Optimal estimation of scalar seismic moment

For any seismic source specified by a frequency-dependent moment-rate tensor M(ω), we define the total moment MT(ω)=∥M∥/√2 and the isotropic moment MI(ω)=|trM|/√6. A method is presented for estimating these scalar seismic moments from noisy seismic data when the source mechanism is uncertain or completely unknown. Our formulation exploits the linear relation between squared moment M2 (total or isotropic) and the product of two seismic spectra; in a particular frequency band, an estimate is constructed as a linear combination of power and cross-spectra integrated across the band. The coefficients yielding an exact estimate from perfect data are the solution to a linear system of equations involving spectral integrals of the transfer functions that relate M to the seismograms. The failure to solve this system exactly induces an error in whose statistics can be calculated from a likelihood function for the source mechanism, which we model using the hyperspherical normal distribution and its Gaussian approximation and generalizations. We also develop expressions for the bias and variance induced in by ambient seismic noise and by transfer-function errors due to aspherical heterogeneity. To optimize the estimate, the coefficients specifying are computed by minimizing a non-negative-definite quadratic form constructed from these statistics. We have applied the method to IDA records of the deep-focus Honshu earthquake of 1978 March 7 and the shallow-focus Oaxaca earthquake of 1978 November 29. For each event, estimates of MT have been obtained with good precision over disjunct 1-mHz bands spanning the frequency interval 1–11mHz; their relative standard deviations range from 10 to 22 per cent. Our best estimate of MT averaged over the entire 1–11mHz interval is 0.43 × 1027 dyne cm for Honshu and 2.8 × 1027 dyne cm for Oaxaca. The isotropic component of the Oaxaca event, as measured by the ratio , is negligibly small (< 0.1). In the case of Honshu, however, this ratio averages about 0.34; all six estimates at frequencies greater than 5 mHz are significantly greater than zero at the 90 per cent confidence level, and four of the six are significant at the 95 per cent level. This observation lends credence to the conjecture made previously by seismologists that isotropic compression accompanies some deep-focus earthquakes.