COUNT

For NMO using nearest neighbor interpolation, count is used to get inverse NMO (Claerbout, 1985). Count forms a diagonal of the matrix NMO^T NMO. We can see the shape of this diagonal in Figure a. Count corresponds to the jacobian in the continuous case. The expression for the jacobian is derived in Appendix C.

Claerbout (1987) introduces count(x,s,t) for velocity analysis as a count of times that a point located on the mesh at (x,t) will be added into slowness s. For the purposes of this section count(t,x) is introduced as a count of times that a point located on the mesh at (x,t) will be added into velocity stack, so that we can write

$$count(t,x)=\sum_s count(x,s,t).$$ (19)

For linear interpolation count(t,x) may be defined as a sum of weights with which a point will be added into velocity stack. For NMO we will get a curve in Figure b. In both cases a theoretical count (jacobian) is drawn through the curves. Count may be expressed with the operator $\bold H$ as (after some normalization)

$$count(t,x)={\bf H^TH}d(t,x),$$ (20)

where

d(t,x)=1

for all t and x. This can be seen from the way adjoint operators are computed (spreading).

Count for a CMP gather is shown in Figure . We can see that the line t=x/v_min is the line of the maximum count. Count on each trace is equal to the sum of NMO counts of this trace for each velocity involved in velocity analysis (Appendix B and C). This is apparent from Figure c.

Velocity analysis was done for fifty velocities. From Figure c we can see that in most of the gather below the line t=x/v_vmin count does not differ much from 50. This implies that count(x,s,t) does not differ much from 1 in this part of the gather. This result corresponds to NMO, where the same result holds tru truee.

The area above the line t=x/v_max is not covered by any hyperbola, so that this part of the gather cannot be restored. It contributes to the singularity of the operator ${\bf H^TH}$.