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Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. We compare the performance of splitting techniques for stable 3D Fourier finite-difference (FFD) migration techniques in terms of image quality and computational cost. The FFD methods are complex Padé FFD and FFD plus interpolation, and the compared splitting techniques are two- and four-way splitting as well as alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. From numerical examples in homogeneous and inhomogeneous media, we conclude that, though theoretically less accurate, alternate four-way splitting yields results of comparable quality as full four-way splitting at the cost of two-way splitting.

Because of its superiority in areas of complex geology, wave-equation migration is substituting Kirchhoff migration in practice. However, while Kirchhoff migration counts on more than 30 years of technological development, wave-equation migration methods still need to be improved in various aspects. One of these aspects is the efficient implementation of three-dimensional wave-equation migration.

The application of a three-dimensional wave-equation migration technique adds the problem of computational cost to those of stability and precision of the chosen migration algorithm. To speed up migration techniques like finite-difference (FD) [

When the splitting is applied to the implicit FD migration operator in such a way that the resulting equations are solved alternatingly in the inline and crossline directions, the resulting FD scheme is known as an alternating-direction-implicit (ADI) scheme. This procedure has the drawback of being incorrect for strongly dipping reflectors, resulting in large positioning errors for this type of reflectors when the dip direction is away from the coordinate directions and thus outside the migration planes. This imprecision leads to numerical anisotropy, that is, a migration operator that acts quite differently in different directions.

To improve this behaviour while retaining the advantages of a rather low computation cost, different procedures have been proposed over the years. Ristow ([

Inverting the idea of Kitchenside [

Collino and Joly [

Zhou and McMechan [

Biondi [

To overcome the problem of instabilities in models with strong lateral velocity contrasts, Biondi [

Another computationally less expensive method to stabilize FFD migration in the presence of strong lateral velocity contrasts was proposed by Amazonas et al. [

In this work, we study possibilities of efficiently implementing these stable FFD migration techniques in 3D. We implemented and compared splitting techniques for FFDPI [

The one-way wave equation [

A well-used possibility for the approximation of the square root in the one-way wave equation (

This approximation is used in most practical FD migration schemes. Depending on the number

However, when the interest is on accurate imaging up to very high propagation angles, approximation (

To overcome this problem, Milinazzo et al. [

The complex Padé approximation (

Joining the two series into one, expanding the fractions into Taylor series, and grouping the terms of equal power leads to

Using this approximation, the one-way wave equation (

As seen above, the theoretical value of

To solve (

Discretizing these differential equations using a Crank-Nicolson FD scheme, we obtain

In the technique called two-way splitting, also known as alternating-directions-implicit (ADI) method [

On the other hand, this procedure also has disadvantages. The biggest one is the introduction of numerical anisotropy into the propagation of the wavefield, because the numerical error increases with the azimuth between the propagation direction and the coordinate directions. This degrades the migrated image, introducing errors in the positioning of steeply dipping reflectors.

To overcome the problems with numerical anisotropy, Ristow and Rühl [

The multiway splitting form of the 2D Padé operators is given by the complex Padé expansion of the square-root operator in (

There are two ways of obtaining the unknown coefficients

In conventional implementations of multiway splitting, operators (

We compare our results of CPFFD migration to another stable FFD migration technique, FFDPI migration [

In 3D, after two-way splitting, the resulting difference equation is approximated by the system

Biondi [

To study the numerical anisotropy of FFD migration operators after splitting, we calculated impulse responses for zero-offset migration in a homogeneous medium with velocity 2.5 km/s. The source pulse was a Ricker wavelet with central frequency 25 Hz, with its center positioned at an arrival time of 1.12 s. The migration grid was

The top part of Figures

Migration impulse responses. (a)–(d): reference result by phase-shift migration using the true medium velocity. (e)–(h): FFD migration using conventional two-way splitting;

Figures

Figures

Figures

Migration impulse responses. (a)–(d): FFD migration using alternating four-way splitting;

Figures

Figures

For a more realistic test of the different splitting techniques for FFD migration, we calculated zero-offset impulse responses for the EAGE/SEG salt model. Here, we used a seismic pulse in the centre of the model, described by a Ricker wavelet with central frequency of 15 Hz, dislocated by

We represent the results by vertical cuts parallel to the

EAGE/SEG salt model. Representation by 4 vertical cuts at

EAGE/SEG salt model. Representation by 4 horizontal cuts at

Figures

Impulse response of FFD migration with two-way splitting. Vertical cuts as in Figure

Impulse response of FFD migration with two-way splitting. Horizontal cuts as in Figure

Impulse response of FFD migration with alternating four-way splitting. Vertical cuts as in Figure

Impulse response of FFD migration with alternating four-way splitting. Horizontal cuts as in Figure

For comparison, Figures

Impulse response of FFDPI migration with two-way splitting using 10 reference velocities. Vertical cuts as in Figure

Impulse response of FFDPI migration with two-way splitting using 10 reference velocities. Horizontal cuts as in Figure

Even for this experiment with 10 reference velocities, we still see some effects of numerical dispersion in Figures

In this paper, we have implemented 3D versions of complex Padé FFD (CPFFD) and FFD plus interpolation (FFDPI), which have proven to be more stable in the presence of strong lateral velocity contrasts than other FFD migration implementations. For CPFFD migration, we have compared the effects of different ways of directional splitting and compared its results to those of FFDPI migration. Alternating four-way splitting, that is, applying the differential operators in the coordinate directions at one depth level and in the diagonal directions at the next depth level, proved to be an improvement over conventional two-way splitting at almost no extra cost. Although this procedure is theoretically less accurate than complete four-way splitting, that is, all four directions applied at all depth levels, our numerical results were of comparable quality. Extensions of the alternating splitting technique can be thought of like eight-way splitting where the remaining directions are covered two by two in the next two depth steps.

From our numerical tests with splitting the CPFFD and FFDPI migration operators, we conclude that FFDPI migration is the most robust of the tested methods. Even implemented only using two-way splitting, it did show only a fair amount of numerical dispersion, but no visible numerical anisotropy. However, for practical use, FFDPI is a rather expensive method because it needs a large number of reference velocities to function with acceptable precision. Thus, for a more economic migration with acceptable image quality, alternating four-way splitting in FFD migration is an interesting alternative.

One minor problem of multiway splitting should be mentioned. The differential operator in the diagonal directions can cause aliasing effects because of the fact that the grid spacing in this direction is by a factor of

This work was kindly supported by the Brazilian agencies CAPES, FINEP, and CNPq, as well as Petrobras and the sponsors of the