At this point a discussion of steering filters is appropriate. Plane waves with a given slope on a discrete grid can be predicted (destroyed) with compact filters Schwab (1997). Inverting such a filter by the helix method, we can create a signal with a given arbitrary slope extremely quickly. If this slope is expected in the model, the described procedure gives us a very efficient method of preconditioning the model estimation problem, fitting goal (2).

How can a plane prediction (steering) filter be created? On the helix surface,
the plane wave *P*(*t*,*x*) = *f* (*t* - *p x*) translates naturally into a
periodic signal with the period of , where *N*_{t} is
the number of points on the *t* trace, and , where is the plane slope,
^{}
and and correspond to the mesh size.
If we design a filter that is two columns long
(assuming the columns go in the *t* direction), then the *plane
prediction* problem is simply connected with the
*interpolation* problem: to destroy a plane wave, shift the
signal by *T*, interpolate it, and subtract the result from the
original signal. Therefore, we can formally write

(6) |

Different choices for the operator in (6)
produce filters with different length and prediction power.
A shifting operation corresponds to the filter with the *Z*-transform
, while the operator corresponds to an
approximation of with integer powers of *Z*. One possible
approach is to expand using the Taylor series around
the zero frequency (*Z*=1). For example, the first-order approximation
is

(7) |

(8) |

(9) |

(10) |

Figure 2

The second-order Taylor approximation yields

which corresponds to the 2-D filter

(12) |

(13) |

If instead of Taylor series in *Z*, we use a rational (Padè)
approximation, the filter will get more than one coefficient in the
first row, which corresponds to an implicit finite-difference scheme.
For example, the [1/1] Padè approximation is

(14) |

(15) |

Figure 3

Other types of interpolations could be used for the steering filters Fomel (1997b) The two bottom panels of Figure 3 show the impulse responses for the filters, based on the tapered sinc interpolation. The filters suffer from high-frequency oscillations, but otherwise also perform well.

It is interesting to note that a space-variant convolution with
inverse plane filters can create signals with different shape, which
remains planar only locally. This situation corresponds to a variable
slowness *p* in the one-way wave equation (9). Figure
4 shows an example: predicting hyperbolas with a 7-point
Lagrangian filter.

Figure 4

10/9/1997