Remote sensing offers an efficient means of mapping bathymetry in river systems, but this approach has been applied primarily to clear-flowing, gravel bed streams. This study used field spectroscopy and radiative transfer modeling to assess the feasibility of spectrally based depth retrieval in a sand-bed river with a higher suspended sediment concentration (SSC) and greater water turbidity. Attenuation of light within the water column was characterized by measuring the amount of downwelling radiant energy at different depths and calculating a diffuse attenuation coefficient, Kd. Attenuation was strongest in blue and near-infrared bands due to scattering by suspended sediment and absorption by water, respectively. Even for red wavelengths with the lowest values of Kd, only a small fraction of the incident light propagated to the bed, restricting the range of depths amenable to remote sensing. Spectra recorded above the water surface were used to establish a strong, linear relationship (R2 = 0.949) between flow depth and a simple band ratio; even under moderately turbid conditions, depth remained the primary control on reflectance. Constraints on depth retrieval were examined via numerical modeling of radiative transfer within the atmosphere and water column. SSC and sensor radiometric resolution limited both the maximum detectable depth and the precision of image-derived depth estimates. Thus, although field spectra indicated that the bathymetry of turbid channels could be remotely mapped, model results implied that depth retrieval in sediment-laden rivers would be limited to shallow depths (on the order of 0.5 m) and subject to a significant degree of uncertainty.
 Remote sensing has emerged as a powerful tool for mapping river morphology and in-stream habitat [Marcus and Fonstad, 2010]. Often-cited advantages of this approach include improved efficiency relative to traditional field-based surveys, and the ability to obtain quantitative, spatially distributed measurements of key channel attributes at high resolution over larger, watershed extents [Marcus and Fonstad, 2008]. Although a number of case studies have demonstrated the potential to characterize river bathymetry via remote sensing [e.g., Winterbottom and Gilvear, 1997; Marcus et al., 2003; Carbonneau et al., 2004; Lejot et al., 2007], most previous research has been empirical and site-specific, lacking generality. More recent work has attempted to overcome this obstacle by seeking to understand the manner in which solar energy interacts with the atmosphere, air-water interface, water column, and substrate, and is ultimately measured by a remote detector. For example, examining the physical basis for retrieving flow depth from remotely sensed data has helped to establish the feasibility of spectrally based bathymetric mapping [Legleiter et al., 2004, 2009], while also acknowledging the limitations associated with these techniques [Legleiter and Roberts, 2005, 2009]. To date, however, only relatively clear-flowing, shallow, gravel bed streams have been considered. The primary objective of this investigation is thus to evaluate the extent to which passive optical remote sensing might provide reliable depth information from larger, sand-bed rivers with greater water turbidity. Here, we review the principles underlying spectrally based depth retrieval and use field spectra and a numerical radiative transfer model to assess the utility of this approach in a more turbid fluvial environment. In a companion paper, we describe the application of these methods to hyperspectral image data from the Platte River, Nebraska, United States [Legleiter et al., 2011].
 In general, passive optical remote sensing of rivers involves measuring visible and near-infrared solar radiation that has been influenced by the Earth's atmosphere, the water flowing in the channel, and, potentially, the stream's bed. The radiance signal recorded by a remote sensor deployed above the river of interest can thus be conceptualized as the sum of four components:
refers to the total at-sensor radiance recorded at a specific wavelength , represents radiance reflected from the bottom, is radiance from the water column itself, and and are additional, confounding sources of radiance reflected from the water surface and scattered into the sensor's field of view along the path through the atmosphere, respectively. The component of interest for bathymetric mapping is , which depends not only on flow depth but also the bottom contrast between the substrate and water column. Legleiter et al.  used this framework to show that accurate depth retrieval is possible when (1) flow depths are shallow; (2) the streambed is relatively bright, with a bottom reflectance significantly higher than the volume reflectance of the water column; (3) scattering by suspended sediment is minimal, such that radiative transfer is dominated by pure water absorption; (4) the illumination and viewing geometry and roughness of the air-water interface do not produce a high reflectance from the water surface; and (5) atmospheric conditions are favorable, such that is negligible relative to the other terms in equation (1). Although this theoretical analysis was substantiated by ground-based spectral measurements and remotely sensed data from a small, gravel bed river, the encouraging results reported by Legleiter et al.  only pertain to a fairly narrow range of conditions highly conducive to depth retrieval. In this paper, we explore a different, potentially far less favorable region of parameter space in which higher suspended sediment concentrations (SSC, denoted by Cs) imply a diminished contribution of to and a larger role for ; under these circumstances, assumptions (1)–(3) above might not be valid. Whereas previous research [Legleiter et al., 2004, 2009] only considered concentrations up to 8 mg L−1, this study examines more turbid conditions with Cs values up to 30 times greater. Our analysis thus represents an important test of the ability of remote sensing techniques to provide bathymetric information in a broader range of riverine environments.
 For water bodies with greater turbidity, image data have more often been used to infer SSC [e.g., Mertes et al., 1993; Pavelsky and Smith, 2009, and references therein] and other aspects of optical water quality [Ritchie et al., 2003; Julian et al., 2008]; for a thorough treatment of these topics, the reader is referred to Bukata et al.  and Mertes et al. . A number of studies have examined the absorption and scattering characteristics of suspended sediments [e.g., Babin et al., 2003; Wozniak and Stramski, 2004; Binding et al., 2008], including recent investigations of the effects of individual particles [e.g., Peng et al., 2009; Effler et al., 2011]. When the optical properties of the water column are of primary interest, becomes the most relevant component of the radiance signal, rather than . In this case, efforts to map SSC or other optically significant constituents are actually hindered by the presence of a shallow, reflective bottom [Tolk et al., 2000]. For example, in a recent study of the Venice lagoon, Volpe et al.  avoided the confounding influence of on by using an independent bathymetric data set to exclude areas shallower than 1.3 m and to provide the depth information needed to invert a bio-optical model. Quantifying the range of conditions under which the bottom is visible, a basic requirement for retrieving water depths or identifying substrate types [Vahtmae et al., 2006], could thus facilitate efforts to characterize the water column as well.
 In fact, the coastal research community has already developed sophisticated methods for simultaneous inference of depth, bottom reflectance, and water column optical properties via spectral optimization [e.g., Lee et al., 1999] or look-up table-based approaches [e.g., Lesser and Mobley, 2007]. These techniques potentially could be applied to inland stream channels as well, but this kind of flexible, physics-based framework has yet to be established for fluvial systems. By examining a more turbid riverine environment, this study represents another incremental step toward this goal. Here, we use field-based spectral measurements from a channel with high SSC to evaluate the potential for measuring the bathymetry of sediment-laden rivers via remote sensing. We also make use of a numerical radiative transfer model to outline some inherent limitations of this approach. More specifically, we pursue the following research objectives:
 1. Characterize the attenuation of light by the water column using measurements of the downwelling spectral irradiance acquired at different depths within the flow.
 2. Establish a relationship between spectral reflectance and water depth using ground-based reflectance data obtained from above the water surface.
 3. Determine the range of depths over which bathymetry can be mapped in highly turbid rivers by performing radiative transfer simulations for a range of suspended sediment concentrations.
 4. Quantify the precision of image-derived depth estimates under such conditions using output from the radiative transfer model and information on the finite sensitivity of remote detectors.
2.1. Study Area and Field Data Collection
 As an initial test of the ability of remote sensing techniques to provide bathymetric information from larger, more turbid rivers, we examined the Platte River in central Nebraska, USA. Field data were collected along two primary study reaches, Cottonwood Ranch and the Rowe Sanctuary, described in more detail in the companion paper [Legleiter et al., 2011]. Here, we focus on water sampling and field spectroscopy conducted at the Rowe Sanctuary. This reach of the Platte is ∼260 m wide, shallow (mean depth = 0.21 m), and braided, with many mobile midchannel sand bars and a median bed material grain size of 0.7 mm [Kinzel and Runge, 2010].
2.1.1. Water Sampling and Suspended Sediment Concentration/Turbidity Measurement
 In this study, we assumed that suspended sediment was the dominant, optically significant constituent of Platte River water and focused on this component in our field data collection and radiative transfer modeling. We characterized SSC within the two study reaches by collecting three water samples from each site. Samples were obtained using a U.S. Geological Survey (USGS) DH-48 suspended sediment sampler lowered and raised through the water column at a uniform rate to provide a depth-integrated measurement of SSC. Concentration values were determined at the USGS Iowa Sediment Laboratory (Iowa City, Iowa) using a standard filtration method [Guy, 1969].
 In addition to these direct measurements of SSC, turbidity values were recorded at 30 min intervals at two USGS gaging stations (available at waterdata.usgs.gov/nwis/) near our study reaches, one located 8.7 km upstream of Cottonwood Ranch (Station 06768000) and the other 16.8 km above the Rowe Sanctuary (Station 06770200). To monitor water quality as part of the Platte River Recovery Implementation Program (available at www.platteriverprogram.org), turbidity data were collected in nephelometric turbidity units (NTU) with a HydroLab MS-5 sonde. Although direct measurements of inherent optical properties of the water column, such as the back-scattering coefficient, would have been preferable, we did not have access to the required instrumentation. Instead, we used turbidity data acquired through an ongoing monitoring program as a surrogate index of temporal variations in water clarity [Davies-Colley and Smith, 2001].
2.1.2. Field Spectroscopy
 Optical characteristics of the Platte River were quantified via ground-based spectral measurements acquired along the Rowe Sanctuary study reach on 16 August 2010. These data were collected using an Analytical Spectral Devices (ASD) FieldSpec3 spectroradiometer with a sampling interval of 1.4 nm and a spectral resolution of 3 nm over the wavelength range from 400 to 900 nm. Prior to each round of measurements, we calibrated the instrument and optimized its integration time by placing the probe above a large, 100% reflectance Spectralon (Labsphere, Inc.) reference panel with a diffuse, Lambertian bidirectional reflectance distribution function (BRDF). Radiance reflected from the Spectralon panel was recorded once while fully illuminated by the solar beam and a second time while shaded from the sun in order to characterize the direct and diffuse components of the radiant energy incident upon the channel. Data were collected around noon local time, with a solar zenith angle of approximately 29°.
 To characterize the attenuation of light within the water column, we measured the downwelling spectral irradiance at different distances z beneath the water surface. These data were acquired by connecting the spectroradiometer to a waterproof, 5 m long fiber optic cable and affixing a foreoptic designed specifically for underwater irradiance measurements to the other end of the cable. This sensor featured a cosine response that served to integrate radiant energy arriving from all directions encompassed within the upper hemisphere. To provide an appropriate viewing geometry, the irradiance probe was attached to a car jack with the sensor facing upward. Measurements of were then obtained at different depths within the water column by lowering the jack incrementally. The 5 m cable length allowed us to position the irradiance probe far enough away to ensure that the sensor had an unobstructed view of the entire upward hemisphere throughout the measurement sequence, which took approximately 10 min to complete. We collected two irradiance profiles, with measurement depths ranging from 0.02 to 0.31 m beneath the water surface, in increments of 0.02–0.03 m. Measurements of were obtained at 9 depths on one of the profiles and 11 on the other, with five spectra recorded at each vertical position on each profile.
 Measurements of spectral reflectance above the water surface were made with a nadir-viewing geometry and an 8° foreoptic that resulted in a 0.21 m field of view when positioned 1.5 m from the bed. For these in-stream spectra, a smaller, Spectralon panel (puck) was used to establish a 100% reflectance standard immediately prior to each sample measurement, which were then obtained in units of reflectance relative to the white reference. This procedure served to account for variable illumination conditions due to the passage of clouds. Taking care to position ourselves so as to avoid casting shadows or creating wakes in the current, we recorded five spectra for each target. These spectra were influenced to some extent by reflectance from the water surface, but we did not attempt to apply any kind of surface reflectance correction due to a lack of data on sky radiance, which would have been required to implement the procedure of Gould et al. . Because all spectra were collected within a 2 hour interval and the roughness of the water surface was not highly variable within this reach, surface reflectance was probably consistent throughout our data set. Moreover, because surface reflectance is spectrally flat, this component of the total reflectance cancels when calculating a ratio of spectral bands [Legleiter et al., 2004], which was the approach used in this study to establish a relationship between depth and reflectance (section 2.3). At each site, flow depth was measured with a ruler and a digital photograph acquired (Figure 1a). In total, reflectance spectra were collected at 24 locations spanning a range of depths from 0.05 to 0.67 m, with a mean and standard deviation of 0.25 and 0.19 m, respectively. In addition, we measured the reflectance of several exposed sand bars to provide information on the bottom reflectance of the channel bed (Figure 1b).
 Postprocessing of the field spectra involved using ASD's ViewSpec Pro software to apply the radiometric calibration for converting raw digital counts recorded by the spectroradiometer to units of reflectance or irradiance. These data were then imported into the Spectral Analysis and Management System [Rueda and Wrona, 2003] and inspected graphically. The five individual spectra from each reflectance measurement location or position along an irradiance profile were plotted together, any obvious outliers discarded, and the remaining spectra averaged to obtain a single mean spectrum for that sample. The resulting mean spectra were then smoothed by twice applying a third-order Savitzky-Golay filter with a 15 nm bandwidth [Legleiter et al., 2009].
2.2. Characterizing Water Column Attenuation
 To quantify the absorption and scattering of incident solar radiation within the Platte River, we used the vertical profiles of described above to compute estimates of the diffuse attenuation coefficient for downwelling irradiance, , defined as [Mobley, 1994]
where is the incremental change in the amount of downwelling radiant energy (integrated over the upper hemisphere) that occurs as light propagates through the incremental distance dz. is an apparent optical property of the water column, depending to some extent on the directional structure of the ambient light field but is strongly related to inherent optical properties, including the absorption, , and backscattering, , coefficients for water and suspended sediment [e.g., Gordon, 1989]. We determined values following the procedure of Mishra et al. . This approach is based on the Beer-Lambert Law, which describes the exponential decrease in the intensity of light with distance traveled through an attenuating medium, such as the water column, as
where is the irradiance at a given depth z, is the irradiance just beneath the surface (i.e., following transmission across the air-water interface), and K is a generic, effective attenuation coefficient with units of m–1. Starting from equation (3), Mishra et al.  derived an expression relating to irradiance measurements at different depths:
where zm is the measurement depth and z1 is a fixed, reference depth close to the water surface. This derivation assumes that is the same for all depth increments dz within the interval between z1 and zm. This assumption is supported by a previous radiative transfer modeling study indicating that did not vary as a function of depth [Legleiter et al., 2004]. Equation (4) has the form of a straight line passing through the origin (i.e., y = mx), so plotting (zm – z1) against for a series of measurement depths zm and determining the slope of the best fit line yields an estimate of the depth-averaged diffuse attenuation coefficient . We used the shallowest depth at which was measured on each profile as the reference depth z1 and performed simple linear regressions for each wavelength to obtain spectra of values.
2.3. Relating Spectral Reflectance to Water Depth by Optimal Band Ratio Analysis
 To map river bathymetry from remotely sensed data, radiometric quantities obtained from an image must be related to observed flow depths. As implied by equation (1), however, the at-sensor radiance signal is influenced by a number of factors, and only certain components of this signal have the potential to yield depth information. In a recent theoretical analysis corroborated by field spectra and hyperspectral image data, Legleiter et al.  argued that, given favorable environmental conditions and an appropriate combination of wavelengths, a simple, ratio-based algorithm can provide an image-derived quantity X that is linearly related to flow depth d:
where K denotes an effective attenuation coefficient for the water column, (RB – RC) represents the bottom contrast between the streambed and optically deep water, and A is a constant that incorporates the irradiance incident upon the channel and accounts for the transmission of light across the air-water interface and through the atmosphere. All of these quantities vary spectrally, but the dependence on wavelength has been replaced by numerical subscripts referring to two different wavelengths. The physical reasoning behind this technique is discussed at length by Legleiter et al. [2004, 2009], based on earlier work in coastal environments [e.g., Lyzenga, 1978; Philpot, 1989; Dierssen et al., 2003; Stumpf et al., 2003; Mishra et al., 2007]. Briefly, because the bottom reflectances of various, typical substrates tend to be within a few percent of one another at a given wavelength in the visible and near-infrared (NIR) portion of the spectrum, different bottom types have similar band ratio values. Absorption by pure water, in contrast, increases by an order of magnitude over this range, so the radiance measured in the longer wavelength band with stronger attenuation decreases more rapidly as depth increases. Consequently, the ratio X increases with depth for K2 > K1 (i.e., when the band with stronger attenuation is used as the denominator) and is insensitive to variations in bottom type. The natural log accounts for the exponential attenuation of light with distance traveled through the water column (equation (3)).
 To retrieve depth from optical image data via a band ratio, a suitable pair of wavelengths must be selected and the coefficients of a linear relation between d and X determined. Legleiter et al.  introduced a simple method, called Optimal Band Ratio Analysis (OBRA), for accomplishing these objectives. This procedure takes as input paired observations of depth and reflectance and performs regressions of d on X for all possible combinations of numerator and denominator wavelengths. The optimal band ratio is that which yields the highest coefficient of determination R2, with the corresponding regression equation providing a calibrated d versus X relation. Moreover, because regressions are performed for all band combinations, the resulting matrix of values can be used to visualize how the strength of the relationship between d and X varies across the spectrum. In this study, we applied OBRA to field-based reflectance measurements to establish the quantitative link between spectral reflectance and water depth needed to map river bathymetry from remotely sensed data.
2.4. Radiative Transfer Modeling
 To determine the range of flow depths that could be detected via remote sensing, and to quantify the precision of image-derived depth estimates, we performed a series of radiative transfer simulations. This analysis made use of the forward image modeling framework described by Legleiter and Roberts , which incorporates the MODTRAN [Berk et al., 1989] and Hydrolight [Mobley, 1994] numerical models to represent the interaction of solar energy with the atmosphere and water column, respectively. For additional detail regarding this approach, the interested reader is referred to Legleiter and Roberts  and the references cited therein. Here, we focus on the application of these models to the Platte River and summarize their parameterization in Table 1. Model inputs were based on atmospheric conditions at the time the hyperspectral image data examined in the companion paper [Legleiter et al., 2011] were acquired, and MODTRAN was used to simulate the downwelling spectral irradiance incident upon the river, with a distinction made between direct and diffuse components. Taking from MODTRAN as input, Hydrolight generated reflectance spectra just above the water surface for water depths ranging from 0.02 to 2 m. Bottom reflectance was defined using a spectrum measured on a sand bar at the Rowe Sanctuary (Figure 1b).
Table 1. Parameterization of the MODTRAN and Hydrolight Models Used to Simulate Radiative Transfer Within the Atmosphere and Water Column, Respectively
From nearest weather station in Kearney, Nebraska
2.0 g cm−2
Little effect in 400–900 nm range
Default climatological value used
Julian day of year
Aug 14, 2010
Solar zenith angle
Rowe site (40.669°N, 98.896°W) at 14:45 GMT
Measured clockwise from North at 0°
Elevation for Rowe site
716 m above terrain to obtain 0.75 m image pixels
R() from Hydrolight
Used to create a reflectance series
0.01 m increments
60–240 mg L−1
20 mg L−1 increments; spans observed range (Table 3)
Sand bar R()
Measured at Rowe site
2 m s−1
Determines roughness of water surface
4 nm increments
 Our representation of the optical properties of the water column was simple: a two-component model consisting of pure water and suspended sediment. The absorption and scattering, , coefficients for pure water were taken from Smith and Baker  and those for suspended sediment were calculated using the brown earth optical cross-sections provided with Hydrolight. This approach allowed us to translate SSC values ranging from 60–240 mg L−1 into inherent optical properties by multiplying the absorption and scattering cross-sections, with units of m2 g−1, by concentrations in g m−3 (equivalent to mg L−1) to yield coefficients in units of m–1. In addition, we used the Petzold average particle phase function included in Hydrolight to determine backscattering coefficients from values.
 These model inputs are illustrated in Figure 2, which shows how the absorption, scattering, and backscattering coefficients of pure water and suspended sediment vary across the spectrum for three different SSC values. Although few data on the optical characteristics of suspended sediment are available for rivers, absorption and scattering are known to depend on the particle size distribution, refractive index, and mineral composition [Babin et al., 2003; Wozniak and Stramski, 2004; Peng and Effler, 2010; Effler et al., 2011]. Selecting a particular optical cross-section thus represents an important assumption that could affect model results [Binding et al., 2005]. For our first-order analysis, a single optical cross-section was used to examine the effect of varying concentrations of a given type of sediment; future studies might examine the influence of particle size and composition via more detailed modeling.
 Similarly, we have not considered the potential influence of chlorophyll, chromophoric dissolved organic matter (CDOM), or detritus on the optical properties of the water column because we were primarily interested in the effects of suspended sediment, which we assumed to be the dominant constituent of highly turbid Platte River water in the red and near-infrared wavelengths we found to be most useful for bathymetric mapping. CDOM levels often vary with SSC, however, and, where present, would impact the amount of light absorbed within the water column and could further constrain depth retrieval, particularly in the shorter blue and green wavelengths where CDOM absorption is strongest. Finally, an additional simplification made in this study was the use of a single, constant SSC throughout the water column, rather than a more realistic vertical profile in which concentrations are greater near the bed [Bridge, 2003].
 In addition to the spectra simulated for a sand bed, we also modeled for a black, zero-reflectance bottom at each output depth and for a hypothetical, infinitely deep water column with the same optical properties. This latter set of spectra allowed us to evaluate the range of depths over which a sandy substrate would be detectable. Each spectrum in this Hydrolight-generated database was propagated through the atmosphere to the flying height of the airborne hyperspectral sensor described by Legleiter et al.  using MODTRAN. Finally, spectra were convolved with the sensor's spectral response function to obtain band-integrated, at-sensor radiance values, , for all 3990 combinations of depth, SSC, and bottom type.
2.4.1. Characterizing Depth Retrieval Dynamic Range
 These radiative transfer simulations provided a means of determining the range of depths over which the bathymetry of a turbid, sand-bed river could be mapped, as well as an indication of the precision of image-derived depth estimates. The key concept in this analysis, articulated by Philpot  and applied to the fluvial environment by Legleiter et al. , is that the streambed can only be distinguished from optically deep water if the difference between the at-sensor radiance measured above a reflective substrate, , at some finite depth d0 and that which would be recorded above an infinitely deep water body with the same optical properties, , exceeds the smallest radiance increment, called the noise-equivalent delta radiance , that can be resolved by a particular sensor. The maximum detectable depth is thus the greatest depth for which this condition holds:
In this study, we used the sand-bed Hydrolight spectra to define for d0 up to 2 m and a model run for a hypothetical, infinitely deep water column to define .
 To assess whether the bottom would be visible at a given depth, we also used Hydrolight spectra modeled for a zero-reflectance, black substrate at the same depths used for the sand bed runs. Previous modeling indicated that the radiance from infinitely deep water differs from that of a water body of finite depth, even if the bottom reflectance is zero, because the volume reflectance due to scattering by suspended sediment is reduced when the water column is truncated [Legleiter et al., 2009]. The greatest depth at which the sand bed could be distinguished was thus determined via analogy with expression (6) by comparing the difference between sand-bed and black-bottom spectra for the same depth to the sensor's
where is the at-sensor radiance for a perfectly absorbing substrate at depth d0.
2.4.2. Characterizing Bathymetric Precision
 We used a similar approach to quantify bathymetric precision, which we defined as the smallest change in depth that could be resolved by an imaging system with a given noise-equivalent delta radiance [Philpot, 1989; Legleiter and Roberts, 2009]. Just as the visibility of the bed depends on the magnitude of the difference in the at-sensor radiance measured above a sand substrate versus that recorded above a black bottom (or infinitely deep water), a change in depth at some base depth d0 can only be detected if the following condition holds
In general, an increase in depth () does not produce the same change in radiance, , as a decrease in depth () of the same magnitude, so expression (8) must be evaluated separately to determine the smallest detectable increase in depth and the smallest detectable decrease in depth [Legleiter and Roberts, 2009]. In this study, we determined these values by comparing the spectrum modeled for the specified base depth d0 to radiance spectra for both greater and lesser depths and taking and to be the smallest (in absolute value) depth increments that satisfied condition (8). Following Legleiter and Roberts , we summarized these results in terms of the total width of a bathymetric contour interval, defined as . The width of these intervals represents the range of depths that cannot be distinguished from the base depth d0 by a sensor with a particular . The quantity thus quantifies the uncertainty inherent to spectrally based depth retrieval from digital image data.
 The noise-equivalent delta radiance serves as an index of the sensor's radiometric resolution and exerts an important control on the range of depths over which depths can be retrieved, as well as the precision of these estimates. In this study, we assumed that , the smallest change in radiance that can be resolved, is the inverse of the number of discrete values possible for an imaging system and does not vary spectrally (i.e., ). For example, for an instrument that records 12-bit data, 212 = 4096 distinct digital numbers are available, and the band-integrated is 1/212 = 0.000244 Wm–2 sr–1. We also considered systems with lesser and greater radiometric resolutions (Table 2). Specifying in this manner yields theoretical, optimal values for and because, in reality, the smallest detectable change in radiance will depend not only on the number of discrete values possible (i.e., 10- versus 12-bit instruments) but also atmospheric conditions, water surface state, sensor technical characteristics, and perhaps other factors that might lead to larger values of . These factors could be summarized in terms of an environmental signal-to-noise ratio that varies with wavelength [e.g., Brando et al., 2009], and although methods for estimating this crucial parameter have been developed for marine settings [e.g., Wettle et al., 2004], these techniques are not directly applicable to fluvial systems. The current investigation thus uses the simple definition of described above to estimate an upper bound for the maximum detectable depth and a lower bound for the bathymetric contour intervals associated with the hypothetical imaging systems listed in Table 2 for the conditions observed along the Platte River.
Table 2. Characteristics of Sensors Considered in Evaluating Maximum Detectable Depth and Precision of Image-Derived Depth Estimates
Distinct Digital Numbers
3.1. Measurements of Suspended Sediment Concentration and Turbidity
 Concentrations of suspended sediment measured in water samples collected from two study reaches along the Platte River are presented in Table 3. Mean values of SSC were 108 and 161 mg L−1 for the Cottonwood and Rowe reaches, respectively. The higher concentrations of suspended sediment at Rowe could be a consequence of finer bed material in this reach, part of an overall gradual downstream-fining trend along the Platte [Kinzel and Runge, 2010].
Table 3. Suspended Sediment Concentration Measurements and Summary Statistics
SSC (mg L−1)
14 Aug 2010
14 Aug 2010
14 Aug 2010
mean ± 1 std. dev.
108 ± 18
16 Aug 16 2010
16 Aug 16 2010
16 Aug 16 2010
mean ± 1 std. dev.
161 ± 56
 In addition to the grab samples analyzed for SSC, turbidity was measured continuously at gaging stations. Time series of turbidity values for the period encompassing both our field-based spectral measurements and the acquisition of the hyperspectral image data described in the companion paper (C. J. Legleiter et al., submitted) are shown in Figure 3. Mean turbidity values for the gauge near Cottonwood Ranch were 85.8 NTU on the day of the remote sensing flight (14 August 2010) and 92.9 NTU on the day we collected field spectra (16 August 2010). The corresponding values for the gauge above Rowe Sanctuary were 49.5 and 47.4 NTU, respectively.
 Although the SSC values recorded at Cottonwood Ranch were somewhat smaller than for the Rowe Sanctuary (Table 3), turbidity measurements from Cottonwood exceeded those for the Rowe site by nearly a factor of 2 (Figure 3). Because turbidity represents an integrated, overall metric of optical water quality, the higher turbidity values at Cottonwood could indicate that constituents other than suspended minerals might have been more significant at this site. Additional components contributing to greater turbidity might include chlorophyll, CDOM, and organic material transported in suspension. These materials could have been more prevalent at the Cottonwood site, where the Platte flows through cattle pastures, than at the Rowe Sanctuary, which might have been less influenced by agricultural activities. Differences in sediment grain size also might have contributed to the contrast in turbidity between the two sites due to the effect of particle size on scattering efficiency [e.g., Binding et al., 2005] and the vertical distribution of suspended sediment within the water column. A detailed accounting of optical water quality and its controlling factors is beyond the scope of this investigation, however.
 In any case, the SSC values of over 100 mg L−1 that we observed along the Platte River exceed the concentrations reported in earlier studies of remote sensing of bathymetry [Legleiter et al., 2009] and bottom type [Vahtmae et al., 2006] by a factor of 20 or more. Similarly, measured turbidities were 10–20 times greater than those reported for two of the four rivers examined by Julian et al.  and up to several times the largest value (27.4 NTU) documented in their study of optical water quality. Both the SSC and turbidity data thus indicate that the conditions we observed along the Platte River were far more turbid than have been considered to date.
3.2. Attenuation of Light Within the Water Column
 Radiant energy transmitted through the air-water interface interacts with water, suspended sediment, and potentially other optically significant constituents, leading to attenuation of the light stream propagating toward the riverbed. To quantify the strength and spectral character of this attenuation, we recorded vertical profiles of the downwelling spectral irradiance at various depths z beneath the water surface at two locations along the Rowe Sanctuary reach. The irradiance spectra in Figure 4 indicate that the amount of radiant energy is greatest just beneath the water surface (z = 0.02 m) and decreases toward the bed (z = 0.31 m) as photons are absorbed and/or scattered within the water column. Also note that, even for a constant spacing between vertical positions along a profile, the difference in between successive measurements is reduced at greater depths, due to the exponential attenuation of light implied by equation (3). The shape of these spectra is determined by the output from the sun, the modification of this energy by Earth's atmosphere (e.g., the pronounced decrease in irradiance at 760 nm due to absorption by oxygen), and various wavelength-dependent radiative transfer processes operating within the water column.
 To summarize these processes, we calculated the diffuse attenuation coefficient for downwelling irradiance for each wavelength, following Mishra et al. . An example of this analysis for a red band is shown in Figure 5a. Using the shallowest depth on the profile as the reference depth z1, we subtracted the logarithm of the irradiance at each subsequent measurement depth zm from the logarithm of the irradiance recorded at z1 and plotted these values against the corresponding differences in depth, zm – z1. The slope of the best-fit line on such a plot yields a depth-averaged estimate of , per equation (4). These slopes were determined via regression, and R2 values for the two irradiance profiles varied spectrally from 0.918 to 0.998, with a mean of 0.973. Consistently strong, linear relationships like that depicted in Figure 5a imply that radiative transfer within the water column was adequately described by equation (3). As dictated by the Beer-Lambert Law, pairs of measurements separated by greater vertical distances encompassed longer paths through the water column and hence experienced greater attenuation of the downwelling light stream, resulting in larger decreases in irradiance between the two measurement depths. For more closely spaced readings, the path length was shorter, less attenuation occurred, and irradiances recorded at the two depths were more similar.
 The spectra resulting from this analysis are shown in Figure 5b. This attenuation coefficient summarizes the rate at which the amount of radiant energy was diminished as the downwelling light stream traversed the water column. Higher values of imply that the photons comprising this light stream experienced absorption and scattering interactions with greater frequency and were thus less likely to reach the bottom of the river. Figure 5b indicates that was greatest at shorter blue and green wavelengths and in the near-infrared (NIR), with values in excess of 10 m–1 at both ends of the spectrum. In general, is related to the inherent optical properties, as implied by Gordon's  approximate analytical expression: , where is the incidence angle of the solar beam in water. Reference to the absorption and back-scattering coefficients shown in Figure 2 indicated that for wavelengths less than 550 nm, high values of resulted from absorption and, to a lesser extent, back-scattering by suspended sediment (Figure 2). Scattering redirected downwelling photons onto new paths, including some oriented back toward the upper hemisphere, and thus imparted a higher volume reflectance to the water column itself. increased with wavelength from 700–800 nm due to a pronounced increase in the absorption coefficient of pure water over this range, decreased around 810 nm due to slightly weaker absorption by water molecules in this band, and then increased at longer NIR wavelengths as absorption of light by water became increasingly efficient. The weakest attenuation occurred from 575 to 650 nm, with taking on approximately constant values of 3.6 and 4.6 for the two profiles. The slight increase in at 675 nm was due to absorption by chlorophyll. To put these values in perspective, Mishra et al.  reported mean values of 0.48 and 0.503 in the red (600–700 nm) for two stations in the clear waters of the Caribbean Sea, a full order of magnitude less than the values we calculated in a much more turbid fluvial environment.
 Because retrieving water depth from remotely sensed data requires that the bottom be detectable, the amount of radiant energy penetrating through the water column to the bed is a crucial factor to consider. To quantify transmission of light through the sediment-laden waters of the Platte River, values calculated from irradiance profiles were used to compute the ratio of the downwelling irradiance at a specified depth z, , to the downwelling irradiance just beneath the air-water interface, . Equation (3) thus becomes . The effective attenuation coefficient was approximated as to account for the two-way path through the water column to the bed and back to the surface [Philpot, 1989; Maritorena et al., 1994]. Technically, this parameterization is oversimplified because the attenuation of the upwelling flux from the bottom and from the water column itself differs from that of the downwelling flux [Maritorena et al., 1994; Dierssen et al., 2003]. The approximation has been used in several previous studies [Philpot, 1989; Maritorena et al., 1994; Legleiter and Roberts, 2009], however, and was deemed adequate for our purposes here.
 The results of this analysis are summarized in Figure 6, in which warmer red and orange tones indicate depths to which a relatively large proportion of the radiant energy entering the river can propagate, whereas cooler blue shades represent depths that receive very little of the incident sunlight. The relatively low values of from 575–700 nm imply that, under the highly turbid conditions we observed at Rowe Sanctuary, penetration of light to the streambed was greatest in the red portion of the spectrum. Even for these more transmissive wavelengths, only 10% of the irradiance just beneath the water surface propagated to a depth of 0.28 m. For blue and NIR bands experiencing stronger attenuation due to scattering and absorption, respectively, the ratio declined to 10% within the first 0.11 m. In general, the high rates of attenuation we documented imply that in deeper areas of the channel very little solar energy would be able to propagate to the bed and back to the water surface. Even for the red bands for which penetration was greatest, only 1%–2% of the incident irradiance reached depths of 0.5 m. The results in Figure 6 are relative in the sense that was scaled by , and the absolute amount of irradiance would depend on and its controlling factors, such as time of year, time of day, and atmospheric/cloud cover conditions. Nevertheless, strong attenuation by the water column reduced the amount of radiant energy that reached the streambed and limited the range of depths over which river bathymetry could be remotely mapped.
3.3. Field Spectra and the Relationship Between Depth and Reflectance
 In order to retrieve information on water depth from optical image data, a quantitative relation between observed flow depths and an image-derived radiometric quantity must first be established. In this study, we sought to obtain such a relation for the highly turbid Platte River by measuring depth and reflectance at 24 locations along the Rowe Sanctuary study reach. These spectra were recorded above the water surface and are illustrated in Figure 7, which indicates that spectra for depths from 0.03 m to 0.67 m exhibited a similar general shape. Reflectance increased with wavelength up to 580 nm as a result of the decrease in attenuation by the water column over this range (Figure 5b), increased backscattering by suspended sediment (Figure 2c), and the steady increase in the bottom reflectance of the sandy substrate throughout the visible and NIR portions of the spectrum (thick dashed line in Figure 7). Reflectance reached a plateau from 580 to 650 nm, the bands in which values were relatively low (Figure 5b), which allowed greater amounts of radiant energy to propagate to the bed and be reflected upward. The spectra declined to a local minimum at 675 nm associated with absorption by chlorophyll; the presence of a similar feature in the sand spectrum used to define the bottom reflectance suggests that some chlorophyll also might be present on the bed in the form of periphyton. On the shoulder of this absorption band, reflectance reached a local maximum near 700 nm, coinciding with the minimum , and then declined rapidly with increasing wavelength in the NIR due to strong absorption by water. Reflectance increased to a local maximum at 810 nm due to a decrease in the absorption coefficient of pure water in this band (Figure 2), and then decreased to very low values in the longest NIR wavelengths we measured.
 To establish a relationship between depth and reflectance, these spectra were subjected to the Optimal Band Ratio Analysis (OBRA) described in section 2.3. This procedure involved performing a regression of depth d against the image-derived quantity X, defined by equation (5), for all possible combinations of numerator () and denominator () wavelengths. The resulting matrix of values is shown graphically in Figure 8a, in which warmer red tones indicate band combinations that yielded strong, linear relationships between d and X, whereas cooler blue tones represent pairs of wavelengths for which X was not a useful predictor of d. The prevalence of red, orange, and yellow colors in Figure 8a implies that the simple log-transformed band ratio X was strongly related to d across a broad range of wavelengths, with values on the order of 0.65 or greater when a NIR denominator band was paired with any numerator band less than 725 nm. The greatest predictive power was achieved when a green numerator band ( nm) was coupled with a longer-wavelength red or NIR denominator band ( nm). The optimal band ratio occurred within this region: a regression of d versus yielded an R2 value of 0.949 (Figure 8b). Many other band combinations had values nearly as high, suggesting that a broad range of wavelengths could be useful for depth retrieval, not just a small, restricted region of the spectrum. More importantly, these results indicate that even in a turbid river with high concentrations of suspended sediment, water depth exerted a primary control on reflectance in this shallow channel. Our field observations thus imply that strong, linear relationships between depth and a simple, radiometric quantity X could be exploited to infer bathymetry from remotely sensed data. This possibility is explored in a companion paper (C. J. Legleiter et al., submitted).
3.4. Radiative Transfer Modeling of the Constraints on Spectrally Based Depth Retrieval
 Though encouraging, the results depicted in Figure 8 were derived from a small number of field spectra acquired from a shallow channel, with d < 0.3 m for two-thirds of the data set. Irradiance profiles recorded at the same site indicated that the downwelling light stream was attenuated rapidly within the water column, resulting in high values of (Figure 5b) and limiting the amount of radiant energy propagating to the bed (Figure 6). If the irradiance incident upon the bed is not sufficient, an imaging system with a fixed, finite sensitivity might not be able to detect the bottom. Even if the bottom can be distinguished, differences in depth might produce such small changes in at-sensor radiance that only relatively large variations in bathymetry could be resolved from remotely sensed data. To evaluate such constraints on spectrally based depth retrieval, we performed a series of radiative transfer simulations using the MODTRAN and Hydrolight numerical models, as described in section 2.4. Model runs were parameterized for our study area along the Platte River (Table 1), but this analysis was more general in the sense that a range of depths, suspended sediment concentrations, and sensor radiometric resolutions (i.e., values) were considered in evaluating both the maximum detectable depth and the precision of image-derived depth estimates. Our results thus provided insight on the inherent limitations associated with remote bathymetric mapping of turbid river channels.
3.4.1. Maximum Detectable Depth
 To estimate flow depth from image data, the streambed must be distinguishable from optically deep water. Such a distinction can only be made if the difference in at-sensor radiance between the river bottom and an infinitely deep water column exceeds the smallest change in radiance, , a given imaging system is capable of resolving. To quantify the range of depths that could be detected in a turbid, sand-bed river such as the Platte, we compared at-sensor radiance values modeled for a sandy bottom, , and for an infinitely deep water column, . The maximum detectable depth dmax was estimated by determining the largest depth for which condition (6) was satisfied. We considered depths from 0.02 to 2 m and repeated these calculations for suspended sediment concentrations from 60 to 240 mg L−1 and values for the hypothetical 10-, 12-, and 14-bit imaging systems described in Table 2.
 The results of this analysis are summarized in Figure 9. The band-integrated, at-sensor radiance spectra for three different suspended sediment concentrations shown in Figures 9a, 9c, and 9e indicate that for depths as shallow as 0.4 m, the difference between the sand bed river and infinitely deep water was quite small, on the order of 0.002 Wm–2 sr–1 or less. The difference was greatest from 580 to 700 nm, where attenuation by the water column was weakest, and was also relatively large in the NIR for shallow depths of 0.1 and 0.2 m. In general, as the flow depth increased, spectra from the sand bed river took on a closer resemblance to infinitely deep water with the same optical properties because the bottom's contribution to the total at-sensor radiance declined while the contribution from the water column itself became more significant. Similarly, a comparison of Figures 9a, 9c, and 9e indicates that, for a given depth d0, decreased as Cs increased due to greater backscattering by suspended sediment, which resulted in higher volume reflectance from within the water column.
 The maximum depth detectable by imaging systems of varying radiometric resolution under these conditions was determined by comparing sand bed spectra modeled for a range of depths to the spectrum. These results are depicted in Figures 9b, 9d, and 9f, which indicates that relatively large values of dmax occurred: (1) in wavelengths with weaker water column attenuation; (2) when Cs was low; and (3) when was low. Even for the wavelength of greatest penetration (694 nm), the smallest Cs considered (60 mg L−1), and the most sensitive instrument (14-bit, corresponding to a band-integrated Wm–2 sr–1), dmax was only 0.72 m. The broken lines in Figures 9b, 9d, and 9f indicate wavelengths for which condition (6) was not satisfied for any of the depths represented in our database of simulated spectra, implying that the bottom was effectively invisible in that band and thus precluding any estimate of depth. Consistent with the widely used rule of thumb that the bottom should be visible down to a depth of approximately , the maximum detectable depth was on the order of 0.5 m in the red portion of the spectrum for which values were approximately 4 m–1. For both shorter and longer wavelengths, dmax was shallower due to stronger attenuation (i.e., higher Kd values) in these bands. Values of dmax were smaller for larger Cs because greater concentrations of suspended sediment produced more scattering and reduced the difference between and for a given depth below the detection limit of typical sensors. Instruments capable of resolving more subtle variations in brightness were characterized by smaller values of and could provide depth information over a somewhat greater dynamic range. In general, however, these results imply that spectrally based bathymetric mapping of turbid rivers might be reliable only in shallow depths on the order of 0.5 m or less, with the actual value of dmax depending on suspended sediment concentration and sensor radiometric resolution.
 Given the relatively shallow depths typical of rivers, particularly wide, braided channels such as the Platte, comparing sand bed spectra to an infinitely deep water column could lead to overly conservative estimates of dmax. An alternative means of evaluating the dynamic range of depth retrieval involved determining the greatest depth at which the sand substrate could be distinguished from a zero-reflectance, black bottom at the same depth. The results of this analysis, summarized in Figure 10, differed from those based on the spectrum because truncating the water column with a zero-reflectance bottom at some finite depth reduced the volume reflectance; smaller amounts of suspended sediment were available to backscatter incident radiation when the thickness of the water column was not as great. As a result, differences in at-sensor radiance between sand bed and black bottom spectra for a given depth were larger than for the infinitely deep case. As before, the distinction between the sand bed and black bottom was reduced in deeper water and for higher concentrations of suspended sediment. The greatest depths at which a sandy substrate would be detectable by sensors of varying radiometric resolution are shown in Figures 10b, 10d, and 10f. Because the presence of the perfectly absorbing substrate would act to reduce the at-sensor radiance and make the river appear darker than if the bottom were composed of sand, condition (7) was satisfied for slightly greater depths than condition (6), up to 0.75 m for a 14-bit instrument at a wavelength of 648 nm. Maximum depths at which the bed would be visible were still quite limited, however, with typical values of 0.5 m or less for the high SSC conditions we considered.
Figure 11 more concisely represents the relationships among maximum detectable depth, suspended sediment concentration, and sensor radiometric resolution in specific green and NIR bands for both the infinitely deep water column and black bottom. For both wavelengths and either hypothetical substrate, an increase in Cs from 60 to 240 mg L−1 resulted in a steady decrease in dmax from as high as 0.6 m for a sensitive, 14-bit instrument in the NIR to as little as 0.04 m for a less sensitive, 10-bit detector in the green portion of the spectrum. For a given Cs, dmax values were shallowest for the sensor with the lowest radiometric resolution (10-bit) because such an instrument would not be capable of distinguishing slight variations in brightness between a sand bed and an infinitely deep water column or black bottom; dmax values for the 12- and 14-bit sensors were more similar to one another. Estimates of dmax were consistently greater when a sand bed spectrum was compared to a black bottom rather than an infinitely deep water column because the reduced volume reflectance associated with the finite-depth, zero-reflectance substrate made the river appear darker and led to larger, more readily detectable differences in at-sensor radiance. Comparison of Figures 11a, 11c, and 11b, 11d indicates that greater depths could be detected in the NIR band than in the green portion of the spectrum, most notably when the sand bed spectra were compared to the black bottom. At a wavelength of 518 nm, radiative transfer within the water column was dominated by scattering by suspended sediment and Cs exerted a primary control on dmax, which declined to very low values on the order of 0.15 m for Cs greater than 100 mg L−1. Figure 2 shows that in the NIR backscattering by sediment was relatively less important and that radiative transfer within the water column was more strongly influenced by pure water absorption, making these bands somewhat less sensitive to Cs and more responsive to variations in depth. For the black bottom, dmax values at a wavelength of 780 nm remained in the range 0.25–0.3 m for Cs up to 240 mg L−1, but strong absorption by the water itself could be a limiting factor in this portion of the spectrum. In any case, our results clearly indicate that depth retrieval from highly turbid rivers could be restricted to shallow depths of 0.5 m or less.
3.4.2. Bathymetric Precision
 Another important factor to consider in evaluating the utility of remotely sensed data for mapping the bathymetry of sediment-laden channels such as the Platte is the precision of image-derived depth estimates. As described in section 2.4.1, a difference in depth can only be resolved if the corresponding change in at-sensor radiance exceeds the sensor's noise-equivalent delta radiance; that is, condition (8) must be satisfied. In this study, we used a database of modeled spectra for depths from 0.02 to 2 m to determine the smallest detectable increase in depth and the smallest detectable increase in depth relative to an initial depth of d0 and then used this information to derive bathymetric contour interval widths, denoted by . These calculations were performed for a range of initial depths, suspended sediment concentrations, and sensor radiometric resolutions and are summarized in Figure 12. In Figure 12, the initial depth d0 is represented by a thick black line and the thinner red and blue lines represent the smallest detectable decrease and increase in depth, respectively, relative to d0. The shaded areas in between the red and blue lines thus indicate the range of depths that could not be distinguished from d0 by the hypothetical 10-, 12-, and 14-bit imaging systems we considered. For a shallow depth of 0.1 m and Cs up to 140 mg L−1, image-derived depth estimates had a precision on the order of 0.01 m for wavelengths greater than 550 nm, even for the least sensitive 10-bit instrument. At shorter wavelengths, however, scattering by suspended sediment dictated that only very large (if any) increases in depth could be detected. For greater base depths of 0.2 and 0.4 m, bathymetric contour intervals became wider, especially for higher values of Cs, and the range of wavelengths with small was restricted to a smaller set of red bands (Figures 12h and 12c). In general, sensors with smaller values of provided more precise depth estimates, but, for d0 = 0.4 m and a higher Cs of 140 mg L−1, even a 14-bit detector could not resolve any increase in depth beyond the 0.4 m base depth, nor could any decrease in depth smaller than 0.1 m be detected. These results imply that, in addition to limiting the range of depths over which bathymetry could be remotely mapped, high concentrations of suspended sediment could subject image-derived depth estimates to a considerable degree of uncertainty.
4. Discussion and Conclusions
 Although remote sensing of rivers affords considerable potential to advance our understanding of fluvial systems and to facilitate various management applications [Marcus and Fonstad, 2008], the utility of this approach has only been evaluated in a few, favorable riverine environments. The motivation for this study was to assess the feasibility of extending spectrally based bathymetric mapping techniques developed for clear-flowing, gravel bed streams to larger, sand-bed rivers that transport significant quantities of sediment in suspension, greatly reducing the clarity of the water they convey. To make this assessment, we employed a two-tiered strategy that involved both field-based measurements of the optical characteristics of a highly turbid channel and numerical radiative transfer models parameterized for this field site but intended to provide more general insight on the constraints associated with depth retrieval from optical image data. These two approaches yielded disparate results: whereas the analysis of field spectra was quite encouraging, our modeling efforts offered a much more sobering perspective on the prospects for remote sensing of sediment-laden stream channels. In this section, we highlight key outcomes arising from this investigation and point out some of the limitations of our study.
 Attenuation of light within the water column is the physical process that allows flow depths to be inferred from optical image data, but attenuation also limits the range of conditions under which reliable bathymetric information can be obtained via remote sensing. In this study, we used ground-based spectral measurements from a shallow, braided reach of the Platte River to characterize attenuation within a turbid stream channel. The amount of downwelling radiant energy was measured at various depths beneath the water surface and used to calculate a diffuse attenuation coefficient, , for visible and NIR wavelengths from 400 to 900 nm. This analysis indicated that radiative transfer was adequately described by a simple formulation (the Beer-Lambert Law) that expresses the exponential attenuation of light with distance traveled through the aquatic medium, implying that, at first order, a more complex treatment of this process was not necessary. Moreover, we demonstrated that radiometric data that can be collected efficiently in the field provide a means of summarizing the absorption and scattering interactions experienced by photons in transit between the water surface and river bed. In this study, values determined from two irradiance profiles were on the order of 10 m–1 for blue and longer-wavelength NIR bands due to scattering by suspended sediment and strong absorption by pure water, respectively. Even for the 575–650 nm spectral region where attenuation was weakest, values of approximately 4 m–1 were an order of magnitude greater than those reported from clear-water coastal environments [e.g., Mishra et al., 2005]. An important implication of the strong attenuation we documented along the Platte was that, even for very shallow depths on the order of 0.5 m or less, only a small fraction of the sunlight transmitted through the air-water interface would propagate through the water column to the bottom and back to the surface.
 A second field data set, consisting of reflectance measurements from above the water surface, was used to establish a quantitative connection between water depth and spectral reflectance. Observations spanning a range of depths from 0.04 to 0.67 m were subjected to an optimal band ratio analysis to identify the wavelength combination that yielded the strongest linear relation between depth, d, and a log band ratio, X (equation (5)). For the 24 spectra we collected along the Platte, numerator and denominator bands of 593 and 647 nm, respectively, were selected via this procedure, and the R2 value of the corresponding d versus X regression was 0.949. Notably, many other band combinations had R2 values nearly as high, implying that a broad range of wavelengths could be used to derive radiometric quantities strongly related to flow depth. These results indicate that, even in a more turbid river with suspended sediment concentrations 20 times greater than those considered in previous investigations, depth remained the primary control on reflectance. Again, this study showed that the information required to link depth and reflectance, and thus enable spectrally based bathymetric mapping, could be obtained via field measurements. This approach would amount to a form of site-specific calibration, however. Compiling a larger database of field spectra from diverse fluvial environments might lead to more general relationships that could be transferred among sites. The results of this study, most notably a strong relation between an easily calculated radiometric quantity and observed flow depths in a highly turbid channel, indicate that such an effort could be fruitful.
 As attempts to expand the remote sensing of rivers to encompass a broader range of channels proceed, however, attention must also be paid to the inherent limitations associated with spectrally based bathymetric mapping. To quantify these constraints, we employed a forward modeling strategy based on numerical simulations of radiative transfer within the atmosphere and water column. In this study, the MODTRAN and Hydrolight models were parameterized for the Platte River and used to characterize the dynamic range and precision of depth retrieval from airborne hyperspectral image data. This analysis involved comparing modeled spectra for a sand-bed river and for an infinitely deep water column with the same optical properties (i.e., SSC). Maximum detectable depths were determined by identifying the greatest depth at which the difference in at-sensor radiance between the sand-bed and infinitely deep spectra exceeded the smallest change in radiance the imaging system was capable of resolving. Maximum depths were greatest in bands with relatively weak attenuation, when sediment concentrations were low, and when the sensor had a high radiometric resolution. For the highly turbid conditions evaluated in this study, however, maximum detectable depths for Cs = 60 mg L−1 did not exceed 0.72 m in any wavelength, even for a relatively sensitive 14-bit instrument. Increased volume reflectance from the water column further reduced dmax for higher sediment concentrations. As an alterative to the infinitely deep water column, which might lead to overly conservative estimates of dmax, we also compared the modeled sand-bed spectra to spectra with a zero-reflectance, black bottom at the same depth. Truncating the water column in this manner reduced volume reflectance, made the water appear darker, and led to greater differences in at-sensor radiance, relative to the infinitely deep case. Nevertheless, the maximum depth at which the sand bed could be distinguished from the black bottom was no greater than 0.75 m for the conditions we considered. Overall, our simulations suggested that bathymetric mapping of sediment-laden stream channels could be limited to shallow depths of less than about 0.5 m, with the actual value varying as a function of SSC and sensor radiometric resolution.
 In addition to this fundamental constraint on the range of depths retrievable from optical image data, the precision of these depth estimates could also be limited by high concentrations of suspended sediment. Bathymetric precision was quantified in terms of contour intervals that represented the sum of the smallest detectable increase and decrease in depth relative to a specified initial depth. Again, the radiometric resolution of the sensor was a key control, with more sensitive 12- or 14-bit instruments capable of resolving smaller variations in depth than a 10-bit imaging system. The width of these bathymetric contour intervals, which served as an index of the uncertainty associated with image-derived depth estimates, increased with depth and SSC. For shallow depths and red or NIR bands, precision was on the order of 0.01 m, but scattering by suspended sediment dictated that only much larger changes in depth could be resolved at shorter wavelengths. In deeper water (0.4 m) and higher concentrations of suspended sediment (140 mg L−1), only decreases in depth of 0.1 m or greater were detectable. In general, our results indicate that spectrally based depth retrieval from turbid rivers could be subject to a significant amount of uncertainty. Because sensor characteristics exert a direct influence on the magnitude of this uncertainty, determining the smallest change in radiance detectable by an imaging system for a given set of environmental conditions is an important consideration for evaluating not only bathymetric contour intervals but also maximum detectable depths.
 The radiative transfer modeling exercise reported herein involved a number of simplifications, and future studies might seek to relax some of these assumptions: (1) water and suspended sediment were the only optically significant constituents of the water column; (2) absorption and scattering coefficients for suspended mineral matter were related to sediment concentrations via optical cross-sections assumed to be representative of the material transported by the river of interest; and (3) suspended sediment was uniformly distributed throughout the water column. Refinements to this framework would thus consider the effects of chlorophyll and chromophoric dissolved organic matter (CDOM), explicitly link inherent optical properties to sediment characteristics, and incorporate a more realistic vertical profile of suspended sediment concentrations. In this study, radiative transfer modeling was used only in a forward mode to simulate spectra for a specified set of environmental parameters. These models could potentially be applied in an inverse manner as well, with the environmental parameters inferred via comparison of simulated and observed spectra. A precedent for this approach has been established in coastal settings, and radiative transfer model inversion in river systems thus represents a viable research topic.
 In summary, the field component of this investigation indicated that spectrally based bathymetric mapping of turbid rivers is feasible, stimulating some optimism regarding the application of remote sensing techniques to larger, sand-bed channels with high concentrations of suspended sediment. A companion paper examines this possibility in detail [Legleiter et al., 2011]. The radiative transfer modeling phase of our study also highlighted some of the challenges that would be involved in such an effort. Although calibration to local conditions remains necessary, and the results reported herein are specific to a particular field site along the Platte River, the approach we have employed is quite general. This study has demonstrated the utility of ground-based spectral measurements for characterizing the attenuation of light within the water column and for establishing relationships between flow depths and radiometric quantities. An initial field effort can thus help to guide a subsequent remote sensing campaign or, should the field data fail to reveal any strong connection between reflectance and depth, to avoid an ill-advised investment in image data. Similarly, radiative transfer modeling can provide useful information on the range of depths that could be mapped via remote sensing, as well as the precision of image-derived depth estimates. This hybrid, field- and modeling-based approach can thus facilitate efficient, informed use of remote sensing techniques across a broad range of riverine environments.
 Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Financial support was provided by the Office of Naval Research Coastal Geosciences Program (grants N0001409IP20057 and N00014-10-1-0873). AISA image data were acquired through the University of NE's CHAMP facility, coordinated by Rick Perk. Mark Dixon assisted with field data collection. Alan Weidemann, Nina Kilham, and Roger Clark provided useful comments.