Characterizing Fracture Spatial Patterns by Using Semivariograms

Semivariogram is applied to fracture data obtained from detailed scanline surveys of nine field sites in western New York, USA in order to investigate the spatial patterns of natural fractures. The length of the scanline is up to 36 m. How both fracture spacing and fracture length vary with distance is determined through semivariogram calculations. In this study, the authors developed a FORTRAN program to resample the fracture data from the scanline survey. By calculating experimental semivariogram, the authors found five different types of spatial patterns that can be described by linear, spherical, reversed spherical, polynomial I (for a<0) and polynomial II (for a>0) models, of which the last three are newly proposed in this study. The well-structured semivariograms of fracture spacing and length indicate that both the location of the fractures and the length distribution within their structure domains are not random. The results of this study also suggest that semivariograms can provide useful infor     

The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets

Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.     

Practically Applicable Central Limit Theorem for Spatial Statistics

Let {Z(s):s∈D⊆ℝ ^d } be a zero mean stationary random field observed at a finite number of locations. Lahiri (Sankhya Ser. A 65:356–388, 2003) proved spatial central limit theorems (CLT) for ∑ _i=1 ⁿ Z(s _i ) assuming a ‘nearly infill domain sampling’. Applications of his results depended on the underlying spatial sampling region and the design in a complicated fashion. The main objective of this paper is to provide CLTs that could be applied easily in practice. We present two main results assuming a ‘nearly infill domain sampling’ defined mainly in terms of dependence. Theorem 1 establishes a CLT for ∑ _i=1 ⁿ Z(s _i ) and Theorem 2 is obtained mainly for applications to density estimates. We report on a simulation study for illustrating a way of applying our results in practice.     

Optimal Interpolation of Gravity Maps Using a Modified Neural Network

This paper proposes an interpolation method based on a modified Kohonen artificial neural network, and is used to interpolate marine gravity data on a regular grid. This method combines accuracy comparable to that of kriging with a much shorter computing time than kriging. It is particularly efficient when both the size of the grid and the quantity of available data are large. Under some hypotheses similar to those of kriging with a trend, the unbiasedness and optimality of the method can be demonstrated. Comparison with kriging with a trend using marine gravity data shows similar results. Although neural interpolation is slightly less efficient, it is more robust outside of the marine data area.     

Applicability of Statistical Learning Algorithms for Spatial Variability of Rock Depth

Two algorithms are outlined, each of which has interesting features for modeling of spatial variability of rock depth. In this paper, reduced level of rock at Bangalore, India, is arrived from the 652 boreholes data in the area covering 220 sq⋅km. Support vector machine (SVM) and relevance vector machine (RVM) have been utilized to predict the reduced level of rock in the subsurface of Bangalore and to study the spatial variability of the rock depth. The support vector machine (SVM) that is firmly based on the theory of statistical learning theory uses regression technique by introducing ε-insensitive loss function has been adopted. RVM is a probabilistic model similar to the widespread SVM, but where the training takes place in a Bayesian framework. Prediction results show the ability of learning machine to build accurate models for spatial variability of rock depth with strong predictive capabilities. The paper also highlights the capability of RVM over the SVM model.     

Identifying the Representative Elementary Volume for Permeability in Heterolithic Deposits Using Numerical Rock Models

Using a range of realistic 3D numerical lithofacies (dm-scale) models of ripple laminated sandstone intercalated with mudstone we evaluate how single-phase permeability varies as a function of sample support. The models represent a range of mudstone content which is typical for tidal deposits. Furthermore, the spatial distribution of flow barriers (i.e. mudstone) is not random, but governed by sedimentological rules giving a variable anisotropy ratio as a function of mudstone content. Both vertical and horizontal permeability are found to vary at small sample volumes, but these fluctuations reduce as the sample volume increases. The vertical permeability increases while the horizontal permeability is nearly constant as a function of sample support for small mudstone contents. For higher mudstone content, the horizontal permeability decreases while the vertical permeability is nearly constant as a function of sample support. We propose a criterion, based on a normalised standard deviation, to determine the Representative Elementary Volume (REV). The size of the REV is dependent on both the property measured (vertical and horizontal permeability) and the correlation lengths of the lithological elements (i.e. lithofacies). Based on this we identify three flow upscaling regimes that each require a different method for upscaling: (1) layered systems where the arithmetic and harmonic averages are appropriate, (2) systems close to the percolation threshold where a percolation model should be used, and (3) discontinuous systems where an effective medium method provides the best estimate of permeability. The work gives, by using numerical experiments on a range of heterogeneous systems, a new insight in determination of the REV for permeability at the lithofacies scale and its relation to sedimentological parameters.     

New U-Pb Geochronological Constraints on Formation and Evolution of the Susong Complex Zone in the Dabie Orogen

<正>Objective The Susong complex zone(SCZ)is a relatively lowgrade metamorphic unit mostly with an epidoteamphibolite facies,located in the southernmost part of the Dabie orogen.However,its rock compositions,ages,metamorphic processes and petrogenesis are still     

A Distance-based Prior Model Parameterization for Constraining Solutions of Spatial Inverse Problems

Spatial inverse problems in the Earth Sciences are often ill-posed, requiring the specification of a prior model to constrain the nature of the inverse solutions. Otherwise, inverted model realizations lack geological realism. In spatial modeling, such prior model determines the spatial variability of the inverse solution, for example as constrained by a variogram, a Boolean model, or a training image-based model. In many cases, particularly in subsurface modeling, one lacks the amount of data to fully determine the nature of the spatial variability. For example, many different training images could be proposed for a given study area. Such alternative training images or scenarios relate to the different possible geological concepts each exhibiting a distinctive geological architecture. Many inverse methods rely on priors that represent a single subjectively chosen geological concept (a single variogram within a multi-Gaussian model or a single training image). This paper proposes a novel and practical parameterization of the prior model allowing several discrete choices of geological architectures within the prior. This method does not attempt to parameterize the possibly complex architectures by a set of model parameters. Instead, a large set of prior model realizations is provided in advance, by means of Monte Carlo simulation, where the training image is randomized. The parameterization is achieved by defining a metric space which accommodates this large set of model realizations. This metric space is equipped with a “similarity distance” function or a distance function that measures the similarity of geometry between any two model realizations relevant to the problem at hand. Through examples, inverse solutions can be efficiently found in this metric space using a simple stochastic search method.     

Reconstruction of Incomplete Data Sets or Images Using Direct Sampling

With increasingly sophisticated acquisition methods, the amount of data available for mapping physical parameters in the geosciences is becoming enormous. If the density of measurements is sufficient, significant non-parametric spatial statistics can be derived from the data. In this context, we propose to use and adapt the Direct Sampling multiple-points simulation method (DS) for the reconstruction of partially informed images. The advantage of the proposed method is that it can accommodate any data disposition and that it can indifferently deal with continuous and categorical variables. The spatial patterns found in the data are mimicked without model inference. Therefore, very few assumptions are required to define the spatial structure of the reconstructed fields, and very limited parameterization is needed to make the proposed approach extremely simple from a user perspective. The different examples shown in this paper give appealing results for the reconstruction of complex 3D geometries from relatively small data sets.     

Lunar Subsurface Mineralogy and Density Profile Modelling based on Moon Mineralogy Mapper Observation

The methods of deriving Fe O and Ti O_2 contents from the Clementine spacecraft data were discussed,and an approach was developed to derive the content from the measurements using the Moon Mineralogy Mapper (M~3) instrument on Chandrayaan-1.The density of lunar bedrock was then modeled on the basis of the derived Fe O and Ti O_2 abundances.The Fe O and Ti O_2 abundances derived from the M~3 data were compared with the previous results of the Clementine data and were in good agreement.The Fe O abundance data also agreed well with the Lunar Prospector data,which were used as an independent source.The previous Clementine and newly M~3 derived abundances were compared with the laboratory measured Fe O and Ti O_2 contents in the Apollo and Luna returned samples.The Clementine derived Fe O content was systematically 1%–2%lower than the laboratory measurements in all the returned samples.The M~3 derived content agreed well with the returned Apollo samples and was within±2.8%of the laboratory measurements.The Clementine derived Ti O_2abundance was systematically 0.1%–4%higher than the laboratory measurements of the returned samples.The M~3 derived Ti O_2 agreed well (±0.6%) with the laboratory measurements of the returned samples,except for samples with high Ti O_2content.However,these results should be carefully interpreted because the error range requires verification.No error analysis was provided with the previous Clementine derived contents.     

Fractal and Multifractal Propertiesof the Spatial Distribution of Natural Fractures——Analyses and Applications

The Cantor's dust theory is applied to investigate the scaling properties of the spatial distribution of natural fractures obtained from detailed scanline surveys of 27 field sites in the Appalachian Plateau of western New York, USA. The results obtained in this study indicate: 1) fracture spacing is characterized by fractal and multifractal properties. On small scales analyses yield an average fractal dimension of 0.15, which suggests a very high degree of clustering. In contrast, on large scales, fractures tend to be more regular and evenly distributed with an average fracture dimension of 0.52; 2) fractal dimension varies with different sets in different orientations, which can be attributed to interactions between pre-existing fractures and younger ones, as well as variations of the intensity of the stresses under which the fractures were formed; 3) a time sequence of fracture set formation can be proposed based on fractal and multifractal analyses, which consists of (from old to young): N-S, NW, EN     

Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the point-support model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in area-to-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum.This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a naïve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.     

Modified Markov Chain Monte Carlo Method for Dynamic Data Integration Using Streamline Approach

In this paper, the Markov Chain Monte Carlo (MCMC) approach is used for sampling of the permeability field conditioned on the dynamic data. The novelty of the approach consists of using an approximation of the dynamic data based on streamline computations. The simulations using the streamline approach allows us to obtain analytical approximations in the small neighborhood of the previously computed dynamic data. Using this approximation, we employ a two-stage MCMC approach. In the first stage, the approximation of the dynamic data is used to modify the instrumental proposal distribution. The obtained chain correctly samples from the posterior distribution; the modified Markov chain converges to a steady state corresponding to the posterior distribution. Moreover, this approximation increases the acceptance rate, and reduces the computational time required for MCMC sampling. Numerical results are presented.     

Scattered Data Approximation on the Bisphere and Application to Texture Analysis

The paper deals with the approximation and optimal interpolation of functions defined on the bisphere \mathbb S²×\mathbb S²\mathbb {S}^{2}\times \mathbb {S}^{2} from scattered data. We demonstrate how the least square approximation to the function can be computed in a stable and efficient manner. The analysis of this problem is based on Marcinkiewicz–Zygmund inequalities for scattered data which we present here for the bisphere. The complementary problem of optimal interpolation is also solved by using well-localized kernels for our setting. Finally, we discuss the application of the developed methods to problems of texture analysis in material science.     

The Tau Model for Data Redundancy and Information Combination in Earth Sciences: Theory and Application

Many decision-making processes in the Earth sciences require the combination of multiple data originating from diverse sources. These data are often indirect and uncertain, and their combination would call for a probabilistic approach. These data are also partially redundant with each other or with all others taken jointly. This overlap in information arises due to a variety of reasons—because the data arises from the same geology, because they originate from the same location or the same measurement device, etc. The proposed tau model combines partially redundant data, each taking the form of a prior probability for the event being assessed to occur given that single datum. The parameters of that tau model measure the additional contribution brought by any single datum over that of all previously considered data; they are data sequence-dependent and also data value-dependent. Data redundancy depends on the sequence in which the data is considered and also on the data values themselves. However, for a given sequence, averaging the tau model parameters over all possible data values leads to exact analytical expressions and corresponding approximations and inference avenues. Information on multiple-point connectivity of permeability arrives from core data, well-test data and seismic data which are defined over varying supports with complex redundancy between these information sources. In order to compute these tau weights for determining connectivity, one needs a model of data redundancy, here expressed as a vectorial training image (Ti) constructed using a prior conceptual knowledge of geology and the physics of data measurement. From such a vectorial Ti, the tau weights can be computed exactly. Neglecting data redundancy leads to an over-compounding of individual data information and the possible risk of making extreme decisions.     

Types, Metallogenic Environments and Characteristics of Temporal and Spatial Distribution of Copper Deposits in China

Based upon a comprehensive study of 123 copper deposits in China. this paper reaches the following conclusions: 1. The crust in China was solidified pretty late; with the accretion of the plate, the copper mineralization moved towards its margin in space and became successively younger in age. 2. The major copper ore types ever found in the world have mostly been discovered in China, in which the porphyry type seems to be the most important and the massive sulfide type in the transitional bed between marine clastic and carbonate rocks is clearly defined. 3. Carbonate strata are widespread in China and their deposition lasted for a long geological period, so the proportion of copper deposits occurring in them is large as compared with other parts of the world. 4. Seven metallogenic epochs can be recognized, in which the Mesozoic one plays the leading role and the middle(late) Palaeozoic and Cenozoic ones are next in importance. Mineralization was comparatively simple in the early geological period. and became diversified later on. In the early epochs copper deposits related to basaltic magmatism and metamorphosed marine sediments predominated, whereas in the later epochs those related to granitic magmatism and continental sedimentation were dominant. 5. There exist in China thirteen metallogenic provinces, of which the Lower Yangtze downwarping belt, Sanjiang fold system, Jiangnan axis and Xikang-Yunnan axis are of greater significance. 6. The crustal mobility in China was rather pronounced, the polycyclic evolution of the crust has resulted in such obvious phenomena as the inheritance of mineralization, the coexistence of various types and the superposition of different genetic types.     

Textures in deforming forsterite aggregates up to 8 GPa and 1673 K

We report results from axisymmetric deformation experiments carried out on forsterite aggregates in the deformation-DIA apparatus, at upper mantle pressures and temperatures (3.1–8.1 GPa, 1373–1673 K). We quantified the resulting lattice preferred orientations (LPO) and compare experimental observations with results from micromechanical modeling (viscoplastic second-order self-consistent model—SO). Up to 6 GPa (~185-km depth in the Earth), we observe a marked LPO consistent with a dominant slip in the (010) plane with one observation of a dominant [100] direction, suggesting that [100](010) slip system was strongly activated. At higher pressures (deeper depth), the LPO becomes less marked and more complex with no evidence of a dominant slip system, which we attribute to the activation of several concurrent slip systems. These results are consistent with the pressure-induced transition in the dominant slip system previously reported for olivine and forsterite. They are also consistent with the decrease in the seismic anisotropy amplitude observed in the Earth’s mantle at depth greater than ~200 km.