Two dimensional inverse scattering from buried magnetic

147

ANNALS OF GEOPHYSICS, VOL. 51, N. 1, February 2008

Key words ground penetrating radar – magneticanomalies – electromagnetic scattering – microwavetomography – linear inverse scattering

1. Introduction

Historically, electromagnetic inverse scat-tering has been focused much more on dielec-tric anomalies than on magnetic anomalies(Colton and Kress, 1992; Chew, 1995). This isbecause in the microwave range «to deal with»a dielectric contrast is certainly more commonthan «to deal with» a magnetic contrast.

However, in some cases magnetic anom-alies can occur too. This can happen, for exam-ple, in the case of buried ceramic pipes orburied brick walls. Also, the study presented inthis paper is of interest in the diagnostics of newmaterials being invented nowadays, some ofwhich show magnetic properties in the mi-crowave range (Jarvis et al., 2004; Chen et al.,2005).

Moreover, in some situations the soil itselfexhibits magnetic more than electric properties.This occurs in presence of some magnetic min-erals (Nabighian, 1987) such as magnetite,hematite, maghemite and/or iron in its freestate, or when the soil is strongly polluted by in-dustrial contaminants.

Finally, the very recent development of theMars exploration programmes, such as MAR-SIS and SHARAD, has aroused a significant in-terest in the influence of magnetic minerals onthe loss and propagation characteristics of elec-tromagnetic waves in the GPR frequency range(Stillman and Olhoeft, 2004, 2006).

For this reason, the scattering from electricand magnetic anomalies in an electric and mag-netic soil is also worth studying. Some recentworks have dealt with inverse scattering frommagnetic anomalies. In particular, Gustafssonand He (2000) tackled the two dimensional in-verse scattering problem related to anomaliesthat show both dielectric and magnetic contraststarting from both the measurement of the elec-tric and magnetic scattered field. In addition,Abubakar and van der Berg (2004) addressed thehomologous three dimensional problem. Com-putational problems related to the forward mod-el were dealt with in Nie et al. (2006). However,Gustafsson and He (2000) and Nie et al. (2006)

Two dimensional inverse scattering from buried magnetic anomalies

Raffaele Persico (1) and Francesco Soldovieri (2)(1) Istituto per i Beni Archeologici e Monumentali (IBAM-CNR), Lecce, Italy

(2) Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA-CNR), Napoli, Italy

AbstractThis paper deals with the problem of the electromagnetic linear inverse scattering from magnetic anomaliesburied in a lossy half space, for a scalar and two dimensional case. First, the formulation of the exact model ofthe electromagnetic scattering is given. Then, the linear inverse problem is solved by resorting to the well as-sessed Singular Value Decomposition tool. The reliability of the solution procedure is tested with synthetic da-ta achieved by a FDTD code.

Mailing address: Dr. Raffaele Persico, Istituto per i Be-ni Archeologici e Monumentali (IBAM-CNR), Via perMonteroni, Campus Universitario, 73100 Lecce, Italy; e-mail: [email protected]

Vol51,1,2008_DelNegro 16-02-2009 21:28 Pagina 147

148

R. Persico and F. Soldovieri

refer to the case of objects embedded in a homo-geneous medium, and Gustafsson and He (2000)and, Abubakar and van den Berg (2004) cast theinverse problem by assuming as datum both theelectrical and magnetic scattered fields.

Unlike from the papers mentioned above,here we consider a half space two-dimensionalgeometry and assume as datum of the problemonly the scattered electric field. This choice isjustified because in GPR prospecting one es-sentially gathers a quantity roughly proportion-al to the electric field in the observation point.

The paper is organised as follows. Section IIintroduces the scattering equations relative to the«magnetic» two dimensional scalar scatteringoperator (providing all the relevant calculationsin Appendix). The same section also proposes aninversion algorithm based on a linear model ofelectromagnetic scattering. Such a simplifiedmodel is similar to the well known Born modelwidely exploited for dielectric objects (Croccoand Soldovieri, 2003; Soldovieri et al., 2007).Section III presents numerical examples wherethe linear inversion is exploited to process exactscattered field data provided by means of anFDTD code. Finally, conclusions follow.

2. Formulation of the problem

The 2D reference scenario is composed oftwo homogeneous half spaces, separated by aplanar interface at z=0 (fig. 1). The upper halfspace is free space, whereas the lower halfspace is characterized by a relative dielectricpermittivity εs and by a relative magnetic per-meability µs. Both εs and µs can be complex toaccount for losses.

We consider the forward and inverse scatter-ing problems in frequency domain. The source isassumed to be infinitely long and invariant alongthe y-axis. The considered measurement config-uration is multi-bistatic (i.e. we consider a B-scan prospecting with a fixed offset between thetransmitting and the receiving antennas) within aprefixed band of frequencies Ω = [ωmin,ωmax].The source-observation point ranges within theobservation domain Σ=[−xM, xM] located at theair/soil interface (z=0).

The objects are assumed infinitely longalong the y-axis, embedded in the lower halfspace and they are assumed to be located in theinvestigation domain D=[−a, a]×[zmin, zmin+2b].

Our aim is to obtain a spatial map of the mag-netic properties of the investigation domain D,and on the basis of this map to infer the presence,location and geometry of the magnetic anom-alies. Thus, the quantity of interest is the relativemagnetic permeability µr(x, z) in the domain D;this drives to assume, as actual unknown of theproblem, the magnetic contrast function

. (2.1)

It is assumed that the magnetic contrast is dif-ferent from zero only inside the investigationdomain D. In general, the magnetic contrast de-pends on the frequency. Here, such a depend-ence will be neglected in the inversion model.

It is supposed that the buried objects do notshow a meaningful electric contrast with thesurrounding soil. The equations of the exactelectromagnetic scattering from buried magnet-ic anomalies are fully derived in the Appendix.The final result is given as

(2.2)

( , ) ( , ) ( , , , )

( , )( , )

( , )( , )

E x z E x z k G x z x z

E x zx z

x zdx dz x z D

1

inc sim

D

m

m

2

4 dχχ

= +

+

l l

l ll l

l ll l

##

( , )( , )

x zx z

ms

r sχ µµ µ

=-

Fig. 1. Geometry of the problem for the two di-mensional case and for the adopted multi-bistaticconfiguration


149


(2.3)

where ks is the wave-number in the soil (lowerhalf-space).

Equation (2.2) accounts for the total electricfield inside the investigation domain D and isgiven as the sum of two contributions: the inci-dent field (i.e., the field in the investigation do-main D in absence of the magnetic anomalies)and the field due to the presence of scatteringobjects in the investigation domain. Thus

denotes the internal magneticdyadic Green’s function (row vector of two ele-ments) where

(2.4)

Equation (2.3) accounts for the scattered elec-tric field on the measurement domain Σ, thatrepresents the datum of an inverse scatteringproblem. The scattered field can be regarded asthe electric field radiated by a magnetic currentdensity. This magnetic current is related to theproduct of the gradient of the electric field in-side the investigation domain times the function

involving the un-known contrast. The denotesthe external magnetic Green’s function (rowvector of two elements) where

[ , ]G G G imzeim

emx e=( , ) ( ( , ))x z x z1m mχ χ+l l l l

= ( ) (

( ) )( ) ( )( ) ( )

(

exp

exp

Gk

sign z z

jk u z zk u k uk u k u

jk

41 1

imzs

zss zo o zs

s zo o zs

zs

2π

µ µµ µ

- -

- - + +-

-

3

3

-

+

( ( ))exp ju x x du-

( )( ))u z z+

l

l

l

l

d

n

#

( ( ) )

( ) ( )( ) ( )

( ( ))

exp

exp

Gk k

ujk u z z

k u k uk u k u

ju x x du

41

imxzs s

zs

o zs s zo

o zs s zo

zs

2π

µ µµ µ

=- - - +

+-

-

3

3

-

+

( ( )( ))exp jk u z z+ - +

l

l

l

d

n

#

[ , ]G Gim

imx imz=G

( , ) ( , , , ) ( , )

( , )

( , )( , )

E x z k G x z x z E x z

x z

x zdx dz x z

1

s sem

D

m

m

2 4

d /χχ

=

+

l l l l

l ll l

l l

##

(2.5)

The gradient vector of the total field is defined interms of the unique component of the electric to-tal field. In formulas, this gradient is defined as

(2.6)

Similarly to the more widely studied case of di-electric buried anomalies (Crocco and Sol-dovieri, 2003; Soldovieri et al., 2007), the scat-tering eqs. (2.2) and (2.3) make the inverse scat-tering problem non-linear and ill-posed.

In order to cope with this difficulty, we adopta simplified model of the scattering that allowsus to deal with a linear inverse scattering prob-lem. The linearization will prevent false solu-tions, intrinsically related to nonlinearity, andwill ensure the stability of the solution by adopt-ing well assessed regularization schemes basedon the Singular Value Decomposition (Berteroand Boccacci, 1998). Conversely, the adoption ofthe linear model of the electromagnetic scatter-ing, analogously to the case of reconstruction ofdielectric anomalies, does not achieve a quanti-tative reconstruction of the buried objects but on-ly retrieves information about their location, sizeand (approximately) shape.

In order to achieve a linear model for theproblem at hand, we assume the hypothesis oflow contrast levels ⎜χm⎜<< 1. In this case we have

(2.7)

and

(2.8)

i.e. the total field inside the investigation do-main can be approximated with the incident

( , ) ( , )E x z E x zinc,

( , )

( , )( , )

x z

x zx z

1 m

m

m,χχ

χ+ l ll l

l l

E

zExE

4

2222

=l

l

J

L

KKKK

N

P

OOOO

-=( ( ) ( ))

( ( ) ) ( ( ) )

( ) ( ( ))

exp exp

exp

Gk k u k u

jk u z jk u z

k u ju x x du

21

emzs s o zs s zo

o o zo zs

zs

2π ε µ µε µ

+-

-3

3

-

+l

l

#

-=( ( ) ( ))

( ( ) ) ( ( ) )

( ( ))

exp exp

exp

Gk k u k u

jk u z jk u z

u ju x x du

21

emxs s o zs s zo

o o zo zs

2π ε µ µε µ

+-

-3

3

-

+l

l

#


150


field. The approximation (2.8) can be «physi-cally» justified by stating that the objects aresmall perturbations with respect to the hostmedium.

At this point, we make the further assump-tion that

(2.9)

Let us state that eq. (2.9) cannot be straightfor-wardly inferred from eq. (2.8). However, in theframework of low level contrast, eq. (2.9) es-sentially amounts to assuming some smooth-ness properties of the incident field. Such an as-sumption is an increasingly reasonable hypoth-esis as far as the point in the investigation do-main is farther and farther from the transmittingantenna.

Now, by substituting eq. (2.7) and eq. (2.9)in eq. (2.3), we obtain the linear inverse scatter-ing model given by

(2.10)

Thus, the problem at hand is reduced to the in-version of the linear integral relation (2.10),where Es represents the datum of the problemwhile χm is its unknown.

In order to completely specify the integralrelationship (2.10), we still have to specify theincident field. Since the primary source is a fil-amentary current, the incident field can be cal-culated in a fashion analogous to the calculationof the Green’s functions provided in the Appen-dix.

The result is

(2.11)

where f is the frequency and Io is the level of thecurrent. Thus, the gradient vector of the inci-dent field is given by

( , , , )

( ( ) ( ))( ( ) ) ( ( ) )

( ( ))

exp exp

exp

E x z x z f I

k kjk z jk z

jv x x dv

inc s o S o

s zo o zs

zs zo

S

µ µ

µ ν µ νν ν

=-

+-

-3

3

-

+

l l

l

l

#

( , ) ( , )

( , )

E x z k G E x z

x z dx dz z

s sem

inc

D

m

2 4

d /χ

= l l

l l l l

##

( , ) ( , )E x z E x zinc4 4,l l l l

. (2.12)

3. Numerical results

This section shows some numerical resultsto back-up the previous formulation. The syn-thetic scattered field data have been obtained bymeans of the FDTD code GPRMAX (Giannop-ulos, 2003). Therefore, the code for the data istotally independent from the inversion code. Inparticular, GPRMAX provides total field datain time domain, therefore we have to pre-process these data to obtain scattered field datain the frequency domain. The pre-processingessentially consists of muting the first part ofthe traces, relative to the answer to the air-soilinterface and to Fourier transform the traces af-ter this muting (Soldovieri et al., 2006). The ze-ro time is chosen in the first maximum of thesimulated traces.

Equation (2.10) has been discretized by ex-ploiting Methods of Moments (Collin, 1985).In particular a point matching is considered indata space, while the magnetic contrast func-tion has been represented thanks to a function-al basis made up of Fourier harmonics along thehorizontal direction (x-direction) and pulsefunction along the depth (z-direction).

The inversion of the linear system obtainedfrom the discretization of eq. (2.10) has beenperformed thanks to the Truncated SVD(TSVD) (Bertero and Boccacci, 1998) thatachieves a stable solution of the problem.

In a first example, we considered a squareobject with sides of 0.2 m, buried in a soil at thedepth of 1.7 m (referred to the upper side). Thedata are gathered in air at 0.01 m from the inter-face with a spatial step of 0.02 m and an offsetbetween the transmitting and receiving anten-nas of 0.1 m. The observation line from the firstto the last source point is 1.88 m long, so thatwe have 95 GPR traces. The investigation do-main is 1.98m large and 2m deep, and starts

( ( ) ( ))( ( ) ) ( ( ) )

( )( ( ))

exp exp

exp

E f I

k kjk z jk z

jv

jk vjv x x dv

inc o S o

o zs s zo

zs zo

ZS

S

µ µ

µ ν µ νν ν

=-

+-

--

-

3

3

-

+l

ld n

#


151


from the air-soil interface. The work frequencyband ranges from 200 MHz to 1 GHz with afrequency step of 25 MHz. The soil exhibits arelative dielectric permittivity equal to 9 and aconductivity equal to 0.01 S/m. which corre-sponds to a complex equivalent permittivityequal to , with εo=8.854××10−12 F/m being the dielectric permittivity ofthe free space. The relative magnetic perme-ability of the soil is µs=1. The homogeneousburied object shows the same equivalent per-mittivity of the soil, whereas its relative mag-netic permeability is µr=5.

We added a white gaussian noise to the syn-thetic data, so that the signal to noise ratio forthe total field was 20 dB.

For the inversion scheme, we chose to retainin the TSVD expansion only the terms forwhich the singular values were larger than 0.01times the maximum singular value.

Figure 2 depicts the amplitude of the re-trieved contrast function normalized with re-spect to its maximum; the reconstruction iscompared with the actual object depicted with asolid line. The tomographic reconstruction ac-

( . )j f9 0 01 2r oε π ε= -

curately locates the upper side of the object anddetermines its horizontal extent.

In order to show possible effects of theshape and mutual interactions between the ob-jects, we propose a further example where theparameters are unchanged with respect to theprevious one except that we have two circularburied objects instead of one square target. Thetwo circular objects are buried at 0.7 m (withrespect to their centres), their radius is 0.1 cmand the distance between the centres is 0.25 m.

The reconstruction is now obtained by con-sidering in TSVD expansion the singular valueslarger than 0.1 the maximum singular value.

The tomographic reconstruction is shown infig. 3 and compared to the actual objects. Sincethe objects are shallower than the previous case,this time we reconstruct both the upper andlower parts of them. However, the reconstruc-tion of the lower parts is deeper than the actualbottom of the objects because the probing wavepropagates in the object more slowly than in thesurrounding soil, and this is not accounted for

Fig. 2. Modulus of the retrieved contrast functionfor a square buried magnetic object at the depth of1.7 m. The reconstruction essentially images the up-per side of the object.

Fig. 3. Modulus of the retrieved contrast functionfor two circular buried magnetic objects at the depthof 0.7 m. The reconstruction essentially images theupper side of the objects and the spot at the dept of 1m arises for the mutual interactions between the twoobjects.


152


within a linear model (Crocco and Soldovieri,2003; Soldovieri et al., 2007). Moreover, themutual electromagnetic interferences (not ac-counted by a linear model) between the two ob-jects add constructively in a point at the depthof about one meter thereby creating an artefact.

Finally, let us stress that the adoption of thetwo different thresholds in TSVD (0.01 for thefirst case and 0.1 for the second case) makes itpossible to investigate regions at different depthin the investigation domain. In particular, bylowering the TSVD threshold, we have the pos-sibility to exploit in the reconstruction a largernumber of singular functions. As a matter offact, this allows us to image deeper objects be-cause the «support» of higher order singularfunctions are located at progressively increas-ing depth.

4. Conclusions

This paper has dealt with the scatteringfrom magnetic buried anomalies. This topic hasbeen rarely addressed in literature, where in

most cases it is assumed that both the soil andthe buried objects do not have any magneticproperty. Some times, however, one can meet asituation where the soil or the buried objects orboth can have meaningful magnetic properties.As future developments we are working on acomplete model of the 2D scattering fromburied objects that show both dielectric andmagnetic properties. Moreover, we will proposean analysis of the errors related to the fact thatone may meet a magnetic object while lookingfor dielectric objects or, conversely, a dielectricobject while looking for magnetic ones.

As further future developments, we wouldalso like to address the question of the choice ofthe type of basis functions exploited to repre-sent the unknowns. In fact, this choice affectsboth the numerical efficiency of the solution al-gorithm and the accuracy in the representationof the singular functions characterizing the lin-ear operator to be inverted.

Finally, let us stress that this paper is onlythe first step towards the application of the ap-proach also in realistic situations: experimentalvalidations are in order.

Appendix

This appendix works out the mathematical formulation of the 2D scattering equations for mag-netic anomalies, the source being constituted by a filamentary electrical current (the reference geom-etry is in fig. 1).

Let us start from Maxwell’s equations

(A.1)

and let us write the total fields as the superposition of the incident (unperturbed) fields and of the scat-tered fields.

(A.2)

. (A.3)

Due to the two-dimensional geometry and to the kind of source, we know a priori that the electrical

H H Hinc S

= +

E E Einc S

= +

j H

H j E J

H

E

E

0

0

0

4#

4#

4$

4$

ωµ

ωε

ε ρ

µ

=-= +==

Z

[

\

]]]

]]]


153


field is directed along the invariance direction, whereas the magnetic field is orthogonal to it. Withrespect to the reference system of fig. 1, therefore, we have

.(A.4)

By substitution of eq. (A.2) and eq. (A.3) in eq. (A.1), after some straightforward passages, we ob-tain Maxwell equations for the scattered fields

(A.5)

being(A.6)

where ∆µ is the difference between the object and the background magnetic permeability. From the third equation in (A.5), we can rewrite the scattered electric field by means of a poten-

tial vector (also reported as Fitzgerald vector; Collin, 1985) F as

(A.7)

This equation also means that, in general, we can write the potential vector as

(A.8)

Substituting (A.7) in the second of Maxwell eq. (A.5) we have

(A.9)

Therefore, we can express the vector (Hs−jωF) as the gradient of a scalar potential function Φ, andso we have

(A.10)

By substitution of this equation in the first of Maxwell eq. (A.5), we obtain.

(A.11)

from which, by means of a well known vector identity (Collin, 1985) we have

(A.12)

kb being the wavenumber of the background medium, equal to

(A.13)( ( ) )Im

kk z

k z D k 0b b b

s s s s

0 0 0 d

d #

/ω µ ε

ω µ ε

ω µ ε= =

==

*

( )F F k F j Jb b b b meq

224 4 4: 4 ωε µ εΦ= - --

( )F j j F J1

bb meq

4# 4# 4ε ωµ ω Φ=- + -

H j F H j Fs s

&4 4ω ωΦ Φ- = = +

( )H j F 0s

4# ω- =

( , ) ( , )F F x z i F x z ix x z z= +

E F1

sb4#ε=

J j Hmeq

ω µ∆=

E j H J

H j E

E

Hj

J

0

1

s b s meq

s b s

s

sb

meq

4#

4#

4$

4$ 4$

ωµ

ωε

ωµ

=- -=

=

=-

Z

[

\

]]]

]]]

( , )

( , ) ( , )

E E x z i

H H x z i H x z i

s sy y

s sx x sz z

== +


154


where ko and ks are the wave number in the free space and in the lower half space, respectively. Fromeq. (A.12) we have

(A.14)

At this point, by exploiting the gauge of Lorentz (Collin, 1985), we can impose

(A.15)

so that eq. (A.14) evolves in

. (A.16)

Moreover, eq. (A.15) substituted in eq. (A.10) arises the expression of the magnetic scattered fieldvs. the only potential vector

. (A.17)

At this point, the problem is recast as the resolution of Helmotz eq. (A.16) with radiating boundaryconditions at infinity. In order to solve it, let us begin by writing eq. (A.16) along its components.

(A.18)

In order to solve the problem, as is customary, we consider the Fourier transform of all quantitiesalong the x-axis. The adopted convention is

(A.19)

Therefore, in the transformed domain (u, z) eqs. (A.18) are rewritten as

(A.20)

being

. (A.21)

The imaginary part of the square roots in eqs. (A.21) is meant to be negative to ensure that the am-plitude of the fields vanishes far from the sources.

Since eqs. (A.20) are linear, we can solve them by adding the solutions of suitable impulsive an-swers, as it is well known, therefore, let us begin to consider impulsive sources placed in some gener-ic (buried) point (x', z'):

(A.22)( , ) ( ) ( )

( , ) ( ) ( )

J x z I x x z z

J x z I x x z z

meqx meqx

meqz meqz

δ δ

δ δ

= - -= - -

l l

l l

k k uk k u z

k k u z Dzb b

z

zs s

2 2 0 02 2

2 2

d

d

/= - =

= -= -

*

( )

( ) .

zF

k u F J

zF

k u F J

xzb x b meqx

zzb z b meqz

2

22

2

22

22

22

ε

ε

+ =

+ =

tt t

tt t

( , ) ( , ) ( ) ,expF u z F x z jux dx i x zi i= - =3

3

-

+

t #

.

xF

zF

k F J

xF

zF

k F J

x xb x b meqx

z zb z b meqz

2

2

2

22

2

2

2

22

22

22

22

22

ε

ε

+ + =

+ + =

( )H j F

j

Fs

b b

4 4$ω ωε µ= -

F k F Jb b meq

2 24 ε+ =

F j 0b b4$ ωµ ε Φ+ =

( )F k F F j Jb b b b meq

2 24 4 4$ ωµ ε εΦ+ = + +


155


Which in the transformed domain becomes

(A.23)

with this sources, eqs. (A.20) are recast as

(A.24)

Because of the radiation condition at infinity (Collin, 1985), both (A.24) equations have a general so-lution that can be written as

(A.25)

In order to find the four functions A, B, C and D we have to impose four conditions. These aregiven as:

1) Continuity of the tangential component of the electric field at the air-soil interface.2) Continuity of the tangential component of the magnetic field at the air-soil interface.3) Continuity of the potential vector at the depth zl of the source4) Integrability of the source and of the potential vector (i.e. of eqs. (A.24)) about the depth of the

source.An important point is the fact that we can look for a solution parallel to the source both with re-

spect to the x-component and to the z-component of the magnetic current density, i.e. we can solvefor two uncoupled problems synthesized as follows:

(A.26)

Thanks to this (not trivial) fact, we can retrieve the eight quantities Ax, Bx, Cx, Dx, Az, Bz, Cz, Dz, fromtwo uncoupled linear systems The calculations are long but straightforward, and the procedure isquite known. Therefore, we can limit to provide the final results of these passages, that are

(A.27)

( ( ) ( ))( ) ( )

( ( ) ) ( ( ) )

( )( )

( ( )( ))( ) ( )( ) ( )

( ( )( ))

( )( )

( ( )( ))( ) ( )( ) ( )

( ( )( ))

expexp exp

expexp exp

expexp exp

F

k k u k uj k u I jux

jk u z jk u z z

k uj I jux


jk u z z z z

k uj I jux


jk u z z z z

0

20

2

<

< <

<

x

zo o zs s zo

o o zs meqx

zo zs

zs

s meqx

zss zo o zs

s zo o zs

zs

zs

s meqx

zss zo o zs

s zo o zs

zs

µ µε µ

εµ µµ µ

εµ µµ µ

=

+-

-

- - + +-

- +

- - - + +-

- +

ll

ll l l

ll l l

t d

d

n

n

=

=

G

G

Z

[

\

]]]]

]]]]

F F i J J i

F F i J J i

x x meq meqx x

z z meq meqz z

,

,

= == =

t t

t t

( ) ( ( ) )

( ) ( ( ) ) ( ) ( ( ) )

( ) ( ( ) )

( , )

exp

exp exp

exp

F

A u jk u z z

B u jk u z C u jk u z z z

D u jk u z z z

i x z

0

0

<

< <

<

i

i z

i zs i zs

i zs

0

= + -

-

=l

l

t

Z

[

\

]]

]]

( ) ( ) ( )

( ) ( ) ( )

exp

exp

zF

k u F I jux z z

zF

k u F I jux z z

xzb x b meqx

zzb z b meqz

2

22

2

22

22

22

ε δ

ε δ

+ = - -

+ = - -

l l

l l

tt

tt

( , ) ( ) ( )

( , ) ( ) ( )

exp

exp

J u z I jux z z

J u z I jux z z

meqx meqx

meqz meqz

δ

δ

= - -= - -

l l

l l

t

t


156


and

(A.28)

From the potential vector, we can express the scattered electric field thanks to eq. (A.7), being thecurl meant in the transformed domain. The result is (after some further straightforward passages)

(A.29)This is the solution in the transformed domain and for an impulsive source. In order to pass to thedistributed solution relevant to the case at hand, we express eq. (A.6) versus the scattered field andalong its components

(A.30)

Therefore, the elementary magnetic currents to be substituted and then integrated in eq. (A.29) aregiven by

(A.31)

After substitution of eq. (A.31) in eq. (A.29) and after integration in the variables and in the inves-

( , ) ( , )( , )

( , ) ( , )

( , ) ( , )( , )

( , ) ( , )

I x z J x z dx dzx z

x z

zE x z

dx dz

I x z J x z dx dzx z

x z

xE x z

dx dz

1

1

meq meqm

m y

meqz meqm

m y

x x

z

22

22

χχ

χχ

= = +

= = +

l l l l l ll l

l lll l

l l

l l l l l ll l

l lll l

l l

( , ) ( , )( , ) ( , )

( , )

( , ) ( , )

( , ) ( , )( , ) ( , )

( , )

( , ) ( , )

J x z j H x z jj z

E x zz

E x z

x z

x z

zE x z

J x z j H x z jj x

E x zx

E x z

x z

x z

xE x z

1

1

1

1

meq xy

b

b

b y

m

m y

meq zy

b

b

b y

m

m y

x

z

22

22

22

22

22

22

ω µ ω µ ωµµ

µ µµµ

χχ

ω µ ω µ ωµµ

µ µµµ

χχ

∆ ∆ ∆

∆

∆ ∆ ∆

∆

= = = + =

= +

= =- = + =

=- +

l l l lll l

ll l

l ll l

ll l

l l l lll l

ll l

l ll l

ll l

( ( )( ))exp jk u z z- ++

( ( ) ( ))( ) ( )

( ( ) ) ( ( ) )

( ( ) ( ))( )

( ( ) ) ( )

( )( ) ( ( ) )

( ) ( )( ) ( )

( )( )

( ( ) )( ) ( )( ) ( )

( ( )( ))

expexp exp

expexp exp

expexp

expexp exp

E

k u k uk u I jux

jk u z jk u z

k u k uu I jux

jk u z jk z z

I juxsign z z jk u z z

k u k uk u k u

k uuI jux


jk u z z z D

2

2

sy

s o zs s zo

o o z meq

zs zo

s o zs s zo

o o meqz

zs zo

meqz

zss zo o zs

s zo o zs

zs

zs

meqz

zso zs s zo

o zs s zo

zs

s x

d

d

/

ε µ µε µ

ε µ µε µ

µ µµ µ

µ µµ µ

=

- +-

- +

+ +-

-

- - - - +-

+ - - - + +-

- +

+

ll

ll

ll l l

ll l

t

d

d

n

n

=

=

G

G

Z

[

\

]]]]]

]]]]]

( ) ( )( )

( ( ) ) ( ( ) )

( )( )

( ( )( ))( ) ( )( ) ( )

( ( )( ))

( )( )

( ( )( ))( ) ( )( ) ( )

( ( )( ))

expexp exp

expexp exp

expexp exp

F

k u k uj I jux

jk u z jk u z z

k uj I jux


jk u z z z z

k uj I jux


jk u z z z z

0

20

2

<

< <

<

z

o zs s zo

o o meqz

zs zo

zs

s meqz

zso zs s zo

o zs s zo

zs

zs

s meqz

zso zs s zo

o zs s zo

zs

µ µε µ

εµ µµ µ

εµ µµ µ

=

+-

-

- - + +-

- +

- - - + +-

- +

ll

ll l l

ll l l

t d

d

n

n

=

=

G

G

Z

[

\

]]]]

]]]]


157


tigation domain D, the scattered field in the domain is obtained. In order to achieve the scattered fieldin the spatial domain an inverse Fourier transform is still required. Again, the passages are long butstraightforward, therefore we only provide the final result of them

A.32

in air and

A.33

in the soil.At this point, since the total field in the investigation domain is given by the sum of the incident

and the scattered field, eqs. ((A.31)-(A.33)) provide quite immediately eqs. (2.1) and (2.2) in the Sec-tion 2.

( , ) ( ) ( ( ) )( ) ( )( ) ( )

( ( )) ( ( ))( , )

( , ) ( , )

( ( ) )( ) ( )( ) ( )

( ( )) ( ( ))( , )

( , ) ( , )

exp

exp exp

exp

exp exp

E x z kk

sign z z jk u z zk u k uk u k u

jk z z ju x x dux z

x z

zE x z

dx dz

kk k

ujk u z z

k u k uk u k u

jk z z ju x x dux z

x z

xE x z

dx dz z

41

1

41

10>

sy ss

zss zo o zs

s zo o zs

D

zsm

m

ss zs

zso zs s zo

o zs s zo

D

zsm

m

22

22

22

22

π µ µµ µ

χχ

π µ µµ µ

χχ

= - - - + +-

- + - + +

+ - - - + +-

- + - +

3

3

3

3

-

+

-

+

l l

l ll l

l lll l

l l

l

l ll l

l lll l

l l

df

df

n

n

n

o

=

=

F

F

###

###

( , )( ( ) ( ))

( ) ( ( ) ) ( ( ) )( ( ))

( , )

( , ) ( , )

( ( ) ( ))

( ) ( ( ) ) ( ( ) )( ( ))

( , )

( , ) ( , )

exp expexp

exp expexp

E x z kk k u k u

k u jk u z jk u zju x x du

x z

x z

zE x z

dx dz

kk k u k u

u u jk u z jk u zju x x du

x z

x z

xE x z

dx dz z

21

1

21

10

sy ss s o zs s zo

o o zs zo zs

D

m

m

ss s o zs s zo

o o zo zs

D

m

m

22

22

22

22

1

π ε µ µε µ

χχ

π ε µ µε µ

χχ

= - +-

-

+ +

+ - +-

-

+

3

3

3

3

-

+

-

+

ll

l ll l

ll l

l l

ll

l ll l

ll l

l l

f

f

p

p

###

###

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Two dimensional inverse scattering from buried magnetic