Enhancing the resolution of sparse rock property measurements using machine learning and random field theory

The travel time of rock compressional waves is an essential parameter used for estimating important rock properties,such as porosity,permeability,and lithology.Current methods,like wireline logging tests,provide broad measurements but lack finer resolution.Laboratory-based rock core measurements offer higher resolution but are resource-intensive.Conventionally,wireline logging and rock core measurements have been used independently.This study introduces a novel approach that integrates both data sources.The method leverages the detailed features from limited core data to enhance the resolution of wireline logging data.By combining machine learning with random field theory,the method allows for probabilistic predictions in regions with sparse data sampling.In this framework,12 parameters from wireline tests are used to predict trends in rock core data.The residuals are modeled using random field theory.The outcomes are high-resolution predictions that combine both the predicted trend and the probabilistic realizations of the residual.By utilizing unconditional and conditional random field theories,this method enables unconditional and conditional simulations of the underlying high-resolution rock compressional wave travel time profile and provides uncertainty estimates.This integrated approach optimizes the use of existing core and logging data.Its applicability is confirmed in an oil project in West China.

ESTIMATION OF AVERAGE DIFFERENTIAL ENTROPY FOR A STATIONARY ERGODIC SPACE-TIME RANDOM FIELD ON A BOUNDED AREA

In this paper,we mainly discuss a discrete estimation of the average differential entropy for a continuous time-stationary ergodic space-time random field.By estimating the probability value of a time-stationary random field in a small range,we give an entropy estimation and obtain the average entropy estimation formula in a certain bounded space region.It can be proven that the estimation of the average differential entropy converges to the theoretical value with a probability of 1.In addition,we also conducted numerical experiments for different parameters to verify the convergence result obtained in the theoretical proofs.

An Interpolation Method for Karhunen-Loève Expansion of Random Field Discretization

In the context of global mean square error concerning the number of random variables in the representation,the Karhunen–Loève(KL)expansion is the optimal series expansion method for random field discretization.The computational efficiency and accuracy of the KL expansion are contingent upon the accurate resolution of the Fredholm integral eigenvalue problem(IEVP).The paper proposes an interpolation method based on different interpolation basis functions such as moving least squares(MLS),least squares(LS),and finite element method(FEM)to solve the IEVP.Compared with the Galerkin method based on finite element or Legendre polynomials,the main advantage of the interpolation method is that,in the calculation of eigenvalues and eigenfunctions in one-dimensional random fields,the integral matrix containing covariance function only requires a single integral,which is less than a two-folded integral by the Galerkin method.The effectiveness and computational efficiency of the proposed interpolation method are verified through various one-dimensional examples.Furthermore,based on theKL expansion and polynomial chaos expansion,the stochastic analysis of two-dimensional regular and irregular domains is conducted,and the basis function of the extended finite element method(XFEM)is introduced as the interpolation basis function in two-dimensional irregular domains to solve the IEVP.

Characterizing large-scale weak interlayer shear zones using conditional random field theory

The shear behavior of large-scale weak intercalation shear zones(WISZs)often governs the stability of foundations,rock slopes,and underground structures.However,due to their wide distribution,undulating morphology,complex fabrics,and varying degrees of contact states,characterizing the shear behavior of natural and complex large-scale WISZs precisely is challenging.This study proposes an analytical method to address this issue,based on geological fieldwork and relevant experimental results.The analytical method utilizes the random field theory and Kriging interpolation technique to simplify the spatial uncertainties of the structural and fabric features for WISZs into the spatial correlation and variability of their mechanical parameters.The Kriging conditional random field of the friction angle of WISZs is embedded in the discrete element software 3DEC,enabling activation analysis of WISZ C2 in the underground caverns of the Baihetan hydropower station.The results indicate that the activation scope of WISZ C2 induced by the excavation of underground caverns is approximately 0.5e1 times the main powerhouse span,showing local activation.Furthermore,the overall safety factor of WISZ C2 follows a normal distribution with an average value of 3.697.

Impact of random and scattered coincidences from outside of field of view on positron emission tomography/computed tomography imaging with different reconstruction protocols

Image quality in positron emission tomography(PET)is affected by random and scattered coincidences and reconstruction protocols.In this study,we investigated the effects of scattered and random coincidences from outside the field of view(FOV)on PET image quality for different reconstruction protocols.Imaging was performed on the Discovery 690 PET/CT scanner,using experimental configurations including the NEMA phantom(a body phantom,with six spheres of different sizes)with a signal background ratio of 4:1.The NEMA phantom(phantom I)was scanned separately in a one-bed position.To simulate the effect of random and scatter coincidences from outside the FOV,six cylindrical phantoms with various diameters were added to the NEMA phantom(phantom II).The 18 emission datasets with mean intervals of 15 min were acquired(3 min/scan).The emission data were reconstructed using different techniques.The image quality parameters were evaluated by both phantoms.Variations in the signal-to-noise ratio(SNR)in a 28-mm(10-mm)sphere of phantom II were 37.9%(86.5%)for ordered-subset expectation maximization(OSEM-only),36.8%(81.5%)for point spread function(PSF),32.7%(80.7%)for time of flight(TOF),and 31.5%(77.8%)for OSEM+PSF+TOF,respectively,indicating that OSEM+PSF+TOF reconstruction had the lowest noise levels and lowest coefficient of variation(COV)values.Random and scatter coincidences from outside the FOV induced lower SNR,lower contrast,and higher COV values,indicating image deterioration and significantly impacting smaller sphere sizes.Amongst reconstruction protocols,OSEM+PSF+TOF and OSEM+PSF showed higher contrast values for sphere sizes of 22,28,and 37 mm and higher contrast recovery coefficient values for smaller sphere sizes of 10 and 13 mm.

CONVERGENCE RATES IN THE STRONG LAWS OF NONSTATIONARYρ~*-MIXING RANDOM FIELDS

By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing random fields.

CONVERGENCERATESIN THESTRONG LAWSOFASYMPTOTICALLY NEGATIVELY ASSOCIATEDRANDOM FIELDS

In this paper, a notion of negative side ρ \|mixing ( ρ\+- \|mixing) which can be regarded as asymptotic negative association is defined, and some Rosenthal type inequalities for ρ\+- \|mixing random fields are established. The complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are also discussed for ρ\+-\| mixing random fields. The results obtained extend those for negatively associated sequences and ρ\+*\| mixing random fields.

Neumann stochastic finite element method for calculating temperature field of frozen soil based on random field theory

To study the effect of uncertain factors on the temperature field of frozen soil, we propose a method to calculate the spatial average variance from just the point variance based on the local average theory of random fields. We model the heat transfer coefficient and specific heat capacity as spatially random fields instead of traditional random variables. An analysis for calculating the random temperature field of seasonal frozen soil is suggested by the Neumann stochastic finite element method, and here we provide the computational formulae of mathematical expectation, variance and variable coefficient. As shown in the calculation flow chart, the stochastic finite element calculation program for solving the random temperature field, as compiled by Matrix Laboratory （MATLAB） sottware, can directly output the statistical results of the temperature field of frozen soil. An example is presented to demonstrate the random effects from random field parameters, and the feasibility of the proposed approach is proven by compar- ing these results with the results derived when the random parameters are only modeled as random variables. The results show that the Neumann stochastic finite element method can efficiently solve the problem of random temperature fields of frozen soil based on random field theory, and it can reduce the variability of calculation results when the random parameters are modeled as spatial- ly random fields.

Some Limit Theorems for Weighted Sums of Random Variable Fields

Let{Xn^-,n^-∈N^d}be a field of Banach space valued random variables, 0 〈r〈p≤2 and{an^-,k^-, （n^-,k^-） ∈ N^d × N^d ,k^-≤n^-} a triangular array of real numhers, where N^d is the d-dimensional lattice （d≥1 ）. Under the minimal condition that {｜｜Xn^-｜｜ r,n^- ∈N^d} is {｜an^-,k^-｜^r,（n^-,k^-）} ∈ N^d ×N^d,k^-≤n^-}-uniformly integrable, we show that ∑（k^-≤n^-）an^-,k^-,Xk^-^（L^r（or a,s,）→0 as ｜n^-｜→∞ In the above, if 0〈r〈1, the random variables are not needed to be independent. If 1≤r〈p≤2, and Banach space valued random variables are independent with mean zero we assume the Banaeh space is of type p. If 1≤r≤p≤2 and Banach space valued random variables are not independent we assume the Banach space is p-smoothable.

Precise asymptotics in the law of the logarithm for random fields in Hilbert space

Consider the positive d-dimensional lattice Z^d（d≥2） with partial ordering ≤, let {XK; K∈Z＋^d} be i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with mean zero and covariance operator ∑ and set partial sums SN =∑K≤nXK,K,N∈Z＋^d. Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε↓ by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of （Gut and Spataru, 2003）.

THE DECISION OF THE OPTIMAL PARAMETERS IN MARKOV RANDOM FIELDS OF IMAGES BY GENETIC ALGORITHM

This paper introduces the principle of genetic algorithm and the basic method of solving Markov random field parameters.Focusing on the shortcomings in present methods,a new method based on genetic algorithms is proposed to solve the parameters in the Markov random field.The detailed procedure is discussed.On the basis of the parameters solved by genetic algorithms,some experiments on classification of aerial images are given.Experimental results show that the proposed method is effective and the classification results are satisfactory.

Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

The thermodynamics and the phase diagram of random field Ising model （RFIM） on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field （B）. A Gaussian distribution of random field （hi） with zero mean and variance （hi2 = H2RF is considered. The free-energy （F）, the magnetization （M） and the order parameter （q） are investigated for several values of coordination number （z）. The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature Tc = J/k and when HRF = 0 for finite z. The free-energy （F） values increase as T increases for different intensities of random field （HRE） and finite z. The internal energy （U） has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field HRF increases, The ferromagnetic （FM） paramagnetic （PM） phase boundary is clearly observed only when z →∞. While FM PM-spin glass （SG） phase boundaries are present for finite z. The magnetic susceptibility （X） shows a sharp cusp at Tc in a small random field for finite z and rounded different peaks on increasing HRF.

Phase diagrams of spin-3/2 Ising model in the presence of random crystal field within the effective field theory based on two approximations

The bimodal random crystal field （A） effects are investigated on the phase diagrams of spin-3/2 Ising model by using the effective-field theory with correlations based on two approximations： the general van der Waerden identity and the approximated van der Waerden identity. In our approach, the crystal field is either turned on or turned off randomly for a given probability p or q = 1 -p, respectively. Then the phase diagrams are constructed on the （A,kT/J） and （p,kT/J） planes for given p and A, respectively, when the coordination number is z = 3. Furthermore, the effect of randomization of the crystal field is illustrated on the （△,kT/J） plane for p = 0.5 when z - 3,4, and 6. All these are carried out for both approximations and then the results are compared to point out the differences. In addition to the lines of second-order phase transitions, the model also exhibits first-order phase transitions and the lines of which terminate at the isolated critical points for high p values.

CONVERGENCE RATES IN THE STRONG LAWS FOR A CLASS OF DEPENDENT RANDOM FIELDS

By using a Rosenthal type inequality established in this paper,the complete convergence rates in the strong laws for a class of dependent random fields are discussed.And the result obtained extends those for ρ --mixing random fields,ρ *-mixing random fields and negatively associated fields.

Simulation of artificial random field considering the temporal-spatial variation

An apparent wave velocity varying with the frequency of seismic wave was adopted rather than an arbitrary one. The phase difference spectrum was introduced to consider the non-stationary properties of frequency contents in simulating artificial random field. The non-stationary random field on hard site considering the temporal-spatial variation was simulated by the way of spectral representation method. This random field can be used as the input of earthquake ground motion of the seismic response analysis of large-span spatial structures considering the effect of multi-supported excitation.

The Rate of Asymptotic Normality of Frequency Polygon Density Estimation for Spatial Random Fields

This paper is to investigate the convergence rate of asymptotic normality of frequency polygon estimation for density function under mixing random fields, which include strongly mixing condition and some weaker mixing conditions. A Berry-Esseen bound of frequency polygon is established and the convergence rates of asymptotic normality are derived. In particularly, for the optimal bin width , it is showed that the convergence rate of asymptotic normality reaches to ?when mixing coefficient tends to zero exponentially fast.

Effects of classical random external field on the dynamics of entanglement in a four-qubit system

We investigate the dynamics of entanglement through negativity and witness operators in a system of four noninteracting qubits driven by a classical phase noisy laser characterized by a classical random external field(CREF).The qubits are initially prepared in the GHZ-type and W-type states and interact with the CREF in two different qubit-field configurations,namely,common environment and independent environments in which the cases of equal and different field phase probabilities are distinguished.We find that entanglement exhibits different decaying behavior,depending on the input states of the qubits,the qubit-field coupling configuration,and field phase probabilities.On the one hand,we demonstrate that the coupling of the qubits in a common environment is an alternative and more efficient strategy to completely shield the system from the detrimental impacts of the decoherence process induced by a CREF,independent of the input state and the field phase probabilities considered.Also,we show that GHZ-type states have strong dynamics under CREF as compared to W-type states.On the other hand,we demonstrate that in the model investigated the system robustness’s can be greatly improved by increasing the number of qubits constituting the system.

Modified Maximum Likelihood Estimation of the Spatial Resolution for the Elliptical Gamma Camera SPECT Imaging Using Binary Inhomogeneous Markov Random Fields Models

In this work a complete approach for estimation of the spatial resolution for the gamma camera imaging based on the [1] is analyzed considering where the body distance is detected (close or far way). The organ of interest most of the times is not well defined, so in that case it is appropriate to use elliptical camera detection instead of circular. The image reconstruction is presented which allows spatially varying amounts of local smoothing. An inhomogeneous Markov random field (M.r.f.) model is described which allows spatially varying degrees of smoothing in the reconstructions and a re-parameterization is proposed which implicitly introduces a local correlation structure in the smoothing parameters using a modified maximum likelihood estimation (MLE) denoted as one step late (OSL) introduced by [2].

Quantum Theory of Mesoscopic Fractional Electric Fields in a Cavity of Viscous Medium

With conjecture of fractional charge quantization (quantum dipole/multiple moments), Fourier transform stretching, twisting and twigging of an electron quanta and waver strings of electron quanta, the mathematical expressions for mesoscopic fractional electron fields in a cavity of viscous medium and the associated quantum dielectric susceptibility are developed. Agreement of this approach is experimentally evidenced on barite and Fanja site molecular sieves. These findings are in conformity with experimental results of 2012 Physics Nobel prize winning scientists, Serge Haroche and David J. Wineland especially for cavity quantum electro-dynamics electron and its associated mesoscopic electric fields. The mover electron quanta strings lead to warping of space and time following the behaviour of quantum electron dynamics.

Research on the Seismic Wave Field of Karst Cavern Reservoirs near Deep Carbonate Weathered Crusts

Fracture and cavern hydrocarbon reservoirs in carbonates are an important pool type worldwide. The karst cavern reservoirs are easiest to identify on seismic reflection data. The prediction, exploration, and development of this type of reservoir require theoretical research on seismic wave fields reflected from complex inhomogeneous media. We compute synthetic seismic sections for fluidfilled cavern reservoirs of various heights and widths using random media models and inhomogeneous media elastic wave equations. Results indicate that even caverns significantly smaller than 1/ 4 wavelength are detectible on conventional band-width seismic sections as diffractions migrated into bead-type events. Diffraction amplitude is a function of cavern height and width. We introduce a width-amplitude factor which can be used to calculate the diffraction amplitude of a cavern with a limited width from the diffraction amplitude computed for an infinitely wide cavern.