DERIVATIVES OF THE TRAVELTIME FIELD

The work considers derivatives of seismic-wave traveltime fields and their application to kinematic problems in seismic exploration. Relations for the first and second derivatives of the traveltime field in inhomogeneous and layered media are derived, together with recurrence formulas that make it possible to recalculate the derivatives along a ray and across layer boundaries. The relationship between traveltime-field derivatives in the two-point source–receiver coordinate system and the corresponding derivatives in Common Depth Point (CDP) coordinates is considered. On this basis, the Normal Incidence Point (NIP) theorem is analyzed, relating the kinematics of reflected waves in CDP observations to waves originating from a reflection point on the boundary. The role of this theorem in the physical and mathematical justification of the CDP method is discussed. The expansion of the traveltime field and of the squared traveltime in a Taylor series up to second-order derivatives is investigated, and the relationship between the expansion coefficients and the propagation direction and curvature of the fundamental-wave wavefronts is established. Dix formulas and their analogues are also considered as a special case of the general scheme for recalculating traveltime-field derivatives in a layered medium. The material represents a self-contained presentation of a chapter devoted to the mathematical foundations of seismic-wave kinematics and is intended for specialists in seismic data processing and interpretation, researchers, graduate students, and students of geophysical disciplines.