Gaële Perrusson

Multistatic Response Matrix of a 3-D Inclusion in Half Space and MUSIC Imaging

Recent work on the retrieval of 3-D bounded dielectric and/or magnetic inclusions in free space is extended to burial in a half-space. Though emphasis is on the case of a single inclusion, enabling the derivation of closed-form mathematical results in illuminating fashion, the approach extends to the case of (an unknown number of) well-separated inclusions. Within an asymptotic field formulation derived from exact contrast-source vector integral formulations satisfied by the time-harmonic fields and using proper reciprocity relationships of the dyadic Greens functions, the multistatic response matrix of the inclusion is constructed from the leading-order term of the fields. Its singular value structure is analyzed in detail for a dielectric or a magnetic contrast, or both contrasts. This is performed in the case of a unique electric dipole array operated in the transmit/receive mode at a single frequency. A multiple signal classification-type algorithm follows from the decomposition, yielding a cost functional the magnitude of which peaks at the inclusion center. Numerical results illustrate the above as a function of the geometric and electromagnetic parameters of the configuration. Imaging of a spherical inclusion is in particular investigated from severely noisy synthetic data, as well as of two inclusions embedded within a non-conductive or conductive half space.

Hybrid Differential Evolution and Retrieval of Buried Spheres in Subsoil

The characterization of conductive obstacles in subsoil is investigated in the induction regime. Validated by numerical experimentation, a simple model is proposed to calculate the main electromagnetic quantities of interest, the interaction between the obstacles being taken into account. The model is based on the extended Born approximation, the Lax-Foldy multiple diffraction theory, and introduction of equivalent spherical obstacles. Those are retrieved via a hybrid algorithm of differential evolution using a communication strategy between groups, which in particular enables separation of coupled obstacles close to one another.

Low-frequency solution for a perfectly conducting sphere in a conductive medium with dipolar excitation

This contribution concerns the interaction of an arbitrarily orientated, time-harmonic, magnetic dipole with a perfectly conducting sphere embedded in a homogeneous conductive medium. A rigorous low-frequency expansion of the electromagnetic field in positive integral powers (jk) to n, k complex wavenumber of the exterior medium, is constructed. The first n = 0 vector coefficient (static or Rayleigh) of the magnetic field is already available, so emphasis is on the calcu- lation of the next two nontrivial vector coefficients (at n = 2 and at n = 3) of the magnetic field. Those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) vector Laplace equations. They are given in compact fashion, as infinite series expansions of vector spherical harmonics with scalar coefficients (for n = 2). The good accuracy of both in-phase (the real part) and quadrature (the imaginary part) vector components of the diffusive magnetic field are illustrated by numerical computations in a realistic case of mineral exploration of the Earth by inductive means. This canonical representation, not available yet in the literature to this time (beyond the static term), may apply to other practical cases than this one in geoelectromagnetics, whilst it adds useful reference re- sults to the already ample library of scattering by simple shapes using analytical methods.

Conductive masses in a half-space Earth in the diffusive regime: fast hybrid modeling of a low-contrast ellipsoid

Electromagnetic three-component magnetic probes at diffusion frequencies are now available for use in slim mineral-exploration boreholes. When a source is operated at or below the surface of the Earth in the vicinity of a conductive orebody, these probes provide, after appropriate processing, the secondary vector magnetic field attributed to this body. Proper inversion of the resulting datasets requires as a first step a clear understanding of the electromagnetic interaction of model signals with model bodies. In this paper, the response of a conductive ellipsoid buried at shallow depth in a half-space Earth is investigate by a novel hybrid approach combining the localized nonlinear approximation and the low frequency scattering theory. The ellipsoidal shape indeed fits a large class of scatterers and yet is amenable to analytical calculations in the intricate world of ellipsoidal harmonics, while the localized nonlinear approximation is known to provide fairly accurate results at least for low contrasts of conductivity between a scattering body and its host medium. In addition, weak coupling of the body to the interface is assumed. The primary field accounts for the presence of the interface, but multiple reflection of the secondary field on this interface is neglected. After analyzing the theoretical bases of the approach, numerical simulations in several geometrical and electrical configurations illustrate how estimators of the secondary magnetic field along a nearby borehole behave with respect to a general-purpose method-of-moments (MoM) code. Perspectives of the investigation and extensions, in particular, to two-body systems, strong coupling to the interface, and high contrast cases, are discussed.

Electromagnetic scattering by a triaxial homogeneous penetrable ellipsoid: Low‐frequency derivation and testing of the localized nonlinear approximation

The field resulting from the illumination by a localized time-harmonic low-frequency source (typically a magnetic dipole) of a voluminous lossy dielectric body placed in a lossy dielectric embedding is determined within the framework of the localized nonlinear approximation by means of a low-frequency Rayleigh analysis. It is sketched (1) how one derives a low-frequency series expansion in positive integral powers of (jk), where k is the embedding complex wavenumber, of the depolarization dyad that relates the background electric field to the total electric field inside the body; (2) how this expansion is used to determine the magnetic field resulting outside the body and how the corresponding series expansion of this field, up to the power 5 in (jk), follows once the series expansion of the incident electric field in the body volume is known up to the same power; and (3) how the needed nonzero coefficients of the depolarization dyad (up to the power 3 in (jk)) are obtained, for a general triaxial ellipsoid and after careful reduction for the geometrically degenerate geometries, with the help of the elliptical harmonic theory. Numerical results obtained by this hybrid low-frequency approach illustrate its capability to provide accurate magnetic fields at low computational cost, in particular, in comparison with a general purpose method-of-moments code.

Electric and magnetic dipoles for geometric interpretation of three-component electromagnetic data in geophysics

In the second sentence of section 5.5 (Inversion of a sphere response) the infinite conductive space is given incorrectly as 5 Ω m. This value should be 50 Ω m.

LOW-FREQUENCY DIPOLAR EXCITATION OF A PERFECT ELLIPSOIDAL CONDUCTOR

This paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results.

The localized nonlinear approximation in ellipsoidal geometry: a novel approach to the low-frequency scattering problem

The localized nonlinear approximation provides a very effective method within the integral equation framework of electromagnetic scattering theory. Existing results in this direction are confined to the spherical geometry alone. In this work, we extend the known results for the sphere to the case of ellipsoidal geometry which can approximate genuine three-dimensional scattering obstacles. Reduction to prolate and oblate spheroids, where rotational symmetry is present, is also discussed.

MUSIC-type Imaging of Dielectric Spheres from Single-Frequency, Asymptotic and Exact Array Data

Imaging of a dielectric sphere from its Multi-Static Response (MSR) matrix at a single frequency of operation is considered herein via a MUSIC-type, non-iterative method. Synthetic data are both asymptotic ones and data calculated by the Coupled Dipole Method (CDM) which, in contrast, models the wavefield in exact fashion. Comparisons of scattered fields, distributions of singular values, and MUSIC images are carried out. In particular, even far beyond the domain of application of the asymptotic modeling (on which the analysis of the MSR matrix is based), it is shown that fair localization of the sphere is achieved from CDM data.

LOW-FREQUENCY ELECTROMAGNETIC MODELING AND RETRIEVAL OF SIMPLE OREBODIES IN A CONDUCTIVE EARTH

As relevant in mineral exploration of the Earth using inductive electromagnetic means, (diffusive) scattering by a perfectly conducting ellipsoidal anomaly placed in a homogeneous conductive medium and illuminated by a time-harmonic magnetic dipole, is considered by means of a low frequency expansion in positive integral powers of (jk), k complex wavenumber of the exterior medium. The aim is to construct a simple yet robust model of the fields to identify the anomaly when magnetic fields are collected nearby. The approach is illustrated by field calculations and inversions of both synthetic and experimental data.