RunSemblance Documentation
Makes a semblance correlation of the two input traces. This is a full (non-gated) semblance.
Usage
The program is used as follows:
runSemblance input1FileName input2FileName
Where the input1FileName and input2FileName are in our standard file format.
Output
The output of the program appears one place:
(1) A copy of the output data is placed in the file lastDataOutput.
The returned floating point array is numbered ,
and the
point represents the value of the semblance
when the middle of the event is lined up with the
point.
Theory
We present here the definition of semblance implemented in the LwsVector class using the method Semblance. We start by reviewing the basic theory of cross correlation. Our implementation of cross correlation is defined in RunXCorr. We summarize below the theory of cross correlation with lag.
Cross Correlation with Lag:
Define:
(1.1)
or in the digital domain:
(1.2)
where l is the lag, and k is the center of the window.
This should be normalized by
(1.3)
When normalized correctly, it will be between -1 and 1. It is 1 when f=g and l=0, and it is -1 when f=-g and l=0. Here is the complete cross correlation formula:
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(1.4) |
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In the frequency domain:
If is the fourier transform of convolution, we
have:
(1.5)
which has lots of phase and DC problems.
Semblance with Lag
Cross correlation is very sensitive to phase differences. A better choice is Semblance, a measure of the coherence between traces. This is not usually computed with lag, but we need that here:
This is approximately the energy of f and G divided by the mean energy of each component. It is between 0 and 1, and is 1 when f=g and l=0.
In our implementation, we assume that the is an event, i.e.
and
have different lengths, and we are interested
in the results of passing
over
.
Let
and
be the lengths of
and
respectively. In our case,
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Then in equation (2.1) and
are extended with 0 values as necessary, and
we have
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that is we sum whenever and
overlap, that is in the region where
and
.
Remember that
is the left hand endpoint of
and so one should add
to
to find the lag corresponding to the center of
the event.
It turns out that when the length of and
are very different, the semblance for where
they partially overlap can be biased. So, instead of (2.3)
we use only those points for which
and
completely overlap, making use of the
assumption (2.2).
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There is no clear frequency domain analogue. If ,
then
.
If
then
.
If
then
.
In general, it can be characterized as the energy of the sum divided by the
mean energy of the components of the sum. This method is more robust with
respect to phase distortions.
For more than two samples, (2.1) can be generalized:
|
(2.5) |
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Writing the equation this way makes it clearer that we are
comparing energy of the sum to the mean energy of the components. However, for
notational convenience the lag, which is different for each j, is applied the before summing.
In any case, we use equation (2.4)
in RunSemblance and in Semblance. To
make the array reference the center of the event, we pad zeros on the beginning and the end of the
resulting array. Then the returned floating point array is numbered
,
and the
point represents the value of the semblance
when the middle of the event is lined up with the
point.