Comparison of wavelet- and FFT-based bathymetry retrieval methods and its application to nearshore X-band radar image sequenses

Abstract

Wavelet- and Fast Fourier Transform (FFT)-based methods for bathymetry retrieval from X-band radar image sequences are compared and analyzed. Both methods utilize the similar idea of the waves' phase shift estimation using cross-spectral analysis. Within the FFT-based approach the corresponding technique is used to determine wave vector's components from the image sequence frequency decomposition. The last means that the time FFT is applied to the original image sequence. Then for each frequency slice the corresponding wavenumber is derived applying the cross-spectrum analysis of the one pixel shifted images in the corresponding spatial direction. In such a way a set of wavevector-frequency (k, ω) pairs are formed and filtered according to a confidence criterion that reflects the stability of the local phase pattern. In the case of a wavelet-based method the corresponding cross-spectral analysis is applied to the 2D Continuous Wavelet Transform (CWT) directional complex spectra for pairs of successive images, resulting in a set of wavevector-celerity (k, c) pairs. Further, the corresponding set of pairs are fitted to the unknown depth using nonlinear least-square method and finite water depth linear dispersion relationship as a model. Weights proportional to the spectral power density and confidence values are used in the fitting process for the wavelet- and FFT-based methods correspondingly. Furthermore, both methods are verified by applying it to stochastic simulations of corresponding shoaling sea elevation image sequences and real X-band radar image sequences collected near the Hofn tidal inlet (Iceland). For the wave simulations, a linear solution of a mild slope equation is utilized. In order to accout for the effects ofthe ambient currents, a ray-tracing technique is applied. As a testing case, the shoaling of an incident JONSWAP spectrum-based wavefields are evaluated both on the following and opposing currents. A radar image model including tilt and shadowing modulations together with speckle noise is further applied to the modeled surface elevations. Both methods are able to reconstruct the original bathymetry for intermediate to shallow water depths (kph<1.2) with plausible accuracy both for all the synthetic cases (with varied probing geometries, bottom topography, ambient current, and sea state conditions) and real radar data case. In the last case, the accuracy of the FFT-based method is on the level 0.7-0.9 m in terms of the mean absolute error value with fairly small bias the standard deviation of the error is also less than 1 m in the whole area studied except the tidal channel, where the depth gradients are significantly larger. The wavelet-based method showes a higher bias with comparable mean absolute error and standard deviation.