Understanding river form and behavior requires an efficient means of measuring channel morphology. This study evaluated the potential to map the bathymetry of two clear-flowing, shallow (<3 m deep) gravel bed rivers <60 m wide from 2 m-pixel WorldView2 satellite images. Direct measurements of water column optical properties were used to quantify constraints on depth retrieval. The smallest detectable change in depth was 0.01–0.04 m and the maximum detectable depth was 5 m in green bands but <2 m in the near-infrared; lower sensor radiometric resolution yields less precise estimates over a smaller range. An algorithm for calibrating a band ratioX to field measurements of depth d proved effective when applied to spectra extracted from images (R2 = 0.822 and 0.594 for the larger and smaller stream, respectively) or measured in the field (R2 = 0.769 and 0.452). This procedure also identified optimal wavelength combinations, but different bands were selected for each site. Accuracy assessment of bathymetric maps produced using various calibration approaches and image types indicated that: 1) a linear d vs. Xrelation provided depth estimates nearly as accurate as a quadratic formulation; 2) panchromatic and pan-sharpened multispectral images with smaller 0.5 m pixels did not yield more reliable depth estimates than the original images; and 3) depth retrieval was less reliable in pools due to saturation of the radiance signal. This investigation thus demonstrated the feasibility, as well as the limitations, of measuring the bathymetry of clear, shallow gravel bed rivers from space.
 The form and behavior of gravel bed rivers reflect complex interactions among morphology, flow, and sediment transport. Understanding connections between form and process is thus a principal objective of fluvial geomorphology, but progress toward this goal is hindered by the difficulty of collecting basic data on topography, flow conditions, and bed material properties. Moreover, the logistical constraints associated with traditional field methods for measuring these attributes often limit investigations to short, isolated study reaches. Although recent advances in instrumentation, such as total stations [Keim et al., 1999], real-time kinematic global positioning systems (RTK GPS) [Brasington et al., 2000], and terrestrial laser scanning [Hodge et al., 2009], have enabled more efficient data collection, most research continues to focus on scales ranging from a few to several tens of channel widths, often with little consideration of the broader watershed context for these detailed surveys. A more synoptic perspective on fluvial systems will require a different approach, and remote sensing is increasingly viewed as a viable alternative [Marcus and Fonstad, 2008, 2010]. Previous studies have demonstrated the feasibility of deriving various types of river information from image data, ranging from measurements of lateral channel migration from historical air photos [e.g., Micheli and Kirchner, 2002] to suspended sediment concentrations inferred from Landsat scenes [Mertes et al., 1993; Kilham et al., 2012]. The most mature application of remote sensing to rivers is retrieval of water depth from passive optical images, primarily multi- or hyperspectral data acquired from aerial platforms [e.g.,Winterbottom and Gilvear, 1997; Marcus et al., 2003; Lejot et al., 2007; Legleiter et al., 2009; Flener et al., 2012]. In this study, we build upon these results by using field measurements and satellite imagery to evaluate the potential for mapping the bathymetry of gravel bed rivers from space.
 Knowledge of water depth is valuable for a number of different purposes. For example, depth is an important parameter for any hydrologic study that involves computing river discharge and/or routing flows. From a geomorphic perspective, the depth, along with the water surface slope, exerts a primary control on the boundary shear stress that in turn drives bed material transport. In an ecological context, the thickness and optical properties of the water column determine the amount of solar energy that propagates to the streambed to fuel primary production by benthic algae. For managers interested in monitoring stream condition and change, depth is one of the principal quantities used to assess habitat quality. In addition, when combined with information on water surface elevations, depth measurements can be used to obtain topographic input data for numerical modeling of flow and sediment transport. Similarly, sequential observations of channel morphology can be used to identify areas of erosion and deposition and hence to infer bed material transfer and storage [e.g., Ashmore and Church, 1998; Ham and Church, 2000]. A capacity to map river bathymetry efficiently, reliably, and over large extents would thus benefit the riverine sciences in many ways.
 Remote sensing could provide such capability. The potential of this approach has been confirmed first through empirical case studies [e.g., Winterbottom and Gilvear, 1997; Marcus et al., 2003] and more recently by considering the underlying physics [Legleiter et al., 2004, 2009]. Passive optical remote sensing of river bathymetry involves measuring the amount of solar radiation reflected from the channel, which depends not only on depth but also the texture of the water surface, the concentration and composition of sediment and organic materials within the water column, and the reflectance of the streambed. Moreover, all of these quantities vary as a function of wavelength λ. To gain insight as to how the processes that govern the interaction of light and water both enable and limit the remote sensing of rivers, Legleiter et al.  used a radiative transfer model to isolate the effects of depth d, suspended sediment concentration, and bottom reflectance RB(λ) on the upwelling spectral radiance L(λ) recorded by a remote detector. For conditions representative of a shallow, clear-flowing gravel bed river, this analysis indicated that depth was the primary control onL(λ) and that taking the logarithm of the ratio of two specific bands yielded an image-derived quantityX linearly related to d. Subsequently, we argued on theoretical grounds that spectrally-based depth retrieval would be feasible whendis small, the substrate is highly reflective relative to the overlying water column, and attenuation of light is dominated by pure water absorption rather than scattering by suspended sediment. Radiative transfer modeling, field-based reflectance measurements, and bathymetric maps derived from hyperspectral image data supported these conclusions and confirmed the utility of a simple band ratio for remote measurement of water depths in certain types of rivers [Legleiter et al., 2009].
 Although previous studies have shown that depth information can be retrieved from aerial images, the feasibility of mapping river bathymetry from space has not been assessed. In this study, we used field measurements and multispectral image data to explore the possibility of measuring gravel bed river depths from a satellite platform. Our research objectives were to: (1) characterize the inherent optical properties of the water column by obtaining novel in situ measurements of absorption and attenuation in a pair of clear-flowing gravel bed streams, (2) establish relationships between depth and reflectance based on field spectra from these rivers, and (3) evaluate different approaches for calibrating image-derived quantities to flow depth and (4) assess the accuracy of depth estimates produced from various types of satellite imagery.
2.1. Study Area
 To evaluate the feasibility of mapping river bathymetry from space, we examined two gravel bed streams in the Rocky Mountains, USA: the Snake River in Grand Teton National Park and Soda Butte Creek (SBC) in Yellowstone National Park (Figure 1). Both watersheds are snowmelt-dominated and generally exhibit clear water conditions during late summer low flows. The two rivers feature wandering planforms with both sinuous, single-thread and more complex multi-thread segments. In addition, both streams are highly dynamic, with extensive morphologic change occurring during spring runoff in many years, including 2011. The steep, glaciated valley of SBC is composed of weak Eocene volcanic rock and provides an abundant sediment supply that drives channel change [Meyer, 2001]. The Snake River is regulated at Jackson Lake, but sediment delivered from tributaries below the dam [Erwin et al., 2011], along with large woody debris, contribute to frequent morphologic adjustments. We selected these rivers for study because the complexity and dynamism of these channels imply that remote sensing might be useful, if not necessary, for characterizing their form, behavior, and evolution.
 For the Snake River we focused on a meander called Rusty Bend, shown in Figure 1c and described in Table 1. The channel curves smoothly to the right through this reach and has a relatively simple morphology consisting of a gravel bar along the inner (right, or north) bank and a vertical to steeply sloping cutbank on the outside of the bend where the Snake River erodes into a high terrace. For the purposes of this investigation, the bar-pool topography of Rusty Bend was attractive because of the broad range of depths encompassed: very shallow over the bar and up to 2.77 m in the pool along the outer bank. Also, a variety of substrates with different reflectance characteristics were present within this reach: clean gravel, bright green benthic algae, and blocks of lighter-colored bedrock (Figure 2b).
Table 1. Channel Characteristics for Study Reaches
Snake River Rusty Bend
Soda Butte Creek Footbridge Reach
The symbol denotes the pixel-scale mean depth determined from field measurements via ordinary block kriging.
The symbol n is the number of values for the reach.
 Soda Butte Creek has been the subject of previous efforts to map river bathymetry from aerial platforms [Marcus et al., 2003; Legleiter et al., 2004, 2009; Legleiter, 2012a] and thus provides a convenient opportunity to assess whether similar information might be derived from satellite images. In this study, we focused on the Footbridge Reach, a site for which ground-based topographic surveys have been conducted each year since 2005 [Legleiter, 2012b]. The reach comprises a sweeping meander and large point bar, which is now being incised by a chute channel that has captured much of the stream's flow and could soon cut off the bend. Because the discharge is divided among multiple channels and the image was acquired under base flow conditions, depths were generally quite shallow (Table 1) but deeper pools were present in the middle of the chute channel, along the west bank above the bend apex, and against the east bank at the lower end of the reach. The smaller size of SBC also implied that mixed pixels along channel margins would be more extensive and potentially more problematic than for the Snake River.
2.2. Field Data Collection
 On the Snake River, we surveyed channel bed topography using a high-precision (2–3 cm, both horizontal and vertical) real-time kinematic (RTK) GPS receiver. Elevations were measured at points arranged along cross-sections that traversed exposed bars and shallow areas of the wetted channel. Depths were determined by subtracting the bed elevation for in-stream survey points from the elevation recorded at the water's edge along each transect [Legleiter et al., 2011a]. For areas that were too deep to wade safely, the GPS receiver was mounted on a cataraft and configured to record water surface elevations while communicating with an echo sounder that measured flow depths. Measurements were obtained along a series of transects across our Rusty Bend study reach as well as longitudinal profiles recorded as the cataraft traveled downstream. Over 22 km of the Snake River was surveyed in this manner during a 10-day period in August and September 2011, resulting in a total of 73,686 echo sounder-based depth measurements. In addition, an acoustic Doppler current profiler (ADCP) deployed from a kayak provided additional depth observations. To ensure that the depth measurements obtained via the three methods were consistent with one another, we compared echo sounder and ADCP readings to the closest wading depth. This analysis indicated that in shallow areas where the data sets overlapped, the echo sounder depths were biased shallow by 7 cm and the ADCP depths biased deep by 3 cm relative to the wading depths. The mean differences between the echo sounder and ADCP depths and the corresponding wading points were used to adjust the echo sounder and ADCP data to match the wading depths. The resulting, combined bathymetric field data set was then used for calibration and validation of image-derived depth estimates.
 For the smaller Soda Butte Creek, all field measurements were obtained by wading, which allowed us to access all but the deepest portions of the channel. Field data collection at the Footbridge Reach involved using the RTK GPS and a robotic total station to complete a detailed, terrain-sensitive survey that emphasized important breaks in slope such as the top and base of banks [e.g.,Lane et al., 1994; Wheaton et al., 2010]. In addition, water surface elevations were measured along channel margins and used to calculate depths for in-stream points as the difference between water surface and bed elevations. An earlier study confirmed that depths determined in this manner agreed closely with depths measured directly with a ruler, which would have been a less efficient means of obtaining these data [Legleiter, 2012c].
2.3. Geostatistical Analysis
 Before using the depth measurements described above to calibrate image-derived depth estimates and assess their accuracy, the original field data were processed using geostatistical techniques. These analyses served two important purposes: 1) up-scaling observations collected at points to the dimensions of an image pixel [Bailly et al., 2010]; and 2) producing continuous depth maps for comparison with the remotely sensed bathymetry. To account for the non-convex geometry of these meandering rivers, which implies that conventional Euclidean distances are not a valid metric [e.g.,Little et al., 1997], data were transformed from the original Cartesian reference frame to an orthogonal curvilinear, channel-centered coordinate system defined by a streamwise axissalong the channel centerline and a cross-stream, or normal,n axis oriented perpendicular to the centerline [Smith and McLean, 1984; Legleiter and Kyriakidis, 2006]. The spatial structure of each reach was then quantified using anisotropic variograms, as described by Legleiter [2012d]. Briefly, variograms summarize the degree to which observations are spatially correlated with one another as a function of the distance and direction (i.e., lag vector) between pairs of points and thus provide information on overall variability as well as the length scales over which this variability is expressed along and across the channel. In this study, de-trending was not necessary because we used depth measurements rather than bed elevations that would have been influenced by the overall slope of the channel. Sample variograms were calculated for thes and n directions by restricting the angular tolerances associated with each lag vector class. Variogram model parameters were estimated first manually using an iterative graphical procedure and then refined using a weighted least squares algorithm that emphasized shorter lag distances with larger numbers of pairs [Zhang et al., 1995; Pardo-Iguzquiza, 1999].
 Although field measurements were collected at discrete points, depth estimates derived from satellite images pertain to larger areas of space, represented by image pixels. To account for this difference in scale, or change of support [e.g., Atkinson and Curran, 1995], we employed a geostatistical algorithm known as ordinary block kriging (OBK), described in general by Goovaerts  and in a remote sensing context by Bailly et al. . Here, we briefly summarize the rationale for and implementation of this procedure. Essentially, we used OBK to upscale point observations to the dimensions of an image pixel and obtain spatially distributed estimates of the pixel-scale mean depth. Also, because some pixels contained multiple points, this analysis served to avoid the redundancy that would have occurred if the original field measurements were paired directly with specific image pixels. Instead, the OBK algorithm yielded a single estimate of the average depth within each pixel, regardless of the number of points present within that area. To compute OBK estimates, we first created a fine-scale grid of discretization points for each reach, consisting of 16 points per pixel, and transformed these prediction locations to the same channel-centered coordinate system as the field data. A depth estimate for each discretization point was then calculated using an ordinary kriging algorithm that incorporated the anisotropic variogram models described above [Legleiter and Kyriakidis, 2008]. Taking the average of the ordinary kriging estimates for the 16 points within each pixel yielded the block kriging estimate of the pixel-scale mean depth [Goovaerts, 1997, p. 152]. For purposes of calibration and validation, OBK depth estimates were computed only for those pixels containing one or more point measurements. In addition, to examine spatial patterns of depth retrieval accuracy, continuous field-based bathymetric maps were generated by estimating via OBK the mean depth for all pixels within each reach.
2.4. Spectral Characteristics of Gravel Bed Rivers: Field Measurements and Data Analysis
 One of our long-term research objectives is to build a more thorough database on the spectral characteristics of fluvial environments, where only a few quantitative observations of water column optical properties and bottom reflectance have been made [Legleiter et al., 2009, 2011b]. In this study, we began the process of compiling a spectral library for rivers by collecting field spectra and measuring the apparent and inherent optical properties of the water column in a pair of clear-flowing gravel bed streams.
 Reflectance spectra were recorded from above the water surface using an Analytical Spectral Devices (ASD) FieldSpec3 spectroradiometer that measured wavelengths from 350–1025 nm with a 1 nm sampling interval; only the 400–850 nm region was considered here due to noise at both ends of the spectrum. A 100% reflectant Spectralon calibration panel was used to establish a white reference prior to each round of measurements. For data collection along the Snake River, the ASD was mounted on a rod extending from the rear of the cataraft and configured to record spectra once each second as we traversed a series of transects across Rusty Bend (Figure 2a). Flow depths were recorded simultaneously using the survey instrumentation described above, providing the paired observations of depth and reflectance needed to develop bathymetric mapping algorithms. Moreover, these data extended the range of river conditions under which spectra have been measured from shallow, wadeable streams [Legleiter et al., 2009] to a deeper, larger channel with diverse bottom types (Figure 2b). For SBC, field spectra were recorded at points accessed by wading and depths measured with a ruler. Additional detail on the acquisition and processing of reflectance data were provided by Legleiter et al. [2011b].
 Because water column optical properties influence the feasibility of inferring depth from image data, we directly measured several attributes of the water within our study streams. Attenuation of light was characterized by measuring the total amount of incident solar radiation, referred to as the downwelling spectral irradiance Ed(λ), at various depths within the water column. These irradiance profiles were collected by connecting the ASD to an upward-facing cosine response detector that integrated radiant energy arriving from all directions within the upper hemisphere to obtainEd(λ); the fore-optic was attached to a waterproof cable and mounted on an adjustable scissor lift to position the sensor at different depths (Figure 2c). These measurements were used to calculate the diffuse attenuation coefficient Kd(λ), an apparent optical property that quantifies the rate at which light is attenuated with distance traveled through the water column, following the procedure outlined by Mishra et al.  and applied to the Platte River by Legleiter et al. [2011b]. In addition, we used a WET Labs ac-9 to directly measure two inherent optical properties of the water column, the absorption and attenuation coefficients,a and c. These optical data were collected on several dates along the Snake River and SBC. Ancillary data in support of these measurements included water samples analyzed for suspended sediment concentration and turbidity readings made with a Eureka Environmental Manta2 multiprobe.
 Measuring water column optical properties allowed us to examine two important constraints on remote sensing of river bathymetry: the precision of spectrally-based depth estimates and the maximum depth detectable by an imaging system. This analysis was based on the early work ofPhilpot , which was revisited in a fluvial context by Legleiter and Roberts . The original publications provide additional detail, but only the key results relevant to this investigation are highlighted herein. Values of Kd(λ) determined from our field measurements were used to calculate bathymetric precision as
where Δd(λ) is the smallest detectable difference in depth at an initial depth d0, ΔLN(λ) is the sensor's noise-equivalent delta radiance (essentially the smallest change in brightness the system can resolve), andLB(λ) is that portion of the total radiance signal that has interacted with the bottom and is thus related to depth. Because ΔLN(λ) and LB(λ) are both spectral radiance values, the units cancel and only the ratio ΔLN(λ)/LB(λ) is significant in equation (1). This ratio serves as a convenient index of the detectability of the bottom for a given river and sensor configuration. In this study, we calculated Δd(λ) values by specifying ΔLN(λ)/LB(λ) = 0.01, based on prior radiative transfer modeling [Legleiter and Roberts, 2009], and using 2Kd(λ) as an effective attenuation coefficient, following Philpot  and Maritorena et al. . Similarly, the maximum detectable depth occurs when the difference between the at-sensor radiance and the radiance from a hypothetical infinitely deep water column is equivalent to the sensor's ΔLN(λ) and was calculated as
In this case, we used ΔLN(λ)/LB(λ) values of 0.1, 0.01, 0.001, and 0.0001 to illustrate the effects of a greater bottom contrast between the substrate and water column and/or a more sensitive detector, either or both of which would correspond to smaller values of ΔLN(λ)/LB(λ).
2.5. Image Data Acquisition and Processing
 To evaluate the feasibility of mapping gravel bed river bathymetry from space, this study used satellite images acquired by the WorldView-2 (WV2) sensor. This imaging system became operational in January 2010 and features a unique combination of high spatial resolution (pixel sizes of 0.5 and 2 m for panchromatic and multiband images, respectively) and multispectral measurement capabilities, with eight bands spanning visible and near-infrared (NIR) wavelengths. In addition to the standard blue, green, red, and NIR bands, WV2 also includes coastal (400–450 nm), yellow (585–625 nm), and red edge (705–745 nm) bands potentially useful for depth retrieval from shallow streams as well as a longer-wavelength NIR band (860–1040 nm) that could be used to discriminate land from water. This satellite also features advanced pointing technology that allows for off-nadir viewing, reduced revisit times, and precise geometric positioning, with nominal geo-referencing accuracies on the order of 4 m (Digital Globe, data available athttp://worldview2.digitalglobe.com/about/, 2012).
 WV2 images of the Snake River and SBC were acquired on 13 September 2011 (Figure 1). Deliverables included geo-referenced multispectral and panchromatic data sets and supporting metadata. In addition to the original images, which were not radiometrically calibrated and consisted of raw digital numbers, we received atmospherically-corrected data processed to units of apparent surface reflectance in-house by DigitalGlobe. Depth retrieval performance was evaluated for the multispectral reflectance images as well as the higher spatial resolution panchromatic data sets. In addition, we considered hybrid, pan-sharpened images that consisted of eight spectral bands, like the original multispectral images, but had a smaller 0.5 m pixel size equivalent to the panchromatic images; these images were generated using the Gram-Schmidt spectral sharpening tool in the ENVI software. The three different types of images (multispectral, panchromatic, and pan-sharpened) allowed us to assess the relative significance of spatial and spectral resolution for remote sensing of river depths.
 Geo-referencing of the WorldView-2 image for the Snake River was highly accurate, with stream banks and other distinctive features in the image closely aligned with our field maps; no further geometric correction of this data set was needed. For SBC, however, the spatial referencing of the original image was not as reliable, with many of the points at which depths were measured plotting outside the wetted channel when overlain on the image. Given the smaller size of this stream and the abrupt variations in depth over the scale of a single, 2 m image pixel, improved geo-referencing was required. The necessary refinement was achieved by digitizing a wetted channel polygon on the image and comparing this feature to a polygon created from water surface elevation points surveyed in the field. The parameters of an affine transformation were then iteratively adjusted so as to maximize the area of overlap between the image- and field-based channel polygons and the original image transformed using the optimal parameters [Legleiter, 2012a]. This algorithm greatly improved agreement between field and image data, with depth points located within the wetted channel on the transformed image.
 An important pre-processing step was the definition of in-stream masks for each image. These masks served to isolate active channels and thus highlight variations in reflectance within the water portion of each scene. In this study, binary, water-only masks were produced by displaying the longest wavelength NIR band as a gray scale image, inspecting the histogram of pixel values, and interactively adjusting the contrast stretch to determine a NIR reflectance threshold that effectively distinguished dark water from brighter terrestrial features. Pixels with NIR reflectance values below this threshold were included in an initial water mask that was refined using image processing operations: morphological opening to remove isolated pixels, interactive segmentation to select in-stream image objects, and morphological closing to fuse small gaps [Legleiter et al., 2011a]. The resulting raster masks then were converted to vector representations that enabled manual editing to remove persistent shadows along steep cut banks, for example. The vector masks were rasterized and applied to the original images. Finally, the resulting in-stream images were spatially filtered using a 3 × 3 pixel Wiener smoothing filter that has been shown to improve depth retrieval performance [Legleiter, 2012c].
2.6. Spectrally-Based Depth Retrieval
 Mapping river bathymetry from satellite images requires a quantitative relationship between water depth d and reflectance R(λ) in one or more spectral bands; radiance or raw digital numbers also could be used for this purpose. Efforts to establish such relationships are complicated, however, by the influence exerted on the remotely sensed signal by several other factors, including variations in bottom reflectance, water column optical characteristics, water surface roughness, and atmospheric conditions. The availability of multiple spectral bands provides some leverage for isolating the effect of depth, and Legleiter et al. showed that under appropriate circumstances and for certain combinations of wavelengths, the image-derived quantity
is linearly related to depth. The physical basis for ratio-based depth retrieval was examined in detail byLegleiter et al. [2004, 2009]and is only summarized herein. The upwelling spectral radiance from a clear-flowing, shallow stream channel is primarily a function of depth and bottom albedo. Whereas the reflectances of various substrates tend to be within a few percent of one another at a given wavelength and thus have similar band ratio values, the absorption coefficient of pure water increases by an order of magnitude from the blue into the NIR portion of the spectrum. As a result, the reflectance in the longer wavelength band with stronger absorption,λ2, decreases more rapidly as depth increases than does R(λ1) and the ratio X increases with depth while remaining relatively insensitive to differences in bottom type. Taking the logarithm of the band ratio accounts for the exponential attenuation of light by the water column.
 This approach to retrieving water depth from remotely sensed data requires: 1) selecting a suitable pair of wavelengths; and 2) calibrating a linear relation between d and X. Both of these objectives can be achieved by Optimal Band Ratio Analysis, or OBRA [Legleiter et al., 2009]. Given paired observations of depth and reflectance, this algorithm performs regressions of d on X for all possible combinations of numerator (λ1) and denominator (λ2) wavelengths and identifies the optimal band ratio as that which yields the highest coefficient of determination R2; the corresponding regression equation provides a calibrated d vs. X relation. Because depth is regressed against X values defined by all possible band combinations, OBRA also is useful for examining spectral variations in the nature and strength of the relationship between d and X, which can be visualized by plotting the matrix of R2(λ1,λ2) values as a matrix.
 In this study, we performed OBRA of both field spectra collected along the Snake River and SBC and image spectra extracted from WV2 images of these streams. Analysis of the field spectra made use of the collocated depths measured with the echo sounder or by wading, whereas OBRA of image spectra used pixel-scale mean depths estimated via OBK. For the larger and deeper Snake River, we also performed a modified version of OBRA that included not only a linearX term but also an X2 term in each regression, based on the finding of Dierssen et al.  that a quadratic equation improved bathymetric retrievals in areas of greater depth.
 To assess whether d vs. X relations derived from field spectra could be used to infer depth from remotely sensed data, we convolved the original field spectra, which were recorded with a sampling interval of 1 nm (Figure 3a), to the specific band passes of the WV2 sensor. The convolved spectra shown in Figure 3bthus had a similar, though less well-resolved, shape as the original measurements, but the convolved field spectra differed from the image spectra in absolute magnitude due to residual atmospheric effects (Figure 3c). To account for this difference, we subtracted the mean reflectance of the image spectra from that of the field spectra for each band and then added this correction factor to the image spectra to force a closer agreement between the field measurements and remotely sensed data (Figure 3d). OBRA was then performed for the convolved field spectra and the resulting d vs. X relation applied to both the original image and the image corrected to better match the field spectra. This analysis thus allowed us to evaluate the performance of bathymetric mapping algorithms calibrated via field spectroscopy.
 Although OBRA exploits the spectral information available from multiband data sets, this procedure is not applicable to panchromatic images that achieve a higher spatial resolution by integrating electromagnetic energy from across the spectrum. To assess whether this additional spatial detail might enable reliable depth retrieval without requiring spectral information, we evaluated the bathymetric mapping capabilities of the 0.5 m-pixel panchromatic WV2 images of the Snake River and SBC. Rather than relatingd to X as for the multispectral data, we used the linear transform, or Lyzenga  algorithm, a more traditional approach to depth retrieval in rivers [e.g., Winterbottom and Gilvear, 1997; Fonstad and Marcus, 2005; Flener et al., 2012]. In this case, the image-derived quantity related to depth is given by
where R represents the reflectance integrated over the WV2 sensor's panchromatic band and R∞is the reflectance from a hypothetical, infinitely deep water column that encompasses the contributions from the water column, water surface, and atmosphere but which lacks any signal from the bottom; the latter term is thus known as the deep-water correction. Computationally, we implemented this algorithm using the raw digital numbers (DN) from each gray scale image and used the minimum pixel value from the in-stream portion of the scene as an estimate ofR∞; the last term on the right is added to avoid taking the logarithm of zero. Pixel-scale mean depths estimated from the original field measurements via OBK were then regressed againstY values for the corresponding locations to establish d vs. Y calibration relations.
 Calibration relationships obtained via OBRA or the linear transform were based on randomly selected subsets (50%) of the pixel-scale mean depths derived from the original field measurements. The remainingdvalues were reserved and used to validate image-derived depth estimates; accuracy assessment involved calculating mean errors (an indication of bias) and root mean square errors (RMSE), as described byLegleiter et al. [2011a], and examining residual maps. In addition, we performed regressions of observed (pixel-scale mean depths obtained via OBK) versus predicted (derived from the image) depths [Pineiro et al., 2008]. We conducted this type of analysis for Rusty Bend and the Footbridge Reach and considered several different calibration approaches and image data types: 1) linear vs. quadratic OBRA for the Snake River; 2) OBRA of convolved field spectra, with or without a correction applied to the image to force better agreement with the field spectra; and 3) panchromatic vs. pan-sharpened WV2 images.
3.1. Optical Characteristics of Gravel Bed Rivers
3.1.1. Water Column Optical Properties
 Ancillary data collected along the Snake River and SBC confirmed our visual impression of exceptional water clarity. Turbidity values were consistently low (2–3 Nephelometric Turbidity Units, or NTU) and suspended sediment concentrations were minimal: 2 mg/L for each of three water samples from the Snake River and one sample from SBC. We characterized the interaction of solar energy with the water column by measuring vertical profiles of downwelling spectral irradiance and calculating values of the diffuse attenuation coefficient Kd(λ) (Figure 4). Data sets from six different dates resulted in similar Kd(λ) spectra that varied little with wavelength through the visible but increased abruptly at 700 nm due to a sharp rise in the absorption coefficient of pure water, denoted by aw(λ), in the NIR. Similarly, the dip in Kd(λ) around 810 nm is associated with a decrease in aw(λ) at this wavelength. Because these streams had only small amounts of suspended sediment or dissolved organic matter, their optical properties were dictated primarily by those of pure water. To illustrate the contrast between these clear-flowing gravel bed rivers and a more turbid sand-bed channel, aKd(λ) spectrum from the Platte River was added to Figure 4 [Legleiter et al., 2011b]. The higher turbidity (49.5 NTU) and suspended sediment concentration (161 mg/L) of the Platte River lead to Kd(λ) values 2–5 times greater than those observed in our study area throughout the visible, with the greatest difference occurring in shorter blue-green wavelengths more susceptible to scattering by suspended sediment. In the NIR, where optical properties were driven primarily by pure water absorption, theKd(λ) spectra for the three rivers converged but remained higher for the Platte.
 The diffuse attenuation coefficient is an apparent optical property influenced by variations in the ambient light field, in addition to the spectral characteristics of the water itself. To isolate the effects of the water on the interaction of light with the river, we made direct measurements of inherent optical properties with an ac-9. In situ observations of the attenuationa(λ) and absorption c(λ) coefficients quantitatively verified the clarity of both streams (Figure 5). Again, the data sets agreed closely with one another, with higher values of a(λ) and c(λ) in the shortest- and longest-wavelength bands and the weakest absorption and attenuation in the green at 555 nm, implying that penetration of solar energy through the river was most efficient at this wavelength. Also included inFigure 5 are the absorption coefficients of pure water and suspended sediment, based on a specified concentration of 2 mg/L (= 2 g/m3) and an optical cross-section included with the HydroLight radiative transfer model [Mobley and Sundman, 2001]. These data are consistent with our measurements of a(λ) for λ > 600 nm, where sediment has little influence on overall water column absorption. For shorter wavelengths more strongly absorbed by suspended mineral matter, the sum of the pure water and suspended sediment absorption coefficients agrees well with our field data, implying that a simple two-component optical model might be a sufficient description of these streams. The difference between the absorption and attenuation coefficients is due to the effects of scattering, primarily by suspended sediment [Legleiter et al., 2011b]. Overall, the relatively simple optical characteristics and great clarity of the Snake River and SBC implied that these channels would be amenable to spectrally-based depth retrieval.
 Our measurements of water column optical properties also allowed us to quantify some of the limitations associated with remote sensing of river bathymetry. Because imaging systems have a finite capacity to detect small changes in the amount of upwelling spectral radiance, truly continuous depth maps cannot be derived from digital image data. Instead, the smallest change in depth a particular sensor can resolve depends on the rate at which light is attenuated by the water column, the amount of radiance reflected from the streambed LB(λ), and the sensor's noise-equivalent delta radiance ΔLN(λ) [Philpot, 1989; Legleiter et al., 2004; Legleiter and Roberts, 2009]. To explore the implications of this important concept, we inserted observed values of Kd(λ) into equation (1) and calculated the smallest detectable change in depth Δd(λ) at a range of initial depths d0; the ratio ΔLN(λ)/LB(λ) was held constant at 0.01 [e.g., Philpot, 1989]. The results of this analysis are illustrated in Figure 6a, based on an irradiance profile from the Snake River; Δd(λ) values for other sites and dates were nearly identical because Kd(λ) values for the various data sets were so similar. For this example, at a wavelength of 700 nm a difference in depth of 0.01 m or less would be detectable at depths up to 0.4 m, and even at a depth of 1 m Δd(λ) remained less than 0.04 m. These calculations imply that if the imaging system is sufficiently sensitive and the bottom is well-illuminated and highly reflective, precise depth estimates could be derived from remotely sensed data. Note, however, that less sensitive instrumentation (i.e., larger ΔLN(λ)), less reflective substrates, and/or smaller amounts of incident radiation (i.e., smaller LB(λ)), would lead to larger Δd(λ) and less precise depth estimates. For example, images acquired at greater solar zenith angles (e.g., early or late in the day, or during the spring or fall) and/or encompassing darker streambed materials (e.g., basalt) will have smaller values of LB(λ) and hence larger values of Δd(λ) for a given sensor.
 The finite sensitivity of imaging systems also dictates that river bathymetry can only be mapped up to some maximum detectable depth. Again, dmax(λ) depends on the optical properties of the water column, summarized in terms of an effective attenuation coefficient 2Kd(λ), and the ratio ΔLN(λ)/LB(λ), where in this case ΔLN(λ), the smallest difference in radiance the imaging system can resolve, is set equal to the difference between the at-sensor radiance and the radiance from optically deep water [Philpot, 1989; Legleiter et al., 2004]. Maximum detectable depths were calculated via equation (2) for a range of ΔLN(λ)/LB(λ) values; the results of this analysis, based on a Kd(λ) spectrum from the Snake River, are summarized in Figure 6b. For ΔLN(λ)/LB(λ) = 0.0001 (i.e., a highly sensitive instrument), dmax(λ) was 10.9 m in the green wavelengths where water column attenuation was weakest. As pure water absorption increased through the red and NIR, dmax(λ) decreased to less than 2 m for λ> 730 nm. For a system with a lower radiometric resolution, corresponding to a two-order of magnitude decrease in ΔLN(λ)/LB(λ) to 0.01 (the value used to calculate the Δd(λ) values in Figure 6a), depths up to 5.45 m could be detected at 560 nm, but dmax(λ) = 0.67 in the NIR at 760 nm. These results implied that a sensor capable of resolving small changes in radiance would be crucial to mapping bathymetry across a broad range of depths. Similarly, multispectral data would allow for selection of bands well-suited for depth retrieval across this range. The maximum detectable depth also would be influenced by the nature of the fluvial environment itself. For a given imaging system (i.e., a fixed ΔLN(λ)), dmax(λ) depends on the bottom contrast between the substrate and water column, implying that highly reflective substrates and/or clear water would favor depth retrieval from deeper channels. Also note that the absolute magnitude of LB(λ) is significant, such that images acquired under less well-illuminated conditions (i.e., early or late in the day, or at high latitudes) would result in smaller values ofdmax(λ). In any case, our field measurements of water column optical properties, together with the analytical framework represented by equations (1) and (2), implied that bathymetry could be mapped remotely with a high degree of precision across the range of depths, typically less than 2.5 m, observed in many gravel bed rivers.
3.1.2. Field Spectroscopy and Relationships Between Reflectance and Water Depth
 A primary objective of this study was to establish quantitative relationships between reflectance and water depth for the two streams we examined. Field spectra measured from above the water surface on the Snake River and SBC were processed following Legleiter et al. [2011b]. Each set of field spectra, together with the corresponding depth measurements, was then used as input to the OBRA algorithm described in Section 2.6. This procedure identified combinations of wavelengths that were sensitive to variations in depth but robust to other factors that might influence reflectance, such as substrate heterogeneity or sun glint from the water surface. The results of this analysis are summarized in Figure 7 using OBRA matrices that represent the strength of the linear relation between d and X for all possible band combinations. Figure 7a indicates that for Rusty Bend, where depths reached up to 3 m, a strong (R2 = 0.887) relation between d and X was obtained for a green numerator band and red denominator band. Moreover, the OBRA matrix shows that a broader range of wavelengths would have resulted in d vs. X relations nearly as strong: numerator bands less than 550 nm resulted in R2 > 0.8 for 575 < λ2 < 720 nm. For this relatively deep reach, the NIR portion of the spectrum was not useful because strong absorption by pure water lead to saturation of the reflectance signal in pools. Similarly, the decrease in predictive power at λ2 = 675 nm was due to chlorophyll absorption by benthic algae present on the streambed; the reduced bottom albedo resulted in a smaller reflectance and a somewhat weaker relation between d and X in this band.
 Results from SBC were similar, but the optimal band ratio produced an even higher regression R2 (0.975) for a pair of longer wavelengths, and a number of other bands would have yielded d vs. X relations nearly as strong (Figure 7b). In contrast to Rusty Bend, NIR wavelengths were effective denominator bands for any numerator λ1 < 730 nm. In this case, the NIR was more useful due to shallower depths along the Footbridge Reach. Strong absorption by pure water at these wavelengths implied that small changes in depth would produce in large changes in reflectance, resulting in very strong d versus X relations. Because depths were so shallow, the saturation that limited the utility of the NIR on the Snake River was less of an issue on SBC. In general, stronger d vs. X relations at longer NIR wavelengths could be expected for shallower streams. Low concentrations of suspended sediment and dissolved organic matter and highly reflective substrates also favor remote bathymetric mapping [Legleiter et al., 2009]. Both rivers examined in this study satisfied these criteria, and analysis of field spectra showed that spectrally-based depth retrieval would not only be feasible but potentially highly accurate.
 An example of this capability is given in Figure 8, which shows data collected on a transect across Rusty Bend. Depths measured by the echo sounder agreed closely with depths calculated from the field spectra using the OBRA relation from Figure 7a. Correspondence between the observed and predicted profiles was best over the shallow bar surface on the right side of the channel (Figure 8, left). Through the middle of the stream, the OBRA relation resulted in slight over-predictions of depth, and the spectrally-based estimates were shallower than the echo sounder data in the pool along the outer bank. These discrepancies were small, however, on the order of 0.2–0.4 m, and this cross-section illustrated the robust performance of the OBRA relation across a range of depths up to 2.75 m. This level of accuracy was noteworthy due to the pronounced differences in bottom reflectance along this transect (Figure 2c). Whereas most of the cross-section consisted of a gravel substrate with some degree of algal coating, several large blocks of clay bedrock were located near the outer bank. This cohesive material was noticeably lighter-colored than the surrounding gravel and resulted in large spikes in reflectance in this part of the channel, as shown inFigure 8for the OBRA denominator band. The OBRA-based depth estimates were shallower at these locations but not to the degree that the abrupt increases in reflectance might seem to dictate. These results thus confirmed that OBRA enabled effective depth retrieval in the presence of highly heterogeneous substrates, cited byLegleiter and Roberts  as one of the primary advantages of this technique. This study supported this conclusion in the context of a deeper river with more variable bottom reflectance than had been considered previously.
 These results, though encouraging, were derived from field spectra that provided essentially continuous reflectance data with a 1 nm sampling interval. To assess whether strong relations between d and X could be derived from image data that integrate reflectance over a smaller number of broader bands, we convolved the original field spectra to match the WV2 sensor response function, as described in Section 2.6 and illustrated in Figures 3a and 3b. The convolved spectra consisted of eight discrete bands and were subjected to OBRA to quantify the extent to which reduced spectral resolution might compromise the ability to retrieve bathymetry from satellite images. The results of this analysis are summarized in Figure 9a, which indicates only a slight decrease in the OBRA R2 to 0.839. For the convolved spectra, the optimal WV2 band combination was a green numerator and yellow denominator, essentially the same wavelengths as for the original field spectra. The OBRA matrix also indicated that the blue band would have been an effective numerator with either the yellow or red band as a denominator. These results implied that the sensor's broader bands contained sufficient spectral information for inferring depth and provided further evidence of the feasibility of mapping river bathymetry from space.
3.2. Mapping River Bathymetry From Satellite Image Data
 Our in situ measurements of water column optical properties, together with visual inspection of WV2 images, implied that flow depths could be inferred from satellite data. In this section, we report depth retrieval results from the Snake River and SBC, with a focus on accuracy assessment of depth estimates obtained using various calibration approaches and image data types.
3.2.1. Snake River
184.108.40.206. Depth Retrieval Calibration Approaches
 The most direct means of mapping bathymetry from remotely sensed data involved extracting image spectra from the locations of field-based depth measurements and using these data to calibrate a relationship betweendand the image-derived quantityX defined by equation (3). The Snake River posed a challenging test of this approach because flow depths ranged up to 3 m, far greater than the shallow streams that have been the subject of our prior remote sensing investigations [Legleiter et al., 2009, 2011a]. In larger gravel bed rivers, saturation of the radiance signal might lead to a non-linear relationship betweend and X and could preclude depth retrieval from pools. To account for this possibility we performed both linear OBRA, with a single X term in the regression against field measurements of d, and a quadratic OBRA in which an X2 term also was included in the regression, based on a similar study by Dierssen et al. . The resulting OBRA matrices are presented in Figures 9b and 9c, which indicate that the same band combination, a blue numerator with a yellow denominator, was optimal for both linear and quadratic formulations. The simple d vs. X regression yielded a strong (R2 = 0.827) linear relationship between depth and the band ratio, but adding an X2 term only slightly increased predictive power (R2 = 0.864). Closer inspection of the OBRA matrices indicated similar spectral patterns, but the NIR bands became more useful when the X2term was included. To an extent, allowing for a non-lineard vs. X relation accounted for saturation of the radiance signal in deeper water in the NIR. Nevertheless, the marginal improvement provided by the quadratic term implied that the additional complexity of this approach was not justified.
 Bathymetric maps generated from the WV2 satellite image of Rusty Bend using linear and quadratic OBRA relations are shown in Figures 10a and 10b, respectively. The maps were very similar to one another, with the linear formulation yielding slightly shallower depth estimates in the pool along the outer bank and over the bar surface on the right (north) side of the channel, relative to the quadratic OBRA. Both techniques produced realistic depictions of the gross morphology, with an asymmetric cross-sectional shape extending from above the apex through the middle of the bend before the thalweg shifted toward the right at the lower end of the reach. Depth retrieval accuracy was assessed via comparison with pixel-scale mean depths obtained by OBK of the field survey data. The resulting residual maps are shown inFigures 11a and 11b, where negative residuals (bright red tones) represent over-predictions of depth and positive residuals (darker blue tones) represent under-predictions of depth. Again, spatial patterns for the two versions of OBRA were similar, with the lineard vs. X relation leading to more extensive underestimates of depth along the outer bank in the upper end of the reach and a greater number of overestimates along the left bank past the bend apex. Most depth retrieval residuals were on the order of 0.25 m but locally ranged as high as 0.6 m, nearly half the mean depth of 1.23 m, in the thalweg above the apex and on the outer bank through the lower portion of Rusty Bend.
 For the most part, depth estimates from both linear and quadratic OBRA were reliable, as indicated by high R2 values for observed vs. predicted (OP) regressions based on field measurements set aside for validation (Figure 12 and Table 2). Moreover, small mean errors and OP intercept and slope values near 0 and 1, respectively, implied that depth estimates were unbiased on average. The OP regression plot in Figure 12arevealed a curved trend for linear OBRA, however, with systematic under-prediction of depth in both shallow and deep water. This trend arose from the non-linearity of the relationship betweend and X and was effectively removed by including an X2 term in the OBRA regression (Figure 12b). Nevertheless, the depth retrieval RMSE and OP regression standard errors were only slightly less for the quadratic vs. linear OBRA, and a typical error of 0.22 m would be less than 20% of the reach-averaged mean depth. Because the quadratic formulation did not significantly improve accuracy, the standard linear OBRA appeared to be well-suited for bathymetric mapping in larger, deeper gravel bed rivers such as the Snake, although anX2 term could prove useful in streams with greater depths.
Table 2. Depth Retrieval Accuracy Assessment for Various Calibration Methods and Image Types
 An alternative strategy for remote mapping of river bathymetry would involve making direct measurements of depth and reflectance and using field spectra to develop d versus X relations that could then be applied to remotely sensed data. If the spectral response function of the sensor were known, the field spectra could be convolved to match the instrument's band passes, as shown in Figures 3a and 3b for the WV2 satellite. Ideally, OBRA of the convolved field spectra would result in a calibrated relationship between d and X that could be applied directly to images of the river from which the field spectra were collected and potentially other streams as well. To assess the feasibility of this approach, we measured depth and reflectance on transects across Rusty Bend, convolved the spectra to the WV2 bands, and performed OBRA. This analysis is summarized in Figure 9a, which indicated a strong linear relation (R2 = 0.839) between d and X, defined as the logarithm of the ratio of the green and red bands. This R2 value was nearly as high as that associated with the original field spectra (Figure 7a) and slightly better than that associated with the linear OBRA of image spectra (Figure 9b). Applying the OBRA relation from Figure 9a to the WV2 image resulted in the bathymetric map in Figure 10c, which has the same color scaling as the maps produced via OBRA of image spectra. A comparison of these maps indicated that the pool along the outer bank was not well-resolved by depth estimates based on convolved field spectra, and depths on the shallow bar surface tended to be over-predicted. This pattern was evident in the residual map inFigure 11c, which featured underestimates on the order of 0.6 m in the pool above the bend apex and overestimates of similar magnitude at the lower end of the reach. The OP regression for these depth estimates yielded a lower R2 value of 0.75, a large negative intercept term, and a slope much greater than 1 (Figure 12c), implying that predictions based on convolved field spectra were biased.
 In an effort to account for this bias, we adjusted the image data to better match the field spectra as described in Section 2.6 and illustrated in Figure 3. This adjustment, which could be considered a simple means of radiometric calibration and atmospheric correction, served to modify the image so that the d vs. X relation derived from the convolved field spectra (Figure 9a) could be applied directly to the WV2 scene. The resulting bathymetric map, shown in Figure 9d, featured greater depth estimates than did the original, uncorrected image but still failed to resolve the full depth of the pool along the outer bank. On the opposite side of the channel, depths tended to be under-predicted, including some negative estimates. The spatial pattern of these errors is illustrated inFigure 11d, which shows that even after modifying the image data to better match the convolved field spectra, pool depths were underestimated by 0.5 m or more throughout much of the bend; overestimates of 0.5 m or more occurred at the lower end of the reach. Accuracy assessment using the validation subset of the field survey resulted in OP regression intercept and slope values closer to 0 and 1, respectively, implying that the image correction was effective in removing the bias associated with calibration based on field spectra, but the OP R2 improved only marginally, from 0.75 to 0.77. The RMSE and OP regression standard errors for both depth retrieval methods based on convolved field spectra were greater than those for OBRA of image spectra; typical errors on the order of 0.3 m would be approximately 25% of the mean depth for the reach. These results imply that calibration via field spectroscopy was a plausible strategy, especially when the image was corrected to better match the field spectra, but this approach was less reliable than OBRA of image spectra.
220.127.116.11. Image Data Types
 Although depth retrieval via OBRA requires multiple spectral bands, bathymetric maps also can be generated from single-band or panchromatic gray scale images using the linear transform algorithm introduced byLyzenga . In the context of satellite imagery, this capability is potentially significant because many spaceborne sensors feature a panchromatic band with a greater spatial resolution than the individual multispectral bands; for the WV2 system, the pixel sizes for the panchromatic and multispectral images are 0.5 m and 2 m, respectively. If spectral information is not essential for accurate depth retrieval, application of the linear transform (equation (4)) might enable more effective mapping of smaller streams and enhanced spatial detail in larger rivers. If spectral information does prove critical, a hybrid approach based on a pan-sharpened image that features the high spatial resolution of the panchromatic image but also incorporates the multispectral data could prove to be most useful.
 We explored this possibility and evaluated the relative significance of spectral and spatial resolution by producing bathymetric maps from 0.5 m-pixel panchromatic and pan-sharpened WV2 images of the Snake River. These maps are included inFigures 10e and 10f, along with the bathymetry inferred from the original, 2 m-pixel multispectral images. For the panchromatic scene, depth retrieval via the linear transform algorithm captured the general morphology of the reach but the map had a grainy appearance due to high-frequency noise that persisted despite the application of a smoothing filter. The area of deeper flow along the outer bank was narrower in the bathymetric map derived from the panchromatic data than in the map based on the original multispectral image (Figure 10a), suggesting that spectral information was needed to resolve greater depths. The corresponding residual map (Figure 11e) highlighted this pattern, with depth underestimates on the order of 0.6 m extending across a greater fraction of the channel width over the upper half of the reach. Past the bend apex, depths were over-predicted in a large area along the left side of the channel. Accuracy assessment of the linear transform-derived depths indicated a weaker, but still fairly strong agreement between observations and predictions (R2 = 0.67), and the OP regression equation did not imply any systematic bias. Closer examination of Figure 12erevealed that this agreement deteriorated considerably in deeper water, with the image-derived estimates failing to increase as rapidly as the field-surveyed depths ford > 1 m.
 The bathymetric map derived from the pan-sharpened image (Figure 10f) included a broader area of deep water along the outer bank than did the map produced from the panchromatic image, suggesting that the additional spectral information resulted in a more accurate representation of the thalweg. The grainy texture of the panchromatic image persisted in the pan-sharpened data, however, and appeared to be even more pronounced over the point bar on the right side of the channel at the lower end of the reach. The residual map inFigure 11fwas similar to that associated with the panchromatic image, but the large underestimates in the upper portion of the bend were less extensive. Accuracy assessment of depth estimates from the pan-sharpened multispectral image, which had a pixel edge dimension four times smaller than the original multispectral data, yielded an OP regressionR2 of 0.80 (Figure 12f), which was a significant improvement over the panchromatic image but was less than that associated with the 2 m-pixel multispectral image. This result implied that greater spectral information content was more important than enhanced spatial detail for depth retrieval from this relatively large gravel bed river. The noise introduced by fusing the panchromatic image with the multispectral data also might have contributed to the greater depth retrieval errors associated with the pan-sharpened image. In any case, our analysis suggested that the additional image processing required to generate the pan-sharpened image was not justified and that depth retrieval from the original multispectral images might be both simpler and more accurate in this setting.
3.2.2. Soda Butte Creek
18.104.22.168. Depth Retrieval Calibration Approaches
 The Footbridge Reach presented a different type of challenge for remote sensing of river bathymetry: for this small gravel bed stream, the pixel size was a significant fraction of the mean channel width, particularly under the low-flow conditions when the WV2 image was acquired. Moreover, at this discharge, SBC split into two separate channels, each of which was further divided by mid-channel bars. As a result, the flow was quite shallow (Table 1) and deriving information on channel form thus required an imaging system capable of resolving subtle variations in depth. Based on the results for the Snake River, and because SBC was so much shallower, we focused on the standard linear formulation of the OBRA algorithm and did not pursue the quadratic alternative. Similarly, because our analysis of convolved field spectra from the Snake River indicated that depth estimates would be biased unless the image data were adjusted to match the field spectra, we considered only the latter approach for SBC.
 OBRA results obtained using spectra extracted from the WV2 image or measured directly in the field are shown in Figures 13a and 13b, respectively. For the image spectra, the highest d vs. X regression R2 occurred for a green numerator and NIR numerator (the sensor's ‘red edge’ band) but was only 0.585, significantly less than the 0.827 R2 on the Snake River. The OBRA matrix in Figure 13a indicated that this NIR band was by far the most useful denominator wavelength with any visible numerator; other denominator bands produced much weaker d vs. X relations. For the convolved field spectra, the OBRA results were more encouraging, with an optimal R2 value of 0.962 for a red numerator band and the same 725 nm denominator as for the image spectra. Unlike the image spectra, however, a broader range of bands produced strong, linear relationships between d and X, including longer NIR wavelengths. This result was expected because strong absorption by pure water in the NIR caused small changes in depth to be expressed as relatively large changes in reflectance; because depths in this reach were so shallow, saturation of the NIR reflectance signal was less of an issue than along the deeper Snake River. OBRA of the two Footbridge Reach data sets suggested that bathymetric mapping based on field spectra might be more effective than extracting spectra directly from an image. Whereas many of the image spectra were from mixed pixels contaminated by radiance from adjacent terrestrial features, each of the field spectra sampled a smaller area entirely within the wetted channel. Atmospheric effects present in the satellite image data but absent from the field measurements also might have contributed to the inferior OBRA results for the image spectra.
 Bathymetric maps produced using these two calibration methods are presented in Figures 14a and 14b, both of which have a common color scale with an upper limit set to the maximum depth observed in the field. In general, both maps captured the gross morphology of the reach, with deeper flow along the left (east) bank at the upper end of our study area, in a pool on the far right channel near the entrance to the bend, in the lower half of the left chute channel, and at the bottom of the reach where the channels converge and the stream enters a curve to the right. Aside from these areas, depths were very shallow, on the order of 0.2 m. Although both OBRA of image spectra and application of a d vs. X calibration relation derived from field spectra to the corrected WV2 image produced reasonable spatial patterns, the absolute magnitudes of the depth estimates were less reliable, especially for calibration based on field spectra. The bathymetric map shown in Figure 14b exceeded the specified color limits at both the shallow and deep ends of the spectrum, resulting in the dark red tones most notable over the point bar near the apex and the dark blue tones at the upper and lower ends of the reach. Depths estimated from image spectra, in contrast, fell within the range of observed depths and were more accurate.
 Differences between the two approaches also were highlighted by the residual maps shown in Figures 15a and 15b. For the image-based algorithm, most of the residuals were near zero, with a tendency to over-predict depth in shallow areas and under-predict in pools. When the OBRA relation from the field spectra was applied to the corrected image, the overestimates of depth in shallow riffles and bars were more pronounced, as was the under-prediction in the left chute channel and along the outer banks at both the upper and lower ends of the reach. Large positive residuals near steep cutbanks suggested that the presence of both bright terrestrial features and deep water within mixed pixels might have lead to unreliable depth estimates along channel margins and that sensor spatial resolution might have been a limiting factor for this small stream. Similarly, the inability of either method to detect the deep pool in the middle of the left chute channel, evident as a dark blue area on the residual maps, also implied that the 2 m pixel size might not have been adequate for mapping SBC under low-flow conditions.
 The accuracy assessment summarized in Figures 16a and 16b yielded further insight regarding discrepancies between bathymetric maps produced from field vs. image spectra. Regression of observed vs. predicted depths yielded a stronger correlation (R2 = 0.59) for depth retrieval based on image spectra than when using the convolved field spectra for calibration, which resulted in an R2of only 0.45. More importantly, however, the image-based estimates were unbiased, with an OP regression intercept near 0 and slope near 1. In contrast, depth estimates derived using the calibration relation for the convolved field spectra resulted in a positive intercept term nearly as large as the mean depth for the reach and a much smaller slope, implying biased estimates. This bias lead to over-predictions of depth in shallow areas and under-predictions in deep areas and persisted despite the correction applied to the image to force closer agreement with the field spectra. Although the simple image adjustment outlined inSection 2.6allowed for reasonably accurate depth retrieval for the Snake River, this technique was not effective on SBC, implying that more sophisticated radiometric calibration and atmospheric correction might be required before relationships derived from field spectra can be applied to remotely sensed images. Another factor that might have contributed to biased estimates was the use of a near-infrared band that might have been more strongly affected by residual atmospheric effects not fully accounted for in the reflectance retrieval algorithm used by DigitalGlobe.
22.214.171.124. Image Data Types
 Given the small size of SBC and the modest depth retrieval performance of the 2 m-pixel WV2 data, we evaluated whether images with greater spatial resolution might prove more useful for bathymetric mapping. Depth maps produced from 0.5 m-pixel panchromatic and pan-sharpened images are shown inFigures 14c and 14d, with the same color scaling as used for the maps generated from the coarser-resolution data. Overall spatial patterns were similar, but the smaller pixel size revealed some details of the morphology that were not evident inFigures 14a and 14b. For example, the depth map generated by applying the linear transform to the panchromatic image captured shoaling of the flow onto the point bar in the middle of the three channels present at the bend apex, a feature that was not resolved by any of the other image types. The linear transform failed to detect the full depth of pools located in the left chute channel and at the lower end of the reach, however. The pan-sharpened multispectral image did not provide quite as much detail but yielded greater depth estimates in the pools than did the panchromatic data. The residual maps inFigure 15indicated that depth retrieval errors from the higher resolution images were of a similar magnitude as those associated with the original multispectral image. Over-predictions of depth in the riffle at the upper end of the reach and approaching the bar at the bend apex and under-predictions of depth throughout the lower portion of the reach were salient in the residual maps for both the panchromatic and pan-sharpened images. The accuracy assessment summarized inFigures 16c and 16dindicated that depth estimates were unbiased in both cases but were more reliable for the pan-sharpened than for the panchromatic image. OP regressionR2 values of 0.50 and 0.41, respectively, were significantly less than the 0.59 obtained using image spectra from the original multispectral scene. Typical errors of 0.11 m were half the mean flow depth for the reach. These results imply that even in a smaller stream such as SBC, the additional spatial detail provided by the panchromatic image did not translate into improved depth retrieval performance. Instead, spectral information was essential for reliable bathymetric mapping.
 Efforts to characterize the complex interactions among flow, morphology, and sediment transport that shape alluvial river channels have often been compromised by the difficulty of acquiring basic measurements of channel form. As a result, field studies typically have examined only short reaches in isolation. Important, increasingly interdisciplinary research opportunities thus exist at larger segment and watershed scales [e.g., Fausch et al., 2002; Carbonneau et al., 2011]. For example, geomorphologists might seek to examine how the detailed process mechanics documented through reach-scale studies self-organize within a catchment to create emergent patterns in channels and landscapes that have tended to be described only in conceptual or theoretical terms [e.g.,Church, 2002; Benda et al., 2004]. Similarly, greater insight on the distribution of in-stream habitat within a watershed would help biologists to understand the movement of fish throughout their life histories [e.g.,Ganio et al., 2005]. In an applied context, a more synoptic perspective on fluvial systems would help to provide a holistic, integrated context for the planning, implementation, and monitoring of river restoration projects [e.g., Beechie et al., 2010; Downs et al., 2011]. For all of these purposes, the lack of an efficient means of mapping river morphology has impeded progress and the development of new techniques could thus stimulate significant advances in each of these fields [Fonstad and Marcus, 2010]. Motivated by this prospect, this investigation demonstrated the feasibility of mapping the bathymetry of clear-flowing gravel bed rivers from satellite image data. Using field measurements and WV2 imagery from two streams in the northern Rocky Mountains, we showed that information on flow depth can be retrieved from multispectral data across a range of channel sizes from approximately 20–60 m in width and 0.2–1.25 m in mean depth. A simple band ratio-based algorithm, calibrated using either spectra extracted from the image or measured directly in the field, provided reliable depth estimates. The results of this study thus imply that satellite-based mapping of river bathymetry could become a viable tool for river research and management.
4.1. Constraints on Spectrally-Based Bathymetric Mapping
 Although the potential for remote sensing methods to contribute to the riverine sciences is justifiable cause for excitement [Marcus and Fonstad, 2010], we advocate a cautious approach that acknowledges the constraints as well as the capabilities associated with this new technology. The retrieval of water depth from passive optical image data, for example, is subject to some important caveats. The most significant constraint on the spectrally-based approach is the limited range of stream conditions under which bathymetry can be mapped reliably. For rivers having greater depths (on the order of several meters) and/or more turbid water due to higher amounts of suspended sediment and/or organic matter, accurate depth estimates are less likely, and much of the channel might exceed the maximum detectable depth [e.g.,Legleiter et al., 2011b, 2011a]. Overhanging vegetation, shadows, mixed pixels along channel margins, dark substrates (resulting in low LB(λ) values), sun glint from the water surface, and hazy atmospheric conditions can also compromise, if not preclude, effective depth retrieval [Legleiter et al., 2009].
 In addition to these environmental factors, the extent to which an image data set can satisfy the information requirements of a particular investigation also depends on the sensor itself. For example, the imaging system must have sufficient spatial resolution to detect the channel features of interest. Similarly, instruments with greater spectral resolution might enable more accurate bathymetric mapping by providing radiance measurements in a larger number of narrower wavelength bands, some of which could be highly responsive to changes in depth. Radiometric resolution refers to a sensor's ability to detect subtle differences in radiance, such as those associated with small variations in water depth, and thus exerts a primary control on both the precision of depth estimates and the maximum detectable depth. Forward image modeling, which involves simulating an image “from the streambed up” given information on the morphology and optical characteristics of the channel and the technical specifications of the sensor, provides a means of determining a priori the accuracy, precision, and dynamic range of image-derived depth estimates. This approach allows tradeoffs among spatial, spectral, and radiometric resolution to be evaluated in the context of a specific river of interest [Legleiter and Roberts, 2009].
 In this study, rather than resort to modeling, we made direct measurements of water column attenuation in a pair of gravel bed rivers and used these data to quantify the limitations of spectrally-based depth retrieval. Vertical profiles of downwelling spectral irradiance were used to calculate values of the diffuse attenuation coefficientKd(λ) that were in turn used to determine the smallest detectable change in depth and maximum detectable depth via the theoretical framework established by Philpot ; this analysis was performed for hypothetical imaging systems with specified levels of radiometric resolution. Our results indicated that for these clear-flowing streams, a typical instrument with a ΔLN(λ)/LB(λ) ratio of 0.01 would yield depth estimates with a precision on the order of 0.01–0.04 m and a maximum detectable depth ranging from 5 m in the green portion of the visible spectrum to 1.3 m in the NIR. More sensitive instrumentation would yield more precise estimates and increase the dynamic range of depth retrieval, but, conversely, less sophisticated sensors would provide estimates subject to greater uncertainty over a more restricted range of depths. These theoretical calculations were consistent with our analysis of actual satellite image data and corresponding field-based depth measurements. Typical errors of 0.238 and 0.094 m for the larger and smaller of our study streams, respectively, corresponded to 20% and 42% of the mean flow depths for the two reaches examined in detail. Depth estimates calibrated using field measurements representative of the overall distribution of depths within each reach were less reliable in deeper water, so the maximum detectable depth was not as well-constrained by our data, but pools over 2.5 m deep were captured by satellite-based bathymetric maps. A combination of in situ observations of water column optical properties, theoretical calculations, and careful accuracy assessment will be most effective in defining the constraints associated with remote sensing of river bathymetry.
4.2. Alternative Approaches to Remote Measurement of River Morphology
 This study focused on retrieving water depth from passive optical image data using a single, relatively simple band ratio algorithm, but several alternative strategies for remote sensing of river morphology are available as well. For example, the same type of forward image modeling described in Section 4.1forms the basis of more advanced spectrum-matching methods now favored by the coastal research community for mapping depth, bottom composition, and concentrations of various optically significant constituents of the water column [e.g.,Lee et al., 1999; Mobley et al., 2005]. In theory, these techniques could be adapted for application to riverine environments, but this prospect has yet to be explored. Implementing such an approach would require additional data on the spectral characteristics of different fluvial substrates and water types, as well as significant modeling effort. In this study, we made some initial progress toward this goal by making field measurements of reflectance and apparent and inherent optical properties of the water column for a pair of clear-flowing gravel bed rivers. If a suitable database can be developed, this physics-based approach could allow for greater generality and thus eliminate the need to coordinate remotely sensed data collection with field measurements to establish relationships between image-derived quantities and observed flow depths. Continued reliance upon this type of empirical calibration, which involves regressing in situ depth measurements against image pixel values, would undermine one of the principal advantages of remote sensing.
 A complementary technology for characterizing river morphology is light detection and ranging, or LiDAR. Although LiDAR has become a preferred method of measuring topography [Slatton et al., 2007], typical LiDAR systems cannot measure submerged bed elevations because water strongly absorbs the NIR laser pulses emitted by these instruments. To obtain a complete, hybrid representation of the fluvial environment, LiDAR topography from exposed bars and floodplains can be combined with channel bathymetry derived from passive optical image data, although such data fusion involves various technical challenges [Legleiter, 2012a]. Newly developed, water-penetrating green LiDAR systems provide a more direct solution for measuring both subaerial and submerged surfaces [e.g.,McKean et al., 2008]. Originally developed for coastal applications, these bathymetric LiDARs provide a relatively large laser spot size and low point density and tend to over-estimate bed elevations in riverine settings [Kinzel et al., 2007]. In addition, laser returns from the water surface, water column, and streambed can be difficult to distinguish in shallow channels [Kinzel et al., 2012]. Thus, although green LiDAR has outstanding potential for measuring riverine topography, these systems remain experimental and have yet to achieve operational status as a viable monitoring tool.
 Passive optical remote sensing, in contrast, has become routine, with image data acquired on a regular basis and made freely available to the public. For example, aerial photography acquired through the National Agriculture Imagery Program is accessible online and has been shown to provide reasonably accurate bathymetric information for clear-flowing gravel bed rivers. The low radiometric resolution of these basic image data resulted in saturation of the radiance signal, however, and the full depth of pools could not be detected [Legleiter, 2012c]. For focused scientific investigations or any application for which precise estimates across a broad range of depths are necessary, acquiring task-specific data with more advanced multi or hyperspectral sensors might be more appropriate. This type of airborne data collection requires careful planning and can be complicated by a number of different factors. In our experience, logistical coordination of pilots, planes, and instrumentation can prove difficult, particularly if image acquisition is to be synchronized with field-based measurements for purposes of calibration and validation. Weather, mechanical problems, and other unforeseen circumstances can derail even the most well-thought out missions. In sensitive areas, such as the National Parks where we conducted this investigation, flight permits must be secured and restrictions on flying height above terrain can compromise spatial resolution and sensor signal-to-noise. For larger study areas or channels that do not follow a straight course, multiple flight lines might be needed, resulting in multiple images that must be geo-referenced and mosaicked and might not be consistent with one another in terms of atmospheric conditions and viewing and illumination geometry. For these reasons, airborne data might not be optimal for remote sensing of rivers.
4.3. Potential Advantages of Mapping River Bathymetry From Space
 Satellite imagery could prove to be a viable alternative with several distinct advantages. A number of commercial satellites, such as QuickBird, GeoEye, and WorldView, now provide the kind of spatial resolution required to map small- to medium-sized gravel bed rivers, with pixel sizes of 2 m or less for multispectral images and 0.5 m for panchromatic images. These sensors also feature off-nadir pointing capabilities that provide greater flexibility for tasked data collection during an acquisition window specified by the user. Because the platform is already in orbit, logistical problems associated with remotely sensed data collection are less likely and obtaining flight permits is not an issue; planning can instead focus on field activities. Satellite images typically encompass a larger area than could be acquired along an aerial flight line, so longer river segments or even entire watersheds can be captured in a single scene with uniform radiometry, rather than a mosaic of multiple flight strips. For cases where the entire image is not relevant, commercial providers often market data on a per km2basis to match the user's area of interest, so satellite imagery might be more cost-effective as well. Because these data products often are geo-referenced and atmospherically corrected, satellite images can be used as delivered, without the need for extensive pre-processing that could require specialized software and expertise. In addition, obtaining repeat coverage to characterize channel change is more feasible because orbiting satellites can be tasked on an as-needed basis in response to a geomorphically significant event, whereas acquiring post-flood aerial data would involve a second round of complex logistical arrangements. For application-oriented users more concerned with information derived via remote sensing and less interested in the details of flight planning and image processing, satellite data might provide a simpler, more consistent solution.
 Efforts to characterize and understand the morphology and dynamics of alluvial river channels are often compromised by the difficulty of measuring their form and behavior via conventional, ground-based field methods. This study explored the potential to map river bathymetry from passive optical images acquired from spaceborne satellite platforms that provide high resolution data, such as the 2 m-pixel WorldView2 images evaluated here. Our results indicate that water depths in clear-flowing, mid- to large-sized gravel bed rivers with depths <3 m and widths <60 m can be estimated reliably from multispectral data. Direct measurements of water column optical properties were used to quantify some of the key constraints associated with such spectrally-based depth retrieval. For typical levels of sensor radiometric resolution, the smallest detectable change in depth was calculated to be on the order of 0.01–0.04 m and the maximum detectable depth to vary with wavelength from >5 m for green bands to <2 m in the NIR. A simple, band ratio-based algorithm for selecting appropriate combinations of wavelengths and calibrating relationships between field measurements of depth and image-derived quantities was shown to be effective when applied to spectra extracted from an image or recorded directly in the field, although a different pair of bands was selected for each of the streams examined. Adding a quadratic term to these relationships provided only a marginal improvement in bathymetric accuracy in a deeper river. Similarly, neither panchromatic nor pan-sharpened multispectral images with greater spatial resolution yielded more accurate depth estimates, even in a smaller stream for which the pixel size was 10% of the mean channel width. These results implied that spectral information content was crucial to reliable depth retrieval. The principal conclusion of this investigation is that, under appropriate circumstances that include clear water conditions and shallow to moderate depths, river bathymetry can be mapped from space, provided that field measurements of depth are available for calibration. By exploiting this capability, while also acknowledging the limitations associated with remote sensing, scientists might gain novel insight on the dynamics of certain types of fluvial systems.
 This investigation was supported by a grant from the Office of Naval Research Littoral Geosciences and Optics Program (grant N000141010873). The National Park Service granted permission to conduct research in Yellowstone and Grand Teton National Parks. The University of Wyoming-National Park Service Research Center and the Yellowstone Ecological Research Center provided logistical support. The ac-9 instrument was borrowed from the Naval Research Laboratory and the United States Geological Survey loaned suspended sediment sampling equipment and processed water samples. Gregory Miecznik of DigitalGlobe coordinated acquisition of the WorldView-2 images used in this study. C.L. Rawlins and Floyd Legleiter assisted with field work. The editor, associate editor, and three anonymous reviewers provided useful comments on an earlier version of this paper.