Introduction Directory UMM :Data Elmu:jurnal:J-a:Journal Of Applied Geophysics:Vol43.Issue2-4.2000:

Ž . Journal of Applied Geophysics 43 2000 271–280 www.elsevier.nlrlocaterjappgeo A matched-filter approach to wave migration Carl Leuschen a, , Richard Plumb b,1 a Radar Systems and Remote Sensing Laboratory, Department of Electrical Engineering and Computer Science, The UniÕersity of Kansas, 2291 IrÕing Hill Road, Lawrence, KS, USA b Department of Electrical Engineering, State UniÕersity of New York at Binghamton, Binghamton, NY 13902-6000, USA Received 20 October 1998; received in revised form 14 April 1999; accepted 13 May 1999 Abstract Wave migration is a technique in which the reflectivity of the Earth is interpreted by extrapolating the fields measured on the surface into the ground. The motivation of this paper is to develop a generalized imaging algorithm based on a matched-filter that shows a mathematical connection between currently used migration techniques. The filter is determined by estimating the received signal when a specific test target exists in the ground. To keep the method general, a point scatterer is used as this target, while distributed objects are modeled without changing the filter characteristics by a collection of independent point scatterers. Also, the specific forms of the Green’s functions, which describe wave propagation in the ground, are not included in the formation of this approach leaving more freedom in the implementation. When the filter is applied to measured data of a monostatic survey, the resulting method becomes a forward scattering problem in which these data become time-reversed current sources. Next, specific forward scattering techniques are applied to this matched-filter approach and the resulting methods are compared to traditional migration techniques. In doing so, we find that the general form of most migration techniques can be shown using a matched-filter, while the major differences lie in the actual interpretation of the wave propagation that is used to implement the filter. The similarities of the matched-filter-based approaches to traditional techniques are used to show a connection and general overview of wave migration. Finally, these methods are applied to data collected over pipes buried in sand. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Ground-penetrating radar; Imaging; Migration; Scattering

1. Introduction

Wave migration, traditionally applied to seis- mic applications and recently to ground- penetrating radar, includes a number of tech- Corresponding author. Tel.: 1-785-864-7739; fax: 1- 785-843-7789; e-mail: leuschenrsl.ukans.edu 1 Tel.: 1-607-777-4846; fax: 1-607-777-4464; e-mail: rplumbbinghamton.edu. niques that are used to transform the scattered- field response from the x–t domain or image space into the x–z domain or object space ŽClaerbout, 1971; Berkhout, 1981, 1982; Gazdag . and Sguazzero, 1984 . Due to the large beamwidths of the receiving elements, images displayed in the x–t domain usually contain energy dispersed across much of the receiving array resulting in directional ambiguities. As a result, the only information that can be directly 0926-9851r00r - see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S 0 9 2 6 - 9 8 5 1 9 9 0 0 0 6 4 - 6 interpreted from a single time response is range information when depth is desired. The transfor- mation into the x–z domain redistributes the energy in such a manner that these scattering events are migrated to their true location in space where depth can now be interpreted. The main goal of this paper is to provide a general overview of wave migration for ground-penetrating radar using a matched-filter approach. The motivation is to provide a theo- retical connection to traditional seismic tech- niques so their use can be extended to radar applications. Using a generalized matched-filter Ž . definition Leuschen and Plumb, 1998 and spe- cific solutions and approximations to the wave equation, this paper outlines four methods for transforming a monostatic radar survey from the x–t domain to the x–z domain. Also, three of these methods are associated to traditional mi- gration algorithms, not only showing the con- nection between a matched-filter and migration but also a common link between existing tech- niques. This paper will begin by examining the matched-filter technique and the general equa- tion for an image. The basic steps of the formu- lation will be presented along with the neces- sary approximations and assumptions. Next, four forward scattering techniques will be applied to this image equation. These methods include ba- sic spherical spreading, propagation in the f–k domain, a far-field approximation of the wave equation, and the finite-difference time-domain Ž . FDTD method. Finally, these methods will be related to existing techniques and applied to measurements collected over four air-filled PVC pipes buried in sand as a comparison.

2. The matched-filter technique