scholarly journals Statistical simulation of seismic noise in a multidimensional area in determining frequency characteristics of geological media

2014 ◽  
pp. 62-68
Author(s):  
Z. Vyzhva ◽  
K. Fedorenko ◽  
A. Vyzhva
Keyword(s):  
Random Fields ◽  
Spectral Decomposition ◽  
Seismic Noise ◽  
Statistical Simulation ◽  
Frequency Characteristics ◽  
Partial Sums ◽  
Multidimensional Space ◽  
Mean Square ◽  
Spatial Variables ◽  
The Mean

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

About approximation of 3-D random fields and statistical simulation

2003 ◽  
Vol 11 (3) ◽  
Author(s):  
Zoya O. Vyzhva
Keyword(s):  
Random Fields ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Statistical Simulation ◽  
Gaussian Random Fields ◽  
Statistical Characteristics ◽  
Mean Square ◽  
The Mean ◽  
Isotropic Random ◽  
Three Dimensional Space

The estimator of the mean-square approximation of 3-D homogeneous and isotropic random field is investigated. The problem of statistical simulation of realizations of random fields in threedimensional space is considered. The algorithm for the receiving of this realization has been formulated, which has been constructed on the base the mean-square approximation of random fields estimator. It has been constructed the statistical model for the Gaussian random fields in three-dimensional space, which has been given by its statistical characteristics.

scholarly journals ABOUT METHODS OF RANDOM FIELDS STATISTICAL SIMULATION ON THE SPHERE BY THE AIRCRAFT MAGNETOMETRY DATA

2018 ◽  
pp. 107-113
Author(s):  
Z. Vyzhva ◽  
V. Demidov ◽  
A. Vyzhva
Keyword(s):  
Random Fields ◽  
Spectral Decomposition ◽  
Spline Approximation ◽  
Statistical Simulation ◽  
Partial Sums ◽  
Observation Data ◽  
Random Component ◽  
Geophysical Research ◽  
Random Components ◽  
Map Accuracy

There have been developed universal methods of statistical simulation (Monte Carlo methods) of geophysical data for generating random fields on the sphere on grids of required detail and regularity. Most of the geophysical research results are submitted in digital form, which accuracy depends on various random effects (including equipment measurement error). The map accuracy problem occurs when the data cannot be obtained with a given detail in some areas. ²t is proposed to apply statistical simulation methods of random fields realizations, to solve the problems of conditional maps, adding of data to achieve the necessary precision, and other similar problems in geophysics. Theorems on the mean-square approximation of homogeneous and isotropic random fields on the sphere have been proved by special partial sums. A spectral coefficients method was used to formulate algorithms of statistical simulation by means of these theorems. A new effective statistical technique has been devised to simulate random fields on the sphere for geophysical problems. Statistical simulation of random fields on the sphere based on spectral decomposition has been introduced in order to enhance map accuracy by the example of aeromagnetic survey data in the Ovruch depression. It is divided into deterministic and random components for data analysis. The deterministic component is proposed to approximate by cubic splines and the random component is proposed to modeling on the basis of random fields on the sphere by spectral decomposition. Model example – the aircraft magnetometry data. According to the algorithm we received random component implementations on the study area with twice detail for each profile. When checking their adequacy we made the conclusions that the relevant random components histogram has Gaussian distribution. The built variogram of these implementations has the best approximation by theoretical variogram which is connected to the Bessel type correlation function. The final stage was the imposing array of random components on the spline approximation of real data. As a result, we received more detailed implementation for the geomagnetic observation data in the selected area.

scholarly journals Fractional random fields associated with stochastic fractional heat equations

2005 ◽  
Vol 37 (01) ◽  
pp. 108-133 ◽  
Author(s):  
M. Ya. Kelbert ◽  
N. N. Leonenko ◽  
M. D. Ruiz-Medina
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Evolution Equations ◽  
Heat Equations ◽  
Mean Square ◽  
Fractional Evolution Equations ◽  
The Mean

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

scholarly journals Piecewise-Multilinear Interpolation of a Random Field

2013 ◽  
Vol 45 (04) ◽  
pp. 945-959 ◽  
Author(s):  
Konrad Abramowicz ◽  
Oleg Seleznjev
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Asymptotic Approximation ◽  
Field Observations ◽  
Mean Square ◽  
Local Behavior ◽  
Approximation Accuracy ◽  
Dimensional Cube ◽  
The Mean ◽  
Fractional Brownian Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

All-Temperature Partition Function from Variationally Optimized Canonical Density Matrix

1983 ◽  
Vol 38 (12) ◽  
pp. 1373-1382
Author(s):  
R. Baltin
Keyword(s):  
Density Matrix ◽  
Ground State ◽  
Order Differential Equation ◽  
Bloch Equation ◽  
Mean Square ◽  
Spatial Variables ◽  
One Dimensional ◽  
Exactly Solvable ◽  
The Mean ◽  
The One

Abstract For the canonical density matrix C(r, r0,β) a variational ansatz C̄̄f = (1 - f̄) Ccl + f̄ Cgr is made where Ccl and Cgr are the classical and the ground state expressions which are exact in the high temperature (β → 0) and in the low-temperature limits (β → + ∞), respectively, and f̄ is a trial function subject to the restriction that f̄ → 0 for β → 0 and f̄ → 1 for β → ∞. With the approximation that f̄ be dependent only upon β, not upon spatial variables, the mean square error arising when Cf is inserted into the Bloch equation is made a minimum. The Euler equation for this variational problem is an ordinary second order differential equation for f̄=f(β) to be solved numerically. The method is tested for the exactly solvable case of the one dimensional harmonic oscillator.

scholarly journals Fractional random fields associated with stochastic fractional heat equations

2005 ◽  
Vol 37 (1) ◽  
pp. 108-133 ◽  
Author(s):  
M. Ya. Kelbert ◽  
N. N. Leonenko ◽  
M. D. Ruiz-Medina
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Evolution Equations ◽  
Heat Equations ◽  
Mean Square ◽  
Fractional Evolution Equations ◽  
The Mean

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

scholarly journals Piecewise-Multilinear Interpolation of a Random Field

2013 ◽  
Vol 45 (4) ◽  
pp. 945-959
Author(s):  
Konrad Abramowicz ◽  
Oleg Seleznjev
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Asymptotic Approximation ◽  
Field Observations ◽  
Mean Square ◽  
Local Behavior ◽  
Approximation Accuracy ◽  
Dimensional Cube ◽  
The Mean ◽  
Fractional Brownian Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

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