2-D Niblett-Bostick magnetotelluric inversion - MTNet

J. RODRÍGUEZ et al. 2-D Niblett-Bostick
( t)
( t)
σ = σ .
(15)
j w l ( j+
l)
l=
−n
of the data in such a way that the averaging functions
resemble boxcar functions for the averages to have the usual
meaning. The resulting models are not intended to produce
responses that fit the data, but to address the non-uniqueness
character of the inverse problem. Within the limitations of
linearization, they can be called true averages. More rigorous
approaches for computing averages have been developed for
the 1-D magnetotelluric problem (e.g. Weidelt, 1992), but their
extension to higher dimensions are far from trivial.
Simulated averages
The shortcut that we take consists of computing models
that are averages of themselves everywhere with a given
neighborhood, and that at the same time their responses fit
the data at an optimal level. Consider that
(15)
The dynamics of the HANN in (14) transforms to
(16)
Following Zhang and Paulson (1997), we will refer to
this as the regularized version of the HANN or RHANN.
The updated conductivities sj
Geologica Acta, 8(1), 15-30 (2010)
19
DOI: 10.1344/105.000001513
(t+l) are obtained from averages
of the original values sj (t) . The choice of the appropriate
filter depends on the application. Zhang and Paulson (1997)
found binomial filters useful to stabilize their solution.
In our case, we use boxcar functions, for this leads to the
simplest and most intuitive type of averages. The filter, with
2n+1 uniform weights wk = (2n+1) -1 its performance in 1-D, and compare its results with
existing methods of average estimations. In the following
lines, we test the performance of equation (16) as it applies
to the solution of equation (6). The elements of the matrix
are computed using equation (7) which represents the 1-D
version of the proposed approximation. The synthetic data
for the tests, shown in Fig. 1A, correspond to apparent
conductivity responses of the model shown in Fig. 1B.
The tests are intended to demonstrate that the proposed
approach for the computation of averages has the same
properties as the simple formula for averages in Gómez-
Treviño (1996). The formula is
(18)
n
( t)
( t)
σ where represents the average of conductivity
j = ∑ w lσ
( j+
l)
.
(15)
1 2
n
l=
−n
between zi ( t)
( t)
= 0.707da(Ti),i = 1,2. The averages are assigned
σ j = ∑ w lσ
( j+
l)
.
(15) to the mean geometrical depth z = √ z z . Any two data
1 2
l=
−n
points can be used in the formula regardless of how far
1 1 ⎧ N n
⎫
( t+
1)
( t)
apart they are in the sounding curve. If the points are
i = + sgn⎨
∑ Tij[ ∑ wlσ
( j+
l)
] + I i ⎬.
(16)
2 2 ⎩ j≠i1
= 1 1 l=
−n
⎧ N n
⎭ ⎫
( t+
1)
( t)
contiguous, the windows are narrow and the average
σ i = + sgn⎨
∑ Tij[ ∑ wlσ
( j+
l)
] + I i ⎬.
(16)
2 2
model, the average of the real earth, has the highest
⎩ j≠i
= 1 l=
−n
⎭
possible resolution. If, on the other hand, the chosen
t+
1)
points are wide apart, the windows are themselves wide,
t )
for z and z tend to separate. As the windows widen, the
1 2
− n,..., n
models tend to flatten, as they should. This intuitively
1
= , k = −n,...,
n
appealing feature of spatial averages is built into equation
2n
+ 1
(18) in spite of its simplicity. Figures 1C and 1D show
N
n
1 1
( t)
( t)
+ sgn{
this feature developing when considering averages taken
∑ [( 1−
b ) Tijσ
j + β ∑ Tijwlσ
( j+
l)
] + I i}.
(17)
2 2
N
n
from data values 1 and 5 periods apart, respectively, for
( t+
1)
j≠i
= 1 1
l=
−n
( t)
( t)
σ i = + sgn{ ∑ [( 1−
b ) Tijσ
j + β ∑ Tijwlσ
( j+
l)
] + I i}.
(17)
2 2
the synthetic data presented in Fig. 1A.
j≠i
= 1
l=
−n
σ
, k = -n,..., n (square
a ( T2
) δ a ( T2
) −σ
a ( T1
) δ a ( T1
)
σ ( z1,
z2
) >= window), can be of different widths. , We introduce (18) a further A corresponding sequence of full filtered models (β =1)
parameter β δ ato
( Taccommodate
2 ) σ−
δ ( T )
a ( Ta
2 ) 1 δ a ( Ta
2traditional
) −σ
a ( T1
) trade-off δ a ( T1
)
< σ ( z
between is presented in Figs. 1E and 1F when the RHANN algorithm
1,
z2
) >=
, (18)
two contrasting features. δWe
modified the dynamics to is applied to the same set of data. It can be observed that
a ( T2
) −δ
a ( T1
)
the behavior of the averages follows that of the averages
i ), ( i = 1,
2 1, 2 ) >
shown in Figs. 1C and 1D. There are some differences
that can be traced back to basic differences between the
two approaches. One is that equation (18) is based on the
(17) original Niblett-Bostick approximation that assumes a
boxcar function as the kernel of equation (5). The RHANN
When β =1 Hwe
( xget
, 0)
the full filtered solution. On the other hand, algorithm uses the kernel as indicated in equation (5) which
x 2
| when σ a | = ωµ β =0 0 | we return to | , the original unfiltered solution (A.1) given includes a small negative sidelobe (Gómez-Treviño, 1987b)
E y ( x,
0)
by equation (14). In this way, by varying this parameter we and extends to infinity. This is a somewhat less restrictive
can gradually control the effect of the averaging process. approximation than the rather simpler boxcar function. The
δE
( x,
0)
H ( x,
0)
other difference originates in the fact that the windows in
y δ x
| σ a | = −2
| σ a | Re[ − ]. (A.2)
each approach are necessarily different, because in the first
E ( x,
0)
H ( x,
0)
APPLICATIONS y
x
case they cannot be uniform, for they are determined by the
data, and in the second case they are uniform for all depths.
1-D averages
The overall effect is a somewhat better performance for
the approximation given by equation (5) judging by the
Before proceeding to the application in 2-D of the above resemblance of the average models to the original model
numerical averaging technique, it is convenient to illustrate from which the data were drawn.
2
∇ E y − iωµ
0σ
( x,
z)
E y = 0,
(A.3)
z
− 2i δ
E y ( x,
z)
= E(
0)
e ,
(A.4)
2 2
δ = .
(A.5)
ωµ 0σ
z z
= 0 . 707δ
a ( Ti ), i = 1,
2
1 2 z z
n
( t)
( t)
σ j = ∑ w lσ
( j+
l)
.
(15)
l=
−n
1 1 ⎧ N n
⎫
( t+
1)
( t)
σ i = + sgn⎨
∑ Tij[ ∑ wlσ
( j+
l)
] + I i ⎬.
(16)
2 2 ⎩ j≠i
= 1 l=
−n
⎭
( t+
1)
σ i
(t )
σ j
1
wk = , k = −n,...,
n
2n
+ 1
N
n
( t+
1)
1 1
( t)
( t)
σ i = + sgn{ ∑ [( 1−
b ) Tijσ
j + β ∑ Tijwlσ
( j+
l)
] + I i}.
(17)
2 2 j≠i
= 1
l=
−n
σ a ( T2
) δ a ( T2
) −σ
a ( T1
) δ a ( T1
)
< σ ( z1,
z2
) >=
, (18)
NEXES
δ a ( T2
) −δ
a ( T1
)
< ( 1, 2 ) >
H x ( x,
0)
2
| σ a | = ωµ 0 | | ,
(A.1)
E y ( x,
0)
x , z)
δE
y ( x,
0)
δH
x ( x,
0)
δ | σ a | = −2
| σ a | Re[ − ]. (A.2)
E y ( x,
0)
H x ( x,
0)
( x , z)
x ( x,
z)
y ( x,
z)
2
∇ E y − iωµ
0σ
( x,
z)
E y = 0,
(A.3)
z
− 2i δ
E y ( x,
z)
= E(
0)
e ,
(A.4)
2 2
δ = .
(A.5)
ωµ 0σ
( x , z)
z z σ
zi = 0 . 707δ
a ( Ti
), i = 1,
2
1 2 z z
1 1 ⎧ N n
⎫
( t+
1)
( t)
σ i = + sgn⎨
∑ Tij[ ∑ wlσ
( j+
l)
] + I i ⎬.
(16)
2 2 ⎩ j≠i
= 1 l=
−n
⎭
( t+
1)
σ i
(t )
σ j
1
wk = , k = −n,...,
n
2n
+ 1
N
n
( t+
1)
1 1
( t)
( t)
σ i = + sgn{ ∑ [( 1−
b ) Tijσ
j + β ∑ Tijwlσ
( j+
l)
] + I i}.
(17)
2 2 j≠i
= 1
l=
−n
σ a ( T2
) δ a ( T2
) −σ
a ( T1
) δ a ( T1
)
< σ ( z1,
z2
) >=
,
δ a ( T2
) −δ
a ( T1
)
(18)
< ( 1, 2 ) >
n
( t)
( t)
σ j = ∑ w lσ
( j+
l)
.
(15)
l=
−n
( 1)
1 1 ⎧ N n
⎫
t+
( t)
i = + sgn⎨
∑ Tij[ ∑ wlσ
( j+
l)
] + I i ⎬.
(16)
2 2 ⎩ j≠i
= 1 l=
−n
⎭
− n,..., n
N
n
1 1
( t)
( t)
+ sgn{ ∑ [( 1−
b ) Tijσ
j + β ∑ Tijwlσ
( j+
l)
] + I i}.
(17)
2 2 j≠i
= 1
l=
−n
σ a ( T2
) δ a ( T2
) −σ
a ( T1
) δ a ( T1
)
σ ( z1,
z2
) >= , (18)
δ a ( T2
) −δ
a ( T1
)
z =
3
i ), i = 1,
2
ANNEXES
H x ( x,
0)
2
| σ a | = ωµ 0 | | ,
(A.1)
E y ( x,
0)
σ ( x,
z)
H x ( x,
0)
2
| σ a | = ωµ 0 | | ,
(A.1)
E y ( x,
0)
δE
y ( x,
0)
δH
x ( x,
0)
δ | σ a | = −2
| σ a | Re[ − ]. (A.2)
E y ( x,
0)
H x ( x,
0)
δE
y ( x,
0)
δH
x ( x,
0)
| σ a | = −2
| σ a | Re[ − ]. (A.2)
E y ( x,
z)
E y ( x,
0)
H x ( x,
0)
H x ( x,
z)
E y ( x,
z)
z z σ
zi = 0 . 707δ
a ( Ti
), i = 1,
2
1 2 z z z =
ANNEXES
H x ( x,
0)
2
| σ a | = ωµ 0 | | ,
(A.1)
E y ( x,
0)
σ ( x,
z)
δE
y ( x,
0)
δH
x ( x,
0)
δ | σ a | = −2
| σ a | Re[ − ]. (A.2)
E y ( x,
0)
H x ( x,
0)
E y ( x,
z)
H x ( x,
z)
E y ( x,
z)
2
∇ E y − iωµ
0σ
( x,
z)
E y = 0,
(A.3)
z
− 2i δ
E y ( x,
z)
= E(
0)
e ,
(A.4)
2 2
δ = .
(A.5)
ωµ 0σ
H x ( x,
z)
∇
× E = −iωµ
0H
n
∑