Although this is for one horizon, it will be useful to solve this equation for all horizons simultaneously. This would allow the problem to be regularized and its solution to be truly global. In particular, it would no longer be necessary to iterate to remove residual structure in cases of the dip changing with time.
The Euler-Lagrange equation is used in calculus of variations Farlow (1993) to find a function (t) that will minimize a functional (J). It is analogous to the calculus expression for finding x to minimize the function f(x) in:
Figure a is a cartoon image of the horizon to be flattened. Figure b shows the dip field of that horizon. Figure c is the output time field. In this case, we are looping over the output field. Each value in Figure c contains time values of where to sample the data in order to flatten it. Figure d shows the time shift values from the output time field for this horizon.
Equation (13) is non-linear and therefore tricky to solve because dip functions,px and py, are dependent on the unknown, t, and the last term is taking a derivative with respect to the unknown, t.
Incidentally, dropping the last term from equation 13, the equation can be simplified to: