Nonlinear signal comparison and high-resolution measurement of seismic or acoustic wave dispersion
The present disclosure relates generally to signal comparison in seismic/acoustic imaging and data processing. In particular, this disclosure relates to a new nonlinear signal comparison (NLSC) approach which obtains a uniform resolution across a frequency band. The overall resolution in NLSC can be controlled over the frequency band by an adjustable parameter.
1. A method for comparing seismic signals for dispersion analyses for use in analyzing properties of a material, comprising:
propagating a wave through a material, wherein a first receiver and a second receiver are located in the material, wherein the second receiver is located at a distance x from the first receiver, wherein the wave propagates through the first receiver and the second receiver, and wherein the first receiver and the second receiver perform measurements of the wave;
assigning a first time series waveform d 1 (t) for the wave propagating through the material at the first receiver;
assigning a second time series waveform d x (t) for the wave propagating through the material at the second receiver, wherein the first and second time series waveforms relate to pressure, particle displacement, or velocity;
calculating a non-linear signal comparison measurement S NL (ω, V ph ) using collected data and the equation:
S
NL
(
ω
,
V
ph
)
=
1
T
∫
0
T
exp
(
-
[
d
_
1
(
t
;
ω
)
-
d
_
x
(
t
+
x
V
ph
;
ω
)
]
2
4
ω
2
π
-
2
σ
2
)
dt
,
wherein T is a selected length of time for a window of interest,
d 1 (t;ω) is a variance-normalized seismic waveform for d 1 (t) filtered around frequency ω and is equal to σ 1 −1 d 1 (t;ω) wherein σ 1 is variance or an energy-related measure of the collected data,
d
_
x
(
t
+
x
V
ph
;
ω
)
is a variance-normalized seismic waveform for d x (t) filtered around frequency ω and is equal to
σ
x
-
1
d
_
x
(
t
+
x
V
ph
;
ω
)
wherein V p h is phase velocity equal to x divided by t and wherein σ x is variance or an energy-related measure of the collected data, and
σ is an overall continuous nonnegative parameter for controlling overall resolution;
calculating a background measurement S π , for the non-linear signal comparison measurement S NL (ω,V ph ) using the equation:
S π =I 0 ( b ) e −b
wherein σ is a modified Bessel function of zero-th order, and
b
=
π
2
σ
2
ω
2
T
;
and
calculating a normalized and scaled non-linear signal comparison measurement S NLSC (ω, V ph ;σ) using the equation:
S
NLSC
(
ω
,
V
ph
;
σ
)
=
S
NL
(
ω
,
V
ph
)
-
S
π
1
-
S
π
,
wherein σ is chosen from zero to infinity, and wherein S NLSC (ω, V ph ;σ) measures sensitivity to a time shift and velocity and provides signal comparison information for the wave at the first and second receivers; and
using the signal comparison information in dispersion analyses to calculate properties, characteristics, or structures of the material.
2. The method of claim 1 , wherein the first receiver and the second receiver are part of a set of multiple receivers and the steps of the method are repeated to provide signal comparison information for the wave at additional receivers.
3. The method of claim 1 , further comprising the step of using the signal comparison information in borehole acoustic logging.
4. The method of claim 1 , wherein the signal comparison information is used in seismic imaging and migration, wherein the first time series waveform is assigned as a downgoing forward propagated wavefield, and wherein the second time series waveform is assigned as a backward extrapolated receiver wavefield.
5. The method of claim 1 , wherein the signal comparison information is used in seismic waveform inversion, wherein the first time series waveform is assigned as a modeled waveform, and wherein the second time series waveform is assigned as an actually recorded wavefield.
6. The method of claim 1 , wherein σ is chosen to be in the range of 0.001 to 10.