ProposalBathymetry from surface waves
Evaluation of georegistered imagery
for estimating water depth
William D. Philpot
CornellUniversity, Ithaca, NY14853
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 atleast reasonable accuracy using only a short image sequence. Abileah reported results using 2-4images. Leu et al. use only one image, but require that a deep water area also be imaged. Bothreport accuracies of 10% or better in the depth retrieval; however, Ablieah did not describe hismethodology in sufficient detail to duplicate his approach, and Leu et al. did not account forsurface 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.
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
(1)
where is the wave frequency, the gravitational acceleration, the wavenumber vector, the water depth, and 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 ), Eq. (1) reduces simply to:
.(2)
Similarly, for shallow water waves (), also in the absence of a surface current, we have:
(3)
where 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 tomake 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.
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, shallow-water 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 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.
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
- Review existing imagery to locate a set of promising images to test the proposed methodology.
- Register image pairs.
- 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.
- Evaluate the accuracy of the retrieved depths and the spatial resolution achieved.
- 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.
Estimated time and funding:
July2010 – Jun 2011: $75,616
July 2011 – Jun 2012: $81,364
Total budget: $156,979
References
Abileah, R. (2006a). Mapping shallow water depth from satellite. ASPRS 2006, Reno, Nevada, 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 Technology47(6): 10-+.
Balci, M. and H. Foroosh, Subpixel estimation of shift directly in the Fourier domain. IEEE Transactions on Image Processing, 2006. 15(7):1965-1972.
Baumker, M. and F.J. Heimes, New Calibration and Computing Method for Direct Georeferencing of Image and Scanner Data Using the Position and Angular Data of an Hybrid Inertial Navigation System, in OEEPE-Workshop Integrated Sensor Orientation. 2001: Hannover, Germany.
Dugan, J. P., G. J. Fetzer, et al. (2001). Airborne optical system for remote sensing of ocean waves. Journal of Atmospheric and Oceanic Technology18: 1267-1276.
Dugan, J. P., C. C. Piotrowski, et al. (2001). "Water depth and surface current retrievals from airborne optical measurements of surface gravity wave dispersion." Journal of Geophysical Research-Oceans106(C8): 16903-16915.
Grilli, S.T. and J. Skourup, Depth inversion for nonlinear waves shoaling over a barredbeach, in Coastal Engineering 1998. 1998, American Society of Civil Engineers: Copenhagen, Denmark.
Grodecki, J. and G. Dial, Block adjustment of high-resolution satellite images described by rational polynomials. Photogrammetric Engineering and Remote Sensing, 2003. 69:59-68.
Leu, L. G., Y. Y. Kuo, et al. (1999). Coastal bathymetry from the wave spectrum o f SPOT images. Coastal Engineering Journal41: 21-41.
Wozencraft, J. and D. Millar, Airborne lidar and integrated technologies for coastal mapping and nautical charting. Marine Technology Society Journal, 2005. 39(3):27-35.