Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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267<br />
modifies the<br />
î7õ<br />
, and the deconvolutional operator has a factor that scales the<br />
<br />
”<br />
desired it is necessary that the<br />
lengths of the source-pulse and of the temporal window for truncation and smoothing<br />
do not contain the first multiple at distance . Consequently, the elongated operator<br />
ö-<br />
g<br />
The first term of the series is , where the variance<br />
Õ Õ g<br />
desired segment<br />
¢ïóï<br />
g<br />
ù<br />
ùg<br />
¢ïóï<br />
. From this we conclude that for windowing ¢ïóï<br />
with deconvolves the multiple of period , and the operator with length<br />
deconvolves ö- the multiples and , and so ö- on. (see Figure 2 and 3)<br />
ú ög<br />
ö- ö<br />
The application of the KBC solution in discrete form ù<br />
to a seismic trace, ,<br />
consists in a sequence of point-to-point operations organized in a definite sequence. ü<br />
g <br />
The matrix computed as:<br />
¹<br />
g<br />
g<br />
g<br />
g<br />
g<br />
(13)<br />
ý×ú<br />
S©<br />
¹ 6<br />
S©<br />
D©<br />
ú£º<br />
D©<br />
(14)<br />
g<br />
g<br />
g<br />
g<br />
g<br />
by:<br />
where ú<br />
g<br />
\©<br />
6 ý ¹ ®<br />
¹<br />
is the state transition matrix. The gain matrix ®<br />
g<br />
is calculated<br />
ß 6 - è (15)<br />
g<br />
g<br />
g<br />
¸ g<br />
The state vector is calculated by:<br />
ý ¹<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
ø 6 ú£® ûÀ ø 6<br />
ý«À<br />
And the output has the expression: + ü<br />
the multiple-free sismogram.<br />
g<br />
g<br />
ýÑú<br />
g <br />
S©<br />
ø 6<br />
S©<br />
. The strategy is to recover<br />
We start identifying the variable with the non-stationary model.<br />
1ý<br />
The<br />
ýL<br />
seismic<br />
pulse is represented by the matrix:<br />
. The selection of the state vector<br />
is nonunique, and in this case the is defined as:<br />
$ ý þ<br />
š§ è (17)<br />
óÇü<br />
ü (16)<br />
g<br />
g<br />
g<br />
èBè è g<br />
The dynamic equations of the system to establish the recursive process of generation<br />
of the state vector is completed by the following model:<br />
ý ¤<br />
D©<br />
þ ú‚©<br />
(18)<br />
g<br />
g<br />
g<br />
g <br />
is theoretically considered as a white stochastic process.<br />
This equation<br />
projects the trace forward through a weighed sum of previous values in a<br />
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similar<br />
g þ<br />
formalism to the WHL. The coefficients are defined by a chosen model and experimentation,<br />
and in the present case has c#$ been the exponential model. The state is<br />
written as<br />
è (19)<br />
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c#$<br />
g<br />
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