Summary. 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.

A new method for computing synthetic seismograms

IN view of the number of different lines of approach, a borderline subject is apt to amass a vast literature, some of it difficult of access to investigators This is particularly true of seismology ; the phenomena of earthquakes have attracted the attention of many famous geologists, and the recording of earthquake tremors involves the methods of experimental physics, while the interpretation of the records of earthquakes has provided a great stimulus to the development of the theory of the propagation of disturbances in elastic media and has ultimately thrown considerable light on the composition of the interior of the earth Many books have been written on seismology in its various aspects, but many of the most interesting modern geophysical developments have hitherto been available only in the original papers It is therefore of considerable interest to read the elegant exposition that Prof K E Bullen has presented in the book under review An Introduction to the Theory of Seismology By Prof K E Bullen Pp xiv + 276 (Cambridge: At the University Press, 1947) 18s net

An Introduction to the Theory of Seismology

Seismic data fron the four stations of the Apollo passive seismic network have been analyzed to obtain the velocity structure of the moon. Analysis of body wave phases from artificial impacts of known impact time and position yields a crustal section. In the Mare Cognitum region the crust is about 60 km thick and is layered. In the 20-km-thick upper layer, velocity gradients are high and microcracks may play an important role. The 40-km-thick lower layer has a nearly constant 6.8-km/sec velocity. There may be a thin high-velocity layer present beneath the crust. The determination of seismic velocities in the lunar mantle is attempted by using natural impacts and deep moonquakes. The simplest model that can be proposed for the mantle consists of a 'lithosphere' overlying an 'asthenosphere'.

Structure of the Moon

We have developed a method for estimating travel times as a function of epicentral range and hypocentral depth which is faster than ray tracing or direct evaluation of ray integrals yet more compact and general than the interpolation of traditional travel-time tables. In this method, delay or intercept time (tau) as a function of ray parameter and source depth is tabulated. We show that direct manipulation of delay time yields travel time as an explicit function of range eliminating the iteration required to determine the proper ray parameter in ray tracing or the evaluation of ray integrals. Travel-time versus range tables exhibit the same explicit dependence by virtue of having eliminated the common independent variable, ray parameter. However, delay-time branches are monotonic, single-valued, and have fixed ray parameter limits. In contrast, travel-time branches may be nonmonotonic and multi-valued and generally have range limits which vary with source depth. Thus, delay-time tables are simpler to generate and interpolate than are travel-time tables and so result in travel-time estimates which are both more reliable and more precise for a given table size. Further, by retaining the explicit ray parameter dependence and hence a more direct relationship with the ray integrals, the delay-time method offers more flexibility in organizing suites of related phases and in tailoring the algorithm to meet specific computational requirements than does the travel-time table approach. Additionally, we show how the delay-time method can be extended to laterally inhomogeneous media.

The computation of seismic travel times

The usual methods of interpolating velocity-depth profiles cause discontinuities in the velocity gradient and anomalous geometrical ray amplitudes. Numerical methods are presented for interpolating a velocity profile and evaluating the ray integrals which avoid these difficulties. They allow smooth amplitude curves to be calculated directly. An example is given for the Herrin P velocity profile.

On the computation of seismic ray travel times and amplitudes

The “shadow” from an Epstein “sech 2 ” profile is caused by exponential decay of amplitude. The decay constant, which is independent of frequency, is identical with that obtained from the classical ray theory. The spectral amplitudes and theoretical seismograms evaluated numerically show the effect of the reversal on the acoustic wave propagation.

The “shadow” from a velocity reversal

Some numerical experiments are performed in the application of the classical Herglotz-Wiechert method to the inversion of discrete travel-time data. The problem is restricted to the simplest form of travel-time curve and it is illustrated how these restrictions can be included in the more general methods developed recently. An envelope containing all possible models is obtained. The velocity model can be averaged over a range of ray parameters. The envelope of the averaged models (which is different from the averaged envelope of models) is dependent on the averaging window. Hence, a trade-off curve is obtained between the averaging window and the envelope width (velocity uncertainty).

C. H. Chapman

Papers

On the computation of seismic ray travel times and amplitudes

The “shadow” from a velocity reversal

Some numerical experiments on the inversion of discrete travel-time data