The computation of theoretical seismograms for models in which the elastic parameters and density vary only with depth (in a plane, cylindrical or spherical geometry) reduces to the solution of an ordinary differential equation plus the evaluation of inverse transformations. In principle, the problem is straightforward. In practice, many techniques and approximations can be used at each stage and many combinations and variants are possible. In this paper, we discuss a new method of evaluating the inverse transforms. Any method can be used to solve the differential equation and we only discuss a few analytic approximations to illustrate the new method. The inverse transformations are a frequency and wavenumber integral. Essentially four techniques can be used to evaluate these depending on the order of integration and whether the wavenumber integral is distorted from the real axis. Three of these have been widely used, but the technique of evaluating the frequency integral first and keeping the wavenumber real is new. In this paper, we discuss some of the advantages of this combination.

Fundamentals of Seismic Wave Propagation, published in 2004, presents a comprehensive introduction to the propagation of high-frequency body-waves in elastodynamics. The theory of seismic wave propagation in acoustic, elastic and anisotropic media is developed to allow seismic waves to be modelled in complex, realistic three-dimensional Earth models. This book provides a consistent and thorough development of modelling methods widely used in elastic wave propagation ranging from the whole Earth, through regional and crustal seismology, exploration seismics to borehole seismics, sonics and ultrasonics. Particular emphasis is placed on developing a consistent notation and approach throughout, which highlights similarities and allows more complicated methods and extensions to be developed without difficulty. This book is intended as a text for graduate courses in theoretical seismology, and as a reference for all academic and industrial seismologists using numerical modelling methods. Exercises and suggestions for further reading are included in each chapter.

Summary Seismic traveltimes are the most widely exploited data in seismology. Their Fréchet or sensitivity kernels are important tools in tomographic inversions based on the Born or single‐scattering approximation. The current study is motivated by a paradox posed by two seemingly irreconcilable observations in the numerical calculations for the sensitivity kernels of the traveltime perturbations. Calculations of kernels for 2‐D media by the normal‐mode approach indicate that traveltimes are most sensitive to the structure on and around the geometrical ray paths corresponding to the seismic arrivals, whereas calculations for 3‐D media by geometrical ray theory predict exactly zero sensitivities on the ray paths. In the current work, we employ these two completely different wave‐propagation approaches, the more efficient geometrical ray theory and the more accurate normal‐mode theory, to investigate the 3‐D sensitivities of the delay times to shear‐wave speed variations. Expressions for the delay‐time Fréchet kernels are presented for both methods, and extensive numerical experiments are conducted for various types of seismic phases as well as for different reference earth models. The results show that the contradictory observations are but two examples of a wide range of behaviours in the delay‐time sensitivity. For most of the seismic phases in realistic reference models with multiple discontinuities, wave‐speed gradients and low‐velocity zones, the wavefields are highly complicated and ray theory, which describes the response by the contributions of a few geometrical rays between the source and receiver, produces qualitatively different delay‐time kernels from those obtained by the normal‐mode theory, which includes essentially all contributions.

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