Minimum/maximum autocorrelation factors applied to grade estimation

There are many situations in the mining industry where grade estimation of multiple correlated variables is required. The resulting model is expected to reproduce the data correlation, but there is no guarantee that the correlation observed among data will be reproduced by the model if the variables are independently estimated by kriging, and the correlation is not explicitly taken into account. The best geostatistical approach to address this estimation problem is to use co-kriging, which requires both cross and direct covariance modeling of all variables. However, the co-kriging method is labor-intensive when the problem involves more than three attributes. An alternative is to decorrelate the variables and estimate each one independently, using, for instance, the minimum/maximum autocorrelation factors (MAF) approach. This method involves the application of a linear transformation to the correlated variables, transforming the original data into a space where they are uncorrelated. The resulting transformed data can be individually estimated using kriging, avoiding the use of the linear model of coregionalization. Once the kriging has been performed, the MAF estimates are back-transformed to the original data space, re-establishing their correlation.The methodology is illustrated in a case study where there are two variables with correlation coefficient, ρ = ?0.98. The MAF transformation was applied in combination with ordinary kriging (herein denoted as KMAF). Co-kriging was performed to provide a benchmark for comparing the results obtained through KMAF. The results obtained by co-kriging and KMAF showed less than 1 % average deviation between the two block models.     

Simulation of intrinsic random fields of order <Emphasis Type="Italic">k</Emphasis> with a continuous spectral algorithm

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matérn generalized cross-covariance.     

An approximate method for joint sequential simulation of multiple spatial variables

Multivariate simulation is an important longstanding problem in geostatistics. Fitting a model of coregionalization to many variables is intractable and often not permitted; however, the matrix of collocated correlation coefficients is often well informed. Performing a matrix simulation with LU decomposition of the correlation matrix at each step of sequential simulation is implemented in some software. The target correlation matrix is not reproduced because of conditioning to local data and the particular variable ordering in the sequential/LU decomposition. A correction procedure is developed to calculate a modified correlation matrix that leads to reproduction of the target correlation matrix. The theoretical and practical aspects of this correction are developed.     

A fully non-stationary linear coregionalization model for multivariate random fields

This paper introduces an extension of the traditional stationary linear coregionalization model to handle the lack of stationarity. Under the proposed model, coregionalization matrices are spatially dependent, and basic univariate spatial dependence structures are non-stationary. A parameter estimation procedure of the proposed non-stationary linear coregionalization model is developed under the local stationarity framework. The proposed estimation procedure is based on the method of moments and involves a matrix-valued local stationary variogram kernel estimator, a weighted local least squares method in combination with a kernel smoothing technique. Local parameter estimates are knitted together for prediction and simulation purposes. The proposed non-stationary multivariate spatial modeling approach is illustrated using two real bivariate data examples. Prediction performance comparison is carried out with the classical stationary multivariate spatial modeling approach. According to several criteria, the prediction performance of the proposed non-stationary multivariate spatial modeling approach appears to be significantly better.     

Multivariate grid-free geostatistical simulation with point or block scale secondary data

A novel grid-free geostatistical simulation method (GFS) allows representing coregionalized variables as an analytical function of the coordinates of the simulation locations. Simulation on unstructured grids, regridding and refinement of available realizations of natural phenomena including, but not limited to, environmental systems are possible with GFS in a consistent manner. The unconditional realizations are generated by utilizing the linear model of coregionalization and Fourier series-based decomposition of the covariance function. The conditioning to data is performed by kriging. The data can be measured at scattered point-scale locations or sampled at a block scale. Secondary data are usually used in conjunction with primary data for the improved modeling. Satellite imaging is an example of exhaustively sampled secondary data. Improvements and recommendations are made to the implementation of GFS to properly assimilate secondary exhaustive data sets in a grid-free manner. Intrinsic cokriging (ICK) is utilized to reduce computational time and preserve the overall quality of the simulation. To further reduce the computational cost of ICK, a block matrix inversion is implemented in the calculation of the kriging weights. A projection approach to ICK is proposed to avoid artifacts in the realizations around the edges of the exhaustive data region when the data do not cover the entire modeling domain. The point-scale block value representation of the block-scale data is developed as an alternative to block cokriging to integrate block-scale data into realizations within the GFS framework. Several case studies support the proposed enhancements.     

Borehole radar velocity inversion using cokriging and cosimulation

A new radar velocity tomography method is presented based on slowness covariance modeling and cokriging of the slowness field using only measured travel time data. The proposed approach is compared to the classical LSQR algorithm using various synthetic models and a real data set. In each case, the proposed method provides comparable to or better results than LSQR. One advantage of this approach is that it is self-regularized and requires less a priori information. The covariance model also allows stochastic imaging of slowness fields by geostatistical simulations. Stable characteristics and uncertain features of the inverted models can then be easily identified.     

Stochastically conditioned flow paths and travel time distribution in highly heterogeneous synthesized aquifers / Chemins d’écoulement et distribution des temps de parcours au sein d’aquifères synthétiques hautement hétérogènes sous conditions stochastiques

Abstract

Abstract Characterization of heterogeneity at the field scale generally requires detailed aquifer properties such as transmissivity and hydraulic head. An accurate delineation of these properties is expensive and time consuming, and for many if not most groundwater systems, is not practical. As an alternative approach, stochastic representation of random fields is used and presented in this paper. Specifically, an iterative stochastic conditional simulation approach was applied to a hypothetical and highly heterogeneous pre-designed aquifer system. The approach is similar to the classical co-kriging technique; it uses a linear estimator that depends on the covariance functions of transmissivity (T), and hydraulic head (h), as well as their cross-covariances. A linearized flow equation along with a conditional random field generator constitutes the iterative process of the conditional simulation. One hundred equally likely realizations of transmissivity fields with pre-specified geostatistical parameters were generated, and conditioned to both limited transmissivity and head data. The successful implementation of the approach resulted in conditioned flow paths and travel-time distribution under different degrees of aquifer heterogeneity. This approach worked well for fields exhibiting small variances. However, for random fields exhibiting large variances (greater than 1.0), an iterative procedure was used. The results show that, as the variance of the ln[T] increases, the flow paths tend to diverge, resulting in a wide spectrum of flow conditions, with no direct discernable relationship between the degree of heterogeneity and travel time. The applied approach indicates that high errors may result when estimation of particle travel times in a heterogeneous medium is approximated by an equivalent homogeneous medium.     

随机介质非均匀地质体尺度研究(英文)

为了定量地反映复杂非均匀介质非均匀地质体的尺度大小,本文利用统计学方法建立了能够很好地描述复杂非均匀介质特征的随机介质模型,模型参量自相关长度描述了非均匀介质横向和纵向上非均匀体的平均尺度。基于所建立的随机介质模型通过速度的横向变化和速度标准差分别探讨了自相关长度与非均匀体尺度之间的关系。对速度横向变化的研究表明:随机介质内速度具有一定均值和方差并呈随机扰动特征;随着模型自相关长度的增大,非均匀体尺度也随之增大。通过速度标准差的研究得出自相关长度与非均匀体尺度之间关系的拟合公式,利用此公式可以定量地获取非均匀体尺度的实际大小。     

Kinematic transformations for planar multi‐directional pseudodynamic testing

The pseudodynamic (PSD) test method imposes command displacements to a test structure for a given time step. The measured restoring forces and displaced position achieved in the test structure are then used to integrate the equations of motion to determine the command displacements for the next time step. Multi‐directional displacements of the test structure can introduce error in the measured restoring forces and displaced position. The subsequently determined command displacements will not be correct unless the effects of the multi‐directional displacements are considered. This paper presents two approaches for correcting kinematic errors in planar multi‐directional PSD testing, where the test structure is loaded through a rigid loading block. The first approach, referred to as the incremental kinematic transformation method, employs linear displacement transformations within each time step. The second method, referred to as the total kinematic transformation method, is based on accurate nonlinear displacement transformations. Using three displacement sensors and the trigonometric law of cosines, this second method enables the simultaneous nonlinear equations that express the motion of the loading block to be solved without using iteration. The formulation and example applications for each method are given. Results from numerical simulations and laboratory experiments show that the total transformation method maintains accuracy, while the incremental transformation method may accumulate error if the incremental rotation of the loading block is not small over the time step. A procedure for estimating the incremental error in the incremental kinematic transformation method is presented as a means to predict and possibly control the error. Copyright © 2009 John Wiley & Sons, Ltd.     

Fast algorithms for automatic mapping with space-limited covariance functions

In this paper we discuss a fast Bayesian extension to kriging algorithms which has been used successfully for fast, automatic mapping in emergency conditions in the Spatial Interpolation Comparison 2004 (SIC2004) exercise. The application of kriging to automatic mapping raises several issues such as robustness, scalability, speed and parameter estimation. Various ad-hoc solutions have been proposed and used extensively but they lack a sound theoretical basis. In this paper we show how observations can be projected onto a representative subset of the data, without losing significant information. This allows the complexity of the algorithm to grow as O(n m ²), where n is the total number of observations and m is the size of the subset of the observations retained for prediction. The main contribution of this paper is to further extend this projective method through the application of space-limited covariance functions, which can be used as an alternative to the commonly used covariance models. In many real world applications the correlation between observations essentially vanishes beyond a certain separation distance. Thus it makes sense to use a covariance model that encompasses this belief since this leads to sparse covariance matrices for which optimised sparse matrix techniques can be used. In the presence of extreme values we show that space-limited covariance functions offer an additional benefit, they maintain the smoothness locally but at the same time lead to a more robust, and compact, global model. We show the performance of this technique coupled with the sparse extension to the kriging algorithm on synthetic data and outline a number of computational benefits such an approach brings. To test the relevance to automatic mapping we apply the method to the data used in a recent comparison of interpolation techniques (SIC2004) to map the levels of background ambient gamma radiation.

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Admissible nested covariance models over spheres cross time

Nested covariance models, defined as linear combinations of basic covariance functions, are very popular in many branches of applied statistics, and in particular in geostatistics. A notorious limit of nested models is that the constants in the linear combination are bound to be nonnegative in order to preserve positive definiteness (admissibility). This paper studies nested models on d-dimensional spheres and spheres cross time. We show the exact interval of admissibility for the constants involved in the linear combinations. In particular, we show that at least one constant can be negative. One of the implications is that one can obtain a nested model attaining negative correlations. We provide characterization theorems for arbitrary linear combinations as well as for nonconvex combinations involving two covariance functions. We illustrate our findings through several examples involving nonconvex combinations of well-known parametric families of covariance functions.     

A reservoir skeleton-based multiple point geostatistics method

Traditional stochastic reservoir modeling, including object-based and pixel-based methods, cannot solve the problem of reproducing continuous and curvilinear reservoir objects. The paper first dives into the various stochastic modeling methods and extracts their merits, then proposes the skeleton-based multiple point geostatistics (SMPS) for the fluvial reservoir. The core idea is using the skeletons of reservoir objects to restrict the selection of data patterns. The skeleton-based multiple point geostatistics consists of two steps. First, predicting the channel skeleton (namely, channel centerline) by using the method in object-based modeling. The paper proposes a new method of search window to predict the skeleton. Then forecasting the distributions of reservoir objects using multiple point geostatistics with the restriction of channel skeleton. By the restriction of channel centerline, the selection of data events will be more reasonable and the realization will be achieved more really. The checks by the conceptual model and the real reservoir show that SMPS is much better than Sisim (sequential indicator simulation), Snesim (Single Normal Equation Simulation) and Simpat (simulation with patterns) in building the fluvial reservoir model. This new method will contribute to both the theoretical research of stochastic modeling and the oilfield developments of constructing highly precise reservoir geological models.

     

Modeling of Wide-Angle Seismic Attributes Using the Gaussian Beam Method

In this study, observed seismic attributes from shot gather 11 of the SAREX experiment are used to derive a preliminary velocity and attenuation model for the northern end of the profile in southern Alberta. Shot gather 11 was selected because of its prominent Pn arrivals and good signal to noise ratio. The 2-D Gaussian beam method was used to perform the modeling of the seismic attributes including travel times, peak envelope amplitudes and pulse instantaneous frequencies for selected phases. The preliminary model was obtained from the seismic attributes from shot gather 11 starting from prior tomographic results. The amplitudes and instantaneous frequencies were used to constrain the velocity and attenuation structure, with the amplitudes being more sensitive to the velocity gradients and the instantaneous frequencies more sensitive to the attenuation structure. The resulting velocity model has a velocity discontinuity between the upper and lower crust, and lower velocity gradients in the upper and lower crust compared to earlier studies. The attenuation model has Q _p ^-1 values between 0.011 and 0.004 in the upper crust, 0.0019 in the lower crust and a laterally variable Q _p ^-1 in the upper mantle. The Q _p ^-1 values are similar to those found in Archean terranes from other studies. Although the results from a single gather are non-unique, the initial model derived here provides a self-consistent starting point for a more complete seismic attribute inversion for the velocity and attenuation structure.     

A geostatistically based inverse model for three-dimensional variably saturated flow

We present a geostatistically based inverse model for characterizing heterogeneity in parameters of unsaturated hydraulic conductivity for three-dimensional flow. Pressure and moisture content are related to perturbations in hydraulic parameters through cross-covariances, which are calculated to first-order. Sensitivities needed for covariance calculations are derived using the adjoint state sensitivity method. Approximations of the conditional mean parameter fields are then obtained from the cokriging estimator. Correlation between parameters and pressure – moisture content perturbations is seen to be strongly dependent on mean pressure or moisture content. High correlation between parameters and pressure data was obtained under saturated or near saturated flow conditions, providing accurate estimation of saturated hydraulic conductivity, while moisture content measurements provided accurate estimation of the pore size distribution parameter under unsaturated flow conditions.     

Three‐term inversion of prestack seismic data using a weighted l2, 1 mixed norm

We present a new inversion method to estimate, from prestack seismic data, blocky P‐ and S‐wave velocity and density images and the associated sparse reflectivity levels. The method uses the three‐term Aki and Richards approximation to linearise the seismic inversion problem. To this end, we adopt a weighted mixed l_{2, 1}‐norm that promotes structured forms of sparsity, thus leading to blocky solutions in time. In addition, our algorithm incorporates a covariance or scale matrix to simultaneously constrain P‐ and S‐wave velocities and density. This a priori information is obtained by nearby well‐log data. We also include a term containing a low‐frequency background model. The l_{2, 1} mixed norm leads to a convex objective function that can be minimised using proximal algorithms. In particular, we use the fast iterative shrinkage‐thresholding algorithm. A key advantage of this algorithm is that it only requires matrix–vector multiplications and no direct matrix inversion. The latter makes our algorithm numerically stable, easy to apply, and economical in terms of computational cost. Tests on synthetic and field data show that the proposed method, contrarily to conventional l₂‐ or l₁‐norm regularised solutions, is able to provide consistent blocky and/or sparse estimators of P‐ and S‐wave velocities and density from a noisy and limited number of observations.     

目标函数叠前保幅偏移方法与应用

将理论反射率与偏移反射率的差作为目标函数,给出一种迭代振幅补偿保幅偏移方法.把偏移看作一个反问题,寻找反问题中的最优解.偏移算子是正演算子的伴随共轭,其伴随矩阵非对角占优.通过预条件优化伴随矩阵,使Hessian矩阵准对角化.依据地震波传播稳定相位理论,计算反射点,以反射点为中心、菲涅耳带为半径偏移.考虑振幅几何扩散补偿、散射角度补偿,在迭代反演过程求出最优解,即保幅偏移解.本文给出了一个保幅数值模拟结果和一个实际地震数据实例.     

On the impact of salinity observations on state estimates in Ems Estuary

The hydrodynamics of Ems Estuary are dominated by tides and their interaction with buoyancy forcing. Such an environment is challenging for any effort to bring together observations and model results. In this study, we investigate how salinity measurements in the Ems Estuary affect the reconstruction of the salinity field. Similar to the traditional observing system experiments, the impact of specific observational arrays is simulated in the framework of statistical experiments. The experimental algorithm mainly relies on the model covariance matrix. Each experiment results in an estimate of the reconstruction error. The analysed observation configurations involve single and multiple, as well as stationary and non-stationary observing arrays. Generally, the reconstruction of the ocean state improves with increasing the density of observations. It appears that certain locations are more favourable for reconstruction than others. In fact, the regions separating the main dynamical realms resist strongest to the reconstruction effort. Extending the covariance matrix by the temporal cross-covariances between the model grid points enables to evaluate the impact of observations taken from a moving platform. This approach further improves the outcome of the experiments, resulting in reconstruction errors near zero with the exception of the tidal river. The cross-covariance information is able to tackle even the irregular dynamics arising on the border between the different physical regimes.     

A geostatistically based inverse model for three-dimensional variably saturated flow

We present a geostatistically based inverse model for characterizing heterogeneity in parameters of unsaturated hydraulic conductivity for three-dimensional flow. Pressure and moisture content are related to perturbations in hydraulic parameters through cross-covariances, which are calculated to first-order. Sensitivities needed for covariance calculations are derived using the adjoint state sensitivity method. Approximations of the conditional mean parameter fields are then obtained from the cokriging estimator. Correlation between parameters and pressure – moisture content perturbations is seen to be strongly dependent on mean pressure or moisture content. High correlation between parameters and pressure data was obtained under saturated or near saturated flow conditions, providing accurate estimation of saturated hydraulic conductivity, while moisture content measurements provided accurate estimation of the pore size distribution parameter under unsaturated flow conditions.     

联合多点地质统计学与序贯高斯模拟的随机反演方法

岩相和储层物性参数是油藏表征的重要参数,地震反演是储层表征和油气藏勘探开发的重要手段.随机地震反演通常基于地质统计学理论,能够对不同类型的信息源进行综合,建立具有较高分辨率的储层模型,因而得到广泛关注.其中,概率扰动方法是一种高效的迭代随机反演策略,它能综合考虑多种约束信息,且只需要较少的迭代次数即可获得反演结果.在概率扰动的优化反演策略中,本文有效的联合多点地质统计学与序贯高斯模拟,并结合统计岩石物理理论实现随机反演.首先,通过多点地质统计学随机模拟,获得一系列等可能的岩相模型,扰动更新初始岩相模型后利用相控序贯高斯模拟建立多个储层物性参数模型;然后通过统计岩石物理理论,计算相应的弹性参数;最后,正演得到合成地震记录并与实际地震数据对比,通过概率扰动方法进行迭代,直到获得满足给定误差要求的反演结果.利用多点地质统计学,能够更好地表征储层空间特征.相控序贯高斯模拟的应用,能够有效反映不同岩相中储层物性参数的分布.提出的方法可在较少的迭代次数内同时获得具有较高分辨率的岩相和物性参数反演结果,模型测试和实际数据应用验证了方法的可行性和有效性.

     

Studying instrumental linear polarization at the Large Solar Vacuum Telescope (LSVT)

Impact linear polarization in solar flares is studied with the Large Solar Vacuum Telescope (LSVT) using the spectral polarimetric method. This method makes it possible to minimize the effect of instrumental polarization with an error of up to 10⁻² owing to the normalization of the spectral line intensity to the continuum spectrum intensity with negligible linear polarization. As a result, the Hα line intensity in two orthogonally polarized spectral stripes coincides in the absence of solar polarization. However, in the presence of linear polarization in a flare, the spectral polarimetric method does not rule out that the error can be present in determining the Stokes parameters Q and U because of their possible relative “leakage.” Linear instrumental polarization of LSVT has been performed using polaroid rotation before the major mirror. Twelve elements of a telescope matrix, characterizing linear polarization, have been determined. The usage of a matrix makes it possible to specify the observed Q and U values accurate to 10⁻³ of their magnitude.