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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 (*x*_{1},*x*_{2},*x*_{3},*x*_{4}, and *x*_{5}) 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
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Stanford Exploration Project

10/21/1998