Ž .
Journal of Applied Geophysics 43 2000 199–213 www.elsevier.nlrlocaterjappgeo
Ground penetrating radar inversion in 1-D: an approach for the estimation of electrical conductivity, dielectric permittivity and
magnetic permeability
1
O. Lazaro-Mancilla , E. Gomez-Trevino
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Departamento de Geofısica Aplicada, Centro de InÕestigacion Cientıfica y de Educacion Superior de Ensenada CICESE ,
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Km 107 Carretera Tijuana-Ensenada, Ensenada, B.C. 22800, Mexico Received 15 September 1998; received in revised form 22 February 1999; accepted 15 March 1999
Abstract
This paper presents a method for inverting ground penetrating radargrams in terms of one-dimensional profiles. We resort to a special type of linearization of the damped E-field wave equation to solve the inverse problem. The numerical algorithm
for the inversion is iterative and requires the solution of several forward problems, which we evaluate using the matrix propagation approach. Analytical expressions for the derivatives with respect to physical properties are obtained using the
self-adjoint Green’s function method. We consider three physical properties of materials; namely dielectrical permittivity, magnetic permeability and electrical conductivity. The inverse problem is solved minimizing the quadratic norm of the
residuals using quadratic programming optimization. In the iterative process to speed up convergence we use the Levenberg–Mardquardt method. The special type of linearization is based on an integral equation that involves derivatives
of the electric field with respect to magnetic permeability, electrical conductivity and dielectric permittivity; this equation is the result of analyzing the implication of the scaling properties of the electromagnetic field. The ground is modeled using
thin horizontal layers to approximate general variations of the physical properties. We show that standard synthetic radargrams due to dielectric permittivity contrasts can be matched using electrical conductivity or magnetic permeability
variations. The results indicate that it is impossible to differentiate one property from the other using GPR data. q 2000 Elsevier Science B.V. All rights reserved.
Keywords: Ground penetrating radar; Frechet derivatives; Inverse problem; Parameter estimation; Quadratic programming
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Corresponding author. Department of Applied Geo- physics, CICESE, P.O. Box 434843, San Diego, CA
92143-4843, USA.
Fax: q1-526-174-4880;
e-mail: olazarocicese.mx
1
Paper presented at Seventh International Conference on Ground Penetrating Radar, May 27–30, 1998. Univer-
sity of Kansas, Lawrence, KS, USA.
1. Introduction
Ž .
Given a ground penetrating radar GPR sig- nal or radargram, it is of interest to determine
all the possible information about the subsurface features that gave origin to the signal. In terms
of a physical model of the ground, we need to estimate the parameters of the model. This rep-
resents an inverse problem. For the case of
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 5 9 - 2
GPR, the problem is nonlinear and besides it requires to invert for the three physical proper-
ties of materials; namely magnetic permeability, electric conductivity and dielectric permittivity,
all as functions of depth. The purpose of this paper is to demonstrate that an integral equation
formulation of the one-dimensional GPR in- verse problem can be used to recover any of the
three physical properties.
The scaling properties of Maxwell’s equa- tions allow the existence, for the case of quasi-
static fields, of simple integral equations for Ž
. electrical conductivity Gomez-Trevino, 1987 .
´ ˜
These equations were developed in an attempt to reduce the generality of linearization to the
exclusive scope of electromagnetic problems. The reduction is achieved when the principle of
Ž similitude for quasi-static fields neglecting dis-
. placement currents
is imposed on linearized forms of the field equations. The combination
leads to exact integral relations that represent a unifying framework for the general electromag-
netic inverse problem. The constructed formula- tion is exact and preserves the nonlinear depen-
dence of the model and the data. Iterating the integral equations, it is possible to recover a
conductivity distribution from a given set of data. This was demonstrated by Esparza and
Ž .
Gomez-Trevino 1996; 1997 for magnetotel-
´ ˜
luric and direct current resistivity measure- Ž
. ments. In the work of Gomez-Trevino 1999 ,
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the equations for quasi-static fields are general- ized by the inclusion of displacement currents.
Our inverse procedure is based on the general equations that involve the three electromagnetic
properties of materials.
We consider 1-D profiles and assume a plane wave approximation. The algorithm is iterative
and for the multiple evaluation of the forward problem we use the matrix propagation ap-
proach as described by Lazaro-Mancilla and
´
Ž .
Gomez-Trevino 1996 . We present analytic
´ ˜
derivatives of the electric field with respect to the three physical properties. The self-adjoint
Green’s function method as described by Ž
. McGillivray and Oldenburg 1990 is applied in
the derivations. The inverse problem is solved using quadratic programming. In this we follow
Ž .
Gill et al. 1986 . To ensure convergence we apply
the Levenberg–Mardquardt
method Ž
. Levenberg, 1944; Mardquardt, 1963 .
We present some examples of parameter esti- mation and show how a synthetic radargram due
to dielectric permittivity variations can be inter- preted as if it were produced by magnetic per-
meability contrasts which, although not impos- sible to find in nature, they may in practice be
very unlikely to occur. Magnetic permeability variations are commonly neglected because in
most GPR applications the magnetic character- istics of geological materials are seldom differ-
ent from those of free space. However, signifi- cant permeability variations can be associated to
horizontal layers with high concentrations of magnetic minerals in a paleo-beach environ-
ment, enriched in hematite or limonite which are produced by chemical alterations of mag-
netite or titanomagnetite, heavy minerals whose relative permeabilities may vary from 1.0064 to
Ž .
6.5 Rzhevsky and Novik, 1971 . The radar- grams due to permittivity variations may also be
explained by electrical conductivity contrasts. In this case, the ambiguity is more likely to occur
in practice given the wide range of variation of this property.
2. Theory
