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 for a regular pattern of substantial 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. Without giving details of how this is done, 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]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 this formula:
[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.
Another restriction on the use of Eq. (1) are the assumptions underlying linear wave theory. These assumptions are: (i) irrotational wave flow, (ii) [math]H/h \lt \lt 1[/math] and (iii) [math]H / \lambda \lt \lt 1[/math]. However, these assumptions are not satisfied in the nearshore zone where waves become skewed and asymmetric.
If weak nonlinearity is assumed in the shoaling zone (prior to wave breaking) nonlinear Stokes theory can be applied if the Ursell number [math]U_r = kH / (kh)^3[/math] is small. In this case the dispersion relation can be approximated by^{[13]}
[math]c = \Large\frac{gT}{2 \pi }\normalsize \, \sigma \Big( 1 + \Large\frac{9  10 \sigma^2+9 \sigma^4}{32 \sigma^4}\normalsize (kH)^2 \Big) + O[(kH)^4] , \qquad \sigma = \tanh kh .\qquad (3)[/math]
From this formula [math]\sigma[/math] and [math]h[/math] can be determined by a numerical inversion procedure.
Field observations of wave height and wave celerity show that the shallow water linear dispersion relation underestimates the wave speed at wave breaking and inside the surf zone. Measured celerity values can be 20% higher than predicted by the linear dispersion relation^{[14]} or even more^{[15]}. Weak nonlinearity cannot be assumed in the zone where waves are breaking. If the wave after breaking is surfing onshore like a bore, the bore formula for the celerity can be applied (see Tidal bore dynamics, Eq. (1) ),
[math]c = \sqrt{gh} \, \sqrt{(1+\large\frac{H}{h}\normalsize)(1+\large\frac{H}{2h}\normalsize)} . \qquad (4)[/math]
An alternative approach is applying cnoidal wave theory. This gives^{[14]}
[math]c \approx \sqrt{g h} \, \sqrt{1 + \alpha \large\frac{H}{h}\normalsize } , \qquad (5)[/math]
where [math]\alpha[/math] is a function of the Ursell number with value close to 1. Empirical evidence^{[16]} suggests [math]c \approx \sqrt{g h} \, \sqrt{1 + 0.45 \large\frac{H_s}{h}\normalsize } [/math], where [math]H_s[/math] is the significant wave height.
A physicsbased approach uses a modified dispersion relation according to the Boussinesq theory that describes the propagation of weakly nonlinear and weakly dispersive waves for Ursell numbers of order unity ([math]O[H/h] \sim O[(kh)^2] \lt \lt 1[/math]). In this theory, nonlinear interactions between resonant triads of frequencies ([math]\omega, \pm \omega', \omega \mp \omega'[/math]) lead to the growth of forced highfrequency components that modify the wave shape in shallow water. The resulting dispersion relation to order [math](kh)^2[/math] is^{[17]}^{[18]}
[math]c(\omega) = \Large\frac{\omega}{k(\omega)}\normalsize = \sqrt{gh} \Big[ 1 + \Large\frac{h \omega^2}{3g} + \frac{h^2 \omega^4}{36g^2}  \frac{1}{h}\normalsize \gamma_{am} \Big]^{1/2} , \qquad \gamma_{am} = \Large\frac{3}{2  \hat{\eta}(\omega)^2}\normalsize \, \int_{\infty}^{\infty} \Re \big( \hat{\eta}(\omega') \hat{\eta}(\omega  \omega') \hat{\eta}^*(\omega) \big) d \omega' , \qquad (6)[/math]
where [math]\hat{\eta}(\omega) = \hat{\eta}(k, \omega)[/math] is the Fourier transform of the surface elevation [math]\eta (x, t) = \int \int \hat{\eta}(k, \omega) \exp(i(kx\omega t)) dk d\omega[/math]. Application of this dispersion relation requires datasets of the free surface elevation with high space and time resolution from which [math]\hat{\eta}(\omega)[/math] can be determined. Lidars currently offer the most robust and practical solution for collecting such highlyresolved surface elevation data in the field. The depth [math]h[/math] can be determined from Eq. (6) by a leastsquares fit to values of [math]c(\omega)[/math] around the peak wave frequency^{[18]}. Using this theory, reasonable agreement was found (within 10%) in laboratory experiments between the real depth profiles in the shoaling and surf zones and depth profiles derived from the interpretation of water surface patterns using the dispersion relation Eq. (6) ^{[18]}.
Symbols
Variable  Description  Variable  Description  Variable  Description 

[math]c = \omega / k[/math]  wave celerity  [math]H[/math]  wave height  [math]x[/math]  crossshore coordinate 
[math]g[/math]  gravitational acceleration  [math]t[/math]  time  [math]\eta(x,t)[/math]  wave surface elevation 
[math]k = 2 \pi / \lambda[/math]  wave number  [math]T[/math]  wave period  [math]\lambda[/math]  wavelength 
[math]h = h(x)[/math]  local still water depth  [math]U_r = kH / (kh)^3[/math]  Ursell number  [math]\omega[/math]  radial wave frequency 
Related articles
 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
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 ↑ 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
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 ↑ Holland, T.K. 2001. Application of the linear dispersion relation with respect to depth inversion and remotely sensed imagery. IEEE Trans. on Geos. and Rem. Sens. 39: 20602071
 ↑ Herbers, T.H.C., Elgar, S., Sarap, N.A. and Guza, R.T. 2002. Nonlinear dispersion of surface gravity waves in shallow water. Journal of Physical Oceanography 32: 1181–1193
 ↑ ^{18.0} ^{18.1} ^{18.2} Martins, K., Bonneton, P., de Viron, O., Turner, I.L., Harley, M.D. and Splinter, K. 2022. New Perspectives for Nonlinear DepthInversion of the Nearshore Using Boussinesq Theory. Geophysical Research Letters 50, e2022GL100498
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