Dip Estimation

To estimate two local slopes p₁ and p₂, we treat vector $\bf r$ 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:

$$\sum \bold D_{xx} \bold D_{xt} = \sum_{i=1}^n \bold D^i_{xx} \bold D^i_{xt}.$$

Expanding $\bold r^T \bold r$ from equation (10) and collecting terms yields a nonlinear function of p₁ and p₂, which we denote Q(p₁,p₂):

$$\begin{eqnarray} Q(p_1,p_2) &=& \sum \bold D_{xx}^2 + p_1 \cdot 2\sum \bold ... ...bold D_{xt} \bold D_{tt} + p_1^2 p_2^2 \cdot \sum \bold D_{tt}^2.\end{eqnarray}$$ (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.

$$\begin{eqnarray} \frac{\partial Q(p_1,p_2)}{\partial p_1} = f(p_1,p_2) &=& \sum ... ...1^2 \bold D_{xt} \bold D_{tt} + p_1^2 p_2 \sum \bold D_{tt}^2 = 0\end{eqnarray}$$ (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:

$$\begin{eqnarray} p_{1,k+1} &=& p_{1,k} + \frac{ -f(p_{1,k},p_{2,k}) g_{p_2}(p_{1... ...,k},p_{2,k}) -f_{p_2}(p_{1,k},p_{2,k}) g_{p_1}(p_{1,k},p_{2,k}) }\end{eqnarray}$$ (14) (15)

The estimated slopes at iteration k are p_1,k and p_2,k. f_p1(p_1,k,p_2,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 (p_1,0,p_2,0), and furthermore, may converge to a local minimum. In practice, however, the method converges to machine precision within 3-5 iterations.