Irregular-geometry Dip Estimation Methodology

Figure illustrates a simplified conceptual model of seismic data. Given two traces, we assume that a seismic event passes through each trace location at times t₁ and t₂, and that the event takes the form of a plane in the neighborhood of the two traces.

Imagine that we wanted to measure t₂-t₁, given the local dip of the plane, p_x and p_y. The total time shift is simply the sum of the time shifts along the x and y planes, going first from x₁ to x₂ and then from y₁ to y₂. We can write an equation directly:

 

t₂-t₁ = p_x(x₂-x₁) + p_y(y₂-y₁) (1)

However, in actuality we do not know p_x and p_y, but using Claerbout's puck method, we can measure t₂-t₁. Implemented on a computer, the puck method computes the time shift (in pixels) between two traces that optimally aligns a seismic event on the two traces as a function of time. In other words, the method measures  

$$p_{21} = \frac{t_2-t_1}{\Delta t},$$ (2)

where $\Delta t$ is the time sampling of the traces. We can now rewrite equation accordingly:  

$$p_{21} = \frac{p_x(x_2-x_1)}{\Delta t} + \frac{p_y(y_2-y_1)}{\Delta t}$$ (3)

Equation (3) describes the linear relationship between the computer dip measured between traces 2 and 1, p₂₁, and the local 3-D reflector dip. We require two equations to obtain a unique estimate of the parameters, but the noise and incoherency inherent to real data make it desirable to use more than two traces and exploit the statistical ``smoothing'' of least-squares estimation.

If trace 1 is the ``master'' trace and traces 2 through n its neighbors, let us define the forward modeling operator $\bold A$ 

$$\bold A = \left[ \begin{array} {cc} \frac{x_2 - x_1}{\De... ...}{\Delta t} & \frac{y_n - y_1}{\Delta t} \end{array} \right]$$ (4)

and the data vector $\bold d$ 

$$\bold d = \left[ \begin{array} {c} p_{21} \\ p_{31} \\ \vdots \\ p_{n1} \end{array} \right].$$ (5)

The estimated p_x and p_y at trace location 1 are then the solution to the normal equations:  

$$\left[ \begin{array} {c} p_x \\ p_y \end{array} \right] = \left( \bold A^T \bold A \right)^{-1} \bold A^T \bold d$$ (6)

Inversion of the $2 \times 2$ matrix $\bold A^T \bold A$ is trivial. The result of the process is a pair of dip measurements (p_x and p_y) at each trace location. These dip measurements must be interpolated to fill the entire model grid. In this paper, I use an expanding-window smoothing algorithm to accomplish the task. If reflectors are continuous, their dips should be somewhat smooth in space, so to some extent, spatial smoothing is justified.