9.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Seismic array | 3/3 | https://en.wikipedia.org/wiki/Seismic_array | reference | science, encyclopedia | 2026-05-05T09:43:55.448067+00:00 | kb-cron |
E
(
ω
,
k
0
−
k
)
=
1
2
π
∫
−
∞
∞
|
w
¯
(
ω
)
|
2
⋅
|
C
(
k
0
−
k
)
|
2
d
ω
{\displaystyle E(\omega ,k_{0}-k)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|{\bar {w}}(\omega )|^{2}\cdot |C(k_{0}-k)|^{2}d\omega }
, where
C
(
k
0
−
k
)
=
1
M
∑
j
=
1
M
e
i
w
r
j
(
k
0
−
k
)
{\displaystyle C(k_{0}-k)={\frac {1}{M}}\sum _{j=1}^{M}e^{iwr_{j}(k_{0}-k)}}
This equation is called the transfer function of an array. If the slowness difference is zero, the factor
|
C
(
k
0
−
k
)
|
2
{\displaystyle |C(k_{0}-k)|^{2}}
becomes 1.0 and the array is optimally tuned for this slowness. All other energy propagating with a different slowness will be suppressed.
=== Slowness estimation === Slowness estimation is a matter of forming beams with different slowness vectors and comparing the amplitudes or the power of the beams, and finding out the best beam by looking for the vapp and backazimuth combination with the highest energy on the beam.
==== f-k analysis ==== Frequency-wavenumber analysis is used as a reference tool in array processing for estimating slowness. This method was proposed by Capon in 1969 and further developed to include wide-band analysis, maximum-likelihood estimation techniques, and three-component data in the 1980s. The methodology exploits the deterministic, non-periodic character of seismic wave propagation to calculate the frequency-wavenumber spectrum of the signals by applying the multidimensional Fourier transform. A monochromatic plane wave w(x,t) will propagate along the x direction according to equation
w
(
x
,
t
)
=
A
e
i
2
π
(
f
0
t
−
k
0
x
)
{\displaystyle w(x,t)=Ae^{i2\pi (f_{0}t-k_{0}x)}}
It can be rewritten in frequency domain as
W
(
k
x
,
f
)
=
A
δ
(
f
−
f
0
)
δ
(
k
x
−
k
0
)
{\displaystyle W(k_{x},f)=A\delta (f-f_{0})\delta (k_{x}-k_{0})}
which suggests the possibility to map a monochromatic plane wave in the frequency-wavenumber domain to a point with coordinates (f, kx) = (f0, k0). Practically, f-k analysis is performed in the frequency domain and represents in principle beamforming in the frequency domain for a number of different slowness values. At NORSAR slowness values between -0.4 and 0.4 s/km are used equally spaced over 51 by 51 points. For every one of these points the beam power is evaluated, giving an equally spaced grid of 2601 points with power information.
==== Beampacking ==== A beampacking scheme was developed at NORSAR to apply f-k analysis of regional phases to data of large array. This algorithm performs time-domain beamforming over a predefined grid of slowness points and measures the power of the beam. In practice the beampacking process gives the same slowness estimate as for the f-k analysis in the frequency domain. Compared to the f-k process, the beampacking process results in a slightly (about 10%) narrower peak for the maximum power.
==== Plane wave fitting ==== Another way of estimating slowness is to pick carefully times of the first onset or any other common distinguishable part of the same phase (same cycle) for all instruments in an array. Let ti be the arrival time picked at site i, and tref be the arrival time at the reference site, then τi = ti − tref is the observed time delay at site i. We observe the plane wave at M sites. With M ≥ 3. The horizontal components (sx, sy) of the slowness vector s can be estimated by
s
^
=
m
i
n
s
∑
j
=
1
M
(
τ
j
−
r
j
⋅
s
)
2
{\displaystyle {\hat {s}}={\underset {s}{min}}\sum _{j=1}^{M}(\tau _{j}-r_{j}\cdot s)^{2}}
Plane wave fitting requires interactive analyst's work. However, to obtain automatic time picks and thereby provide a slowness estimate automatically, techniques like cross-correlation or just picking of peak amplitude within a time window may be used. Because of the amount of required computations, plane wave fitting is most effective for arrays with a smaller number of sites or for subarray configurations.
== Applications == Current seismic arrays worldwide:
=== Gräfenberg === The Gräfenberg array is the first digital broadband array that has a continuous data history from 1976 until today. This array consists of 13 broadband stations in the Fränkische Alb. It extends approximately 100 kilometers north-south and approximately 40 kilometers east-west.
=== YKA === YKA or Yellowknife Seismological Array is a medium size seismic array established near Yellowknife in the Northwest Territories, Canada, in 1962, in cooperative agreement between the Department of Mines and Technical Surveys (now Natural Resources Canada) and the United Kingdom Atomic Energy Authority (UKAEA), to investigate the feasibility of teleseismic detection and identification of nuclear explosions. YKA currently consists of 19 short period seismic sensors in the form of a cross with an aperture of 2.5 km, plus 4 broadband seismograph sites with instruments able to detect a wide range of seismic wave frequencies.
=== LASA ===
LASA or Large Aperture Seismic Array is the first large seismic array. It was built in Montana, USA, in 1965.
=== NORSAR === NORSAR or Norwegian Seismic Array was established at Kjeller, Norway in 1968 as part of the Norwegian-US agreement for the detection of earthquakes and nuclear explosions. It has been an independent, not-for-profit, research foundation within the field of geo-science since 1999. NORSAR was constructed as a large aperture array with a diameter of 100 km. It is the largest stand-alone array in the world.
=== NORES and ARCES === NORES was the first regional seismic array constructed in southern Norway in 1984. A sister array ARCES was established in northern Norway in 1987. NORES and ARCES are small aperture arrays with a diameter of only 3 km.
=== GERES === GERES is a small aperture array built in the Bavarian Forest near the border triangle of Germany, Austria and Czechia, in 1988. It consists of 25 individual seismic stations arranged in 4 concentric rings with radius of 200m, 430m, 925m and 1988m.
=== SPITS === SPITS is a very small aperture array at Spitsbergen, Norway. It was originally installed in 1992 and upgraded to IMS standard in 2007 by NORSAR.
== See also == Seismometer Array processing
== References ==