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 ,
´ ˜
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
2.1. Forward problem Consider a plane wave at normal incidence
upon an n-layer earth model as illustrated in Fig. 1. The three electromagnetic properties vary
from layer to layer; each layer is linear, homo- geneous and isotropic. A dependence of the
form e
i v t
is assumed for the fields, where t is time and v angular frequency. It is further
assumed that the physical properties do not depend on frequency. The governing differential
Ž . equations for the electric E z and magnetic
x
Fig. 1. Layered earth model whose three electromagnetic properties 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: d
2 2
E z y g E
z s 0, 1
Ž . Ž .
Ž .
x i
x 2
d z d
2 2
H z y g H
z s 0, 2
Ž . Ž .
Ž .
y i
y 2
d z where:
1r2 2
g s im s v y m ´ v 3
Ž .
Ž .
i i
i i
i
is the propagation constant of the layer. The quantities m , s
and ´ represent magnetic
i i
i
permeability, electrical conductivity and dielec- tric permittivity, respectively. The solution is
obtained in the standard form of a boundary value problem where we consider 2 n boundary
conditions and 2 n unknowns. The electric field in the ith layer is expressed in terms of an
outgoing and a reflected wave, i.e.:
E s A e
yg
i
z
q B e
g
i
z
, 4
Ž .
x i
i
i
where A and B are coefficients that need to be
i i
determined. Applying continuity of the fields at each boundary and employing standard propaga-
tion matrices, we can find the fields in one layer in terms of the fields in the next layer. Setting
the amplitude of the incident wave A s 1, the electric field at the surface is given simply as:
E 0 s 1 q B . 5
Ž . Ž .
x
The radargram is computed as in Lazaro-
´
Ž .
Mancilla and Gomez-Trevino 1996 by apply-
´ ˜
ing the inverse Fourier transform to the product of the electric field and the Ricker’s pulse spec-
Ž . trum S f , i.e.:
y1
E s FFT E
f S f .
6
Ž . Ž . Ž .
x
A variety of examples are presented by Ž
. Lazaro-Mancilla and Gomez-Trevino 1996 that
´ ´
˜
illustrate the signatures produced by variations in m, s and ´. By trial and error methods it
was possible to match radargrams due to varia- tions in one property with those produced by the
other two. In the present work, we reconsider this 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 magnetic field 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:
E s G
m r
X
d
3
r
X
y G
s r
X
d
3
r
X
Ž . Ž .
H H
E, m E , s
X X
V V
y
G ´ r
X
d
3
r
X
. 7
Ž . Ž .
H
E, ´
X
V
G , G
and G represent the functional or
E,m E, s
E, ´
Frechet derivatives of E with respect to m, s
´
and ´ , respectively. E represents the data and the 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 be reduced to an algebraic form for piecewise uni-
form media. To this end, it is convenient to write the functional derivatives over the ith
region in terms of partial derivatives. The inte-
gration of the functional derivatives of E with respect to m, s and ´ can be written as:
EE
X 3
s G
d r , 8
Ž .
H
E, m
X
Em
V i
i
EE
X 3
s G
d r , 9
Ž .
H
E, s
X
Es
V i
i
EE
X 3
s G
d r , 10
Ž .
H
E, ´
X
E´
V i
i
where V
X
is the volume of the ith region. Eq.
i
Ž . 7 becomes:
n
EE EE
EE
E s m y
s y ´ .
11
Ž .
Ý
i i
i
ž
Em Es
E´
i i
i is1
We base the solution of the inverse problem on Ž
. Eq. 11 .
2.3. Analytic Frechet deriÕatiÕes
´
A fundamental step in the solution of most nonlinear inverse problems is to establish a
relationship between changes in a proposed model and corresponding changes in the data.
Once this relationship is established, it becomes possible to refine an initial model to obtain an
improved fit to the observations. In traditional linearized analysis, the Frechet derivative is the
´
connecting link between changes in the model Ž .
and changes in the data. In Eq. 7 , the same derivative 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 in
the response to a change in the model. We derive the analytic derivatives using the self-ad-
joint Green’s function method. The wave equa- tion for the case of arbitrary variations in the
physical
properties can
be written
from Maxwell’s equations as:
X X
2
hE y g hE s 0,
12
Ž .
Ž .
Ž
2
.
1r2
where m s 1rh and g s imsv y m´v .
Ž .
Strictly speaking, Eq. 12
should include a source term. However, for the purpose of the
perturbation analysis that follows its inclusion is unnecessary because we consider perturbations
of only the physical properties and the fields.
Ž .
Perturbing Eq. 12 , keeping only first order
terms in the resulting perturbations and using Ž
. Eq. 12 again there results:
h
X
h
X Y
X X
2
d E q d E y g d E s yd
E q 2g Edg .
ž
h h
13
Ž .
In the self-adjoint form: h
X X
X X
2
hd E y g hd E s yhd
E q 2g hEdg
Ž .
ž
h s w z .
14
Ž . Ž .
Ž .
The solution of Eq. 14
can be obtained through the use of the Green’s function for the
Ž .
layered model. Let G z, z be this function and
Ž .
d z, z the corresponding Dirac delta function.
Ž .
Ž .
Considering Eq. 12 , G z, z must obey:
X X
2
hG z , z y g hG z , z
s d z , z .
15
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
y q
Integrating Eq. 15 between z and z
in the usual manner, one obtains:
h z
q
G
X
z
q
, z y h z
y
G
X
z
y
, z s 1.
Ž . Ž
. Ž
. Ž .
16
Ž .
We use this equation to obtain a quantitative Ž
. Ž .
relationship between G z, z s 0 and E z . For Ž
. Ž
. the moment consider that G z, z
s G z , z . That is, G can be interpreted either as the field
at z due to a source at z , or as the field at z due to a source at z. For convenience we
choose to consider G as the field at z due to a source at z
because this field is simply the physical field within the layered model. This
Ž .
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 is
irrelevant whether the source is placed right on
Ž .
Ž . top or above the model. Thus, G z,0 and E z
are the same, except for a multiplying factor to
account for physical dimensions and differences in source strength. That is:
E z
Ž .
G z s ,
17
Ž . Ž .
C where C is independent of depth and we have
Ž .
Ž . written G z, z s 0 s G z .
Ž .
Ž .
Substituting Eq. 17 into Eq. 16 we must have:
X X
hE y hE
s C. 18
Ž .
q y
The discontinuity is produced by the source Ž
X
. at z s 0. The fact that yhE ri v is simply
the magnetic field of the source provides the clue for what the value of C must be. The
magnetic fields produced by a plane source have the same intensity on both sides of the
plane but are of different signs. Furthermore, the intensity of the magnetic field is equal to
one-half the strength of the source. Using Eq. Ž .
4 we can compute the magnetic field just on top of the model. The result can be written as
Ž
y
.
X
Ž
y
. Ž
y
. Ž
. h 0
E 0 s h 0
g B y 1 ,
where we
have kept A s 1 as for the computation of the electric field. The above-arguments then lead to:
C s yh 0
y
E
X y
. 19
Ž .
Ž .
Ž .
We must then have that: m E z
Ž .
G z s .
20
Ž . Ž .
g B y 1
Ž .
We get the Frechet derivative for the 1-D GPR
´
problem through:
`
d E 0 s G z w z d z.
21
Ž . Ž . Ž .
Ž .
H
Ž . Ž
. Substituting w z from Eq. 14 , G from Eq.
Ž .
Ž .
20 and developing Eq. 21 we have:
`
m d
m z
Ž .
X 2
d E 0 s y
E z
d z
Ž . Ž .
H
2
g B y 1
m z
Ž .
Ž .
` 2
2
y E
z v d´ z d z
Ž . Ž .
H
` 2
q E
z i v ds d z . 22
Ž . Ž .
H
The last two integrals are more direct to obtain than the first. To obtain the first integral
Ž it is necessary to use the wave equation Eq.
Ž ..
12 in the intermediate steps.
Ž . In terms of Eq. 7 , we have:
m E
X 2
G s
y ,
23
Ž .
E, m 2
ž
g B y 1
m z
Ž .
Ž .
m
2
G s
E i v , 24
Ž .
Ž .
E, s
g B y 1
Ž .
m
2
G s
yE v . 25
Ž .
Ž .
E, ´
g B y 1
Ž .
Ž . Ž
. Ž .
Substituting Eqs. 23 – 25
into Eqs. 8 –
Ž .
10 , respectively, one obtains expressions for the partial derivatives in terms of the electric
field 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 m
Ž .
s y Em
g B y 1
Ž .
i
=
2
1 g A
i i
y2g z y2g z
i iq 1
i i
y e
y e
Ž .
2
2 m
i
2g
2
A B
i i
i
y z
y z
Ž .
iq1 i
2
m
i 2
1 g B
i i
2g z 2g z
i iq 1
i i
q e
y e ,
26
Ž .
Ž .
2 m
i
For the nth layer we have: EE 0
m 1 g A
2
Ž .
n n
y2g z
n n
s y e
. 27
Ž .
2
Em g
B y 1 2 m
Ž .
n n
For electrical conductivity the derivatives are: EE 0
m
Ž .
s i v
Es g
B y 1
Ž .
i
=
2
1 A
i y2g z
y2g z
i iq 1
i i
y e
y e
Ž .
2 g
i
q2 A B z y z
Ž .
i i
iq1 i
2
1 B
i 2g z
2g z
i iq 1
i i
q e
y e ,
28
Ž .
Ž .
2 g
i
For the nth layer: EE 0
m 1
Ž .
2 y2g z
n n
s i v A
e .
n
ž
Es g
B y 1 2g
Ž .
n n
29
Ž .
For permittivity the results are: EE 0
m
Ž .
2
s y v
E´ g
B y 1
Ž .
i
=
2
1 A
i y2g z
y2g z
i iq 1
i i
y e
y e
Ž .
2 g
i
1 B
2 i
q2 A B z y z q
Ž .
i i
iq1 i
2 g
i
=
2g z 2g z
i iq 1
i i
e y e
, 30
Ž .
Ž .
and the expression for the nth layer: EE 0
m A
2
Ž .
n 2
y2g z
n n
s y v
y e
.
ž
E´ g
B y 1 2g
Ž .
n n
31
Ž .
The values of the derivatives in the time domain are obtained by applying the Fourier
transform as already explained in relation to Eq. Ž .
6 . We use a layered model that can be made up of many thin layers to simulate arbitrary
variations of the physical properties. The corre- sponding integral expressions reduce to systems
of 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- Ž
mulated for linear problems see for example .
Gill et al., 1986 . For example, for magnetic Ž
. permeability we rearrange Eq. 11 as:
n n
n
EE EE
EE E q
s q ´ s
m . 32
Ž .
Ý Ý
Ý
i i
i
Es E´
Em
i i
i is1
is1 is1
The set of unknown permeability values is placed on the right side of equation. The con-
ductivity and permittivity values are assumed to be known and are placed on the left side of the
equation together with the electric field data. There is one equation for each data point of the
radargram. The partial derivatives are computed by assuming a model that includes values for all
the physical properties of the layers. The left-
Ž .
hand side of Eq. 32 then reduces to a column vector of numbers; the right-hand side can be
written as the product a matrix of numbers and a column vector that contains the unknowns.
The elements of the matrix are simply the par- tial derivatives of the electric field with respect
to the unknowns. To each data point there cor- responds a different row of the matrix.
Ž .
We use the same Eq. 11
for estimating electrical conductivity. What changes is the ar-
rangement of the different terms. The equation is in this case:
n n
n
EE EE
EE yE q
m y ´ s
s .
Ý Ý
Ý
i i
i
Em E´
Es
i i
i is1
is1 is1
33
Ž .
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 derivatives
are with respect to electrical conductivity. The column 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
n
EE EE
EE yE q
m y s s
´ .
Ý Ý
Ý
i i
i
Em Es
E´
i i
i is1
is1 is1
34
Ž .
The last three equations have the form:
y s A x , 35
Ž .
where y is the vector containing the radargram data. y also contains partial derivatives and the
parameters that are not considered to be un- knowns. The matrix A contains the partial
derivatives with respect to the parameter that is considered the unknown, where in all cases is
represented by x. The problem is obviously nonlinear: E and its partial derivatives depend
on all three physical properties. This means that the sets of equations cannot be solved in a
single iteration using linear analysis. We apply linear analysis iteratively by assuming every-
thing known, except the parameters on the right side of the equations.
Quadratic programming consist of minimiz- ing the quadratic norm of the residuals subject
to lower and upper bound in each parameter, that is:
1
2
5 5
Minimize F x s y y A x
, 36
Ž . Ž .
2
subject to x F x F x . The vector x represents
l u
l
a lower limit imposed on the properties of the layers; x
is a corresponding upper limit. The
u
inequality must be understood as applying to corresponding components of the vectors x , x
l
and x . The algorithm finds a solution for x
u
such as the square of the residuals is a mini- mum, with the additional constraint that the
parameters must fall within the upper and lower bounds previously established. The function to
Ž .
minimize in Eq. 36 can be represented as: 1
T T
F x s c x q x S x
37
Ž . Ž .
2
subject to x F x F x ,
l u
T T
T
where c s y A y 38
Ž .
Ž .
and S s A
T
A is the symmetric Hessian matrix.
Ž .
Strictly speaking, Eq. 37 should include a term Ž
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 the
Hessian a term l I, with l 1. We calculate the Hessian in a way that the diagonal is unitar-
ian, so the process is modified to minimize:
1 1
2 2
X X
X
5 5
5 5
F x s y y A x
q l x
, 39
Ž . Ž .
2 2
A
X
s AV, 40
Ž .
x
X
s V
y1
x , 41
Ž .
1 Õ
s ,
42
Ž .
ii n
2
A
Ý
ji js1
and Õ s 0, i j. Finally, when we have posed
i j
the problem and have the vectors and the matri- ces in accordance with the last equation, we
proceed to get x
X
, and consider that x s Vx
X
. To improve convergence in the iterative process
we use the Levenberg–Mardquardt method Ž
. Levenberg, 1944; Mardquardt, 1963 .
On the other hand, the rms error of estima- tion is calculated at each iteration using:
rms error
256
1
2
E t y E t
Ž . Ž .
Ý
h i
e i
256
is1
s 100,
43
Ž .
2
E
h
where:
256
1
2
E s E
t ,
44
Ž . Ž .
Ý
h h
i
256
is1
Ž . and where E t
represents the value of the
h i
hypothetical radargram for the time t . There are
i
Ž . 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 E
h
Ž .
as defined in Eq. 44 to normalize the rms of Ž
. 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 for
h i
Ž . radargrams since E t s 0 for many t . On the
h i
i
other hand, when noise is added to radargrams this is done as a percentage of E . In this way
h
all values of the radargram are contaminated with noise, not only those that are different
Ž . from zero. Using percentages of E t
directly
h i
does not add any noise to the quiet sections of the radargram.
3. Results
