Singularity Maps Research Articles

Mineral exploration and regional surface geochemical datasets: An anomaly detection and k-means clustering exercise applied on laterite in Western Australia

A comprehensive geochemical survey was conducted in the western Yilgarn Craton, Western Australia, in 2007, collecting 3142 surface samples of regolith. Our study used this data to target potential sites for undiscovered buried or concealed Cu-Zn-Pb, and NiCu deposits. The core approach used the singularity mapping technique for detecting anomalies at the local scale. This work proposes a procedure to create a composite multi-element singularity map by linearly combining individual element singularity maps, using element-to-element correlation coefficients as weights for the linear combination process. Furthermore, the k-means clustering algorithm was applied to combinations of sub-sets of data and singularity values. Expert validation indicated that the k-means clustering approach yielded the best results when using 4 or 5 clusters, separating the distinct sites of the MINEDEX database. In either case, the incorporation of the singularity values provided the most accurate outcomes, with a dominant cluster correctly classifying up to 60 to 80 % of identified Cu and Ni deposits and mines, respectively.Based on these results and on the range of computed singularity values, simple rules were established to identify sampled data points satisfying the following criteria: (i) meeting the defined threshold singularity value and belonging to the k-means cluster that include the mines and (ii) not being in the neighbourhood of any known mineralization site from the MINEDEX database. These locations thus represent potential mineralization sites that warrant further investigation and exploration follow-up. The outcomes of this study strongly support the efficiency of anomaly detection and k-means clustering method applied on a regional surface geochemical dataset for mineral exploration to detect and target mineral systems.

Exotic image formation in strong gravitational lensing by clusters of galaxies – II. Uncertainties

ABSTRACT Due to the finite amount of observational data, the best-fitting parameters corresponding to the reconstructed cluster mass have uncertainties. In turn, these uncertainties affect the inferences made from these mass models. Following our earlier work, we have studied the effect of such uncertainties on the singularity maps in simulated and actual galaxy clusters. The mass models for both simulated and real clusters have been constructed using grale. The final best-fitting mass models created using grale give the simplest singularity maps and a lower limit on the number of point singularities that a lens has to offer. The simple nature of these singularity maps also puts a lower limit on the number of three image (tangential and radial) arcs that a cluster lens has. Hence, we estimate the number of galaxy sources giving rise to the three image arcs, which can be observed with the James Webb Space Telescope (JWST). We find that we expect to observe at least 20–30 tangential and 5–10 radial three-image arcs in the Hubble Frontier Fields cluster lenses with the JWST.

Exotic image formation in strong gravitational lensing by clusters of galaxies – I. Cross-section

ABSTRACT In a recent paper, we have discussed the higher order singularities in gravitational lensing. We have shown that a singularity map, comprising A3- lines and unstable (point) singularities (A4 and D4), is a compact representation of high magnification regions corresponding to a given lens model for all possible source redshifts. It marks all the optimal locations for deep surveys in the lens plane. Here, we present singularity maps for 10 different clusters lenses selected from the Hubble Frontier fields (HFF) and the Reionization Lensing Cluster Survey (RELICS) surveys. We have identified regions in the lens plane with a high magnification for sources up to redshift 10. To determine the dependence of unstable (point) singularities on lens mass model reconstruction techniques, we compared singularity maps corresponding to the different mass models (provided by various groups in the HFF survey) for each cluster lens. We find that the non-parametric (free-form) method of lens mass reconstruction yields the least number of point singularities. In contrast, mass models reconstructed by various groups using a parametric approach have a significantly larger number of point singularities. We also estimate the number of galaxies lying near these unstable (point) singularities, which can be observed with the James Webb Space Telescope (JWST). We find that we expect to get at least one hyperbolic umbilic and one swallowtail image formation for a source at z > 1 for every five clusters with JWST. These numbers are much higher than earlier estimates.

Mineral prospectivity modelling using singularity mapping and multifractal analysis of stream sediment geochemical data from the auriferous Hutti-Maski schist belt, S. India

The regional stream sediment geochemical survey has become an all the more important exploration technique for delineating potential mineral zones to target exploration. However, the effective appraisal of pathfinder elements has remained a fundamental challenge in exploration geochemistry. In this regard, the singularity mapping technique based on modern nonlinear theory and fractal-multifractal geometry is a new efficient method to distinguish thresholds of anomalies based on stream sediment geochemical data. The auriferous Hutti-Maski schist belt in Eastern Dharwar Craton (EDC), S. India has been taken up for delineation future targets using local singularity analysis (LSA) of stream sediment geochemical data. Ordinary Singularity Mapping (OSM) and Spatially Weighted Singularity Mapping (SWSM) are employed for mapping singularity of gold pathfinder elements (Au, As, Ag, Hg, Sb, Se, Cu, and Zn). In SWSM, a distance-based fuzzy weighting method is used to estimate the spatial weighting factor with respect to ore controlling shear zones. The local isotropic singularity index and the local anisotropic singularity index are estimated for each of the eight elements. The Student’s t-statistics based on Weight of Evidence (WOE) model is used to determine the threshold singularity index value to separate geochemical anomaly from background. Principal Component Analysis (PCA) is applied to singularity maps of multiple elements derived using OSM and SWSM techniques. The first principal component (PC-1) scores estimated by the OSM-PCA and SWSM-PCA lead to comprehensive geochemical anomaly maps. Results of two different singularity mapping techniques are compared using Geochemical Mineralization Probability Index (GMPI), Concentration-Area (C-A) fractal analysis and Prediction-Area (P-A) plots. It is observed that the SWSM method results in a smaller target with higher predictive ability compared to OSM method that delineates a larger target area.

Multi-parameter Analysis of Local Singularity Mapping and Its Application to Identify Geochemical Anomalies in the Xishan Gold Deposit, North China

Identification of geochemical anomalies from geological background is of great significance in the exploration of complex mineralization systems. For a 2D problem, the window-based local singularity mapping has been used widely to identify the distribution patterns of geochemical data. However, the optimal window parameters for calculating the singularity index are hard to determine. Previous studies commonly use the default parameters when applying singularity mapping. In this study, window-based local singularity mapping was performed and improved by comprehensive analysis of multiple parameters to explore geochemical anomalies associated with gold mineralization in the Xishan deposit, North China, with the aim of revealing undiscovered mineralization. By using Au anomalies as an example, the parameters that may influence the result of window-based local singularity processing have been analyzed and discussed to improve the mapping result. The parameters include the average concentration calculation algorithm, the shape of the sliding windows, the window size increment and the number of windows. Success-rate curves and area under the success-rate curve have been used to assess the spatial correlation of the singularity map with the known mineral occurrences. While square sliding window is the most regularly used window shape, circular and elliptical windows can be alternative choices. We found that the directions of major axis of the ellipses parallel or quasi-parallel to geological strike fit the locations of ore deposits better if the mineralization system is controlled by regional faults. After taking the influence of different parameters into account, geochemical anomalies were successfully separated from background and have been enhanced compared to anomalies identified solely from concentration values. Singularity–quantile analysis has been applied to recognize and separate multiple geochemical anomaly populations based on the singularity map in frequency and spatial domain. While the Au concentration map shows quite scattered strong and weak geochemical anomalies, the linear regions of positive singularity resolved by singularity–quantile analysis coincide well with the location of regional faults and alteration zones, which might indicate footprints of ore-forming fluids. Based on the singularity maps of multi-elements (Au, Ag, Cu and Pb), we resolved two prospect areas of mineralization bounded by regional faults, hydrothermally altered rocks and lamprophyres with positive singularity, which warrant further investigation for undiscovered mineralization.

Entropy Mapping Approach for Functional Reentry Detection in Atrial Fibrillation: An In-Silico Study.

Catheter ablation of critical electrical propagation sites is a promising tool for reducing the recurrence of atrial fibrillation (AF). The spatial identification of the arrhythmogenic mechanisms sustaining AF requires the evaluation of electrograms (EGMs) recorded over the atrial surface. This work aims to characterize functional reentries using measures of entropy to track and detect a reentry core. To this end, different AF episodes are simulated using a 2D model of atrial tissue. Modified Courtemanche human action potential and Fenton–Karma models are implemented. Action potential propagation is modeled by a fractional diffusion equation, and virtual unipolar EGM are calculated. Episodes with stable and meandering rotors, figure-of-eight reentry, and disorganized propagation with multiple reentries are generated. Shannon entropy (), approximate entropy (), and sample entropy () are computed from the virtual EGM, and entropy maps are built. Phase singularity maps are implemented as references. The results show that and maps are able to detect and track the reentry core of rotors and figure-of-eight reentry, while the results are not satisfactory. Moreover, and consistently highlight a reentry core by high entropy values for all of the studied cases, while the ability of to characterize the reentry core depends on the propagation dynamics. Such features make the and maps attractive tools for the study of AF reentries that persist for a period of time that is similar to the length of the observation window, and reentries could be interpreted as AF-sustaining mechanisms. Further research is needed to determine and fully understand the relation of these entropy measures with fibrillation mechanisms other than reentries.

Singularity maps applied to a vegetation index

Agricultural drought quantification is one of the most important tasks in the characterisation process of this natural hazard. Recently, several vegetation indexes based on remote-sensing data have been applied to quantify it, being the Normalized Difference Vegetation Index (NDVI) the most widely used. Some index-based drought insurances define a drought event through the comparison of actual NDVI values in a given period with a NDVI threshold based on historical data of that period extrapolating this result spatially to the surrounded areas. Hence, the spatial statistical approach is very relevant and has not been deeply studied in this context. Drought can be highly localised, and several authors have recognised the critical role of the spatial variability. Therefore, it is important to delimit areas that will share NDVI statistical distributions and in which the same criteria can be applied to define the drought event. In order to do so, we have applied for the first time in this context the method of singularity maps commonly used in localisation of mineral deposits. The NDVI singularity maps calculated for each season and different years are shown and discussed in this context. For this study we have selected a region that includes the whole Autonomous Community of Madrid (Spain). The resulting singularity maps show that areas where the NDVI follows theoretically a spatial normal/log-normal distribution (α≅2) are widely scattered in the area of study and vary across seasons and years. Therefore, the extrapolation of normal/log-normal NDVI statistics should be applied only inside these areas.

Local 3D segmentation of soil pore space based on fractal properties using singularity maps

Over the last decade, major technological advances in X-ray computed tomography (CT) have allowed for the investigation and reconstruction of three-dimensional (3D) natural porous media architectures at very fine scales. Soil scientists can use the internal structure information to develop predictive models for a range of physical, chemical and biological processes in soil. Image segmentation and thresholding are crucial steps when applying these methods to extract complex pore space geometry information from images. Traditional thresholding algorithms face challenges related to the heterogeneity of soil samples, noise and artefacts introduced during the image acquisition process.This study proposes a new segmentation method using local greyscale value (GV) concentration variabilities based on fractal concepts. Singularity maps were created to measure the GV concentration at each point. The C-V method was combined with the singularity map approach (Singularity-CV method) to define thresholds that can be applied to binarize CT images.This study also introduces a new method for creating 3D synthetic soil images based on truncated multifractals that simulate low-contrast and non-bimodal GV histograms. A synthetic soil image was created with the objective to compare traditional segmentation methods (Otsu and maximum entropy) with the Singularity-CV method. We obtained better results in porosity and more amount of pores at all scales than traditional methods, although some small pores were incorrectly identified due to the ability to amplify every anomalous GVs. Misclassification error (ME) was low and similar to Otsu.Two different 3D CT soil images were also used in this analysis, corresponding to samples of the same soil with 1.2 and 1.6gcm−3 bulk densities. After applying the Singularity-CV method to the GV images, the results were compared with the aforementioned traditional segmentation methods. The image comparison was based on the porosity, pore size distribution (PSD) and cumulative pore size distribution (CPSD). The Otsu method achieves a higher porosity than the Singularity-CV method because it defines the largest pores. However, the Singularity-CV method detected more pores at all sizes. The maximum entropy method always yielded the lowest porosity.

New segmentation method based on fractal properties using singularity maps

To improve our understanding of the factors that control a number of critical soil processes, it is important to properly identify pore space zones. Computed tomography (CT) images provide increasingly reliable information about the geometry of pores and solids in soils at very small scales and are a non-invasive technique. However, the low contrast between voids and pores and the non-bimodal greyscale value (GV) histograms observed in the CT images present a problem with respect to pore identification.In this work, we propose a new segmentation method based on the local GV concentration variability when using singularity maps. We also introduce a new method for creating synthetic CT soil images based on truncated multifractals that simulate low-contrast and non-bimodal GV histograms.From three different bidimensional CT soil images and one synthetic soil image, singularity maps were created to measure the GV concentration at each point by defining areas with self-similar properties that were shown as power-law relationships in Concentration-Area plots (C-A method). The C-A method together with a singularity map (the “Singularity-CA” method) was used to define thresholds that were applied to binarize the CT images. Promising results were obtained when the greyscale images were overlapped with the borders of the binarized images.Once these black and white images were obtained, we compared them with commonly used segmentation methods: the Otsu, Iterative and Maximum Entropy methods. The image comparison was based on the total porosity, cumulative pore size distribution (CPSD), cumulative pore size–area distribution (CPSAD) and misclassification error (ME). We found that the Otsu and the Iterative methods overestimated the size of large pores compared to the ground-truth image, which overestimated the porosity. Conversely, the Singularity-CA method extracted medium and large pores with a good fit compared to the ground-truth image. It was observed that whereas the Otsu and Iterative methods obtained the large pores, the “Singularity-CA” method detected an aggregate of small pores, although many of these small pores were incorrectly identified. This was due to the latter's high sensitivity when detecting anomalous GV values. The Maximum Entropy method always resulted in the lowest porosity in the real CT soil samples and the greatest porosity in the synthetic soil image samples.

Application of singularity analysis for mineral potential identification using geochemical data — A case study: Nanling W–Sn–Mo polymetallic metallogenic belt, South China

The Nanling W–Sn–Mo polymetallic belt located in southern China is endowed with considerable mineral resources, especially W–Sn–Mo polymetallic deposits. Most of these deposits were formed within a short time interval at the age of Yanshanian. In this paper, singularity analysis within a context multifractal was used to identify weak anomalies hidden within geochemical background. The C-A method was applied in the determination of threshold values of three elements (W, Sn and Mo) using derived geochemical singularity indices. Using the threshold values, each of the three singularity maps was classified into three levels based on singularity indices, and then classified maps were integrated by the raster-value overlay method. The resultant W–Sn–Mo polymetallic prospectivity map generated not only highlights known areas of W–Sn–Mo mineralization, but also identifies high potential areas with no known mineral occurrences.

Multifractal analysis of geochemical anomalies: A tool for assessing prospectivity at the SE border of the Ossa Morena Zone, Variscan Massif (Spain)

The aim of this paper is to demonstrate discrimination of geochemical anomalies by application of multifractal analysis techniques to a stream sediment geochemical data (774 samples) from a geologically complex area at the boundary between the Ossa Morena (OMZ) and South Portuguese (SPZ) zones of the Iberian Variscan Massif. In this area, which is poorly investigated up to now, small to medium size iron–oxide–copper–gold (IOCG-type) and PbZnCu mineralizations are known to exist. The use of multifractal techniques highlighted weak anomalies not manifested by using conventional methods. The use of the C–A method has been very helpful for establishing thresholds separate anomalies of elements from the regional background, but local anomalies could not be identified as clearly as with the singularity mapping method. The characterization of selected elements (Fe, Cu, Pb, and Zn) is based on the use of multifractal spectrum curves f(α) and indices such as R, Δα and M (multifractality). These have proved to be very useful for characterizing different geochemical behaviors (e.g. depletion or enrichment of elements) as well as spatial distribution and singularity. The singularity (α) maps of the studied elements show areas more accurately compared to conventional geochemical maps, and reveal that local anomalies are not only spatially associated with known occurrences but also are disposed along trends consistent with structures known to be linked to mineralization. Therefore, singularity maps used as spatial evidence for integration with other layers of spatial evidence in mineral prospectivity mapping to provide a useful input for decision-making about further mineral exploration in the study area.

Obtaining configuration space and singularity maps for parallel manipulators

The aim of this paper is to describe a general methodology to obtain the entire set of positions that a parallel manipulator can reach and the workspace regions where the robot is controllable. The workspace is computed using a hybrid analytical-discrete procedure. Next the singularity maps are traced by carrying out a kinematic analysis of the positions obtained. To perform the latter a systematic method has been introduced to obtain the corresponding Jacobian matrices. The result of the whole process is the computation of singularity-free workspace regions, associated with certain working and assembly modes. After that, strategies to enlarge the accessible space are easier to plan and implement. This methodology is based on disassembling the manipulator into a mobile platform and a set of kinematic chains.

Quantitative Panoramic Imaging of Epicardial Electrical Activity

Fluorescent imaging with voltage- and/or calcium-sensitive dyes has revolutionized cardiac physiology research. Here we present improved panoramic imaging for optically mapping electrical activity from the entire epicardium of the Langendorff-perfused rabbit heart. Combined with reconstruction of the 3D heart surface, the functional data can be conveniently visualized on the realistic heart geometry. Methods to quantify the panoramic data set are introduced by first describing a simple approach to mesh the heart in regular grid form. The regular grid mesh provides substrate for easy translation of previously available non-linear dynamics methods for 2D array data. It also simplifies the unwrapping of curved three-dimensional surface to 2D surface for global epicardial visualization of the functional data. The translated quantification methods include activation maps (isochrones), phase maps, phase singularity, and electric stimulus-induced virtual electrode polarization (VEP) maps. We also adapt a method to calculate the conduction velocities on the global epicardial surface by taking the curvature of the heart surface into account.