CHOOSING SMOOTHING DIRECTIONS

Prediction-error filters work best on predicting local plane waves Canales (1984); Claerbout (1992c). With non-stationary filters, it is possible to predict data with variable slopes. For preconditioning the filter estimation problem, such filters can be smoothed along the direction where the slope stays locally constant.

To put this principle into a mathematical form, let us denote the monodop data as P(x,y), where x and y are the coordinate values. On a seismic data section, the y coordinate would have the meaning of time, but here we would like to develop a general method that would work on different kinds of data. The local dip field of the data can be defined by the formula  

$$D(x,y) = - \frac{P_x}{P_y}\;,$$ (4)

where P_x and P_y denote the first partial derivatives: $P_x = \frac{\partial P}{\partial x}$, $P_y = \frac{\partial P}{\partial y}$.To validate formula (4), consider a plane-wave model with the slope s:  

$$P(x,y) = P_0(y - s x)\;.$$ (5)

Substituting (5) into formula (4), we can see that the D(x,y) indeed produces an estimate of s Claerbout (1992a). In the general case, D(x,y) corresponds to the tangent of the local plane wave angle, measured from the x axis in the direction of the y axis. Bednar (1997) describes an application of formula (4) for computing coherency attributes.

Instead of using formula (4) explicitly, we intend to estimate prediction-error filters that would destroy local plane waves in the data Claerbout (1992c); Schwab (1998). To precondition the filter estimation problem we can smooth the filters in the direction of the least change in the slope. By analogy with (4), the smoothing direction can be defined as follows:  

$$S(x,y) = - \frac{D_x}{D_y}\;,$$ (6)

or, substituting formula (4),  

$$S(x,y) = - \frac{P_x P_{xy} - P_y P_{xx}}{P_y P_{xy} - P_x P_{yy}}\;,$$ (7)

where P_xx, P_yy, and P_xy are the corresponding second-order partial derivatives. An important analytical test case is a constant-velocity CMP gather, composed of reflection hyperbolas:  

$$P_{\mbox{hyper}}(x,y) = P_0\left(\sqrt{y^2 - s^2 x^2}\right)\;.$$ (8)

Substituting (8) into formula (7) leads to the expression  

$$S_{\mbox{hyper}}(x,y) = \frac{y}{x}\;,$$ (9)

which suggests smoothing the estimated prediction-error filters along radial lines on the $\{x,y\}$ plane Crawley et al. (1998).

 

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Figure 4 Synthetic model from Basic Earth Imaging (left), its estimated dip field (center), and estimated smoothing directions (right).

 

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Figure 5 Seismic shot gather (left), its estimated dip field (center), and estimated smoothing directions (right).

Figure 4 and 5 illustrate a practical application of formulas (4) and (6) on a synthetic reflectivity model from Basic Earth Imaging Claerbout (1995) and on a shot gather from the Yilmaz collection Yilmaz (1987). In both cases the first- and second-derivative operators were computed with simple finite-difference schemes. To avoid a non-stable division in formulas (4) and (6), we solve the regularized least-square system  

$$\left\{\begin{array} {rcl} \bold{D} \bold{x} & \approx & \... ...on \nabla \bold{x} & \approx & \bold{0} \end{array} \right.\;,$$ (10)

where $\bold{D}$ and $\bold{N}$ denote the denominator and the numerator respectively, $\epsilon$ is the scalar regularization parameter, and $\bold{x}$ is the estimated regularized ratio. Our simple two-point finite-difference scheme does not handle correctly the aliased dips on the seismic gather in Figure 5. Nevertheless it produces a reasonable output, which we can use as a rough estimate of the smoothing directions.