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## Dip Estimation

To estimate two local slopes p1 and p2, we treat vector in equation (10) as a familiar prediction error, and find the p1 and p2 which minimize the squared norm of the prediction error. First we define the following shorthand:

Expanding from equation (10) and collecting terms yields a nonlinear function of p1 and p2, which we denote Q(p1,p2):
 (11)
To find the least-squares-optimal p1 and p2, we compute the partial derivatives of Q(p1,p2), set them equal to zero, and solve a system of two equations.
 (12) (13)
I use Newton's method for two variables to compute the optimal slopes by updating estimates of p1 and p2 with the following iteration:
 (14) (15)
The estimated slopes at iteration k are p1,k and p2,k. fp1(p1,k,p2,k) is, for example, the partial derivative of f(p1,p2) with respect to p1. While intimidating, equations (14) and (15) result simply from the inversion of a 2-by-2 matrix of second derivatives (of Q(p1,p2)), the so-called Hessian matrix. Since, the partial derivatives of f and g are non-constant, the problem is non-quadratic, which implies that Newton's method may diverge for certain initial guesses (p1,0,p2,0), and furthermore, may converge to a local minimum. In practice, however, the method converges to machine precision within 3-5 iterations.

Next: Slope estimation tests Up: The Method Previous: Discretizing the problem
Stanford Exploration Project
11/11/2002