OBJECTIVE FUNCTIONS

The formulation of an optimal estimation usually consists of two steps. The first step is to construct an objective function that depends on the unknown parameter to be determined. This function measures the quality of the estimation. The second step is to estimate the unknown parameters through the extremization of the objective function, depending on type of the quality measure. In this section, I use dip estimation of seismic events as an example to describe two types of quality measures.

Let us suppose a seismic section P(t,x) contains events reflected from subsurface boundaries. We want to estimate the local dips of these events. An event with local dip p at point (t,x) follows the trajectory  

$$\hat{t} = t+p(\hat{x}-x)$$ (1)

at that point. For each point on the section, we can construct a $2L_t\times 2L_x$ subsection by applying linear moveout corrections to a window of data  

$$P_{ij}=P(t+i\Delta t+jp\Delta x,x+j\Delta x).$$ (2)

where $-L_t \le i \le L_t$ and $-L_x \le j \le L_x$. The symbols $\Delta t$ and $\Delta x$ are the temporal and spatial sampling intervals, respectively. If the local dip p used for the linear moveout correction is equal to the true dip at the location (t,x), then P_ij is composed of horizontal events. Otherwise, the events in P_ij are slanted. With this observation, we can define the objective functions.