PUBLICATIONS
Water Resources Research
RESEARCH ARTICLE
10.1002/2014WR016624
Key Points:
New algorithms enable depth
retrieval without field data for
calibration
Image data linked to channel
hydraulics through a flow resistance
equation
Provides bathymetric accuracy
comparable to direct, field-based
calibration
Supporting Information:
Software S1
Supporting Information S1
Correspondence to:
C. J. Legleiter,
[email protected]
Citation:
Legleiter, C. J. (2015), Calibrating
remotely sensed river bathymetry in
the absence of field measurements:
Flow REsistance Equation-Based
Imaging of River Depths (FREEBIRD),
Water Resour. Res., 51, 2865–2884,
doi:10.1002/2014WR016624.
Received 4 NOV 2014
Accepted 25 MAR 2015
Accepted article online 28 MAR 2015
Published online 26 APR 2015
Calibrating remotely sensed river bathymetry in the absence of
field measurements: Flow REsistance Equation-Based Imaging
of River Depths (FREEBIRD)
Carl J. Legleiter1
1
Department of Geography, University of Wyoming, Laramie, Wyoming, USA
Abstract Remote sensing could enable high-resolution mapping of long river segments, but realizing
this potential will require new methods for inferring channel bathymetry from passive optical image data
without using field measurements for calibration. As an alternative to regression-based approaches, this
study introduces a novel framework for Flow REsistance Equation-Based Imaging of River Depths (FREEBIRD). This technique allows for depth retrieval in the absence of field data by linking a linear relation
between an image-derived quantity X and depth d to basic equations of open channel flow: continuity and
flow resistance. One FREEBIRD algorithm takes as input an estimate of the channel aspect (width/depth)
ratio A and a series of cross-sections extracted from the image and returns the coefficients of the X versus d
relation. A second algorithm calibrates this relation so as to match a known discharge Q. As an initial test of
FREEBIRD, these procedures were applied to panchromatic satellite imagery and publicly available aerial
photography of a clear-flowing gravel-bed river. Accuracy assessment based on independent field surveys
indicated that depth retrieval performance was comparable to that achieved by direct, field-based calibration methods. Sensitivity analyses suggested that FREEBIRD output was not heavily influenced by misspecification of A or Q, or by selection of other input parameters. By eliminating the need for simultaneous field
data collection, these methods create new possibilities for large-scale river monitoring and analysis of channel change, subject to the important caveat that the underlying relationship between X and d must be reasonably strong.
1. Introduction
Remote sensing has emerged as an efficient means of measuring river bathymetry at high spatial resolution
over segment scales in certain circumstances [Marcus and Fonstad, 2010]. Under appropriate conditions, primarily shallow depth and clear water [Legleiter et al., 2009], channel morphology, and in-stream habitat can
be mapped by relating flow depth d to an image-derived quantity X. An important limitation of current
methods of inferring river bathymetry from passive optical image data is the need for field measurements
of depth to calibrate this X versus d relation. Typically, pixel values are regressed against in situ depths to
estimate the coefficients of a linear equation linking imagery to bathymetry
d5b0 1b1 X
where the intercept b0 and slope b1 comprise the coefficient vector b5½b0
C 2015. American Geophysical Union.
V
All Rights Reserved.
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(1)
b1 0 .
This method, though simple and widely applied [e.g., Winterbottom and Gilvear, 1997; Lejot et al., 2007; Williams et al., 2014], is subject to a number of disadvantages. First, field measurements ideally would be
obtained at the same time the image is acquired, particularly in dynamic channels and/or where changes in
flow stage are significant. If field and image data are not collected simultaneously, changes in depth, if not
the morphology of the channel itself, can result in biased estimates of b. Coordinating field crews with flight
operations involves a number of logistical challenges, however, and perfect timing is difficult to achieve in
practice. Second, the X versus d coefficients obtained via regression against field measurements are scenespecific and might not be applicable to other data sets, particularly if the images have not been radiometrically calibrated. Third, pairing field-based measurements collected at discrete points with specific image pixels requires precise georeferencing; any positioning error will cause incorrect associations of pixel values
with in situ depth observations. A more subtle but related issue is that of scale, as depths measured in the
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field with conventional surveying equipment pertain to points, whereas image pixels represent some larger
area. Ideally, the X versus d relation would be established using pixel-scale mean depths, which can be
derived from point measurements via simple spatial averaging or more advanced geostatistical upscaling
techniques [Bailly et al., 2010; Legleiter and Overstreet, 2012]. Similarly, the sampling strategy employed in
the field dictates the range of depths included in the calibration data set, implying that a lack of measurements from the deepest portions of the channel could lead to systematic underprediction of depth. Conversely, failure to survey very small depths in the field could result in negative depth estimates along
shallow channel margins.
Finally, and most importantly, calibration by regression is only feasible when field measurements are available, which often precludes application of this technique to remotely sensed data acquired through monitoring programs such as the National Agricultural Imagery Program (NAIP), obtained by high-resolution
commercial satellites (e.g., WorldView2), or compiled in historical archives. Moreover, this type of empirical
calibration undermines some of the main advantages of remote sensing: the ability to collect data from
inaccessible areas and the capacity to map long river segments in a matter of minutes rather than weeks, as
might be required by a field crew. Continued reliance upon simple, regression-based approaches will
impede the scientific advances that might be made via remote sensing of rivers and inhibit the application
of these methods to resource management. To enable further progress toward these goals, a means of
deriving depth information from remotely sensed data in the absence of field measurements is needed.
A more practical, river-specific approach to calibrating remotely sensed bathymetry incorporates principles
of open channel flow. Fonstad and Marcus [2005] introduced a pair of algorithms for hydraulically assisted
bathymetry (HAB) that enable depth retrieval from passive optical image data when simultaneous field data
are not available. Instead, the HAB-1 technique combines the discharge Q recorded at a gage located within
the scene, widths measured for a series of cross-sections extracted from the image, and channel bed slopes
measured from maps or digital elevation models with the continuity equation for Q, Manning’s flow resistance equation, and an assumed frequency distribution of depths to estimate minimum, mean, and maximum depths for each cross-section (XS). Further assuming that pixel brightness is inversely related to d, the
minimum, mean, and maximum depths calculated for each XS are linked to the maximum, mean, and minimum pixel values along the corresponding image transect. The resulting brightness-depth relation is then
applied throughout the image to produce a bathymetric map. The HAB-2 algorithm uses an expression
describing the attenuation of light with distance traveled through the water column (Beer’s law) in an iterative manner along with the hydraulic relations to identify an attenuation coefficient that yields depth estimates that reproduce, via the continuity and Manning equations, the gage discharge.
Although initial results reported by Fonstad and Marcus [2005] were only moderately encouraging, subsequent testing of the HAB-2 algorithm on the McKenzie River demonstrated strong agreement between
HAB-estimated depths and field-based measurements for 0.5 m resolution, film-based aerial photography
[Walther et al., 2011]. A more recent study on the River Tana in Finland found that both HAB algorithms performed nearly as well as an empirical calibration approach [Flener et al., 2012]. The HAB technique thus has
demonstrated potential to retrieve depth without direct calibration to field measurements. Knowledge of
discharge and assumptions regarding cross-sectional shape, flow resistance, and the nature of radiative
transfer are still required, however. The most important contribution of Fonstad and Marcus [2005] might
not be the HAB algorithms per se but rather the general concept of using principles of open channel flow
to help constrain image-derived depth estimates. This concept is explored further herein.
Building upon the work of Fonstad and Marcus [2005] and previous research on remote sensing of discharge
by Bjerklie et al. [2003, 2005a, 2005b], this paper introduces algorithms for Flow REsistance Equation-Based
Imaging of River Depths (FREEBIRD) that exploit hydraulic principles to calibrate remotely sensed river
bathymetry in the absence of field measurements. A conceptual overview of this approach is presented in
Figure 1, which illustrates the inputs, algorithms, and outputs, as well as the connections between them.
Unlike the HAB techniques, FREEBIRD does not require a priori information on Q, selection of a flow resistance coefficient, or any assumptions regarding the frequency distribution of d. The only inputs essential to
FREEBIRD are an initial estimate of the channel aspect (width/depth) ratio A at the flow level of interest and
the water surface slope S, along with a small number of algorithm-specific parameters. Moreover, the FREEBIRD algorithms not only retrieve d from image data but also yield estimates of Q; one procedure uses
knowledge of Q to help constrain the bathymetry. The following sections: (1) describe the FREEBIRD
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Figure 1. Overview of Flow REsistance Equation-Based Imaging of River Depths (FREEBIRD), with the different components of the diagram distinguished in the legend at the bottom.
Also included are insets illustrating (a) inverse and (b) direct relations between the image-derived quantity X and the flow depth d. The nature of the X versus d relation is a required initial input to the depth retrieval algorithms.
algorithms; (2) apply these methods to satellite image data and aerial photography; (3) assess depth
retrieval accuracy using detailed field measurements from a gravel-bed river; (4) evaluate the sensitivity of
the algorithms to their input parameters; and (5) discuss the potential utility of FREEBIRD for remote sensing
of rivers.
2. Algorithms for Flow Resistance Equation-Based Imaging of River Depths
Even if field measurements of d are not available to directly calibrate remotely sensed bathymetry, b can be
estimated by invoking geometric and hydraulic constraints. Equation (1) represents a critical assumption: X
is linearly related to d. For the restricted set of rivers for which this this assumption is valid [Legleiter et al.,
2009], determining the coefficients of the X versus d relation allows bathymetric maps to be produced by
applying (1) to each pixel within the channel. X can be defined in a number of different ways, such as a linear transform [Lyzenga, 1981] or an appropriate band ratio [Legleiter et al., 2009]. Given this framework for
passive optical depth retrieval, the first step in FREEBIRD involves creating a series of N cross-sections (XS’s)
and extracting X values for each of the M pixels along each of these transects. The XS’s should encompass
the full range of d present in the river of interest and represent a variety of channel shapes. Applying FREEBIRD to a set of transects distributed throughout a reach, rather than a single cross-section, makes the algorithm robust to local variations in channel geometry and establishes an appropriate length scale for
defining hydraulic slope. Similarly, averaging over a number of cross-sections reduces variability and leads
to more reliable estimates of depth and discharge.
The foundation of FREEBIRD and the core of each algorithm described below is a flow resistance equation
developed by Bjerklie et al. [2005b] for the purpose of estimating in-bank discharges. Based on Manning’s
equation, the Bjerklie et al. [2005b] relation features a smaller exponent on S, 0.33, than the value used
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traditionally, 0.5; this modification reduced the variance associated with estimating flow resistance (i.e.,
selecting a value of Manning’s n). Instead, a single, constant conductance coefficient k was included in a
modified form of Manning’s equation that was calibrated and validated using a large database of 1037 discharge measurements from 103 rivers. The original expression of [Bjerklie et al., 2005b, equation (7)] was
intended for estimating Q as the product of width w, mean flow depth d (assumed equivalent to the
hydraulic radius), and cross-sectional average velocity v . For FREEBIRD, a more convenient form of the flow
resistance equation is obtained by dividing Q by w and d to obtain an expression for velocity
0:67
v 5k d S0:33
(2)
For units of meters and seconds, the conductance coefficient k 5 7.14.
The critical connection underlying FREEBIRD is a substitution of the X versus d relation (1) into the flow
resistance equation (2), which links the image data to the channel hydraulics. For each incremental lateral
distance, Dw (i.e., pixel) along a XS extracted from the image, the unit discharge q is calculated as the product of the image-derived depth estimate d for that pixel and the local depth-averaged velocity v calculated
^ is obtained by summing
from equation (2) based on that depth estimate. The total estimated discharge Q
unit discharges across the channel. These relationships are summarized by
^
Q5
M
X
j51
M
M
X
X
Dwj dj vj 5
Dwj ½b0 1b1 Xj ½kðb0 1b1 Xj Þ0:67 S0:33 5DwkS0:33
ðb0 1b1 Xj Þ1:67
j51
(3)
j51
where the subscript j indexes the M pixels along a XS. Points along the transect are assumed to be equally
spaced such that the width increment is constant: Dwj 5Dw.
Before describing FREEBIRD algorithms in greater detail, several key assumptions underlying this framework
must be acknowledged: (1) A reasonably strong relation exists between X and d. Without such a relation,
FREEBIRD, or even empirical calibration of image-derived depths, is futile. (2) Moreover, the X versus d relation is linear across the full range of depths present in the river and consistent throughout the scene so that
the same b can be applied to the entire image, although a separate X versus d relation must be established
for each image to which the algorithm is applied. (3) The modified Manning equation developed by Bjerklie
et al. [2005b] is applicable to the river of interest and provides a reliable estimate of v for a given d. Other,
similar flow resistance equations have been proposed, for example by Lopez et al. [2007a] and Rupp and
Smart [2007], but equation (2) performed nearly as well as these regression- or theory-based formulations
when evaluated using an independent data set [Lopez et al., 2007a, 2007b]. (4) The conductance coefficient of the flow resistance equation [k 5 7.14, Bjerklie et al., 2005b] is appropriate for the river interest
and is constant, obviating the need to estimate a resistance coefficient such as Manning’s n. (5) The modified Manning equation applies not only to the XS as a whole [i.e., for estimating Q after Bjerklie et al.,
2005b, equation (7)] but also to each individual width increment across the channel so that equation (2)
can be used to estimate the local depth-averaged flow velocity v for a given local depth d and water surface slope S. (6) Finally, the FREEBIRD algorithms are intended for, but not necessarily restricted to, singlethread channels, or braided streams treated one channel at a time. With this flow resistance equationbased framework in place, and subject to the preceding caveats, the following subsections, along
with Figure 1, outline two algorithms for FREEBIRD. MATLAB code implementing these procedures is
included in supporting information associated with this article and posted on the author’s web site: www.
fluvialremotesensing.org.
2.1. Channel Aspect Ratio (CAR)
The primary input to the first FREEBIRD algorithm is an initial estimate of the channel aspect ratio A5w=d.
Blodgett [1986] provides a useful compilation of aspect ratios and other characteristics of channel geometry
for rivers in the western U.S. By geometric reasoning, the CAR algorithm associates the shallowest depth d0
along a XS with either the minimum or maximum X value along the corresponding image transect. Two
cases must be distinguished. If X and d are inversely related such that b1 < 0, d0 is assigned to the largest
value of X along the transect (Figure 1a). Conversely, if X and d are directly related such that equation (1)
has a positive slope (i.e., b1 > 0), d0 is linked to the smallest value of X along the XS (Figure 1b). To ascertain
the nature of the X versus d relation, the user can inspect the X image as shown in Figures 1a and 1b: if shallow areas along the channel margins correspond to larger values of X, the relation is inverse, whereas
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smaller values of X in shallow water imply a direct relationship. The extreme value of X along an image transect thus can be defined as
(
max ðXj Þ; if b1 < 0
X0;i 5
(4)
min ðXj Þ; if b1 > 0
where j indexes the pixels along the ith XS. Substituting into equation (1) yields
d0 5b0 1b1 X0;i
(5)
Similarly, the mean depth for the XS is related to the mean pixel value X i along the XS:
d i 5b0 1b1 X i
(6)
Equations (5) and (6) can be manipulated algebraically to obtain a pair of expressions for the intercept and
slope coefficients specific to the ith XS:
d i 2d0 b0;i 5d0 2 Xi
X i 2X0;i
(7)
d i 2d0
b1;i 5 X i 2X0;i
(8)
The d0 parameter can be assumed to be 0 or set to a small value (e.g., 5 cm) for coarser-resolution images
for which the pixel-scale mean d > 0 along channel margins.
In the absence of field data, a plausible initial estimate of d i can be obtained by specifying the channel
aspect ratio A, measuring the wetted channel width wi for each XS, averaging these values to obtain a
w=A,
and solving for the reach-averaged mean depth as d5
reach-averaged mean width w,
where the lack
of a subscript on d indicates a reach-averaged value rather than the mean for an individual XS. A is a basic
geometric descriptor of river form that depends on channel gradient, bed material grain size, and bank
composition and an appropriate value can be assigned based on field experience or regional relationships
[e.g., Blodgett, 1986]; sensitivity of the CAR algorithm to the input A is examined in section 4.3.1. Equation
and d,
(1) is thus calibrated by specifying values of A and d0, using a set of XS’s to obtain reach-averaged w
and then calculating b0;i and b1;i coefficients separately for each XS via equations (7) and (8). The resulting
estimates of bi are then
N pooled and averaged to obtain a single set of coefficients representative of the
X
^ i for each XS,
entire reach: b51=N
bi . Substituting b into equation (3) yields an estimated discharge Q
i51
^
which can then be averaged to obtain an estimate of Q, where again the lack of a subscript indicates a
reach-averaged value. Estimates of b from the CAR procedure also serve as initial values for a second,
optimization-based algorithm.
2.2. Matching Known Discharge (MKD)
If Q at the time of image acquisition is known from gage records, this information can be used within the
FREEBIRD framework to calibrate remotely sensed bathymetry so as to match the known Q. In that sense,
the MKD algorithm is analogous to the HAB techniques proposed by Fonstad and Marcus [2005] but
involves a different computational approach. MKD minimizes the disagreement between discharges estimated via equation (3) and the known Q by modifying the coefficients of the X versus d relation. More spe^ i for
cifically, a hypothesized value of b, initially that derived from CAR, is used in equation (3) to estimate Q
each XS. The N estimated discharges are compared to the known value and the root mean squared error
(RMSE) calculated as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u N
u1 X
eQ 5 t
ðQ^i 2QÞ2
(9)
N i51
^ based on image-derived depths serves as the
This measure of disagreement between the known Q and Q
objective function for a numerical routine that minimizes eQ by adjusting b.
Equation (1) can yield d < 0 along channel margins if the aquatic signal is contaminated by radiance from
adjacent terrestrial features or if b is misspecified. Within the FREEBIRD framework, negative depth
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estimates are avoided by assigning a velocity of 0 to any pixels for which the current b results in d < 0 so
^ To prevent the MKD algorithm from returning a b that would result
that these areas do not contribute to Q.
in d < 0 for a large number of pixels, the user specifies the fraction of depth estimates allowed to be negative. For each iteration of the optimization routine, the objective function (9) is set to a very large value if
the current b yields an unacceptably high proportion of negative depth estimates, forcing the optimization
toward coefficients that result in d > 0.
The MKD algorithm is applicable to historical images that include gages with streamflow records, or any
case for which Q at the time of image acquisition is known. In addition to Q, inputs to MKD include: the A
and d0 parameters for CAR, which provides the initial estimate of b required by the optimization routine;
the water surface slope S; and the negative depth fraction. Sensitivity to these inputs, including Q, is examined in section 4.3.2. The algorithm attempts to match the known Q and returns the eQ value for which the
RMSE between the N discharges inferred from the image and the known Q was minimized, as well as the
final b, which can be used to produce a bathymetric map.
3. Data and Methods
FREEBIRD is intended to be a flexible means of retrieving depth information from readily available image
data in the absence of field measurements. As an initial assessment of the degree to which this objective
was achieved, the algorithms outlined above were applied to a pair of images from the Snake River in
Grand Teton National Park (Figure 2a), each subset to a meander bend where a detailed survey had been
conducted close to the time of image acquisition. The locations of these reaches are indicated in Figure 2b
and their characteristics summarized in Table 1, along with image metadata. Field data from these surveys
were used to assess the accuracy of depth retrieval via the new, hydraulically-assisted approach outlined
herein. Further information on the study area, field data collection, and image processing methods was provided by Legleiter and Overstreet [2012] and Legleiter et al. (Comparative evaluation of hyperspectral imaging and bathymetric LiDAR for measuring channel morphology across a range of river environments,
submitted to Earth Surface Processes and Landforms, 2014) and the following sections include only a brief
summary.
3.1. Remotely Sensed Data
A 0.5 m pixel, panchromatic image of Rusty Bend (Figure 2d) acquired by the WorldView-2 (WV2) satellite
was used to evaluate the feasibility of depth retrieval when neither field measurements nor spectral information is available. In this case, an image-derived quantity X linearly related to d was obtained via a linear
transform [Lyzenga, 1981]. X values were regressed against in situ depth observations made within 2 weeks
of image acquisition. This Field-Calibrated Linear Transform (FCLT) approach established a baseline to which
bathymetry inferred via FREEBIRD independent of any ground-based measurements could be compared.
The National Agricultural Imagery Program (NAIP) provides continuous aerial photography of the coterminous U.S. on a 3 year cycle. These data are freely available and could be useful for mapping fluvial systems;
previous research on the Snake River demonstrated that reasonably accurate (R2 50:64) depth estimates
can be derived from even a compressed, county-level mosaic [Legleiter, 2013]. In this study, the most recent
NAIP data, an uncompressed digital orthophoto comprised of 1 m pixels, was used to assess whether
bathymetry could be mapped from public domain, multispectral images in the absence of field data. More
specifically, a four-band NAIP image of Swallow Bend (Figure 2c) was used to produce bathymetric maps
using an established, field-based calibration technique [Optimal Band Ratio Analysis, or OBRA, Legleiter
et al., 2009], and the new FREEBIRD algorithms. Accuracy assessment involved comparing depths estimates
from each method to field measurements collected within 4 days of image acquisition.
3.2. Field Data
Field campaigns on Rusty and Swallow Bends provided direct measurements of depth for the dual purposes
of calibrating traditional depth retrieval algorithms (FCLT and OBRA) and assessing the accuracy of depths
estimated by empirical methods and via FREEBIRD. Field data were collected using a combination of wading
surveys with a real-time kinematic (RTK) GPS, an echo sounder deployed from a cataraft, and an acoustic
Doppler current profiler (ADCP) mounted on a kayak. Depths recorded by these three instruments were
cross-calibrated and then all point measurements located within a given image pixel were spatially
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Figure 2. (a) Location study area in the context of the western U.S. (b) 2011 WV2 satellite image overview of the Snake River from Jackson
Lake (top left) to Deadman’s Bar (bottom left); flow is from north to south. Rectangles indicate the two primary study reaches. (c) 2012
NAIP image and field-based depth measurements on Swallow Bend; flow is from east to west. (d) 2011 WV2 satellite image and fieldbased depth measurements on Rusty Bend; flow is from east to west.
Table 1. Study Reach Characteristics, Field Data Summary, and Image
Metadata
Study Reach
Rusty Bend
Swallow Bend
a
57.6 6 12.4 (23)
1.19 6 0.58 (8179)
88.2 6 3.42 (36)
46.8
0.0035
30 Aug to 7 Sep2011d
WorldView-2 satellite
1: panchromatic
0.5
13 Sep 2011
56.0 6 14.5 (49)
1.18 6 0.52 (7654)
70.9 6 4.9 (75)
47.5
0.0011
14 Aug to 19 Aug 2012e
NAIPf aerial photography
4: blue, green, red, NIR
1
18 Aug 2012
Width (m)
Depthb (m)
Dischargec (m3/s)
Channel aspect ratio
Channel bed slope
Field data collection
Image data source
Image bands
Pixel size (m)
Image date
a
Mean 6 standard deviation (number of image cross-sections) for wetted
channel widths.
b
Mean 6 standard deviation (number of measurements) for pixel-scale
mean depths.
c
Mean 6 standard deviation (number of ADCP transects).
d
Discharge transects recorded on 30 Aug, 31 Aug, and 3 Sep2011.
e
Discharge transects recorded on 18 Aug and 19 Aug 2012.
f
National Agricultural Imagery Program.
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averaged to obtain pixel-scale mean
depths [Legleiter and Overstreet, 2012].
Summary statistics for these depth
measurements are listed in Table 1.
In addition to depth, the SonTek RiverSurveyor S5 ADCP recorded velocity
profiles along a series of XS’s distributed throughout each bend. The velocity data, sampled at 1 Hz, were
integrated over the vertical cells comprising each profile and laterally across
the channel to yield direct measurements of river discharge. Discharges
were calculated separately for backand-forth traverses along the close,
regularly spaced (0:25w) XS’s evident
in Figures 2c and 2d. The discharge
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^ estimated via CAR and to specmeasurements summarized in Table 1 were used to assess the accuracy of Q
ify the input Q for MKD.
3.3. Image Processing and Empirical Depth Retrieval
Initial processing of the WV2 and NAIP data involved established methods reported by [Legleiter and Overstreet, 2012]. Masks highlighting the in-stream portion of each image were produced by band thresholding,
morphological operations, and minor manual editing. Masked images were smoothed with a 3 3 3 m Wiener spatial filter. Neither image was radiometrically calibrated and digital numbers were used throughout
the analysis.
For the panchromatic WV2 scene from Rusty Bend, the image-derived quantity X was calculated using the
linear transform [Lyzenga, 1981]
X5ln ½DN2min ðDNÞ11
(10)
Subtracting the minimum digital number DN within the channel from each in-stream pixel is referred to as
a deep-water correction and accounted for radiance contributed from the atmosphere, water surface, and
water column. The last term was included to avoid taking the log of 0. The resulting pixel values were then
regressed against depths measured in the field to establish an X versus d relation for bathymetric mapping.
For the NAIP image of Swallow Bend, X was defined by Optimal Band Ratio Analysis [OBRA Legleiter et al.,
2009]. For certain combinations of wavelengths, and under appropriate environmental conditions, the logarithm of the ratio of the DN measured in two spectral bands yields an image-derived quantity linearly
related to depth:
DN1
(11)
X5ln
DN2
where the subscripts 1 and 2 identify the numerator and denominator bands, respectively. OBRA selects the
best bands for retrieving depth from a particular image by calculating X for all possible band combinations,
performing a regression of X against field measurements of d for each pair, and identifying the optimal
band ratio as that which yields the highest R2 value. The corresponding regression equation serves to calibrate X to d.
For both data sets, the image-derived quantity X defined by equation (10) or (11) was calibrated using field
measurements spatially averaged to obtain pixel-scale mean depths. Half of the field data were used to calibrate the X versus d relations while the other half was retained for validation in the form of observed versus
predicted regressions. Bathymetric maps were produced by inserting the calibrated b into equation (1) and
applying this expression to the masked images. The remotely sensed bathymetry produced via these empirical methods were used to compare depths retrieved by FREEBIRD to the depths that would have been
inferred from the same images if the field data had been used for calibration. This test was particularly
robust because the field surveys used in this study provided very thorough coverage, with a sampling density unlikely to be matched by most operational field efforts, and thus represented a best case scenario for
direct calibration.
3.4. Application of FREEBIRD Algorithms to the Snake River
The first steps in the FREEBIRD workflow illustrated in Figure 1 are to identify an image-derived quantity X
linearly related to depth and determine whether this relationship is inverse or direct. For the panchromatic
satellite image of Rusty Bend, only one band was available and X was defined unambiguously via equation
(10). Moreover, the linear transform implied that X was inversely related to d; larger values of X occurred in
shallower water (Figure 1a). For multispectral images, however, selecting an appropriate X and specifying
the nature of the X versus d relation is less straightforward. Whereas OBRA uses field measurements to identify the optimal band ratio for depth retrieval, a user lacking field data must select an appropriate combination of bands based on more qualitative criteria. In this study, the Swallow Bend subset was used to
produce a separate X image for each possible band combination, a total of six for this four-band data set.
The resulting images were displayed side-by-side with a common color scale and visually inspected to
select the X image that appeared most strongly related to depth. The primary cues guiding this choice were
the dynamic range and spatial coherence of each version of X. Similarly, without a priori knowledge of the
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relationship between DN and d for
each band, particularly since the
data were not radiometrically calibrated, the selected X image was
examined visually to determine the
nature of the X versus d relation.
The next user action required by
FREEBIRD is selecting a number of
XS’s from the X image. Although
image processing and GIS software
provide tools for sampling raster
data along manually digitized
transects, a more automated
approach was developed for this
study. An algorithm for transforming coordinates from a Euclidean to
a channel-centered frame of reference [Legleiter and Kyriakidis, 2006]
was augmented to return the orientation of normal vectors at a regular, user-specified interval along a
digitized channel centerline (Figure
3a). The normal vectors were then
used to create lines extending
toward each bank from the centerline. Intersecting these lines with a
channel boundary polygon proFigure 3. Automated extraction of image cross-sections. (a) Cross-section (XS) orientation is determined from centerline normal vectors. XS end points are identified as the
duced from the in-stream mask
intersection of the normal vectors with the channel boundary polygon. Normal vectors
defined the end-points of each XS.
are created at a specified interval along the centerline to produce a series of regularly
In this study, only the first (closest
spaced XS’s like those shown in (b).
to centerline) intersection with the
channel polygon was retained under the assumption of a single-thread morphology, although the procedure could be generalized to accommodate braided rivers as well. A streamwise sampling interval of 20 m
resulted in 23 XS’s for Rusty Bend and 49 for Swallow Bend (Figure 3b), which were then overlain on the X
image for each reach and used to extract pixel values along each transect.
CAR is the simpler of the two FREEBIRD methods and provides an initial estimate of b for MKD. In this study,
the input aspect ratio for each reach was defined by dividing the mean width of the XS’s by the mean of
the pixel-scale mean depths derived from the field surveys. This approach ensured that A was specified as
can be determined from an image in the absence of field data, in pracaccurately as possible. Although w
and thus A, will not be known a priori. The impact of uncertainty in A was evaluated via the sensitivity
tice d,
analysis described below. The other parameter required by CAR, the minimum depth d0, was set to 0.05 m
to account for mixed pixels along the banks.
For the optimization-based MKD algorithm, the same A and d0 values were used to obtain an initial b
via CAR. The water surface slope for each reach was calculated from RTK GPS surveys. The input Q for
MKD was specified as the mean Q for the ADCP transects measured along each reach and the negative depth fraction was set to 0.05. For both reaches and both FREEBIRD algorithms, bathymetric maps
were produced by applying the final, calibrated intercept and slope coefficients, b0 and b1, to the X
images.
3.5. Accuracy Assessment
The accuracy of FREEBIRD was evaluated by comparing image-derived depth estimates to field measurements. To facilitate this analysis, an ordinary kriging procedure developed specifically for application to river
channels [Legleiter and Kyriakidis, 2008] was used to predict (interpolate) the depth at the location of each
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Table 2. Parameter Values for Sensitivity Analysis
Parameter
Channel Aspect Ratio (CAR)
Aspect ratio A
Bed slope S
Minimum depth d0 (m)
Cross-section spacing Ds (m)
Matching Known Discharge (MKD)
Aspect ratio A
Bed slope S
Known discharge Q (m3/s)
Negative depth fraction
Cross-section spacing Ds (m)
Base Value
Minimum
Maximum
Increment
A 5 46.8
S 5 0.0011
0.05
20
0.75A
0.75S
0
10
1.25A
1.25S
0.1
50
0.05A
0.05S
0.01
10
A 5 46.8
S 5 0.0011
Q 5 70.9
0.05
20
0.75A
0.75S
0.75Q
0
10
1.25A
1.25S
1.25Q
0.1
50
0.05A
0.05S
0.05Q
0.01
10
image pixel. This approach enabled direct comparison of field-based depths and image-derived estimates.
Inspecting depth maps and XS’s produced from field data by kriging and from remotely sensed data via
direct calibration and by FREEBIRD provided a qualitative impression of depth retrieval accuracy and differences among algorithms. Including the FCLT-derived and OBRA-derived depth estimates allowed FREEBIRD
to be considered in the context of the bathymetry that would be inferred from the same images if field
data were used for calibration.
Quantitative accuracy assessment involved characterizing the error associated with remotely sensed
bathymetry via three metrics. Depth retrieval residuals (errors) were defined as [Legleiter and Roberts, 2009]
eðsÞ5df ðsÞ2d^r ðsÞ
(12)
where df ðsÞ is the field-based depth predicted via kriging at location s and d^r ðsÞ refers to the depth estimated from the remotely sensed data. A positive residual thus implied an underestimation of depth
whereas eðsÞ < 0 indicated that the depth inferred from the image exceeded that observed in the field.
Equation (12) was used to obtain residuals for each pixel along the XS’s used as input to FREEBIRD. The distribution of eðsÞ along each of these transects was summarized in terms of the mean error, an indication of
bias, and the root mean square error (RMSE), a measure of precision. Streamwise profiles of mean error and
RMSE for the various techniques were used to examine along-reach spatial variations in bathymetric accuracy and precision. Reach-averaged mean and RMS errors were obtained by averaging values for the individual XS’s.
3.6. Sensitivity Analysis
The objective of FREEBIRD is to enable remote sensing of river bathymetry in the absence of field measurements and with minimal user intervention. The algorithms do require a limited number of inputs, however,
and the impact of uncertainty in these inputs on the resulting depth estimates was evaluated by performing a sensitivity analysis for the NAIP image of Swallow Bend. The sensitivity of each algorithm to its input
parameters was examined by varying one parameter at a time between 625% of the known value in 5%
increments while holding the other parameters constant at the base case (known) values listed in Table 2.
In addition, different XS spacings were specified to quantify the influence of sample size. The FREEBIRD procedures were rerun for each combination of parameter values and XS spacing and an accuracy assessment
completed for each of the resulting bathymetries. Sensitivity analysis involved plotting the estimated X versus d coefficients, estimated discharges, and mean and RMS depth retrieval errors against the parameter of
interest. Depth estimates extracted along an example XS also were used to visualize how uncertainty in the
input parameters influenced the output bathymetry.
4. Results
4.1. Imaging River Depths From Panchromatic Satellite Image Data
Legleiter and Overstreet [2012] demonstrated the potential to map gravel-bed river bathymetry from space
and identified some advantages of acquiring image data from orbital platforms. This study evaluated the
possibility of inferring depth from satellite images, even if simultaneous field data are not available for calibration, via the FREEBIRD algorithms introduced herein. Moreover, examining a panchromatic WV2 scene
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Figure 4. Depth maps for Rusty Bend derived from (a) field observations via kriging, (b) calibration of the image-derived quantity X directly
to field data, and two FREEBIRD algorithms: (c) channel aspect ratio, and (d) match known discharge. A common color scale, from 0 to the
99th percentile of the pixel-scale mean depths measured in the field, is used for all four maps. Depth estimates are compared along the
cross-section in (e), the location of which is shown by a solid line on the maps. The locations of other cross-sections are indicated by point
symbols.
assessed the potential to map bathymetry from a single brightness value (DN) for each pixel, rather than
more detailed spectral information from multiple bands.
To establish a baseline to which the FREEBIRD techniques could be compared, an established depth
retrieval method was applied to the WV2 image of Rusty Bend. The field-calibrated linear transform (FCLT)
produced an observed versus predicted (OP) R2 of 0.67, but the OP scatter plot revealed a bimodal distribution of residuals, possibly related to variations in substrate type [Flener et al., 2012], and indicated that d
tended to be underpredicted in deeper water. The moderate strength of the X versus d relation limited the
range of estimated depths and the FCLT-generated bathymetric map in Figure 4b showed relatively little
spatial variation compared to the map produced from field data via kriging (Figure 4a). More specifically, d^ r
ðsÞ < d f ðsÞ along the outside of the bend, most notably above the apex, whereas d^ r ðsÞ was overpredicted
along the shallow margins of the point bar on the right (north) side of the channel.
The CAR algorithm produced depth estimates that were similar to those from FCLT in shallow areas but
deeper in the pool (Figure 4c), including greater overestimates relative to d f ðsÞ adjacent to the outer bank,
as illustrated by the example XS shown in Figure 4e. Overall, the CAR-derived bathymetry exhibited greater
variation than that produced by FCLT, mainly in the form of a deeper, more extensive pool along the outside of the bend. Mean and RMS CAR depth retrieval errors calculated for each of the 23 XS’s (indicated by
points in Figure 4c) varied spatially along the channel. CAR-based depths tended to be shallower than the
field data in the upper portion of Rusty Bend, leading to positive mean errors, but deeper than d f ðsÞ below
the bend apex, consistent with the visual impression given by Figures 4a and 4c. The RMSE, an index of
bathymetric precision, was more stable throughout the reach. Averaged over the 23 XS’s, the mean CAR
depth retrieval error was nearly zero (Table 3), implying that, overall, d^ r ðsÞ was unbiased for this algorithm.
Similarly, the reach-averaged RMSE of 0.32 m for CAR was comparable to that of the FCLT method; this
amount of error represented 627% of the mean flow depth for the reach. In addition to depth retrieval,
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Table 3. Flow Resistance Equation-Based Imaging of River Depths (FREEBIRD) Results
^
Algorithm
b0
b1
Q
Objective Functiona
Rusty Bend WV2: Panchromatic
FCLT
2.63
CAR
2.95
MKD
2.92
Swallow Bend NAIP: log(Blue/Red)
OBRA
20.077
CAR
20.315
MKD
20.302
a
Mean Error (m)
RMSE (m)
20.611
20.727
20.751
–
97.79
88.22
–
–
8.17E-11
0.03
20.02
0.069
0.319
0.32
0.332
22.92
23.47
23.67
–
62.36
70.9
–
–
1.99E-11
0.0079
0.0064
20.093
0.257
0.281
0.308
Objective function for MKD is the RMSE between known and estimated discharge.
^ from CAR was 10.1% greater than measured
CAR used equation (3) to estimate discharge. For Rusty Bend, Q
in the field, mainly due to overestimation of depth in the lower half of the reach. This level of agreement
between FREEBIRD-inferred and directly measured Q was surprisingly good considering the only riverspecific inputs to CAR were the channel aspect ratio and water surface slope.
Field measurements of Q also enabled evaluation of MKD, a FREEBIRD technique that calibrates imagederived depth estimates by iteratively adjusting the coefficients of the X versus d relation to match the
known discharge. The bathymetric map generated by MKD (Figure 4d) was very similar to that produced by
CAR but comparison of the XS’s in Figure 4e indicated that MKD-based depths were slightly shallower than
those from CAR across the entire channel. The reach-averaged mean depth retrieval error for MKD was
0.069 m, 3.5 times greater than for CAR but opposite in sign; the positive mean error implied a shallow bias.
The RMSE of MKD depths was only 0.01 m greater than FCLT or CAR, implying a similar level of precision.
Table 3 indicates that the algorithm lived up to its name and exactly matched the known discharge. MKD
^ from CAR was in error by more than
converged on the user-specified input Q even though the initial Q
^
^ by
10%. In this case, the CAR-based Q exceeded the known Q, so the MKD optimization routine reduced Q
adjusting b to produce smaller depths. The known Q thus was reproduced at the expense of a shallow bias
for d^ r ðsÞ.
4.2. Imaging River Depths From Publicly Available Aerial Photography
The most recent NAIP coverage for Wyoming was acquired in 2012, within a few days of the field surveys
on Swallow Bend, and these two data sets were used to assess the feasibility of inferring bathymetry from
publicly available images. As a benchmark for evaluating FREEBIRD results, an existing, empirical means of
retrieving depth from multispectral data, OBRA [Legleiter et al., 2009], was applied to the NAIP subset from
Swallow Bend. For this data set, the optimal ratio was blue/red and using these bands in equation (11)
resulted in a strong linear relationship between X and d. OP regression yielded an R2 of 0.75 and intercept
and slope coefficients of 0 and 1, indicating that d^ r ðsÞ was unbiased on average. The negative slope of the
OBRA X versus d relation was opposite that predicted by radiative transfer theory for these wavelengths
because the image was not radiometrically calibrated. Had the image been in radiance rather than DN, b1
would have been positive. The OBRA-based bathymetric map in Figure 5b showed clear variations in d and
effectively captured the morphology of the reach. Relative to the field-based map in Figure 5a, however,
OBRA underestimated d f ðsÞ in deeper areas along the north bank in the middle of the reach and on the
outside of the bend where the channel curved north; this shallow bias was consistent with previous studies
[Legleiter, 2013] and theoretical expectations [Legleiter and Roberts, 2009].
Whereas OBRA selects the optimal band ratio by performing regressions against field measurements of d,
the objective of FREEBIRD is to enable depth retrieval in the absence of such data. To identify an appropriate definition of X, separate images were produced for all six possible band combinations and inspected visually to determine which one appeared most strongly related to depth. The X image created by inserting
the blue and red bands into equation (11) was spatially coherent and exhibited strong gradients in brightness across the channel that resembled the pattern of depths expected for bar-pool morphology. On the
basis of these qualitative criteria, the blue/red X image was selected for depth retrieval, consistent with the
quantitative, field-calibrated OBRA results. In addition, visual examination of the image indicated small values of X in the thalweg and large X values on shallow bar margins, implying an inverse relation between X
and d.
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Figure 5. Depth maps for Swallow Bend derived from (a) field observations via kriging, (b) optimal band ratio analysis calibrated using
field data, and two FREEBIRD algorithms: (c) channel aspect ratio, and (d) match known discharge. A common color scale, from 0 to the
99th percentile of the pixel-scale mean depths measured in the field, is used for all four maps. Depth estimates are compared along the
cross-section in (e), the location of which is shown by a solid line on the maps. The locations of other cross-sections are indicated by point
symbols.
Applying CAR to the Swallow Bend NAIP image resulted in depth estimates that were grossly similar to
those from OBRA but deeper in pools and shallower over bar margins, where CAR produced more extensive
negative depth estimates (Figure 5c). Overall, however, the CAR bathymetry closely resembled the fieldbased depth map. Although the mean and RMS depth retrieval errors computed for the 49 XS’s extracted
from the image varied slightly over the course of Swallow Bend, the reach-averaged mean error was essentially zero (Table 3), suggesting a lack of systematic bias. The average RMSE of 0.28 m was only 0.02 m
greater than for the OBRA-based depth map and represented 24% of the mean depth for the reach. The dis^ might have
charge estimated by CAR was 12% less than Q measured in the field. The reach-averaged Q
underestimated Q because the extracted XS’s only encompassed a single channel and thus did not account
for flow in the secondary channel at the upper end of the reach.
If the discharge is known a priori, the MKD algorithm can use this information to constrain depth retrieval.
In this case, depths derived by MKD were greater than those from OBRA or CAR and more closely
^ from CAR was less
approached d f ðsÞ in the pool on the left side of the example XS (Figure 5e). Because Q
than the known Q, the MKD optimization routine matched Q by increasing the absolute value of the coefficient on X to produce greater depths and therefore velocities per (2). Inflating d^ r ðsÞ to reproduce Q resulted
in a negative reach-averaged mean error because the MKD-based depths tended to exceed d f ðsÞ. The
RMSE for MKD was only 0.02 m greater than CAR and 0.04 m greater than OBRA, however, implying that
the deep bias did not compromise bathymetric precision. MKD again exactly matched the known Q, implying that the algorithm was sufficiently flexible to reproduce Q for a range of channel configurations. Had
^ for those XS’s, depth
flow in the secondary channel at the entrance to Swallow Bend been included in Q
retrieval might have been more accurate.
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Figure 6. Sensitivity of the channel aspect ratio (CAR) algorithm to the input aspect ratio A, varied from 75% to 125% of the known value A0 with all other parameters held constant.
Effects of variation in A on (a) X versus d intercept and slope coefficients, (b) discharge estimated via the CAR algorithm, and (c) depth retrieval mean and root mean square errors. (d)
Illustration of the effects of variation in A on depth estimates for the example cross-section shown in Figure 5. The different lines in (d) represent multiples of the known input aspect
ratio A0.
4.3. Sensitivity Analysis of FREEBIRD Algorithms
FREEBIRD enables remote sensing of river bathymetry in the absence of field measurements but does
require a limited number of inputs. A sensitivity analysis was performed to assess how misspecification of
these inputs might affect depths inferred by each algorithm. All of the parameters listed in Table 2 were
perturbed as described in section 3.6, but most of these inputs had minimal impact on depth estimates.
The following sections thus focus on the subset of parameters that might exert a greater influence on
FREEBIRD.
For the Swallow Bend NAIP image upon which this analysis was based, the X versus d relation was inverse,
and the negative sign on b1 complicated interpretation of X versus d slope coefficients. For b1 < 0, an algebraically smaller (but larger in absolute value) b1 lead to greater depth estimates for a given X and b0. If the
X versus d relation had been direct and b1 were positive, the opposite logic would apply and an algebraically larger value of b1 would result in greater depth estimates.
The water surface slope S is a critical input to both CAR and MKD that must be specified on the basis of field
data, topographic maps, or DEM’s with varying levels of reliability. Although uncertainty in S could propagate through the FREEBIRD algorithms to affect the output bathymetry, varying S from 625% of the known
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value S0 indicated a negligible impact on CAR-estimated depths. For MKD, varying S across this range had a
relatively minor effect on depth estimates and the known Q was consistently reproduced. For Swallow
Bend, S > S0 lead to shallower depth estimates from MKD, mainly by increasing (decreasing the absolute
value of) b1. As dictated by the flow resistance equation (2), a larger slope translated into higher velocities
such that the input Q was matched with shallower depths. In this case, overestimating S reduced the mean
and RMS depth retrieval errors by making depth estimates smaller and thus decreasing the magnitude of
overestimates that inflated mean error. A potential source of uncertainty in S is the distance over which S is
measured. In this study, a reach-averaged hydraulic slope was obtained by dividing the difference in water
surface elevation between the first and last XS’s by the streamwise distance between the two transects.
Although the definition of a reach depends on local context, in general, reaches should be sufficiently long
to provide a representative value of slope, as well as a large number of XS’s for input to FREEBIRD.
4.3.1. CAR
The most important input for FREEBIRD is the channel aspect ratio A, which must be specified based on field
experience [e.g., Blodgett, 1986], measurements at gaging stations, or hydraulic geometry relations [e.g.,
Parker et al., 2007]. CAR uses A to estimate coefficients of the X versus d relation and the resulting b is used
as an initial value for the optimization-based MKD algorithm. The input A thus could influence both FREEBIRD procedures but primarily affects CAR. The impact of uncertainty in A on CAR is summarized in Figure 6,
with A varying between 625% of the known value A0 calculated from field data. As A increased, estimated
depths decreased because a larger A implied a wider, shallower morphology. Conversely, A < A0 implied a
deeper, narrower channel and thus resulted in greater depth estimates. Variations in A were manifested as
an increase (decrease in absolute value) in b1 for larger A (Figure 6a), leading to smaller depth estimates.
^ decreasing from 101 to
The reduction in d for large A also decreased the estimated discharge, with Q
42.9 m3/s, relative to a field-measured Q of 70.9 m3/s, as A increased from 0:75A0 to 1:25A0 (Figure 6b).
Depth retrieval accuracy also was affected by uncertainty in A. Although mean error was 0 when the true
value A0 was used as input, error increased to 0.24 m for 1:25A0 and decreased to 20.39 m for 0:75A0 . Bathymetric precision was less sensitive to A, particularly for A > 0:9A0 (Figure 6c). Positive mean error for A
> A0 indicated a shallow bias due to the broad, shallow morphology implied by a larger A, whereas a deep
bias occurred for A < A0 due to the imposition of a narrower, deeper channel form. These effects are illustrated by the XS in Figure 6d, where the greatest dr ðsÞ was associated with the smallest A and the shallowest dr ðsÞ with the largest A.
A second input parameter required by CAR is the minimum depth d0 along a XS; sensitivity analysis indi^ but these effects were
cated that larger d0 lead to slightly shallower depth estimates and thus smaller Q,
negligible. These results indicated that CAR would not be significantly affected by misspecification of d0; in
practice, a default value of d0 50:05 m should be adequate. Similarly, experimenting with different XS spacing indicated that CAR was robust to the number of XS’s provided as input, without any consistent trends in
^ as a function of XS spacing.
estimated d or Q
4.3.2. MKD
By definition, MKD seeks to reproduce a known, user-specified discharge, and the algorithm exactly
matched Q for both Rusty and Swallow Bends. Nevertheless, MKD requires several input parameters that
could influence estimates of d and Q. For example, the aspect ratio A was involved indirectly as a means of
obtaining an initial b via CAR and thus had a very minor effect on MKD output. For the Swallow Bend NAIP
image, a larger A lead to slightly shallower depths and smaller discharges, but to a much lesser degree than
for CAR. The MKD algorithm matched Q regardless of the input A, suggesting that the optimization routine
achieved its objective even if the initial b from CAR was inaccurate. One reason for the lack of sensitivity to
A was that a smaller (larger in absolute value) b1 translated into greater d^ r ðsÞ in deep areas but also more
^ Excluding pixels
numerous negative depth estimates in shallow areas, which thus did not contribute to Q.
^
with d < 0 from Q provided an internal buffering mechanism that made the MKD algorithm less sensitive to
misspecification of A but was also the rationale for including the fraction of pixels allowed to have negative
depth estimates as an additional input parameter.
As the name implies, an essential input to MKD is the known discharge to match. Misspecification of Q had
a significant effect on the coefficients of the X versus d relation and resulting depth estimates. For Q > Q0 ,
b0 became less negative and b1 more negative (larger in absolute value; Figure 7a), leading to greater depth
estimates. Conversely, Q < Q0 lead to shallower depth estimates. Figure 7b indicates that Q was exactly
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Figure 7. Sensitivity of the matching known discharge (MKD) algorithm to the input discharge Q, varied from 75% to 125% of the known value Q0 with all other parameters held constant. Effects of variation in Q on (a) X versus d intercept and slope coefficients, (b) the root mean square error of estimated discharges for the sampled cross-sections relative to the
specified input discharge (note units of vertical scale), and (c) depth retrieval mean and root mean square errors. (d) Illustration of the effects of variation in Q on depth estimates for the
example cross-section shown in Figure 5. The different lines in (d) represent multiples of the known input discharge Q0.
matched for 0:75Q0 21:25Q0 , but errors in Q degraded bathymetric accuracy. Mean depth retrieval error
increased to 0.12 m when d was biased shallow for the smallest Q and decreased to 20.285 m for the
largest Q due to overestimation of d. Bathymetric precision was less sensitive to Q but RMSE increased
to 0.43 cm for Q51:25Q0 . In essence, MKD reproduced the known Q regardless of whether that input
was accurate. The XS in Figure 7d indicates that Q was matched by increasing depth (and hence velocity) when the input Q was too large, or by decreasing depths (and velocities) when the input Q was too
small.
^ one means of matching Q might be to set a value of b
Because pixels with d < 0 do not contribute to Q,
that leads to large areas with d < 0. To avoid this scenario, an additional input to MKD is the fraction of pixels allowed to have d < 0. Sensitivity analysis for this parameter indicated a minor effect on depth retrieval.
The estimated b0 and b1 exhibited a stepped response when varying the negative depth fraction from 0 to
0.1, with a threshold at 0.03 as certain areas with d < d0 were first included. The objective function (9) had
a binary response because eQ was set to a large value when the allowable fraction of negative depth estimates was exceeded, in which case MKD defaulted to the initial b from CAR. The effect of XS spacing also
was examined but had a very minor effect on depth retrieval.
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5. Discussion
Although Marcus and Fonstad [2010] championed the potential for remote sensing of rivers to foster scientific advances and facilitate resource management, progress toward these goals has been hindered by continued reliance upon empirical methods that require field measurements for calibration. An alternative
approach, introduced by Fonstad and Marcus [2005] and explored further in this study, retrieves depth information from passive optical image data by incorporating principles of open channel flow. Linking a linear
relationship between an image-derived quantity X and depth d to a pair of fundamental hydraulic relations—the continuity equation for discharge and a flow resistance equation that predicts velocity as a function of depth and slope—allows for remote sensing of river bathymetry even when field measurements are
not available for calibration. This connection between radiative transfer and fluid mechanics is the conceptual basis for Flow REsistance Equation-Based Imaging of River Depths (FREEBIRD).
Initial testing of two FREEBIRD algorithms, one based on a user-specified channel aspect (width/depth) ratio
and one designed to match a known discharge, produced accurate depth estimates from a panchromatic
satellite image and multispectral aerial photography of the gravel-bed Snake River. These results suggest
that FREEBIRD could play a key role in expanding the application of remote sensing methods to certain
types of rivers. For example, the capacity to map bathymetry from space without field measurements for
calibration could facilitate river research and management because satellite images can be obtained from
data archives or by tasking a sensor to collect a new image from an area of interest. Similarly, applying
FREEBIRD methods to NAIP data, which are acquired on a 3 year cycle, could become a useful, practical tool
for stream monitoring.
FREEBIRD offers a number of advantages over traditional, field-based calibration of image-derived depth
estimates. At a practical level, the logistics of data collection are simplified because field campaigns do not
need to be coordinated with image acquisition. Because simultaneous field data are not required, FREEBIRD
can be applied to existing images acquired through monitoring programs such as the NAIP or complied in
historical archives, thus creating new possibilities to examine channel change over time. Moreover, this
alternative, hydraulic approach might provide more reliable bathymetry than regression of individual pixel
values against colocated depth measurements. Automated extraction of close, regularly spaced crosssections provides even, thorough sampling of river morphology that would be difficult to achieve in the
field and can thus avoid the calibration biases that might result from unbalanced sampling designs. The
results of this study suggest that FREEBIRD yields depth estimates that are nearly as accurate and precise as
depths calibrated directly to field measurements, perhaps obviating, or at least reducing, the need for
ground-based data collection. Also, whereas empirical calibration only yields depth estimates, incorporating
a flow resistance equation provides plausible estimates of discharge in addition to spatial information on
river bathymetry.
FREEBIRD is not without limitations, however; prospective users of these methods must bear in mind several
caveats. Most importantly, image-derived depth estimates are only as valid as the underlying X versus d
relation; if this relationship is weak or nonlinear, reliable bathymetric information cannot be obtained via
FREEBIRD, or by direct calibration. The nature and strength of the X versus d relation depends on the radiative transfer processes at work within the river of interest, as well as the technical characteristics of the
imaging system. FREEBIRD cannot circumvent issues related to turbidity, greater depths, shadows, atmospheric conditions, or sensors with limited radiometric resolution that might undermine any effort to
remotely map river bathymetry.
The hydraulic basis of FREEBIRD also must be considered critically. The foundation of these methods is a
very simple, essentially empirical representation of open channel hydraulics given by equation (2), a modified version of Manning’s equation. Strictly speaking, this formulation pertains only to steady, uniform flow
conditions, relatively simple channel geometries, and within-bank discharges. In addition, d is assumed
equivalent to the hydraulic radius, which is a reasonable approximation for wide, shallow channels but less
so for smaller aspect ratios. Similarly, this type of flow resistance equation is not intended for streams where
the bed material grain size is a significant fraction of the mean flow depth (i.e., low relative submergence).
Given this hydraulic foundation, FREEBIRD is thus most applicable to mid-sized, broad, rectangular or trapezoidal, single-thread channels observed at low to intermediate discharges. In more complex environments
and/or at higher flows, the algorithms presented herein might not be appropriate.
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Other limitations more specific to FREEBIRD include the need to specify, at a minimum, an estimate of the
channel aspect ratio and the water surface slope. The algorithm designed to match a known discharge also
requires a priori information on discharge from streamflow records and is thus likely to be restricted to
images that include gaging stations. In addition to these river-specific inputs, the FREEBIRD algorithms also
require other parameters (minimum depth for CAR and fraction of depth estimates allowed to be negative
for MKD) to which the resulting depth estimates might be sensitive, although initial testing indicated that
this was not the case. More importantly, the FREEBIRD workflow involves some subjective interpretation to
assess the nature—inverse or direct—of the X versus d relation. In most cases, this determination should be
straightforward, but for multispectral image data a greater challenge is to identify an appropriate band ratio
for defining X via equation (11). Candidate images can be generated by calculating X for all possible band
combinations, but the user must visually inspect these images and employ various qualitative criteria to
select the version of X that appears most strongly related to d. In this study, the inverse X versus d relation
for the blue/red band ratio was inconsistent with radiative transfer theory, which would have predicted a
stronger, direct relation for green/red. These results highlight the need to carefully inspect images to assess
the nature and strength of the essential relation between X and d.
Further complications might arise in attempting to apply FREEBIRD to longer river segments. This study
evaluated the performance of CAR and MKD in two short reaches for which detailed field data were available, but the potential to map bathymetry over several to tens of km remains untested. In principle, coefficients of the X versus d relation inferred at a local reach scale should also pertain to greater extents along
the same river, provided that water column optical characteristics are uniform (i.e., no significant sources of
suspended sediment) and image radiometric calibration is consistent. Moreover, the hydraulic approximations associated with the simple flow resistance equation (2) must be reasonable throughout the area of
interest: discharge and slope should remain approximately constant, without tributary inflows or abrupt
changes in channel gradient. Although the FREEBIRD algorithms were highly efficient at the reach scale considered herein, requiring only a few seconds to run on a standard personal computer, computational costs
could become a more important consideration in applying this approach to larger numbers of XS’s distributed over longer river segments.
By enabling depth retrieval in the absence of field measurements, FREEBIRD liberates the user from the
issues associated with calibration by regression. Although this freedom from field work is appealing, a lack
of field data can be limiting in other ways. Most importantly, if depth measurements are not made at the
time the image is acquired, validation of image-derived depth estimates will not be possible, or at least not
via traditional methods of comparison to field data. A lack of validation data does not preclude application
of FREEBIRD, of course, but without some means of accuracy assessment the user must have a high degree
of confidence in the algorithms. That confidence must be justified, however. Although the initial results
reported in this study indicate that FREEBIRD can provide reliable bathymetric information from a clearflowing gravel-bed river, further testing in a broader range of river environments and with various types of
image data is needed to assess the potential of this technique for wider application.
6. Conclusions
Although remote sensing has outstanding, demonstrated potential to provide river information at high spatial resolution and over long river segments, bathymetric mapping from passive optical image data has
been hindered by empirical methods that require field measurements for calibration. This study introduced
an alternative framework for Flow REsistance Equation-Based Imaging of River Depths (FREEBIRD) that enables depth retrieval in the absence of field data. Instead, bathymetry is inferred by combining hydraulic
principles with a linear relation between an image-derived quantity X and depth d. One algorithm requires
only an initial estimate of the channel aspect (width/depth) ratio while another uses streamflow records to
better constrain the bathymetry by matching a known discharge. In this study, FREEBIRD was applied to a
panchromatic satellite image and a publicly available multispectral air photo from a gravel-bed river. Comparison of the resulting depth estimates to detailed field surveys indicated that bathymetric accuracy and
precision were comparable to depth estimates directly calibrated to field data. Moreover, sensitivity analyses implied that the FREEBIRD algorithms were not overly sensitive to misspecification of the input aspect
ratio and known discharge and were robust to the selection of a limited number of additional input
parameters.
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By eliminating the need for simultaneous field data collection, FREEBIRD creates new possibilities to expand
scientific knowledge of river systems and facilitate water resource management. For example, the algorithms presented herein could allow information on channel morphology and in-stream habitat to be efficiently extracted from readily available image data sources. Similarly, because these methods can be
applied retrospectively to image time series, FREEBIRD could be used to examine changes in not only channel planform but also cross-sectional shape, from which volumes of erosion and deposition could be
inferred. By liberating users from the myriad issues associated with direct calibration of image-derived
depth estimates to field data, FREEBIRD could lead to more widespread application of remote sensing to rivers and thus play a key role in unlocking the potential for detailed, regular mapping of long river segments,
provided imaging conditions are suitable.
Notation
d
X
b0
b1
b5½b0
Q
S
n
N
M
k
w
d
b 1 0
v
Dw
q
v
^
Q
Acknowledgments
This investigation was supported by a
grant from the Office of Naval
Research Littoral Geosciences and
Optics Program (grant
N000141010873). Brandon Overstreet
and C.L. Rawlins assisted with field
work. The National Park Service
granted permission to conduct
research in Grand Teton National Park.
The University of Wyoming-National
Park Service Research Center provided
logistical support. Graham Sander, an
Associate Editor, and two anonymous
reviewers provided insightful
comments. The field and image data
used in this study are available from
the author upon request. Software
developed for implementing the
methods described herein is provided
as supporting information and is
posted on the author’s web site at
www.fluvialremotesensing.org.
LEGLEITER
j
A
d0
i
X0;i
d i
X i
w
eQ
e
s
df
d^r
flow depth.
image-derived quantity.
intercept of X versus d relation.
slope of X versus d relation.
vector of coefficients.
river discharge.
water surface slope.
Manning’s flow resistance coefficient.
number of cross-sections (XS’s).
number of pixels along a XS.
conductance coefficient.
wetted channel width.
reach-averaged mean flow depth.
cross-sectional mean velocity.
width increment along a XS.
discharge per unit width.
local depth-averaged velocity.
estimated total river discharge.
index of M pixels along a XS.
aspect (width/depth) ratio.
minimum depth along a XS.
index of N XS’s along reach.
min or max X value along XS.
mean depth of i th XS.
mean X value for ith XS.
reach-averaged wetted width.
RMSE of estimated discharges.
depth retrieval residual (error).
location coordinate vector.
field-based pixel-mean depth.
remotely sensed depth estimate.
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