Long-Period Surface Wave Seismology: Love Wave Phase Velocity and Polar Phase Shift

Abstract

Due to continuing improvement in instrumentation and data processing techniques, increasingly longer periods are being obtained in seismic surface wave measurements. Three aspects of surface wave propagation on a sphere, which become important at long periods, are treated in some detail: (1) the dependence of the spherical phase velocity on the relative positions of epicentre and station, (2) the importance of the polar component of Love waves, relative to the azimuthal component, and (3) the accuracy of the common practice of treating surface wave data as though they represent a continuous-frequency phenomenon.The original work on polar phase shift theory (Brune et al.) predated derivations of the response of a sphere to realistic models of the earthquake mechanism. We update the original theory by basing the development on these later derivations.I and em on I and 8. Long-period surface wave seismology pm(j3 0) = arg (&?/marg (Gm). 41 1 and define (2.15) Also, from the first term of the expansion for xp, for large 1, we have rm t

Measurements of mantle wave velocities and inversion for lateral heterogeneities and anisotropy: 3. Inversion

“…At long period the approximation made in NA2 that the phase velocity is sampled uniformly over the great circle path, the zeroth-order approximation, breaks down. The first-order approximation has been derived by Schwab and Kausel (1976) for Love waves and by Wielandt (1980) for Rayleigh waves. The first-order correction increases with the increase of period and with the decrease of distance between receiver and poles and depends on the source focal mechanism.…”

Regionalization of the Arctic Region, Siberia and Eurasian Continental Area

“…For a treatment of the situations which require the use of (2.03), (2.04), or their sum, see Schwab and Kausel (1976). In this same reference, the justification is given for our major departure from previously reported computations of Rayleigh wave dispersion on a spheret Strictly speaking, Rayleigh waves only exist on a sphere at the discrete set of frequencies corresponding to integral values of the polar order number 2 .…”

“…A summary of the general methods we have applied in our computations is given by Knopoff et al (1974). An elaboration of, and certain justifications for these procedures have recently been given by Schwab and Kausel (1976 --!1 contract, are contained in Nakanishi, Schwab and Kausel (1976), and Nakanishi, Schwab and Knopoff (1976). These manuscripts formed the necessary bridge between first generation and second generation dispersion programs.…”

“…However, fixing 9'and evaluating the corresponding angular frequency w does not yield the dispersion data at equal frequency intervals, which we desire to use in the usual numerical technique for obtaining time series by inverse Fourier transformation. Schwab and Kausel (1976) have shown that, for most practical applications of propagating surface waves, non-integral Z at equally-spaced frequencies…”

“…This computation corresponds to about 50,000 free oscillating modes. Theoretical seismograms in the time domain are then obtained by inverse Fourier transformation as described by Kausel and Schwab (1973) and Schwab and Kausel (1976). The cost of computing the volume of frequency-domain information, which is required to construct the synthetic seismograms down to a period as short as 10 sec, has been prohibitively high up to the present time: but, with our current optimization we have succeeded m reducing the computation time to about 6 min on our IBM 360/91 computer (an expenditure of about $50) for a given earth structure.…”