Dale P. Winebrenner

Inverse electromagnetic scattering models for sea ice

Inverse scattering algorithms for reconstructing the physical properties of sea ice from scattered electromagnetic field data are presented. The development of these algorithms has advanced the theory of remote sensing, particularly in the microwave region, and has the potential to form the basis for a new generation of techniques for recovering sea ice properties, such as ice thickness, a parameter of geophysical and climatological importance. Moreover, the analysis underlying the algorithms has led to significant advances in the mathematical theory of inverse problems. In particular, the principal results include the following. (1) Inverse algorithms for reconstructing the complex permittivity in the Helmholtz equation in one and higher dimensions, based on layer stripping and nonlinear optimization, have been obtained and successfully applied to a (lossless) laboratory system. In one dimension, causality has been imposed to obtain stability of the solution and layer thicknesses can be obtained from the recovered dielectric profile, or directly from the reflection data through a nonlinear generalization of the Paley-Wiener theorem in Fourier analysis. (2) When the wavelength is much larger than the microstructural scale, the above algorithms reconstruct a profile of the effective complex permittivity of the sea ice, a composite of pure ice with random brine and air inclusions. A theory of inverse homogenization has been developed, which in this quasistatic regime, further inverts the reconstructed permittivities for microstructural information beyond the resolution of the wave. Rigorous bounds on brine volume and inclusion separation for a given value of the effective complex permittivity have been obtained as well as an accurate algorithm for reconstructing the brine volume from a set of values. (3) Inverse algorithms designed to recover sea ice thickness have been developed. A coupled radiative transfer-thermodynamic sea ice inverse model has accurately reconstructed the growth of a thin, artificial sea ice sheet from time-series electromagnetic scattering data.

Layer stripping for the Helmholtz equation

We develop a new layer stripping technique for the inverse scattering problem for the one-dimensional Helmholtz equation on the half line. The technique eliminates the use of “trace formulas,” relying instead on a nonlinear Plancherel equality that provides a simple and precise characterization of the reflection data. We prove both convergence of the algorithm and wellposedness of the forward and inverse scattering problems.

Linear and nonlinear inverse scattering

In this paper we discuss one-dimensional scattering and inverse scattering for the Helmholtz equation on the half-line from the point of view of the layer stripping. By full or nonlinear scattering, we mean the transformation between the index of refraction (actually half of its logarithmic derivative) and the reflection coefficient. We refer to this mapping as nonlinear scattering, because the mapping itself is nonlinear. Another appropriate name is multiple scattering, as this model includes the effects of multiple reflections.By linear scattering we mean the Born, or single scattering, approximation. This is the Frechet derivative of the full scattering transform at the constant index of refraction, which can be calculated to be exactly the Fourier transform. In [J. Sylvester, D. P. Winebrenner, and F. Gylys-Colwell, SIAM J. Appl. Math., 56 (1996), pp. 736--754], we introduced a variant of layer stripping based on causality and the Riesz transform, rather than on trace formulas---see [A. Brickstein and...

Laboratory measurements of sea ice: connections to microwave remote sensing

The connections between laboratory measurements and remote-sensing observations of sea ice are explored. The focus of this paper is on thin ice, which is more easily simulated in a laboratory environment. The authors summarize results of C-band scatterometer measurements and discuss how they may help in the interpretation of remote-sensing data. They compare the measurements with observations of thin ice from ERS and airborne radar data sets. They suggest that laboratory backscatter signatures should serve as bounds on the interpretation of remote-sensing data. They examine these bounds from the perspective of thin ice signatures, the effect of temperature, and surface processes, such as frost flowers and slush on these signatures. Controlled experiments also suggest new directions in remote-sensing measurements. The potential of polarimetric radar measurements in the retrieval of thickness of thin ice is discussed. In addition to the radar results, the authors discuss the importance of low-frequency passive measurements with respect to the thickness of thin ice.