Abstract
A new method to invert X-band radar images for linear shoaling conditions is proposed. The commonly used approach for this type of inverse problems is the Fourier transform. Unlike in deep water conditions, in the shoaling region, waves are modulated both in terms of wavelength and amplitude. However, Fourier analysis assumes spacial and temporal periodicity, and homogeneity limiting its applicability to this region. In order to overcome these limitations, a wavelet based technique is developed. The proposed technique treats every spatial radar image within the time sequence individually, so no information on the dispersion relation is required. For validation purposes, surface elevation range-time shoaling realizations based on the mild slope equation are prepared. A radar imaging model including tilt and shadowing modulations, speckle noise, and the radar equation is applied to these realizations to provide modeled grazing incidence radar images. The inversion process starts with the application of the continuous wavelet transform independently for each spacial image. The procedure continues with employing a successive range independent modulation transfer function to the wavelet spectra in the wavenumber domain. Then, after a phase shift correction, an inverse continuous wavelet transform is applied. The procedure is finalized by a calibration of the retrieved maps. After the calibration, a thorough comparison between the original and the reconstructed surface elevations is performed. It shows high efficiency of the proposed method in treating wave number and amplitude modulated signals, as well as in addressing local phase shifts due to tilt modulation and noise contamination. The new inversion method is proven to have high accuracy in inhomogeneous conditions. It shows high potential to be implemented for individual wave reconstruction using real aperture radars.
Highlights
Incoherent nautical X-band radars have been increasingly used for oceanographic measurements in the last decades
X 0 kj (ξ) j=1 where ζ(x, t) is the surface elevation, aj is the amplitude of the wave component with angular frequency ωj and wave number kj that is uniformly distributed over [0, 2π) initial random phase φj
To reduce the computational costs, the traditional definition of the continuous wavelet transform (CWT) is slightly changed to rewrite Equation (6) using the identity W(a, b) = F −1 {F {W(a, b)}} (F is a Fourier transform), which results in the following:
Summary
Incoherent nautical X-band radars have been increasingly used for oceanographic measurements in the last decades. Their ability to probe continuously in time enables the study of both the spatial and temporal evolution of the sea surface elevation with reasonably good resolution. We refer to this work as the conventional method of radar image inversion Their main idea was to use a dispersion relation shell filter in a 3D Fourier space and to subsequently apply an empirical modulation transfer function in order to fit the spectrum of modulations to the actual wave spectrum. As for the signal processing approach for non-stationary or non-homogeneous data, several methods have been developed in the past few decades, such as the windowed Fourier transform and, more recently, the wavelet transform [14] The latter enables localization in the space or time-frequency domain through translation and dilation of the so-called mother wavelet.
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