Next: Amplitude equivalence Up: 9: INTEGRAL OPERATORS OF Previous: Formulas for amplitudes (3D

## 2D case

If we use the same type of considerations as in the 3D case, we receive

where is a smooth function,

Let us expand the class of discontinuities

with arbitrary (noninteger) q, and

R(-)q(t) = R(+)q(-t).

We must also expand the notion of q-equivalence (which was introduced in the Chapter 1) to a noninteger q: if .The operator of noninteger differentiation was considered above for q=1/2 (see Chapter 8). For arbitrary q it can be defined by spectra response .

It can be shown that

It is easy to see that

and

(it is proposed that d(-)>0).

If d(-)<0, then

and general formula

where .

Let d(-)=0. We introduce the order of touching of curves and :the order = p if

and

If the order p is even, then

If the order is uneven

The point is a special one if, for given and ,.Each special point of the order p=2 is the point on caustics.

Next: Amplitude equivalence Up: 9: INTEGRAL OPERATORS OF Previous: Formulas for amplitudes (3D
Stanford Exploration Project
1/13/1998