A velocity filter removes
interfering
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extr ...
signals by exploiting the difference between the travelling velocities of desired
seismic wave
A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic energy. S ...
form and undesired interfering signals.
Introduction
In
geophysical
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
applications sensors are used to measure and record the seismic signals. Many filtering techniques are available in which one output waveform is produced with a higher
signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
than the individual sensor recordings. Velocity filters are designed to remove interfering signals by exploiting the difference between the travelling velocities of desired seismic waveform and undesired interfering signals. In contrast to the one dimensional output produced by multi-channel filtering, velocity filters produce a two-dimensional output.
Consider an array of ''N'' sensors that receive one desired and ''M'' undesired broadband interferences. Let the measurement of ''n''
th sensor be modeled by the expression:
: x
n(t)=
a
mns
m(t-T
mn)+ŋ
n(t)___(1)
where n=1,2,...,N; m=0,1,...,M; s
m(t) are signals travelling across the array, and ŋ
n(t) represents zero-mean white random noise at the ''n''
th sensor, uncorrelated from sensor to sensor. The parameters a
mn and T
mn are amplitude gain and time delays of the signal s
m(t) when received at ''n''
th sensor.
Without loss of generality, we shall assume that s
0(t) is the desired signal and s
1(t),s
2(t),...,s
M(t) are the undesired interferences. Additionally we shall assume that T
0n=0, and a
0n =1. This essentially means that the data has been time shifted to align the desired seismic signal so that it appears on all sensors at the same time and balanced so that the desired signal appears with equal amplitudes. We assume that the signals are digitized prior to being recorded and that the length K of time sequences of recorded data is large enough for the complete delayed interfering waveforms to be included in the recorded data. In the discrete frequency domain, the ''n''
th trace can be expressed as:
: X
n(k)=S
0(k) +
a
mnS
m(k)e
−jwkTmn + N
n(k)___(2)
where k=0,1,...,K-1; w
k=(2π/K) is the sampling angular frequency.
Using matrix notation, (2) can be expressed in the form:
: X(k) = A(k) S(k) + N(k)___(3)
Velocity filtering
Frequency domain multichannel filters F
1(k), F
2(k), ..., F
N(k) can be applied to the data to produce one single output trace of the form,:
: Y(k) =
F
N(k)X
n(k) ___(6)
In matrix form, the above expression can be written as:
: Y(k) = X'(k)F(k) = S'(k) A'(k) F(k) + N'(k) F(k) ___(7)
where F(k) is an N x 1 vector whose elements are the individual channel filters. That is,
: F(k) =
1(k), F2(k), ..., FN(k)">1(k), F2(k), ..., FN(k)___(8)
By following the procedure discussed in, an optimum filter vector F(k) can be designed to attenuate, in the least square sense, the undesired coherent interferences S
1(k),S
2(k),...,S
M(k) while preserving the desired signal S
0(k) in Y(k). This filter can be shown
to be of the form:
![Eq 9](https://upload.wikimedia.org/wikipedia/commons/3/3a/Eq_9.jpg)
where ''h'' is an arbitrary N x 1 nonzero vector, ''u'' =
,0,...,0 ''I'' is the unit matrix, ''B
r(k)'' is a submatrix of the matrix obtained by dropping all linearly dependent rows, and L(k) is a lower triangular matrix satisfying:
:
r(k)">(k) Br(k) r(k)">(k) Br(k)sup>H = I
![Eq 10](https://upload.wikimedia.org/wikipedia/commons/a/ab/Eq_10.jpg)
The multichannel processing scheme described by
1produces one dimensional output trace. A velocity filter, on the other hand, is a two-dimensional filter which produces a two-dimensional output record.
A two-dimensional record can be generated by a procedure which involves repeatedly applying multichannel optimum filters to a small number of overlapping subarrays of the input data,.
![Subarray](https://upload.wikimedia.org/wikipedia/commons/1/11/Subarray.jpg)
More specifically, consider a subarray of W channels, where W<
th trace of the output record, and traces N-W+1,N-W+2,...,N of the input record produce the (N-W+1)st trace, which is the last trace, of the output record. For a large N and small W, as is typically the case in geophysical data, the output record can be viewed as comparable in dimensions to the input record. Clearly for such a scheme to work effectively W must be as small as possible; while at the same time it must be large enough to provide the necessary attenuation of the undesired signals. Note that a maximum of W-1 undesired interferences can be totally suppressed by such a scheme.[Magdy T, Hanna, "Velocity Filters for multiple interference attenuation in Geophysical Array Data",IEEE Trans. on geoscience and remote sensing, Vol 26,No. 6,pp. 741-748, November 1998.]
References
{{reflist, refs
Seismology