GeoScienceWorld
Volume

Modern Spectral Analysis with Geophysical Applications

By Markus Båth
Edited by Michael R. Cooper

Abstract

This book provides a bibliography of the material available concerning geophysical applications of spectral analysis. There are in all 1,483 numbered references. In addition to methodical developments, this bibliography includes geophysical applications.

  1. Page 5
    Abstract

    1. Abatzoglou, T.J., 1985. A fast maximum likelihood algorithm for frequency estimation ofa sinusoid based on Newton’s method. IEEE Trans. Acoust., Speech, Signal Process., ASSP-33: 77–89.

    A fast algorithm to estimate the frequency of a sinusoid is presented here. It is based on Newton’s method for finding the root of an equation, and it is shown that under easily met conditions, the root mean-square error (RMSE) of the estimator is practically equal to the Cramér-Rao bound after only two iterations of Newton’s method for all signal-to-noise ratios (SNRs) above threshold. The estimator’s probability density function is computed analytically, and the RMSE is calculated for one and two iterations of Newton’s method. Its computational load is shown to be significantly less than other conventional algorithms. - It should be noted that FFT methods form an essential part of the development.

  2. Page 27
    Abstract

    1. Bacon, L.D. and Post, R.E., 1984. A limitation of the Kumaresan-Prony algorithm in direction-finding applications. IEEETrans. Acoust., Speech, Signal Process., ASSP-32: 912–914.

    A modification of the forward-backward linear prediction method of spectral analysis due to Tufts and Kumaresan was evaluated for use in direction finding with an array of sensors. Their modifications increase the resolution and allow it to function properly at SNR below the usual threshold. A special case of this method, the Kumaresan-Prony (KP) case, is especially attractive due to its easeof computation. It provides resolution nearly as good as the optimum case of the general method with a significant saving in computer time. However, this case was found to exhibit blind angles, where performance was severely degraded in spite of wide spacing of the sources. These blind angles are a serious obstacle to the application of this method. This problem appears to affect only the KP case.

  3. Page 66
    Abstract

    1. Cadzow, J. A., 1978. Improved spectral estimation from incomplete sampled-data observations. Proc. RADC Spectrum Estimation Workshop 1978: 109–123. Also in ASSP-27: 4–12 (1979).

    In this paper, a particularly efficient procedure for achieving improved spectral estimations from incomplete observations of data sequences is presented. The essence of the method is that of. appropriately estimating the unobserved (or missing) data by extrapolation/interpolation and using the enlarged data base to generate the improved spectrum estimate. Clearly, the effectiveness of this method will be dependent on how well the missing data can be estimated. Empirical evidence accumulated to date indicates that this paper’s procedure is effective as well as being computationally efficient.

  4. Page 116
    Abstract

    1. Dahlhaus, R., 1984. Parameter estimation of stationary processes with spectra containing strong peaks. In: Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics 26,1- Franke, W. Häardle and D. Martin, Editors, Springer, New York, pp. 50–67.

    The advantage of using data tapers in parameter estimation of stationary processes is investigated. Consistency and a central limit theorem for quasi maximum likelihood parameter estimates with tapered data are proved, and data tapers leading to asymptotically efficient parameter estimates are determined. Finally the results of a Monte Carlo study on Yule-Walker estimates with tapered data are presented. When the spectrum contains high peaks the spectrum at the other frequencies may be overestimated markedly, and thus masked. Tapering is introduced to avoid this leakage effect.

  5. Page 132
    Abstract

    1. Eby, LM., 1978. Estimation and prediction for certain models of spatial time series. Ph.D. thesis, Dept. Statist., Univ. Florida. Also, Statist. Tech. Rep. No. 44, 62M10-2 by L.M. Eby, J.T. McClave and R.L. Scheaffer, Dept. Math. Comp. Sci. & Statist., Univ. South Carolina, Columbia (1979).

  6. Page 139
    Abstract

    1. Fahlman, G.G. and Ulrych, T.J., 1982. A new method for estimating the power spectrum of gapped data. Mon. Not. Roy. Astron. Soc., 199: 53–65.

    The problem of estimating the power spectrum of a discrete time series which consists of individual segments separated by long gaps is considered. A maximum entropy approach is used to fill the gaps with a prediction based on the observed data segments. An important aspect of the ME spectral analysis is its relationship to the autoregressive (AR) representation of a time series. It is this relationship that is used to fill the gaps. While the ME-AR model is essential for the prediction part of the problem, the final analysis of the spectral content of the data may be performed by standard Fourier techniques, thus with amplitudes and phases directly obtainable. This is the preferable approach when the total time spanned by the data is long compared to any periods of interest.

  7. Page 163
    Abstract

    1. Gabel, R.A. and Roberts, R.A., 1980. Signals and Linear Systems. Wiley, New York, 415 pp.

    Among the seven chapters in this book, we note particularly Ch. 3, pp. 110–168, on State variables*, something that is frequently met with in the modem literature. This chapter presents state variable formulation and solution for discretetime systems, followed by the corresponding treatment with regard to continuoustime systems.

    *State variables=important properties.

  8. Page 183
    Abstract

    1. Haddad, J., 1980. Spectral estimation of noisy signals using a parallel resonator model and state-variable techniques. M.S. thesis, Univ. Minnesota, Minneapolis, MN.

  9. Page 203
    Abstract

    1. Ibrahim, M.K., 1987. New high-resolution pseudospeetrum estimation method. IEEE Trans. Acoust., Speech, Signal Process., ASSP-35: 1071–1072.

    One of the most popular techniques for high-resolution power spectrum estimation is the Capon maximum likelihood method (MLM). It has been observed and verified analytically for large data records that the spectral estimate of MLM exhibits less variance, but also less resolution, than the AR spectral estimate. A new pseudospectrum estimation is presented here. Simulation results show an increase in the resolution and no significant increase in the variance of the new spectra when compared to MLM. Although MLM was derived originally for wavenumber estimation, it is stated here in a real-time series context. But the treatment can be extended easily to wavenumber estimation.

  10. Page 206
    Abstract

    1. Jaarsma, D., 1969. The theory of signal detectability: Bayesian philosophy, classical statistics, and the composite hypothesis. Tech. Rep. TR-200, Cooley Electronics Lab., Univ. Michigan, Ann Arbor, Mich.

  11. Page 224
    Abstract

    1. Kanai, H., Abe, M., and Kido, K., 1987. Accurate autoregressive spectrum estimation at low signal-to-noise ratio using a phase matching technique. IEEE Trans. Acoust., Speech, Signal Process., ASSP-35: 1264–1272.

    When noise is added to a signal under analysis, the signal is not described by the AR process, even if the signal is the response of an AR system because of the introduction of the spectral zeros by the additive noise. A number of methods for estimating AR parameters of such AR-plus-noise models have been proposed in the literature. However, accurate AR parameters are not obtained by such methods. This paper develops a new method of estimating AR parameters accurately, even when the signal is contaminated by white noise. It is confirmed with computer simulations and experiments that the proposed method is useful for accurate estimation of the AR parameters.

  12. Page 263
    Abstract

    1. Lacoss, R.T., 1971. Data adaptive spectral analysis methods. Geophysics, 36: 661–675. Also in Modem Spectrum Analysis, D.G. Childers, Editor, IEEE Press, New York, pp. 134–148 (1978).

    Two new methods (MLM of Capon and MEM of Burg) for power spectral density estimation of scalar time series have been experimentally investigated. Both methods, unlike conventional methods, adapt to the actual characteristics of the process under study. The new techniques are particularly valuable if the process contains one or more narrow peaks in frequency which are to be resolved. In this case, the output peaks from MEM are proportional to the square of the power in the narrow peaks but the area is equal to power. The peak values of the :MLM reflect power directly. - This paper, frequently referred to in the literature, is especially valuable because of the comparisons made between conventional methods (Bartlett window), MLM, and MEM, partly for artificial samples (sine waves plus white noise), partly for actual records (long-period seismic noise).

  13. Page 288
    Abstract

    1. MacDonald, G.J., 1989. Spectral analysis of time series generated by nonlinear processes. Rev. Geophys., 27: 449–469.

    For nonlinear regions prior to the onset of chaos, the periodogram offers many advantages in analyzing the spectral characteristics of a time series. The periodogram is an optimal detector, in the maximum likelihood sense of determining the frequencies of interacting oscillators. Periodogram analysis can be extended to situations in which samples are taken at uneven intervals. - The methods of spectral analysis based on generalized linear models such as maximum entropy are inappropriate for dealing with deterministic systems in the presence of noise. The statistical validity of peaks obtained by linear, parametric modeling is not well understood. - The first half of the paper gives a theoretical review, while the second half presents periodogram analyses of Earth’s obliquity, sunspots, and atmospheric carbon dioxide levels (obliquity=the angle of tilt of the Earth’s axis of rotation on the ecliptic).

  14. Page 325
    Abstract

    1. Nadeu, C. and Bertran, M., 1983. Two methods of spectral estimation with rational log spectrum. Proc. 2nd Europ. Signal Process. Conf., EUSIPCO-83: 471–474. Cf. also Signal Process., 10: 7–18 (1986).

    Two methods being a compromise between 1) spectral matching of rather flat spectra and 2) resolution are presented [MEM is good at 2) but not at 1); conventional methods are good at 1) but not at 2)]. An efficient iterative algorithm allows one to find the spectral estimate. Some results, extracted from an extensive empirical study, are presented. On the one hand, the methods show a clear enhancement, with respect to the MEM, when the correct spectrum has no prominent peaks or is not AR. On the other hand, the resolution is higher than in the conventional methods and even than in the maximum-likelihood method (MLM) of Capon.

  15. Page 342
    Abstract

    1. Oakley, O.H. and Lozow, J.B., 1977. The resolution of directional wave spectra using the maximum likelihood method. Rep. 77–1, Dept. Ocean Eng., MIT, Cambridge, MA.

  16. Page 355
    Abstract

    1. Pack, D.J., 1977. A computer program for the analysis of time series models using the Box-Jenkins philosophy. Automatic Forecasting Systems Inc., P.O. Box 563, Hatboro, PA 19040. and Dept. Comput. Sci., Ohio State Univ., Columbus, Ohio.

  17. Page 387
    Abstract

    1. Quinn, B.G., 1980. Order determination for a multivariate autoregression. J. Roy. Statist. Soc., B 42: 182–185.

    The discussion of the determination of the order for univariate autoregressions has been extensive in the literature. In this paper, the procedure proposed by Hannan and Quinn (1979) for determining the order of univariate autoregressions is extended to multivariate autoregressions. The procedure is shown to be strongly consistent.

  18. Page 38
    Abstract

    1. Radoski, B.R., Fougere, P.F., and Zawalick, E.J., 1974. A high resolution power spectral estimate: The maximum entropy method Tech. Rep. AFCRL-TR-74-0088, Air Force Cambridge Research Labs., 21 pp.

    This paper discusses three methods of spectral analysis: Blackman-Tukey, Levinson, and Burg. The first is an improvement ofthe usual Blackman-Tukey method, resulting in a smooth spectrum and higher resolution. The Levinson spectrum is a maximum-entropy spectral estimate, calculated by an algorithm due to Levinson. It provides improved resolution over the modified Blackman-Tukey method. The third spectrum is also called a BPEC (Burg prediction error coefficients) spectrum.. It is also a maximum-entropy estimate but calculated according to Burg’s technique. It is an improvement over the Levinson spectrum and produces a sharp spectrum with high resolution. The spectra are compared for constructed as well as real examples (geomagnetic micropulsations, sunspot numbers). - The paper is very clear and tutorial.

  19. Page 405
    Abstract

    1. Sabri, M.S. and Steenaart, W., 1978. An approach to band-limited signal extrapolation: The extrapolation matrix. IEEE Trans. Circuits and Systems, CAS-25: 74–78. Also in Proc. IEEE Int. Symp. Circuits Syst., pp. 674–677 (1977). Comments by the authors in ASSP-28: 254 (1980).

    Several extrapolation and spectrum estimation techniques have been presented in the literature, among them the maximum-entropy method. This paper presents a fast band-limited signal extrapolation technique where the total extrapolation process is achieved by a single matrix operation. The proposed technique and its implementation has many advantages over known extrapolation techniques in terms of computational savings and accuracy of the results. A realization is presented which allows adjustment to any given signal bandwidth. This realization can work on a real-time basis. The development is based on Fourier methods, but is still included here because of its relations to :MEM analysis.

  20. Page 461
    Abstract

    1. Tacconi, G., Editor, 1977. Aspects. of Signal Processing with Emphasis on Underwater Acoustics. NATO Adv. Study Inst., Ser. C, Vol. 33, Reidel, Dordrecht, Part 1+Part 2, 787 pp.

    The content is divided into six sections: (1) Properties of time/space variant transmission channel and underwater communications (8 papers). (2) Detection estimation and tracking techniques. (11 papers). (3) Time-space processing, adaptive processing and normalization, quantization methods (17 papers). (4) Displays, pattern recognition, human decision (2 papers). (5) Relevant inputs from other fields (7 papers). (6) Modern processor architecture and techniques (5 papers). Three papers are reported in this bibliography in more detail under the respective authors (O.L. Frost, RT. Lacoss, A.B. Baggeroer),

  21. Page 487
    Abstract

    1. Ulrych, T.J., 1972. Maximum entropy power spectrum of truncated sinusoids. J. Geophys. Res., 77: 1396–1400.

    The frequency shifts observed in the power spectra of truncated sinusoids, when the spectra are computed by using the periodogram, are avoided when the power spectra are determined by means of a maximum-entropy algorithm. However, cf. Chen and Stegen (1974). Furthermore, as demonstrated by the examples given, the :MEM resolution of the spectral peak is very striking. After a brief review of :MEM, it is stated that the significant point about this technique is that theprediction-error-filter coefficients are determined by using only the available data. No null extension to the data is required for estimating the autocorrelation values for lags greater than zero.

  22. Page 491
    Abstract

    1. Vaccaro, R.J., 1984. On adaptive implementations of Pisarenko’s harmonic retrieval method Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., ICASSP 84: 6.1.1-6.1.4.

    The problem of locating spectral lines (sinusoids) in white noise has been studied intensively in the past few years, especially since the work of Pisarenko. Recently, adaptive implementations ofPisarenko’s method have been developed in which the estimates can be updated as new data is observed, and so these algorithms have the ability to track slowly time-varying processes. In this paper, a new adaptive algorithm based on the inverse power method is developed and compared by simulation to other methods. Conclusions are drawn which affect any adaptive implementation of Pisarenko’s method.

  23. Page 497
    Abstract

    1. Wakefield, G.B., 1985. High-ftdelity estimation of continuous spectra. Ph.D. thesis, Dept. Elect. Eng., Univ. Minnesota, Minneapolis, MN.

  24. Page 512
    Abstract

    1. Xinhua, L., 1989. Determination of amplitude spectrum by maximum entropy estimation. Geophys. Prospecting for Petroleum (Res. Inst. Geophys. Prospect. for Petroleum, Ministry of Geology and Minerals, c/o China Int Book Trading Corp., P.O. Box 2820, Beijing, PRC), 28: 73–80 (in Chinese with English abstract).

    An approach to determine the amplitude using MEM (maximum-entropy method) is presented in this paper. It is based on the interrelation between MEM and MLM (maximum-likelihood method). Simulation test results show it is easily operated and works well. The paper presents formulas for MEM and MLM spectra and the relation between them.

  25. Page 513
    Abstract

    1. Yakovlev, A.P., 1983. Detection of the earth’s natural oscillations by laser deformographs. Izvestiya, Acad. Sci. USSR, 19: 318–321.

    The problem concerns detection of the earth’s natural oscillations ex,cited by earthquakes with M < 8.0 by new apparatus. Results are presented for the earth’s natmal oscillations excited by an earthquake of M=7.6 (Mexico, March 14, 1979). Selection of the oscillations was carried out by means of discrete Fomier transformation, by the smoothed window of Blakeman-Harris (autoperiodogram), and by the method of maximum entropy. The last method exhibits higher resolving power.

  26. Page 515
    Abstract

    1. Zagalsky, N.R., 1967. Exact spectral representation of truncated data. Proc. IEEE, 55: 117–118.

    A method is presented for the improvement of the spectral resolution. The proposed technique evaluates the spectrum S(ω) by expanding S(ω) as a series sum of orthogonal functions, and solves for the coefficients of these functions in terms of the computed truncated spectrum ST(ω). This paper deals exclusively with Fourier spectral analysis, but is included here because it gives a useful development of the Fourier technique in offering a higher spectral resolution than the conventional method - in a way, forerunning the recent developments. A numerical application could have been instructive.

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