Fractals in Geology and Geochemistry

The identification of geochemical anomalies in oil and gas indicators is a fundamental task in oil and gas exploration, as the process of oil and gas accumulation is a low probability event. Machine learning algorithms for anomaly detection are applicable to the identification of oil and gas geochemical anomalies related to oil and gas accumulation. However, when using oil and gas indicators for anomaly detection, the diversity of these indicators often leads to the influence of the indicator redundancy on the identification of such features. Therefore, it is particularly important to select appropriate oil and gas indicators for anomaly detection. In this study, a hybrid model combining unsupervised machine learning methods and singularity analysis methods was used to evaluate oil and gas indicator anomalies using geochemical data from the Taiwan Strait Basin. The models used in this study include the singularity index model, the principal component model combined with the singularity index model, and the cluster analysis combined with the principal component model and the singularity index model. PCA models can reduce the dimensions of the data and retain as much information as possible. CLA divides data samples into different groups, so that samples within the same group are more similar and samples between different groups are less similar. LSP is mainly used for measuring the setting and singular degree of local anomalies in multi-scale geochemistry, geophysics, and other types of local anomalies, and it has a unique advantage in extracting low and weak anomalies and nonlinear characteristics. The results of the study show that the results obtained using the CLA-PCA-LSP hybrid model are very similar to those obtained by performing PCA on the entire index and then calculating the singularity index. This also verifies that, for the study areas of the Jiulongjiang Depression and Jinjiang Depression, we can select oil and gas indicators that are favorable for exploration analysis, without including all indicators in the analysis scope, thereby improving the computational efficiency. The application of a singularity analysis method and generalized self-similarity principle in extracting the geochemical information of oil and gas indicators in the Taiwan Strait Basin highlights key technologies such as the identification of weak anomalies, decomposition of composite anomalies, and integration of spatial information. The combination anomalies delineated by the singularity analysis method and S-A method not only reflect the spatial relationship with known oil and gas reservoir distribution, but also show the multiple combination anomalies in unknown areas, providing favorable guidance for the next exploration direction in the Taiwan Strait Basin. Full article

AI-driven mineral prospectivity mapping (MPM) is a valid and increasingly accepted tool for delineating the targets of mineral exploration, but it suffers from noisy and unrepresentative input features. In this study, a set of fractal and multifractal methods, including box-counting calculation, concentration–area fractal modeling, and multifractal analyses, were employed to excavate the underlying nonlinear mineralization-related information from geological features. Based on these methods, multiple feature selection criteria, namely prediction–area plot, K-means clustering, information gain, chi-square, and the Pearson correlation coefficient, were jointly applied to rank the relative importance of ore-related features and their fractal representations, so as to choose the optimal input feature dataset readily used for training predictive AI models. The results indicate that fault density, the multifractal spectrum width (∆α) of the Yanshanian intrusions, information dimension (D₁) of magnetic anomalies, correlation dimension (D₂) of iron-oxide alteration, and the D₂ of argillic alteration serve as the most effective predictor features representative of the corresponding ore-controlling elements. The comparative results of the model assessment suggest that all the AI models trained by the fractal datasets outperform their counterparts trained by raw datasets, demonstrating a significant improvement in the predictive capability of fractal-trained AI models in terms of both classification accuracy and predictive efficiency. A Shapley additive explanation was employed to trace the contributions of these features and to explain the modeling results, which imply that fractal representations provide more discriminative and definitive feature values that enhance the cognitive capability of AI models trained by these data, thereby improving their predictive performance, especially for those indirect predictor features that show subtle correlations with mineralization in the raw dataset. In addition, fractal-trained models can benefit practical mineral exploration by outputting low-risk exploration targets that achieve higher capturing efficiency and by providing new mineralization clues extracted from remote sensing data. This study demonstrates that the fractal representations of geological features filtered by multi-criteria feature selection can provide a feasible and promising means of improving the predictive capability of AI-driven MPM. Full article

The degradation of soil bonding, which can be described by the evolution of bond degradation variables, is essential in the constitutive modeling of cemented soils. A degradation variable with a value of 0/1.0 indicates that the applied stress is completely sustained by bonded particles/unbounded grains. The discrete element method (DEM) was used for cemented soils to analyze the bond degradation evolution and to evaluate the degradation variables at the contact scale. Numerical cemented soil samples with different bonding strengths were first prepared using an advanced contact model (CM). Constant stress ratio compression, one-dimensional compression, conventional triaxial tests (CTTs), and true triaxial tests (TTTs) were then implemented for the numerical samples. After that, the numerical results were adopted to investigate the evolution of the bond degradation variables B_N and B₀. In the triaxial tests, B₀ evolves to be near to or larger than B_N due to shearing, which indicates that shearing increases the bearing rate of bond contacts. Finally, an approximate stress-path-independent bond degradation variable B_σ was developed. The evolution of B_σ with the equivalent plastic strain can be effectively described by an exponential function and a hyperbolic function. Full article

Detrital zircon geochronology plays a crucial role in provenance analysis, serving as one of the fundamental strategies. The age spectrum of detrital zircons collected from the sedimentary unit of interest is often compared or correlated with that of potential source terranes. However, biases in the age data can arise due to factors related to detrital sampling, analysis techniques, and nonlinear geological mechanisms. The current study reviewed two sets of detrital zircon datasets established in 2011 and 2021 to discuss the origins of the Tibetan Plateau. These datasets collected from different media effectively demonstrate a progressive understanding of provenance affinity among the main terranes on the Tibetan Plateau. This highlights issues regarding weak and unclear temporal connections identified through analyzing the age spectrum for provenance analysis. Within this context, a local singularity analysis approach is currently employed to address issues associated with unclear and weak provenance information by characterizing local variations in nonlinear behaviors and enhancing detection sensitivity towards subtle anomalies. This new graphical approach effectively quantifies temporal variations in detrital zircon age populations and enhances identification of weak provenance information that may not be readily apparent on conventional age spectra. Full article

The quantification of the irregular morphology and distribution pattern of mineral grains is an essential but challenging task in ore-related mineralogical research, allowing for tracing the footprints of pattern-forming geological processes that are crucial to understanding mineralization and/or diagenetic systems. In this study, a large model, namely, the Segmenting Anything Model (SAM), was employed to automatically segment and annotate quartz, lepidolite and albite grains derived from Yichun rare-metal granite (YCRMG), based on which a series of fractal and multifractal methods, including box-counting calculation, perimeter–area analysis and multifractal spectra, were implemented. The results indicate that the mineral grains from YCRMG show great scaling invariance within the range of 1.04~52,300 μm. The automatic annotation of mineral grains from photomicrographs yields accurate fractal dimensions with an error of only 0.6% and thus can be utilized for efficient fractal-based grain quantification. The resultant fractal dimensions display a distinct distribution pattern in the diagram of box-counting fractal dimension (D_b) versus perimeter–area fractal dimension (D_PA), in which lepidolites are sandwiched between greater-valued quartz and lower-valued albites. Snowball-textured albites, i.e., concentrically arranged albite laths in quartz and K-feldspar, exhibit characteristic D_b values ranging from 1.6 to 1.7, which coincide with the fractal indices derived from the fractal growth model. The zonal albites exhibit a strictly increasing trend regarding the values of fractal and multifractal exponents from core to rim, forming a featured “fractal-index banding” in the radar diagram. This pattern suggests that the snowball texture gradually evolved from rim to core, thus leading to greater fractal indices of outer zones, which represent higher complexity and maturity of the evolving system, which supports a metasomatic origin of the snowball texture. Our study demonstrates that fractal analyses with the aid of a large model are effective and efficient in characterizing and understanding complex patterns of mineral grains. Full article

Exploration of mineral resources in the deep sea has become an international trend. However, deep-sea mineral exploration faces challenges such as complex offshore drilling and the weak and mixed signals of ore deposits. Therefore, studying methods for identifying weak and mixed anomalies and extracting composite information in the deep sea is crucial for innovative prediction and evaluation of deep-sea mineral resources. In this study, the Central Pacific Ocean, Northwestern Pacific Ocean, and Eastern Pacific Ocean were selected as research areas. Drawing upon the fractal self-similarity exhibited by rare earth minerals in the deep-sea sediments within the Pacific Ocean, we conducted an analysis and comparison of the fractal geochemical characteristics in various regions of the Pacific Ocean’s deep-sea sediments. Thereafter, we studied the spatial distribution of rare earth elements (REEs) in deep-sea sediments in these regions to explore the mechanisms responsible for rare earth enrichment in the Pacific Ocean. The results revealed that the geochemical fractal characteristics of deep-sea sediments in the Northwestern Pacific Ocean Basin and the Central Pacific Ocean Basin were similar, whereas there were slight differences in the fractal characteristics observed in the Eastern Pacific Ocean Basin. By calculating the singularity index of CaO/P₂O₅, it was found that the singularity index in the Central and Northwestern Pacific Ocean basins was lower than that in the Eastern Pacific Ocean Basin, suggesting that the phosphorus content in the Eastern Pacific Ocean Basin was lower than that in the Central and Northwestern Pacific Ocean basins. In the Eastern Pacific Ocean, we found that phosphorus content in deep-sea sediments was the primary controlling factor for REE enrichment. Conversely, in the Central and Northwestern Pacific Ocean, both the phosphorus and calcium content in deep-sea sediments played significant roles in REE enrichment. Full article