Authors' reply to comments by Takuya Sakamoto, Shouhei Kidera, and Toru Sato on "seabed algorithm and comments on 'modeling and migration of 2-D georadar data: A stationary phase approach"

Copyright © 2008 IEEES. A. Greenhalgh and L. Maresco

Modeling and migration of 2-D georadar data: A stationary phase approach

Copyright © 2006 IEEE This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.This paper presents the basic kinematic and dynamic imaging and migration equations for zero-offset two-dimensional georadar profiling. The kinematic equations are derived from simple considerations of spatial impulse responses and a generating function. The dynamic equations follow from a multidimensional stationary phase approximation to the infinite spectral integrals. They show how the radar signal (amplitude and phase) depends on the shape and curvature of the reflector. The imaging equations are evaluated for the special cases of a point scatterer, a continuous reflector, and a terminating reflector. A general formula is developed by which to migrate an arbitrary shaped event of variable amplitude on the georadar section.Stewart A. Greenhalgh and Laurent Maresco

Electric Potential and Fréchet Derivatives for a Uniform Anisotropic Medium with a Tilted Axis of Symmetry

In this paper we develop analytic solutions for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green's function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσl, dG/dσt, dG/dθ0) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real-time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic eart