Next: Pseudounitary NMO with linear Up: UNITARY OPERATORS Previous: Meaning of B'B

## Unitary and pseudounitary transformation

A so-called unitary transformation conserves energy. In other words, if , then ,which requires .Imagine an application where the transformation seems as if it should not destroy information. Can we arrange it to conserve energy? The conventional inversion
 (27) (28)
can be verified by direct substitution. Seeking a more symmetrical transformation between and than the one above, we define
 (29)
and the transformation pair
 (30) (31)
where we can easily verify that by direct substitution. In practice, it would often be found that is a satisfactory substitute for ,and further that the unitary property is often a significant advantage.

Is the operator unitary? It would not be unitary for NMO, because equation (19) is not invertible. Remember that we lost (x1,x2,x3,x4, and x5) in (17). is unitary, however, except for lost points, so we call it pseudounitary." A trip into and back from the space of a pseudounitary operator is like a pass through a bandpass filter. Something is lost the first time, but no more is lost if we do it again. Thus, ,but for any .Furthermore, ,but .In mathematics the operators and are called idempotent" operators. Another example of an idempotent operator is that of subroutine advance() 1

Next: Pseudounitary NMO with linear Up: UNITARY OPERATORS Previous: Meaning of B'B
Stanford Exploration Project
10/21/1998