Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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266<br />
The formulation has for basis expressing the response of any system by an ordinary<br />
differential equation of order µáD©<br />
ãâ Ô–<br />
è (10)<br />
Ô–<br />
Ô–<br />
– <br />
6 -<br />
É<br />
î <br />
ý‚ä<br />
ý5© -<br />
The transformation to the state <br />
Ô– Ô–<br />
variable å and is by substituting the higher<br />
derivatives of —– . The resulting dynamic ø state equation in the general â (continuous,<br />
time-variant) compact form are:<br />
—–<br />
—–<br />
—–<br />
x —–<br />
Ô–<br />
(system)<br />
<br />
(11)<br />
(output)è (12)<br />
ý’æ<br />
—–<br />
â —–<br />
—–<br />
—–<br />
Ô–<br />
—–<br />
úèç<br />
únç<br />
—–<br />
, Ô– and —– are matrices with variable elements in – Ô–<br />
; x —–<br />
ä<br />
function that generates the state;<br />
ü<br />
structure of the matrix <br />
is the forcing<br />
<br />
is the selected form for the output given by the<br />
<br />
is the addtive noise present.<br />
<br />
—– ; ç<br />
—–<br />
The continuous form solution is given by the system of three coupled equations:<br />
, (state estimation differential equa-<br />
ÒJé (1)<br />
tion)<br />
ê % ìë<br />
Ô–<br />
(2) ®<br />
—–<br />
n®<br />
—–<br />
—–<br />
?ó<br />
ø À<br />
—–<br />
ú_®<br />
, (the gain matrix);<br />
—–<br />
—–<br />
—–<br />
—–<br />
(3) å<br />
ý ¹<br />
ú ¹<br />
¹<br />
, (the Ricatti non-linear differential equation).<br />
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aí<br />
Ô–<br />
—–<br />
—–<br />
—–<br />
6 -<br />
Ô–<br />
—–<br />
Ô–<br />
Ô–<br />
—–<br />
—–<br />
Ô–<br />
aí<br />
x —–<br />
6 -<br />
—–<br />
DEVELOPMENT OF THE ALGORITHMS WHL AND KBC<br />
The WHL solution in discrete form to the problem under analysis is a modified classical<br />
prediction operator for multiple attenuation. We resume as:<br />
(a) The desired output: ¿<br />
, where<br />
ö<br />
is the prediction distance;<br />
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g<br />
¾ g<br />
g<br />
g<br />
QÐ<br />
ý‚‘<br />
(b) The WHL equation in parametric form: î ¨<br />
q§<br />
the events to be suppressed is represented by the equation î<br />
ýò¢ïóï<br />
.<br />
¤ ¢ïQï<br />
QÐ<br />
(c) The WHL prediction operator with a rectangular window ô<br />
to select out<br />
¾ g<br />
ŸÐ<br />
ÄÉ<br />
ŸÐ<br />
g<br />
An example intentionally simple is the case of multiples not ” accounted for in ,<br />
represented as a delayed pulse of units ( = layer thickness/layer velocity) with a<br />
ö ö , which results<br />
¢3ÏqÏ<br />
generalization through the following function: • g <br />
îöõ<br />
g<br />
.<br />
¢3ÏqÏ<br />
in an autocorrelation of the form: ¢3ÏqÏ<br />
g<br />
2§ .<br />
g<br />
g<br />
î7õ<br />
$<br />
ö$ ú ö<br />
÷»<br />
î7õ2ùÉ<br />
…(2<br />
$ þ<br />
ö$<br />
¢øNø<br />
¢ïóï
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