Coherence measure

Another kind of quality measure for an estimation process is a coherence measure:  

$$C(t,x,p) = \sum^{L_t}_{i=-L_t}W_t(i)\left[\sum^{L_x}_{j=-L_x}W_x(j)P_{ij} \right]^2.$$ (5)

Coherence measures have been previously used in seismic data processing, for example, in velocity analysis (Biondi, 1990). Its normalized version is called semblance:

 

$$C_n(t,x,p) = \sum^{L_t}_{i=-L_t}W_t(i){\left[ \displaystyle{... ...ght]^2 \over \displaystyle{\sum^{L_x}_{j=-L_x}}W_x(j)P^2_{ij}}.$$ (6)

It can be shown that both C(t,x,p) and C_n(t,x,p) reach their maximum values when the traces in the subsection P_i,j are identical, which implies that the estimated p is equal to the true dip.