Acta Geophysica
vol. 61, no. 3, Jun. 2013, pp. 569-582
DOI: 10.2478/s11600-012-0099-4
Increased Resolution of Subsurface Parameters
from 1D Magnetotelluric Modeling
Mothe SHIREESHA1 and Tirumalashetty HARINARAYANA1,2
1
National Geophysical Research Institute,
Council of Scientific and Industrial Research, Hyderabad, India
e-mail: [email protected] (corresponding author)
2
GERMI Research Centre, Pandit Deendayal Petroleum University,
Gujarat, India; e-mail: [email protected]
Abstract
In electric and electromagnetic techniques, it is well known that the
principle of equivalence poses a problem in the interpretation of subsurface layers. This means the inversion problem can provide the conductivity-thickness product more confidently than the individual parameters –
conductivity and thickness – separately. The principle of equivalence
corresponds to the middle layer in a three-layer earth structure. In order
to resolve this problem, we have touched upon the different formulae of
apparent resistivity proposed by earlier workers considering the real and
imaginary parts of the impedance tensor and designed a new formula to
compute apparent resistivity for different models. We observed that the
application of our new formula for apparent resistivity using the combination of real and imaginary parts of the impedance has a better resolution as compared to earlier conventional formulae of apparent resistivity.
These results have been demonstrated through both forward and inverse
modeling schemes.
Key words: electromagnetics, magnetotellurics, apparent resistivity,
impedance, resolution.
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1.
M. SHIREESHA and T. HARINARAYANA
INTRODUCTION
Magnetotellurics (MT) is a passive exploration technique that utilizes
a broad spectrum of naturally occurring electromagnetic fields in the earth to
determine the electrical resistivity of the subsurface. Apparent resistivity is
traditionally defined as the resistivity of a homogeneous half-space which
will produce the same response as that measured over the real earth with the
same acquisition parameters (position, transmitted current, etc.). Cagniard
(1953) has given the conventional formula to compute apparent resistivity.
Since then, several alternative formulae of apparent resistivity have been put
forward. Different apparent resistivity curves are obtained with conventional
frequency-domain processing and time-domain processing of MT data
(Kunetz 1972).
When several measurements are made with different orientations at one
site, the apparent resistivity is described as a tensor, and it is multivalued
depending upon its direction. Tensor apparent resistivities are routinely calculated for MT methods (Vozoff 1972). Many investigations have already
been carried out to evaluate the relative viability of the different formulae as
a useful apparent resistivity parameter (Spies and Eggers 1986, Kant et al.
1991) by analyzing the apparent resistivity curves from forward models.
Another new definition (Basokur 1994) was proposed by using frequency
normalized impedance function and using model curves computed for a 1D
layered earth model. Garcia and Jones (2008) have developed an algorithm
to perform the continuous wavelet transform, using Morlet wavelets, of
high-frequency audio magnetotelluric data, focusing on the superior estimates in the AMT dead band of 1 through 5 kHz. These works have yielded
physical insights into the MT method and have demonstrated alternate
approaches, in some cases superior to the conventional methods for delineating the subsurface parameters.
2.
METHODOLOGY
Apparent resistivity calculations have been carried out for three-layer models
using real, imaginary and absolute values of surface impedance. The apparent resistivity behavior shows characteristic features of the layered parameters using different definitions of apparent resistivity. It is concluded that the
use of various definitions of the apparent resistivity can be useful in qualitative interpretation in the field. The new formula proposed in the presented
study considers both real and imaginary components of the impedance with a
new combination in defining the apparent resistivity of both conductive and
resistive middle layers that includes models named K, H, A, and Q type ones
in 1D environment.
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In electrical and electromagnetics methods, the measured data are presented as apparent resistivity curves, pseudo-sections, and maps. The conventional formula (Cagniard 1953) of apparent resistivity has traditionally
been used for presentation of magnetotelluric data as detailed below
( ).
ρ a , Z (Absolute) = (1 / ωμ ) z
2
(1)
Spies and Eggers (1986) have pointed out the possibility of alternative
definitions of apparent resistivity which give a better approximation to the
true resistivities of subsurface layers as given below
ρ a ,Re( Z )(Re al) = ( 2 / ωμ ) ( Re( z ) 2 ) ,
ρ a ,Im( Z )(Im aginary) = ( 2 /ωμ ) ( Im( z ) 2 ) .
(2)
(3)
Since the apparent resistivity is a normalizing procedure with physical
significance, it is possible to formulate new definitions which produce better
results for some specific applications than those produced using the conventional definition. For this purpose, we utilized the real and imaginary parts of
impedance in an innovative way in order to enhance the signal. The new
formula of apparent resistivity was introduced in the present study to give
a better description of the magnetotelluric response of a layered-earth model.
For a uniform half-space, the observed surface impedance is related to the
true resistivity by
Z (ω ) = Ex(ω ) Hy (ω ) ,
where E(ω) and H(ω) are orthogonal horizontal components of the electric
and magnetic fields. Following Cagniard’s lead, the conventional practice
has been to define the apparent resistivity in terms of an amplitude of the
observed impedance. In order to normalize the impedance versus frequency,
a new equation is formed as follows:
{
ρ a ( New ) = ( 2 /ωμ ) ( Re( z ) 2 )
3
( Im( z ) ) } ,
2 2
(4)
where ω is the angular frequency, and µ is the magnetic permeability of free
space.
There are definite rules to be followed for defining the response functions. The well known rules for the definition of response functions are:
‰ the response function should asymptote to the surface layer resistivity for higher frequencies,
‰ the response function should asymptote to the lower half space resistivity (bottom layer) for lower frequencies,
‰ depending on the nature of middle layer resistivity, the response
function may vary for intermediate frequencies.
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M. SHIREESHA and T. HARINARAYANA
The proposed new formula obeys these three fundamental rules of
response function. The computed response using the new formula is compared with earlier formulae (Spies and Eggers 1986) and we observed that
the proposed new formula is superior to earlier ones, as shown clearly with
a few examples. For this purpose, different three-layer models are considered and discussed in the following.
3.
RESULTS AND DISCUSSIONS
In order to test the application of the new formula for apparent resistivity, we
have taken different models named K, H, A, and Q type ones and computed
the apparent resistivity using four different formulae (Eqs. (1)-(4)) of abso-
Fig. 1. Plots of apparent resistivity versus period obtained for different K-type (resistive middle layer) models: (a) K1, (b) K2, and (c) K3 with varying thickness of the
middle layer using formulae (1) to (4) of apparent resistivity; the models are shown
on the top right-hand side.
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INCREASED RESOLUTION OF SUBSURFACE PARAMETERS
573
lute, real, imaginary, and the present new formula of apparent resistivity.
Different thicknesses of the middle layer in each case are given. The relative
resolution capability of all the four formulae is discussed in each case below.
3.1 K-type models (ρ1 < ρ2 > ρ3)
Figure 1 represents plots of apparent resistivity versus period obtained for
different K-type (resistive middle layer) models (K1, K2, and K3) with varying thickness of the middle layer using formulae (1) to (4) of apparent resistivity. The models are shown on the top right-hand side of Fig. 1. A higher
resolution for the new formula proposed in the presented study can be
clearly seen.
Fig. 2. Plots of apparent resistivity versus period obtained for different H-type models: (a) H1, (b) H2, and (c) H3 using formulae (1) to (4) of apparent resistivity; the
model considered is shown on the bottom right-hand side.
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M. SHIREESHA and T. HARINARAYANA
3.2 H-type models (ρ1 > ρ2 < ρ3)
Next, we consider conductive middle layer (H-type) models for resolution
analysis.
Figure 2 represents plots of apparent resistivity versus period obtained
for different H-type models (H1, H2, and H3) using formulae (1) to (4) of
apparent resistivity. The model considered is shown on the bottom righthand side. As before, the proposed new formula has shown greater resolution
to the buried middle layer compared to conventional and earlier formulae.
3.3 A-type models (ρ1 < ρ2 < ρ3)
Next, we examine the resolution analysis of three layers with increasing
resistivity (A-type).
Fig. 3. Plots of apparent resistivity versus period obtained for different A-type models: (a) A1, (b) A2, and (c) A3 using formulae (1) to (4) of apparent resistivity; the
model considered is shown on the bottom right-hand side.
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Figure 3 represents plots of apparent resistivity versus period obtained
for different A-type models (A1, A2, and A3) using different formulae (1) to
(4) of apparent resistivity. The model considered is shown on the bottom
right-hand side. Also in this case, Eq. (4) is proved to be more sensitive
compared to Eqs. (1)-(3).
3.4 Q-type models (ρ1 > ρ2 > ρ3)
Finally, we examine the resolution when the subsurface resistivity continuously decreases with depth (Q-type).
Figure 4 represents plots of apparent resistivity versus period obtained
for different Q-type models (Q1, Q2, and Q3) using formulae (1) to (4) of
Fig. 4. Plots of apparent resistivity versus period obtained for different Q-type models: (a) Q1, (b) Q2, and (c) Q3 using formulae (1) to (4) of apparent resistivity; the
model considered is shown on the right-hand side.
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M. SHIREESHA and T. HARINARAYANA
apparent resistivity. The model considered is shown on the right-hand side.
Also in this case, Eq. (4) is proved to be superior to other equations, as can
be seen clearly from the figure.
After testing the resolution of all four types of models, a comparative
study of the effect of change in thickness is made for different formulae (1)(4) and presented in the form of percentage variation on a log scale.
Figure 5 represents the effect of change in thickness using percentage
variation of models 2 and 3 with respect to model 1 for all four K, H, A, and
Q types.
In all the above cases, the effect of the change in thickness of the intermediate layer is sharply distinguishable with the proposed new formula
compared to those obtained using the other conventional formulae. All the
above examples shown are based on forward modeling. Nowadays, for all
practical purposes the data are processed mostly using inverse modeling
LOG % VARIATION
3
ABSOLUTE
REAL
IMAGINARY
NEW
2.5
2
1.5
-2
-1
0
1
2
3
4
LOG PERIOD(sec)
Fig. 5a. Percentage variation as a function of period for models K1 and K2. The percentage variation is calculated using the formula K2/K1 × 100, where K1 and K2 are
the magnetotelluric response functions of the K-type models 1 and 2, respectively
(Fig. 1a, b).
LOG % VARIATION
3
ABSOLUTE
REAL
IMAGINARY
NEW
2.5
2
1.5
-2
-1
0
1
LOG PERIOD(sec)
2
3
4
Fig. 5b. Percentage variation as a function of period for models K1 and K3. The percentage variation is calculated using the formula K3/K1 × 100, where K1 and K3 are
the magnetotelluric response functions of the K-type models 1 and 3, respectively
(Fig. 1a, c).
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LOG % VARIATION
INCREASED RESOLUTION OF SUBSURFACE PARAMETERS
3
2
1
0
ABSOLUTE
REAL
IMAGINARY
NEW
-1
-2
-4
-3
-2
-1
0
1
LOG PERIOD(sec)
2
3
4
5
LOG % VARIATION
Fig. 5c. Percentage variation as a function of period for models H1 and H2. The percentage variation is calculated using the formula H2/H1 × 100, where H1 and H2 are
the magnetotelluric response functions of the H-type models 1 and 2, respectively
(Fig. 2a, b).
3
2
1
0
ABSOLUTE
REAL
IMAGINARY
NEW
-1
-2
-4
-3
-2
-1
0
1
LOG PERIOD(sec)
2
3
4
5
Fig. 5d. Percentage variation as a function of period for models H1 and H3. The percentage variation is calculated using the formula H3/H1 × 100, where H1 and H3 are
the magnetotelluric response functions of the H-type models 1 and 3, respectively
(Fig. 2a, c).
LOG % VARIATION
2.5
2
1.5
ABSOLUTE
REAL
IMAGINARY
NEW
1
0.5
-2
-1
0
1
2
LOG PERIOD(sec)
3
4
5
Fig. 5e. Percentage variation as a function of period for models A1 and A2. The percentage variation is calculated using the formula A2/A1 × 100, where A1 and A2 are
the magnetotelluric response functions of the A-type models 1 and 2, respectively
(Fig. 3a, b).
schemes. In what follows we demonstrate our results using the inverse modeling too.
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M. SHIREESHA and T. HARINARAYANA
LOG % VARIATION
3
2
1
0
ABSOLUTE
REAL
IMAGINARY
NEW
-2
-1
0
1
2
3
LOG PERIOD(sec)
4
5
Fig. 5f. Percentage variation as a function of period for models A1 and A3. The percentage variation is calculated using the formula A3/A1 × 100, where A1 and A3 are
the magnetotelluric response functions of the A-type models 1 and 3, respectively
(Fig. 3a, c).
4
LOG % VARIATION
ABSOLUTE
REAL
IMAGINARY
NEW
3
2
1
-4
-3
-2
-1
0
LOG PERIOD(sec)
1
2
3
4
Fig. 5g. Percentage variation as a function of period for models Q1 and Q2. The percentage variation is calculated using the formula Q2/Q1 × 100, where Q1 and Q2 are
the magnetotelluric response functions of the Q-type models 1 and 2, respectively
(Fig. 4a, b).
LOG % VARIATION
ABSOLUTE
REAL
IMAGINARY
NEW
4
3
2
1
-4
-3
-2
-1
0
1
2
3
4
LOG PERIOD(sec)
Fig. 5h. Percentage variation as a function of period for models Q1 and Q3. The percentage variation is calculated using the formula Q3/Q1 × 100, where Q1 and Q3 are
the magnetotelluric response functions of the Q-type models 1 and 3, respectively
(Fig. 4a, c).
4.
CASE STUDY
A model is constructed based on the borehole data available for a location
near Lodhika, Saurashtra region, Gujarat. The use of joint inversion has been
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LOG RHO(ohm-m)
INCREASED RESOLUTION OF SUBSURFACE PARAMETERS
4
ABSOLUTE
REAL
IMAGINARY
NEW
3
2
Layer 1
Layer 2
Layer 3
Layer 4
1
-4
-3
-2
-1
0
1
2
R (ohm-m) Thickness(m)
100
1350
25
1600
50
550
1000
-3
4
5
LOG PERIOD(sec)
Fig. 6. Apparent resistivity versus log time period for the borehole data using the
four formulae of apparent resistivity.
demonstrated for the MT data at this location (Harinarayana 1999). The geological strata consist of Deccan traps, a sedimentary formation followed by
high-resistivity basement. Accordingly, 100 ohm·m is assumed for Deccan
traps, 25 and 50 ohm·m for sedimentary formation and 1000 ohm·m resistivity for the basement. The above subsurface section is considered based on
the empirical data of Lodhika borehole. Firstly, forward modeling has been
carried out to generate the data. Later, the inversion modeling has been carried out for the data generated using all the four formulae of absolute, real,
imaginary and new apparent resistivity as given below.
Figure 6 represents the plot of apparent resistivity versus log time period
for the borehole data using four formulae of apparent resistivity.
Table 1
Lodhika borehole actual model strata and the four inverse models
taking the four formulae (absolute, real, imaginary, and new) of apparent resistivity
Lodhika
borehole
strata
Resistivity [Ohm-m]
ρ1
100
ρ2
25
ρ3
50
ρ4
1000
Thickness [m]
h1
1350
h2
1600
h3
550
Depth [m]
d1
1350
d2
2950
d3
3500
Inverse model Inverse model Inverse model Inverse model
taking real taking imaginary taking new
taking absolute
formula
formula
formula
formula
96.4
20.1
47.9
956.9
98.1
22.4
48.9
973.4
98.1
22.0
48.8
980.5
98.2
24.0
49.4
973.3
1180.0
1275.8
519.1
1262.8
1418.9
533.8
1235.0
1402.6
534.0
1308.3
1519.1
544.5
1235.0
2637.6
3171.6
1308.3
2827.4
3371.8
1180.0
2455.7
2974.8
1262.8
2681.7
3215.5
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M. SHIREESHA and T. HARINARAYANA
Fig. 7. Resistivity versus depth for the actual borehole strata and the four other
inverse models:
The proposed new formula showed better resolution than all the other
formulae of apparent resistivity, as presented in Fig. 6. Table 1 also shows
the inverse model using the new formula. The values derived using the new
formula of the present study have shown the values of the model parameters
assumed for the borehole strata as compared to other formulae of apparent
resistivity.
Figure 7 shows the plots of resistivity versus depth for the actual borehole strata and the four other inverse models.
5.
CONCLUSIONS
The new formula has shown higher resolution in identifying subsurface layers. This is clearly seen by varying the thickness and resistivity of the middle
layer. Thus it can be concluded that subsurface geological structures can
more accurately be estimated using our formula than by using conventional
formula of apparent resistivity. Specifically, one can say that:
‰ The new formula for apparent resistivity has better detectability
compared to other formulae, as clearly shown in Figs. 1-4.
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‰ The new formula for apparent resistivity is more sensitive to thin
layers compared to conventional formulae, as shown in Fig. 5.
‰ Once the apparent resistivity is computed using the new formula,
(Eq. 4), the method could be effectively implemented to interpret the data.
‰ The new formula for apparent resistivity has indicated the subsurface
structure more clearly in terms of both the thickness and resistivity of the
subsurface layers.
‰ The new formula for apparent resistivity is very much useful in oil,
geothermal and mineral exploration for delineating thin structures below the
surface as it is well known that oil, gas, and mineral zones are sources of
anomalous conductive or resistive structure compared to the background.
‰ The superiority of higher resolution of the proposed new formula is
also demonstrated using inverse algorithm considering the strata for a location near Lodhika, Gujarat.
‰ Although the presented study is concentrated on 1D modeling, we
are working on extending it onto2D and 3D models.
A c k n o w l e d g e m e n t . We are thankful to the Director of NGRI for
supporting this research and giving permission to publish this paper.
References
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M. SHIREESHA and T. HARINARAYANA
Spies, B.R., and D.E. Eggers (1986), The use and misuse of apparent resistivity in
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Received 25 January 2012
Received in revised form 25 September 2012
Accepted 2 November 2012
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