Objective function and solution

We can apply the approximation in equation (10) to the objective functions. For example, from equations (3) and (7) we can derive  

$$\begin{array} {lll} E^0(t,x,\Delta p) & = & \displaystyle{\s... ...rtial t}j\Delta p \Delta x \right)\right]^2 \right\}\end{array}$$ (11)

The optimal estimate of $\Delta p$ is the minimizer of this function. Because the objective function is a quadratic function of the unknown $\Delta p$, one can use the standard least-squares techniques to find the solution of this linear optimization problem:

$$\Delta p = \displaystyle{\displaystyle{\sum^{L_t}_{i=-L_t}}W... ...ystyle{\partial P \over \partial t}j\Delta x \right)^2 \right]}$$ (12)

Once $\Delta p$ is found for each t, the pick can be improved by using equation (9).