Travel times, body waves, and normal modes of the earth

Abstract

abstract This paper describes the physical relationship between the normal mode and ray representations of body waves, and develops equations and experimental procedures for analyzing body waves in terms of normal modes. For the simplest body waves considered, the horizontally-polarized shear waves S H and ( ScS ) H , the following equation relating the travel time curve to the phase velocity versus period curves corresponding to the various normal modes is derived: T = ( t − Δ d t / d Δ ) { I − 1 + [ ( T ) φ 2 − ( T ) φ 1 ] / 2 π } where T is the period corresponding to the phase velocity dΔ/ dt on the dispersion curve for the I th normal mode; t is the travel time corresponding to a distance Δ; and ϕ 1 ( T ) and ϕ 2 ( T ) are the spectral phases of a body wave at two successive points of reflection a distance Δ apart on a given ray path. The group velocity is given by: U = Δ t − { d [ ( T ) φ 2 − ( T ) φ 1 ] / d ω } I = const where ω is the angular frequency, 2π/ T . These equations are derived from a constructive phase interference requirement for successive reflections of body waves. They may also be applied to S v and PS waves over certain phase velocity ranges. T may be determined experimentally from the phase spectra of successive surface reflections of body waves of a given phase velocity (e.g., ScS at a distance Δ and ScSScS at a distance 2Δ). Analysis of S H and ( ScS ) H waves determines points on the torsional higher mode dispersion curves; and analysis of the S v or PS waves determines points on the spheroidal higher mode dispersion curves. Theoretical seismograms of multiply reflected ScS waves and actual seismograms of S , ScS and PS waves are analyzed to test and illustrate the method. Results are given for phase velocities ranging from 8 to 37 km/sec and for mode numbers ranging from 3 to more than 10. The limited results are in agreement with the shear velocity distribution of the Gutenberg model for a Caracas-Halifax path (Atlantic Ocean region) and with that of the Jeffreys model for a Bermuda-Alert path (partly Atlantic Ocean and partly Canadian shield). A method for rapid computation of approximate theoretical dispersion curves, based on the above equations, is presented.