On the pseudo cross-variogram

Normal cross-variograms cannot be estimated from data in the usual way when there are only a few points where both variables have been measured. But the experimental pseudo cross-variogram can be computed even where there are no matching sampling points, and this appears as its principal advantage. The pseudo cross-variogram may be unbounded, though for its existence the intrinsic hypothesis alone is not a sufficient stationarity condition. In addition the differences between the two random processes must be second order stationary. Modeling the function by linear coregionalization reflects the more restrictive stationarity condition: the pseudo cross-variogram can be unbounded only if the unbounded correlation structures are the same in all variograms. As an alternative to using the pseudo cross-variogram a new method is presented that allows estimating the normal cross variogram from data where only one variable has been measured at a point.     

Pseudo-cross variograms, positive-definiteness, and cokriging

Cokriging allows the use of data on correlated variables to be used to enhance the estimation of a primary variable or more generally to enhance the estimation of all variables. In the first case, known as the undersampled case, it allows data on an auxiliary variable to be used to make up for an insufficient amount of data. Original formulations required that there be sufficiently many locations where data is available for both variables. The pseudo-cross-variogram, introduced by Clark et al. (1989), allows computing a related empirical spatial function in order to model the function, which can then be used in the cokriging equations in lieu of the cross-variogram. A number of questions left unanswered by Clark et al. are resolved, such as the availability of valid models, an appropriate definition of positive-definiteness, and the relationship of the pseudo-cross-variogram to the usual cross-variogram. The latter is important for modeling this function.     

Multivariable spatial prediction

For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.     

Comparison between different kriging estimators

Six different geostatistical estimators (linear kriging, lognormal kriging, and disjunctive kriging, each with and without a nonbias, i.e., universality condition) were compared using data from a polymetallic deposit in Algeria. The differences between estimators with and without the nonbias condition were far more pronounced than between the different kriging methods. This highlights the importance of choosing an appropriate stationarity model for the data. The criterion concerning kriging weight of the mean in simple kriging, proposed by Remacre (1984, 1987) and Rivoirard (1984) was found to be helpful for determining blocks where the choice of the stationarity hypothesis was critical.     

The Variance-Based Cross-Variogram: You Can Add Apples and Oranges

The variance-based cross-variogram between two spatial processes, Z₁ (·) and Z₂ (·), is var (Z₁ ( u ) – Z₂ ( v )), expressed generally as a bivariate function of spatial locations uandv. It characterizes the cross-spatial dependence between Z₁ (·) and Z₂ (·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z₁ ( u ) – Z₁ ( v ), Z₂ ( u ) – Z₂ ( v )). One concern with the variance-based cross-variogram has been that Z₁ (·) and Z₂ (·) might be measured in different units (apples and oranges). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z₁ (·) and Z₂ (·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of uandv only through the difference u – v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.     

Generalized cross-covariances and their estimation

Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations. They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models.     

Multiple-Point Simulation with an Existing Reservoir Model as Training Image

The multiple-point simulation (MPS) method has been increasingly used to describe the complex geologic features of petroleum reservoirs. The MPS method is based on multiple-point statistics from training images that represent geologic patterns of the reservoir heterogeneity. The traditional MPS algorithm, however, requires the training images to be stationary in space, although the spatial distribution of geologic patterns/features is usually nonstationary. Building geologically realistic but statistically stationary training images is somehow contradictory for reservoir modelers. In recent research on MPS, the concept of a training image has been widely extended. The MPS approach is no longer restricted by the size or the stationarity of training images; a training image can be a small geometrical element or a full-field reservoir model. In this paper, the different types of training images and their corresponding MPS algorithms are first reviewed. Then focus is placed on a case where a reservoir model exists, but needs to be conditioned to well data. The existing model can be built by process-based, object-based, or any other type of reservoir modeling approach. In general, the geologic patterns in a reservoir model are constrained by depositional environment, seismic data, or other trend maps. Thus, they are nonstationary, in the sense that they are location dependent. A new MPS algorithm is proposed that can use any existing model as training image and condition it to well data. In particular, this algorithm is a practical solution for conditioning geologic-process-based reservoir models to well data.     

我国东海及邻近海域气体水合物可能的分布范围

由气体水合物稳定存在的温度-压力条件,根据我国东海及其邻近海域的海底温度、地壳热流和地温梯度的分布,指出了该海区气体水合物可能的分布范围。主要在东海陆坡即冲绳海槽的西坡、冲绳海槽的东坡、和琉球海沟的西坡、琉球海沟和菲律宾海盆。分析指出,目前并不能肯定东海陆架区不存在气体水合物稳定存在的区域。由于冲绳海槽的热流值较高,特别是冲绳海槽的中部轴区,那里有较高的地温梯度,因而,海槽两坡的气体水合物主要分布在南部冲绳海槽.     

Distribution of kriging error and stationarity of the variogram in a coal property

If a particular distribution for kriging error may be assumed, confidence intervals can be estimated and contract risk can be assessed. Contract risk is defined as the probability that a block grade will exceed some specified limit. In coal mining, this specified limit will be set in a coal sales agreement. A key assumption necessary to implement the geostatistical model is that of local stationarity in the variogram. In a typical project, data limitations prevent a detailed examination of the stationarity assumption. In this paper, the distribution of kriging error and scale of variogram stationarity are examined for a coal property in northern West Virginia.     

Rainfall flood hazard at nuclear power plants in India

Flood hazard evaluation is an important input for Nuclear Power Plants external events safety studies. In the present study, flood hazard at various nuclear sites in India due to rainfall has been evaluated. Hazard estimation is a statistical procedure by which rainfall intensity versus occurrence frequency is estimated from historical records of rainfall data and extrapolated with asymptotic extreme value distribution. Rainfall data needed for flood hazard assessment are daily annual maximum rainfall (24?h data). The observed data points have been fitted using Gumbel, power law and exponential distribution, and return period has been estimated. To study the stationarity of rainfall data, a moving window estimate of the parameters has been performed. The rainfall pattern is stationary in both coastal and inland regions over the period of observation. The coastal regions show intense rainfall and higher variability than inland regions. Based on the plant layout, catchment area and drainage capacity, the prototype fast breeder reactor (PFBR) site is unlikely to be flooded.     

Kriging and Spatial Design Accelerated by Orders of Magnitude: Combining Low-Rank Covariance Approximations with FFT-Techniques

Computational power poses heavy limitations to the achievable problem size for Kriging. In separate research lines, Kriging algorithms based on FFT, the separability of certain covariance functions, and low-rank representations of covariance functions have been investigated, all three leading to drastic speedup factors. The current study combines these ideas, and so combines the individual speedup factors of all ideas. This way, we reduce the mathematics behind Kriging to a computational complexity of only $\mathcal{O}(dL^{*} \log L^{*})$ , where L ^? is the number of points along the longest edge of the involved lattice of estimation points, and d is the physical dimensionality of the lattice. For separable (factorized) covariance functions, the results are exact, and nonseparable covariance functions can be approximated well through sums of separable components. Only outputting the final estimate as an explicit map causes computational costs of $\mathcal{O}(n)$ , where n is the number of estimation points. In illustrative numerical test cases, we achieve speedup factors up to 10⁸ (eight orders of magnitude), and we can treat problem sizes of up to 15 trillion and two quadrillion estimation points for Kriging and spatial design, respectively, within seconds on a contemporary desktop computer. The current study assumes second-order stationarity and simple Kriging on a regular, equispaced lattice, without working with restricted neighborhoods. Extensions to many other cases are straightforward.     

涡动相关仪观测数据的处理与质量评价研究

涡动相关仪能够较准确地直接测量地表—大气间的湍流交换,在世界范围内得到了广泛的应用。但它的使用是有条件限制的,如果不进行必要的修正,得到的通量就可能有较大的误差。以密云观测站一年的涡动相关仪观测数据为例进行分析。结果表明：野点值剔除、坐标旋转以及超声温度订正对地表感热、潜热等通量的测量结果影响均在±1%之内,但坐标旋转对动量通量影响较大,必须对潜热和CO2通量进行空气密度效应订正;湍流谱在惯性副区基本满足-2/3次方定律,协谱基本满足-4/3次方定律。经过对观测数据的筛选和处理后,约75%的观测数据质量较好,2%的数据需要剔除。通量贡献源区分析表明,全天和白天均有超过70%的通量源区落在感兴趣区域内,超过90%的通量贡献最大点落在感兴趣区域内。     

Mineral equilibria in a six-component seawater system,Na-K-Mg-Ca-SO4-Cl-H2O,at 25°C

Phase relations in the 6-component system Na-K-Mg-Ca-SO₄-Cl-H₂O have been calculated for halite saturation, 25°C and 1 atm pressure. Using a Jänecke projection with the apices Ca-Mg-K₂-SO₄, 27 stable invariant points have been located which are connected by 69 univariant curves. Polyhalite is the only quaternary solid, but anhydrite occupies the bulk of the interior tetrahedral space. Consequently, 24 of the invariant points lie very close to the Ca-free base, Mg-K₂-SO₄. The remaining three points involve tachyhydrite and/or antarcticite. All points but two (20,27) represent peritectic conditions. Metastable equilibria have been calculated for the Ca-free system and yield relations corresponding to the solar diagram.Seawater lies in the subspace anhydrite-halite-carnallite-kieserite-bischofite (point 20) and its evaporation has been discussed for conditions of equilibrium and fractional crystallization. After gypsum is converted to anhydrite, halite precipitates. The next phase, under equilibrium conditions, is glauberite, crystallizing at the expense of anhydrite. Continued evaporation leads to glauberite resorption and eventual replacement by polyhalite. Then follow the magnesium sulfates epsomite, hexahydrite and kieserite, which are joined by carnallite. Polyhalite is replaced by anhydrite and bischoflte is added at the final invariant condition. Kainite does not appear as a primary phase under equilibrium conditions, but it is an important phase during fractional crystallization, where Ca-phases are not allowed to back-react with the brine.Up to the appearance of glauberite, thickness ratios of halite: anhydrite couplets (equilibrium or fractionation) can vary from 0 to 7, the relative amount of halite increasing with more intense evaporation. During evaporation, the activity of H₂O decreases from 0.98 (seawater) to 0.34 (final invariant brine). The data provided can be used to evaluate the effects of mineral precipitation, evaporation and brine mixing for a wide variety of natural brines.     

Derivatives of Spatial Variances of Growing Windows and the Variogram

Geostatistical analysis of spatial random functions frequently uses sample variograms computed from increments of samples of a regionalized random variable. This paper addresses the theory of computing variograms not from increments but from spatial variances. The objective is to extract information about the point support space from the average or larger support data. The variance is understood as a parametric and second moment average feature of a population. However, it is well known that when the population is for a stationary random function, spatial variance within a region is a function of the size and geometry of the region and not a function of location. Spatial variance is conceptualized as an estimation variance between two physical regions or a region and itself. If such a spatial variance could be measured within several sizes of windows, such variances allow the computation of the sample variogram. The approach is extended to covariances between attributes that lead to the cross-variogram. The case of nonpoint sample support of the blocks or elements composing each window is also included. A numerical example illustrates the application of this conceptualization.     

Time-series analysis in cyclic stratigraphy: An example from the Cretaceous of the Southern Alps, Italy

The importance of time-series analysis in cyclic stratigraphy is evaluated by comparing three different methods (adaptive multiple taper spectral analysis, auto-/cross-correlation analysis, cova functions) applied to stratigraphic time series from the Early Cretaceous Cismon section in northern Italy. Carbonate content and magnetic susceptibility vary in a quasi-cyclic fashion in this pelagic limestone section and are almost perfectly negatively correlated. The spectral technique requires a high degree of preprocessing of the original data (interpolation and resampling at a regular interval, filtering, inversion) which introduces smoothing and rounding errors. The statistical correlation analysis also requires evenly and (for cross-correlation) correspondingly spaced series. The geostatistical cova functions (a generalization of the cross-variogram) prove to be the most versatile and robust of the methods compared. Cova functions can be calculated from unevenly and noncor-respondingly spaced time series without any preprocessing. This method also retains relatively more of the signal if noise and extreme outliers obscure the picture. The periodicities detected in the Cismon time series fall in the range of Milankovitch cycles. Cycle periods of 45 cm and 80 cm likely correspond to dominant precession and obliquity cycles. Due to the inaccuracy of the Cretaceous time scale periods cannot be matched exactly yet, but cycle ratios are close to expected ratios so that there is great potential for future cyclostratigraphic work to contribute to a substantial improvement of the geologic time scale.     

Moving Surface Spline Interpolation Based on Green’s Function

Some commonly used interpolation algorithms are analyzed briefly in this paper. Among all of the methods, biharmonic spline interpolation, which is based on Green’s function and proposed by Sandwell, has become the mainstream method for its high precision, simplicity and flexibility. However, the minimum curvature method has two flaws. First, it suffers from undesirable oscillations between data points, which is solved by interpolation with splines in tension. Second, the computation time is approximately proportional to the cube of the number of data constraints, making the method slow for situations with dense data coverage. Focusing on the second problem, this paper introduces the moving surface spline interpolation method based on Green’s function, and the interpolation error equations are deduced. Because the proposed method only chooses the nearest data points by using the merge sort algorithm for interpolating, the computation time is greatly decreased. The optimal number of the nearest points can be determined by using the interpolation error estimation equation. No matter how many data points there are, this method can be implemented without difficulty. Examples show that the proposed method can obtain high interpolation precision and high computation speed at the same time.     

西安黄土土性参数随机场模型的适用性探讨

能否用齐次正态随机场模型来模拟土体性质的空间分布特性,其关键在于土性随机场模型是否具有平稳性和各态历经性。国内大多数研究都是在假定土性随机场模型是平稳随机场的条件下研究应用Vanmarcke 模型的。关于黄土的土性参数的空间特性研究亦是如此,未对其土性剖面随机场的平稳性及各态历经性进行检验。本文以西安市曲江某项目29个钻孔的双桥静力触探(CPT)数据为研究样本,首先讨论了关于原始数据的齐次化处理,即趋势分量的消除方法。然后对该场地黄土土性剖面随机场的平稳性及各态历经性进行了检验,检验结果表明:西安黄土梁洼地貌上的Q3黄土层、Q3古土壤层和Q2黄土层其土性随机场模型具平稳性和各态历经性。故Vanmarcke 随机场模型适用于模拟西安黄土土性剖面。     

Statistical estimation of the apex of a sediment distribution system from paleocurrent data

Precise statistical models are proposed for the distribution of paleocurrent directions in distributary systems. These models are based on a von Mises distribution for directions of paleocurrents in a system. The method of maximum likelihood is used to obtain an estimated position and confidence regions for the apex of the system. Both axial data (where only the trend of the paleocurrent is known) and directional data (where the direction of paleoflow is known) can be used. These models have been applied to two fluvial distributary systems in flat-lying Miocene deposits in the Ebro Basin, northern Spain. The estimate of the position of the apex of one of these systems is in good agreement with the position at the basin margin predicted from sedimentary facies information. The estimate of the position of the apex of the second system lies outside of the sedimentary basin; this result is considered to indicate that the distributary system had several feeder points along the basin margin, a situation which is predicted also from facies distributions. This statistical procedure can be used successfully on small data sets of around a hundred paleocurrent readings provided that they are scattered evenly across the area or lie in a wide arc.     

A modified randomized maximum likelihood for improved Bayesian history matching

Randomized maximum likelihood is known in the petroleum reservoir community as a Bayesian history matching technique by means of minimizing a stochastic quadratic objective function. The algorithm is well established and has shown promising results in several applications. For linear models with linear observation operator, the algorithm samples the posterior density accurately. To improve the sampling for nonlinear models, we introduce a generalized version in its simplest form by re-weighting the prior. The weight term is motivated by a sufficiency condition on the expected gradient of the objective function. Recently, an ensemble version of the algorithm was proposed which can be implemented with any simulator. Unfortunately, the method has some practical implementation issues due to computation of low rank pseudo inverse matrices and in practice only the data mismatch part of the objective function is maintained. Here, we take advantage of the fact that the measurement space is often much smaller than the parameter space and project the prior uncertainty from the parameter space to the measurement space to avoid over fitting of data. The proposed algorithms show good performance on synthetic test cases including a 2D reservoir model.     

Statistical regression applied to borehole strain measurements data analysis

Summary In mining and geotechnical engineering, it is usually necessary to carry out field measurements in order to obtain information. Parameters are often measured indirectly and calculated based on certain relationships to the measured quantities. More often, the number of measurements taken is greater than the minimum required, in order to increase the reliability of results. However, some data points are less reliable than others for reasons such as measurement errors; a solution which best fits the measurement data is obtained accordingly. As a result, there is a residual or a difference between the individual quantities measured and those predicted from the best-fit solution. This brings about a question of how big a residual is acceptable for a solution to be reliable. It is also important to know whether the data point with the largest residual is the most erroneous, whether those data points with large residuals should be deleted and how many of them should be deleted. Standard deviation may provide a measure of the data divergence but it is questionable if this parameter can be used as a measure of the reliability of solution. In order to solve these problems, the author has done extensive study in this area, especially as part of geotechnical data analysis. In this paper, the statistical multiple regression method is introduced to analyse the measurement data. The method is applied to the analysis ofin situ stress measurement and can be easily adopted to analyse data from other field measurements and laboratory tests. An example is included which illustrates the analysis procedure and shows the advantages of the method.