Power Spectrum of Hard-Limited Gaussian Processes

Hall, Harry M.

doi:10.1002/j.1538-7305.1969.tb01204.x

Cited by 4 publications

(2 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We seek an exact or approximate link allowing to recover the information about the seismic noise, Cs, from the modulated correlation function C H(x) of the modulated signal which is corrupted by the tectonic sources. Motivated by the earlier work in Rice (1944); Middleton (1948); Hall (1969), we express the correlation function C H(x) of the modulated signal via an integral involving the characteristics of the modulator and the joint characteristic function of the input signals. Exact analytical or approximate formulas for C H(x) can be derived if the joint characteristic function of the input signals can be represented as a series of appropriately factorized terms; such a factorized series representation is possible for a large class of Gaussian and non-Gaussian processes, as shown below.…”

Section: Appendix A: Fourier Conventionmentioning

confidence: 99%

“…The study of correlations of digitized stochastic Gaussian-distributed processes has a long history, stretching back almost 80 years. In particular, we draw from two articles, van Vleck & Middleton (1966) and Hall (1969), in which the former calculate the correlation function of digitized Gaussian processes while the latter estimate the signal-to-noise degradation incurred due to the application of the one-bit filter. In the field of seismology, it was discussed by e.g., Tomoda (1956); Aki (1957), but is not used in contemporary noise tomography and its properties have not been carefully investigated for relevant problems.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Interpreting cross-correlations of one-bit filtered seismic noise

Hanasoge

Branicki

2013

Geophysical Journal International

View full text Add to dashboard Cite

Seismic noise, generated by oceanic microseisms and other sources, illuminates the crust in a manner different from tectonic sources, and therefore provides independent information. The primary measurable is the two-point cross-correlation, evaluated using traces recorded at a pair of seismometers over a finite-time interval. However, raw seismic traces contain intermittent large-amplitude perturbations arising from tectonic activity and instrumental errors, which may corrupt the estimated cross-correlations of microseismic fluctuations. In order to diminish the impact of these perturbations, the recorded traces are filtered using the nonlinear one-bit digitizer, which replaces the measurement by its sign. Previous theory shows that for stationary Gaussian-distributed seismic noise fluctuations one-bit and raw correlation functions are related by a simple invertible transformation. Here we extend this to show that the simple correspondence between these two correlation techniques remains valid for non-stationary Gaussian and a very broad range of non-Gaussian processes as well. For a limited range of stationary and non-stationary Gaussian fluctuations, we find that one-bit filtering performs at least as well as spectral whitening. We therefore recommend using one-bit filtering when processing terrestrial seismic noise, with the substantial benefit that the measurements are fully compatible with current theoretical interpretation (e.g., adjoint theory). Given that seismic records are non-stationary and comprise small-amplitude fluctuations and intermittent, large-amplitude tectonic/other perturbations, we outline an algorithm to accurately retrieve the correlation function of the small-amplitude signals.

show abstract

Section: Appendix A: Fourier Conventionmentioning

confidence: 99%

mentioning

confidence: 99%