Weighting functions

Generally, whatever weight is chosen for the forward operator, is also applied to the adjoint. However, because we have changed the way we design our operator, we must rethink our weighting. We can derive the proper weighting function for our Kirchoff and moveout operators in a continuum and apply it to our operator (see Jedlicka (1989)). This means that our approximate pull adjoint is an exact adjoint in a continuum.

We can think of a single trace of data $\tilde{d} (t)$, a model trace $m(\tau )$, scalar weights w and $\tilde{w}$,and an operator which performs the change of variables (this could be NMO, for example):

$$m(\tau ) = wd\, (t\, (\tau ))$$ (1)

$$\tilde{d} (t) = \tilde{w} \tilde{m} \, (\tau \, (t))$$ (2)

We can multiply the two equations and integrate both sides with respect to model variable $\tau$:

$$\int m\tilde{m} \, d\tau = \int [\frac{w}{\tilde{w}} ] d\tilde{d} \, d\tau$$ (3)

and change the variable of integration:

$$\int m\tilde{m} \, d\tau = \int [\frac{w}{\tilde{w}} \frac{d\tau }{dt} ] d\tilde{d} \, dt$$ (4)

The statement of the dot product test in a continuum is

$$\int m\tilde{m} \, d\tau = \int d\tilde{d} \, dt$$ (5)

which means that

$$\frac{w}{\tilde{w}} \frac{d\tau }{dt} = 1$$ (6)

From our NMO and Kirchoff equations,

$$t^2 = \tau ^2 + \frac{x^2}{v^2}$$ (7)

$$t dt = \tau d\tau$$ (8)

$$\frac{dt}{d\tau }=\frac{\tau }{t}$$ (9)

This tells us the ratio of one weighting function to the other. However, some experimentation may be required to find the optimal choices. I find good results using w = z/t, $\tilde{w} = 1$, but there may be more effective combinations.