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Magnitude and neural networks
Published in H. Ogasawara, T. Yanagidani, M. Ando, Seismogenic Process Monitoring, 2017
The well-known algorithm called Fast Fourier Transform (FFT) is the basis for spectral calculations. But it has some shortcuts, which result from the fact that the real input time series are of finite length. Such time series can be treated as a result of multiplication in the time domain (which means convolution in the frequency domain) of an infinite time series - needed to calculate theoretically the exact Fourier transform - and a rectangular window, with zeros outside the range covered by presented time series and ones inside that range. But a rectangular window has very poor spectral properties, which distorts the calculated spectra. Sometimes one can use a single window better than a rectangular one, such as Hanning, Hamming or cosine, but improvement is still limited. To avoid poor performances of the single windows the use of a multitaper (Thomson 1982, Park et al. 1987) is often suggested instead. The use of a multitaper is a simple procedure: one needs to apply several different windows to the input time series, using FFT every time, and then the resulted spectrum is calculated as the average of these particular results. Instead of a simple average, a most sophisticated method exists, where the final spectrum is calculated in an iterative process (Park et al. 1987). The multitaper windows – called prolate spheroidal windows – are the eigenvectors vof the matrix eigenvalue problem: () C*ν=λ(N,W)*ν
Non-parametric modelling and simulation of spatiotemporally varying geo-data
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2022
Yu Wang, Yue Hu, Kok-Kwang Phoon
For decades, spectral representation (e.g. Shinozuka and Deodatis 1991, 1996; Hu and Schiehlen 1997; Cressie 2015) has been widely used as an effective tool for spatiotemporal data modelling, leading to numerous advancements in various fields and applications, e.g. oceanography (e.g. Thomson and Emery 2014; Wortham and Wunsch 2014), climate data series analysis (e.g. Yiou, Baert, and Loutre 1996; Bosshard et al. 2011), geological data interpretation (e.g. Mugglestone and Renshaw 1998; Asadzadeh and de Souza Filho 2016), structural dynamics and seismic analysis (e.g. Shinozuka 1972; Huang et al. 2004; Gao et al. 2012; Wu et al. 2014). Classical spectral representation is based on the classical Fourier transform and uses trigonometric functions with different frequencies to characterise the data in the frequency domain, and hence, it enables a non-parametric, flexible, and concise representation of many spatiotemporally varying geo-data (e.g. Båth 1974; Stranneby 2004; Smith 2013). The key step of spectral representation techniques is to properly establish a power spectral density (PSD) function (e.g. Welch 1967; Percival and Walden 1993; Stoica and Moses 2005) for which a series of non-parametric methods have already been developed, such as periodogram estimate (e.g. Bartlett 1950; Oppenheim, Buck, and Schafer 2001), Welch’s method (e.g. Welch 1967), multitaper methods (e.g. Babadi and Brown 2014). The non-parametric branch of spectral representation methods makes it possible to obtain insight into the spectrum of the concerned spatiotemporally varying geo-data directly from measurements.
Assessing seismicity in Bangladesh: an application of Guttenberg-Richter relationship and spectral analysis
Published in Geomatics, Natural Hazards and Risk, 2023
Abu Reza Md. Towfiqul Islam, Mst. Yeasmin Akter, Sumaia Amanat, Edris Alam, Mst. Laila Sultana, Shamsuddin Shahid, Arnob Das, Susmita Datta Peu, Javed Mallick
The multitaper spectrum analysis approach was first presented by Thomson (1982) and has been extensively used in seismogram analysis (Park 1987). In a multitaper analysis, the data are multiplied by some leakage-resistant tapers rather than just one. As a result, one record produces a very tapered time series. Each of these time series’ DFTs (discrete Fourier transforms) yields several ‘eigen spectra,’ which are then averaged to create a single spectral estimate. Many different multitapers have been suggested, such as Slepian tapers, discrete prolate spheroidal sequences, sinusoidal tapers, etc., to name a few of them. The fundamental concept of this multitaper technique is that, given moderate circumstances, the spectral estimations would be independent of each other at every frequency, provided the data tapers are constructed as suitably orthogonal functions. The design of the multitapers maximizes resistance to spectral leakage while allowing each taper to sample the time series differently. The second taper partly recovers the statistical data that the first taper abandoned; the third taper partially recovers the data that the first two tapers partially discarded, and so on. The higher-order tapers allow for an unacceptably large amount of spectral leakage; hence, only a few lower-order tapers are used. The trade-off between leakage and variation that plagues single-taper estimates may be avoided by using these tapers to construct an estimate. Group of orthonormal tapers, including harmonically connected sinusoidal tapers. These tapers are also known as minimal bias tapers or sinusoidal tapers. The continuous time minimum bias tapers are given in Equation (8)