9.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Seismic array | 2/3 | https://en.wikipedia.org/wiki/Seismic_array | reference | science, encyclopedia | 2026-05-05T09:43:55.448067+00:00 | kb-cron |
Assuming that the noise nj(t) has a normal amplitude distribution with zero mean and variance σ2 at all sites, then the variance of the noise after summation is
σ
s
2
=
M
σ
2
{\displaystyle \sigma _{s}^{2}=M\sigma ^{2}}
and the standard deviation is
M
σ
2
{\displaystyle {\sqrt {M}}\sigma ^{2}}
. That means the standard deviation of the noise is multiplied by
M
{\displaystyle {\sqrt {M}}}
while the coherent signal is multiplied by
M
{\displaystyle M}
. The theoretical improvement of the SNR by beamforming (aka array gain) will be
G
=
M
{\displaystyle G={\sqrt {M}}}
for an array containing M sites.
==== The N-th root process ==== N-th root process is a non-linear method to enhance the SNR during beamforming. Before summing up the single seismic traces, the N-th root is calculated for each trace retaining the sign information. signum{wj(t)} is a function defined as -1 or +1, depending on the sign of the actual sample wj(t). N is an integer that has to be chosen by the analyst
B
N
(
t
)
=
∑
j
=
1
M
n
j
(
t
+
τ
j
)
N
⋅
s
i
g
n
u
m
{
w
j
(
t
)
}
{\displaystyle B_{N}(t)=\sum _{j=1}^{M}{\sqrt[{N}]{n_{j}(t+\tau _{j})}}\cdot signum\{w_{j}(t)\}}
Here the value of the function
s
i
g
n
u
m
{
w
j
(
t
)
}
{\displaystyle signum\{w_{j}(t)\}}
is defined as ±1 depending on the sign of the actual sample wj(t). After this summation, the beam has to be raised to the power of N
b
(
t
)
=
|
B
N
(
t
)
|
N
⋅
s
i
g
n
u
m
{
w
j
(
t
)
}
{\displaystyle b(t)=|B_{N}(t)|^{N}\cdot signum\{w_{j}(t)\}}
The N-th root process was first proposed by K. J. Muirhead and Ram Dattin in 1976. With the N-th root process, the suppression of uncorrelated noise is better than with linear beamforming. However, it weighs the coherency of a signal higher than the amplitudes, which results in a distortion of the waveforms.
==== Weighted stack methods ==== Schimmel and Paulssen introduced another non-linear stacking technique in 1997 to enhance signals through the reduction of incoherent noise, which shows a smaller waveform distortion than the N-th root process. Kennett proposed the use of the semblance of the signal as a weighting function in 2000 and achieved a similar resolution. An easily implementable weighted stack method would be to weight the amplitudes of the single sites of an array with the SNR of the signal at this site before beamforming, but this does not directly exploit the coherency of the signals across the array. All weighted stack methods can increase the slowness resolution of velocity spectrum analysis.
==== Double beam technique ==== A cluster of earthquakes can be used as a source array to analyze coherent signals in the seismic coda. This idea was consequently expanded by Krüger et al. in 1993 by analyzing seismic array data from well-known source locations with the so-called "double beam method". The principle of reciprocity is used for source and receiver arrays to further increase the resolution and the SNR for small amplitude signals by combining both arrays in a single analysis.
=== Array transfer function === The array transfer function describes sensitivity and resolution of an array for seismic signals with different frequency contents and slownesses. With an array, we are able to observe the wavenumber
k
=
2
π
/
λ
=
2
π
⋅
f
⋅
s
{\displaystyle k=2\pi /\lambda =2\pi \cdot f\cdot s}
of this wave defined by its frequency f and its slowness s. While time-domain analog-to-digital conversion may give aliasing effects in the time domain, the spatial sampling may give aliasing effects in the wavenumber domain. Thus the wavelength range of seismic signals and the sensitivity at different wavelengths must be estimated. The difference between a signal w at the reference site A and the signal wn at any other sensor An is the travel time between the arrivals at the sensors. A plane wave is defined by its slowness vector so
w
n
(
t
)
=
w
(
t
−
r
n
⋅
s
0
)
{\displaystyle w_{n}(t)=w(t-r_{n}\cdot s_{0})}
, where
r
n
{\displaystyle r_{n}}
is the position vector of site n The best beam of an array with M sensors for a seismic signal for the slowness so is defined as
b
(
t
)
=
1
M
∑
j
=
1
M
w
j
(
t
+
r
j
⋅
s
0
)
{\displaystyle b(t)={\frac {1}{M}}\sum _{j=1}^{M}w_{j}(t+r_{j}\cdot s_{0})}
If we calculate all time shifts for a signal with the slowness so with respect to any other slowness s, the calculated beam becomes
b
(
t
)
=
1
M
∑
j
=
1
M
w
j
(
t
+
r
j
⋅
(
s
0
−
s
)
)
{\displaystyle b(t)={\frac {1}{M}}\sum _{j=1}^{M}w_{j}(t+r_{j}\cdot (s_{0}-s))}
The seismic energy of this beam can be calculated by integrating over the squared amplitudes
E
(
t
)
=
∫
−
∞
∞
b
2
(
t
)
d
t
=
∫
−
∞
∞
[
1
M
∑
j
=
1
M
w
j
(
t
+
r
j
⋅
(
s
0
−
s
)
)
]
2
d
t
{\displaystyle E(t)=\int _{-\infty }^{\infty }b^{2}(t)dt=\int _{-\infty }^{\infty }[{\frac {1}{M}}\sum _{j=1}^{M}w_{j}(t+r_{j}\cdot (s_{0}-s))]^{2}dt}
This equation can be written in the frequency domain with
w
¯
(
ω
)
{\displaystyle {\bar {w}}(\omega )}
being the Fourier transform of the seismogram w(t), using the definition of the wavenumber vector k = ω⋅ s