Coupled-mode surface wave seismograms

We extend and implement the variational method for calculating accurate long-period surface seismograms on a smooth laterally heterogeneous earth. This method accounts fully for multiplet coupling along a dispersion branch due to an elastic perturbation in three-dimensional model parameters. The effect of an anelastic perturbation on multiplet coupling is investigated and found to be insignificant. The subspace projection method, which approximates the time-consuming variational method using the coupling strength as a small parameter, is enhanced to include the second order corrections. With these accurate and efficient methods, we examine the accuracy of the path-integral approximation for forward modelling of surface waveforms and for finding global three-dimensional velocity structure of the earth's upper mantle. This path-integral approximation produces less than 4% misfit variances of phase anomalies of Rayleigh wavepackets R1-R3 calculated by summing fundamental coupled modes between 100-600 s for model M84A + 1066A. Phase anomalies measured on coupled-mode wavepackets R1-R3 are inverted for a degree-8 phase velocity model in the period range 119-235 s. This inversion achieves more than 98% variance reduction in data; the correlation between input and output models in 0.99, and the amplification is in the range 0.96-1.03. We demonstrate that source amplitude variations, due to the deviation in takeoff azimuth of the ray at the source, can be of comparable magnitude to the amplitude anomalies caused by geometric focusing and defocusing. Amplitude anomalies measured on coupled-mode waveforms for R1-R3 are also inverted using the expression including source and focusing effects valid to first order in the lateral heterogeneity. This inversion yields 81-94% variance reduction with the model correlation in the range 0.92-0.97 and the amplication in the range 0.97-1.04. These results suggest that if lateral heterogeneity of the earth's upper mantle is dominated by long-wavelength structure similar to that of model M84A, tomographic inversion of phase anomalies is accurate and even inversion of amplitude anomalies might be possible.