UNITARITY

Believe it or not, we often have problems where we are unsure of the precise form of $\bold F$.This can arise when we know a travel time but are uncertain about an amplitude. It also arises when we are unsure about a source radiation pattern. It also arises when we scan a parameter like in a velocity spectrum.

We begin from an operator $\bold F$ that is somehow wrong in that it should really have pre and post diagonal operators applied, say $\bold G = \bold D_d \bold F \bold D_m$.Although we may be unsure how to define the $\bold D_d$ and the $\bold D_m$,we are quite sure we will be happy with an operator that conserves the energy in the original data. Thus we seek approximate idempotence,

$$\bold G'\bold G \quad \approx \quad (\bold G'\bold G )^2$$ (10)