Can a Training Image Be a Substitute for a Random Field Model?

In most multiple-point simulation algorithms, all statistical features are provided by one or several training images (TI) that serve as a substitute for a random field model. However, because in practice the TI is always of finite size, the stochastic nature of multiple-point simulation is questionable. This issue is addressed by considering the case of a sequential simulation algorithm applied to a binary TI that is a genuine realization of an underlying random field. At each step, the algorithm uses templates containing the current target point as well as all previously simulated points. The simulation is validated by checking that all statistical features of the random field (supported by the simulation domain) are retrieved as an average over a large number of outcomes. The results are as follows. It is demonstrated that multiple-point simulation performs well whenever the TI is a complete (infinitely large) realization of a stationary, ergodic random field. As soon as the TI is restricted to a limited domain, the statistical features cannot be obtained exactly, but integral range techniques make it possible to predict how much the TI should be extended to approximate them up to a prespecified precision. Moreover, one can take advantage of extending the TI to reduce the number of disruptions in the execution of the algorithm, which arise when no conditioning template can be found in the TI.     

Estimation of Sill Matrices in the Linear Model of Coregionalization

When using least squares to fit the linear model of coregionalization to multivariate geostatistical data, the sill matrices for the different regions must be estimated, subject to the constraint that they be non-negative definite. In 1992, Goulard and Voltz proposed and empirically examined an iterative algorithm for doing this. Although no proof was given for its convergence or for the uniqueness of the solution to the problem, the algorithm has subsequently been extensively and successfully used. In this paper, we prove that the minimization problem, in fact, has a unique solution and that the algorithm is guaranteed to converge to it from any starting point. We also discuss the effect of the starting point on the speed of convergence.     

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.     

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.     

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.     

Assessment of flood mitigation through riparian detention in response to a changing climate – a case study

Considering that urban areas may suffer more substantial losses than riparian farmlands during floods, diverting floodwater into riparian areas for temporal detention is expected to mitigate flood damage in downstream urban areas. In this study, an assessment has been conducted to evaluate the effect of flood mitigation through riparian detention in response to a changing climate in the Tou-Chien River basin of Taiwan. An integrated 1D–2D flow model was used to simulate the movement of flood wave in the main stream and the overbank flow inundating into the nearby lowlands. Based on the numerical simulation results, the flooding extents in the basin corresponding to different return periods of flood using existing flood prevention infrastructures were investigated. A detention strategy by lowering the levee along the riparian farmlands was proposed to avoid severe flooding in the densely populated urban areas of the basin. Research findings showed that the proposed detention measure can completely protect the downstream areas from overbank flooding when a flood having 20-yr period occurs, and can effectively alleviate the downstream flooding area from 27.4 to \(7.6\,\hbox {km}^{2}\) for a flood possessing 200-yr period.     

U-Pb Ages of the Late Early Cretaceous Magmatic Rocks in the Zhalantun Area of Inner Mongolia

A great deal of early-to-mid Early Cretaceous magmatic activities have been recorded in the Zhalantun area of Inner Mongolia,while the late Early to Late Cretaceous magmatic rocks have been barely reported(Guo et al.,2018;Zhang Xiangxin et al.,2017).At present,only a few Late Cretaceous magmatic activities were reported in the Arongqi area,such as volcanic rocks of the Gushanzhen Formation.However,the Gushanzhen Formation lacks accurate isotopic age,and contemporaneous intrusive rocks has not been reported yet.In this work,we collected the volcanic rocks from the Gushanzhen Formation and contemporaneous intrusive rocks in the Zhanlantun and nearby,and aim to figure out the formation ages of volcanic rocks of the Gushanzhen Formation and accompanied intrusive rocks by analyzing zircon U-Pb isotopes(Fig.1).     

Continental ca 1.7 – 1.69 Ga Fe-rich metatholeiites in the Curnamona Province,Australia: a record of melting of a heterogeneous,subduction-modified lithospheric mantle

Mg-rich and Fe-rich metatholeiites intruded the Willyama Supergroup of the southern Australian Curnamona Province in the Late Palaeoproterozoic at ca 1700 Ma and 1685 Ma, respectively. Intrusion of the Fe-rich metatholeiites occurred during a period of punctuated extension in the Willyama basin. Major-element concentrations are variable (SiO₂ 45.4 – 56.5 wt%; Fe₂O₃? 8.5 – 20.7; TiO₂ 0.46 – 2.52 wt%; Mg# 70.5 – 29.1) and, in conjunction with trace-element data, support near-closed-system fractionation of a mantle-derived melt with little or no replenishment. Fractionation produced progressively Fe-rich derivative melts. Crystallising phases were dominated by clinopyroxene and olivine, whereas Fe – (Ti) oxide crystallisation was hindered. Primitive mantle-normalised immobile trace elements are characterised by variable Th, Nb, Sr, P and Ti anomalies. Chondrite-normalised rare-earth element patterns for the most primitive, Mg-rich samples from the western Broken Hill Domain have La_N/Sm_N < 1, whereas the most evolved Fe-rich samples from the Olary Domain have ratios of La_N/Sm_N > 1. Initial εNd values range between – 2.2 and + 2.7 for the majority of the samples, with the isotopic compositions showing no correlation with differentiation or assimilation. The combined geochemical and isotopic data suggest that the southern Curnamona Province metatholeiites were extracted from a depleted mantle in the western Broken Hill Domain, and a variably enriched, heterogeneous subcontinental lithospheric mantle in the Olary Domain. Magmatism most likely occurred in a backarc basin or intracontinental setting. It is speculated that the geochemically enriched mantle component was derived from subduction-related processes, probably related to pre-Willyama basin accretionary processes along the southern and eastern margins of the North Australian Craton.     

Directional Wavelets and a Wavelet Variogram for Two-Dimensional Data

This paper describes two new approaches that can be used to compute the two-dimensional experimental wavelet variogram. They are based on an extension from earlier work in one dimension. The methods are powerful 2D generalizations of the 1D variogram that use one- and two-dimensional filters to remove different types of trend present in the data and to provide information on the underlying variation simultaneously. In particular, the two-dimensional filtering method is effective in removing polynomial trend with filters having a simple structure. These methods are tested with simulated fields and microrelief data, and generate results similar to those of the ordinary method of moments variogram. Furthermore, from a filtering point of view, the variogram can be viewed in terms of a convolution of the data with a filter, which is computed fast in O(NLogN) number of operations in the frequency domain. We can also generate images of the filtered data corresponding to the nugget effect, sill and range of the variogram. This in turn provides additional tools to analyze the data further.     

Log-Ratio Analysis Is a Limiting Case of Correspondence Analysis

It is common practice in compositional data analysis to perform the log-ratio transformation in order to preserve sub-compositional coherence in the analysis. Correspondence analysis is an alternative approach to analyzing ratio-scale data and is often contrasted with log-ratio analysis. It turns out that if one introduces a power transformation into the correspondence analysis algorithm, then the limit of the power-transformed correspondence analysis, as the power parameter tends to zero, is exactly the log-ratio analysis. Depending on how the power transformation is applied, we can obtain as limiting cases either Aitchison’s unweighted log-ratio analysis or the weighted form called “spectral mapping”. The upshot of this is that one can come as close as one likes to the log-ratio analysis, weighted or unweighted, using correspondence analysis.     

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     

Using Simultaneous Diagonalization to Identify a Space–Time Linear Coregionalization Model

Although there are multiple methods for modeling matrix covariance functions and matrix variograms in the geostatistical literature, the linear coregionalization model is still widely used. In particular it is easy to check to ensure whether the matrix covariance function is positive definite or that the matrix variogram is conditionally negative definite. One of the difficulties in using a linear coregionalization model is in determining the number of basic structures and the corresponding covariance functions or variograms. In this paper, a new procedure is given for identifying the basic structures of the space–time linear coregionalization model and modeling the matrix variogram. This procedure is based on the near simultaneous diagonalization of the sample matrix variograms computed for a set of spatiotemporal lags. A case study using a multivariate spatiotemporal data set provided by the Environmental Protection Agency of Lombardy, Italy, illustrates how nearly simultaneous diagonalization of the empirical matrix variograms simplifies modeling of the matrix variograms. The new methodology is compared with a previous one by analyzing various indices and statistics.     

Origin of a late Neoproterozoic (605 ± 13 Ma) intrusive carbonate–albitite complex in Southern Sinai, Egypt

New geochemical, isotopic, and geochronological data and interpretations are presented for late Neoproterozoic intrusive carbonates and related rocks of southern Sinai, Egypt (northernmost Arabian–Nubian Shield). The Tarr carbonates are coarsely crystalline and related to explosive emplacement of hypabyssal and volcanic albitite at 605 ± 13 Ma. The carbonates associated with the albitites are divisible into two types: primary dolomitite and secondary breunneritite (Fe-rich magnesite). The dolomitite was clearly intrusive but differs from classic igneous carbonatites, containing much lower abundances of incompatible elements, such as REE, U, Th, Rb, Nb, Y, P, Sr, Zr, Ba, and total alkalies. The breunneritite is a secondary replacement of dolomitite, probably marking the roots of a vigorous hydrothermal system. Albitites show pristine abundances of major and trace elements and were not subjected to a major metamorphic overprint. They are relatively more fractionated, alkaline and related to within-plate A-type magmas, were emplaced in an extensional or non-compressive tectonic regime in the cupola of high-level A-type granite. Tarr albitites may represent residual magma remaining after near-total crystallization of an A-type granite pluton at depth, forcibly emplaced into the roof above the cooling pluton. The intrusive dolomitite exsolved from highly differentiated albitite melt, in the apical regions of a still-buried alkaline “A-type” granite pluton that was rich in CO₂; these volatiles migrated upwards and towards the cooler margins of the magma body. Late NNE-SSW extension allowed a shallow-level cupola to form, into which albitite melts and carbonate fluids migrated, culminating in explosive emplacement of albitite breccia and intrusive carbonate. Isotopic compositions of Tarr dolomitite and albitite indicate these are consanguineous and ultimately of mantle origin. Magmatic volatiles fenitized the wall rock, while submarine hydrothermal activity transformed some of the dolomitite into breunneritite. Recognition of Tarr-type should encourage similar hypabyssal complex intrusions to be sought for in association with A-type granitic plutons elsewhere.     

Functional Methods for Classification of Different Petrographic Varieties by Means of Reflectance Spectra

The need for improved product quality in the aggregates industry is driving the search for greater automation in rock type identification. In practice, reflectance spectra in visible and near-infrared light may reliably be used for the classification of rock classes and their variants. Previous studies introduced statistical classification of six rock variants by means of infrared spectra. The present investigation extends these studies to cover twelve rock types and variants of worldwide economic importance. These were measured by visible and near-infrared light. Statistical classification of these spectra is highly challenging due to the high number of groups and the high dimensionality of the data. In functional data analysis, spectra are regarded as curves instead of vectors of characteristics. To obtain a compact form that is more susceptible to further analysis, the spectra are represented by a B-spline basis. Two functional versions of linear support vector machines and penalized functional discriminant analysis are considered for classification. The multiclass problem is addressed by margin trees and by considering all one-against-one classifications combined with a voting strategy for testing. Since classification error estimated by 5-fold cross-validation is very low, in particular for penalized discriminant analysis, we conclude that the rock types can be classified reliably.     

Hierarchical Geostatistical Analysis of an Experimental Stratigraphy

A hierarchical geostatistical analysis is conducted on a high-resolution, multiscale hydraulic conductivity (ln K) map, created by scaling up an experimental stratigraphy. Unlike a previous study which evaluates ln K variograms within individual depositional environments, this study analyzes deposits (or samples) that incorporate multiple depositional environments. Based on conductivity cutoffs selected from a global ln K histogram, an indicator map is created to divide the deposits into 4 categories: sand, silty sand, clayey silt, and clay (Hierarchy-I). Based on facies and facies assemblage types selected using geological criteria, two more indicator maps are created at a higher hierarchy (Hierarchy-II) to divide the deposits into 14 units and 2 units, respectively. For each sample, its experimental ln K variogram is decomposed into 4 auto- and cross-transition component variograms. The decomposition characteristics are then evaluated against the underlying heterogeneity and specific division rule. The analysis reveals that: (1) ln K cutoffs (sand contents of the physical stratigraphy) can be used to distinguish the shifts in dominant deposition mode; (2) sample univariate modes depend on the choice of hierarchical division; (3) sample variograms exhibit smooth-varying correlation structures (exponential-like variograms are observed in samples with a large variance in mean facies length); (4) the decomposition characteristics are sensitive to the division based on conductivity cutoffs, but not sensitive to the division based on depositional environment (For all samples, with appropriate division, the sample variogram is closely approximated by the sum of the cross-transition component variograms.); and (5) at the Hierarchy-II level, the 2-unit division gives similar decomposition characteristics as the 14-unit division. For the select samples, parsimony in hierarchical division is achieved at the facies assemblage scale.     

New SHRIMP geochronology for the Western Fold Belt of the Mt Isa Inlier: developing a 1800 – 1650 Ma event framework ∗

The integration of detrital and magmatic U – Pb zircon SHIRMP geochronology with facies analysis has allowed the development of a chronostratigraphic framework for the Leichhardt and Calvert Superbasins of the Western Fold Belt, Mt Isa Inlier. This new event chart recognises three supersequences in the Leichhardt Superbasin: the Guide, Myally and Quilalar Supersequences. The Guide Supersequence spans the interval ca 1800 – 1785 Ma and includes the Bottletree Formation and the Mt Guide Quartzite. Sequence relationships suggest that this sedimentary package represents an asymmetric second-order cycle, recording a thickened transgressive suite of deposits and a comparatively thin second-order highstand. The overlying Myally Supersequence spans the interval ca 1780 – 1765 Ma and includes the Eastern Creek Volcanics and syndepositional Lena Quartzite, and the Myally Subgroup. This package represents a second-order supersequence cycle in which mafic volcanism was initiated during a phase of east – west extension. Following the cessation of volcanism, transgression led to the deposition of the Alsace Quartzite and deeper water Bortala Formation. An increase in the rate of sediment supply over accommodation resulted in progradation and deposition of the Whitworth Quartzite and redbed playa facies of the Lochness Formation as accommodation closed. The Quilalar Supersequence spans the interval ca 1755 – 1740 Ma. Sequence analysis in the eastern part of the Leichhardt River Fault Trough identifies a transgressive suite of facies at the base of this supersequence. Black shales from the upper part of the transgressive deposits characterise the condensed section for this supersequence. Facies analysis indicates that deposition took place in a series of storm-, tide- and wave-dominated shelfal marine depositional systems. Although there are no new depositional age constraints for the younger Bigie Formation, field relationships suggest that it is coeval with, or immediately preceded, the ca 1710 Ma Fiery magmatic event. Therefore, a separate supersequence is defined for the Bigie Formation, the Big Supersequence, even though it may be more genetically related to the Fiery magmatic event. The Big Supersequence, together with the ca 1690 Ma Prize Supersequence, comprise the Calvert Superbasin. The evolution of the Leichhardt and Calvert Superbasins are temporally and spatially related to magmatism. In particular, the new maximum depositional ages for the Guide and Myally Supersequences refine the age of the Eastern Creek Volcanics to ca 1780 – 1775 Ma. The new age for the Weberra Granite is within error of the age for the Fiery Creek Volcanics, indicating that they are both part of the ca 1710 Ma Fiery event. New ages for the Sybella Granite confirm that magmatism associated with this magmatic event is refined to 1680 – 1670 Ma, and is followed by deposition of the Gun Supersequence. Combining the new geochronological constraints with previous work now provides a detailed stratigraphic event framework between 1800 and 1575 Ma for the Western Fold Belt of the Mt Isa Inlier, and allows detailed comparisons and correlations with the Eastern Fold Belt and other Proterozoic terranes.     

Reaction rim growth in the system MgO-Al2O3-SiO2 under uniaxial stress

We synthesize reaction rims between thermodynamically incompatible phases in the system MgO-Al₂O₃-SiO₂ applying uniaxial load using a creep apparatus. Synthesis experiments are done in the MgO-SiO₂ and in the MgO-Al₂O₃ subsystems at temperatures ranging from 1150 to 1350 °C imposing vertical stresses of 1.2 to 29 MPa at ambient pressure and under a constant flow of dry argon. Single crystals of synthetic and natural quartz and forsterite, synthetic periclase and synthetic corundum polycrystals are used as starting materials. We produce enstatite rims at forsterite-quartz contacts, enstatite-forsterite double rims at periclase-quartz contacts and spinel rims at periclase-corundum contacts. We find that rim growth under the “dry” conditions of our experiments is sluggish compared to what has been found previously in nominally “dry” piston cylinder experiments. We further observe that the nature of starting material, synthetic or natural, has a major influence on rim growth rates, where natural samples are more reactive than synthetic ones. At a given temperature the effect of stress variation is larger than what is anticipated from the modification of the thermodynamic driving force for reaction due to the storage of elastic strain energy in the reactant phases. We speculate that this may be due to modification of the physical properties of the polycrystals that constitute the reaction rims or by deformation under the imposed load. In our experiments rim growth is very sluggish at forsterite-quartz interfaces. Rim growth is more rapid at periclase-quartz contacts. The spinel rims that are produced at periclase-corundum interfaces show parabolic growth indicating that reaction rim growth is essentially diffusion controlled. From the analysis of time series done in the MgO-Al₂O₃ subsystem we derive effective diffusivities for the Al₂O₃ and the MgO components in a spinel polycrystal as ${\rm D}_{MgO} = 1.4 \pm 0.2 \cdot 10^{-15}$  m²/s and ${\rm D}_{Al_2O_3} = 3.7 \pm 0.6 \cdot 10^{-16}$  m²/s for T?=?1350 °C and a vertical stress of 2.9 MPa.