US 11,237,283 B2Granted

Visco-pseudo-elastic TTI FWI/RTM formulation and implementation

AssigneeExxonMobil Upstream Research Company

InventorsPraveen Nakshatrala, Anoop A. Mullur, Hong Zhao, Valeriy V. Brytik

Life of this patent

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Abstract

A method, including: obtaining, with a computer, an initial geophysical model; modeling, with a computer, a forward wavefield based on the initial geophysical model with wave equations including a second order z-derivative in a rotated coordinate system that accounts for a tilted transverse isotropic (TTI) medium; modeling, with a computer, an adjoint wavefield with adjoint wave equations including a second order z-derivative in a rotated coordinate system that accounts for a tilted transverse isotropic (TTI) medium, wherein the wave equations and the adjoint wave equations include relaxation terms accounting for anelasticity of earth in an update of a primary variable and an evolution relationship for the relaxation terms; and obtaining, with a computer, a gradient of a cost function based on a combination of a model of the forward wavefield and a model of the adjoint wavefield.

Claims11

Dependent claims

A method, comprising:

obtaining, with a computer, an initial geophysical model;
determining a stiffness ratio (al);
modeling, with a computer, a forward wavefield based on the initial geophysical model with wave equations including a second order z-derivative in a rotated coordinate system that accounts for a tilted transverse isotropic (TTI) medium:
modeling, with a computer, an adjoint wavefield with adjoint wave equations including a second order z-derivative in a rotated coordinate system that accounts for a tilted transverse isotropic (TTI) medium,
wherein the wave equations and the adjoint wave equations include relaxation terms accounting for anelasticity of earth in an update of a primary variable and an evolution relationship for the relaxation terms, and further wherein the evolution relationship is
∂
R
l
∂
t
=
a
l
⁢
σ
.
-
ω
l
⁢
R
l
,
in which Rl is a relaxation term, {dot over (σ)} is a first derivative of stress, al is the stiffness ratio, and ωl is a characteristic relaxation frequency, and wherein:
the wave equations are
∂
σ
->
∂
t
=
σ
->
.
+
S
σ
→
-
∑
l
=
1
NMECH
⁢
ω
l
⁢
m
l
→
,
⁢
∂
σ
->
.
∂
t
=
C
⁢
⁢
σ
->
,
and
∂
m
l
→
∂
t
=
-
ω
l
⁢
m
l
→
+
a
l
⁢
σ
->
.
,
and
the adjoint wave equations are
-
∂
σ
←
∂
t
=
σ
←
.
+
∑
l
=
1
NMECH
⁢
a
l
⁢
m
l
←
,
⁢
-
∂
σ
←
.
∂
t
=
C
⁢
⁢
σ
←
+
S
σ
←
,
and
∂
m
l
←
∂
t
=
-
ω
l
⁢
m
l
←
+
a
l
⁢
σ
←
.
,
with t being time, {dot over (σ)} being a first derivative of stress, al being the stiffness ratio, and ωl being a characteristic relaxation frequency, σ being stress, ml being a memory response, C being material properties, and Sσ being application of a source to stress;
obtaining, with a computer, a gradient of a cost function based on a combination of a model of the forward wavefield and a model of the adjoint wavefield;
and
updating the initial geophysical model, with the computer, with an adjustment determined from the gradient of the cost function to obtain an updated geophysical model.

158 more elements

A method, comprising:

obtaining, with a computer, an initial geophysical model:
determining a stiffness ratio (al);
modeling, with a computer, a forward wavefield based on the initial geophysical model with wave equations including a second order z-derivative in a rotated coordinate system that accounts for a tilted transverse isotropic (TTI) medium;
modeling. with a computer, an adjoint wavefield with adjoint wave equations including a second order z-derivative in a rotated coordinate system that accounts for a tilted transverse isotropic (TTI) medium;
wherein the wave equations and the adjoint wave equations include relaxation terms accounting for anelasticity of earth in an update of a primary variable and an evolution relationship for the relaxation terms, and further wherein the evolution relationship is
∂
R
l
∂
t
=
a
l
⁢
σ
.
-
ω
l
⁢
R
l
,
in which Rl is a relaxation term, {dot over (σ)} is a first derivative of stress, al is the stiffness ratio, and ωl is a characteristic relaxation frequency, and wherein:
the wave equations are
∂
σ
->
∂
t
=
σ
->
.
+
S
σ
→
-
∑
l
=
1
NMECH
⁢
ω
l
⁢
m
l
→
,
⁢
∂
σ
->
.
∂
t
=
C
⁢
⁢
σ
->
,
and
∂
m
l
→
∂
t
=
-
ω
l
⁢
m
l
→
+
a
l
⁢
σ
->
.
,
and
the adjoint wave equations are
-
∂
σ
←
∂
t
=
σ
←
.
+
∑
l
=
1
NMECH
⁢
a
l
⁢
m
l
←
,
⁢
-
∂
σ
←
.
∂
t
=
C
⁢
⁢
σ
←
+
S
σ
←
,
and
∂
m
l
←
∂
t
=
-
ω
l
⁢
m
l
←
+
a
l
⁢
σ
←
.
,
with t being time, {dot over (σ)} being a first derivative of stress, al being the stiffness ratio, and ωl being a characteristic relaxation frequency, a being stress, m, being a memory response, C being material properties, and Sσ being application of a source to stress;
obtaining, with a computer, an imaging condition based on a combination of a model of the forward wavefield and a model of the adjoint wavefield;
and
generating a subsurface image from the imaging condition.

158 more elements

Dependencies are inferred from the claim text.

Facts

Application: 16/000,960Filed 6 Jun 2018 · us provisional application
Priority: 13 Jul 2017
Published: US 2019/0018155 A117 Jan 2019
Granted: 1 Feb 2022
Term ends: Not estimatedThe source records don’t support an estimate.

Classification8 codes

G01V 1/282
Processing seismic data, e.g. for interpretation or for event detection: application of seismic models, synthetic seismograms
G01V 1/303
Analysis: for determining velocity profiles or travel times
G01V 1/306
Analysis: for determining physical properties of the subsurface, e.g. impedance, porosity or attenuation profiles
G01V 2210/614
Analysis by combining or comparing a seismic data set with other data: synthetically generated data