Three-dimensional seismic depth migration

One-pass 3-D modeling and migration for poststack seismic data may be implemented by replacing the traditional 45{dollar}spcirc{dollar} one-way wave equation (a third-order partial differential equation) with a pair of second and first order partial differential equations. Except for an extra correction term, the resulting second order equation has a form similar to Claerbout's 15{dollar}spcirc{dollar} one-way wave equation, which is known to have a nearly circular horizontal impulse response. In this approach, there is no need to compensate for splitting errors. Numerical tests on synthetic data show that this algorithm has the desirable attributes of being second-order in accuracy and economical to solve. A modification of the Crank-Nicholson implementation maintains stability.; Absorbing boundary conditions play an important role in one-way wave extrapolations by reducing reflections at grid edges. Clayton and Engquist's 2-D absorbing boundary conditions for one-way wave extrapolation by depth-stepping in the frequency domain are extended to 3-D using paraxial approximations of the scalar wave equation. Internal consistency is retained by incorporating the interior extrapolation equation with the absorbing boundary conditions. Numerical schemes are designed to make the proposed absorbing boundary conditions both mathematically correct and efficient with negligible extra cost. Synthetic examples illustrate the effectiveness of the algorithm for extrapolation with the 3-D 45{dollar}spcirc{dollar} one-way wave equation.; Frequency-space domain Butterworth and Chebyshev dip filters are implemented. By regrouping the product terms in the filter transfer function into summations, a cascaded (serial) Butterworth dip filter can be made parallel. A parallel Chebyshev dip filter can be similarly obtained, and has the same form as the Butterworth filter; but has different coeffcients. One of the advantages of the Chebyshev filter is that it has a sharper transition zone than that of Butterworth filter of the same order. Both filters are incorporated into 3-D one-way frequency-space depth migration for evanescent energy removal and for phase compensation of splitting errors; a single filter achieves both goals. Synthetic examples illustrate the behavior of the parallel filters. For a given order of filter, the cost of the Butterworth and Chebyshev filters is the same. A Chebyshev filter is more effective for phase compensation than the Butterworth filter of the same order, at the expense of some wavenumber-dependent amplitude ripples.; An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis. Under this expression, geometrical spreading can be determined only by the anisotropic parameters in the first layer, the traveltime derivatives, and source-receiver offset. An explicit, numerically feasible expression for geometrical spreading can be further obtained by considering some of the special cases of transverse isotropy, such as weak anisotropy or elliptic anisotropy. Therefore, with the techniques of non-hyerbolic moveout for transverse isotropic media, geometrical spreading can be calculated by using picked traveltimes of primary P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading.