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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.

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Stanford Exploration Project

1/13/1998