The multivariate (co)variogram as a spatial weighting function in classification methods

The multivariate variogram and the multivariate covariogram are used as spatial weighting functions for forming spatially homogeneous groups automatically. The groups are created after either deflating similarities between distant samples with the multivariate covariogram or by inflating dissimilarities between distant samples with the multivariate variogram. These approaches can be seen as generalization of the Oliver and Webster proposal. Two data sets show the efficiency of the two weighting functions when compared to the classical approach which does not take spatial information into account. In one case study, the weighting of similarities by the multivariate covariogram showed more interpretable results than the weighting of dissimilarities by the multivariate variogram.     

Numeric examination of multivariate soil samples

Numerical methods for the examination of multivariate soil samples are presented in geometric terms. Techniques of coordinate representation by principal components, by nonmetric scaling, and by a new method are discussed, as are techniques for agglomerative hierarchic cluster analysis. These are illustrated by two sets of previously published data.     

Spatial analysis of two-dimensional orientation data

A frequently encountered problem in structural geology is the analysis and interpretation of directional data with a geographical distribution. Such data may arise, for example, as samples of fracture strikes collected at a number of locations (rocky outcrops, road cuttings, etc.), trends of geological structures mapped in underground mines where they are exposed in the tunnel roof, or as lineaments annotated on aerial or satellite photographs. From such data, it is of interest to determine the structural domains in the area, that is, to partition the area into subareas of homogeneous structural pattern. This paper presents a method for performing such an analysis, and applies it to a data set collected from an underground coal mine.     

An empirical assessment and illustration of some multivariate morphometric techniques

An artificial data set is used to illustrate the morphologic properties of some common multivariate techniques and consideration of three common situations. The first concerns a sample showing no obvious groupings. In this situation principal components (or coordinates) and factor analyses give a logical ordination of form variation; cluster analysis produces sizedominated groups. The second situation considers an homogeneous sample where size and shape have important implications. Principal components are tested for association with size and shape, both of which can be isolated if isometry exists; if allometry is present, isolation of shape is possible only by size elimination, e.g., conversion to ratios. The third situation examines a sample of unknown groupings in which shape variation is the only interest. Aside from ratios, two other methods which produce shape-dominant clusters are assessed. Some of the options available in cluster analysis are also examined.     

Use of indicator variograms for an enhanced spatial analysis

Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general.     

基于地质统计学反演方法的洛带气田储层预测

研究区域为洛带气田遂宁组,该储层具有厚度薄、低孔隙度、低渗透层等特征,单一的约束稀疏脉冲反演难以达到精确刻画有利相带。地质统计学反演适用于复杂储层的地震预测和描述,与常规波阻抗反演相比,具有较高的垂向分辨率,能更有效地反映厚度薄的储层,反演结果更接近实际地质特征,能够很好地弥补约束稀疏脉冲反演缺陷,实现对储层特征的精细描述。针对研究区地质特点,根据储层发育特征和地质规律,基于地质统计学理论,采用岩性划分及统计学参数分析等技术方法进行储层预测,有效地预测了洛带气田的砂体厚度、平均孔隙度平面以及储层厚度。     

A method for seafloor classification using directional variograms, demonstrated for data from the western flank of the Mid-Atlantic Ridge

Seafloor classification is aimed at quantitatively characterizing seafloor properties such as roughness and anisotropy, and at using such spatial characteristics to distinguish geological provinces automatically. From geostatistical principals, a variogram method is developed for seafloor classification and it is demonstrated for data from the western flank of the Mid-Atlantic Ridge at 25°45N to 26°40N. This study uses HYDROSWEEP bathymetric data which have been ping-edited to flag erroneous data records, and navigation corrected. The classification method can handle the resultant data gaps inside the survey swaths as well as interpret data from several swaths. For a suite of test areas representative of different geological provinces, directional variograms are calculated, and characteristic parameters are extracted for the classification. Examples include a sediment pond, abyssal hill terrain in several segments and of variable spacing, inside and outside corners of ridge discontinuities, and mixed morphological forms. The dependency of the results on random or regular subsampling and on the size of the test area is investigated.     

Numerical classification by association analysis of multivariate mineralogic, petrographic, presence-absence data from some Sub-Himalayan carbonates; Northwest India

Fourteen mineralogical and petrographic variables for 82 samples of Sirban Limestone of Riasi, Jammu, and Kashmir State, and 18 mineralogical and petrographic variables for 36 samples of Bilaspur Limestone, Himachal Pradesh, India, were numerically classified based on presence-absence data using association analysis. The samples were collected along vertical profiles. The variables for the Sirban Limestone of Riasi were classified into three groups: (a) dolomite (b) micrites, and (c) terrigenous material represented by clay minerals. These three groups correspond to (a) near-shore sabkha dolomites, (b) deep water micrites, and (c) terrigenous contribution of clay minerals (except corrensite) to the carbonate depositional basin. Bilaspur Limestone samples were classified also into the same three groups. The results of the present study corroborate the earlier conclusions of the writer that the Sub-Himalayan carbonates of Northwest India generally represent rocks of shallow carbonate depositional basins, showing transition from near shore sabkha dolomites to basinal limestones and shales.     

Choosing the level of detail for depicting two-variable spatial relationships

A method for determining the most appropriate scale for presentation of spatial relationships between two variables combines the principle of entropy decomposition with Phipps' entropy-based method for determining mutual information of landscape patterns. The method enables determination of the optimal level of detail for maps, data matrices, or other depictions of two spatially associated phenomena. An example from wetland mapping is given.     

A PDF-based analysis of the spatial structure of retailing

The present paper proposes a new method for analyzing the spatial structure of retailing, using microscale locational data of individual retail stores. The method is based on the probability density function (PDF) of stores estimated from their locational data, and consequently it is applicable to both micro- and macro-scale retail analyses. The PDF approach provides a set of quantitative methods that permit us (1) to measure the degree of agglomeration, (2) to classify spatial patterns of store location, (3) to analyze the relationship between the size and function of retail agglomerations, and (4) to analyze the spatial structure of retail agglomeration. An empirical study is performed to test the validity of the method, and some empirical findings are shown. This revised version was published online in July 2006 with corrections to the Cover Date.     

Soil erosion susceptibility assessment and validation using a geostatistical multivariate approach: a test in Southern Sicily

A certain number of studies have been carried out in recent years that aim at developing and applying a model capable of assessing water erosion of soil. Some of these have tried to quantitatively evaluate the volumes of soil loss, while others have focused their efforts on the recognition of the areas most prone to water erosion processes. This article presents the results of a research whose objective was that of evaluating water erosion susceptibility in a Sicilian watershed: the Naro river basin. A geomorphological study was carried out to recognize the water erosion landforms and define a set of parameters expressing both the intensity of hydraulic forces and the resistance of rocks/soils. The landforms were mapped and classified according to the dominant process in landsurfaces affected by diffuse or linear water erosion. A GIS layer was obtained by combining six determining factors (bedrock lithology, land use, soil texture, plan curvature, stream power index and slope-length factor) in unique conditions units. A geostatistical multivariate approach was applied by analysing the relationships between the spatial distributions of the erosion landforms and the unique condition units. Particularly, the density of eroded area for each combination of determining factors has been calculated: such function corresponds, in fact, to the conditional probability of erosion landforms to develop, under the same geoenvironmental conditions. In light of the obtained results, a general geomorphologic model for water erosion in the Naro river basin can be depicted: cultivated areas in clayey slopes, having fine-medium soil texture, are the most prone to be eroded; linear or diffuse water erosion processes dominate where the topography is favourable to a convergent or divergent runoff, respectively. For each of the two erosion process types, a susceptibility map was produced and submitted to a validation procedure based on a spatial random partition strategy. Both the success of the validation procedure of the susceptibility models and the geomorphological coherence of the relationships between factors and process that such models suggest, confirm the reliability of the method and the goodness of the predictions.     

Use of discriminant analysis for classification of strata in sedimentary successions

The technique of instrumental neutron activation analysis (INAA) has been used to analyze chip samples of geological material for 12 elements. Discriminant analysis has been used to classify the unknown chip samples to the correct stratum in a sedimentary succession.     

Significance tests for multivariate normality of clusters from branching patterns in dendrograms

A significance test is presented for whether, based on levels of branches in a dendrogram, a cluster is from a multivariate normal distribution. The method compares the observed cumulative graph of number of branches with a graph derived from a simple logistic function. Provided the number of objects or variables is not small, the difference between graphs can be tested by the Kolmogorov-Smirnov, Cramér-von Mises, and Lilliefors statistics.Logistic functions were obtained by simulation and are available for three similarity measures: (1) Euclidean distances, (2) squared Euclidean distances, and (3) simple matching coefficients, and for five cluster methods: (1) WPGMA, (2) UPGMA, (3) single linkage (or minimum spanning trees), (4) complete linkage, and (5) Ward's increase in sums of squares. For simple matching coefficient, the mean intracluster similarity also is required.The method allows a test of whether the dendrogram could be from a cluster of smaller dimensionality due to character correlations. Good fit of the data to abnormally large or small dimensionality provides an important warning to interpretation of the dendrogram. Quantiles of test statistics were found by simulation to be well-approximated by logistic functions. The Lilliefors test is recommended for general use; if a conservative test is required, the two-tailed Kolmogorov-Smirnov test is most suitable. The method is suitable for use with a hand calculator, and a computer program for it is available from the author.     

A size classification for debris flows

A 10-fold classification for debris flow size is proposed based on total volume, peak discharge and area inundated by debris. Size classes can be used for regional overview studies where detailed site investigations are either unnecessary, too costly or where the highest hazard and risk creeks need to be identified for further study. They are also useful to compare the regional impact between affected areas and the effects of rainstorms, and they allow lay-people to obtain an understanding of debris flow magnitude and consequences. Finally, different size classes allow the estimation of travel times to points of interest based on empirically derived equations. It is proposed that agencies concerned with debris flows should establish a documentation of debris flow size according to this classification, which serves as a data base for hazard and risk planning.     

Extensions to Spatial Factor Methods with an Illustration in Geochemistry

Many applications involving spatial data require several layers of information to be simultaneously analyzed in relation to underlying geography and topographic detail. This in turn generates a need for forms of multivariate analysis particularly oriented to spatial problems and designed to handle spatial structure and dependency both within and between spatially indexed multivariate responses. In this paper we focus on one group of such methods sometimes referred to as spatial factor analysis. Use of these techniques has so far been mostly restricted to applications in the geosciences and in some forms of image processing, but the methods have potential for wider use outside these fields. They are concerned with identifying components of a multivariate data set with a spatial covariance structure that predominantly acts over a particular spatial range or zone of influence. We review the various forms of spatial factor analysis that have been proposed and emphasize links between them and with the linear model of coregionalization employed in geostatistics. We then introduce extensions to such methods that may prove useful in exploratory spatial analysis, both generally and more specifically in the context of multivariate spatial prediction. Application of our proposed exploratory techniques is demonstrated on a small but illustrative geochemical data set involving multielement measurements from stream sediments.     

A multivariate sedimentary environmental analysis of Great South Bay and South Oyster Bay, New York

A multivariate statistical strategy for classifying paleoenvironments is effective for studying modern sedimentary processes in western Great South Bay and South Oyster Bay, New York. The 13 whole phi weight percent variables were tested for redundancy with R-mode cluster analysis. The samples were partitioned statistically into five environmentally significant facies using Q-mode cluster analysis: (A)sandy gravel, (B)sandy silt, (C)silty sand, (D)slightly gravelly sand, and (E)fine sand. An ordination depicted gradational relationships among the samples and the facies. It was used to evaluate the environmental and textural parameter gradients within the sample space. Interpretations obtained in this manner and by examination of the grain-size curves suggest that these sediments were deposited by waves and currents on beaches and in wave zones (facies C and E),shoal areas (facies B),and tidal channels (facies A and D).Tidal currents, wave action, and eelgrass control the distribution of sediments within the two bays.