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To completely use large incident angle seismic data and improve the accuracy of amplitude-variation-with-offset inversion, especially for density information, we derived second-order approximate expressions of the P- and SV-wave reflection coefficients at the solid/solid media interface. Vectors and Hessian matrices were used to describe the first- and second-order terms of the approximate expressions. The second-order approximations were then rewritten in a simple form. Furthermore, different pseudosecond-order approximations with different power series of the ray parameter p were derived based on the Taylor expansion of the second-order approximation of the P-wave reflection coefficient for the incident P wave (PP-wave reflection coefficient) according to the ray parameter p. By establishing four different single-phase medium models, we verified that the accuracies of the second-order approximate expressions were remarkably higher than those of the first-order approximations. Furthermore, the pseudo p2 second-order approximation of the PP-wave reflection coefficient was highly consistent with the exact analytical expression.

Ordovician carbonate reservoirs in the Shunbei area of the Tarim Basin are characterized by large burial depth, small vertical fault displacement of the strike-slip fault zone, small karst development scale, strong reservoir heterogeneity, complex and diverse seismic reflection types, which lead to unclear seismic response characteristics of fault zone and reservoir, and further restrict the fine description of ultra-deep (>7300 m) fault controlled reservoirs. On the basis of high-precision 3D seismic data analysis, combined with drilling, logging, and comprehensive geological interpretation data in the Shunbei area, seismic geological models of different types of reservoirs and fault zones are established. The changes in seismic response characteristics of fault zones and reservoirs are simulated and analyzed by changing reservoir size, scale, fault and fracture cavity combination modes, as well as other factors. On this basis, two seismic identification models of fault-controlled reservoirs in the Shunbei area are summarized and established. The seismic identification mode of seismic reflection marker wave of fault-controlled reservoir is mainly “fault seismic response characteristics & weak reflection” and red trough anomaly under T74. The internal seismic identification modes are mainly “fault seismic response characteristics & random reflection”, “fault seismic response characteristics & beading”, “fault seismic response characteristics & linear weak reflection”, and “fault seismic response characteristics & random reflection”. Among them, except for “fault seismic response characteristics & random reflection”, the coincidence rate of reservoir prediction of other three types of seismic response characteristics is more than 70%. The coincidence rate of the other three types of seismic response characteristics of a reservoir is more than 70%. The research results provide a reference for the prediction and description of ultra-deep carbonate fault-controlled reservoirs.