Spatial variability analysis of precipitation in northwest Iran

The spatial variability of precipitation was investigated in the northwestern corner of Iran using data collected at 24 synoptic stations from 1986 to 2015. Principal component analysis (PCA) and cluster analysis (CA) were used to regionalize precipitation in the study area. Eleven precipitation variables were averaged and arranged as an input matrix for the R-mode PCA to identify the precipitation patterns. Results suggest that the study area can be divided into four spatially homogeneous sub-zones. In addition, the spatial patterns of annual precipitation were identified by applying the T-mode PCA and CA to the annual precipitation data. The delineated spatial patterns revealed three distinct sub-regions. The resultant maps were compared with the spatial distribution of the rotated principal components (PCs). Results pointed out that the delineated clusters are characterized by different precipitation variability; and using different precipitation parameters can lead to different spatial patterns of precipitation over northwest Iran.     

Application of geostatistical methods in gold geochemical anomalies identification (Montemor-O-Novo, Portugal)

The study described herein concerns the application of geostatistical methods to data soil from Montemor-O-Novo area (Southern Portugal). In the area, the gold mineralised zones (Banhos, Caeiras, Falés, Gamela, Malaca and Monfurado) are characterised by different geological settings and mineralogical assemblages. A total of 1211 soil samples were collected in Montemor-O-Novo area and analysed for Cu, Pb, Zn, As, Ba and Au by atomic absorption spectrometry.To account for spatial structure, simple and cross variograms were computed for the main directions of the grid sampling. From the experimental variograms a linear model of coregionalization composed of three structures, a nugget effect and two anisotropic spherical structures, was fitted to each of the six variables. The coregionalization matrices deduced from the theoretical model show the relationships between the variables at different scales. These matrices were compared with those obtained by principal component analysis (PCA).This methodology was the basis for estimating the corresponding spatial components (Y0, Y1 and Y2) using factorial kriging analysis (FKA). Maps of raw data, Y0, Y1 and Y2 were made for each variable.The use of multivariate analysis permit the study of the spatial structure intrinsic to geochemical data and the identification and refinement of significant anomalies related to Au-bearing mineral deposits.     

Identifying spatial heterogeneity of coal resource quality in a multilayer coal deposit by multivariate geostatistics

This study presents a new geostatistical approach to characterization of the geometry and quality of a multilayer coal deposit using the data of seam thickness as a geometric property and the contents of ash, sodium, total sulphur, and the heating value as quality properties. A coal deposit in East Kalimantan (Borneo), Indonesia, which has a synclinal geological structure, was chosen as the study site. Semivariogram analysis clarified the strong dependence of heating value on ash content in the top and bottom parts of each seam and the existence of a strong correlation with sodium content over the sub-seams in the same location. The correlations between the geometry and quality of the seams were generally weak. A linear coregionalization model was used to derive the spatial correlation coefficients of two variables at each scale component from the single- and cross-semivariogram matrices. Because the data were correlated spatially in the same seam or over different seams, multivariate techniques (ordinary cokriging and factorial cokriging) were mainly used and the resultant spatial estimates were compared to those derived using a univariate technique (ordinary kriging). A factorial cokriging was effective to decompose the spatial correlation structures with different scales. Another important characteristic was that the sodium content shows distinct segregation: the low zones are concentrated near the boundary of the sedimentary basin, while the high zones are concentrated in the central part. The main component of sodium originates from the abundance of saline water. Therefore, it can be inferred that seawater had stronger effects on the coal depositional process in the central basin than in the border part. The geostatistical modeling results suggest that the thicknesses of all the major seams were controlled by the syncline structure, while the coal qualities chiefly were originated from the coal depositional and diagenetic processes.     

结构方程模型在地球化学数据分析中的应用

为了实现通过确定地球化学组合元素来反映成矿异常, 本文在主成分分析模型的基础上, 引入了新的结构方程模型(SEM).与主成分所不同的是, 结构模型综合了经典统计方法中的因子分析和路径分析方法, 以与研究对象具有较好的拟合度为标准来确定最优解, 并通过模型最优解来确定新的成分组合, 因此结构模型所确定的成分变量不一定是具有最大变化性, 而是与研究对象最接近的因子变量, 该因子能够更好地反映研究对象.介绍了结构方程模型方法的原理, 并利用加拿大Nova Scotia省西南部湖泊沉积物地球化学数据建立了与热液型金矿有关的地球化学元素结构方程模型, 研究了结构方程模型所给出的组合变量空间分布规律以及与金矿床的关系.与主成分分析方法所给出的计算结果进行对比发现, 结构模型所计算的与金矿相关的组合变量与矿床的空间相关性较高, 并且对金矿床(矿点)也具有较好的预测性.      

Geochemical characterization of heavy metal contaminated area using multivariate factorial kriging

This paper describes a geostatistical method, known as factorial kriging analysis, which is well suited for analyzing multivariate spatial information. The method involves multivariate variogram modeling, principal component analysis, and cokriging. It uses several separate correlation structures, each corresponding to a specific spatial scale, and yields a set of regionalized factors summarizing the main features of the data for each spatial scale. This method is applied to an area of high manganese-ore mining activity in Amapá State, North Brazil. Two scales of spatial variation (0.33 and 2.0 km) are identified and interpreted. The results indicate that, for the short-range structure, manganese, arsenic, iron, and cadmium are associated with human activities due to the mining work, while for the long-range structure, the high aluminum, selenium, copper, and lead concentrations, seem to be related to the natural environment. At each scale, the correlation structure is analyzed, and regionalized factors are estimated by cokriging and then mapped.     

Regionalized favorability theory for information synthesis in mineral exploration

This paper presents a regionalized method for the estimation of a favorability function through generalization of all relevant variables (explanatory and target) into random functions. The new method allows the use of cross-covariance functions in addition to ordinary covariances for extracting spatial joint information, which is virtually overlooked in the conventional analyses. The optimal weights for a favorability equation are derived from solving a generalized eigen-system established by the maximization of covariances between a favorability function and the principal components of a set of pre-selected target variables. Various correlation coefficients may be computed to assist in interpretation of the favorability estimates. Both favorability functions and correlation coefficients may be estimated for a point or a block. The regionalized favorability theory can be compared to cokriging in that both use the sample-sample covariances to account for the sample-sample relations and the point-sample covariances to account for the point-sample configurations. The new technique is demonstrated on a test case study, which involves the integration of geochemical, airborne-geophysical, and structural data sets for the target selection of hydrothermal gold-silver deposits.     

Factorial kriging with external drift: a case study on the Penedono Region,Portugal

Factorial kriging has been used in geochemical exploration for the estimation and cartography of the spatial components of the variables, helping with the identification and interpretation of geochemical anomalies. Those spatial components appear by the decomposition of the variables in its several structural components, given by the variograms. In this paper a new form of factorial kriging is introduced, by using the geologic information as an external drift. This was achieved considering as an external variable (external drift) sample co-ordinates on the first axis resultant from a principal component analysis (PCA), interpreted as a lithological factor. With this type of geostatistical technique each point appears in the resultant maps as a combination of geochemical and geological information, attending the geographic localization of the samples. This technique was tested on a set of 2450 sediment samples collected on a 640 km² area, between the Trás-os-Montes e Alto Douro and the Beira Alta regions. From the 34 initial elements analyzed (10 major elements P₂O₅, SiO₂, Al₂O₃, Fe₂O₃, MgO, CaO, Na₂O, K₂O, TiO₂ and MnO expressed in oxide percentage and 24 elements expressed in ppm As, Bi, Ag, Sb, W, B, Cu, Pb, Zn, Sn, Nb, Li, Be, Zr, Y, La, Ba, Cd, Mo, V, Cr, Co, Ni and Sr) only the results obtained for 2 of them are presented in this work. The first was Sn, which is associated with some mineralisation in this region and the other one was Zn, which shows similar behavior in the whole area, with the exception of a small region.     

Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity

This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.     

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.     

Exploring the joint compositional variability of major components and trace elements in the Tellus soil geochemistry survey (Northern Ireland)

The complexity of modern geochemical data sets is increasing in several aspects (number of available samples, number of elements measured, number of matrices analysed, geological-environmental variability covered, etc), hence it is becoming increasingly necessary to apply statistical methods to elucidate their structure. This paper presents an exploratory analysis of one such complex data set, the Tellus geochemical soil survey of Northern Ireland (NI). This exploratory analysis is based on one of the most fundamental exploratory tools, principal component analysis (PCA) and its graphical representation as a biplot, albeit in several variations: the set of elements included (only major oxides vs. all observed elements), the prior transformation applied to the data (none, a standardization or a logratio transformation) and the way the covariance matrix between components is estimated (classical estimation vs. robust estimation). Results show that a log-ratio PCA (robust or classical) of all available elements is the most powerful exploratory setting, providing the following insights: the first two processes controlling the whole geochemical variation in NI soils are peat coverage and a contrast between “mafic” and “felsic” background lithologies; peat covered areas are detected as outliers by a robust analysis, and can be then filtered out if required for further modelling; and peat coverage intensity can be quantified with the %Br in the subcomposition (Br, Rb, Ni).     

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

Comparison of mineralization pattern of geochemical data in spatial and position-scale domain using new DWT- PCA approach

In the current research to determine the mineralization pattern and discuss the mineralization components, the information of position - scale domain of geochemical data has been analyzed. A new method is proposed based on coupling discrete wavelet transforms (DWT) and principal component analysis (PCA) for mineralization elements forecasting applications. The results of this study indicate the potential of DWT–PCA method for geochemical data processing. Wavelet transform (WT), as a multi-spectral analysis method, can decompose the spatial and temporal signals into different frequencies. The features of mineralization can be identified using the position - scale domain of geochemical data that may not be achievable in spatial domain. The geochemical data from the Dalli region have been processed in the spatial domain using PCA. The surface geochemical data of 30 elements have been transformed to position–scale domain using two-dimensional discrete wavelet transform (2DDWT). Wavelet functions (WFs) of Haar, Coiflet2, Biorthogonal3.3 and Symlet7 have been applied separately to decompose the geochemical data to high and low frequencies in one level. To obtain more accurate and complete information of mineralization, a new index has been presented based on wavelet coefficients. Based on this new index, significant results have been obtained by using PCA of the index. The coefficients distribution map (CDM) as a new exploratory criterion has been generated based on 2DDWT to show the geochemical distribution map (GDM). Finally, the results of WT have been compared with the results of spatial domain and the best method of wavelet for interpretation of geochemical data has been introduced. The results of geochemical data analysis by DWT–PCA approach have been confirmed by the exploratory drillings in the study area.     

Relationship of geological and geothermal field properties: Midcontinent area, USA, an example

Quantitative approaches to data analysis in the last decade have become important in basin modeling and mineral-resource estimation. The interrelation of geological, geophysical, geochemical, and geohydrological variables is important in adjusting a model to a real-world situation. Revealing the interdependences of variables can contribute in understanding the processes interacting in sedimentary basins. It is reasonably simple to compare spatial data of the same type but more difficult if different properties are involved. Statistical techniques, such as cluster analysis or principal components analysis, or some algebraic approaches can be used to ascertain the relations of standardized spatial data. In this example, structural configuration on five different stratigraphic horizons, one total sediment thickness map, and four maps of geothermal data were copared. As expected, the structural maps are highly related because all had undergone about the same deformation with differing degrees of intensity. The temperature gradients derived (1) from shallow borehole logging measurements under equilibrium conditions with the surrounding rock, and (2) from non-equilibrium bottom-hole temperatures (BHT) from deeper depths are mainly independent of each other. This was expected and confirmed also for the two temperature maps at 1000 ft which were constructed using both types of gradient values. Thus, it is evident that the use of a 2-point (BHT and surface temperature) straightline calculation of a mean temperature gradient gives different information about the geothermal regime than using gradients from temperatures logged under equilibrium conditions. Nevertheless, it is useful to determine to what a degree the larger dataset of nonequilibrium temperatures could reflect quantitative relationships to geologic conditions. Comparing all maps of geothermal information vs. the structural and the sediment thickness maps, it was determined that all correlations are moderately negative or slightly positive. These results are clearly shown by the cluster analysis and the principal components. Considering a close relationship between temperature and thermal conductivity of the sediments as observed for most of the Midcontinent area and relatively homogeneous heat-flow density conditions for the study area these results support the following assumptions: (1) undifferentiated geothermal gradients, computed from temperatures of different depth intervals and differing sediment properties, cannot contribute to an improved understanding of the temperature structure and its controls within the sedimentary cover, and (2) the quantitative approach of revealing such relations needs refined datasets of temperature information valid for the different depth levels or stratigraphic units.     

Universal cokriging under intrinsic coregionalization

Under the intrinsic coregionalization model if both primary and secondary measurements are available at all sample locations, the conventional geostatistical wisdom is that cokriging provides exactly the same solution as univariate kriging on the primary process alone. However, recent eamples have been given where nonzero secondary cokriging weights have accurred under this spatial dependence structure. This note identifies the conditions under which secondary information is useful under the assumption of intrinsic coregionalization. An illustration is given using a dataset of plutonium and americium concentrations collected from a region of the Nevada Test Site.     

Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains

The continuous-lag Markov chain provides a conceptually simple, mathematically compact, and theoretically powerful model of spatial variability for categorical variables. Markov chains have a long-standing record of applicability to one-dimensional (1-D) geologic data, but 2- and 3-D applications are rare. Theoretically, a multidimensional Markov chain may assume that 1-D Markov chains characterize spatial variability in all directions. Given that a 1-D continuous Markov chain can be described concisely by a transition rate matrix, this paper develops 3-D continuous-lag Markov chain models by interpolating transition rate matrices established for three principal directions, say strike, dip, and vertical. The transition rate matrix for each principal direction can be developed directly from data or indirectly by conceptual approaches. Application of Sylvester's theorem facilitates establishment of the transition rate matrix, as well as calculation of transition probabilities. The resulting 3-D continuous-lag Markov chain models then can be applied to geo-statistical estimation and simulation techniques, such as indicator cokriging, disjunctive kriging, sequential indicator simulation, and simulated annealing.     

Assessing spatial variability in soil characteristics with geographically weighted principal components analysis

Dimensionality reduction methods such as principal components analysis (PCA) provide a means of identifying trends in soil characteristics which may be represented by a wide range of variables. However, these characteristics may be highly spatially variable and so the results from PCA represent, in some sense, an “average” of locally distinct characteristics. One approach to account for these local differences is to introduce a geographical weighting scheme into the PCA process. In this paper, such an approach is assessed in the exploration of soil characteristics in the state of Pennsylvania, USA. Data from 878 georeferenced soil profiles which include different soil parameters (n = 12) were extracted from the National Soil Survey Center database. Where data are parts of compositions (e.g., percentages of sand, silt, and clay), analysis using raw data is not appropriate and such data were transformed using log ratios (specifically, balances). Single variables (i.e., those which are not parts of compositions) were logged. The first two principal components explain over 50% of the variance. The mapped values suggest marked spatial variation in soil characteristics, but it is not possible to assess which of these variables explain most variation in particular regions from the simple maps of raw variables. Geographically weighted PCA (GWPCA) provides additional information which is obscured by PCA, and it also provides a set of component scores and loadings at all data locations. The soil variable with the largest loading at most locations of Pennsylvania is the logged base saturation (BSln), and this supports the findings of the conventional PCA analysis. While BSln loads most highly in most of the eastern third, the middle and the south west of the state, the northwest is less spatially consistent in terms of the variables which explain most variation. For GWPC 1, the variable with the second largest loading at most locations (i.e., primarily the south and west) is CEC.B1 (the log ratio of Ca, Mg, and Na to K and EXACID), while CEC.B2 (the log ratio of Ca and Mg to Na), pHln (logged pH) and BSln dominate in other areas. The GWPCA results suggest that there is marked spatial variation in multivariate soil characteristics across Pennsylvania state and that results from standard PCA obscure this considerable variation.     

Incorporation of principal component analysis,geostatistical interpolation approaches and frequency-space-based models for portraying the Cu-Au geochemical prospects in the Feizabad district,NW Iran

Delineation of mineralization-related geochemical anomalies of stream sediment data is an essential stage in regional geochemical exploration. In this study, principal component analysis (PCA) was applied to 12 selected elements to acquire a multi-element geochemical signature associated with Cu-Au mineralization in Feizabad district, NE Iran. The spatial distribution of enhanced multi-element geochemical signature of the second component (PC2) was modeled by different geostatistical procedures including variogram calculation, ordinary kriging (OK) and inverse distance weighting (IDW) interpolation techniques. Concentration-area (C-A) fractal and U-spatial statistics models were then applied to the continuous-value interpolated models for delineation of geochemical anomalies. Quantitative comparison of results based on the known mineral occurrences in the study area was carried out using normalized density index and success-rate curves. All generated models represent a high positive relation with known Cu (±Au) deposits in the study area, although, comparison of the results revealed that the OK-based U-spatial statistics model was superior to the rest of models. Besides, the low, moderate and high-intensity anomalies are spatially associated with geological-structural features in the study area.     

Joint simulation of multiple variables with a Markov-type coregionalization model

Many applications are multivariate in character and call for stochastic images of the joint spatial variability of multiple variables conditioned by a prior model of covariances and cross- covariances. This paper presents an algorithm to perform cosimulation of such spatially intercorrelated variables. This new algorithm builds on a Markov-type hypothesis whereby collocated information screens further away data of the same type, allowing cosimulation without the burden of a full cokriging. The proposed algorithm is checked against a synthetic multi-Gaussian reference dataset, then against a multi-Gaussian cosimulation approach using full cokriging. The results indicate that the proposed algorithm perform as well as the full cokriging approach in reproducing the univariate and bivariate statistics of the reference set, yet at less cpu cost.     

The control of maturity and kerogen type on quantitative analytical pyrolysis data

Quantitative pyrolysis-gas chromatography has been performed on 96 kerogen samples isolated from 17 wells on the Norwegian Continental shelf. Petrographic and bulk geochemical measurements were also performed on the samples, and a combined data set of 117 variables for each sample was analysed using principal components analysis (PCA). This approach provides an objective and reproducible means of kerogen characterisation, which can be easily automated. In addition to objective kerogen characterisation and facile visualisation of facies and maturity related chemical trends, the method has the potential to allow objective prediction of key geochemical parameters such as maturity level from pyrogram data.     

Indicator principal component kriging

An alternative to multiple indicator kriging is proposed which approximates the full coindicator kriging system by kriging the principal components of the original indicator variables. This transformation is studied in detail for the biGaussian model. It is shown that the cross-correlations between principal components are either insignificant or exactly zero. This result allows derivation of the conditional cumulative density function (cdf) by kriging principal components and then applying a linear back transform. A performance comparison based on a real data set (Walker Lake) is presented which suggests that the proposed method achieves approximation of the conditional cdf equivalent to indicator cokriging but with substantially less variogram modeling effort and at smaller computational cost.