Weighted Curvature Approximation Research Articles

Doppler Simulation and Analysis for 2-D Sea Surfaces Up to Ku-Band

The numerical simulation of Doppler spectrum for 2-D sea surfaces at high microwave bands requires a huge number of sample points for the sea surface, which leads to a huge amount of computation and requires a large computer memory. As a result, the Doppler simulation for 2-D sea surfaces at high microwave bands is considered to be a great challenge. In this paper, the Fourier spectrum of 2-D sea surface is decomposed into multiple blocks, and the sea surface corresponding to the complete Fourier spectrum is generated block by block. Then, the weighted curvature approximation method is adopted and the electromagnetic (EM) scattering fields of the 2-D time-evolving sea surfaces are obtained after calculating the scattering contributions of the blocks separately. In this way, the Doppler spectra of 2-D sea surfaces up to the X-band and Ku-band are simulated. To illustrate the behavior of Doppler spectrum for 2-D sea surfaces, Doppler spectra for 2-D linear and choppy wave model sea surfaces are simulated and analyzed under different radar and environment parameters. Conclusion for the evolution of Doppler spectra with incident angle, wind direction, and radar frequency is obtained. Moreover, the numerically simulated Doppler characteristics are compared with the results of analytical prediction and the results for 1-D sea surfaces. Although similar behaviors are observed in the comparisons, the details are different.

Revisiting the Short-Wave Spectrum of the Sea Surface in the Light of the Weighted Curvature Approximation

Existing models for the short-wave spectrum of the sea surface are not consistent with microwave satellite data when multi-bands and multi-incidence data sets are considered. We devise a simple parametric model for the short-wave omnidirectional spectrum of the sea surface on the basis of a three-band (C, Ku, and Ka) and multi-incidence (low, moderate, and large) data set and an improved analytical scattering model, namely the non-Gaussian Weighted Curvature Approximation. This spectrum is also constrained by several optical measurements which provide a priori conditions on the total and filtered mean-square slopes. It is compared with classical models such as Elfouhaily and Kudryavtsev unified curvature spectra. Significant differences are observed at wave numbers corresponding to the range of decimeter scales. The new spectrum is by construction fully consistent with the omnidirectional normalized radar cross section of the multi-band data set.

Analytical Techniques for the Doppler Signature of Sea Surfaces in the Microwave Regime—I: Linear Surfaces

This paper is the first in a series of two papers on the use of combined improved hydrodynamic and electromagnetic analytical models for the simulation of the ocean Doppler spectrum at microwave frequencies. Under a linear assumption for the sea surface, we derive statistical expression for the main Doppler characteristics according to asymptotic scattering models. We consider classical models such as the Kirchhoff approximation and the two-scale model, as well as the more recent weighted curvature approximation (WCA). We recover two salient features of Doppler signature in the microwave regime. First, the Doppler characteristics are very sensitive to polarization, with higher mean Doppler shift in horizontal polarization. This is correctly rendered by the WCA but not the classical models. Second, the first two moments of the Doppler spectrum exhibit a nontrivial dependence on incidence angle. Results compare favorably with rigorous numerical computations for 1-D surfaces published in the literature. The simplicity and accuracy of the analytical models provide a valuable tool for the Doppler analysis of 2-D sea surfaces.

The weighted curvature approximation in scattering from sea surfaces

A family of unified models in scattering from rough surfaces is based on local corrections of the tangent plane approximation through higher-order derivatives of the surface. We revisit these methods in a common framework when the correction is limited to the curvature, that is essentially the second-order derivative. The resulting expression is formally identical to the weighted curvature approximation, with several admissible kernels, however. For sea surfaces under the Gaussian assumption, we show that the weighted curvature approximation reduces to a universal and simple expression for the off-specular normalized radar cross-section (NRCS), regardless of the chosen kernel. The formula involves merely the sum of the NRCS in the classical Kirchhoff approximation and the NRCS in the small perturbation method, except that the Bragg kernel in the latter has to be replaced by the difference of a Bragg and a Kirchhoff kernel. This result is consistently compared with the resonant curvature approximation. Some numerical comparisons with the method of moments and other classical approximate methods are performed at various bands and sea states. For the copolarized components, the weighted curvature approximation is found numerically very close to the cut-off invariant two-scale model, while bringing substantial improvement to both the Kirchhoff and small-slope approximation. However, the model is unable to predict cross-polarization in the plane of incidence. The simplicity of the formulation opens new perspectives in sea state inversion from remote sensing data.

Two families of non-local scattering models and the weighted curvature approximation

There are several nonlocal scattering models available in the literature. Most of them are given with little or no mention of their expected accuracy. Moreover, high- and low-frequency limits are rarely tested. The most important limits are the low-frequency or the small perturbation method (SPM) and the high-frequency Kirchhoff approximation (KA) or the geometric optics (GO). We are interested in providing some insight into two families of non-local scattering models. The first family of models is based on the Meecham–Lysanov ansatz (MLA). This ansatz includes the non-local small slope approximation (NLSSA) by Voronovich and the operator expansion method by Milder (OEM). A quick review of this first family of models is given along with a novel derivation of a series of kernels which extend the existing models to include some more fundamental properties and limits. The second family is derived from formal iterations of geometric optics which we call the ray tracing ansatz (RTA). For this family we consider two possible kernels. The first is obtained from iteration of the high-frequency Kirchhoff approximation, while the second is an iteration of the weighted curvature approximation (WCA). In the latter case we find that most of the required limits and fundamental conditions are fulfilled, including tilt invariance and reciprocity. A study of scattering from Dirichlet sinusoidal gratings is then provided to further illustrate the performance of the models considered.

Weighted curvature approximation: numerical tests for 2D dielectric surfaces

The weighted curvature approximation (WCA) was recently introduced by Elfouhaily et al [7] as a unifying scattering theory that reproduces formally both the tangent-plane and the small-perturbation model in the appropriate limits, and is structurally identical to the former approximation with some different slope-dependent kernel. Due to the simplicity of its formulation, the WCA is interesting from a numerical point of view and the aim of the present paper is to establish its accuracy on some representative test cases. We derive statistical formulae for the coherent field and the cross-section in the case of stationary Gaussian random surfaces. We then specialize to the case of isotropic Gaussian spectra and perform numerical comparisons against rigorous method of moments (MoM)-based results on 2D dielectric surfaces. We show that the WCA remains extremely accurate in a roughness range where other first-order classical approximations (small-slope and Kirchhoff) clearly fail, at the same computational cost. (Some figures in this article are in colour only in the electronic version)