Proposal
Bathymetry from surface waves
Evaluation of georegistered imagery
for estimating water depth
William D. Philpot
Cornell University, Ithaca, NY 14853
Abstract
A procedure for extracting water depth using a single pair of images of surface water
waves as collected using the CHARTS system is proposed. The procedure requires careful
registration of the two images. Once that has been accomplished, the images are transformed to
the frequency domain. The phase shift of the deep water waves between the two frequency
domain images is then related to the surface current velocity and used to correct the solution for
the total velocity of the shallow water waves. The velocity of the shallow water waves is then
used to solve for the local water depth.
Introduction
Water depth may be derived using the dispersion relationship of shallow water gravity
waves in a shoaling environment if the velocity of both deep water and shallow water gravity
waves is known locally (Grilli and Skourup, 1998). Since the wave velocities may be
determined from a time sequence of images of the area in question, this suggests that it should be
possible to determine depth from imagery collected from aircraft if an appropriate sequence of
images can be acquired ( Abileah 2006a; Abileah 2006b; Dugan, 2001a; Dugan et al. 2001b;
Leu, Kuo et al. 1999). In all of these methods, accurate depths require a high accuracy in the
determination of the wave speed. The most robust (and elegant) method was that described by
Dugan et al. (2001b)]. However, this requires long periods of staring at a single location on the
water surface, a requirement that is incompatible with the operational demands of the CHARTS
system.
Only methods that require a short time series of images are really appropriate for use with
CHARTS even though there is a danger of a loss of the accuracy and precision relative to the
method of Dugan et al. (2001a). Both Abileah (2006a) and Leu et al. (1999) have demonstrated
that it is feasible to construct solutions to the dispersion equation with at least reasonable
accuracy using only a short image sequence. Abileah reported results using 2-4 images. Leu et al.
use only one image, but require that a deep water area also be imaged. Both report accuracies of
10% or better in the depth retrieval; however, Ablieah did not describe his methodology in
sufficient detail to duplicate his approach, and Leu et al. did not account for surface currents in
their solution. Nonetheless, these methods can be implemented with a short time series of images
and provide a basis for a procedure that would be appropriate for use with the CHARTS system.
The overall goal is to implement and refine a method for extracting accurate depth
estimates from a short sequence of frame camera images that could be collected using the
CHARTS system. The value of these data would be to fill in gaps in the CHARTS bathymetric
lidar data. These gaps typically occur where the water is too optically thick (turbid) for the
bathymetric lidar to operate effectively. While the resolution of the gravity wave solution will be
very course, it will still provide a measure of continuity for the data set.
Proposal
Bathymetry from surface waves
Theory
Wave kinematics bathymetry (WKB) is a simple approach to estimate local water depth from
remotely-sensed images based on the observation of ocean wave velocity (Abileah, 2006a). The
fundamental principle underlying this method is the linear dispersion relationship
 2  gk tanh(kh)  U  k ,
(1)
where  is the wave frequency, g the gravitational acceleration, k the wavenumber vector, h
the water depth, and U the current velocity vector. In the absence of a background current, and
in water for which the depth is much greater than the wavelength (the deep water limit kh 1),
Eq. (1) reduces simply to:
 2  gk .
Similarly, for shallow water waves ( kh
(2)
1), also in the absence of a surface current, we have:
c  gh ,
(3)
where c   k is the wave speed. Clearly, in the shallow water limit, the wave speed of shallow
water waves is determined by the water depth and is independent of the wavelength.
The task is to obtain the wave speed and the surface current from images showing the surface
water waves and then to calculate the local water depth by inverting the linear dispersion
relationship (Eq. (1)).
Since only long waves (shallow water waves) “feel” the bottom, the surface current
velocity is first estimated from the deep water waves and then the local depth is found based on
the shallow water wave dispersion relationship.
The process of extracting water depth from the surface water wave images consists of the
following tasks:
(1) Pre-processing: before any performing any computation, all images must be
accurately georeferenced and orthorectified.
(2) Solve for the current velocity based on deep water waves
(3) Solve for the wave speed based on the shallow water waves
(4) Solve for the depth using the linear dispersion relationship
While this would best be done using a relatively long time series of images, it is feasible to make
depth estimates using a short series of images. The challenge in using the CHARTS system is to
make an accurate estimate of wave velocity from only a pair of images taken over a relatively
short period of time (1 second) using a procedure adapted from that used by Balci and Foroosh
(2006).
The accuracy of depth estimates from gravity waves depends critically on the accuracy of
the determination of the wave velocities, and the wave velocities are critically dependent on the
resolution, field of view, and precision of the registration of the image set. Fortunately, a
procedure to produce georegistered imagery now exists a result of a recent effort to produce
mosaics of the imagery. The goal of this short project is intended to evaluate the utility of the
existing georegistered imagery for extracting bathymetry based on the surface waves.
Proposal
Bathymetry from surface waves
While it is feasible to derive water depth from a single pair of overlapping frame
camera images. There are several problems to be overcome in accomplishing this. First of all, it
is necessary to simultaneously resolve the shallow and deep water waves. The resolution of the
present camera (functionally no better than 0.4 m pixels at 400 m) is only sufficient to resolve
deep water waves in waters deeper than ~4 m. At the same time, resolving the longer, shallowwater waves is constrained by the area of overlap of an image pair. In general one can only
count on ~120 m of overlap which is only sufficient to resolve shallow-water waves
corresponding to depths of 6 m or less. Under optimal conditions (wave direction perpendicular
to the flight line) the FOV may be sufficient to resolve waves characteristic of depths up to 15 m.
(Figure 1)
Camera FOV and resolution
The images used in this trial were taken using the DT4000 RGB frame camera during a
mission just off the coast of Ft. Lauderdale on 5 Jul 2005. Due to the dynamic roll, pitch and yaw
errors, every individual image must be adjusted independently in order to produce georeferenced
and orthorectified images. Here we use a Rational Polynomial Coefficient (RPC) camera model
to perform the image orthorectification. This model requires knowledge of the interior and
exterior camera orientation. Interior orientation, which transforms the pixel coordinate system to
the camera coordinate system, includes focal length and the pixel size of the camera. All these
parameters are known from the basic camera description [6]. As for the exterior orientation,
which determines the position and angular orientation parameters associated with the image, we
require three rotations: (Omega, Phi, Kappa). However, we are given yaw, pitch, roll and the
flight altitude. Following Baumker and Heimes (2001), a transformation from yaw, pitch and roll
to Omega, Phi and Kappa can be performed providing all the information required for RPC.
To improve the performance of the RPC model, the block adjustment technique
suggested by Grodecki and Dial [8] can be used. In their modified RPC model, ground control
points (GCPs) can be added to increase the accuracy. They demonstrated that even if only one
GCP is used (located at the center of the image), the average error can be reduced from roughly
5m to 2m.
Even assuming that the images are perfectly registered, implementation of this method is
fundamentally limited by the resolution and field of view of the camera. Both shallow water and
deep water wave velocities must be derived from a single set of images. This means that both
must be resolvable within the overlap area of the two images. For a given water depth, a single
image must simultaneously have sufficient resolution to resolve the deep water waves and the
field of view (FOV) to span at least one wavelength of the longer, shallow water waves. The
conditions are illustrated in Figure 1 along with the parameters of the DT4000 camera
(Wozencraft & Miller, 2006).
Summary
It is feasible to derive water depth from a single pair of overlapping frame camera
images. This would provide data in turbid water conditions where the SHOALS lidar is
ineffective with data that is collected routinely as part of the CHARTS mission. While there are
several problems to be overcome in accomplishing this and there will be distinct limits to the
Proposal
Bathymetry from surface waves
precision of the resulting depth estimates, the bathymetry derived from the frame camera
imagery would extend the bathymetric mapping capability into areas not currently accessible by
lidar.
10000
Wavelength / FOV / pixel size [m]
Minimum FOV needed to image
2 cycles of shallow water waves
1000
Optimal
camera FOV
100
Shallow water wave limit
l ≥ 10 h
10
Typical
camera FOV
deep water wave limit
l≤5h
1
Pixel size needed to critically
resolve deep water waves.
Camera resolution
0.1
0
2
4
6
8
10
12
Depth (h) [m]
14
16
18
20
Figure 1: Illustration of the constraints on the image. For a given water depth, a single image must
simultaneously have sufficient resolution to resolve the deep water waves and the field of view to span at least
one wavelength of the longer, shallow water waves. The FOV and resolution of the existing camera when
flown at an altitude of 400 m are also shown. The gray area illustrates the effective resolution range of the
DT400 camera for resolving deep water waves. The green area represents the effective resolution ot the
DT400 camera for shallow water waves given the overlap area that might be expected from an arbitrary set
of images. The red, cross-hatched region represents the effective resolution of shallow water waves under
optimal conditions.
Tasks
1. Review existing imagery to locate a set of promising images to test the proposed
methodology.
2. Register image pairs.
3. Find the surface current and shallow water wave velocities from the registered image
pairs, solve for the depth and compare to the lidar soundings of the area.
4. Evaluate the accuracy of the retrieved depths and the spatial resolution achieved.
5. Make recommendations for a) the continuing to use this procedure (or a modified version
of it) and b) equipment or procedural changes that would enhance the effectiveness of
the method.
Proposal
Bathymetry from surface waves
Estimated time and funding:
July 2010 – Jun 2011: $75,616
July 2011 – Jun 2012: $81,364
Total budget: $156,979
References
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ASPRS.
Abileah, R. (2006b). Shallow-water bathymetry with commercial satellite - A technique for more
than 100 years has matured into a capability for rapid shallow water depth mapping. Sea
Technology 47(6): 10-+.
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Baumker, M. and F.J. Heimes, New Calibration and Computing Method for Direct
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