Robust statistics and geochemical data analysis

Investigation by spectral gamma-ray (SGR) and inductively coupled plasma-mass spectrometry (ICP-MS) geochemical analyses of the Upper Jurassic Arab-D reservoir analogue (central Saudi Arabia) revealed a strong correlation between the SGR response of the outcrop lithofacies and their elemental content. The two units of the reservoir (the Upper Jubaila Member of the Jubaila Formation and the Arab-D Member of the Arab Formation) showed distinctive SGR log profiles controlled mainly by their lithofacies associations. The geochemical analysis revealed four groups of chemical associations. Group 1 includes SiO2, Al3O2, Fe2O3, K2O, TiO2, Zr and Zn. This group has a strong relationship with the radioactive elements U, K and Th (Group 4). The reservoir facies exhibit high concentrations of chemicals from these two clusters. Group 2 includes CaO and Sr. A high concentration of chemicals from this group indicates a tendency towards pure carbonate facies and fewer siliciclastic impurities. A high concentration of MgO, the only chemical included in Group 3, marks dolomitic zones. Group 4 contains the radioactive elements. The boundary between the Upper Jubaila Member and the Arab-D Member is clearly defined from vertical SGR log profiles, vertical geochemical data logging and cross-plots of Group 1 chemicals with the radioactive elements in Group 4. The geochemical data for the Upper Jubaila Member show a very low concentration of U, K and Th. Consequently, the SGR response of the lithofacies was very low. All of the reservoir lithofacies showed high concentration of Group 1 and Group 4 components compared with the non-reservoir lithofacies. The Th/U ratio indicates a general shoaling upwards following the same trend of the outcrop lithofacies. A high Th/U ratio characterizes reservoir lithofacies, whereas a low ratio characterizes non-reservoir lithofacies. The lithofacies and the SGR log motifs were related in the measured sections. This study provides a method for predicting lithofacies from SGR log motifs within a high-resolution stratigraphic framework integrated with the geochemical data analysis. Here, we introduce an exploration guide for subsurface reservoir zonation and the identification of formation tops.

Adjustment of geochemical background by robust multivariate statistics

In this paper, robust statistical procedures in chemometrics have been reviewed. Two robust multivariate calibration methods: robust Kalman filter and robust principal component analysis via projection pursuit, are proposed.

Principal component analysis (PCA) is widely used to extract features and reduce dimensionality in various computer vision and image/video processing tasks. Conventional approaches either lack robustness to outliers and corrupted data or are designed for one-dimensional signals. To address this problem, we propose a robust PCA model for two-dimensional images incorporating structured sparse priors, referred to as structured sparse 2D-PCA. This robust model considers the prior of structured and grouped pixel values in two dimensions. As the proposed formulation is jointly nonconvex and nonsmooth, which is difficult to tackle by joint optimization, we develop a two-stage alternating minimization approach to solve the problem. This approach iteratively learns the projection matrices by bidirectional decomposition and utilizes the proximal method to obtain the structured sparse outliers. By considering the structured sparsity prior, the proposed model becomes less sensitive to noisy data and outliers in two dimensions. Moreover, the computational cost indicates that the robust two-dimensional model is capable of processing quarter common intermediate format video in real time, as well as handling large-size images and videos, which is often intractable with other robust PCA approaches that involve image-to-vector conversion. Experimental results on robust face reconstruction, video background subtraction data set, and real-world videos show the effectiveness of the proposed model compared with conventional 2D-PCA and other robust PCA algorithms.

Simulation experiments have been conducted to examine the potential usefulness of R-mode and Q-mode factor methods in the analysis and interpretation of geochemical data. The R-mode factor analysis experiment consisted of constructing a factor model, using the model to generate a correlation matrix, and attempting to recover the model by R-mode techniques. The techniques were successful in determining the number of factors in the model, but the factor loadings could not be estimated even approximately on the basis of mathematical procedures alone. Q-mode factor methods were successful in recovering all of the properties of a model used to generate hypothetical chemicaldata on olivinesamples, but it was necessary to use a correction previously regarded as unimportant.

Robust principal component analysis (PCA) has been proved to be an efficient multivariate statistical method for extraction of a multivariate structure of geochemical data containing outlier values. Furthermore, because of the compositional characteristics of geochemical data, logratio transformation approaches also have to be implemented to transform the data from the simplex space into real space. In order to model the extracted principal component of transformed data, we used sequential Gaussian simulation (SGS) to overcome the drawbacks associated with the traditional kriging method, e.g. bias conditionally relevant to underestimation and overestimation, the smoothing effect and so on. The results of implementing robust PCA on both untransformed and logratio transformed data, which have been acquired from an orogenic gold deposit, indicate that robust PCA analysis of untransformed data is not capable of certifying element association of gold occurrences. However, in robust PCA analysis of transformed data, both varieties of gold mineralization, associated with pyrite mineral and quartz veins, have been revealed. The first variety of gold occurrence is detected in an integrated PC obtained from PC1, PC2 and PC3, which are associated with Pb, As, Hg, Mo, S and Sb elements in quartz–sulphide veins. The second one, PC4, could be correlated with quartz veins. Furthermore, the results of the geostatistical SGS approach are compatible with the detected anomaly in the Alut gold deposit.

Geochemical data are frequently collected from mineral exploration drill-hole samples to more accurately define and characterise the geological units intersected by the drill hole. However, large multi-element data sets are slow and challenging to interpret without using some form of automated analysis, such as mathematical, statistical or machine learning techniques. Automated analysis techniques also have the advantage in that they are repeatable and can provide consistent results, even for very large data sets. In this paper, an automated litho-geochemical interpretation workflow is demonstrated, which includes data exploration and data preparation using appropriate compositional data-analysis techniques. Multiscale analysis using a modified wavelet tessellation has been applied to the data to provide coherent geological domains. Unsupervised machine learning (clustering) has been used to provide a first-pass classification. The results are compared with the detailed geologist’s logs. The comparison shows how the integration of automated analysis of geochemical data can be used to enhance traditional geological logging and demonstrates the identification of new geological units from the automated litho-geochemical logging that were not apparent from visual logging but are geochemically distinct. KEY POINTS To reduce computational complexity and facilitate interpretation, a subset of geochemical elements is selected, and then a centred log-ratio transform is applied. The wavelet tessellation method is used to domain the drill holes into rock units at a range of scales. Several clustering methods were tested to identify distinct rock units in the samples and multiscale domains for classification. Results are compared with geologist’s logs to assess how geochemical data analysis can inform and improve traditional geology logs.

Chemical exchange saturation transfer (CEST) MRI requires the acquisition of multiple saturation-weighted images and can last several minutes. Misalignments among these images, which are often due to the inevitable motion of the subject, will corrupt CEST contrast maps and result in large quantification errors. Therefore, the registration of the CEST series is critical. However, registration is challenging since common intensity-based registration algorithms may fail to differentiate CEST signals from motion artifacts. Herein, we studied how different patterns of motion affect CEST quantification and proposed a cascaded two-step registration scheme by utilizing features extracted from the entire Z-spectral image series instead of direct registration to a single image. The proposed approach is conducted in two stages: during the first coarse registration, the Z-spectral image series is decomposed by robust principal component analysis (RPCA) to separate CEST contrast from motion. The recomposed image series using only the low-rank component, which contains minimized motion, are averaged to generate a reference for the alignment of the image series. To further remove residual misalignments, the coarse registration is followed by a refinement stage, which uses PCA iteratively to generate motionless synthetic reference series with the first few principal components (PCs) that correspond to CEST contrast. In the end, the quality check is performed to exclude the images with unsuccessful registration. The proposed registration scheme (RPCA + PCA_R) was assessed by both phantom experiments and in vivo data of tumor-bearing mouse brain, with simulated random rigid motion in different patterns applied to the acquired static Z-spectral image series. For comparison, previous correction schemes using an explicit image [either S0 or Ssat(∆ω)] as registration reference were also performed, named as S0_R and Ssat_R respectively. To illustrate the advantage of combination of RPCA and PCA, registration was also exploited using either only the RPCA-based method (RPCA_R) or only the PCA-based one (PCA_R). Compared with the above four methods, RPCA + PCA_R allowed for more accurate correction of the corrupted Z-spectral images, exhibiting smaller MTRasym(∆ω) error maps and lower residual Z-spectra referring to the static data. Among all the five correction methods, the corrected Z-spectral image series by RPCA + PCA_R and the resulting MTRasym(∆ω) maps achieved the highest correlation coefficients (CC) with respect to the static ones. The registration scheme of RPCA + PCA_R provides robust motion correction between two specific Z-spectral images and among an entire image series, through extraction of the static component from the entire Z-spectra set and inclusion of a PCA-based refinement step. Therefore, this method can help improve CEST acquisition and quantification.

This paper presents a novel algorithm - robust kernel principal component analysis (robust KPCA), on the basis of the research of kernel principal component analysis (KPCA) and robust principal component analysis (RPCA). First, this algorithm sets the radius of the images of the training samples in the feature space using kernel tricks, then determines whether the samples are outliers or not, and finally analyzes the training samples which have eliminated the outliers using KPCA algorithm. The improved KPCA algorithm not only retains the non-linearity property of KPCA algorithm but also gets better robustness. Because the effects of outliers are eliminated, robust KPCA algorithm gets higher accuracy than KPCA algorithm for data analysis. The simulation experiments show that the robust KPCA algorithm developed is better than the KPCA algorithm.

In this paper, we propose an efficient singing voice extraction method from a music sound consisting of a singing voice and a drum sound. The proposed method is based on robust principal component analysis (RPCA) which is a technique to separate a given matrix into a sparse matrix and a low rank matrix. Dealing with a spectrogram of the music sound as the given matrix, RPCA gives an extracted singing voice spectrogram as a sparse spectrogram. We improve the capability of RPCA method by removing a residual drum sound from the sparse spectrogram. The residual drum sound is estimated as spectral components which repeatedly arise in the sparse spectrogram. Removing the estimated residual drum sound spectrogram from the sparse spectrogram gives an improved singing voice spectrogram. Simulation results show that the proposed method can improve 10dB of GNSDR as compared with the conventional RPCA singing voice extraction method.

Separating vocal and accompaniment signals from a monaural music signal is a challenging task. Recently, robust principal component analysis (RPCA) has been proposed for use in the magnitude spectrogram domain to separate the low-rank and sparse residual matrices, which are assumed to represent the accompaniment and vocal signals, respectively. In this paper, we propose two extended methods based on the RPCA algorithm for more effective vocal separation. First, we extend the conventional RPCA and propose to use in the spectrogram decomposition framework Schatten p- and lp-norms, which are generalized versions of the nuclear norm and l1-norm used in RPCA, respectively. Second, we apply proper scale compression to the magnitude spectrogram, making it a more appropriate representation for the decomposition. Experiments using the MIR-1K dataset show that the proposed methods yield significantly better separation performance than the conventional RPCA.

Elemental geochemistry to complement stable isotope data of fossil travertine: Importance of digestion method and statistics

Prediction of true classes of surficial and deep earth materials using multivariate spatial data is a common challenge for geoscience modelers. Most geological processes leave a footprint that can be explored by geochemical data analysis. These footprints are normally complex statistical and spatial patterns buried deep in the high-dimensional compositional space. This paper proposes a spatial predictive model for classification of surficial and deep earth materials derived from the geochemical composition of surface regolith. The model is based on a combination of geostatistical simulation and machine learning approaches. A random forest predictive model is trained, and features are ranked based on their contribution to the predictive model. To generate potential and uncertainty maps, compositional data are simulated at unsampled locations via a chain of transformations (isometric log-ratio transformation followed by the flow anamorphosis) and geostatistical simulation. The simulated results are subsequently back-transformed to the original compositional space. The trained predictive model is used to estimate the probability of classes for simulated compositions. The proposed approach is illustrated through two case studies. In the first case study, the major crustal blocks of the Australian continent are predicted from the surface regolith geochemistry of the National Geochemical Survey of Australia project. The aim of the second case study is to discover the superficial deposits (peat) from the regional-scale soil geochemical data of the Tellus Project. The accuracy of the results in these two case studies confirms the usefulness of the proposed method for geological class prediction and geological process discovery.

Quantitative relationship between the resistivity distribution of the by-product plume and the hydrocarbon degradation in an aged hydrocarbon contaminated site

Traditional robust principle component analysis (RPCA) has a high computational cost because RPCA needs to calculate the singular value decomposition of large matrices. To address this issue, this paper proposes a matrix-factorization-based RPCA (MFRPCA) model. MFRPCA has high computation efficiency while improving the robustness and flexibility of traditional RPCA using a non-convex low-rank approximation. Experiment results on challenging datasets demonstrate superior performance of MFRPCA compared with several advanced low-rank reconstruction methods.