US 2026/0276850 A1 · MULTI-PHASE FREE-FIELD INVERSION METHOD FOR NEAR-FAULT SITE BASED ON SPHERICAL/CYLINDRICAL DIFFUSION PROPERTIES

Abstract

A multi-phase free-field inversion method for near-fault sites based on spherical/cylindrical diffusion properties. A dynamic stiffness matrix method is used to construct a multi-phase free-field inversion framework for near-fault sites, and then transforms an inversion problem into a multi-objective optimization problem for solving an “epicentral distance” and a “hypocenter depth”. The spatial-temporal difference between free-field displacement waveforms is quantified by a dynamic WPD, and objective functions are constructed according to the criterion of minimizing the differences between three-dimensional displacement waveforms at two adjacent points. At the same time, constraint conditions are set in combination with propagation and attenuation characteristics of seismic waves, thereby establishing a multi-objective optimization model for near-fault inversion. Subsequently, a NSGA-II is introduced to find a Pareto optimal solution of optimization variables, and the optimal solution is substituted into the inversion framework to complete the multi-phase free-field inversion for near-fault sites.

A multi-phase free-field inversion method for near-fault sites based on spherical/cylindrical diffusion properties, comprising the following steps:

step 1:
establishing a multi-phase free-field inversion framework for near-fault layered-medium sites in a cylindrical coordinate system based on spherical diffusion properties of body waves and cylindrical diffusion properties of surface waves;
step 2:
taking unknown “epicentral distance rHO” and “hypocenter depth H” in the multi-phase free-field inversion framework for near-fault layered media as optimization variables to transform an inversion problem into a multi-objective optimization problem;
and constructing objective functions according to the criterion of minimizing the differences between three-dimensional displacement waveforms at two adjacent positions, and setting constraint conditions according to propagation and attenuation characteristics of seismic waves, thereby establishing a multi-objective optimization model for near-fault inversion;
step 3:
using a Non-dominated Sorting Genetic Algorithm II (NSGA-II) to find a Pareto optimal solution of the optimization variables, and substituting the optimal solution into the inversion framework to complete multi-phase free-field inversion for near-fault sites;
the process of step 1 is as follows:
step 1.1:
assuming a hypocenter is located in an underlying bedrock half-space, and establishing a cylindrical coordinate system r-θ-Z with the position of an epicenter as an origin, the direction of a line connecting the epicenter and a surface station as a radial direction, and the direction of the epicenter pointing to the hypocenter as a vertical direction;
step 1.2:
using a dynamic stiffness matrix method to calculate Green's functions of spherical diffusion P waves, SV waves and SH waves at an arbitrary position (r, θ, z) in the space, respectively:
{
U
¯
r
P
,
j
(
r
,
θ
,
z
)
=
∫
0
∞
u
¯
r
P
,
j
(
k
,
r
,
z
)
⁢
kdk
,
U
¯
Z
P
,
j
⁢
(
r
,
θ
,
z
)
=
∫
0
∞
u
¯
Z
P
,
j
⁢
(
k
,
r
,
z
)
⁢
kdk
U
¯
r
SV
,
j
(
r
,
θ
,
z
)
=
∫
0
∞
u
¯
r
SV
,
j
(
k
,
r
,
z
)
⁢
kdk
,
U
¯
Z
SV
,
j
⁢
(
r
,
θ
,
z
)
=
∫
0
∞
u
¯
Z
SV
,
j
⁢
(
k
,
r
,
z
)
⁢
kdk
U
¯
θ
SH
,
j
(
r
,
θ
,
z
)
=
∫
0
∞
u
¯
θ
SH
,
j
(
k
,
r
,
z
)
⁢
kdk
(
1
)
where k is a horizontal radial wavenumber;
u
¯
r
P
,
j
⁢
and
⁢
u
¯
Z
P
,
j
are wavenumber-domain displacements of particles in the jth layer of layered media induced by unit-amplitude P waves along radial and vertical directions, respectively;
u
¯
r
SV
,
j
⁢
and
⁢
u
¯
Z
SV
,
j
are wavenumber-domain displacements of particles in the jth layer of layered media induced by unit-amplitude SV waves along radial and vertical directions, respectively;
u
¯
θ
SH
,
j
is a wavenumber-domain displacement of particles in the jth layer of layered media induced by unit-amplitude SH waves along a circumferential direction;
and
U
¯
r
P
,
j
,
U
¯
Z
P
,
j
,
U
¯
r
SV
,
j
,
U
¯
Z
SV
,
j
⁢
and
⁢
U
¯
θ
SH
,
j
are spatial-domain displacement Green's functions obtained by Hankel inverse transformation of the above wavenumber-domain displacements, respectively;
step 1.3:
taking j=1 in formula (1) and calculating Green's function solutions of the three types of spherical diffusion body waves at a surface station position (rHO, 0, 0):
{
U
¯
r
P
,
1
(
r
=
r
HO
,
θ
=
0
,
z
=
0
)
=
∫
0
∞
u
¯
r
P
,
1
(
k
,
r
HO
,
0
)
⁢
kdk
U
¯
Z
P
,
1
(
r
=
r
HO
,
θ
=
0
,
z
=
0
)
=
∫
0
∞
u
¯
Z
P
,
1
(
k
,
r
HO
,
0
)
⁢
kdk
,
P
⁢
waves
(
2
)
{
U
¯
r
SV
,
1
(
r
=
r
HO
,
θ
=
0
,
z
=
0
)
=
∫
0
∞
u
¯
r
SV
,
1
(
k
,
r
HO
,
0
)
⁢
kdk
U
¯
Z
SV
,
1
(
r
=
r
HO
,
θ
=
0
,
z
=
0
)
=
∫
0
∞
u
¯
Z
SV
,
1
(
k
,
r
HO
,
0
)
⁢
kdk
,
SV
⁢
waves
(
3
)
U
¯
θ
SH
,
1
(
r
=
r
HO
,
θ
=
0
,
z
=
0
)
=
∫
0
∞
u
¯
θ
SH
,
1
(
k
,
r
HO
,
0
)
⁢
kdk
,
SH
⁢
waves
(
4
)
where the Green's function solutions depend on the unknown quantities “epicentral distance rHO” and “hypocenter depth H”;
step 1.4:
using a normalized inner product (NIP) method to extract body-wave displacement components
U
r
Body
,
1
,
U
θ
Body
,
1
⁢
and
⁢
U
Z
Body
,
1
collected at the surface station, and inverting incident amplitudes
A
P
Body
,
A
SV
Body
⁢
and
⁢
A
SH
Body
of the P waves, the SV waves and the SH waves based on formula (5):
(
A
P
Body
A
SV
Body
A
SH
Body
)
=
[
U
¯
r
P
,
1
U
¯
r
SV
,
1
0
U
¯
Z
P
,
1
U
¯
Z
SV
,
1
0
0
0
U
θ
SH
,
1
]
-
1
⁢
(
U
r
Body
,
1
U
Z
Body
,
1
U
θ
Body
,
1
)
(
5
)
step 1.5:
multiplying the incident wave amplitudes obtained from formula (5) by the Green's functions shown in formula (1), respectively, to construct displacement free fields
U
r
Body
,
j
,
U
Z
B
⁢
ody
,
j
⁢
and
⁢
U
θ
B
⁢
ody
,
j
induced by the spherical diffusion body waves at an arbitrary position in a spatial domain:
(
U
r
Body
,
j
U
Z
Body
,
j
U
θ
Body
,
j
)
=
[
U
¯
r
P
,
j
U
¯
r
SV
,
j
0
U
¯
Z
P
,
j
U
¯
Z
SV
,
j
0
0
0
U
θ
SH
,
j
]
[
U
¯
r
P
,
1
U
¯
r
SV
,
1
0
U
¯
Z
P
,
1
U
¯
Z
SV
,
1
0
0
0
U
θ
SH
,
1
]
-
1
⁢
(
U
r
Body
,
1
U
Z
Body
,
1
U
θ
Body
,
1
)
(
6
)
step 1.6:
performing time and space partial differential calculations on the body-wave displacement free fields, respectively, to construct corresponding velocity free fields and stress free fields and complete body-wave free-field inversion;
step 1.7:
extending and applying the dynamic stiffness matrix method for the body waves in the cylindrical coordinate system to surface waves to deduce spatial-domain free-field displacements
U
r
Ray
,
j
⁢
and
⁢
U
Z
Ray
,
j
induced by cylindrical diffused Rayleigh waves:
{
U
r
Ray
,
j
⁢
(
r
,
θ
,
z
)
=
U
X
Ray
,
j
⁢
(
z
)
⁢
∫
0
∞
[
u
r
Ray
,
j
⁢
(
k
,
r
)
]
⁢
kdk
=
U
X
Ray
,
j
⁢
(
z
)
⁢
∫
0
∞
{
∑
α
∂
[
J
α
(
k
⁢
r
)
]
∂
r
}
⁢
d
⁢
k
U
Z
Ray
,
j
⁢
(
r
,
θ
,
z
)
=
U
Z
Ray
,
j
⁢
(
z
)
⁢
∫
0
∞
[
u
Z
Ray
,
j
(
k
,
r
)
]
⁢
k
⁢
dk
=
U
Z
Ray
,
j
⁢
(
z
)
⁢
∫
0
∞
[
-
i
⁢
∑
α
J
α
⁢
(
k
⁢
r
)
]
⁢
k
⁢
d
⁢
k
(
7
)
U
X
Ray
,
j
(
z
)
=
l
x
Ray
,
j
[
A
P
Ray
,
j
⁢
exp
⁡
(
ik
⁢
s
~
Ray
,
j
⁢
z
)
+
B
P
Ray
,
j
⁢
exp
⁡
(
-
ik
⁢
s
~
Ray
,
j
⁢
z
)
]
-
m
x
Ray
,
j
⁢
t
˜
Ray
,
j
[
A
SV
Ray
,
j
⁢
exp
⁡
(
ik
⁢
t
~
Ray
,
j
⁢
z
)
-
B
SV
Ray
,
j
⁢
exp
⁡
(
-
ik
⁢
t
˜
Ray
,
j
⁢
z
)
]
(
8
)
U
Z
Ray
,
j
(
z
)
=
-
l
x
Ray
,
j
⁢
s
~
Ray
,
j
[
A
P
Ray
,
j
⁢
exp
⁡
(
ik
⁢
s
~
Ray
,
j
⁢
z
)
-
B
P
Ray
,
j
⁢
exp
⁡
(
-
ik
⁢
s
~
Ray
,
j
⁢
z
)
]
-
m
x
Ray
,
j
[
A
SV
Ray
,
j
⁢
exp
⁡
(
ik
⁢
t
~
Ray
,
j
⁢
z
)
+
B
SV
Ray
,
j
⁢
exp
⁡
(
-
ik
⁢
t
˜
Ray
,
j
⁢
z
)
]
(
9
)
where
⁢
u
r
Ray
,
j
⁢
and
⁢
u
z
Ray
,
j
are wavenumber-domain displacements of particles in the jth layer of layered media induced by the Rayleigh waves along radial and vertical directions of the cylindrical coordinate system, respectively;
U
X
Ray
,
j
⁢
and
⁢
U
Z
Ray
,
j
are spatial-domain displacements induced by the Rayleigh waves in a plane coordinate system, respectively;
i is an imaginary number symbol, satisfying i2=−1;
Jα(kr) is an α-order Bessel function of the first kind;
A
P
Ray
,
j
⁢
and
⁢
B
P
Ray
,
j
are amplitudes of upgoing and downgoing inhomogeneous P waves in the jth layer of layered media, respectively;
A
SV
Ray
,
j
⁢
and
⁢
B
SV
Ray
,
j
are amplitudes of upgoing and downgoing inhomogeneous SV waves in the jth layer of layered media, respectively;
k{tilde over (s)}Ray,j and k{tilde over (t)}Ray,j are vertical wavenumbers of the inhomogeneous P waves and the inhomogeneous SV waves, respectively;
l
x
Ray
,
j
is the ratio of the P-wave velocity to the Rayleigh-wave velocity in the jth layer of layered media;
and
m
x
Ray
,
j
is the ratio of the S-wave velocity to the Rayleigh-wave velocity in the jth layer of layered media;
step 1.8:
similarly, deducing a free-field displacement
U
θ
L
⁢
ove
,
j
induced by cylindrical diffused Love waves:
U
θ
Love
,
j
(
r
,
θ
,
z
)
=
U
Y
Love
,
j
(
z
)
⁢
∫
0
∞
[
u
θ
Love
,
j
(
k
,
r
)
]
⁢
kdk
=
U
Y
Love
,
j
(
z
)
⁢
∫
0
∞
{
∑
α
∂
[
J
α
(
k
⁢
r
)
]
∂
r
}
⁢
dk
(
10
)
U
Y
Love
,
j
(
z
)
=
A
S
⁢
H
Love
,
j
⁢
exp
⁡
(
ik
⁢
t
~
Love
,
j
⁢
z
)
+
B
S
⁢
H
Love
,
j
⁢
exp
⁡
(
-
ik
⁢
t
~
Love
,
j
⁢
z
)
(
11
)
is a wavenumber-domain displacement of particles in the jth layer of layered media induced by the Love waves along the circumferential direction of the cylindrical coordinate system;
U
Y
L
⁢
ove
,
j
is a spatial-domain displacement induced by Love waves in the plane coordinate system;
A
S
⁢
H
L
⁢
ove
,
j
⁢
and
⁢
B
S
⁢
H
L
⁢
ove
,
j
are amplitudes of upgoing and downgoing inhomogeneous SH waves in the jth layer of layered media, respectively;
and k{tilde over (t)}Love,j is a vertical wavenumber of the inhomogeneous SH waves;
step 1.9:
conducting mode decoupling to dispersion surface waves, and deducing attenuation factors of single-mode surface-wave free-field displacements along a vertical direction by formula (12) based on mode shapes:
R
r
,
m
R
⁢
a
⁢
y
(
z
)
=
U
X
,
m
Ray
,
j
(
z
)
U
X
,
m
Ray
,
1
(
z
=
0
)
,
R
Z
,
m
R
⁢
a
⁢
y
(
z
)
=
U
Z
,
m
Ray
,
j
(
z
)
U
Z
,
m
Ray
,
1
(
z
=
0
)
,
(
12
)
R
θ
,
m
L
⁢
o
⁢
v
⁢
e
(
z
)
=
U
Y
,
m
Love
,
j
(
z
)
U
Y
,
m
Love
,
1
(
z
=
0
)
where the subscript m represents the mth-order mode of the dispersion surface waves;
step 1.10:
constructing single-mode surface-wave free fields at the surface station position (rHO, 0, 0, and an arbitrary surface reference point (r, 0, 0) according to formulae (7)-(11);
and further deducing attenuation factors of single-mode surface-wave free-field displacements along a horizontal radial direction by formula (13) based on the consistency criterion of vertical motions between two points:
R
r
,
m
R
⁢
a
⁢
y
(
r
)
=
∫
0
∞
[
u
r
,
m
Ray
,
1
(
k
,
r
HO
)
]
⁢
kdk
∫
0
∞
[
u
r
,
m
Ray
,
1
(
k
,
r
)
]
⁢
kdk
,
R
Z
,
m
Ray
(
r
)
=
∫
0
∞
[
u
Z
,
m
Ray
,
1
(
k
,
r
HO
)
]
⁢
kdk
∫
0
∞
[
u
Z
,
m
Ray
,
1
(
k
,
r
)
]
⁢
k
⁢
d
⁢
k
,
(
13
)
R
θ
,
m
L
⁢
o
⁢
v
⁢
e
(
r
)
=
∫
0
∞
[
u
θ
,
m
L
⁢
ove
,
1
(
k
,
r
HO
)
]
⁢
kdk
∫
0
∞
[
u
θ
,
m
L
⁢
ove
,
1
(
k
,
r
)
]
⁢
k
⁢
d
⁢
k
step 1.11:
using the NIP method to extract surface-wave displacement components collected at the surface station, and multiplying decoupled single-mode displacements
U
r
,
m
Ray
,
1
,
U
Z
,
m
Ray
,
1
⁢
and
⁢
U
θ
,
m
Love
,
1
by the corresponding horizontal radial attenuation factors and vertical attenuation factors, respectively, to construct single-mode surface-wave displacement free fields at the arbitrary position (r, θ, z) in the spatial domain:
{
U
r
,
m
Ray
,
j
(
r
,
θ
,
z
)
=
U
r
,
m
Ray
,
1
(
r
HO
,
0
,
0
)
⁢
R
r
,
m
Ray
(
r
)
⁢
R
r
,
m
Ray
(
z
)
U
Z
,
m
Ray
,
j
⁢
(
r
,
θ
,
z
)
=
U
Z
,
m
Ray
,
1
⁢
(
r
HO
,
0
,
0
)
⁢
R
Z
,
m
Ray
(
r
)
⁢
R
Z
,
m
Ray
(
z
)
U
θ
,
m
Lowe
,
j
⁢
(
r
,
θ
,
z
)
=
U
θ
,
m
Lowe
,
1
⁢
(
r
HO
,
0
,
0
)
⁢
R
θ
,
m
Love
(
r
)
⁢
R
θ
,
m
Love
(
z
)
(
14
)
where the single-mode surface-wave displacement free fields depend on the unknown quantity “epicentral distance rHO”;
step 1.12:
performing time and space partial differential calculations on the single-mode surface-wave displacement free fields, respectively, to construct corresponding single-mode velocity and stress free fields;
step 1.13:
based on the mode superposition principle of surface waves, superimposing all single-mode free fields to form multi-mode free fields, thereby completing surface-wave free-field inversion;
and
step 1.14:
superimposing the body-wave free fields and the surface-wave free fields to construct multi-phase total free fields of near-fault sites.

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Multi-phase free-field inversion method for near-fault site based on spherical/cylindrical diffusion properties

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Multi-phase free-field inversion method for near-fault site based on spherical/cylindrical diffusion properties

Life of this patent

Abstract

Claims2