4. Sensitivity
A common approach to derive the sensitivity is to Ž
use the reciprocity of the Green’s functions Weidelt, 1975; Pellerin et al., 1993; Mackie and Madden,
. 1993 . If a 3-D conductivity structure is perturbed
over a 3-D region, Õ, by an amount ds , then the resulting magnetic field H at a receiving point r is
t
given by
H r s H r q G
H
r , r
X
P E r
X
d s r
X
dÕ
X
,
Ž . Ž .
Ž .
Ž . Ž .
H
t t
Õ
9
Ž .
where H is the magnetic field for the unperturbed model and E is the electric field for the perturbed
t H
Ž
X
.
model. The tensor Green’s function G r, r
is the
magnetic field at r for the unperturbed model, due to a unit electric dipole source at r
X
. Under the Born approximation, the perturbation of the magnetic field,
d H, is approximated by
d H f G
H
r , r
X
P E r
X
d s r
X
dÕ
X
, 10
Ž .
Ž . Ž .
Ž .
H
Õ
where E is the electric field for the unperturbed model. Considering the components of the magnetic
Ž .
field, say, the x-component, Eq. 10 reduces to d H s
G
H
r , r
X
E r
X
q G
H
r , r
X
E r
X
Ž .
Ž . Ž
. Ž .
Ž
H
x x x
x x y
y Õ
qG
H
r , r
X
E r
X
d
s r
X
dÕ
X
, 11
Ž .
Ž . Ž .
Ž .
.
x z z
H
Ž
X
. where, e.g., G
r, r is the x-component of the
x y
magnetic field at r, due to the y-directed unit elec- tric dipole source at r
X
. Using the reciprocity rela- tion,
G
H
r , r
X
s G
H
r , r
X
, 12
Ž .
Ž .
Ž .
i j ji
H
Ž
X
. we can interpret G
r, r as representing the y-
x y
component of the electric field at r
X
due to the x-directed magnetic dipole source placed at the re-
ceiving point r. Replacing the magnetic Green’s
Ž .
function with the reciprocal electric field, Eq. 11 becomes
y1
X X
X X
M x
d H s E
r P E r ds r dÕ ,
13
Ž . Ž .
Ž . Ž
.
H
x
i vm
Õ
where E
M x
denotes the electric field due to the x-directed magnetic dipole source with unit moment
placed at the receiving point. If the region Õ corre- sponds to the mth block and its conductivity is s ,
m
we obtain EH
y1
x X
X X
M x
s E
r P E r dÕ .
14
Ž . Ž .
Ž .
H
Es i vm
Õ m
The sensitivities of the other components of the EM fields are derived in the same way. For example, the
sensitivity of the y-component of the electric field is given by
EE
y X
X X
J y
s E
r P E r dÕ ,
15
Ž . Ž .
Ž .
H
Es
Õ m
where E
J y
denotes the electric field due to the y-directed unit electric dipole source at the receiving
Ž .
Ž .
point. Eqs. 14 and 15 show that the sensitivities can be obtained by doing forward modelings with
the fictitious and actual sources and by integrating the dot product of the electric fields. Notice that the
fictitious sources are placed at each receiving point instead of in each model block, which results in a
significant saving in computation time. This proce- dure is referred to as the adjoint-equation method by
Ž .
McGillivray et al. 1994 . Another approach to arrive at the above procedure
Ž . is to begin with the matrix Eq. 8 . Differentiating
Ž . Eq. 8 with respect to the block conductivity s
m
yields
E f EK
K s y
f . 16
Ž .
Es Es
m m
Ž . By analogy with Eq. 8 , the derivative of the field
with respect to s can be interpreted as the field due
m
to a collection of sources described by the right-hand Ž
. side of Eq. 16 . This new source term is the product
of the derivative of K and the column vector of the field for the model. The derivative of the element of
K is zero unless it contains s . This means that the
m
sources are present only in the block of interest. In the finite-difference approximation, the elements of
K that contain the conductivity are the diagonal elements, and they have the form i vms DÕ , where
i i
s is the conductivity of an elemental volume DÕ
i i
Ž .
e.g., Newman and Alumbaugh, 1995 . Thus, the sources in the block have the strength of DÕ multi-
i
plied by the electric field E . Using the reciprocity, it
i
turns out that the derivative of the field is equal to a weighted sum of the fields over the block due to a
unit source placed at the receiving point, where the weights are given by DÕ E . This weighted summa-
i i
Ž .
tion amounts to the integration in Eq. 15 . In the examples below, EM data are inverted for
the logarithms of resistivities. The sensitivities are computed through the relation
E f E f
s ys ,
17
Ž .
m
Eln r Es
m m
Ž .
where r is resistivity i.e., r s 1rs .
m m
m
5. Verification of forward solution
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