Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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264<br />
A discrete description of the 1D convolutional model for the representation of seismic<br />
data, , independent of the horizontal ray parameter , is given by:<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
g<br />
" ú<br />
g <br />
Where represents the effective ”<br />
source-pulse,<br />
<br />
<br />
g<br />
þ<br />
is the signal-message, and<br />
"<br />
" ú<br />
g <br />
is the reflectivity function, ‘<br />
is the eternal temptative to be described as the additive<br />
<br />
signal-noise not accounted for ” in .<br />
Š“@”<br />
è (1)<br />
ý’‘<br />
g<br />
The source time history is represented by the Berlage function:<br />
and in þ<br />
g<br />
where • ý<br />
—– •<br />
¡ r<br />
ý‚˜<br />
r Þ<br />
Hz å and<br />
ý ¢ <br />
—–<br />
rd.<br />
–š Q¡<br />
þ 6œ›^ŸžN ç<br />
We aim at to construct the seismic reflection trace by the convolutional model<br />
based on Betti's theorem. The physics of propagation is governed by the equation of<br />
particle motion <br />
in q§ the 1-D acoustic form: . The<br />
phenomenon is of an incident vertical plane wave on a medium formed ø %e¤¦¥ ø by horizontal,<br />
homogeneous and isotropic layers. The boundary conditions of displacement (or<br />
pressure) and stress continuity …(¨ result in defining the reflection, , and the transmission,<br />
, coefficients for the interface between the layers and , that results<br />
– —ª ¨ g g Ÿª<br />
¡Wª Ÿª<br />
g …(¨ in: , ¨<br />
, where – …j¨ and – ¨ are real<br />
ú©<br />
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¨ …(¨ +- d3…(¨¬d&R+-<br />
numbers, and R ¨7R . The physical problem now has<br />
+been<br />
transformed to a physics of interfaces, and the events making the seismic – Þ trace<br />
ý«© <br />
are considered as primary and secondary reflections (multiples).<br />
The relation between the descendent, 4<br />
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between the top,<br />
propagator:<br />
g<br />
, and the bottom, g ý;® ú©<br />
ú£¢ <br />
, and the ascendent, <br />
(2)<br />
, waves, and <br />
, is expressed by the matricial<br />
layers is given by:<br />
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µ - µ <br />
4 <br />
°²<br />
è è è<br />
è (3)<br />
·±<br />
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Wµ°<br />
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The reflection transfer function for a system of ®<br />
6 -<br />
©´<br />
©³<br />
è (4)<br />
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ý<br />
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The ¨<br />
denominator has the property of being of minimum-phase.<br />
The numerator … ¹ is not necessarily of minimum-phase, what makes<br />
º<br />
¸<br />
be or not to be of minimum-phase. The polynomial division is ilimited, but<br />
A¹ ’º<br />
<br />
numerically made to correspond to the number of layers . ®<br />
4 <br />
In special conditions for analyses we can ” ¼»½… admit , or yet ” that<br />
is composed of the primary incident field and of the secondary<br />
spread out field, and the figures show how the secondary field can<br />
<br />
gradually<br />
. A total response ”
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