A Comparative Overview of Geophysical Methods

This report was prepared with support from the Air Force Research Laboratory, under contract FA8718-07-C-0021.The shallow subsurface structure of the Earth is important to understand for many economic and safety reasons. The problem is usually difficult due to complexity of the earth’s subsurface processes especially near the surface. A number of geophysical methods are used for this purpose using different physical characteristics of the Earth materials. A particular geophysical method illuminates part of the problem, but a reliable solution can only be found by combining results of different methods. In order to synthesize information from different geophysical methods, it is important to understand their similarities and differences. The aim of this study is to correlate the basic principles of geophysical methods side-by-side starting from fundamental equations. This study reveals that many analogies exist among these methods both in their mathematical formulation, and sometimes, in ways they are used in the geophysical applications

Elastic scattering by unbounded rough surfaces

We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators

Elastic scattering by unbounded rough surfaces

We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators

Inverse electromagnetic scattering models for sea ice

Journal ArticleInverse 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

Uniqueness in inverse scattering of elastic waves by three-dimensional polyhedral diffraction gratings

Literaturverz. S. 35 We consider the inverse elastic scattering problem of determining a three-dimensional diffraction grating profile from scattered waves measured above the structure. In general, a grating profile cannot be uniquely determined by a single incoming plane wave. We completely characterize and classify the bi-periodic polyhedral structures under the boundary conditions of the third and fourth kinds that cannot be uniquely recovered by only one incident plane wave. Thus we have global uniqueness for a polyhedral grating profile by one incident elastic plane wave if and only if the profile belongs to neither of the unidentifiable classes, which can be explicitly described depending on the incident field and the type of boundary conditions. Our approach is based on the reflection principle for the Navier equation and the reflectional and rotational invariance of the total field

An inverse method for obtaining the attenuation profile and small variations in the sound speed and density profiles of the ocean bottom

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution May 1985The acoustic properties of marine sediments have a direct effect on the propagation of sound in the ocean. In the frequency range of interest (50 - 500 Hz) the sediment can be modelled as a fluid. Assuming horizontal stratification of the ocean bottom, the acoustic parameters of interest are the compressional wave speed, the compressional wave attenuation and density as a function of depth. An inverse method based on a perturbation technique is presented in this thesis for the determination of these parameters. A monochromatic source experiment is proposed because of the desirability of such an experiment for determining the acoustic properties of an anelastic medium. The input information is the plane wave reflection coerricent as a function of the angle of incidence at a fixed frequency. A nonlinear integral equation relating the variations of these acoustic properties from a known reference value to the plane wave reflection coefficient is derived. This is then linearised using the Born approximation. The region of validity of the Born approximation is derived and based on this the optimum angular aperture for the input data is obtained. The linearised integral equation is a Fredholm integral equation of the first kind. An acceptable stable solution of the integral equation is obtained by imposing a priori constraints on the solution. The inversion method is tested using synthetic data and inversions are carried out for various examples of the attenuation coefficient profile and the sound speed profile. The results obtained with noise free data show good agreement between the true profiles and the reconstructed profiles. The resolution obtainable with the data set is studied using the resolving power theory of Backus and Gilbert and the inversion method is shown to provide adequate resolution. The effect of additive noise in data is examined and inversions performed with noisy data yielded stable acceptable results.I acknowledge the financial support provided by the education office in the Woods Hole Oceanographic Institution and the Office of Naval Research

Exact reconstruction of ocean bottom velocity profiles from monochromatic scattering data

Submitted in partial fulfillment of the requirements for the degree of Doctor of Science at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution January 1987This thesis presents the theoretical and computational underpinnings of a novel approach to the determination of the acoustic parameters of the ocean bottom using a monochromatic source. The problem is shown to be equivalent to that of the reconstruction of the potential in a Schrodinger equation from the knowledge of the plane-wave reflection coefficient as a function of vertical wavenumber, r(kz) for all real positive k z. First, the reflection coefficient is shown to decay asymptotically at least as fast as (1/kz2) for large kz and is therefore inteqrable. The Gelfand-Levitan inversion procedure is extended to include the case of basement velocity higher than the velocity of sound in water. The neglect of bound states is shown to be justified in both clayey silt and silty clay at the 220 Hz frequency of operation. Three methods for the numerical solution of the integral equation are investigated. The first one is an "Improved Born approximation" wherein the solution is given as a series expansion the first term of which is the Born approximation while the second term represents a substantial and yet easy to implement improvement over Born. The two other methods are based on a discretization of the Gelfand-Levitan integral equation, and both avoid a matrix inversion: one by employing a recursive procedure, and the other by coupling the Gelfand-Levitan equation with a partial differential equation. Bounds are obtained on errors in the solution due either to discretization or to data inaccuracy. These methods are tested on synthetic data obtained from known geoacoustic models of the ocean bottom. Results are found to be very accurate particularly at the top of the sediment layer with resolution of less than the wavelength of the acoustic source in the water. Several effects are investigated, such as sampling, attenuation, and noise. Also examined is the gradual restriction of the reflection coefficient to a finite range of vertical wave numbers and the consequent progressive deterioration of the reconstruction. The analysis shows how to reconstruct velocity profiles in the presence of density variation when the experiment is conducted at two frequencies. Our results provide a good understanding of the issues involved in conducting a monochromatic deep ocean bottom experiment and constitute a promising technique for processing the experimental data when it becomes available.Support for this thesis was provided in part by the education office at WHOI and by the Office of Naval Research

Uniqueness in inverse scattering of elastic waves by three-dimensional polyhedral diffraction gratings

We consider the inverse elastic scattering problem of determining a three-dimensional diffraction grating profile from scattered waves measured above the structure. In general, a grating profile cannot be uniquely determined by a single incoming plane wave. We completely characterize and classify the bi-periodic polyhedral structures under the boundary conditions of the third and fourth kinds that cannot be uniquely recovered by only one incident plane wave. Thus we have global uniqueness for a polyhedral grating profile by one incident elastic plane wave if and only if the profile belongs to neither of the unidentifiable classes, which can be explicitly described depending on the incident field and the type of boundary conditions. Our approach is based on the reflection principle for the Navier equation and the reflectional and rotational invariance of the total field