Search Results (48)

History matching traditionally assumes a spatially uniform clay adsorption coefficient Kd, overlooking the fractal heterogeneity of clay mineral distributions in real reservoirs. This study generalizes the classical advection–dispersion framework by treating Kd as a non-uniform field whose logarithmic fluctuations follow fractional Brownian motion (fBm) statistics with Hurst exponent H, and by coupling the resulting heterogeneous retardation to a fractional advection–dispersion equation through the continuous-time random walk bridge β = 2/(2H + 1). A Karhunen–Loève (K-L) expansion compresses the field into a small set of spectral modes, and an adjoint-gradient inversion recovers the clay field from tracer data. The main operational limitation of the K-L approach is that sub-grid variability below the retained spectral cutoff is smoothed; within this boundary the inversion is efficient and numerically stable. Two independent validation paths are pursued. First, a cross-fitting experiment shows that when the true physics is sub-diffusive (β = 0.85), the classical advection–dispersion equation yields a negative coefficient of determination (R2 = −0.927), whereas the correctly specified fractional model fits the data, establishing fractional derivatives as physically necessary rather than mathematically optional. Second, the publicly available EGS Collab DNA-tracer dataset independently confirms statistically significant clay adsorption (retardation R = 1.250 ± 0.009, p < 0.0001). The recovered clay field attains pixel-scale accuracy R2η = 0.844 (Spearman ρ = 0.923) and R2 > 0.94 at reservoir-relevant scales of five cells or coarser. These results indicate that fractal-aware clay-field inversion provides a physically verifiable, rather than purely empirical, pathway for history matching, with direct implications for field-scale contaminant transport prediction and environmental subsurface management, and the evaluation of flue gas enhanced thermal recovery in thin-layer heavy oil reservoirs. Full article

Tunnel seepage in heterogeneous ground can trigger hydrogeological hazards such as concentrated water inflow, groundwater depletion, deformation of surrounding structures, and subsequent eco-environmental degradation. However, these processes are still commonly evaluated using deterministic models that neglect the spatial variability of hydrogeological parameters. To address this limitation, this study develops a stochastic hydro–geo–mechanical–ecological framework that integrates random field theory with physics-informed neural networks (PINNs) for hazard evaluation and rapid prediction of tunnel seepage responses. The spatial variability of key parameters, including permeability and porosity, is characterized using the Karhunen–Loeve expansion and embedded into coupled governing equations for unsaturated–saturated seepage, seepage–stress interaction, and groundwater–soil–vegetation responses. A PINN surrogate model with random-field inputs is then constructed to predict hydraulic head, tunnel inflow, displacement, groundwater depth, vegetation coverage, and soil physicochemical indices, while simultaneously quantifying uncertainty. A karst tunnel case in Chongqing, China, is used to demonstrate the proposed framework. The results show that spatial heterogeneity promotes preferential flow paths and intensifies seepage-induced hazards compared with deterministic mean simulations, leading to larger groundwater drawdown, stronger ecological degradation, and greater overall response variability. The proposed PINN achieves high predictive accuracy (R² > 0.97) and reduces single-case computational time from hours to seconds, enabling efficient multi-scenario evaluation and uncertainty-aware risk assessment. This framework provides a physically consistent and computationally efficient tool for evaluating water-related hazards and long-term environmental impacts in underground engineering. Full article

The purpose of our paper is to provide a family of bilinear orthogonal expansions all based upon the same general pattern that is valid for a wide class of special functions. Our first family involves Jacobi, Laguerre, and Hermite polynomials. We give a discrete analogue of these bilinear expansions, the three families of classical orthogonal polynomials being replaced by zonal spherical functions associated with regular distance graphs. Such expansions playing a key role in the field of mathematical statistics, we show how our results apply to this field. We provide generalizations of the well-known Cramér–von Mises and Watson’s statistics, based upon an interpretation of their kernel in terms of the circular Laplacian. The product formula, well-known for zonal functions on Lie groups, is stated for distance-regular graphs, providing an elegant tool for proofs. Examples involving Hahn, q-Hahn, and Krawtchouk polynomials are given. Full article

Given the escalating impacts of global climate change and extreme weather events, the accurate stability assessment of rainfall-induced landslides necessitates a comprehensive consideration of both seepage processes and the inherent spatial variability of soils. Traditional deterministic and two-dimensional (2D) analyses often fail to capture the multi-dimensional kinematic features of slope failures and the stochastic nature of soil heterogeneity, thereby leading to inaccurate risk assessments. This study proposes a three-dimensional (3D) slope reliability analysis framework. Within this framework, a 3D slope geometric model is constructed using GeoStudio 2025.1.0 software, and seepage analysis is conducted by the SEEP3D module. To account for soil spatial variability, the Karhunen–Loève (K-L) expansion method is employed to discretize key shear strength parameters (effective cohesion and effective angle of internal friction). The factor of safety (F_s) is evaluated using the 3D simplified Bishop method, which is then coupled with Monte Carlo simulations to determine the probability of failure (P_f). The results show that rainfall infiltration causes progressive dissipation of shallow matric suction and a significant rise in the groundwater table near the slope toe, resulting in reduced effective stress in the critical resistance zone. As rainfall intensity increases, the F_s decreases approximately linearly from 1.14 to 0.90, whereas the P_f increases nonlinearly from nearly 0 to 98.36%. Under the rainstorm condition, although the F_s remains above unity at 1.063, the corresponding P_f reaches 23%, indicating that deterministic evaluation based only on the F_s may underestimate the actual failure risk. The proposed framework provides a quantitative tool for evaluating rainfall-induced slope instability by integrating transient hydraulic response, three-dimensional spatial variability, and probabilistic reliability assessment. Full article

As a rigorous and comprehensive foundation for electromagnetic information theory (EIT), we develop a general theory that elucidates the universal stochastic structure of radiated electromagnetic (EM) fields and induced currents in generic EM information transmission systems. The framework encompasses arbitrary random scatterers, input information fields, and EM mutual coupling. The system is modeled as a multiply connected, arbitrary Riemannian manifold within the language of differential geometry. Our approach exploits exact Green’s functions (GFs) on manifolds to construct a novel electromagnetic random field theory (EM-RFT). Interpreted as response functions localized on the surfaces of transceivers and scatterers, the GFs allow us to treat the internal physical details of the EM system as a black box, redirecting analytical attention toward external input–output relations in line with signal processing and communication theory. This integration of random fields (RFs), electromagnetics, and GFs yields a unified framework for deriving and characterizing the stochastic structure of arbitrary EM information transmission systems. We rigorously establish that EM random fields satisfying Maxwell’s equations can always be constructed using system GFs driven by external information fields. The theory further decouples stochastic input RFs from random fluctuations associated with the communication medium (e.g., scatterers), and introduces general correlation propagators valid for arbitrary EM links. Using the Karhunen–Loève expansion, all EM random fields are represented as sums of random variables, providing both a simulation framework for arbitrary EM RFs and a basis for evaluating mutual information between input and output spatial domains at arbitrary locations in the system. Full article

Inverse scattering problems seek anomalies in a medium given data measured after the interaction with emitted waves. Due to noise, predictions about the nature of these inclusions should be complemented with uncertainty estimates. To this end, we propose a progressive framework for inverse scattering from low- to high-dimensional Bayesian formulations depending on the prior information and the problem complexity. We aim to reduce computational costs by exploiting educated prior information. When we look for a few well-separated inclusions in a known medium with information about their number, we resort to low-dimensional parameterizations in terms of a few random variables representing their shape and material constants. We test this approach detecting anomalies in tissues and deposits in stratified subsoils. In more complex situations where the anomalies may overlap, we propose high-dimensional parameterizations obtained from Karhunen–Loève (KL) or Fourier expansions of the density and velocity fields. We employ these methods to characterize oil and gas reservoirs in a salt dome configuration, where the screening effect of the dome cap prevents the obtention of adequate prior information. We characterize the posterior probability by means of affine invariant ensemble and functional ensemble MCMC samplers depending on dimensionality. This provides information on configurations with the highest a posteriori probability and the uncertainty around them, identifying factors that could reduce the uncertainty. In high-dimensional setups, techniques based on KL developments are more effective and stable. A recurring issue is the choice of the a priori covariance (which strongly affects the results) and the choice of its hyperparameters. Here, we use educated choices. Formulations that include them as additional parameters could be a next step at a higher cost. Full article

This paper presents a stochastic framework for the inverse identification of structural material degradation (SMD) in cantilever beams. The method combines the Karhunen–Loéve (KL) expansion for the efficient parameterisation of spatially varying material decay with experimental Frequency Response Function (FRF) data within a Bayesian inference scheme. This approach employs a low-dimensional spectral parameterisation via the KL expansion, which mitigates the curse of dimensionality inherent in element-wise model updating, and provides a full-field probabilistic description of SMD. A two-phase constraint strategy was developed to address the fundamental tension between physical plausibility and algorithmic stability of the inverse identification algorithm: (1) physical regularisation during identification stabilises the ill-posed inverse problem, and (2) post-convergence selective regularisation eliminates physically impossible stiffness enhancements (exceeding 1.1 × baseline) that arise from measurement and modelling uncertainties. This phased approach prevents the algorithm distortion that occurs when constraints are applied too stringently during iteration, while ensuring final results respect fundamental physical principles. The framework is experimentally validated on a steel cantilever beam with a symmetric open-edge cut. Laser vibrometry measurements under swept-sine excitation demonstrate successful localisation and quantification of SMD, with the 95% credible interval accurately capturing the damaged region after physical constraint application. The adaptive constraint strategy resolves the delicate balance between mathematical stability and physical plausibility in inverse identification. Full article

Rainfall infiltration is one of the main factors inducing slope instability, while the spatial heterogeneity and uncertainty of soil parameters have profound impacts on slope response characteristics and stability evolution. Traditional deterministic analysis methods struggle to reveal the dynamic risk evolution process of the system under heavy rainfall. Therefore, this paper proposes an uncertainty analysis framework combining Karhunen–Loève Expansion (KLE) random field theory, Polynomial Chaos Kriging (PCK) surrogate modeling, and Monte Carlo simulation to efficiently quantify the probabilistic characteristics and spatial risks of rainfall-induced slope instability. First, for key strength parameters such as cohesion and internal friction angle, a two-dimensional random field with spatial correlation is constructed to realistically depict the regional variability of soil mechanical properties. Second, a PCK surrogate model optimized by the LARS algorithm is developed to achieve high-precision replacement of finite element calculation results. Then, large-scale Monte Carlo simulations are conducted based on the surrogate model to obtain the probability distribution characteristics of slope safety factors and potential instability areas at different times. The research results show that the slope enters the most unstable stage during the middle of rainfall (36–54 h), with severe system response fluctuations and highly concentrated instability risks. Deterministic analysis generally overestimates slope safety and ignores extreme responses in tail samples. The proposed method can effectively identify the multi-source uncertainty effects of slope systems, providing theoretical support and technical pathways for risk early warning, zoning design, and protection optimization of slope engineering during rainfall periods. Full article

Post-disaster underground (UG) mine environments are characterized by complex and rapidly changing conditions, adding extra attenuation to propagating electromagnetic (EM) waves. One such complex condition is the extreme generation of dust and sudden rise in humidity contributing to extra attenuation effects to propagating waves, especially under varying airborne humidity and dust levels. The existing wave propagation prediction models, especially those that factor in the effect of dust particles, are deterministic in nature, limiting their ability to account for uncertainties, especially during emergency conditions. In this work, the vector parabolic equation (VPE) model is modified to include dust attenuation effects. Using the complex permittivity of dust as a random variable, the Karhunen–Loève (KL) expansion is used to generate random samples of permittivity along the drifts for which each realization is solved using deterministic VPE method. The model is validated using a modified Friis method and experimentally obtained data from literature. The findings show that accounting for dust and humidity effects stochastically captures the extra losses that would have otherwise been lost using deterministic methods. The proposed framework offers key insights for designing resilient underground wireless systems, strengthening miner tracking, and improving safety during emergencies. Full article

Effective containment of contaminant plumes in heterogeneous aquifers is critically challenged by the inherent uncertainty in hydraulic conductivity (K). Conventional, deterministic optimization approaches for pump-and-treat (P&T) system design often fail when confronted with real-world geological variability. This study proposes a novel robust simulation-optimization framework to design reliable hydraulic containment systems that explicitly account for this subsurface uncertainty. The framework integrates the Karhunen–Loève Expansion (KLE) for efficient stochastic representation of heterogeneous K-fields with a Genetic Algorithm (GA) implemented via the pymoo library, coupled with the MODFLOW groundwater flow model for physics-based performance evaluation. The core innovation lies in a multi-scenario assessment process, where candidate well configurations (locations and pumping rates) are evaluated against an ensemble of K-field realizations generated by KLE. This approach shifts the design objective from optimality under a single scenario to robustness across a spectrum of plausible subsurface conditions. A structured three-step filtering method—based on mean performance, consistency (pass rate), and stability (low variability)—is employed to identify the most reliable solutions. The framework’s effectiveness is demonstrated through a numerical case study. Results confirm that deterministic designs are highly sensitive to the specific K-field realization. In contrast, the robust framework successfully identifies well configurations that maintain a high and stable containment performance across diverse K-field scenarios, effectively mitigating the risk of failure associated with single-scenario designs. Furthermore, the analysis reveals how varying degrees of aquifer heterogeneity influence both the required operational cost and the attainable level of robustness. This systematic approach provides decision-makers with a practical and reliable strategy for designing cost-effective P&T systems that are resilient to geological uncertainty, offering significant advantages over traditional methods for contaminated site remediation. Full article

The Karhunen–Loève (KL) expansion decomposes a stochastic process into a set of orthogonal functions with random coefficients. The basic idea of the decomposition is to solve the Fredholm integral equation associated with the covariance kernel of the process. The KL expansion is a powerful mathematical tool used to represent stochastic processes in a compact and efficient way. It has wide applications in various fields including signal processing, image compression, and complex systems modeling. The KL expansion has high computational complexity, especially for large data sets, since there is no single unique transformation for all stochastic processes and there are no fast algorithms for computing eigenvalues and eigenfunctions. One way to solve this problem is to use parametric models for the covariance function. In this paper, an explicit analytical solution of the KL expansion for a parametric model of oscillating covariance function with a small number of parameters is proposed. This model approximates the covariance functions of real images very well. Computer simulation results using a real image are presented and discussed. Full article

To describe the fouling characteristics of compressor blades, fouling is categorized into dense and loose layers to characterize thickness and rough structures. An uncertainty model for dense fouling layer thickness distribution is constructed using the numerical integration and the Karhunen–Loève (KL) expansion method, while the Fouling Longuet-Higgins (FLH) model is proposed to address the uncertainty of loose fouling layer roughness. The FLH model effectively simulates the morphology characteristics of actual blade fouling and elucidates how parameters influence fouling roughness, morphology, and randomness. Based on the uncertainty modeling method, models for dense fouling layer thickness and loose fouling layer morphology are constructed, followed by numerical calculations and aerodynamic performance uncertainty quantification. Results indicate a 75.8% probability of aerodynamic performance degradation due to a dense fouling layer and a 97.2% probability related to the morphology uncertainty of a loose fouling layer when the roughness is 50 μm. This underscores that a mere focus on roughness is inadequate for characterizing blade fouling, and a comprehensive evaluation must also incorporate the implications of rough structures on aerodynamic performance. Full article

A novel partially functional linear regression model with random effects is proposed to address the case of Euclidean covariates and functional covariates. Specifically, the model assumes that the random effects follow a Gaussian process prior to establish the linkage structure between Euclidean covariates and scalar responses. For functional covariates, a linear relationship with scalar responses is assumed, and the functional covariates are approximated using the Karhunen–Loève expansion. To enhance the robustness of the predictive model, a cross-validation-based ensemble strategy is employed to optimize the proposed method. Results from both simulation studies and real-world data analyses demonstrate the superior performance and competitiveness of the proposed approach in terms of prediction accuracy and model stability. Full article

This paper presents a new algorithm for assessing the reliability of three-dimensional (3D) slope stability considering the spatial variability of soil based on the Particle Swarm Optimization (PSO) algorithm. First, a 3D random field is generated using the Karhunen–Loève (K-L) expansion method. Then, the simplified Bishop method of limit equilibrium is coupled with the PSO algorithm to calculate safety factors of the slope. Finally, the failure probability of the slope is determined using the Monte Carlo Simulation method. After validating the rationality of the proposed method through a typical case study, this paper offers an in-depth examination of how soil spatial variability affects the stability of 3D slopes. It is observed that, given identical soil correlation lengths, slope geometric parameters, and failure surface widths, the failure probability is positively correlated with soil spatial variability parameters, while the mean safety factor demonstrates an inverse relationship with these variability parameters. Additionally, the failure probability tends to increase as the soil correlation lengths increase, and it also escalates with the expansion of the failure surface width. In contrast, the mean safety factor exhibits an upward trend with the augmentation of the horizontal correlation length, while it diminishes progressively as the vertical correlation length grows, and it also shows a decline with the widening of the failure surface width. The proposed algorithm significantly improves computational efficiency while ensuring accuracy, making it suitable for the reliability analysis of three-dimensional slopes. Full article

Systems may depend on parameters that can be controlled, serve to optimise the system, are imposed externally, or are uncertain. This last case is taken as the “Leitmotiv” for the following discussion.A reduced-order model is produced from the full-order model through some kind of projection onto a relatively low-dimensional manifold or subspace. The parameter-dependent reduction process produces a function mapping the parameters to the manifold.One now wants to examine the relation between the full and the reduced state for all possible parameter values of interest. Similarly, in the field of machine learning, a function mapping the parameter set to the image space of the machine learning model is learned from a training set of samples, typically minimising the mean square error. This set may be seen as a sample from some probability distribution, and thus the training is an approximate computation of the expectation, giving an approximation of the conditional expectation—a special case of Bayesian updating, where the Bayesian loss function is the mean square error. This offers the possibility of having a combined view of these methods and also of introducing more general loss functions. Full article