Ground penetrating radar inversion in 1-D: an approach for

Ž .Journal of Applied Geophysics 43 2000 199–213www.elsevier.nlrlocaterjappgeo

Ground penetrating radar inversion in 1-D: an approach for theestimation of electrical conductivity, dielectric permittivity and

magnetic permeability 1

O. Lazaro-Mancilla ), E. Gomez-Trevino´ ´ ˜( )Departamento de Geofısica Aplicada, Centro de InÕestigacion Cientıfica y de Educacion Superior de Ensenada CICESE ,´ ´ ´ ´

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 resortto a special type of linearization of the damped E-field wave equation to solve the inverse problem. The numerical algorithmfor the inversion is iterative and requires the solution of several forward problems, which we evaluate using the matrixpropagation approach. Analytical expressions for the derivatives with respect to physical properties are obtained using theself-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 theresiduals using quadratic programming optimization. In the iterative process to speed up convergence we use theLevenberg–Mardquardt method. The special type of linearization is based on an integral equation that involves derivativesof the electric field with respect to magnetic permeability, electrical conductivity and dielectric permittivity; this equation isthe result of analyzing the implication of the scaling properties of the electromagnetic field. The ground is modeled usingthin horizontal layers to approximate general variations of the physical properties. We show that standard syntheticradargrams due to dielectric permittivity contrasts can be matched using electrical conductivity or magnetic permeabilityvariations. The results indicate that it is impossible to differentiate one property from the other using GPR data. q 2000Elsevier Science B.V. All rights reserved.

Keywords: Ground penetrating radar; Frechet derivatives; Inverse problem; Parameter estimation; Quadratic programming´

) Corresponding author. Department of Applied Geo-physics, CICESE, P.O. Box 434843, San Diego, CA92143-4843, USA. Fax: q1-526-174-4880; e-mail:[email protected]

1 Paper presented at Seventh International Conferenceon 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 determineall the possible information about the subsurfacefeatures that gave origin to the signal. In termsof a physical model of the ground, we need toestimate 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: S0926-9851 99 00059-2

( )O. Lazaro-Mancilla, E. Gomez-TreÕinorJournal of Applied Geophysics 43 2000 199–213´ ´ ˜200

GPR, the problem is nonlinear and besides itrequires 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 thispaper is to demonstrate that an integral equationformulation of the one-dimensional GPR in-verse problem can be used to recover any of thethree 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 attemptto reduce the generality of linearization to theexclusive scope of electromagnetic problems.The reduction is achieved when the principle of

Žsimilitude for quasi-static fields neglecting dis-.placement currents is imposed on linearized

forms of the field equations. The combinationleads to exact integral relations that represent aunifying 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 theintegral equations, it is possible to recover aconductivity distribution from a given set ofdata. 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 ,´ ˜the equations for quasi-static fields are general-ized by the inclusion of displacement currents.Our inverse procedure is based on the generalequations that involve the three electromagneticproperties of materials.

We consider 1-D profiles and assume a planewave approximation. The algorithm is iterativeand for the multiple evaluation of the forwardproblem 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 tothe three physical properties. The self-adjointGreen’s function method as described by

Ž .McGillivray and Oldenburg 1990 is applied in

the derivations. The inverse problem is solvedusing quadratic programming. In this we follow

Ž .Gill et al. 1986 . To ensure convergence weapply the Levenberg–Mardquardt methodŽ .Levenberg, 1944; Mardquardt, 1963 .

We present some examples of parameter esti-mation and show how a synthetic radargram dueto 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 bevery unlikely to occur. Magnetic permeabilityvariations are commonly neglected because inmost 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 tohorizontal layers with high concentrations ofmagnetic minerals in a paleo-beach environ-ment, enriched in hematite or limonite whichare produced by chemical alterations of mag-netite or titanomagnetite, heavy minerals whoserelative permeabilities may vary from 1.0064 to

Ž .6.5 Rzhevsky and Novik, 1971 . The radar-grams due to permittivity variations may also beexplained by electrical conductivity contrasts. Inthis case, the ambiguity is more likely to occurin practice given the wide range of variation ofthis property.

2. Theory

2.1. Forward problem

Consider a plane wave at normal incidenceupon an n-layer earth model as illustrated inFig. 1. The three electromagnetic properties varyfrom layer to layer; each layer is linear, homo-geneous and isotropic. A dependence of theform e iv t is assumed for the fields, where t istime and v angular frequency. It is furtherassumed that the physical properties do notdepend on frequency. The governing differential

Ž .equations for the electric E z and magneticx

( )O. Lazaro-Mancilla, E. Gomez-TreÕinorJournal of Applied Geophysics 43 2000 199–213´ ´ ˜ 201

Fig. 1. Layered earth model whose three electromagneticproperties vary from layer to layer. The plane wave propa-gates downward in the positive z-direction.

Ž .H z fields propagating in the z-direction are,y

for the ith layer:

d22E z yg E z s0, 1Ž . Ž . Ž .x i x2d z

d22H z yg H z s0, 2Ž . Ž . Ž .y i y2d z

where:

1r22g s im s vym ´ v 3Ž .Ž .i i i i i

is the propagation constant of the layer. Thequantities m , s and ´ represent magnetici i i

permeability, electrical conductivity and dielec-tric permittivity, respectively. The solution isobtained in the standard form of a boundaryvalue problem where we consider 2n boundaryconditions and 2n unknowns. The electric fieldin the ith layer is expressed in terms of anoutgoing and a reflected wave, i.e.:

E sA eyg i z qB egi z , 4Ž .x i ii

where A and B are coefficients that need to bei i

determined. Applying continuity of the fields ateach boundary and employing standard propaga-tion matrices, we can find the fields in one layerin terms of the fields in the next layer. Setting

the amplitude of the incident wave A s1, the0

electric field at the surface is given simply as:

E 0 s1qB . 5Ž . Ž .x 0

The radargram is computed as in Lazaro-´Ž .Mancilla and Gomez-Trevino 1996 by apply-´ ˜

ing the inverse Fourier transform to the productof the electric field and the Ricker’s pulse spec-

Ž .trum S f , i.e.:

y1EsFFT E f S f . 6Ž . Ž . Ž .x

A variety of examples are presented byŽ .Lazaro-Mancilla and Gomez-Trevino 1996 that´ ´ ˜

illustrate the signatures produced by variationsin m, s and ´ . By trial and error methods itwas possible to match radargrams due to varia-tions in one property with those produced by theother two. In the present work, we reconsiderthis issue from the point of view of an auto-matic fitting process.

2.2. InÕerse problem

Ž .In Gomez-Trevino 1999 , the differential´ ˜equations for the electric field E and magneticfield B are transformed to integral equations for

Ž .magnetic permeability m , electrical conductiv-Ž . Ž .ity s and dielectric permittivity ´ . For the

electric field, the equation is:

Es G m rX d3rX y G s rX d3rXŽ . Ž .H HE,m E ,sX XV V

y G ´ rX d3rX . 7Ž . Ž .H E,´XV

G , G and G represent the functional orE,m E,s E,´

Frechet derivatives of E with respect to m, sánd ´ , respectively. E represents the data andthe unknown functions are m, s and ´ . The

Ž .integral Eq. 7 is nonlinear because the func-tional derivatives depend on the physical prop-

Ž .erties. In our particular case, Eq. 7 can bereduced to an algebraic form for piecewise uni-form media. To this end, it is convenient towrite the functional derivatives over the ithregion in terms of partial derivatives. The inte-


gration of the functional derivatives of E withrespect to m, s and ´ can be written as:

EEX3s G d r , 8Ž .H E,m

XEm Vi i

EEX3s G d r , 9Ž .H E,s

XEs Vi i

EEX3s G d r , 10Ž .H E,´

XE´ Vi i

where V X is the volume of the ith region. Eq.iŽ .7 becomes:

n EE EE EEEs m y s y ´ . 11Ž .Ý i i iž /Em Es Eí i iis1

We base the solution of the inverse problem onŽ .Eq. 11 .

2.3. Analytic Frechet deriÕatiÕes´

A fundamental step in the solution of mostnonlinear inverse problems is to establish arelationship between changes in a proposedmodel and corresponding changes in the data.Once this relationship is established, it becomespossible to refine an initial model to obtain animproved fit to the observations. In traditionallinearized analysis, the Frechet derivative is thećonnecting link between changes in the model

Ž .and changes in the data. In Eq. 7 , the samederivative links directly the data to the model.

The usual approach for developing expres-sions for the Frechet derivatives is to perturb the´governing differential equation and thereby for-mulate a new problem that relates the change inthe response to a change in the model. Wederive the analytic derivatives using the self-ad-joint Green’s function method. The wave equa-tion for the case of arbitrary variations in thephysical properties can be written fromMaxwell’s equations as:

XX 2hE yg hEs0, 12Ž . Ž .Ž 2.1r2where ms1rh and gs imsvym´v .

Ž .Strictly speaking, Eq. 12 should include a

source term. However, for the purpose of theperturbation analysis that follows its inclusion isunnecessary because we consider perturbationsof only the physical properties and the fields.

Ž .Perturbing Eq. 12 , keeping only first orderterms in the resulting perturbations and using

Ž .Eq. 12 again there results:

hX hX

Y X X2dE q dE yg dEsyd E q2g Edg .ž /h h

13Ž .

In the self-adjoint form:

hX

XX X2hdE yg hdEsyhd E q2g hEdgŽ . ž /h

sw z . 14Ž . Ž .Ž .The solution of Eq. 14 can be obtained

through the use of the Green’s function for theŽ .layered model. Let G z, z be this function and0

Ž .d z, z the corresponding Dirac delta function.0Ž . Ž .Considering Eq. 12 , G z, z must obey:0

XX 2hG z , z yg hG z , z sd z , z . 15Ž . Ž . Ž . Ž .Ž .0 0 0

Ž . y qIntegrating Eq. 15 between z and z in the0 0

usual manner, one obtains:

h z q GX z q, z yh z y GX z y, z s1.Ž . Ž . Ž . Ž .0 0 0 0 0 0

16Ž .

We use this equation to obtain a quantitativeŽ . Ž .relationship between G z, z s0 and E z . For0

Ž . Ž .the moment consider that G z, z sG z , z .0 0

That is, G can be interpreted either as the fieldat z due to a source at z , or as the field at z0 0

due to a source at z. For convenience wechoose to consider G as the field at z due to asource at z because this field is simply the0

physical field within the layered model. ThisŽ . Ž .comes from the fact that Eqs. 12 and 15 are

one and the same equation, except for the loca-tion of the source. However, since we are con-sidering the case of a plane wave source, it isirrelevant whether the source is placed right on

Ž . Ž .top or above the model. Thus, G z,0 and E zare the same, except for a multiplying factor to


account for physical dimensions and differencesin source strength. That is:

E zŽ .G z s , 17Ž . Ž .

C

where C is independent of depth and we haveŽ . Ž .written G z, z s0 sG z .0

Ž . Ž .Substituting Eq. 17 into Eq. 16 we musthave:

X < X <hE yhE sC. 18Ž .q y

The discontinuity is produced by the sourceŽ X .at z s0. The fact that yhE riv is simply0

the magnetic field of the source provides theclue for what the value of C must be. Themagnetic fields produced by a plane sourcehave the same intensity on both sides of theplane but are of different signs. Furthermore,the intensity of the magnetic field is equal toone-half the strength of the source. Using Eq.Ž .4 we can compute the magnetic field just ontop of the model. The result can be written asŽ y. XŽ y. Ž y. Ž .h 0 E 0 s h 0 g B y 1 , where we0 0

have kept A s1 as for the computation of the0

electric field. The above-arguments then lead to:

Csyh 0y EX 0y . 19Ž . Ž . Ž .We must then have that:

m E zŽ .0G z s . 20Ž . Ž .

g B y1Ž .0 0

We get the Frechet derivative for the 1-D GPR´problem through:

`

dE 0 s G z w z d z . 21Ž . Ž . Ž . Ž .H0

Ž . Ž .Substituting w z from Eq. 14 , G from Eq.Ž . Ž .20 and developing Eq. 21 we have:

`m dm zŽ .0 X2dE 0 s y E z d zŽ . Ž .H 2g B y1 m zŽ . Ž .00 0

`2 2y E z v d´ z d zŽ . Ž .H

0

`2q E z ivds d z . 22Ž . Ž .H

0

The last two integrals are more direct toobtain than the first. To obtain the first integral

Žit is necessary to use the wave equation Eq.Ž ..12 in the intermediate steps.

Ž .In terms of Eq. 7 , we have:

m EX20

G s y , 23Ž .E,m 2ž /g B y1 m zŽ . Ž .0 0

m0 2G s E iv , 24Ž . Ž .E,sg B y1Ž .0 0

m0 2G s yE v . 25Ž . Ž .E,´g B y1Ž .0 0

Ž . Ž . Ž .Substituting Eqs. 23 – 25 into Eqs. 8 –Ž .10 , respectively, one obtains expressions forthe partial derivatives in terms of the electricfield within a given layer. Considering the ex-plicit expressions for the electric field given by

Ž .Eq. 4 , we can write for the ith layer:

EE 0 mŽ . 0sy

Em g B y1Ž .i 0 0

=

21 g Ai i y2g z y2g zi iq1 i iy e yeŽ .22 mi

2g 2A Bi i iy z yzŽ .iq1 i2mi

21 g Bi i 2g z 2g zi iq1 i iq e ye , 26Ž . Ž .2 mi

For the nth layer we have:

EE 0 m 1 g A2Ž . 0 n n y2g zn nsy e . 27Ž .2Em g B y1 2 mŽ .n 0 0 n

For electrical conductivity the derivatives are:

EE 0 mŽ . 0s iv

Es g B y1Ž .i 0 0

=

21 Ai y2g z y2g zi iq1 i iy e yeŽ .2 gi

q2 A B z yzŽ .i i iq1 i

21 Bi 2g z 2g zi iq1 i iq e ye , 28Ž . Ž .2 gi


For the nth layer:

EE 0 m 1Ž . 0 2 y2g zn ns iv A e .n ž /Es g B y1 2gŽ .n 0 0 n

29Ž .For permittivity the results are:

EE 0 mŽ . 0 2sy vE´ g B y1Ž .i 0 0

=

21 Ai y2g z y2g zi iq1 i iy e yeŽ .2 gi

1 B2i

q2 A B z yz qŽ .i i iq1 i 2 gi

= 2g z 2g zi iq1 i ie ye , 30Ž . Ž .

and the expression for the nth layer:

EE 0 m A2Ž . 0 n2 y2g zn nsy v y e .ž /E´ g B y1 2gŽ .n 0 0 n

31Ž .The values of the derivatives in the time

domain are obtained by applying the Fouriertransform as already explained in relation to Eq.Ž .6 . We use a layered model that can be madeup of many thin layers to simulate arbitraryvariations of the physical properties. The corre-sponding integral expressions reduce to systemsof simultaneous equations as explained below.

2.4. Numerical method

The numerical recovery of the physical prop-erties of the different layers is posed as follows.

Ž .We use Eq. 11 which permits us to use di-rectly quadratic programming techniques as for-

Žmulated for linear problems see for example.Gill et al., 1986 . For example, for magnetic

Ž .permeability we rearrange Eq. 11 as:n n nEE EE EE

Eq s q ´ s m . 32Ž .Ý Ý Ýi i iEs E´ Emi i iis1 is1 is1

The set of unknown permeability values isplaced on the right side of equation. The con-

ductivity and permittivity values are assumed tobe known and are placed on the left side of theequation together with the electric field data.There is one equation for each data point of theradargram. The partial derivatives are computedby assuming a model that includes values for allthe physical properties of the layers. The left-

Ž .hand side of Eq. 32 then reduces to a columnvector of numbers; the right-hand side can bewritten as the product a matrix of numbers anda column vector that contains the unknowns.The elements of the matrix are simply the par-tial derivatives of the electric field with respectto the unknowns. To each data point there cor-responds a different row of the matrix.

Ž .We use the same Eq. 11 for estimatingelectrical conductivity. What changes is the ar-rangement of the different terms. The equationis in this case:

n n nEE EE EEyEq m y ´ s s .Ý Ý Ýi i i

Em E´ Esi i iis1 is1 is1

33Ž .Again, the left-hand side reduces to a column

vector of numbers by assuming a given model.The same applies to the matrix on the right side.The difference is that now the partial derivativesare with respect to electrical conductivity. Thecolumn vector on the right side is now com-posed of the unknown conductivity values.

In the case of dielectric permittivity, we fol-Ž .low the same arrangement. In this case Eq. 11

is written as:n n nEE EE EE

yEq m y s s ´ .Ý Ý Ýi i iEm Es Eí i iis1 is1 is1

34Ž .The last three equations have the form:

ysA x , 35Ž .where y is the vector containing the radargramdata. y also contains partial derivatives and theparameters that are not considered to be un-knowns. The matrix A contains the partialderivatives with respect to the parameter that isconsidered the unknown, where in all cases is


represented by x. The problem is obviouslynonlinear: E and its partial derivatives dependon all three physical properties. This means thatthe sets of equations cannot be solved in asingle iteration using linear analysis. We applylinear analysis iteratively by assuming every-thing known, except the parameters on the rightside of the equations.

Quadratic programming consist of minimiz-ing the quadratic norm of the residuals subjectto lower and upper bound in each parameter,that is:

125 5Minimize F x s yyA x , 36Ž . Ž .

2

subject to x FxFx . The vector x representsl u l

a lower limit imposed on the properties of thelayers; x is a corresponding upper limit. Theu

inequality must be understood as applying tocorresponding components of the vectors x , xl

and x . The algorithm finds a solution for xu

such as the square of the residuals is a mini-mum, with the additional constraint that theparameters must fall within the upper and lowerbounds previously established. The function to

Ž .minimize in Eq. 36 can be represented as:

1T TF x sc xq x S x 37Ž . Ž .

2

subject to x FxFx ,l u

TT Twhere c sy A y 38Ž .Ž .and SsATA is the symmetric Hessian matrix.

Ž .Strictly speaking, Eq. 37 should include a termŽ T .of the form y y . However, the inclusion of

this term does not intervene in the minimiza-tion. The process is stabilized by adding to theHessian a term l I, with l<1. We calculatethe Hessian in a way that the diagonal is unitar-ian, so the process is modified to minimize:

1 12 2X X X5 5 5 5F x s yyA x q l x , 39Ž . Ž .

2 2

AX sAV, 40Ž .xX sVy1x , 41Ž .

1Õ s , 42Ž .ii n

2AÝ ji)js1

and Õ s0, i/ j. Finally, when we have posedi j

the problem and have the vectors and the matri-ces in accordance with the last equation, weproceed to get xX, and consider that xsVxX.To improve convergence in the iterative processwe use the Levenberg–Mardquardt methodŽ .Levenberg, 1944; Mardquardt, 1963 .

On the other hand, the rms error of estima-tion is calculated at each iteration using:

rms error %2561 2

E t yE tŽ . Ž .Ý h i e i256 is1s 100, 43Ž .) 2Eh

where:

25612E s E t , 44Ž . Ž .Ý)h h i256 is1

Ž .and where E t represents the value of theh i

hypothetical radargram for the time t . There areiŽ .256 data points. On the other hand, E t repre-e i

sents the radargram computed from the esti-mated model. Notice that we use the average Eh

Ž .as defined in Eq. 44 to normalize the rms ofŽ .Eq. 43 rather than the individual values of the

radargram. This is because a direct normaliza-Ž .tion by E t may no always be meaningful forh i

Ž .radargrams since E t s0 for many t . On theh i i

other hand, when noise is added to radargramsthis is done as a percentage of E . In this wayh

all values of the radargram are contaminatedwith noise, not only those that are different

Ž .from zero. Using percentages of E t directlyh i

does not add any noise to the quiet sections ofthe radargram.

3. Results

As a test of the effectiveness of the approachpresented here we have applied the method to


one-dimensional radargrams computed byŽ .Lazaro-Mancilla and Gomez-Trevino 1996 .´ ´ ˜

The radargrams consist of 256 data points andare generated by considering variations in elec-trical permittivity, magnetic permeability andelectrical conductivity. The object of the numer-ical experiments is to recover these variationsfrom the radargrams. We consider variations ofeach property separately, and invert the corre-sponding radargrams in terms of the propertyused for their generation. The purpose is toillustrate the applicability of the present ap-proach to the quantitative interpretation ofradargrams, and to show that it represents aviable alternative to existing inversion methods.The experiments demonstrate that there is anintrinsic ambiguity in the interpretation of radar-grams, in the sense that variations of one prop-erty can be genuinely mistaken by variations inthe other two, as originally reported by Lazaro-´

Ž .Mancilla and Gomez-Trevino 1996 who used´ ˜trial and error methods for matching radar-grams. Of particular interest is that a reflectionproduced by a discontinuity in electrical permit-tivity can be reproduced by a double discontinu-ity in electrical conductivity.

3.1. Estimation of permittiÕity

First, we consider the case of a permittivityradargram and its corresponding permittivity es-timation. The process is summarized in Fig. 2.The case is associated to a four-layer soil model

Ž .with permittivity variations Fig. 2a and uni-form conductivity and permeability. The hypo-thetical radargram was contaminated with a levelof random noise of 5% and is shown in Fig. 2b.In the estimation process, the initial model hassix layers with the following parameters: ´ si

6´ , s s0.001 Srm, m sm for is1 to 60 i i 0Ž .Fig. 2c . The conductivity and permeabilityprofiles are kept fixed to their original anduniform values. The permittivity of each of thesix layers is allowed to vary from iteration toiteration until they converge to the model shown

in Fig. 2d. The history of the process on its wayto convergence is shown in Fig. 2e. In thepresent case, the final model is obtained in nineiterations with a rms errors5.0%. The differ-ence between the estimated and hypotheticalradargrams is shown in Fig. 2f. The fact that theestimated and the hypothetical models are al-most identical, and that the residuals are of thelevel of the added noise illustrates the effective-ness of the approach for the case of permittivityestimation.

3.2. Estimation of conductiÕity

We now consider the case of conductivityestimation using as data a conductivity radar-gram. The process is summarized in Fig. 3. Themodel in this case is made up of three conduc-tive thin layers embedded in a relatively less

Ž .conductive medium of 0.001 Srm Fig. 3a .The hypothetical radargram was contaminatedwith a level of random noise of 2% and isshown in Fig. 3b. The model is uniform in

Ž . Žpermittivity ´s10´ and permeability ms0.m . These quantities were not allowed to change0

in the inversion process, only the conductivitiesof the thin and thick layers took part in theoptimization. The initial model has seven layerswith the following properties ´ s10´ , s si 0 i

0.001 Srm, m sm for is1 to 7 Fig. 3c; thei 0

estimating process tends to recover the modelwith an rms errors2.1% after relative fewiterations. The rather sharp drop in rms error inthe very first iteration is something that called

Ž .our attention Fig. 3e . We believe that this is anindication that the inverse problem for conduc-tivity is nearly linear when it is posed in termsof isolated thin layers. This is understandablefrom a physical point of view considering thatthere is almost no change in velocity whengoing from the initial to the final model.

3.3. Estimation of permeability

The third example corresponds to the estima-tion of permeability using as data a permeability


Ž . Ž .Fig. 2. Permittivity estimation from a permittivity radargram. a Hypothetical model with permittivity variations. bŽ . Ž .Hypothetical radargram used as data with 5% random noise added. c Initial model in the inversion process. d Final

Ž . Ž .estimated model with its rms error. e History of the process on its way to convergence. f Difference between theestimated and the hypothetical radargrams.

radargram. The process begins by consideringas data the radargram shown in Fig. 4b, whichin this case has a noise level of 5% and endswith the estimated model represented in Fig. 4d.In this case, the initial model is made up of 10

layers with the following parameters: ´ s2´ ,i 0

s s0.0004 Srm, m sm for is1 to 10 Fig.i i 0

4c. The rms errors4.8% was reached in 13iterations. Only the values of permeability wereallowed to change from iteration to iteration.


Ž . Ž .Fig. 3. Conductivity estimation from a conductivity radargram. a Hypothetical model with conductivity variations. bŽ .Synthetic radargram with 2% random noise added. c Conductive halfspace used as starting model in the inversion process.

Ž . Ž .d Recovered model with its corresponding rms error. e History of convergence. Notice that the process converges fasterŽ .than for the case of permittivity Fig. 2 . This happens especially around the first iterations and indicates that the problem is

Ž .nearly linear for conductivity. f Difference between the estimated and the hypothetical radargrams.

3.4. Cross-estimation

We have considered so far the estimation of agiven property from radargrams computed usingthe same property. We have done this for per-mittivity, permeability and conductivity. Results

illustrate that the method is capable of recover-ing a desirable model and of matching a givenhypothetical radargram with random noiseadded. As stated earlier, this was the first objec-tive of the present work. Strictly speaking, theexamples that we present do not fully demon-


Ž . Ž .Fig. 4. Permeability estimation from a permeability radargram. a Hypothetical model with permeability variations. bŽ . Ž .Hypothetical radargram with 5% random noise added. c Halfspace used as starting model in the inversion process. d

Ž . Ž .Final permeability model with its rms error. e History of convergence. f E-fields difference between the radargram of thefinal model and the hypothetical radargram.

strate that the method can handle discontinuitiesat arbitrary depths. Nevertheless, the above-testssuffice for a confident application of the methodto the second objective of the work, which is todemonstrate that a given radargram can be inter-preted in terms of any of the three electromag-

netics properties. This is the aim of the exam-ples discussed below.

The first case is presented in Fig. 5. Weconsider as data the radargram of Fig. 5b whichhas a noise level of 2%. The radargram wascomputed by assuming a permeability disconti-


Ž .Fig. 5. Permittivity estimation from a permeability radargram. a Hypothetical two-layer model that is uniform inŽ .conductivity and permittivity but not in permeability. b Hypothetical radargram used as data with 2% random noise added.

Ž . Ž .c Dielectric halfspace used as starting model in the inversion process. d Final permittivity estimated model with itscorresponding rms error. The reflection in the radargram is due to the discontinuity in permeability but it is interpreted as a

Ž . Ž .discontinuity in permittivity. e History of the process on its way to convergence. f Residuals.

nuity as shown in Fig. 5a. Instead of attemptingto recover the jump in permeability, we con-sider that the corresponding reflection is due toa discontinuity in permittivity, and proceed toestimate the equivalent model which is shownin Fig. 5d. The initial model, with five possible

discontinuities, consisted of six layers with thefollowing parameters: ´ s 6´ , s s 0.001i 0 i

Srm and m sm , is1 to 6 Fig. 5c. The rmsi 0

errors2.7% was arrived at in 17 iterations.The second case is presented in Fig. 6. We

now consider as data a radargram computed


Ž .Fig. 6. Permeability estimation from a permittivity radargram. a Hypothetical model that is uniform in conductivity andŽ . Ž .permeability but not in permittivity. b Hypothetical radargram used as data with 2% random noise added. c Halfspace

Ž .used as starting model in the inversion process. d Final permeability model with its rms estimation error. The reflection inŽ .the radargram is due to the discontinuity in permittivity but it is interpreted as a discontinuity in permeability. e History of

Ž .convergence. f E-fields difference between the radargram of the final model and the hypothetical radargram.

considering a discontinuity in permittivity onlyand it is interpreted in terms of permeability.The estimated model is shown in Fig. 6d. In thiscase, the initial model consisted of 10 layerswith the following parameters: ´ s4´ , s si 0 i

0.0004 Srm, and m sm for is1 to 10 Fig.i 0

6c. The rms errors2.4% was reached in 10iterations. The hypothetical radargram was con-taminated with a level of noise of 2%.

As a final case we estimate a conductivitymodel that explains data obtained from a per-mittivity radargram. The process is summarized


in Fig. 7. In this case, the equivalent of onepermittivity discontinuity is a double discontinu-ity in conductivity or thin conductive layer. Thehypothetical model is shown in Fig. 7a. We

explain the data with a model containing twothin conductive layers, one right on the surfaceand the other at depth. The first accounts for thesurface discontinuity and the other for the per-

Ž .Fig. 7. Conductivity estimation from a permittivity radargram. a Hypothetical model that is uniform in conductivity andŽ . Ž .permeability but not in permittivity. b Synthetic radargram with 5% random noise added. c Conductivity halfspace used

Ž .as starting model in the iterative inversion process. d Final estimated model with its corresponding rms estimation error.The reflections in the radargram, including the first reflection right from the surface, are due to the discontinuities inpermittivity. In this case, both reflections are interpreted in terms of conductivity variations. The equivalent model consistsof two thin conductive layers, one right on the surface to account for the contrast in permittivity between air and soil, and the

Ž . Ž .other to account for the permittivity discontinuity at depth. e History of convergence. f Residuals show the E-fieldsdifference.


mittivity boundary at depth. The initial modelconsisted of seven layers with the followingparameters: ´ s6´ , s s0.001 Srm, and mi 0 i i

sm for is1 to 7 Fig. 7c. The rms errors0

4.7% was reached in eight iterations. The hypo-thetical radargram was contaminated in this casewith a level of noise of 5%.

4. Conclusion and discussion

At first sight, the results presented in thispaper might appear as pessimistic, in the sensethat they point out the impossibility of recover-ing jointly the three electromagnetic propertiesfrom GPR data. Before reaching for this conclu-sion, one must take into consideration that ourapproach to the problem is perhaps the simplestpossible. It is true that for a plane wave withhorizontal fields this impossibility exists, asdemonstrated by the present work. The possibil-ity of differentiation must come from a furthercomplication of the physical model, by includ-ing for instance frequency dependent propertieson one hand, or by considering vertical electricfields on the other. This may change the picturedrastically. As they stand, the present resultsshould be seen as an ideal limit case that di-rectly points out to the impossibility of differen-tiation.

Acknowledgements

The first author wishes to express his grati-tude to the CONACYT, Facultad de Ingenierıa´

de la UNAM, CICESE and the Gran Frater-Ž .nidad Universal, Lınea Solar, A.C. REDGFU´

for the support during the development of thisproject, and to the reviewers of this paper.

References

Esparza, F.J., Gomez-Trevino, E., 1996. Inversion of mag-´ ñetotelluric soundings using a new integral forms of theinduction equation. Geophys. J. Int. 127, 452–460.

Esparza, F.J., Gomez-Trevino, E., 1997. 1-D inversion of´ ˜resistivity and induced polarization data for the least

Ž .number of layers. Geophysics 62 6 , 1724–1729.Gill, P.E., Hammarling, S.J., Murray, W., Saunders, M.A.,

ŽWright, M.H., 1986. User’s guide for LSSOL Version.1.0 : a FORTRAN package for constrained linear least

squares and convex quadratic programming, TechnicalReport Sol 86-1. Department of Operations Research,Standford University, CA, USA, 38 pp.

Gomez-Trevino, E., 1987. Nonlinear integral equations for´ ˜Ž .electromagnetic inverse problems. Geophysics 52 9 ,

1297–1302.Gomez-Trevino, E., 1999. Reworking Maxwell’s equations´ ˜

in search of alternative approaches for inversion. Sub-mitted to Geophysical Journal International.

Lazaro-Mancilla, O., Gomez-Trevino, E., 1996. Synthetic´ ´ ˜radargrams from electrical conductivity and magneticpermeability variations. J. Appl. Geophys. 34, 283–290.

Levenberg, K., 1944. A method for the solution of certainnonlinear problems in least squares. Q. Appl. Math. 2,164–168.

Mardquardt, D.W., 1963. An algorithm for least squaresestimation of nonlinear parameters. SIAM J. 11, 431–441.

McGillivray, P.R., Oldenburg, D.W., 1990. Methods forcalculating Frechet derivatives and sensitivities for theńonlinear inverse problem: a comparative study. Geo-phys. Prospect. 38, 499–524.

Rzhevsky, V., Novik, G., 1971. The Physics of Rocks. MirPublishers, Moscow.

Documents

Ground penetrating radar inversion in 1-D: an approach for