Abstract
Consider the situation where a plane wave signal is received by a spatial arrangement of recorders. Information derived from observations on such a process can be used to determine the speed and direction of the signal together with properties of the medium through which the signal is being propagated. Certain models for the case where the signal velocity can be regarded as stochastic and where the array is irregular are investigated and estimation procedures proposed. A major practical property of these models is that, unlike their deterministic counterparts, coherence decays to zero as distance between recorders increases.
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References
Ash, R. B. and Gardner, M. F. (1975). Topics in Stochastic Processes, Academic Press, New York.
Brillinger, D. R. (1974). Fourier analysis of stationary processes, Proceeding of the IEEE, 62, 1628–1643.
Brillinger, D. R. (1975). time Series, Data Analysis and Theory, Holt, Rinehart and Winston, New York.
Brillinger, D. R. (1985). A maximum likelihood approach to frequency-wavenumber analysis, IEEE Trans. Acoust., Speech and Signal Process., 33, 1076–1085.
Cameron, M. A. and Hannan, E. J. (1978). Measuring the properties of plane waves, Math. Geol., 10, 1–22.
Cameron, M. A. and Thomson, P. J. (1985). Measuring attenuation, Time Series in the Time Domain, Handbook of Statistics 5 (eds. E. J.Hannan, P. R.Krishnaiah and M. M.Rao), 363–367, North-Holland, Amsterdam.
Capon, J. (1969). High resolution frequency-wave number spectrum analysis, Proceeding of the IEEE, 57, 1408–1418.
Davey, F. J. and Smith, E. G. C. (1982). The Wellington seismic reflection survey: Phase 1 1979, Report No. 187, Geophysics Division, Department of Scientific and Industrial Research, Wellington, New Zealand.
Hamon, B. V. and Hannan, E. J. (1974). Spectral estimation of time delay for dispersive and non-dispersive systems, J. Roy. Statist. Soc. Ser. C, 23, 134–142.
Hannan, E. J. (1970). Multiple Time Series, Wiley, New York.
Hannan, E. J. (1975). Measuring the velocity of a signal, Perspectives in Probability and Statistics (ed. J. M.Gani), 227–237, Applied Probability Trust, Sheffield.
Hannan, E. J. and Thomson, P. J. (1988). Time delay estimation, J. Time Ser. Anal., 9, 21–33.
Hinich, M. J. (1981). Frequency-wave number array processing, J. Acoust. Soc. Amer., 69, 732–737.
Hinich, M. J. (1982). Estimating signal and noise using a random array, J. Acoust. Soc. Amer., 71, 97–99.
Hinich, M. J. and Shaman, P. (1972). Parameter estimation for an r-dimensional plane wave observed with additive independent Gaussian errors, Ann. Math. Statist., 43, 153–169.
Jones, R. H. (1981). Fitting a continuous time autoregression to discrete data, Applied Time Series Analysis II (ed. D. F.Findley), 651–682, Academic Press, New York.
Robinson, P. M. (1977). Estimation of a time series model from unequally spaced data, Stochastic Process. Appl., 6, 9–24.
Sato, H. (1984). Attenuation and envelope formation of three-component seismograms of small local earthquakes in randomly inhomogeneous lithosphere, Journal of Geophysical Research, 89, 1221–1241.
Shumway, R. H. (1983). Replicated time-series regression: an approach to signal estimation and detection, Time Series in the Frequency Domain, Handbook of Statistics 3 (eds. D. R.Brillinger and P. R.Krishnaiah), 383–408, North-Holland, Amsterdam.
Thomson, P. M. (1982). Signal estimation using an array of recorders, Stochastic Process. Appl., 13, 201–214.
Ziskind, I. and Wax, J. (1988). Maximum likelihood localisation of multiple sources by alternating projection, IEEE Trans. Acoust. Speech Signal Process., 36, 1553–1560.
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Thomson, P.J. Signal estimation using stochastic velocity models and irregular arrays. Ann Inst Stat Math 44, 13–25 (1992). https://doi.org/10.1007/BF00048667
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DOI: https://doi.org/10.1007/BF00048667