1887
Volume 8 Number 2
  • E-ISSN: 1365-2478

Abstract

A

Our purpose is to give a short summary of the theory of synthetic seismograms, including all multiple reflections and to show the method of their construction with the use of an electronic computer.

The waves to be considered in reflection seismic being approximately plane and horizontal it is generally admitted that in most cases the propagation phenomena can be described with the equation

(, displacement; velocity; p, density). Moreover, for all practical purposes, the velocity is not a continuous function of the depth In fact, the earth can be divided up into more or less thin layers, with constant velocity inside each layer and sudden variations at each interface. It is therefore reasonable to substitute to the single equation (1) a series of simple propagation equations with constant coefficients

provided a set of boundary conditions is adjoined to them in order to ensure the continuity of displacement and tension.

As with all seismic problems, this is essentially a transient system and a very convenient method to resolve equation (2) is to resort to the Laplace transformation, by writing

Then the general integral of (2) is:

and being two constants. This expression is valid inside of a layer, including the two faces. If the propagation velocity in an adjoining layer is V we get an equation similar to (4), say (4′), but with different constants and At a point on the interface, both expressions are valid. Consequently, if we take three points and into consideration, respectively at the depths V T, and we can write four expressions (4) and

(4′). The continuity of the tension gives a fifth expression. The constants and can be eliminated from these equations. The result of the elimination is

is the reflection coefficient. Going back to the original functions we find a recurrence expression with four terms

In order to make use of this expression, we set out from curve

which divides the plane into two domains:

1) a domain contiguous to the axis where is identical to zero;

2) the remainder of the plane where for o) =–“the signal”–is given in a narrow interval in the proximity of the origin.

The numerical calculation is carried out at the intersection points of two sets of straight lines:

1) equidistant parallels to (with spacing x) and

2) parallels to through the intersection of the curve with the straights of the first family.

Computations having been carried out for all points of the second domain, they lead finally to the values of the function at the surface, outside the interval where the signal is given. This function is the requested synthetic seismogram.

The shape of the signal enters into the calculations. As a matter of fact it is always necessary to try several signals, hence to construct several synthetic seismograms. However, the operation consisting in the modification of the response is simpler than the calculation of the initial film. This leads to the notion of the synthetic impulse seismogram, which is constructed by assuming that the signal is a pure impulse. This impulse seismogram being calculated, it is easy to construct as many synthetic records as there are signals to be taken into account.

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2006-04-27
2020-04-04
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  • Article Type: Research Article
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