# US 10,371,841 B2 · Device and method for calculating 3D reverse time migration in tilted orthorhombic media

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- Number: US 10,371,841 B2
- Document ID: US10371841B2
- Status: Granted
- Kind: B2
- Application number: 13/442,158
- Filed: 2012-04-09
- Published: 2012-10-18
- Granted: 2019-08-06
- Earliest claimed: 2011-04-12
- Assignees: Cgg Services SAS
- Applicants: Yu Zhang
- Inventors: Yu Zhang
- Web page: https://patentgrep.com/patent/US10371841B2

## CPC

- G01V 1/30: Analysis (G01V1/50 takes precedence)
- G01V 2210/51: Migration
- G01V 2210/679: Reverse-time modeling or coalescence modelling, i.e. starting from receivers
- G01V: GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS (means for indicating the location of accidentally buried, e.g. snow-buried, persons A63B29/02)

## Abstract

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.

## Claims

### Claim 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.

### Claim 2 (depends on 1)

The method of claim 1, wherein the wave propagations are related to pressure wave-fields.

### Claim 3 (depends on 1)

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.

### Claim 4 (depends on 3)

The method of claim 3, wherein the first-order derivative D is given by:
D=diag(R1T∇,R2T∇,R3T∇),
where Ri are column vectors related to a transformation matrix R and ∇ is a derivative operator.

### Claim 5 (depends on 4)

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.

### Claim 6 (depends on 5)

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.

### Claim 7 (depends on 3)

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 ε1, ε2, δ2 and δ3 are dimensionless parameters.

### Claim 8 (depends on 3)

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 Vp02 is a p-wave velocity, σ is the stress tensor, and DT is the adjoint of D.

### Claim 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.

### Claim 10 (depends on 9)

The computing device of claim 9, wherein the wave propagation is related to a pressure wave-field.

### Claim 11 (depends on 9)

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.

### Claim 12 (depends on 11)

The computing device of claim 11, wherein the first-order derivative D is given by:
D=diag(R1T∇,R2T∇,R3T∇),
where Ri are column vectors related to a transformation matrix R and ∇ is a derivative operator.

### Claim 13 (depends on 12)

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.

### Claim 14 (depends on 13)

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.

### Claim 15 (depends on 11)

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 ε1, ε2, δ2 and δ3 are dimensionless parameters.

### Claim 16 (depends on 11)

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 Vp02 is a p-wave velocity, σis the stress tensor, and DT is the adjoint of D.

### Claim 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.

### Claim 18 (depends on 17)

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.

### Claim 19 (depends on 18)

The medium of claim 18, wherein the first-order derivative D is given by:
D=diag(R1T∇,R2T∇,R3T∇),
where Ri are column vectors related to a transformation matrix R and ∇ is a derivative operator.

### Claim 20 (depends on 19)

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 ε1, ε2, δ1, δ2 and δ3 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 Vp02 is a p-wave velocity, σ is the stress tensor, and DT is the adjoint of D.