Fractals are widely used in different fields of science. The main characteristic of a fractal set is its fractal dimensions. The presented work addresses the accuracy of estimation of the correlation fractal dimension D-2 and corresponding confidence interval. For example, in geosciences D-2 is applied for characterizing surfaces or the spatial distribution of fracture systems. Two qualitatively different problems are studied. The first problem is related to the analysis of the fractal sets assigned on regular grids. The fractals consisting of a family of irregular points are considered in the second problem. The Monte Carlo method is proposed to estimate the confidence intervals of correlation fractal dimension for the first problem. This approach is based on the statistical generation of the fractal sets realizations with a given D-2. Numerical testing using synthetic models showed the reliability of this method. A comparison with PBM and BCA bootstrap methods is performed. The paper also investigates the accuracy of the D-2 and the corresponding confidence intervals estimate for the second studied problem. An algorithm employing the jackknife method is suggested. Its accuracy has been studied and proved numerically.

The work addresses the development of a statistical model of a discrete fracture network (DFN) using core, outcrop, and seismic data. We consider the DFN model defined by the fractal distribution of fracture centers and a power-law distribution of fracture lengths. We perform the statistical analysis of model realization on different spatial scales to investigate the possibility to evaluate the corresponding correlation fractal dimension and power exponent. Reproducing the statistical parameters estimated from seismic images is investigated by comparison with the parameters of the original model. We conclude that the analysis of the finite subdomain of the considered model of fractures system makes it possible to estimate the model's statistical characteristics with a decrease in the system's linear size by about one order of magnitude. On the other hand, the system built on the base of seismic images does not reproduce the power-law probability density of fracture length distribution. Also, the distribution of fracture centers in this system does not reflect the fractal nature and is uniform. The results obtained in this work show that data obtained from outcrops and core samples can be used to build a statistical DFN model. On the other hand, an explicit DFN model cannot be recovered based on seismic data.

In this paper, we study the effect of the interface roughness on the elastic parameters of layered media. We consider the three-dimensional models of a layered medium with two different elastic materials inside and outside the layer. We generate the first class of models, where the interfaces between the layers are rough, and the elastic parameters of the inner layers are fixed. Then, the numerical upscaling technique is applied to estimate the effective stiffness tensor. Next, we downscale the stiffness tensor to reconstruct the new elastic parameters of the inner layer for the model of second class with flat interfaces; that is the uncertainty of the model geometry is mapped to the uncertainty of the stiffness tensor component for a fixed geometry of the model. After that, we propose an algorithm for extending the results of restoring the elastic tensors for arbitrary parameters of uncertainty applying the bilinear regression with respect to interface rough parameters and bilinear interpolation using two nearest points with respect to the physical parameters of the inner layers. Verification of the algorithm shows that the errors in the recovering covariance matrix do not exceed 7%; that is, it can be used to statistically simulate models of the second class with a flat interface by arbitrary values of the interface roughness and the physical parameters of the layers in the first class of models.

The work is devoted to the development and study of a statistical model of a discrete network of cracks at different spatial scales. The distribution of fracture centers is a fractal set, which is determined by a given value of the correlation fractal dimension D 2 . The fracture lengths are described by a power-law probability distribution with the corresponding value of the exponent α . The possibility to estimate these model parameters based on the corresponding seismic images is investigated.

In the paper, the effect of interface roughness on the elastic parameters of a layered medium is studied. Three-dimensional models of a layered medium with different material inside and outside the layer are considered. Initially, the first class of models is statistically generated with rough interfaces between the layers and fixed elastic parameters of the inner layers. Then the effective stiffness tensor is estimated for the equivalent homogeneous model. After that, new elasticity parameters of the inner layer are reconstructed for second class of models with flat interfacies. Thus, the roughness of the model interfacies is mapped into a statistical distribution of the stiffness tensor components for a fixed model geometry. In addition, an algorithm for extending the results of reconstructing the elastic tensors for arbitrary interface roughness parameters using bilinear regression of covariance matrices is proposed. The algorithm verification shows that the error in restoring the covariance matrix does not exceed 7%; i.e., one can use it in statistical modeling the second class of models with flat interfacies for the arbitrary interface roughness values and given physical parameters of layers in the first class of models.

The present study addresses the sensitivity analysis of particle concentration dispersion in the turbulent flow. A stochastic spectral model of turbulence is used to simulate the particle transfer. Sensitivity analysis is performed by estimations of Morris and Sobol indices. This study allows to define the significant and nonsignificant model parameters. It also gives an idea of the qualitative behavior of the stochastic model used.

Abstract We address the problem of statistical simulation of a scalar real Gaussian random field inside the unit 3D ball. Two different methods are studied: (i) the method based on the known homogeneous isotropic power spectrum developed by Meschede and Romanowicz [M. Meschede and B. Romanowicz, Non-stationary spherical random media and their effect on long-period mantle waves, Geophys. J. Int. 203 2015, 1605–1625] and (ii) the method based on known radial and angular covariance functions suggested in this work. The first approach allows the extension of the simulation technique to the inhomogeneous or anisotropic case. However, the disadvantage of this approach is the lack of accurate statistical characterization of the results. The accuracy of considered methods is illustrated by numerical tests, including a comparison of the estimated and analytical covariance functions. These methods can be used in many applications in geophysics, geodynamics, or planetary science where the objective is to construct spatial realizations of 3D random fields based on a statistical analysis of observations collected on the sphere or within a spherical region.

A method for generating synthetic fields of turbulent velocity fluctuations based on a randomized spectral method (RSM) is proposed. The generated fields have zero divergence, as well as the specified properties of anisotropy and spatial inhomogeneity. The calculation of the canonical problem of a developed turbulent flow in a channel using the generated turbulent fields is in close agreement between the results and the data of direct numerical simulation.

On the base of Randomized Spectral Method (RSM), a new stochastic algorithm for the generation of homogeneous anisotropic turbulent velocity fields has been developed. This technique provides incompressibility of the obtained velocity field realizations simultaneously with the two main required statistical properties—the Reynolds-stress tensor and the turbulent energy spectrum. The method is computationally efficient and can easily be extended to the case of inhomogeneous turbulence.

We describe a method to perform a constrained lithospheric-scale inversion of satellite gravity gradient data. The a priori constraints include: i) data covariance matrix; ii) prior model covariance matrix including a model for spatial variability of mantle heterogeneity;

A method for the numerical generation of anisotropic turbulent velocity fields is presented. The proposed technique is based on the spectral method (SM) [1]. The traditional adaptation of isotropic field generated with spectral methods uses a Cholesky decomposition of Reynolds stresses tensor. After this adaptation the resulted field loses the property of incompressibility provided in the isotropic case. We have modified this method to use it in the anisotropic case and guarantee the incompressibility of generated turbulent field. Comparison of the results of IDDES simulation of canonical turbulent flow using inlet boundary conditions based on modified and non modified spectral methods are presented.

We present a numerical study of the effect of the interfaces' roughness on layered media's upscaled elastic parameters. First, we consider a layered model with two types of elastic materials, assuming that the interfaces are not flat but rough, and apply the numerical upscaling technique to estimate the effective elastic properties of such models. After that, we apply a downscaling technique to reconstruct a layered media with flat interfaces but with uncertainties in elastic moduli of the layers. Next, we compute the covariance of the elements of the reconstructed stiffness matrix and prove that the logarithm of this matrix is linearly related to the logarithms of the standard deviation and the correlation length of the interfaces of the original problem. Finally, we use generated dataset to estimate covariance matrices of the stiffness matrix for arbitrary interface roughnesses.

The paper addresses a global sensitivity analysis of complex models. The work presents a generalization of the hierarchical statistical models where uncertain parameters determine the distribution of statistical models. The double randomization method is applied to increase the efficiency of the Monte Carlo estimation of Sobol indices. Numerical computations are provided to study the accuracy and efficiency of the proposed technique. The issue of optimization of the suggested approach is considered.

This paper presents the results of a sensitivity analysis of the characteristics of discrete fracture network (DFN) connectivity. The sizes of the maximum and mean clusters comprising the DFN and the distribution of the connectivity index were estimated. Attention was primarily directed to the global sensitivity analysis of these characteristics. Sobol sensitivity indices were used to determine the relative contribution of the DFN model parameter uncertainty to the total uncertainty of the studied problem. Dominant Sobol indices for different DFN models and each fracture connectivity characteristic were determined. The analysis of the results allowed the identification of the parameters whose relevant probability distributions are most important for the correct description of the studied statistical model.

The work is devoted to three-dimensional modeling of fractal sets of points. Additional constraints in the form of probability density caused by the frequency of the generated points' spatial distribution are considered. The suggested method for statistical simulation allows reproducing both the given probability distribution defining the spatial position of the generated points and the required fractal dimension. Performed numerical computations confirm the accuracy and efficiency of the proposed method for the considered test models.

A new technique of stochastic generation of anisotropic turbulent velocity fields is based on Randomized Spectral Method (RSM). The algorithms provide incompressibility of the fields constructed simultaneously with the required statistical properties of turbulence, including the Reynolds-stress tensor and the energy spectrum.

A new version of the synthetic turbulent velocity generator is proposed for simulation of turbulent flows. The method is fully stochastic and generates a statistically anisotropic and nonhomogeneous random field, which provides the initial and boundary conditions for the deterministic eddy-resolving model of turbulence. The simulation method has been tested on a three-dimensional problem of developed turbulent channel flow.

О р д е н а Л е н и н а ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ имени М.В.Келдыша Р о с с и й с к о й а к а д е м и и н а у к А.В.Александров, Л.В.Дородницын, Д.Р.Колюхин Стохастический алгоритм генерации бездивергентного анизотропного однородного поля

A workflow for recovering fracture network characteristics from seismic data is considered. First, the presented discrete fracture modeling technique properly describes fracture models on the seismic scale. The key procedure of the workflow is 3D diffraction imaging based on the spectral decomposition of different combinations of selective images. Selective images are obtained by the prestack asymmetric migration procedure, whereas spectral decomposition occurs in the Fourier domain with respect to the spatial dip and the azimuth angles. At the final stage, we performed a topological analysis based on the construction of a merge tree from the obtained diffraction images. The results of the topological algorithm are modeling parameters for the discrete fractures. To analyze the effectiveness of our workflow, a statistical comparison of the recovered parameters and true model parameters was conducted. We used the Kolmogorov-Smirnov test for the statistical analysis of the fracture lengths, whereas the behavior of the Morisita index indicates the statistical distribution of the modeled fracture corridors. Numerical examples with synthetic realistic models provide a detailed, reliable reconstruction of the statistical characteristics of the fracture corridors.