The inverse MagnetoTelluric (MT) operator is approximated by means of the Neural Network (NN). The
methodology of the NN interpretation in classes of the geoelectrical sections described by the hundreds of parameters
is proposed. Error of the NN inversion and field misfit are evaluated. A rapid NN algorithm solving the
inverse problem and detecting changes of time-dependent dynamic parameters of the section is applied to 2D
synthetic data

Gavryushov, S.

Obornev, M. A.

Shimelevich, M. I.

Earth-prints Repository

105
ANNALS  OF  GEOPHYSICS, VOL.  50, N.  1, February  2007
Key  words magnetotelluric inverse problem – neu-
ral network – EM monitoring
1. Introduction
In recent years some examples of the Neur-
al Network (NN) technique applications for so-
lution of the magnetotelluric inverse problems
(Poulton, 2002) and rapid inverse problems
(Zhdanov and Chernyavskiy, 2004) have been
presented. It was shown that this method is ef-
fective when the number of the geoelectrical
section parameters is about 10 (Spichak and
Popova, 2000; Spichak et al., 2002; Shimele-
vich et al., 2002), i.e. the solution is sought
within a narrow class of models. An important
peculiarity of almost instant NN inversion
makes this approach attractive for the real-time
monitoring of electromagnetic parameters of
the medium. It should be added that the NN can
be efficiently applied to the monitoring of a few
parameters of the known section using a rar-
efied set of measurements (Shimelevich et al.,
2003). In the present paper the NN technique is
shown to be effective to reconstruct the conduc-
tivity distribution and its time variations in case
of hundreds of parameters within a wide class
of geoelectrical models.
2. Metodology of the NN inversion in a class
of the geoelectrical sections
The inverse MT problem of the evaluation
of the vector of geoelectrical section parameters
γ=(γ1, ..., γN) according to the MT data β=
=(β1, ..., βM) observed on the Earth surface can
be reduced to a solution of the non-linear oper-
ator equation Ak γ=β , γ∈Γ k; where Γ k its set
associated with the given class Gk , k is the class
number and Ak is the MT forward problem op-
erator, defined on the subset Γ k.
The NN approach is an approximation of
the inverse operator Sk =Ak−1 by a superposition
of non-linear functions of the given type. One
of the widespread NN approximations is the
multilayer perceptron (Raiche, 1991). In this
Rapid neuronet inversion 
of 2D magnetotelluric data for monitoring
of geoelectrical section parameters
Mikhail I. Shimelevich (1), Eugeny A. Obornev (1) and Sergei Gavryushov (2)
(1) Moscow State Geoprospecting University, Moscow, Russia
(2)  Engelhardt Institute of Molecular Biology, Moscow, Russia
Abstract
The inverse MagnetoTelluric (MT) operator is approximated by means of the Neural Network (NN). The
methodology of the NN interpretation in classes of the geoelectrical sections described by the hundreds of pa-
rameters is proposed. Error of the NN inversion and field misfit are evaluated. A rapid NN algorithm solving the
inverse problem and detecting changes of time-dependent dynamic parameters of the section is applied to 2D
synthetic data.
Mailing address: Dr. Mikhail I. Shimelevich, Moscow
State Geoprospecting University, 123001, Sadovaya Ku-
drinskaya str., 22, building 1, Moscow, Russia; e-mail: shi-
melevich@ecc.ru
106
Mikhail I. Shimelevich, Eugeny A. Obornev and Sergei Gavryushov
case, a solution of the inverse problem is sought
as: γ= f(Vf(Wβ)), where the sigmoid functions
are f(x)=1/(1+exp(−x)). To calculate the un-
known matrices V, W the method of NN learn-
ing with a teacher is used. The NN is learned,
using a database of samples that are pairs of
vectors {γp, βp}, obeying the equation: Ak γp=βp,
γp∈Γ k, p=1, ..., P (P is a number of samples).
Vector γp is randomly chosen from the set Γ k. 
A numerical implementation of the method based
on the error Back Propagation algorithm (Raiche,
1991). To estimate its interpolation properties and
quality of the inversion, the independent testing
set is used. There are calculated parameter errors
∆ k=(∆ k1, ..., ∆ kN) and the field misfit δk aver-
aged for the testing set. This independent set of
samples is used to verify the accuracy of the in-
version. This accuracy is increased at growth of
the number of samples P (Shimelevich et al.,
2001). To obtain a practically applicable accura-
cy of the inversion (about a few percent), the
database can include up to hundreds of thousand
of examples. Parallel algorithms are used to ob-
tain such a database.
Learned neuronet Sk determines an approxi-
mate inverse operator Ak−1 for the model class
Gk of the geoelectrical sections, the parameter
errors ∆ k=(∆ k1, ..., ∆ kN) and the field misfit δk
obtained as the result of the NN testing allow us
to estimate the error of the interpretation, and to
solve a problem of a correspondence between
measured data and medium classes Gk (Shime-
levich and Obornev, 1999). To evaluate the im-
pact of the noise, learned neural networks were
tested for data with added noise of various
kinds and magnitudes. Results suggested that
relative errors of the inversion led to field devi-
ation not exceeding the norm of the noise
(Shimelevich et al., 2001).
3. The classes of the parameterized
geoelectrical sections
In practice, an interpreter can deal with sev-
eral hypotheses on the conductivity distribution
model, so the solution of the inverse problem
should be sought in a few classes Gk of the
medium. In this case, it is necessary to have a
set of neuronets S0, S1, ..., corresponding to dif-
ferent classes G0, G1, ... of the parameterized
geoelectrical sections. The parameterized class
Gk is the set of all possible geoelectrical sec-
tions which conductivity can be described as:
σ(y, z) = fk(y, z, γ1, ..., γN), y, z ∈Ω, γnmin≤γn≤
≤ γnmax, n=1, ..., Nk. Here Ω is a domain of R2,
where the electromagnetic problem is formulat-
ed; γnmin, γnmax are the limits of the γn parameter
variation, the fk is the function of the parameter-
ization of the given class and Nk is the number
of parameters which depends on the complexi-
ty of the class Gk.
The generalized example of the section pa-
rameterization is shown in fig. 1. The parame-
ters of the structure are the layer thicknesses de-
fined at the fixed nodes, the conductivities of
the blocks within the layers and the conductivi-
ties at the fixed nodes of the rectangular grid.
For the forward problem solving the parameter-
ized conductivity of the medium is interpolated
between the nodes. The number of sought pa-
rameters of such structures is about a few hun-
dreds, whereas the finite-difference scheme di-
mension of the corresponding boundary value
problem is of tens of thousands.
We will consider three particular examples
of medium classes. 
Fig. 1. The generalised example of the geoelectri-
cal section parameterization. The model can include
N layers (1), (2), (3) and 2D conductivity grid. The
conductivities at the nodes of the 2D grid, thickness-
es and conductivities of the layers in the horizontal
direction are varied.
107
Rapid neuronet inversion of 2D magnetotelluric data for monitoring of geoelectrical section parameters
Class G0 (Grid) – It is the most general class
of the geoelectrical sections. The conductivity
can change within a fixed range at all nodes of
the 2D grid. Between the nodes the conductivi-
ty is interpolated by a 2D spline. 
Class G1 (Layer+Grid) – The upper layer of
the section is described explicitly, whereas the
rest of the section is parameterized as in class
G0. The layer thickness varies within a given
range at the fixed nodes in the horizontal direc-
tion. Between the nodes the layer boundaries
are interpolated by a 1D spline. The layer con-
ductivity varies linearly between the centers of
the blocks in the horizontal direction. 
Class G2 (3 layers+Grid) – The upper three
layers of the section are described explicitly. The
layer boundaries are parameterized as in class G1.
The layer conductivity varies linearly between
the centers of the blocks in the horizontal direc-
tion. A total number of parameters of the de-
scribed conductivity classes is from 233 to 336.
For these classes neuronets S0, S1, S2 were learned
according to the described above principles.
4. NN interpretation of the 2D synthetic data
The structure used to generate the synthetic
data is shown in fig. 2a. This structure (Model
1) includes three layers of the changeable thick-
ness and the underlying stratum. The thickness
of the upper layer varies from 100 to 3900 m.
Its conductivity varies in the horizontal direc-
tion from 0.185 to 0.008 S/m. The conductivi-
ties of the second, third layers and of the under-
lying half-space are constant and equal σ 2=
= 0.0004 S/m, σ 3= 0.2 S/m, σ 4= 0.0002 S/m.
The thickness of the second and third layers
vary from 0 to 13.4 km and from 2 to 4.8 km.
The apparent resistivities and phases for TE and
Fig. 2a-d.  The results of the synthetic data interpretation. a) Model 1 used to generate the synthetic data; b) in-
terpretation, step1 (NN S0 is applied); c) interpretation, step2 (NN S1 is applied); d) interpretation, step3 (NN S2
is applied). The thin solid lines are true boundaries of the layers of Model 1.
a
b
c
d
108
Mikhail I. Shimelevich, Eugeny A. Obornev and Sergei Gavryushov
TM modes were calculated at 13 periods from
0.01 to 1000 s in 126 sites along the profile.
To interpret the generated synthetic data a
set of learned neuronets S=(S0, S1, S2) was
used. At the first stage of the interpretation
there is no hypothesis about the model class of
the section. So, we have to use neuronet S0 cor-
responding to the widest class G0 of geoelectri-
cal sections. The result of the first inversion
(step 1) is shown in fig. 2b. At this stage of the
interpretation, one can draw the only conclu-
sion that there might be a thin superficial con-
ductive layer. As is seen from averaged field
misfits given in fig. 3, all results of imaging re-
lated to the deeper part of the section are unre-
liable on this first step of inversion. A relative-
ly low misfit can be observed only at high fre-
quencies, which suggest that one can only trust
conductivity distribution close to the surface.
Taking this conclusion as a starting point, one
can use neuronet S1 corresponding to the class
(Layer+Grid). The result of the second inver-
sion (step 2) is shown in fig. 2c. The upper lay-
er is seen better and its lower boundary is prac-
tically identical to the true boundary. Three lay-
ers can be seen under it and the conductivity of
the third layer is about 0.1 S/m. At the same
time the field misfit has decreased in the whole
frequency range (see fig. 3). This proves the
correctness of using neuronet S1 on step 2. Tak-
ing into account this fact and the image of the
section in fig. 2c one can try to use neuronet S2
corresponding to the class G2 (3 layers+Grid) at
the next stage of the interpretation. The result of
the inversion (step 3) is shown in fig. 2d. The
second conductive layer is seen better and the
field misfit at the range T≤25 has dropped to
the level of the mean field misfits obtained at
the testing of neuronet S2.Thus, one can see that
the inversion with using neuronet S2 gives a sat-
isfactory variant of the interpretation except for
the deepest part of the section.
An application of the NN requires knowl-
edge of the model class of the resistivity section.
If there is no such information, an interpreter has
to invert data using NN S0, S1, which justifies the
final application of the NN S2. It is important to
note that application of S0 does not required any
information since it is based on the most general
class G0. The sequence of inversions S0, S1 illus-
trates an application of the NN at the lack of a
priori information about the section.
5. Rapid detection of the geoelectrical
parameter variations
To estimate the resolving power of the NN
inverse operator the boundary of the third layer
of Model 1 was changed in three zones marked
by the dashed line in fig. 4. After forward mod-
eling for the modified section, the new synthet-
ic MT data were interpreted using neuronet S2.
The difference between the results of the inter-
pretations is shown in fig. 4. As one can see the
difference conductivity map localizes the zones
of the conductivities changes with a good accu-
racy. Note, that we can apply the same learned
neuronet for inversion, since the modified sec-
tion belongs to the same model class as the ini-
tial section. In this case the data interpretation
can be done very quickly (several seconds on a
single computer). This example shows a high
Fig. 3. Field misfits averaged over three period
ranges: T<1 s; 1<T<25 s; T>25 s at the three NN in-
versions (solid lines). Dotted lines are averaged field
misfits obtained at the neuronet testing.
109
Rapid neuronet inversion of 2D magnetotelluric data for monitoring of geoelectrical section parameters
sensitivity of the NN inverse operators to the
geoelectrical parameter variations. It illustrates
a possibility of a real-time monitoring of resis-
tivity section parameters if these changes can
be detected from the measured fields.
6. Conclusions
1)  A representative set of the learned neu-
ronets allows us to interpret a 2D geoelectrical
sections described by hundreds of parameters
and to estimate the accuracy of the interpretation
results.
2)  Due to rapidity of NN inversion and high
resolution the neuronet technology could be used
for the monitoring and localization of small vari-
ations of the conductivity distribution in a real
time scale.
A clear disadvantage of the method is a large
number of the trained NN for different classes of
the geoelectrical sections and difficulty to
choose a true NN at the lack of information.
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Fig. 4. The difference between the results of the interpretation of the initial and modified geoelectrical sections.
The changed boundaries are marked by the dashed line. The solid lines are the boundaries of Model 1.


Rapid neuronet inversion of 2D magnetotelluric data for monitoring of geoelectrical section parameters

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Rapid neuronet inversion of 2D magnetotelluric data for monitoring of geoelectrical section parameters

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