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
As a test of the effectiveness of the approach presented here we have applied the method to
one-dimensional radargrams
computed by
Ž .
Lazaro-Mancilla and Gomez-Trevino 1996 .
´ ´
˜
The radargrams consist of 256 data points and are generated by considering variations in elec-
trical permittivity, magnetic permeability and electrical conductivity. The object of the numer-
ical experiments is to recover these variations from the radargrams. We consider variations of
each property separately, and invert the corre- sponding radargrams in terms of the property
used for their generation. The purpose is to illustrate the applicability of the present ap-
proach to the quantitative interpretation of radargrams, and to show that it represents a
viable alternative to existing inversion methods. The experiments demonstrate that there is an
intrinsic ambiguity in the interpretation of radar- grams, in the sense that variations of one prop-
erty can be genuinely mistaken by variations in the 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 reflection
produced 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 permittivity
radargram 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 level
of random noise of 5 and is shown in Fig. 2b. In the estimation process, the initial model has
six layers with the following parameters: ´ s
i
6´ , s s 0.001 Srm, m s m for i s 1 to 6
i i
Ž .
Fig. 2c . The conductivity and permeability profiles are kept fixed to their original and
uniform values. The permittivity of each of the six layers is allowed to vary from iteration to
iteration until they converge to the model shown in Fig. 2d. The history of the process on its way
to convergence is shown in Fig. 2e. In the present case, the final model is obtained in nine
iterations with a rms error s 5.0. The differ- ence between the estimated and hypothetical
radargrams is shown in Fig. 2f. The fact that the estimated and the hypothetical models are al-
most identical, and that the residuals are of the level of the added noise illustrates the effective-
ness of the approach for the case of permittivity estimation.
3.2. Estimation of conductiÕity We now consider the case of conductivity
estimation using as data a conductivity radar- gram. The process is summarized in Fig. 3. The
model 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 contaminated
with a level of random noise of 2 and is shown in Fig. 3b. The model is uniform in
Ž .
Ž permittivity ´ s 10´
and permeability m s
. m . These quantities were not allowed to change
in the inversion process, only the conductivities of the thin and thick layers took part in the
optimization. The initial model has seven layers with the following properties ´ s 10´ , s s
i i
0.001 Srm, m s m for i s 1 to 7 Fig. 3c; the
i
estimating process tends to recover the model with an rms error s 2.1 after relative few
iterations. The rather sharp drop in rms error in the very first iteration is something that called
Ž .
our attention Fig. 3e . We believe that this is an indication that the inverse problem for conduc-
tivity is nearly linear when it is posed in terms of isolated thin layers. This is understandable
from a physical point of view considering that there is almost no change in velocity when
going 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. b Ž .
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 the estimated and the hypothetical radargrams.
radargram. The process begins by considering as data the radargram shown in Fig. 4b, which
in this case has a noise level of 5 and ends with the estimated model represented in Fig. 4d.
In this case, the initial model is made up of 10 layers with the following parameters: ´ s 2 ´ ,
i
s s 0.0004 Srm, m s m for i s 1 to 10 Fig.
i i
4c. The rms error s 4.8 was reached in 13 iterations. Only the values of permeability were
allowed to change from iteration to iteration.
Ž . Ž .
Fig. 3. Conductivity estimation from a conductivity radargram. a Hypothetical model with conductivity variations. b Ž .
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 faster
Ž .
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 a
given property from radargrams computed using the 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 given hypothetical radargram with random noise
added. As stated earlier, this was the first objec- tive of the present work. Strictly speaking, the
examples that we present do not fully demon-
Ž . Ž .
Fig. 4. Permeability estimation from a permeability radargram. a Hypothetical model with permeability variations. b Ž .
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 the final model and the hypothetical radargram.
strate that the method can handle discontinuities at arbitrary depths. Nevertheless, the above-tests
suffice for a confident application of the method to the second objective of the work, which is to
demonstrate 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. We consider as data the radargram of Fig. 5b which
has a noise level of 2. The radargram was computed by assuming a permeability disconti-
Ž . Fig. 5. Permittivity estimation from a permeability radargram.
a Hypothetical two-layer model that is uniform in
Ž . 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 its corresponding 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 attempting to recover the jump in permeability, we con-
sider that the corresponding reflection is due to a discontinuity in permittivity, and proceed to
estimate the equivalent model which is shown in Fig. 5d. The initial model, with five possible
discontinuities, consisted of six layers with the following parameters: ´ s 6´ ,
s s 0.001
i i
Srm and m s m , i s 1 to 6 Fig. 5c. The rms
i
error s 2.7 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 and
Ž . 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 in Ž .
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 only and it is interpreted in terms of permeability.
The estimated model is shown in Fig. 6d. In this case, the initial model consisted of 10 layers
with the following parameters: ´ s 4´ , s s
i i
0.0004 Srm, and m s m for i s 1 to 10 Fig.
i
6c. The rms error s 2.4 was reached in 10 iterations. The hypothetical radargram was con-
taminated with a level of noise of 2. As a final case we estimate a conductivity
model that explains data obtained from a per- mittivity radargram. The process is summarized
in Fig. 7. In this case, the equivalent of one permittivity discontinuity is a double discontinu-
ity in conductivity or thin conductive layer. The hypothetical model is shown in Fig. 7a. We
explain the data with a model containing two thin conductive layers, one right on the surface
and the other at depth. The first accounts for the surface discontinuity and the other for the per-
Ž . Fig. 7. Conductivity estimation from a permittivity radargram. a Hypothetical model that is uniform in conductivity and
Ž . 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 in
permittivity. In this case, both reflections are interpreted in terms of conductivity variations. The equivalent model consists of 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-fields difference.
mittivity boundary at depth. The initial model consisted of seven layers with the following
parameters: ´ s 6´ , s s 0.001 Srm, and m
i i
i
s m for i s 1 to 7 Fig. 7c. The rms error s
4.7 was reached in eight iterations. The hypo- thetical radargram was contaminated in this case
with a level of noise of 5.
4. Conclusion and discussion
