Theory of time/depth delays estimation

First consider a data volume d(x,y,z) where x and y are the horizontal axes and z is the depth or time axis. Building on Lomask (2003b), a vertical (time or depth) delay function $\tau(x,y,z)$ is estimated by minimizing the following functional $J(\tau)$: 

$$J(\tau) = \int \int \left[ \left(p_x(x,y,z;\tau)-\frac{\part... ...u)-\frac{\partial \tau}{\partial y}\right)^2 \right] \,dx\,dy,$$ (1)

where p_x is the local step-out vector estimated in the x direction and p_y is the local step-out vector estimated in the y direction. Both vectors depend on $\tau$, which makes the problem of finding the time/depth delays non-linear.

In this paper, we propose solving for $\tau(x,y,z)$ with a quasi-Newton method called L-BFGS Guitton and Symes (2003). The quasi-Newton method is an iterative process where the solution to the problem is updated as follows:

$$\tau_{k+1}=\tau_k - \lambda_k \bold{H}_k^{-1}\nabla J(\tau_k),$$ (2)

where $\tau_{k+1}$ is the updated solution at iteration k+1, $\lambda_k$ the step length computed by a line search that ensures a sufficient decrease of $J(\tau)$ and $\bold{H}_k$ an approximation of the Hessian (or second derivative.) One important property of L-BFGS is that it requires the gradient of $J(\tau)$ only to build the Hessian. The gradient $\nabla J(\tau)$ of $J(\tau)$can be found by introducing the Euler-Lagrange equation and is given by:  

$$\nabla J(\tau)= -\frac{\partial^2 \tau}{\partial x^2} -\fr... ...tial \tau} +\frac{1}{2} \frac{\partial{p_y}^2}{\partial \tau}$$ (3)

The 2-D case is a simple extension of this result where the terms in y are dropped. In practice, the last four terms of the gradient in equation (3) can be precomputed and evaluated at $\tau_k(x,y,z)$ when needed for the BFGS update. This saves a lot of computational effort. Note that the relative vertical (time or depth) delays are computed with respect to a reference trace chosen a priori in the data volume. A weight that would throw-out fitting equations at fault locations can also be incorporated easily in both the gradient and objective function.

The most important components of this time/depth delay evaluation technique are the dip fields p_x and p_y. In our implementation, we use the method of Fomel (2002) to estimate both. This technique estimates local dips from adjacent traces without slant-stacking. It also gives one dip value only for each data point. Next, 2-D and 3-D data examples illustrate the flattening technique.