2-D Niblett-Bostick magnetotelluric inversion - MTNet

a
0
σ
−2
a σ a ( x,
ω)
= ωµ 0 | Z(
x,
ω)
| ,
(1)
−2
σ ( , ) | ( , ) | ,
J. RODRÍGUEZ et al. a x ω = ωµ 0 Z x ω
(1)
2-D Niblett-Bostick
µ
sa(x, T) with respect to s(x, z). The integration is defined
over the entire lower half-space. The recovery of s(x, z) from
sa(x, T) is clearly a nonlinear problem since F depends on
s. Otherwise, the integral equation could be readily solved
using any of the available methods of linear analysis. It
is still possible to apply linear methods sequentially, as
in traditional linearization, simply by updating a starting
model on the right-hand side of the equation. Esparza and
Gómez-Treviño (1996), working with the 1-D problem,
showed that reasonably good results can be obtained in a
single iteration using an adaptive approximation. In 1-D,
the approximation is
(4)
F(z’,s,T) on the left hand-side represents the true Fréchet
derivative for an arbitrary conductivity distribution, and
Fh(z’,sa,T) on the right stands for the much simpler
Fréchet derivative of a homogeneous half-space, whose
conductivity is known and equal to the measured apparent
conductivity. Fh is simply an attenuated cosine function
that gradually vanishes with depth. The factor (1-m) drops
out of the approximation when substituting expression (4)
in equation (2). Using the Fréchet derivative Fh(z’,sa,T)
for a homogeneous half-space (Gómez-Treviño, 1987b),
the approximation in 1-D can be written as
(5)
where . In the original Niblett-Bostick
integral equation, the upper limit of integration is 0.707da
and the kernel is simply (0.707sa) -1 ( x,
w)
0
a
2π σ a ( x,
w)
=
ω
2π
By analogy, making the same type of assumptions and
T =
approximations in 2-D, equation (2) can be written as
a
1 ω
σ a ( x, T) = F( x, x', z', σ, T) σ(
x', z') dx'dz', 1−
m ∫ (2)
( x,
w)
σ
a
a
σ (8)
a( x, T ) =
( x, z ) σ ( x,
wd
) logσ
∫ Fh( x, x', z ', σa, T ) σ(
x', z ') dx' dz '. (8)
a
a m = ,
(3)
a
σ ( x, z)
F h
d logT
Analytical expressions for Fh are derived in Appendix A
−2
, T ( ) x,
T ) σ ( x,
ω)
= ωµ
for the traditional TE and TM modes, respectively. In turn,
a
0 | Z(
x,
ω)
| , σ
a a (1)
they are used in Appendix C to derive the corresponding
( x, z ) σ a ( x,
T )
expressions for series and parallel apparent conductivities.
( x,
w)
σ ( x, z)
1
σ j
a
σ a ( x, T) = F( x, x', z', σ, T) σ(
x', z') dx'dz', 1−
m ∫ (2)
2π
1 T In 2-D, to construct equation (6) the half-space is
σ a ( x, T) = F( x, x', ij
=
z', σ, T) σ(
x', z') dx'dz', d logσ
divided into a large number of rectangular elements. The
a1
− m ∫ (2)
ω
m = , σ j , j = 1,...,
N,
integration (3) over the elements can be performed analytically
F( z', σ , T) = (1 − mF ) h( z', d
σ
log
a,
T
T d logσ
a
). m = ˆ σ (4) , as described in Appendix (3)
ak , k = 1,...,
M
B. This is particularly useful for
( xx , ', z', σ , T)
d logT
a ( x,
w)
handling the singularities N at the points of measurement,
and also for the final rectangles on the sides and bottom of
) xa , ( z x)
, T ) Fxx ( , ', z', σ , T)
ˆ σ ak = ∑ akjσ j , k = 1,...,
M.
(9)
( x, z ) σ ( x,
T )
the model. It is worth j=
1 remarking that each of the elements
a
of sa is a weighted average of all the unknown conductivity
)
( x, z ) σ ( x, z)
σ
( x,
T )
ak ,
a
values and, that on virtue of equation (2), the elements
xa , ( z x)
, T ) σ ( x, z)
σˆ ak
of matrix A are dimensionless. Furthermore, the sum of
2 ∞
−2 z '/ δ ⎡ 2z' 2z' ⎤
a
σ ( ) cos( ) sin( ) ( ') ',
the elements of any row of A is identically unity, which
a T = e σ z dz
δ ∫ σ ( ⎢ + ⎥
0 Fx ( , Tz',
)
(5)
M
M
N
a 1 σ , T
a ⎣ δ
) = (1 − mF
a δ
) h( z', σ a,
T).
1(4) 2 1
2
σ a ( x, T) = F( x, x', z', a σ, T⎦) σ(
x', z') dx'dz', is a very useful property for checking the accuracy of the
( z', σ , T)
1−
m ∫ C = (2) ∑ ( σ ˆ ak −σ
ak ) =
F( z', σ , T) = (1 − mF ) h( z', σ a,
T).
(4) ∑ [ σ ak −∑
akjσ
j ] . (10)
T / δ
computations
2 k=
1 involved. Notice
2 k=
1 that although j=
1 equation (6)
a
h( z', σ a,
T)
F( z', σ , T)
d logσ
a m = ,
(3) is a system of linear equations, the model it represents
F is actually nonlinear, for A depends on the unknown
h
h( z', σ a,
T)
N N M
N M
M
d logT
1
1 2
− 1
C = −
( xx distribution σ through the different values of sa.
h( , z', ', z σ ',
a,
T σ , ) T)
F
∑ ∑ [ −∑
akia
kj ] σ iσ
j −∑
[ − ∑ akiσ
i + ∑ akiσ
ak ] σ
h
2 i=
1 j¹
i=
1 k=
1
i=
1 2 k=
1
k=
1
Fh2( zσ', ∞ σ
a = a,
TA)
a ( x,
T )
−2 z σ'/
. δ ⎡ 2z' 2z' ⎤ (6)
a
σa( T) = e ⎢cos( ) + sin( ) ⎥σ(
z ') dz ',
δ ∫ (5)
x, z )
0
HOPFIELD ARTIFICIAL NEURAL NETWORKS
a ⎣ 2 δ ∞
a −2 z '/ δa
δ ⎡
⎦ 2z' 2z' ⎤
M
a
σa( T) = e cos( ) sin( ) σ(
z ') dz ',
δ ∫ ⎢ + ⎥
(5) 1 2
x, z )
0
a
a = 503 ( − T2 z/
/ δ ) 2z
j ( 2z
/ ) 2z
⎣ δa δa
⎦
∑ ( σ ak ) .
(11)
j δ aai
− j + 1 δai
j+
1
The application of artificial 2 k=
1neural
networks to the
a ( xa,
ij T = ) e cos( ) − e cos( ), (7)
. 707δ
δ a = δ503
T / δ a
inversion of MT data has been explored in various
ai
δ
a
ai
F( z', σ , T) = (1 − mF ) h( z', σ a,
T).
(4) directions. One Nway
Nis
to use the multi-layer N feed forward
−1
0.
707σ
a ) 0 . 707δ
1
a
( z', σ , T)
.
neural E = network − ∑ architecture ∑ Tijσ
iσ
(Rummelhart j −∑
I iσ
i , et al., 1986) (12)
−1
( 0.
707σ
) σ = Aσ.
which uses 2a
set i=
1 of j¹
responses i=
1 and models i=
1 presented to the
( z', σ
a
a (6)
a,
T)
To solve equation (5) numerically, we divide the input and output defined neurons, respectively. During a
σ = Aσ.
a half-space into a large number of layers with a (6)
uniform learning M phase, the network back-propagates M
M
1
(through its
2
( z', σ conductivities. σ The result 2 is that the integral equation 2 can neurons and interconnection weights) errors due to misfit
a,
T)
(
a
− 2z
/ ) z j ( 2z
1 / ) z T
j δ ai
− j + δ
ij = −∑
akia
kj and I i = − ∑ akiσ
i + ∑ akiσ
ak . (13)
ai
j+
1
be awritten
ij = e as a matrix cos( equation ) − e cos( ),
kof
= (7) 1the
model and the obtained 2neural
k=
1 model. Learning k=
1 from
2 ∞ δ ( − 2z
/ ) 2z
j
ai
δ ( 2z
/ ) 2z
−2 z '/ δ ⎡ 2z' j δ 2z' ⎤
a
ai
− j + 1 δai
j+
1
σ ( ) acos(
1 cos( ) ai cos( one response-model ), ‘pattern’ is achieved by updating the
a T = e ⎢ij = e ) + sin( ) ⎥σ(
z−') dz e ',
(7)
0
sa = As. δ ∫ (5)
a ⎣ δa δa
⎦ δ (6) inter neuron connection weights according to a gradient
j
ai
δ ai 1 1 ⎛ N
⎞
( t+
1)
( t)
z
descent minimization criteria. The process is then applied
= 503 T / δ
σ sgn⎜
⎟
j
i = +
.
ai
a
The vector sa contains the data for the different to the complete 2 response-model 2 ⎜ ∑ Tijσ j + I i ⎟
(14)
⎝ j≠i
= 1 data set, thereby ⎠ achieving
707 δ periods. δ The ai
a
vector s represents the unknown conductivity a learning epoch.
−1
distribution in its discrete form, and the matrix A contains ( t+
1)
. 707σ
a )
σ i
the weights of the conductivity elements for all the Once the network is trained, it recovers a model in
(t )
available data. The elements σ a = Aof
σthe
. matrix can be evaluated σ j (6) almost no time when provided with a sounding curve. The
analytically as
1
distinctive feature of this learning approach is that there
a
T ij
is very 1 little physics fed into the algorithms. In fact, as far
T
( − 2z
/ ) 2z
j ( 2z
/ ) 2z
j δ ai
− j + 1 δai
j(
+ AA
1 ) ij as the algorithms are concerned, the models and responses
aij
= e cos( ) − e cos( ), (7)
δ
used in the training sessions may or may not be related
ai
δ ai
(7) through any physical link, the learning process is simply
the same. Hidalgo and Gómez-Treviño (1996) explored
where zj is the top depth of the j-th layer, and dai is the skin this approach for the 1-D problem with reasonably good
i
depth of the i-th measurement.
results. However, extending the method to 2-D would be
Geologica Acta, 8(1), 15-30 (2010)
DOI: 10.1344/105.000001513
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2
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