Error measure

The error in estimating p can not be directly evaluated because the true value of p is unknown. However, a function of p that is related to this error can be evaluated:  

$$E(t,x,p) = \sum^{L_t}_{i=-L_t}W_t(i)\sum^{L_x}_{j,k=-L_x}W_x(j)W_x(k) (P_{ij}-P_{ik})^2,$$ (3)

where W_t and W_x are weighting functions. As a function of p, E(t,x,p) measures the differences between the traces within the subsection P_ij. This function reaches its minimum value when the estimated p is equal to the true dip at (t,x). Therefore, it can be used as an error measure.

It may be desirable to have an error measure that is bounded between and 1. This can be easily done through normalization:

 

$$E_n(t,x,p) = \sum^{L_t}_{i=-L_t}W_t(i){ \displaystyle{\sum^{... ...ik})^2 \over \displaystyle{\sum^{L_x}_{j=-L_x}}W_x(j)P^2_{ij}}.$$ (4)

One can estimate p by minimizing either E(x,t,p) or E_n(x,t,p). If the data contain rich low-frequency components, such as a smoothed well log, equation (3) should be used. For seismic data, the normalized error measure should be used.