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Summary: To properly characterize the Z Sandstone reservoir in Saudi Arabia it is imperative to determine how depth trends affect the amplitude variation with offset (AVO) signature. Rock physics using Hertz-Mindlin theory and the Modified Upper and Lower Hashin-Shtrikman bounds is used to model the effects of mechanical (uncemented) and chemical (cemented) compaction. Uncemented sands portray a Class IV AVO anomaly while cemented sands exhibit a Class III anomaly. The transition depth from uncemented to cemented sands is controlled by the regional geothermal gradient. The presence of cement stiffens the rock, which reduces the porosity, and therefore increasing the velocity. On the AI-Vp/Vs ratio crossplot, the onset of cementation substantially diminishes our ability to discriminate between brine and gas saturated Z Sands. This is caused by the shift in the Vp/Vs ratio for brine filled sands to values similar to gas bearing sands. Zones that are affected only by mechanical compaction (uncemented) exhibit a clearer separation in the Vp/Vs ratio for different pore fluids. Due to the complexity and heterogeneity of the Z Sands, these depth dependent AVO trends are used to constrain amplitude variation with angle (AVA) simultaneous inversion, thus reducing the ambiguity in interpreting the inversion results.

The gravity effect produced by two and three‐dimensional bodies with nonuniform density contrast has been treated by several authors. One of the first attempts in this direction made by Cordell (1973), who developed a method to compute the gravity effect due to a two‐dimensional prism whose density decreases exponentially with depth. A different approach was proposed by Murthy and Rao (1979). They extended the line‐integral method to obtain the gravity effect for bodies of arbitrary cross‐sections, with density contrast varying linearly with depth. Chai and Hinze (1988) have derived a wavenumber‐domain approach to compute the gravity effect due to a vertical prism whose density contrast varies exponentially with depth. Recently, Rao (1990) has developed a closed expression of the gravity field produced by an asymmetrical trapezoidal body whose density varies with depth following a quadratic polynomial.

Summary Properly controlled migration amplitudes are essential in amplitude versus offset analysis. Usually this is achieved by incorporating the amplitude weight factors in the Kirchhoff summation. This involves building tables of in-plane and out-ofplane spreading terms, takeoff and emergence angles for each source and receiver pair in addition to the traveltime tables. In practice, these tables are computed in advance when traveltime tables are generated. However, this increases retrieval time significantly. An alternative approach is to compute the traveltimes analytically and compute the amplitude weights on the fly. Along this line of approach, we present approximate amplitude weights that produce high quality images with less cost. We approximate the true amplitude weights for v(z) media by the weights for constant-velocity media, but accurately compute the takeoff and emergence angle terms that would otherwise contribute to a large error. The method is illustrated by synthetic data containing flat and dipping reflectors in depth dependent velocity media.

Fast frequency‐wavenumber migration for a constant‐velocity medium is generalized for a medium with depth‐dependent velocity and density. As in the constant‐velocity case, the migrated section is obtained with one inverse Fourier transformation of the modified spectrum of the observed wave field. The migrated section is exact in a preselected focusing depth (in the case of common‐midpoint and plane‐wave migration) or in a preselected focusing point (in the case of single‐shot migration) and in the vicinity of this depth or point. Elsewhere the migrated section is distorted. The method is in essence a constant‐velocity migration with the velocity [Formula: see text] at the target depth [Formula: see text] (not with an average velocity), extended by a phase‐shift operation which takes account of the true velocity‐depth function v(z) between z = 0 and [Formula: see text]. If the focusing depth or point is located in a layer with constant velocity and density, the migrated section is exact throughout the layer (for CMP and plane‐wave migration) or along the segment of the ray from shot to focusing point, running through the layer (for single‐shot migration). Examples are given for the successful migration of the plane‐wave response of a horizontally layered medium at vertical and oblique incidence and for the single‐shot response of a horizontally layered structure with embedded reflector elements. The method appears to be particularly useful for the investigation of subsurface structures with restricted dimensions; because of its speed, interactive data interpretation is possible.

We have studied scattered pulse shapes by modeling inhomogeneities as a sequence of infinitesimally thin homogeneous layers. With oblique incidence of plane P or SV waves, the reflected‐converted‐transmitted waves are obtained by taking the calculus limit for the sum of primary interactions of the incident wave with all layer boundaries. The resulting scattered waves thus present themselves naturally in the time domain. For an incident impulse, the scattered pulse shape is merely an analytic function of the depth from which scatter has taken place within the inhomogeneity. The direct application of this simple method appears to be new, and we have found it remarkably accurate when compared with methods in which higher‐order boundary interactions are also retained (i.e., Haskell methods and an adaptation in the time domain which also keeps all multiples). In specific studies of P-waves incident (up to 30 degrees away from the vertical) upon a 5 km thick crust‐mantle transition, between materials having impedance ratio 1:2.8, we find the scattered pulse shapes are given adequately by our theory, for the passband of short‐period seismometers. Indeed, the theory remains remarkably accurate even for long periods, being in error by only 8 per cent at zero frequency.