallest change in depththat can be detected at the shallowest depth.

Although theoretical predictions of depth retrieval precisioncapture the trend and generalmagnitudeobtained via direct numericalsimulation, agreement between the two approaches is not as strong forthe scattering-dominated 510 nm band as for 710 nm, whereabsorption by pure water prevails. Eq. (7) under-predicts themagnitude of the smallest detectable depth decrease in shallowwater and over-predicts | Δd (λ) | at greater depths. The greaterdiscrepancy between theory and computation results from aninadequate description of water column optical properties under theframework of Eq. (6). In general, the analytical approach is lessapplicable to scattering-dominated wavelengths and/or high-Csconditions because Cs affects not only K (λ) but also LB (λ), throughits influence on the contrast between the water column and substrate.Moreover, the optical properties of suspended sediment are not nearlyas well characterized as those of pure water (Bukata et al., 1995).Nevertheless, Fig. 8c and d imply that some bathymetric informationcan be derived from scattering-dominated wavelengths in deeperareas where depth retrieval from absorption-dominated bands is not

Fig. 9. Dimensionless representation of bathymetric precision, based on Philpot (1989,Fig. 10). The optical depth weights the geometric depth d0 by the effective attenuationcoefficient K (λ) and is also known as the effective attenuation length. The vertical axisindicates the depth retrieval percent error, defined as the ratio of the minimumdetectable change in depth Δd to the depth d0. The contours specify the ratio of thesensor's noise-equivalent delta radiance ΔLN (λ) to the bottom-reflected radiance LB (λ)and thus represent an index of the detectability of the bottom.


feasible; for a 12-bit sensor, depth decreases of 0.14 m and 0.32 m aredetectable at a depth of 2m for Cs of 4 gm−3 and 8 gm−3, respectively.In essence, Fig. 8a and b complement Fig. 8c and d, with the

Fig. 10. (a) Location of Soda Butte Creek and the Footbridge Reach study area, denoted by Freach, looking downstream. (c) Input to the forward image model consists of bathymetry d

absorption-dominated band providing precise depth estimates inshallow water, where LT (510 nm) is insensitive to depth, and thescattering-dominated wavelength providing some bathymetric infor-mation at greater depths, where LT (710 nm).

3.2. Controls on bathymetric precision: a dimensionless framework

The theory articulated by Philpot (1989) and revisited here yieldspredictions of bathymetric precision that agree reasonably well withcomputationally intensive direct numerical simulations. More impor-tantly, this analytical approach provides insight on the factorscontrolling depth retrieval performance. Starting from Eq. (7), Philpot(1989) showed how the uncertainty inherent to image-derived depthestimates can be expressed in terms of three dimensionlessquantities:

1. Δd (λ)/d0, the smallest detectable change in depth as a proportionof the initial depth, provides a useful metric of bathymetricprecision. If depth estimates are unbiased on average, Δd (λ)/d0can be interpreted as the depth retrieval percent error.

2. K (λ) d0 is a quantity known as the optical depth that combinesthe geometric depth with an index of the river's optical proper-ties to quantify attenuation by the water column. This dimen-sionless product weights the geometric depth by the spectrallyvariable attenuation coefficient K (λ) so that a fixed d0 can have amuch greater optical depth for strongly absorbing wavelengthsthan for bands with greater transmission. Similarly, because K(λ) depends on Cs, the optical depth can vary for a fixed d0 due tovariations in Cs.

3. ΔLN (λ)/LB (λ), the ratio of the sensor's noise-equivalent deltaradiance to the bottom-reflected radiance, quantifies the detect-ability of the streambed. LB (λ) is primarily determined by thebottom contrast Rb (λ)− R∞ (λ) between the substrate and water

B, in the northeastern corner of Yellowstone National Park, USA. (b) Photograph of theerived from ground survey data and a simple classification of bank cover types.

Table 3Footbridge Reach characteristics.

d± σd Max (d) Width Bed slope Drainage area D16 D50 D84

(m) (m) (m) (m/m) (km2) (mm) (mm) (mm)

0.17±0.10 0.49 22 0.0066 239 23 48 90

Width is the reach-averaged wetted width based on the surveyed water surface profile.Dx denotes the percentile of the bed material grain size distribution for which x% arefiner.


column but also depends on the downwelling irradiance andtransmission properties of the air–water interface. ΔLN (λ) is afunction of radiometric resolution and, in practice, instrumentaland environmental noise.

These quantities define a dimensionless parameter space thateffectively summarizes interactions among the primary controls ondepth retrieval precision. Because K (λ), LB (λ), and possibly ΔLN (λ)depend on wavelength, plots like Fig. 9 (based on Philpot's Fig. 10)can also be used to examine how these factors vary across thespectrum. For sensitive instruments and relatively bright substratesfor which ΔLN (λ)/ LB (λ)b0.01, the theory predicts very precisedepth estimates should be possible, with Δd (λ)/d0 on the order of10% or less for up to 3 optical depths. For example, for d0=1m and K(700 nm)=2 m−1, a ΔLN/ LB value of 0.01 translates into a depthretrieval error Δd (700 nm)b0.05 m. If the optical depth is fixed butthe bottom contrast and/or detector sensitivity are modified todecrease ΔLN (λ)/ LB (λ) by an order of magnitude to 0.1, theuncertainty associated with image-derived depth estimates isgreater than 0.6 m. For deeper and/or more strongly attenuatingwater (i.e., a greater number of optical depths) and a constantbottom reflectance (i.e., a fixed LB (λ)), Fig. 9 indicates that abathymetric precision Δd (λ)/d0b0.1 can only be achieved byinstruments with greater radiometric resolution. For example, ifCs=4 g/m−3, LB (700 nm)=0.008 W m−2 sr−1 nm−1 for d0=2 m,implying that ΔLN (700 nm) must be on the order of 0.00005Wm−2

sr−1 nm−1 to achieve a 10% error at 4 optical depths. For the sameconditions but an initial depth of 1m, the situation improves becauseLB (700 nm) increases to 0.011 W m−2 sr−1 nm−1, but a ΔLN(700 nm) of 0.00027 W m−2 sr−1 nm−1 or less would still berequired to achieve a 10% error. To put this value in perspective, anidealized, noise-free sensor with 12-bit quantization has a minimumΔLN (λ) of 0.00024Wm−2 sr−1 nm−1, only 11% smaller than the ΔLN(700 nm) needed to estimate depth with a precision on the order of0.1 m under these conditions. The fact that many imaging systemscannot resolve such subtle changes in radiance places a fundamentallimit on the precision with which river bathymetry can be remotelymapped.

The theory developed by Philpot (1989) provides a convenientframework for defining these limitations, particularly if one con-siders which factors might vary across an image and which can betreated as constant. For a particular scene from some river of interest,the most important imaging conditions are fixed: downwellingirradiance Ed (λ), solar zenith angle, and atmospheric properties.Similarly, two critical river attributes can be assumed relativelyconstant on a reach scale. First, Cs, which influences K (λ) and LB (λ),should not vary appreciably in the absence of tributary inputs, bankfailures, or other sources of fine sediment. Second, bottom reflec-tance, a primary control on LB (λ), might be fairly uniform in gravel-bed rivers lacking submerged aquatic vegetation. For our study area, alarge sample of field spectra indicated that Rb (λ) was fairlyhomogeneous, even for a substrate comprised of a range of grainsizes and lithologies (Legleiter et al., in press). Additional data setsare needed to assess the generality of this result, however. A thirdriver attribute, water surface state, can vary considerably from brokenwater in riffles to a glassy texture over pools, and these variations insurface roughness can influence LB (λ) through their effect onreflectance and transmittance at the air–water interface. Never-theless, field observations and radiative transfer modeling indicatethat flow depth, the channel characteristic of primary interest, is themost important control on themagnitude, spectral shape, and spatialpattern of LT (λ) in clear, shallow rivers (Legleiter et al., in press).Because ΔLN (λ) is a property of the sensor, Fig. 9 reduces to a singleΔLN (λ)/ LB (λ) curve that relates the optical depth to the depthretrieval percent error. If K (λ) and LB (λ) are assumed constant, asimple plot of Δd (λ) as a function of depth d0 can be obtained. The

analytical approach thus provides an efficient first-order assessmentof the uncertainty inherent to spectrally-based depth retrieval, butthe ability to map spatial variations in bathymetric precision in thecontext of the river's morphology represents a distinct advantage ofthe forward image model.

4. Application of the forward image model to a gravel-bed river

In this section, we present a case study in which the FIM is appliedto a gravel-bed river to evaluate the ability of remote sensingtechniques to contribute to an investigation of river morphodynamics.To support our ongoing study of sediment transfer and channelchange in northern Yellowstone National Park, USA, we addressed twoprimary research questions:

1. How does the reliability of image-derived depth estimates varyspatially in relation to the morphology of a complex, naturalchannel?

2. What are the effects of sensor spatial and radiometric resolution onthe accuracy and precision with which river bathymetry can beremotely mapped from hyperspectral image data?

These issues were examined via forward image modeling of SodaButte Creek, a tributary of the Lamar River in the Park's northeasterncorner (Fig. 10a). This dynamic watershed has been the subject ofseveral previous remote sensing studies (Legleiter et al., 2002, inpress; Marcus et al., 2003; Wright et al., 2000), and we sought todetermine the extent to which image data sets acquired by varioussensors could be used to measure changes in channel morphology,estimate volumes of erosion and deposition, and, ultimately, infer bedmaterial transport rates.

To make this assessment, we applied the FIM to the 320-m longFootbridge Reach (Fig. 10b; Table 3), selected as a representativeexample of alluvial channels within our study area. The principalmodel input was a high-resolution depthmap, which we derived fromground-based surveys. Detailed topographic data was obtained with atotal station and geostatistical techniques used to predict bedelevations on a 0.25 m grid (Legleiter & Kyriakidis, 2006, 2008). Awater surface profile surveyed along the margins of the wettedchannel at a discharge of 1.61 m3 s−1 on 4 September 2006 was usedto estimate the water surface elevation at each grid node. Flow depthd was then calculated as the elevation difference between the watersurface and the streambed and the resulting bathymetry used toparameterize the FIM (Fig. 10c). Under these base-flow conditions, themean and maximum depths were 0.17 m and 0.49 m, withpronounced shoaling over the point bar and greater depths whereflow converges downstream of the large mid-channel bar, along theouter bank through the bend, and on the left side of the channel at thelower end of the reach. Thus, although the range of depths waslimited, spatial variations in depth retrieval accuracy and precisioncould still be examined in relation to river morphology. For this casestudy, the FIM was parameterized as described in Section 2 and theresulting Hydrolight/MODTRAN-generated library coupled to thebathymetry of the Footbridge Reach.

Remote mapping of river bathymetry requires the capacity toresolve subtle changes in radiance on a spatial scale appropriate forthe channel of interest, making sensor characteristics an important


consideration. The first question posed above emphasizes spatialvariations in depth retrieval accuracy and precision, and we exa-mined this issue by analyzing a simulated AISA image comprised of2 m pixels and having a noise-equivalent delta radiance of ΔLN (λ)=1/210=0.00098 W m−2 sr−1. Simulated AISA data were also used toaddress the second question, concerning the effects of spatial andradiometric resolution. We simulated AISA images with pixel sizes of1, 2, and 3 m by adjusting the flying height to 955, 1910, and 2865 mabove terrain, respectively, while holding ΔLN (λ) constant at 1/212=0.00024 W m−2 sr−1. To isolate the role of radiometric reso-lution, we evaluated depth retrieval precision for the 2 m image andassumedΔLN (λ) values of 0.00024, 0.00098, and 0.0039Wm−2 sr−1,corresponding to 12-, 10-, and 8-bit quantization.

4.1. Reach-scale spatial variations in depth retrieval accuracy andprecision

A primary advantage of remote sensing is the ability to obtainspatially distributed bathymetric information throughout the reach ofinterest, rather than point measurements at discrete locations.Forward modeling indicated, however, that the reliability of image-derived depth estimates was not uniform but rather varied spatially asa function of river morphology. To examine these effects in the contextof the Footbridge Reach, we extracted pixel-scale mean depths andLT (λ) spectra from the simulated AISA image at 10 locations alongeach of 25 equally-spaced cross-sections. Data from 13 of thesetransects were used to perform OBRA and calibrate a relation betweenthe image-derived quantity X (Eq. (1)) and flow depth d (Fig. 11a); the12 remaining cross-sections were reserved for validation (Fig. 11b).The bathymetry of the reach was mapped by applying the optimalband ratio regression equation to each in-stream pixel, and the

Fig. 11. Remote mapping of river bathymetry from simulated AISA hyperspectral image data fof predicted vs. observed depths for 120 validation points along 12 cross-sections, withbathymetric contour interval width for the band centered at 693 nm.

resulting depth estimates were then compared to the known, pixel-scale mean depths to characterize bathymetric accuracy. Inspectingthe map of residuals shown in Fig. 11c indicated systematic depthretrieval errors with a coherent spatial pattern dictated by channelmorphology. Most notably, depths were consistently under-predictedby 0.03–0.04 m (positive residuals represented by orange and yellowtones in Fig. 11c) in deeper water along the outer bank through thebend, on the left side of the upper mid-channel bar, and toward theleft bank at the lower end of the reach. This finding implied thatsaturation of the radiance signal might have precluded accurateestimation of pool depths. Similarly, if the X vs. d relation was slightlynon-linear, as observed in coastal studies spanning a broader range ofdepths (Dierssen et al., 2003; Mishra et al., 2007), calibration to a dataset consisting primarily of shallow-water observations could haveresulted in biased estimates in deeper areas. For the remainder of thechannel proper, image-derived depths were in close agreement withor slightly over-predicted the true bathymetry.

Stream banks were problematic due to contamination of theaquatic signal by terrestrial radiance. Forward image modeling ofthese mixed pixels was based on the simple bank cover classificationdepicted in Fig. 10c. Negative depth estimates, corresponding to largepositive residuals, occurred on the downstream portions of the mid-channel and point bars, where relatively bright sand produced mixedpixels with very high radiance in the red and near-infrared. Becausethese absorption-dominated wavelengths were strongly inverselyrelated to depth, high LT (λ) from these areas yielded negativeestimates of d; a similar pattern was observed in a previous studybased on actual AISA data (Legleiter et al., in press). Conversely, forbanks composed of gravel or riparian vegetation, depths were over-predicted. Though counterintuitive, this result was a consequence ofthe reflectance properties of the two bank cover types and the

or the Footbridge Reach. (a) R2 (λ1, λ2) matrix from optimal band ratio analysis. (b) Plotperfect agreement indicated by the 1:1 line. (c) Depth retrieval residuals. (d) Total


periphyton used to parameterize Rb (λ) — gravel and vegetation weredarker than periphyton from 575–700 nm (see Legleiter et al., in press,Fig. 1). Due to minimal absorption by the shallow water column, theaquatic radiance was primarily determined by Rb (λ), which wasbrighter than the adjacent bank. Inclusion of relatively dark gravel orvegetation within the pixel thus reduced the composite LT (λ), whichwas mistaken for deeper water. If the banks were characterized byhigher reflectance, or if Rb (λ) were lower, this over-prediction ofdepth would not have occurred.

Because many geomorphic applications require an ability to detectsubtle variations in depth, the precision of image-derived bathymetricinformation is also an important consideration. To quantify theuncertainty inherent to spectrally-based depth retrieval on SodaButte Creek, we used AISA images simulated for a range of depthincrements to determine, for each pixel, the smallest detectableincrease and decrease in depth and hence the total bathymetriccontour interval Δdc (λ). This analysis was performed for λ=693 nm,the optimal denominator band identified via OBRA. As for the depthretrieval residuals, mapping this metric of precision revealeddistinctive patterns that were clearly influenced by the morphology(Fig. 11d). The bathymetric contour interval varied spatially becausethe magnitude of the change in radiance associated with a change indepth was a function not only of Δd but also the original depth d0. In

Fig. 12. Effects of sensor spatial resolution on spectrally-based depth retrieval from simulatedoptimal band ratio analyses are summarized with R2 matrices. Plots of predicted vs. obserindicated by the 1:1 line.

general, for a fixed Δd, ΔLT (λ; d0+Δd) was inversely related to d0,implying that relatively small changes in depth were detectable inshallower areas near the banks and over the point bar whereas onlylarger Δd satisfied condition (4) in deeper water along the outer bank.For the simulated 10-bit AISA image of the Footbridge Reach, Δdc (λ)was 0.02 m in very shallow water along the inside of the bend,typically 0.07–0.08 m for much of the channel, and as high as 0.12–0.13 m in the pools, representing an uncertainty of 25% of the depth.Together with the preceding analysis of depth retrieval accuracy, ourresults implied that this type of remotely sensed data could providehighly accurate, fairly precise depth estimates for shallow areas butthat bathymetric information from pools would be less reliable,subject to both a systematic bias toward under-prediction andinherently lower precision.

4.2. Effects of sensor characteristics on spectrally-based depth retrieval

To effectively map river bathymetry, an imaging systemmust havea ground instantaneous field of view small enough to resolve themorphologic features of interest. Similarly, the system's detectorsmust be sensitive enough to discern subtle changes in radiance, andtherefore depth. Sensor characteristics thus exert a fundamentalcontrol on the utility of image-derived river information, and we used

AISA hyperspectral image data for the Footbridge Reach. For pixel sizes of 1, 2, and 3 m,ved depths are for 120 points along 12 validation cross-sections; perfect agreement is


simulated AISA images to examine the effect of spatial resolution ondepth retrieval accuracy and the influence of radiometric resolutionon bathymetric precision.

4.2.1. Spatial resolutionImages with pixel sizes of 1, 2, and 3 m were simulated for the

Footbridge Reach, a channel with a mean low-flow water surfacewidth of 22 m and a complex morphology featuring abrupt variationsin depth over short horizontal distances. Optimal band ratio analysesof these images were summarized using the R2 (λ1, λ2) matricesillustrated in Fig. 12, along with plots of predicted vs. observed depthsfor 12 validation cross-sections extracted from each image. For aspatial resolution of 1 m, OBRA yielded an R2 value of 0.892 for the(585 nm, 693 nm) band combination, but Fig. 12a indicates that linearX vs. d relations nearly this strong could have been obtained usingseveral other numerator bands, including the near-infrared. Validationof the resulting depth estimates was encouraging, with predicted andobserved depths tightly clustered along the 1:1 line, though thescatter was somewhat greater at shallower depths, including severallarge over-predictions (Fig. 12b). When the pixel size was increased to2m, OBRA indicated that the number of useful bands was significantlyreduced (Fig. 12c); NIR radiance was no longer strongly related todepth. The same (585 nm, 693 nm) wavelength combination wasoptimal, but the corresponding R2 value dropped to 0.709. Similarly,depth retrieval validation indicated greater scatter, with a few largeunder-predictions in deeper water and consistent over-prediction ofthe shallowest depths (Fig. 12d). This trend of decreasing accuracywith increasing pixel size continued for the 3 m image, for which

Fig. 13. Bathymetric maps and corresponding depth retrieval residuals derived from simulatefar right of each row applies to each panel in that row. (For interpretation of the references

OBRA yielded only a few band combinations marginally useful fordepth retrieval; even for the optimal band ratio, R2 was less than 0.5(Fig. 12e). The resulting estimates systematically under-predicted thedepth for dN0.25 m and over-predicted for db0.15 m (Fig. 12f).

These results were primarily a consequence of mixed pixels alongthe margins of the channel. As the spatial resolution of the image wasreduced from 1 to 2 to 3 m, a larger number of pixels included greaterproportions of sand, gravel, or vegetation, in addition towater. For 1 mimage data, relatively few pixels suffered terrestrial contamination,and a large number of bands, including some in the NIR, were stronglyrelated to depth. For coarser pixel sizes, inclusion of bank materialswith spectral properties distinct from those of water reduced therange of wavelengths useful for mapping bathymetry. For example,the reduced utility of the NIR for the 2 m image was due to thepresence within many near-bank pixels of vegetation, which has amuch higher NIR reflectance thanwater. For 3m data, almost all bandswere unrelated to depth because many more pixels included bankmaterial. Nevertheless, the finding that the (585 nm, 693 nm) ratiowas optimal for all three images suggests that this band combinationwas relatively robust to spectral mixtures.

Another limitation of the coarser-resolution images was thepresence of a broader range of depths within larger pixels. Previousresearch indicated that the log-transformed band ratio X provided anunbiased estimate d ̂ of the pixel-scale mean depth d ̄ in the presenceof sub-pixel scale bottom topography (Legleiter & Roberts, 2005). Inthis study, however, we used a more realistic, Gaussian point spreadfunction to aggregate the base radiance field into image pixels ofvarious sizes. This function assigned a greater weight to radiance from

d AISA hyperspectral image data with pixel sizes of 1, 2, and 3 m. The colour scale at theto colour in this figure legend, the reader is referred to the web version of this article.)


the center of the pixel, whereas the mean depth for the pixel wascomputed by equally weighting each cell of the input depth map; d ̂

could thus yield biased estimates of d_. The bathymetric maps and

corresponding depth retrieval residuals in Fig. 13 confirmed thaterrors of this kind became increasingly common as pixel sizeincreased. A depth map derived from the 1 m AISA image (Fig. 13a)closely matched the field-based bathymetry, and the correspondingresiduals confirmed that depth retrieval was highly accurate withinthe channel proper (Fig. 13d). Image-derived estimates agreed closelywith d

_except along the banks, where under- or over-predictions

occurred depending on bank composition. Notably, pool depth wasnot under-predicted from 1 m image data. As spatial resolution wasdegraded to 2 m, however, the spectrally-based bathymetric map tookon a more blurred appearance, and negative depth estimates nearsandy banks became prominent (Fig. 13b). In addition to the errorsassociated with mixed pixels along the banks, the residual map inFig. 13e indicated that although d ̂ was a reliable estimate of d

_for the

shallower portion of the channel, d_was consistently under-predicted

in deeper areas. For the moderate depths observed in the upper leftchannel, errors were 0.01–0.02 m but increased to 0.03–0.04 m for thedeeper pools along the outer bank. For the simulated 3 m image, theseeffects became even more pronounced. Image-derived estimatesfailed to span the full range of observed depths (Fig. 13c), with amaximum d ̂ of 0.34 m, a 30% under-prediction of the true pool depth.Fig. 13f indicated that large under-predictions (over-predictions) ofdepth occurred along banks composed of sand (vegetation or gravel).Within the channel proper, the residuals mimicked the morphology,with depths over-predicted on the shallow point-bar and under-predicted in the pools. For this coarser-resolution image, even theoptimal band ratio was only weakly related to depth (Fig. 12e), andmost predictions were close to the mean depth; the pattern of errorswas thus dictated by the true, more variable bathymetry.

4.2.2. Radiometric resolutionThe effect of radiometric resolution on bathymetric precision was

examined by generating depth-incremented images for Δd valuesranging from −0.15 m to 0.15 m in steps of 0.1 m; pixel size was fixedat 2 m. Subtracting the radiance for each depth-increment from thatbased on the original bathymetry allowed us to calculate ΔLT (λ; d0+Δd) values and determine the smallest detectable increase and decreasein depth for ΔLN (λ) values corresponding to 12-, 10-, and 8-bitquantizations. The uncertainty associated with spectrally-based depth

Fig. 14. Total bathymetric contour interval width for simulated AISA hyperspectral data withterms of the noise-equivalent delta radiance ΔLN (λ) for the band centered at 693 nm.

estimates was then expressed in terms of the bathymetric contourinterval Δdc (693 nm) for the optimal denominator band identified viaOBRA.

MappingΔdc (693nm) for eachΔLN (λ) highlighted the importance ofradiometric resolution as a fundamental control onbathymetric precision(Fig. 14). For the most sensitive detector evaluated, Δdc (693 nm) wastypically 0.02mover the point bar and increased only slightly in the pools(Fig. 14a), implying that a 12-bit instrument could enable fairly precisedepth estimates under shallow, clear-water conditions— over 96% of theFootbridge Reach had Δdc (693 nm≤0.04) m. For a larger assumed ΔLN(λ) of 0.00098Wm−2 sr−1, contour interval width increased noticeablyand the influence of river morphology on bathymetric precision becamemore evident (Fig. 14b). For this level of radiometric resolution, Δdc(693nm) increased from0.02malong the left bank to0.07–0.08macrossthe point bar, andnearly 90%ofΔdc (693 nm) valueswere less than 0.1m.In the deepest portion of the pool, however, the bathymetric contourinterval increased to 0.14 m, and isolated larger values of Δdc (693 nm)were also observed for mixed pixels along the banks. For this type ofimaging system,bathymetricuncertaintycould thusbeupto a thirdof thepool depth, with a nominal estimate of 0.42 m having an associatedconfidence interval (0.35 m; 0.49 m). Further increasing ΔLN (λ) to0.0039 W m−2 sr−1, the fundamental limit achievable with 8-bit data,severely degraded depth retrieval precision. Only within the shallowestportions of the channel did depth increments as great as 0.15 m producesufficiently large changes in radiance to be detectable (Fig. 14c). For thepoint bar, Δdc (693 nm) was typically on the order of 0.2 m, but thecontour interval exceeded the range of depth increments encompassedby our FIM runs for over 71% of the Footbridge Reach; any depth estimatefrom these areas would have an uncertainty of no less than ±0.15 m.These results clearly indicated that for applications seeking to detectsubtle differences in depth, high radiometric resolution is essential.Conversely, an inability to resolve slight differences in radiance canseverely limit the utility of image data for bathymetric mapping.

5. Discussion

The ability to efficiently measure various river attributes con-tinuously and with high resolution over entire watersheds hasgenerated considerable optimism regarding the potential contributionof remote sensing techniques to the study of fluvial systems (Marcus &Fonstad, 2008). Ultimately, the magnitude of this contribution willdepend on the extent to which the information requirements of

a pixel size of 2 m and (a) 12-, (b) 10-, and (c) 8-bit radiometric resolution, expressed in


particular scientific investigations can be satisfied using image dataacquired from specific reaches of interest by the instrumentationavailable. A consistent, quantitative means of assessing the reliabilityof remotely sensed river information thus becomes critical, and theforward image modeling framework outlined in this paper affordssuch a capacity. Our physics-based approach considers the radiativetransfer processes governing the interaction of solar energy with theatmosphere, water column, and substrate, as well as its measurementby a remote detector. The FIM thus allows for a priori characterizationof the accuracy and precision with which river bathymetry can beremotely mapped from passive optical image data. Because themodel's output is spatially explicit, mapping depth retrieval residualscan reveal systematic biases related to channel morphology. Similarly,bathymetric contour intervals highlight spatial variations in theprecision of image-derived depth estimates. This type of analysisallows the capabilities and, perhaps more importantly, the limitationsof remote sensing of rivers to be examined systematically. Forwardmodeling can thus be used: (1) in a planning mode, to define realisticexpectations for the accuracy and precision with which the bathy-metry of a particular field site can be mapped by a given sensor, or toguide the selection of appropriate instrumentation; (2) retrospec-tively, to assess the reliability of depth estimates derived from various,existing image data sets to infer changes in channel morphology; and(3) for research and development, by providing a controlled,numerical environment for evaluating and refining new algorithmsunder a range of known conditions.

The forward image model is subject to a number of importantlimitations that must be borne in mind, however. In this study, bottomreflectance, water column optical properties, and water surface statewere assumed constant. Streambed spectral characteristics along SodaButte Creek were fairly homogeneous (Legleiter et al., in press), but, ingeneral, river substrates are spatially heterogeneous and might havehighly variable spectral properties; additional field data sets areneeded to explore this issue. The optical properties of the watercolumn are not likely to vary within a reach, provided there are notributary inputs or other sources of fine-grained sediment, but ourdescription of the water column was probably incomplete —

absorption and scattering were specified solely in terms of suspendedsediment concentration, but other materials might have beenoptically significant as well. Water surface state does vary spatiallyas a function of flow hydraulics, bed material grain size, and imaginggeometry and can influence reflectance and transmittance at the air–water interface, but ratio-based depth retrieval is fairly robust to theseeffects (Legleiter et al., 2004, in press). With appropriate data, theseassumptions could be relaxed, and the current modeling frameworkcan accommodate variations in bottom reflectance, optical properties,and surface state. The one-dimensional geometry of the FIM is a morefundamental limitation, however; fully accounting for adjacencyeffects and non-linear mixing, both along the banks and within thechannel proper, would be more challenging. Similarly, a thoroughconsideration of the effects of shadows, sensor signal-to-noisecharacteristics, and image pre-processing, all of which could influenceactual image data, would require incorporating new components intothe model.

In practice, using the FIM to quantify the uncertainty associatedwith image-derived depth estimates will require a means ofcharacterizing sensor noise-equivalent delta radiance. This studydemonstrated that ΔLN (λ) exerts a fundamental control on bathy-metric precision, but our results were based on assumed valuesof this critical parameter. Equating ΔLN (λ) with the change inradiance equivalent to one digital number (Philpot, 1989), allowed usto determine the maximum precision (i.e., smallest bathymetriccontour interval) achievable for a given quantization. For actual imagedata, sensor signal-to-noise characteristics and scene-specific envir-onmental conditions (e.g., atmospheric effects and sun glint; Brando& Dekker, 2003) also influence the magnitude of the smallest change

in radiance detectable by an imaging system. Ideally, ΔLN (λ) wouldbe determined directly from each data set, but algorithms forcharacterizing image noise levels based on large expanses of opticallydeep water (e.g., Wettle et al., 2004) or bright, uniform terrestrialtargets (e.g., Green et al., 2003) might not be applicable to fluvialsystems lacking such features. In any case, estimates of bathymetricprecision derived from the FIM are only as reliable as the ΔLN (λ)values used to determine whether a given change in depth producesa detectable change in radiance. Even then, analyses of simulatedimage data yield theoretical upper bounds on depth retrievalaccuracy and precision. FIM results thus represent a first-orderassessment of the feasibility of remote bathymetric mapping andshould not be interpreted as a complete accounting of the un-certainty inherent to spectrally-based depth retrieval from real imagedata.

With this important caveat, forward image modeling providesinsight as to which geomorphic applications would be well-supportedby remote sensing and which, for the time being, would not. Forexample, FIM results from the Footbridge Reach indicated consistentunder-prediction of pool depths as well as reduced precision in deeperwater; depth retrieval from shallower areas was more reliable. Theseresults have important implications for our research on sedimenttransfer and channel change. Because pool scour is more difficult todetect than bar aggradation, erosion volumes estimated from timeseries of image data are subject to greater uncertainty than depositionvolumes, a difference that should be accounted for when inferring bedmaterial transport rates from observations of morphologic change(e.g., Ashmore & Church, 1998; Ham & Church, 2000). If the locationsbut not the full depths of pools can be determined due to inadequateradiometric resolution, information on the spacing of channel featurescould still be used to parameterize travel distance-based sedimentrouting algorithms (e.g., Gaeuman et al., 2003). A more extensivediscussion of potential applications of remote sensing is beyond thescope of this paper and would be, to an extent, inappropriate. One ofour primary conclusions is that because individual studies address avariety of questions by using sensors with a range of characteristics toexamine riverswith distinct morphological and optical properties, anyassessment of the utility of remotely sensed data should not be overlygeneral. Forward modeling instead allows the reliability of image-derived river information to be evaluated quantitatively in the contextof specific investigations.

The forward image model can thus facilitate progress towardresearch-grade remote sensing of rivers by helping to define the typeof image data needed to satisfy the information requirements ofparticular types of studies. For example, the FIM could be inverted todetermine the radiometric resolution required to detect a specifiedchange in depth over some minimum proportion of the channel.Similarly, OBRA of the original, Hydrolight/MODTRAN-generatedspectra could be used to identify specific, narrow ranges ofwavelengths capable of providing bathymetric information of suffi-cient accuracy for multi-dimensional hydraulic modeling. Defining anappropriate spatial resolution is somewhat more difficult because, inaddition to the issues associated with mixed pixels along the banks,depth retrieval is also complicated by sub-pixel scale bathymetricvariability. As a result, a spatial resolution that is adequate for onereach with a relatively simple morphology might be insufficient foranother, more complex segment of the river, even if the mean channelwidth is the same. The ability of a given imaging system tocharacterize the morphology of a particular river thus depends onthe morphology of that river, and Legleiter and Roberts (2005)suggested that the selection of an optimal spatial resolution could beposed as a kind of sample design problem. Provided field data for arepresentative reach are available, the FIM can be used to define site-specific relationships between image pixel size and various metrics ofbathymetric accuracy and precision. Finally, forward modeling cancontribute to remote sensing of rivers by providing a controlled,


numerical environment for developing and evaluating new algo-rithms. For example, simulated images from the FIM present an idealtest-bed for extending powerful spectrum-matching techniques fromcoastal to fluvial environments (e.g., Mobley et al., 2005).

6. Summary and conclusion

Remote sensing methods can contribute to river research only tothe extent that information derived from image data can provide thelevel of accuracy and precision required by specific investigations.Although the potential to advance our understanding of fluvialsystems clearly exists, we believe a cautious approach is needed toensure efficient, informed use of remote sensing technology. Moti-vated by this objective, we have developed a physics-based forwardimage modeling framework that allows for quantitative analysis of thecapabilities and limitations associated with remote mapping of riverbathymetry from passive optical image data. This paper describedmodel components, compared the FIM to an analytical approach, andapplied the FIM to a gravel-bed river to quantify spatial variations inbathymetric accuracy and precision and examine the effects of sensorspatial and radiometric resolution. Among our more salient findingswere the following:

1. A strong overall agreement between observed depths and image-derived estimates masked systematic biases related to rivermorphology. Saturation of the radiance signal implied under-estimation of pool depths, and depth retrieval along channelmargins was unreliable due to terrestrial radiance from the banks,which could result in negative depth estimates.

2. The precision of spectrally-based depth retrieval dependedstrongly on the ability of the imaging system to resolve subtlechanges in radiance. Bathymetric precision also varied spatially as afunction of depth and was reduced in pools, where a given changein depth produced a smaller change in radiance.

3. Optical properties of the water column, parameterized in terms ofsuspended sediment concentration, dictated which wavelengthswere useful for mapping bathymetry. Near the transition fromscattering- to absorption-dominated radiative transfer, changes indepth produced only very small changes in radiance. For longerwavelength red and NIR bands, very precise depth estimates werepossible in shallow areas due to strong absorption by pure water,which also caused saturation at greater depths. In deeper water,some bathymetric information could be inferred from shorterwavelengths, where decreases in depth corresponded to decreasesin radiance due to truncation of a relatively bright water column bya darker substrate. Optimal depth retrieval thus required informa-tion from across the spectrum.

4. An analytical model developed by Philpot (1989) provided anefficient alternative means of characterizing depth retrievalprecision that agreed reasonably well with computationallyintensive direct numerical simulation. A critical parameter wasthe effective attenuation coefficient, which we specified usingexisting data on the optical properties of water and sediment.Theoretical and computational predictions of bathymetric preci-sion agreed closely for strongly absorbing water but diverged asdepth, sediment concentration, and sensor noise-equivalent deltaradiance increased.

5. This theoretical approach yielded some general insight on thecontrols on bathymetric precision, expressed in terms of threedimensionless quantities: depth retrieval percent error, opticaldepth, and an index of the sensor's ability to detect bottom-reflected radiance. This parameter space was used to characterizethe uncertainty inherent to image-derived depth estimates across arange of conditions.

6. Application of the FIM to a natural gravel-bed river indicated thatspatial resolutionwas the primary control on bathymetric accuracy,

due to the greater prevalence in coarser resolution image data ofboth mixed pixels along the banks and sub-pixel scale variations indepth. Bathymetric precision was determined by sensor radio-metric resolution, expressed in terms of the noise-equivalent deltaradiance.

Forward modeling has enabled us to make more informed use ofremotely sensed data in our ongoing research on river morphody-namics. The true value of the FIM framework is its generality,achieved by incorporating the radiative transfer processes thatgovern the interaction of solar radiation with the water column,streambed, and atmosphere. Although in practice remote sensing ofrivers requires a certain degree of empiricism, this flexible, a physics-based approach to the forward problem provides insight on thefactors controlling the reliability of image-derived river informationand allows the value of that information to be assessed in the contextof specific applications.

Notationλ wavelengthEd (λ) downwelling spectral irradiancefd (λ) direct fraction of Ed (λ)zg ground altitudezs sensor altitudeR (λ) irradiance reflectance in air, just above the water surfaced water depthRb (λ) bottom reflectanceCs suspended sediment concentrationU wind speed, used by Hydrolight to parameterize water

surface roughnessLu (λ) total upwelling radiance in air, just above the water surfaceQ (λ) factor accounting for directional structure of the light fieldEu (λ) Upwelling spectral irradianceLP (λ) path radiance scattered by the atmosphereLT (λ) total at-sensor radianceLT (λ; d) total at-sensor radiance for depth ds location within the channeldm minimum depth in Hydrolight databaseLL (λ) total at-sensor radiance from landfw fraction of a cell of the base radiance field comprised of

waterd pixel-scale mean depthLB (λ) bottom-reflected radianceX ratio-based, image-derived variable linearly related towater

depthR2 regression coefficient of determinationd ̂ image-derived depth estimateε depth retrieval residualΔd change in depthd0 initial depthΔLT (λ; d0+Δd) change in at-sensor radiance corresponding to

change in depth Δd at d0ΔLN (λ) sensor noise-equivalent delta radianceΔd+ (λ) minimum detectable increase in depthΔd− (λ) minimum detectable decrease in depthΔdc (λ) total bathymetric contour interval width| ΔL (λ; d0+Δd) |k kth percentile of the distribution of absolute

radiance changesK (λ) effective attenuation coefficientLW (λ) remotely measured radiance over optically deep waterKd (λ) diffuse attenuation coefficient for downwelling irradiancea (λ) absorption coefficientbb (λ) back-scattering coefficienta*s (λ) specific absorption coefficient for suspended sedimentb*s (λ) specific scattering coefficient for suspended sedimentθsw solar zenith angle in water


Acknowledgements

The National Park Service granted permission to collect field dataused in this study. Financial support was provided by the CanonNational Parks Science Scholarship Program, a Doctoral DissertationResearch Improvement Grant from the National Science Foundation'sGeography Program, a Horton Research Grant from the AmericanGeophysical Union, and the California Space Institute. Andrew Marcusloaned a spectroradiometer used to measure field spectra, DavidLageson loaned survey equipment, and the Yellowstone EcologicalResearch Center provided logistical support. Kyle Legleiter, PatriciaJenkins, Ben Lieb, David Speer, and Jake Galyon assisted in field datacollection. Two anonymous reviewers helped to improve an earlierversion of this paper.

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