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## Zero-approximation.

Let us consider the equation (32) for n = 0:

With accordance to equation (23), this is a linear algebraic equation with degenerate matrices. It may be solved only in the case when is an eigenvector. We restrict ourselves with the situation when the eiconal equation for P-waves is valid. In this case, the eigenvector is so vector can be expressed in the form . In order to specify the amplitude A0 let us consider the equation (32) for n = 1:
 (34)

But any vector can be decomposed into the sum where . Both vectors and are eigenvectors connected with eigenvalues and correspondently, therefore,

We see that the left side of the equation (34) is orthogonal to the vector . It means that equation (33) can be resolved if and only if
 (35)

Inserting expression in equation (33) into the last equation, we use the property of the operator , if s is the natural parameter of a ray , then for any scalar function

where means differentiation along the ray that intersects the given point. After simple manipulation we derive from equation (35) the transient equation for the amplitude A = A0/v:
 (36)
which have solution

The well-known notion of geometrical spreading is connected with by relation (S.V. Goldin, 1986)

consequently,
 (37)
Analogous formula is true for S-waves.

The equation (37) describes how the amplitude of the senior discontinuity of a wave is changing along a ray path.

If the value of J is positive in all points of a given ray , then along the ray
 (38)
where and is the tangent-vector of the ray. And if J < 0, then

The situation when J = 0 means that at q'<q0, .

Next: Fundamental solutions Up: 4: CONNECTION WITH RAY Previous: Ray method
Stanford Exploration Project
1/13/1998