Apparatus and method for imaging a subsurface using frequency-domain elastic reverse-time migration
Provided are an apparatus and method for imaging a subsurface using frequency-domain reverse-time migration in elastic medium. The subsurface imaging method represents a frequency-domain imaging condition as convolution of measured data and a partial derivative wavefield. That is, the subsurface imaging method applies a back-propagation algorithm to represent an imaging condition as convolution of a virtual source and a back-propagated wavefield. Then, the subsurface imaging method divides a virtual source vector and the back-propagated wavefield represented by a displacement vector into P- and S-wave potentials through Helmholtz decomposition, thereby providing a new imaging condition.
1. An apparatus of imaging a subsurface, comprising:
a plurality of receivers configured to receive a seismic data from a region to be observed;
a signal processor configured to estimate sources from measured data measured by the receivers, and to process the seismic data so as to generate a migration image signal for imaging a subsurface of the region to be observed, wherein:
the signal processor generates migration images for a P-wave and a S-wave of a seismic signal, by convolving a virtual source vector and a wavefield displacement vector obtained by back-propagating the seismic signal, with a divergence operator and a curl operator, and
imaging conditions for generating the migration images for the P-wave and the S-wave are represented as equations E-1 and E-2, respectively:
(φ k ) PP =∫Re{∇·[( v ke ) T ]∇·[S e −1 {tilde over (d)} e *]}dω and (E-1)
(φ k ) SS =∫Re{∇×[( v ke ) T ]∇×[S e −1 {tilde over (d)} e *]}dω, (E-2)
where (φ k ) PP is the imaging condition for P wave, (φ k ) SS is the imaging condition for S wave, (v ke ) p , (v ke ) s is the virtual source vector for the P and S wave velocity for the kth model parameter respectively, {tilde over (d)} e is the observed data, Re is the real part of the complex number, ∇· is divergence operator, ∇× is curl operator, T is transpose, * is complex conjugate, ω is angular frequency, and S e −1 is the inverse matrix of the complex impedance matrix in an elastic medium.
2. The apparatus of claim 1 , wherein the virtual source vector is defined as equations (E-3) and (E-4) for the P-wave and the S-wave, respectively:
(
v
ke
)
p
=
-
∂
S
e
∂
v
p
u
~
e
,
and
(
E
-
3
)
(
v
ke
)
s
=
-
∂
S
e
∂
v
s
u
~
e
,
where
∂
S
e
∂
v
p
=
2
ρ
v
p
∂
S
e
∂
λ
,
∂
S
e
∂
λ
,
∂
S
e
∂
v
s
-
4
ρ
v
s
∂
S
e
∂
λ
+
2
ρ
v
s
∂
S
e
∂
μ
,
(
E
-
4
)
λ is the damping factor for stability, v p , v s are the P and S wave velocity respectively, ũ e is the modeled data, and S e is the complex impedance matrix in an elastic medium, μ is the Lame's constant, and ρ is density.
3. The apparatus of claim 2 , wherein the signal processor generates final migration images using equations (E-5) and (E-6):
(
ϕ
k
)
pp
=
NRM
[
∫
NRM
[
Re
{
∇
·
[
(
v
ke
)
T
]
·
∇
·
[
S
e
-
1
d
~
e
*
]
}
Re
{
diag
[
(
∇
·
[
(
v
ke
)
T
]
)
T
(
∇
·
[
(
v
ke
)
T
]
)
]
+
λ
I
}
]
ⅆ
ω
]
,
and
(
ϕ
k
)
ss
=
NRM
[
∫
NRM
[
Re
{
∇
×
[
(
v
ke
)
T
]
∇
×
[
S
e
-
1
d
~
e
*
]
}
Re
{
diag
[
(
∇
×
[
(
v
ke
)
T
]
)
T
(
∇
×
[
(
v
ke
)
T
]
)
]
+
λ
I
}
]
ⅆ
ω
]
.
where NRM is the normalization operator, diag is the diagonal component of the matrix, and I is the identity matrix.
4. A method of imaging a subsurface, comprising:
receiving a seismic data from a region to be observed;
estimating sources from data measured by the receivers, and processing the seismic data so as to generate a migration image for imaging a subsurface of the region to be observed, wherein:
the processing of the seismic data comprises generating migration images for a P-wave and a S-wave of a seismic signal, by convolving a virtual source vector and a wavefield displacement vector obtained by back-propagating the seismic signal, with a divergence operator and a curl operator, and
imaging conditions for generating the migration images for the P-wave and the S-wave are represented as equations E-1 and E-2, respectively:
(φ k ) PP =∫Re{∇·[( v ke ) T ]∇·[S e −1 {tilde over (d)} e *]}dω and (E-1)
(φ k ) SS =∫Re{∇×[( v ke ) T ]∇×[S e −1 {tilde over (d)} e *]}dω, (E-2)
where (φ k ) PP is the imaging condition for P wave, (φ k ) SS is the imaging condition for S wave, (v ke ) p , (v ke ) s is the virtual source vector for the P and S wave velocity for the kth model parameter respectively, {tilde over (d)} e is the observed data, Re is the real part of the complex number, ∇· is divergence operator, ∇× is curl operator, T is transpose, * is complex conjugate, ω is angular frequency, and S e −1 is the inverse matrix of the complex impedance matrix in an elastic medium.