Bathymetry from remote sensing wave propagation
Contents
Surface topographic patterns
Water motion produces undulation patterns at the water surface that contain information about the seafloor bathymetry. This offers the possibility to determine the bathymetry through observation of specific features of the water surface topography. Over the past fifty years, various techniques have been developed that provide information about patterns at the water surface. These techniques use remote sensing and require much less measurement effort than traditional measurements carried out with ships. On the other hand, unlike direct shipbased measurements of water depth, remote sensing data provides indirect information that can only be interpreted using advanced analysis techniques. In addition, bathymetry determination based on wave remote sensing is inherently limited by the wavelengths and time in which propagating waves respond to a spatially varying bottom^{[1]}. The most accurate results are obtained when substantial waves form a regular surface pattern, as in the case of large swell waves.
This article discusses some of the processes that determine the relationship between bathymetry and surface undulation patterns. The focus is on the nearshore zone and surface patterns produced by propagating waves.
Several remote sensing techniques can be used to determine wave patterns at the water surface:
 Video camera mounted on a drone^{[2]}^{[3]} or on a fixed tower^{[4]}^{[5]}, see also Argus applications
 LiDAR (Light Detection And Ranging) mounted on an UAV^{[6]}
 Radar microwave Xband radar^{[7]}^{[8]}^{[9]}, see Use of Xband and HF radar in marine hydrography
 Satellite optical imagery^{[10]}, see Satellitederived nearshore bathymetry
In this article we will not discuss the retrieval of surface wave patterns from remote sensing images; information can be found in the articles cited above. In the following we will assume that wavelength, wave period and wave height can be determined from the remote sensing images in every point of the considered nearshore area.
Depth inversion algorithms
The wave dispersion relation is the key to determining bathymetry from the wavelength [math]\lambda[/math], the wave period [math]T = 2 \pi / \omega[/math] and the wave height [math]H[/math]. According to linear wave theory, the wave dispersion relation can be written as
[math]c = u + \Large\frac{gT}{2 \pi }\normalsize \, \tanh kh , \qquad (1)[/math]
where [math]c = \lambda / T = \omega / k[/math] is the wave celerity, [math]k = 2 \pi / \lambda[/math] is the wave number (the length of the wave number vector [math]\vec{k}[/math]), [math]\omega[/math] is the radial wave frequency, [math]g \approx 9.8 \; ms^{2}[/math] is the gravitational acceleration, [math]h[/math] is the still water depth and [math]u[/math] is the surface current velocity in the wave propagation direction. In the following we assume that the surface current velocity is nil or very small, [math]u \lt \lt c[/math]. If this condition is not satisfied but the surface current velocity is known, it can be subtracted from the measured wave celerity [math]c[/math]. In the converse case where the bathymetry is known and the surface current velocity is substantial but unknown, the component [math]u[/math] of the surface current velocity in the direction of the wave propagation velocity can be determined by observation of the wave propagation velocity [math]c[/math] ^{[11]}^{[12]}.
The local water depth [math]h[/math] can be found by inversion of formula (1):
[math]h = \Large\frac{1}{k }\normalsize \, \tanh^{1} \Big( \Large\frac{2 \pi c}{g T}\normalsize \Big) . \qquad (2)[/math]
In shallow water, [math] kh \lt \lt 1[/math], the relationship (1) between water depth and wave celerity becomes [math]c^2 \approx gh[/math]. Knowledge of the wave celerity is sufficient to determine the water depth. In deep water, [math]k h \gt 1[/math], and [math]\tanh kh \approx 1[/math]. According to Eq. (1), the wave celerity in deep water does not depend on the depth, meaning that the depth cannot be determined from inversion of the dispersion relation.
Other restrictions on the use of Eq. (1) are the assumptions underlying linear wave theory. These assumptions are: (i) irrotational wave flow, (ii) the wave amplitude [math]H[/math] is much smaller than the water depth ([math]H/h \lt \lt 1[/math]) and (iii) much smaller than the wavelength ([math]H / \lambda \lt \lt 1[/math]). However, these assumptions are not satisfied in the nearshore zone where waves become skewed and asymmetric and eventually break.
Although equation (2) generally provides a reasonable approximation, it may be necessary to use more accurate formulas that take into account the nonlinearity of wave propagation in shallow water. Dispersion relations for nonlinear wave propagation are given in the article Nonlinear wave dispersion relations. Application of these formulas require knowledge of the wave height [math]H[/math] or even the whole wave field. The water depth [math]h[/math] can be determined by numerical inversion of the dispersion relations.
Related articles
 Nonlinear wave dispersion relations
 Use of Xband and HF radar in marine hydrography
 Satellitederived nearshore bathymetry
 Bathymetry German Bight from Xband radar
 Waves and currents by Xband radar
 Statistical description of wave parameters
References
 ↑ Almar, R., Bergsma, E. W., Thoumyre, G., Baba, M. W., Cesbron, G., Daly, C., Garlan, T. and Lifermann, A. 2021. Global satellitebased coastal bathymetry from waves. Remote Sensing 13, 4628
 ↑ Bergsma, E.W.J., Almar, R., de Almeida, L. P.M. and Sall, M. 2019. On the operational use of UAVs for videoderived bathymetry. Coastal Engineering 152, 103527
 ↑ Lange A.M.Z., Fiedler, J.W., Merrifield, M.A. and Guza, R.T. 2023. UAV videobased estimates of nearshore bathymetry. Coastal Engineering 185, 104375
 ↑ Stockdon, H.F. and Holman, R.A. 2000. Estimation of wave phase speed and nearshore bathymetry from video imagery. Journal of Geophysical Research 105(C9): 22015–22033
 ↑ Aarninkhof, S.G.J., Ruessink, B.G. and Roelvink, J.A. 2005. Nearshore subtidal bathymetry from timeexposure video images. J. Geophys. Res. 110, C06011
 ↑ Fiedler, J.W., Kim, L., Grenzeback, R. L., Young, A.P. and Merrifield, M.A. 2021. Enhanced surf zone and wave runup observations with hovering dronemounted lidar. Journal of Atmospheric and Oceanic Technology 38(11): 1967–1978
 ↑ Bell, P. S. 1999. Shallow water bathymetry derived from an analysis of Xband marine radar images. Coastal Engineering 37: 513527
 ↑ Greidanus, H. 1997. The use of radar for bathymetry in shallow seas. The Hydrographic Journal 83: 13–18
 ↑ Gawehn, M., van Dongeren, A., de Vries, S., Swinkels, C., Hoekstra, R., Aarninkhof, S. and Friedman, L. 2020. The application of a radarbased depth inversion method to monitor nearshore nourishments on an open sandy coast and an ebbtidal delta. Coastal Engineering 159, 103716
 ↑ Almar, R., Bergsma, E.W., Thoumyre, G., Baba, M.W., Cesbron, G., Daly, C., Garlan, T. and Lifermann, A. 2021. Global satellitebased coastal bathymetry from waves. Remote Sensing 13, 4628
 ↑ Streßer, M., Carrasco, R. and Horstmann, J. 2017. Videobased estimation of surface currents using a lowcost quadcopter. Geosci. Rem. Sens. Lett. IEEE 14: 2027–2031
 ↑ Fairley, I., King, N., McIlvenny, J., Lewis, M., Neill, S., Williamson, B.J., Masters, I. and Reeve, D.E. 2024. Intercomparison of surface velocimetry techniques for dronebased marine current characterization. Estuarine, Coastal and Shelf Science 299, 108682
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