WAVE FIELD EXTRAPOLATION TECHNIQUES FOR INHOMOGENEOUS MEDIA WHICH INCLUDE CRITICAL ANGLE EVENTS. PART I: METHODS USING THE ONE-WAY WAVE EQUATIONS*

WAPENAAR, C.P.A. and BERKHOUT, A.J. 1986, Wave Field Extrapolation Techniques for Inhomogeneous Media which Include Critical Angle Events. Part 11: Methods Using the Two-way Wave Equation, Geophysical Prospecting 34, 147-179. In one-way wave field extrapolation downgoing and upgoing waves are treated independently, which is allowed if propagation at small angles against the vertical in (weakly) inhomogeneous media is considered. In practical implementation the slow convergence of the square-root operator causes numerical deficiencies. On the other hand, in two-way wave field extrapolation no assumptions need to be made on the separability of downgoing and upgoing waves. Furthermore, in practical implementation the use of the square-root operator is avoided. To put the two-way techniques into perspective, it is shown that two-way wave field extrapolation could be described in terms of one-way processes, namely: (1) decomposition of the total wave field into downgoing and upgoing waves; (2) one-way wave field extrapolation; (3) composition of the total wave field from its downgoing and upgoing constituents. This alternative description of two-way wave field extrapolation is valid for media which are homogeneous along the z-coordinate as well as for small dip angles in arbitrarily inhomogeneous media. In addition, it is shown that this description is also valid for large dip angles in 1-D (vertically) inhomogeneous media, including critical-angle events, when the WKBJ one-way wave functions discussed in part I of this paper are considered. For large dip angles in arbitrarily inhomogeneous media the two-way wave equation is solved by means of Taylor series expansion. For practical implementation a truncated operator is designed, assuming gentle horizontal variations of the medium properties. This operator is stable and converges already in the first order approximation, also for critical-angle events.

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