FILM SYNTHETIQUE AVEC REFLEXIONS MULTIPLES THEORIE ET CALCUL PRATIQUE*
Authors: BARANOV V. , KUNETZ G. 
Published: April 2006

Abstract

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

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(u, displacement; V(z), velocity; p, density). Moreover, for all practical purposes, the velocity V (z) is not a continuous function of the depth z. 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

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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

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Then the general integral of (2) is:

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A and B 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 C and D. At a point on the interface, both expressions are valid. Consequently, if we take three points M, P and N into consideration, respectively at the depths z– V T, z and z + V T, we can write four expressions (4) and

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

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

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

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which divides the plane Ozt into two domains:

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

2) the remainder of the plane where for z = o, u(t, o) =s(t)–“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 Oz (with spacing x) and

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

Computations having been carried out for all points (z, t) of the second domain, they lead finally to the values of the function u at the surface, u(t, o), outside the interval where the signal is given. This function u(t, o) 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.