Seismic interpretation theory for an elastic earth

Abstract
The seismic interpretation problem for an isotropic spherical earth is analyzed on the basis of elastic theory, under the assumption that the three independent elastic parameters are unknown continuous functions of the depth. It is shown that solutions for these functions may be obtained in the form of Taylor’s series. The problem is treated for three types of symmetrical excitation conditions on the free surface: (1) a shear source of type p rϕ only; (2) a pressure distribution with vanishing surface shear stress; (3) an excitation consisting of pressure in combination with surface shear stress of type p rθ . In each case the excitation functions are arbitrary functions of time. It is assumed that the associated components of surface displacement over the sphere are known from available observations, as functions of time. Thus, the complete information contained in seismic records is used in the proposed interpretation process, without need of selecting, identifying and assigning arrival times to specific events on the records. The two static elastic parameters may theoretically be determined from observations at a single frequency, including the frequency zero, or static case. The determination of the dynamic elastic parameter requires the use of at least two frequencies. Algebraic checks are obtained by comparing the general solutions with the corresponding results for two special cases in which the elastic parameters vary in a prescribed manner in the interior of the sphere. In both these cases treatment by the classical ray-path method of interpretation is excluded, because the wave velocity decreases with depth. Furthermore, the ray-path method (which is essentially a method of geometrical optics) would fail to distinguish between the two examples in any case, since the velocity function is the same in both, although the elastic parameters differ. In contrast to the valuable ray-path method, the analytical procedures in the present solution of the elastic problem are prohibitively cumbersome. Practical application of elastic theory to the direct interpretation of seismograms requires further development of the theory with probable utilization of modern high-speed computing methods.