The L-BFGS-B algorithm

Alternatively, we can use the L-BFGS-B algorithm for imposing very tight constraints on the picked values of the tau field. The L-BFGS-B algorithm seeks to find a vector of model parameters ${\bf \tau}$ such that we minimize  

$$\mbox{ min } f({\bf \tau})\mbox{ subject to }{\bf \tau}\in\Omega,$$ (18)

where

$${\bf \tau}\in\Omega = \{ {\bf \tau} \in \Re^N\mid l_i\leq \tau_i\leq u_i\},$$ (19)

with l_i and u_i being the lower and upper bounds for the model $\tau_i$, respectively. In this case, l_i and u_i are called simple bounds. For the flattening technique, we want to minimize Guitton et al. (2005b); Lomask (2003b)  

$$f(\tau) = \int \int \left[ \left(p_x(x,y,z;\tau)-\frac{\part... ...u)-\frac{\partial \tau}{\partial y}\right)^2 \right] \,dx\,dy,$$ (20)

The L-BFGS-B algorithm combines a quasi-Newton update of the Hessian (second derivative) with a trust-region method. It has been successfully applied for flattening Guitton et al. (2005b) and dip estimation Guitton (2004).

Incorporating the initial ${\bf \tau}$ field is trivial with the L-BFGS-B method: we simply set the bounds where an a-priori value exists:

$$\begin{eqnarray} l_i & = & \tau_{0_i}-\alpha \times \tau_{0_i} \ u_i & = & \tau_{0_i}+\alpha \times \tau_{0_i}, \end{eqnarray}$$ (21) (22)

where $\alpha$ is a small number ($\approx 0.001$). Note that the L-BFGS-B algorithm allows us to optionally activate the constraints for every point of the model space. Note that in equation 20, the objective function incorporates a smoothing in the vertical direction of the ${\bf \tau}$ field as well.

Although not shown in this paper, the results of the L-BFGS-B algorithm are comparable to the Gauss-Newton approach. However at this stage, the L-BFGS-B algorithm converges faster than the Gauss-Newton technique with preconditioning.