Abstract
Two methods are investigated for estimating the phase velocity of diffusely scattered seismic waves simultaneously arriving from different azimuths and recorded by a two-dimensional array of seismometers. The Hankel spectrum is the average of the frequency-wavenumber (FK) spectrum over all azimuths, while the wavenumber spectrum is derived by integrating the FK spectrum around a contour of constant phase velocity, i.e., a circle centered on the origin in the wavenumber plane. If the conventional estimate of the FK spectrum using the covariance matrix of the seismometer signals is integrated, a closed form for both the Hankel spectrum and the wavenumber spectrum may be found; the two spectra are very similar, the wavenumber spectrum being equal to the Hankel spectrum times the wavenumber. In spite of this similarity, however, we find that the two formulations have significantly different behavior for small wavenumbers, i.e., high phase velocities. In both cases there is a highest (true) velocity that can be estimated from the spectral maximum for a given array aperture (“velocity cut-off”). The Hankel spectrum estimates too high a velocity; for true wavenumbers below a certain limit, infinite velocity is estimated. The wavenumber spectrum, on the other hand, estimates too low a velocity, and there is an upper limit on the estimated velocity. An example illustrating these difficulties for the two methods is given for teleseismic P coda of an event recorded at the NORESS array in southern Norway: in spite of the problems, the analysis is able to demonstrate that the coda consists of two components; a coherent P-wave component with a high phase velocity and a diffuse S-wave component of low phase velocity. The cut-off and bias problem are investigated by numerical simulation for the NORESS array using azimuthal averaging and synthetic signals. The results confirm and quantify the cut-off problem at low wavenumbers and indicate that wavenumbers estimated from the Hankel and wavenumber spectra maxima bracket the true wavenumber, with the Hankel spectrum estimate being low (phase velocity too high) and the wavenumber spectrum estimate high. The bias of both methods decreases with increasing wavenumber (decreasing phase velocity) and they are both asymptotically unbiased. The wavenumber spectrum has a superior performance at low wavenumbers (high phase velocity), but the Hankel spectrum gives superior results at high wavenumbers (low phase velocity). The product of the linear wavenumber (= 1/wavelength) and the array aperture define “high” and “low” wavenumbers; for low wavenumbers, the product is 1 or less. In an Appendix, we find absolute lower bounds on the cut-offs analytically. The problems could be mitigated by using high resolution methods.
Applying GPS Time-Differenced Carrier Phase velocity estimation for Calibration of LDV/INS Integrated Navigation Systems in land vehicles