scholarly journals On the Importance of Statistical Homogeneity to the Scaling of Rain

2019 ◽  
Vol 36 (6) ◽  
pp. 1063-1078 ◽  
Author(s):  
A. R. Jameson
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Power Spectrum ◽  
Wide Sense ◽  
Time Data ◽  
Statistical Homogeneity ◽  
Statistical Heterogeneity ◽  
Cross Correlations ◽  
The Fourier Transform ◽  
Universal Scaling Function

AbstractScaling studies of rainfall are important for the conversion of observations and numerical model outputs among all the various scales. Two common approaches for determining scaling relations are the Fourier transform of observations and the Fourier transform of a correlation function using the Wiener–Khintchine (WK) theorem. In both methods, the observations must be wide-sense statistically stationary (WSS) in time or wide-sense statistically spatially homogeneous (WSSH) in space so that the correlation function and power spectrum form a Fourier transform pair. The focus here is on developing an explicit understanding for the requirement. Statistically heterogeneous (either in space or time) data can produce serious scaling errors. This work shows that the effects of statistical heterogeneity appear as contributions from cross correlations among all of the distinct contributing rainfall components using either method so that the correlation function and its FFT do not form a transform pair. Moreover, the transform then also depends upon the time and location of the observations so that the “observed” power spectrum no longer represents a “universal” scaling function beyond the observations. An index of statistical heterogeneity (IXH) defined in previous work provides a way of determining whether or not a set of rain data may be considered to be WSS or WSSH. The greater IXH exceeds the null, the more likely the derived power spectrum should not be used for general scaling. Numerical simulations and some observations are used to demonstrate all of these findings.

scholarly journals Sensing Fractional Power Spectrum of Nonstationary Signals with Coprime Filter Banks

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xiaomin Li ◽  
Huali Wang ◽  
Wanghan Lv ◽  
Haichao Luo
Keyword(s):  
Fourier Transform ◽  
Power Spectrum ◽  
Filter Banks ◽  
Fractional Fourier Transform ◽  
Linear Time ◽  
Sampling Rate ◽  
Fractional Power ◽  
Fourier Domain ◽  
Wide Sense ◽  
Fractional Power Spectrum

The coprime discrete Fourier transform (DFT) filter banks provide an effective scheme of spectral sensing for wide-sense stationary (WSS) signals in case that the sampling rate is far lower than the Nyquist sampling rate. And the resolution of the coprime DFT filter banks in the Fourier domain (FD) is 2π/MN, where M and N are coprime. In this work, a digital fractional Fourier transform- (DFrFT-) based coprime filter banks spectrum sensing method is suggested. Our proposed method has the same sampling principle as the coprime DFT filter banks but has some advantages compared to the coprime DFT filter banks. Firstly, the fractional power spectrum of the chirp-stationary signals which are nonstationary in the FD can be sensed effectively by the coprime DFrFT filter banks because of the linear time-invariant (LTI) property of the proposed system in discrete-time Fourier domain (DTFD), while the coprime DFT filter banks can only sense the power spectrum of the WSS signals. Secondly, the coprime DFrFT filter banks improve the resolution from 2π/MN to 2π sin α/MN by combining the fractional filter banks theory with the coprime theory. Simulation results confirm the obtained analytical results.

scholarly journals On the Detection of Statistical Heterogeneity in Rain Measurements

2018 ◽  
Vol 35 (7) ◽  
pp. 1399-1413 ◽  
Author(s):  
A. R. Jameson ◽  
M. L. Larsen ◽  
A. B. Kostinski
Keyword(s):  
Heuristic Method ◽  
Mean Value ◽  
Small Data ◽  
Wide Sense ◽  
Mean Values ◽  
Statistical Homogeneity ◽  
Second Moments ◽  
Statistical Heterogeneity ◽  
Rain Events ◽  
Global Mean

AbstractThe application of the Wiener–Khintchine theorem for translating a readily measured correlation function into the variance spectrum, important for scale analyses and for scaling transformations of data, requires that the data be wide-sense homogeneous (stationary), that is, that the first and second moments of the probability distribution of the variable are the same at all times (stationarity) or at all locations (homogeneity) over the entire observed domain. This work provides a heuristic method independent of statistical models for evaluating whether a set of data in rain is wide-sense stationary (WSS). The alternative, statistical heterogeneity, requires 1) that there be no single global mean value and/or 2) that the variance of the variable changes in the domain. Here, the number of global mean values is estimated using a Bayesian inversion approach, while changes in the variance are determined using record counting techniques. An index of statistical heterogeneity (IXH) is proposed for rain such that as its value approaches zero, the more likely the data are wide-sense stationary and the more acceptable is the use of the Wiener–Khintchine theorem. Numerical experiments as well as several examples in real rain demonstrate the potential of IXH to identify statistical homogeneity, heterogeneity, and statistical mixtures. In particular, the examples demonstrate that visual inspections of data alone are insufficient for determining whether they are wide-sense stationary. Furthermore, in this small data collection, statistical heterogeneity was associated with convective rain, while statistical homogeneity appeared in more stratiform or mixed rain events. These tentative associations, however, need further substantiation.

Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform. II. Study of Correlations Between Patterns

2020 ◽  
pp. 2150003
Author(s):  
V. R. Chechetkin ◽  
V. V. Lobzin
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Geomagnetic Activity ◽  
Large Scale ◽  
Statistical Significance ◽  
Periodic Patterns ◽  
Pattern Correlation ◽  
Double Fourier Transform ◽  
The Fourier Transform ◽  
Combined Technique

The discrete double Fourier transform (DDFT) was developed to search for large-scale multi-periodic patterns in the presence of noise and is based on detection of the equidistant series of harmonics generated by the periodic patterns in the discrete Fourier transform (DFT) spectra. As DDFT retains all generic features of the Fourier transform, the corresponding pattern correlation function (PCF) related to DDFT can be introduced similarly to the data correlation function (DCF) related to DFT on the basis of the Wiener–Khinchin relationship. Peaks in PCF indicate the number of periodic patterns in a dataset under analysis and have direct correspondence with the counterpart peaks in the DFT spectrum. The close correspondence between positions of the peaks in the PCF and DFT spectra strongly enhances statistical significance of detected periodicities. Similar PCFs can also be defined for the cepstrum transform. The combined DFT–DCF and DDFT–PCF technique was applied to the detection of cycles in geomagnetic activity using disturbance storm-time (Dst) index. In addition to the known 27-day, semiannual and 11-year cycles of geomagnetic activity, we have also found the annual cycle of activity. The results were compared with those obtained by the cepstrum transform. A multiple cross-check makes the combined technique much more efficient and robust in comparison with the detection based on a unique particular method.

On-line electron-optical correlation computing in the CTEM

1978 ◽  
Vol 36 (1) ◽  
pp. 88-89 ◽  
Author(s):  
R. Guckenberger ◽  
W. Hoppe
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Transfer Function ◽  
Inverse Fourier Transform ◽  
Main Peak ◽  
Optical Correlation ◽  
Auto Correlation ◽  
Auto Correlation Function ◽  
On Line ◽  
The Fourier Transform

Light diffractograms of electron micrographs are frequently used to study the transfer function of the microscope. In order to utilize diffractograms for control operations in the microscope, several attempts have been undertaken to obtain on-line diffractograms /1 - 3/. Alternatively correlation functions (CF) may be used /4-8/. In this paper we describe an electron-optical device for the computation of such CF and its on-line operation in a microscope.The auto-correlation function (ACF) is the inverse Fourier transform of the squared modulus of the Fourier transform (diffractogram) of an image. Therefore it also contains the transfer function. It is its zero peak (main peak) which is of particular interest. In noisy images the main ACF-peak of the noise contributes in an unwanted way to the main ACF-peak of the image. This can be avoided if the ACF will be computed of two images which are identical except for noise /9/ (noise-reduced ACF= NRACF).

Classical and Nonclassical Optical Diffusion

1990 ◽  
Vol 195 ◽  
Author(s):  
J. M. Drake ◽  
A. Z. Genack
Keyword(s):  
Fourier Transform ◽  
Diffusion Coefficient ◽  
Correlation Function ◽  
Optical Transmission ◽  
Optical Field ◽  
Photon Absorption ◽  
Absorption Time ◽  
Transmission Data ◽  
The Fourier Transform ◽  
Field Correlation

ABSTRACTWe present a brief summary of our recent work on classical and nonclassical optical diffusion in disorder metal oxides. Results on the optical transmission through a slab of titania spheres is presented and discussed in some detail illustrating the experimental method for obtaining the optical diffusion coefficient and photon absorption time from the transmission data. We conclude that the optical field-field correlation function with frequency is the Fourier transform of the time of flight distribution.

Directional spectral analysis and filtering of geophysical maps

Geophysics ◽  
1988 ◽  
Vol 53 (12) ◽  
pp. 1587-1591 ◽  
Author(s):  
Freyr Thorarinsson ◽  
Stefan G. Magnusson ◽  
Axel Bjornsson
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Power Spectrum ◽  
Two Dimensional ◽  
Directional Filtering ◽  
Gravity And Magnetic ◽  
The Fourier Transform ◽  
Map Data ◽  
Tectonic Features

The detection of linear anomalies in map data is facilitated by studying the two‐dimensional power spectrum, because the directivity of the energy in the map is preserved in the Fourier transform. The lineaments associated with individual peaks in the spectrum are then separated from the map data by directional filtering and studied independently of other map features. Gravity and magnetic maps from an active rift area in southwestern Iceland are analyzed in this manner. The agreement between the filtered maps is good and they fit the observed tectonic features quite well.

The on-line use of histogram data for high-speed correction and alignment of electron microscope images

1984 ◽  
Vol 42 ◽  
pp. 556-557
Author(s):  
William Krakow
Keyword(s):  
Power Spectrum ◽  
High Speed ◽  
Direct Memory Access ◽  
Digital Television ◽  
Contrast Transfer Function ◽  
The Past ◽  
High Resolution Tem ◽  
On Line ◽  
Correction Of Astigmatism ◽  
The Fourier Transform

In the past few years on-line digital television frame store devices coupled to computers have been employed to attempt to measure the microscope parameters of defocus and astigmatism. The ultimate goal of such tasks is to fully adjust the operating parameters of the microscope and obtain an optimum image for viewing in terms of its information content. The initial approach to this problem, for high resolution TEM imaging, was to obtain the power spectrum from the Fourier transform of an image, find the contrast transfer function oscillation maxima, and subsequently correct the image. This technique requires a fast computer, a direct memory access device and even an array processor to accomplish these tasks on limited size arrays in a few seconds per image. It is not clear that the power spectrum could be used for more than defocus correction since the correction of astigmatism is a formidable problem of pattern recognition.

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