## A combined kinematic-dynamic method for fast computations of the
first-arriving waveforms

**Michal Kvasnicka** ** & **
**Jiri Zahradnik**
### Introduction

Ray and grid travel time methods are powerful for kinematic modelling, i.e. for
calculations of the arrival times and rays of seismic body waves.
Elastic finite-difference (FD) methods belong to the most
powerful tools for computing complete waveforms in complex
subsurface structures. However, the FD methods are rather
time-consuming, and the physical interpretation of the FD results
is difficult, in general.

These facts call for a joint application of the two approaches.
The intention of this paper is to show how the two methods, in
a combination, improve our understanding of the wave propagation
processes in complex subsurface structures.

The most important part of the paper is an extensive
investigation of a method allowing an efficient (accurate and
fast) calculation of the first-arrival waveforms. The idea is
very primitive, and consists of eliminating the unnecessary FD
calculations of "zeros" before the true physical arrival of
a wave. Such an approach has been already mentioned as "expanding
grid" by Boore (1972), and most recently re-introduced in
seismology under the name of "speed-up" by Vidale (1988).
Although the method is not new, a detailed
discussion of its practical aspects is lacking in the literature.
Therefore, the main purpose of this paper is to summarize our
experience with coding and testing the speed-up method. We
describe how our previously developed codes for the ray and FD
modelling are technically linked together, and focus on the
following issues: free parameters of the speed-up method (the
time window around the first arrival), accuracy of the resulting
waveforms, and efficiency of the method. By the efficiency we
mean the computer-time saving measured with respect to the full
FD calculation. The performance of the method is illustrated on
model PICROCOL (1994), extended in this paper from the acoustic
to elastic case.

**Related: **
Examples.

### Whole paper

The paper is available in
PDF format (10 635 kB !!!).

In: Seismic Waves in Complex 3-D Structures, Report 4,
pp. 251-286, Dep. Geophys., Charles Univ., Prague, 1996.

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