Published in Geophysical Prospecting, 59, issue 6, p1132-1143
DOI: 10.1111/j.1365-2478.2011.00990.x
Regularization strategy for the layered inversion of airborne TEM data: application to
VTEM data acquired over the basin of Franceville (Gabon).
Julien Guillemoteau1, Pascal Sailhac1 and Mickaël Béhaegel2
1
Institut de Physique du Globe de Strasbourg & EOST, CNRS‐UDS UMR 75‐16, Strasbourg, France.
2
Areva NC, Geoscience Direction, Mining Business Group, La Défense, Paris, France.
Corresponding author: j.guillemoteau@unistra.fr
Abstract
Airborne transient electromagnetic (TEM) is a cost‐effective method to image the distribution of electrical
conductivity in the ground. We consider layered earth inversion to interpret large data sets of hundreds of kilometre.
Different strategies can be used to solve this inverse problem. This consists in managing the a priori information to
avoid the mathematical instability and provide the most plausible model of conductivity in depth.
In order to obtain fast and realistic inversion program, we tested three kinds of regularization: two are based on
standard Tikhonov procedure which consist in minimizing not only the data misfit function but a balanced
optimization function with additional terms constraining the lateral and the vertical smoothness of the conductivity;
another kind of regularization is based on reducing the condition number of the kernel by changing the layout of
layers before minimizing the data misfit function. Finally, in order to get a more realistic distribution of conductivity,
notably by removing negative conductivity values, we suggest an additional recursive filter based upon the inversion
of the logarithm of the conductivity.
All these methods are tested on synthetic and real data sets. Synthetic data have been calculated by 2.5D modelling;
they are used to demonstrate that these methods provide equivalent quality in terms of data misfit and accuracy of
the resulting image; the limit essentially comes on special targets with sharp 2D geometries. The real data case is
from Helicopter‐borne TEM data acquired in the basin of Franceville (Gabon) where borehole conductivity loggings
are used to show the good accuracy of the inverted models in most areas, and some biased depths in areas where
strong lateral changes may occur.
Keyword: Airborne electromagnetic,Transient electromagnetic, Imaging. Inversion. Regularization.
Introduction
Airborne transient electromagnetic (TEM) surveying was
introduced about fifty years ago in the mining industry
to detect shallow conductive targets like graphitic or
sulphide formations. Nowadays, this method is also
useful for groundwater exploration (Auken et al., 2009)
or on‐shore hydrocarbon exploration (Huang and Rudd,
2008).
Thanks to recent improvement in acquisition systems, it
is now possible to image continuously, quickly and
accurately the electrical conductivity distribution in the
ground with the development of new modelling and
inversion strategies. 2D inversion (Wolfgram et al,
2002), 2.5D inversion (Wilson and Raiche, 2006) or 3D
inversion (Cox et al, 2010) starts to become practical
when applied on airborne electromagnetic (AEM) data.
However, less accurate interpretation as layered earth
inversion remains the most useful method to interpret
fastly large amount of data or to provide prior model for
fast 3D inversion.
The first step of this method is to carefully define the 1D
kernel relating the model of conductivity in depth to the
data of apparent conductivity in time or frequency, the
second step is to carefully invert the data. Actually in
most cases, the layered inversion is an ill‐posed problem
which needs regularization. Zhdanov (2009) provided a
detailed description of the recent improvement notably
the minimum support method (Portniaguine and
Zhdanov, 1999) concerning regularization problem in
EM geophysics.
For 1D AEM inverse problem, one can use standard
Tikhonov strategy. Christensen (2002) developed a fast
method called “One Pass Imaging” which uses a
regularization of the z‐variability (in the vertical
direction). Siemon et al. (2009) and Vallée and Smith
(2009) recently published other results obtained by
regularization with horizontal constraints. In
complement to all these approaches, in this paper we
expose and compare three others methods to solve the
1D inverse problem of AEM data.
Figure 1: Description of the procedure for the
interpretation of TEM data. The definition of the
apparent conductivity aims to avoid the configuration
dependant part of the problem. The forward modelling
consists in computing first the apparent conductivity by
using the ABFM procedure. Secondly, the TEM response
is found out by table look up of the homogeneous
response which has been previously computed using
relation (1) for large number of conductivities. The
inversion of TEM data is the reciprocal process.
The first one uses a constraint over the vertical
derivatives and is similar to the One Pass Imaging
developed by Christensen (2002). As AEM data are over‐
sampled along the flight line direction, we developed a
second approach which uses this information to apply
lateral constraint. This method is based upon a
minimum length criterion over the difference with the
results from the previous sounding. We suggest a third
new approach which allows getting a natural
generalized inverse (which mean no regularization) and
which is based upon local analysis of the condition
number to determine the layer layout prior to the
inversion. All our programs use the linear modelling
called Adaptative Born Forward Mapping (ABFM) to
predict the data (Christensen, 2002). The ABFM
procedure consists in solving a linear relationship
between the apparent conductivity and the real
conductivity. It is based on the hypothesis of normal
distribution of conductivities, which is able to result in a
model with some negative values; the better physical
hypothesis which constrains any conductivity to be
positive is that they obey lognormal statistics. Thus, we
suggest an additional step using the kernel for the
logarithm of the conductivities; this avoids negative
conductivity values in the resulting model and allows
sharper boundaries within the resulting
model.
First we compare these methods when
applied on a simple synthetic case. The
artificial measurements have been
generated by 2.5D modelling using the
program “ArjunAir_705” developed by
the P223 EM modelling project (Raich,
2008, Wilson et al., 2006). The quality of
each method is considered in terms of
error (data misfit) and comparison to
the true model. Then, we apply these
methods on real data set acquired over
the basin of Franceville (Gabon) in order
to detect the bottom of an ampelite
layer characterized by relatively high
electrical conductivities. In that real
data case, the quality of each method is considered in
terms of data misfit and comparison to borehole
measurements.
Imaging the electrical conductivity
Description of the problem
Airborne TEM data are provided in terms of magnetic
field
or its time derivative / recorded in the
receiver loop, where t is the time delay after turn off of
the transmitter loop. In this paper, we consider the
vertical magnetic field located at the centre of a
horizontal circular loop transmitter; in the quasi static
domain it is given in the Fourier domain by (Ward and
Hohmann, 1987):
∞
,
where is the radius of the transmitter loop, is its
altitude and is the amplitude of the electrical current
injected, is the horizontal component of the wave
number in the air and
is the reflection coefficient
which depends on the conductivity of the underground
medium. Because the typical transmitter current is a
step current with turn off at time 0, the transient
response in terms of
/ is computed by performing
the inverse Laplace transform of the expression (1).
Then,
can be obtained by integrating
/ .
Finally the response of the system is given by convolving
the step response to the time derivative of the actual
current injected in the transmitter loop.
One way to simplify the problem is first to convert
data into apparent conductivity and then to
compute the layer conductivities by inversion of
the apparent conductivities. The apparent
conductivity
is defined as the conductivity of
the equivalent homogeneous half space which
provides the same response. Therefore,
is the
solution of the following equality:
,
,
,
layered medium with
layers, the apparent
conductivity versus the
time windows is written as a
linear combination of the conductivity versus depth:
,
,
.
,
,
where
is the vector of apparent conductivity, is the
vector of the layer conductivities and
is the kernel
depending on the apparent conductivity and time. The
latter relation constitutes the forward formulation of
our problem. Assuming that the measured response is
always above noise level, the maximum depth of the
layered media is given equal to penetration depth of the
primary field at the largest time window. This depth
can be approximated by the following relationship
introduced by Christensen (2002):
,
,
where c=2.8 is the ad hoc scaling factor that he
obtained by minimizing the squared difference between
the exact and the approximate apparent conductivities
(Equation 3) summed over six layered models with
different parameters.
.
By introducing the apparent conductivity, one can
separate the problem into two subsequent parts:
1‐ one configuration dependent part which is the
relation between the TEM response and the
apparent conductivity characterised by Relation (1)
for a homogeneous half space, 2‐ a configuration
independent part which relies the apparent
conductivity to the layer conductivities. To
interpret our data, we follow the reciprocal
scheme (see Figure 1). For each time window, the
apparent conductivity is inverted by table look‐up
within abacus containing the current system response
or
/ for a large amount of homogeneous
half space. To compute the real conductivities, we use
the ABFM method (Christensen, 2002) which is the
linearized version of Equation (2) in time‐domain. For a
Figure 2: Vertically constrained inversion of synthetic
noisy data (0.1% of random noise added to
) for a
large number of
. The data misfit versus
regularization describes the balance between data
information and constraints based on a priori
information.
In order to follow the decrease of resolution with depth,
we set the grid interfaces in such a way that the
layer is n times thicker than the first one. Depending on
the magnitude of the discretization of the media, the
problem is over determined, mixed determined or
under determined. In most cases, we consider mixed‐
determined problems because we need a compromise
when choosing the number of layers: enough layers are
necessary to get a good vertical resolution, however too
much layers would increase the computation time
above acceptable values for real time applications. In
the following, we discuss the different strategies to
solve such a 1D inverse problem.
all the inversions discussed in this paper since we do not
have any a priori information on the measurements.
is the vertical smoothness matrix which
depends on the first and second order derivative of the
model, is written as:
Inversion with vertical constraints
where
is equal to half of the current layer thickness
in order to compensate the increasing thickness of
layers with depth. Figure 2 shows the inversion and the
relative apparent conductivity misfit of 1D synthetic
data for an increasing smoothness. We tested values of
regularization control parameter
in the range of
[
,
]. In the bottom of Figure 2, the apparent
conductivity misfit is displayed versus the regularization
weight. If
is too small, the regularization is two weak
and the matrix to be inverted is singular. If
increases, the smoothness becomes more important:
this avoids artefact due to the noise of data until an
optimal value ~ . which well reproduces the initial
model. For larger , the regularization becomes
preponderant and covers the information contained in
the data: the smoothness is too large to reproduce real
conductivity changes and data errors increase as well.
The vertical constraint has been suggested to regularize
1D TEM problem by Christensen (2002) in counterpart
to the measure of the length of the solution. In this
study, we consider only the vertical constraint in order
to identify its own effect. The objective function which
has to be minimized is given as follows:
,
The first sum of this expression constitutes the data
misfit (e.g. a weighted least square) and the second
constitute the regularizing part. is a weighting vector
characterizing the importance of each measurement, it
could be related to the inverse of variances if one also
considers independent normal distributions for the
apparent conductivity at each time delay ti;
is a
weighting factor characterising the vertical variability of
the model. The solution which minimizes the function
is written as follows (Menke, 1989):
σ=[FT Wd F+S]
where
factors
Wd σ a ,
is a diagonal matrix containing the weighting
. We took
equal to an identity matrix for
δz
δz
δz
δz
δz
..
..
δzN
δzN
δzN
δzN
,
δzN
Different approaches exist to find the optimal value for
lambda (Hansen, 1992, 2010) or (Oldenburg and Li,
1994; Constable and Parker, 1987) for iterative
methods. We used the simplest criterion similar to the
discrepancy principle (Aster et al., 2005): we set the
optimal parameter at the highest value of
able to
produce a reasonable data misfit. Consequently, one
has to define the threshold of the data misfit regarding
the level of noise before the inversion.
Figure 3: Laterally constrained inversion of noisy (0.1%) 2.5D synthetic data for different values of . The optimal
choice of regularization is around
. .
the solution and the result provided by the previous
sounding
:
Inversion with lateral constraints
Airborne TEM are usually over sampled along the flight
line. Indeed, since the footprint of the method is larger
than the interval between two soundings, the
measurements cannot vary sharply. It is possible to use
this information as a lateral constraint to regularize this
1D inverse problem. Monteiro Santos et al. (2004) for
ground measurements, Auken et al. (2005) and Viezzolli
et al. (2008) for airborne data set, developed 1D
Laterally Constrained Inversion (LCI) based on a
smoothing term which constraints lateral derivatives of
the model. In these methods, one needs to consider
several soundings simultaneously. Christiansen et al.
(2007) applied this method on small data set from
ground‐based measurements. Siemon et al. (2009)
adapted this method for large airborne continuous
measurements. In order to reduce the computational
cost, we propose a method which allows inverting all
the soundings independently with a simple lateral
constraint. This method consists in using the result
provided by the previous sounding as a reference
model. In that case, the objective function that we have
to minimize is composed of the data misfit plus a
second term which minimizes the difference between
Φ=
The solution
,
‐ (
)
,
,
.
of the current sounding is given by:
λL
.
The parameter controls the magnitude of the lateral
constraint. Figure 3 shows the laterally constrained
inversions of 96 synthetic Bz data with 0.1% of noise
performed every 20 meters over a 2.5D conductivity
model. The synthetic data was generated by using the
program ArjunAir with which we simulate pure step
response in coincident loop geometry (with a loop
transmitter of 26m in diameter and a 1.1m diameter
receiver and a nominal clearance of 45m). They are
composed of 27 channels starting from
to
.
. The aim is to reproduce VTEM (Witherly
et al., 2004) configuration. The model is composed of a
conductive layer of conductivity σ=200mS/m embedded
within a host medium of 10 mS/m. The results of the
inversion and their relative misfit are displayed in Figure
Figure 4: Comparison between simple SVD inversion and SVDal inversion of a noisy (0.1%) 2.5D synthetic dataset. The
algorithm SVDal changes the number of layers in order to reduce the condition number and avoid singular values.
3 for four increasing degrees of smoothness (
.
, . , . , . ). Like in the previous section,
setting
too small allows non realistic values of
conductivities generating unstable data misfit. By
comparing to the true conductivity, we can conclude
that the optimal choice of the regularization factor is
around
. . For larger smoothness, the
inversions need more soundings to converge toward the
right model. If a strong lateral variation occurs, the data
misfit increases first and then decreases as slow as
is
large. We conclude that
has to be chosen reasonably
after the consideration of the lateral data sampling.
Indeed, if the sampling rate increases, larger value of
can be efficient since the lateral influence will be more
important.
SVD inversion with adaptative layout ‘SVDal’
Another way to solve an inverse problem is to use the
natural generalised inverse provided by singular value
decomposition (Lanczos, 1961):
,
where
and
are the matrices of the
eigenvectors related to non‐null eigenvalues and
spanning the model space and the data space
respectively. The advantage of this method is that we
do not need any a priori information. Thus, the natural
generalized inverse leads to minimize only the term of
data misfit without any weighting factors:
Φ=
,
‐ (σ)
.
However, for realistic cases which are ill‐conditioned
problems, the eigenvalues smoothly decrease toward
zero so that it is difficult to identify non‐null ones. In
practice, the solution is to cut off the small eigenvalues
or to damp them like Huang and Palacky (1991) or Chen
and Raiche, (1998). Actually, this method is equivalent
to a least square inversion with a weighted constraint
on the length of the solution. In order to keep a natural
solution (which minimizes only the term of data misfit),
we propose a method which consists in designing the
grid of the model before the inversion in such a way
that the problem is well conditioned. The algorithm of
Figure 5: Application of the logarithmic inversion to a synthetic data set. The starting model which has been used is a
smooth result provided by the vertically constrained inversion. The results show a good convergence in term of
resulting image and data misfit.
this method which we call SVDal can be described as
follows:
1‐ Knowing the maximum depth investigation,
we compute the grid layout for a initial
number of layers.
2‐ We compute the condition number which is
defined as the ratio between the highest
and the lowest non‐zero eigenvalues of the
kernel.
3‐ If the condition number is too high, we
change the number of layers and execute
again the previous steps until the condition
number is sufficiently low.
4‐ At last, the solution is computed using the
relation (10).
large condition numbers. By changing the number of
layers in the SVDal algorithm, one reduces these large
condition numbers to a more reasonable value which
has been fixed to
before the SVD
inversion. The application on synthetic data shows a
good misfit associated to the right convergence in the
model space. As the condition number decreases
naturally with the numbers of unknowns, it is important
to understand that the SVDal algorithm do not find the
lowest value of the condition number but the nearest
reasonable one. By choosing a decreasing number of
layers, one reduces obviously the computational cost
and, in a way, raises the smoothness of the resulting
model.
Figure 4 shows the difference between a simple SVD
inversion with 45 layers and the SVDal inversion. For
simple SVD inversion, some soundings may be ill‐
conditioned; the singularities generate infinite values of
conductivity. These strong artefacts are correlated with
Imaging the logarithm of the conductivity
The principal disadvantage of the methods presented
above is that they are fundamentally based on the
assumption that conductivity is normally distributed:
however it is well known that conductivity of rocks
Figure 6: Layered inversion of layer‐shaped and dyke‐shaped 2D conductive target within two different hosting
media. One can see that 1D inversion fails if the conductivity contrast is higher or in case of vertically shaped
formations.
usually follows a log normal distribution (Palacky, 1987).
In order to remove negative values and to get a more
realistic distribution it is therefore more convenient to
write the problem with the logarithm of the
conductivity:
.
,
,
,
,
,
⁄
By setting ,
,
, / , and
this relation leads back to a new formulation of the
linear relation to be inversed:
.
The new kernel
is highly non linear because it
depends on the model explicitly. Therefore, the
problem has to be solved by using a non linear iterative
method. If one knows a model being relatively close to
the solution, one can use the perturbation theory. We
can write the relation as follows:
,
,
with
,
.
,
,
.
This linear inverse problem is still partly undetermined
and needs regularization. We can use the fact that the
Taylor approximation allows only small perturbations.
The level of perturbation can be regularized by
minimising the length of the vector solution , so the
objective function can be written as follows:
Φ=
,
‐
.
Figure 7: Layered inversion of real data set acquired over the basin of Franceville (Gabon). On the left part: resulting
conductivity sections for the three methods VCI, LCI and SVDal with additional logarithmic inversion. The lithologies
FB and FA are superposed to the TEM section at wells positions in the profile. On the upper right part, the results of
the three methods of TEM inversion are superposed to the drill holes measurement.
By this way, the parameter
controls the magnitude
of the perturbation used at each step. Thus, the
electrical conductivity of the layered media is deduced
using the following formula:
λP
.
(17)
is taken equal to the absolute value of the results
If
provided by inversion of the conductivity, the
logarithmic inversion can be used as an additional
recursive filter which provides a realistic distribution of
conductivity.
Figure 5 shows the application of this method on the
synthetic data set. The starting model is the result of a
vertical constrained inversion in which
has been
chosen using the discrepancy principle for one sounding
in the profile (this sounding is taken randomly). The
magnitude of perturbation
has been selected in
order to provide a good convergence of the data misfit.
Limitation of the 1D interpretation
As expected, these applications show that 1D layered
inversion works quite well when imaging tabular
conductive target with a low conductivity contrast. For
this reason, 1D interpretation is relatively well adapted
when applied to characterize most hydrological target.
However, this approach fails to image properly local
high conductivity contrasts which are typically
encountered in mining exploration. Let us demonstrate
this limitation by showing two synthetic cases with
problematic results of the 1D inversion; synthetic data
have been computed by 2.5 modelling using ArjunAir:
1‐ the first case is the end‐border of a horizontally
shaped formation with conductivity larger than
the host,
2‐ the second case is vertically shaped like a dyke
formation with conductivity larger than the
host.
The conductivity of both targets is set equal to
.
S/m. The inversion is applied for two hosting media
S/m and
S/m. The results (Figure
6) show fake structures which seem to be dipping
conductive slabs: this clearly illustrates how 1D
interpretation is limited in cases of high conductivity
contrast or vertically shaped formations. Consequently,
more time‐consuming method as 2D or 3D inversion
with different regularization strategy (Portniaguine and
Zhdanov, 1999) would be a more appropriate way to
image these kinds of structures. This is also the
conclusions of Ley‐Cooper et al. (2010) who illustrated
the limits of the LCI on synthetic 2D structures.
Application to a real data set
Let us consider the application of the layered inversion
using these regularization methods on real helicopter‐
borne TEM data set (VTEM) acquired over the basin of
Franceville in Gabon for mining exploration. We
interpret the vertical component of the magnetic field
which consists in 1500 soundings of 27 channels starting
from
s to
. ms and acquired every
5m. The transmitter is a four turns loop with a diameter
of 26 m. The electrical current of 200 A is injected
during 8.32 ms before it is turned off; the
pulse
repetition rate is 25 Hz. The apparent conductivity is
computed by table look up of pure step responses
convolved with the measured waveform.
The basin is made of Precambrian sediments which can
host mineralization of uranium. Usually, mineralization
areas are found at the contact between two horizontal
lithologies in the basin: FA sandstone and FB ampelites
where uranium in solution has been precipitated thanks
to the presence of organic matter. FB lithology is
characterized by relatively high electrical conductivity
that constitutes the top of a proterozoic reservoir and
superposes on FA formation that is, composed of coarse
grain size sediments characterized by relatively low
electrical conductivity. This developed contrasts of
conductivity of about three orders of magnitude which
are often located at depths less than 400 m; therefore it
is possible to detect it by using TEM imaging. The
geology of the area has a tabular geometry with low
dips that justifies the use of layered inversion as a first
realistic approximation.
Robustness of the methods is clear on Figure 7 where
we show the comparison between the conductivity
obtained from the layered inversion of an airborne TEM
line and the conductivity measured from logging into
four boreholes in the same area. The topographic map
at the bottom right corner of figure 7 shows the
locations of boreholes and the TEM line considered. On
Figure 7, the geological logs for each borehole are
plotted on TEM conductivity sections. They are also
displayed as background on the right part of the figure
where TEM and borehole conductivity are compared.
Whatever the 1D TEM conductivities, it seems that the
results are slightly more conductive than the
conductivity measured in borehole (up to a factor of 2
within the first hundred meter depths). Similar
considerations are found when comparing borehole
conductivity with other kinds of EM data like CSEM. In
case of TEM, this result may be due to the fact that TEM
eddy currents are mostly horizontals while borehole
conductivities are measured with vertical current lines.
Tabular media are characterized by vertical transverse
isotropy which shows larger conductivity in the
horizontal directions than in the vertical direction.
Therefore, TEM soundings, which are sensitive to the
horizontal conductivity, should produce larger
conductivity than borehole measurements. In addition,
borehole conductivity is measured over a few
centimetres samples of ground while TEM soundings
integrate a larger area of several meters. This scaling
change can generate non negligible differences
between the two methods. Nevertheless, in a
qualitative point of view, borehole 2 exhibits very good
accordance between the different outcomes. The
results at borehole 3 and 4 are coherent as well, except
that we over‐estimate the conductivity of the second
layer in the TEM results (to factor of 10‐20). The two
latter boreholes are situated in the slope which
separates the shelf from the valley. Since the shelf is
supposed to have a thicker conductive layer on its
surface, we suggest that the over‐estimation of the
second layer is the consequence of the topography
which causes bias in the 1D TEM interpretations. We
think that this effect occurs pathologically for borehole
1 which is situated at the top of the shelf border.
Indeed, the TEM sounding does not detect the contact
between the two lithologies.
Conclusion
All three effective ways which we have exposed to
invert the electrical conductivity of a layered medium
allow fast interpretation of a large volume of data
during the survey. They make the regularization
parameters more readable for geophysicists by using
physical considerations as much as possible. Besides, it
is important to note that all the different approaches
provide similar images of conductivity. Therefore, one
can think that the present methods should be used
more as tools to avoid the mathematical instability due
to data uncertainties than specific ways to manage the
theoretical non‐uniqueness (due to equivalent models).
The first method is the One Pass Imaging developed by
Christensen (2002) in which we have simplified the
parameterization by removing the term constraining the
length of the solution. The second method is based on
lateral constraints; it can be considered as a
unidirectional constrained inversion since the a priori
information comes from the previous sounding only and
not from a set of surrounding ones. The major practical
difference with those already reported in the literature
is that soundings are inverted one by one; as a
consequence, the computational cost is reduced. The
third method allows the user to provide a model of
conductivity which does not contain additional
information given by regularization. The user still has to
set one parameter to the program, the maximum value
of the condition number.
Besides, we propose an iterative method which handles
the problem with the logarithm of the conductivity; it is
an additional step which starts from the result given by
one of the three processes described previously. This
method avoids the presence of negative values in the
resulting conductivity model and lends realism to the
conductivity distribution. Especially in case of tabular
lithologies, resulting models show good accordance
with the true model and the borehole measurements
when applied on synthetic and real data respectively.
Not only does this method provide good accuracy as
shown by evaluation on synthetic dataset and real
VTEM survey, but it is really cost effective. We have
written a dual C#/Matlab compiled program; it allows
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Acknowledgement
This paper results from a Joint Research Project
between AREVA‐NC and CNRS UMR 7516. The authors
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