Derivative and integral operators

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Problem 9.31a

The operator is called the “derivative operator”; explain why.


In terms of finite differences (see problem 9.29), the derivative of a function is approximately equal to the change in values between two adjacent points divided by the distance between the two points. Convolving with gives values of the finite-difference derivative of for different values of . To show this, we find the term of . From equation (9.2b), we have

Figure 9.30a.  Seismic section. (i) Unmigrated; (ii) migrated.
Figure 9.30b.  Interpreted migrated seismic section.

Each term (except the first) equals at . Taking , we see that is minus the finite difference derivative at .

Problem 9.31b

What is the integral operator?


The integral, , for continuous functions becomes a summation for digital functions, the equivalent of the above integral being . If we take the operator where both and have terms, we obtain the following for the terms of (omitting ):

The terms are equivalent to the integrals between 0 and for . After the value is reached, the upper limit remains while the lower limit is successively 1, 2, . . . , . Thus we conclude that the operator [1, 1, 1, . . . , 1] is an integral operator. By taking with fewer terms than , we can get the equivalent of an integral over selected parts of .

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