Device and method for calculating 3D reverse time migration in tilted orthorhombic media
A computing device, computing medium and method for generating an image of a tilted orthorhombic medium. The method includes receiving seismic data related to the tilted orthorhombic medium; calculating a wave propagation with a processing device by applying a second-order equation for reverse time migration to the seismic data to generate a tilted orthorhombic wave propagation; and generating the image of the tilted orthorhombic medium based on the tilted orthorhombic wave propagation.
1. A method for generating an image of a tilted orthorhombic medium, the method comprising:
receiving seismic data related to the tilted orthorhombic medium;
propagating wave-fields with a processing device by applying a second-order equation for reverse time migration to the seismic data to obtain tilted orthorhombic wave propagation; and
generating the image of the tilted orthorhombic medium based on the tilted orthorhombic wave propagation.
2. The method of claim 1 , wherein the wave propagations are related to pressure wave-fields.
3. The method of claim 1 , wherein the second-order equation includes a tilted first-order derivative D, an anisotropy parameter matrix N and partial time derivatives.
4. The method of claim 3 , wherein the first-order derivative D is given by:
D =diag( R 1 T ∇,R 2 T ∇,R 3 T ∇),
where R i are column vectors related to a transformation matrix R and ∇ is a derivative operator.
5. The method of claim 4 , wherein the transformation matrix R transforms the stress tensor from a local system to a global system based on three different angles.
6. The method of claim 5 , wherein the transformation matrix R is given by:
[
cos
ϕ
-
sin
ϕ
0
sin
ϕ
cos
ϕ
0
0
0
1
]
[
cos
θ
0
sin
θ
0
1
0
-
sin
θ
0
cos
θ
]
[
cos
β
-
sin
β
0
sin
β
cos
β
0
0
0
1
]
where θ and φ define a vertical axis in the tilted orthorhombic medium and β describes a rotation of the stress tensor in the local plane.
7. The method of claim 3 , wherein the anisotropy parameter matrix N is given by:
N
=
[
1
+
2
ɛ
2
(
1
+
2
ɛ
2
)
1
+
2
δ
3
1
+
2
δ
2
(
1
+
2
ɛ
2
)
1
+
2
δ
3
1
+
2
ɛ
1
1
+
2
δ
1
1
+
2
δ
2
1
+
2
δ
1
1
]
where ε 1 , ε 2 , δ 2 and δ 3 are dimensionless parameters.
8. The method of claim 3 , wherein the second-order equation is given by:
1
V
p
0
2
∂
2
σ
∂
t
2
=
ND
T
D
σ
,
where V p0 2 is a p-wave velocity, σ is the stress tensor, and D T is the adjoint of D.
9. A computing device for generating an image of tilted orthorhombic medium, the computing device comprising:
an interface configured to receive seismic data related to the tilted orthorhombic medium; and
a processor connected to the interface and configured to,
calculate wave propagation by applying a second-order equation for reverse time migration to the seismic data, and
generate the image of the tilted orthorhombic medium based on the wave propagation.
10. The computing device of claim 9 , wherein the wave propagation is related to a pressure wave-field.
11. The computing device of claim 9 , wherein the second-order equation includes a tilted first-order derivative D, an anisotropy parameter matrix N and partial time derivatives.
12. The computing device of claim 11 , wherein the first-order derivative D is given by:
D =diag( R 1 T ∇,R 2 T ∇,R 3 T ∇),
where R i are column vectors related to a transformation matrix R and ∇ is a derivative operator.
13. The computing device of claim 12 , wherein the transformation matrix R transforms the stress tensor from a local system to a global system based on three different angles.
14. The computing device of claim 13 , wherein the transformation matrix R is given by:
[
cos
ϕ
-
sin
ϕ
0
sin
ϕ
cos
ϕ
0
0
0
1
]
[
cos
θ
0
sin
θ
0
1
0
-
sin
θ
0
cos
θ
]
[
cos
β
-
sin
β
0
sin
β
cos
β
0
0
0
1
]
where θ and φ define a vertical axis in the tilted orthorhombic medium and β describes a rotation of the stress tensor in the local plane.
15. The computing device of claim 11 , wherein the anisotropy parameter matrix N is given by:
N
=
[
1
+
2
ɛ
2
(
1
+
2
ɛ
2
)
1
+
2
δ
3
1
+
2
δ
2
(
1
+
2
ɛ
2
)
1
+
2
δ
3
1
+
2
ɛ
1
1
+
2
δ
1
1
+
2
δ
2
1
+
2
δ
1
1
]
where ε 1 , ε 2 , δ 2 and δ 3 are dimensionless parameters.
16. The computing device of claim 11 , wherein the anisotropy parameter matrix N is given by:
1
V
p
0
2
∂
2
σ
∂
t
2
=
ND
T
D
σ
,
where V p0 2 is a p-wave velocity, σis the stress tensor, and D T is the adjoint of D.
17. A non-transitory computer readable medium including computer executable instructions, wherein the instructions, when executed by a processor, implement a method for generating an image of a tilted orthorhombic medium, the instructions comprising:
receiving seismic data related to the tilted orthorhombic medium;
calculating a wave propagation with a processing device by applying a second-order equation for reverse time migration to the seismic data; and
generating the image of the tilted orthorhombic medium based on the wave propagation.
18. The medium of claim 17 , wherein the second-order equation includes a tilted first-order derivative D, an anisotropy parameter matrix N and partial time derivatives.
19. The medium of claim 18 , wherein the first-order derivative D is given by:
D =diag( R 1 T ∇,R 2 T ∇,R 3 T ∇),
where R i are column vectors related to a transformation matrix R and ∇ is a derivative operator.
20. The medium of claim 19 , wherein the transformation matrix R transforms the stress tensor from a local system to a global system based on three different angles and it is given by:
[
cos
ϕ
-
sin
ϕ
0
sin
ϕ
cos
ϕ
0
0
0
1
]
[
cos
θ
0
sin
θ
0
1
0
-
sin
θ
0
cos
θ
]
[
cos
β
-
sin
β
0
sin
β
cos
β
0
0
0
1
]
where θ and φ define a vertical axis in the tilted orthorhombic medium and β describes a rotation of the stress tensor in the local plane,
wherein the anisotropy parameter matrix N is given by:
N
=
[
1
+
2
ɛ
2
(
1
+
2
ɛ
2
)
1
+
2
δ
3
1
+
2
δ
2
(
1
+
2
ɛ
2
)
1
+
2
δ
3
1
+
2
ɛ
1
1
+
2
δ
1
1
+
2
δ
2
1
+
2
δ
1
1
]
where ε 1 , ε 2 , δ 1 , δ 2 and δ 3 are dimensionless parameters, and
wherein the second-order equation is given by:
1
V
p
0
2
∂
2
σ
∂
t
2
=
ND
T
D
σ
,
where V p0 2 is a p-wave velocity, σ is the stress tensor, and D T is the adjoint of D.