Bayesian Compressive Sampling Research Articles

A fused sampling method integrating geotechnical and geophysical data for assessing three-dimensional soil liquefaction-induced damage capacity

Combining intrusive geotechnical site investigations with non-intrusive geophysical surveys is a cost-effective approach to producing data with varying levels of accuracy, uncertainty, and different spatial scales to better characterize the site's liquefaction properties. Moreover, the demand for three-dimensional (3D) subsurface models in geotechnical engineering is increasing, but the models contain uncertainties and spatial variability associated with the use of relevant stochastic and geostatistical methods, and it remains a challenging task to obtain reliable liquefaction assessment results and the corresponding damage capacity. This study proposes a data-driven and non-parametric form of 3D multi-source fusion Bayesian compressive sampling (3D MSF-BCS) method for assessing 3D soil liquefaction-induced damage capacity. It consists of three main components: (i) 3D MSF-BCS fusing sparse geotechnical data (e.g., cone penetration test (CPT)) and geophysical data (e.g., multi-channel analysis of surface waves (MASW)) for 3D site modeling, (ii) quantifying the accuracy and uncertainty of 3D MSF-BCS, and (iii) Soil liquefaction-induced damage capacity analysis in 3D space. The method was applied to numerical examples and a real case study at the Cresselly Place site, and the results showed that the proposed method performs well.

Fusion of sparse non-co-located measurements from multiple sources for geotechnical site investigation

A profile of geotechnical properties is often needed for geotechnical design and analysis. However, site-specific data might be characterized as MUSIC-X (i.e., Multivariate, Uncertain and Unique, Sparse, Incomplete, and potentially Corrupted with “X” denoting the spatial/temporal variability), posing a significant challenge in accurately interpreting geotechnical property profiles. Different sources, or types, of data are commonly available from a specific site investigation program, and they are usually cross-correlated, and thus can provide complementary information. This leads to an important question in geotechnical site investigation: how to integrate multiple sources of sparse data for enhancing the profiling of different geotechnical properties. To address this issue, this study proposes a novel method, called fusion Bayesian compressive sampling (Fusion-BCS), for integrating sparse and non-co-located geotechnical data. In the proposed method, the auto- and cross-correlation structures of different sources of data are exploited in a data-driven manner through a joint sparse representation. Then, profiles of different geotechnical properties are jointly reconstructed from all measurements under a framework of compressive sampling/sensing. The proposed method is illustrated using simulated and real geotechnical data. The results indicate that the accuracy of the interpreted geotechnical property profiles may be significantly improved by integrating multiple sources of site investigation data.

Erratum for “Data-Driven Development of Three-Dimensional Subsurface Models from Sparse Measurements Using Bayesian Compressive Sampling: A Benchmarking Study”

Erratum for “Data-Driven Development of Three-Dimensional Subsurface Models from Sparse Measurements Using Bayesian Compressive Sampling: A Benchmarking Study”

Data-Driven Development of Three-Dimensional Subsurface Models from Sparse Measurements Using Bayesian Compressive Sampling: A Benchmarking Study

With the rapid development of computing and digital technologies recently, three-dimensional (3D) subsurface models for accurate site characterization have received increasing attention, for example, with various data-driven methods developed for 3D subsurface modeling. This leads to a need for validating the 3D modeling results obtained from each method and comparing the performance of different methods in a fair and consistent manner. To address this need, a benchmarking study, which is often used in machine learning (ML), is presented in this study to compare the performance of different 3D subsurface modeling methods in four aspects, including accuracy, uncertainty, robustness, and computational efficiency. A suite of performance metrics is proposed for the four aspects above. Multiple sets of real cone penetration test (CPT) data are compiled in the benchmarking study for quantifying performance of 3D modeling methods using sparse measurements as input, a typical scenario in geotechnical practice. The benchmarking study is illustrated using an in-house software package called Analytics of Sparse Spatial Data based on Bayesian compressive sampling/sensing (ASSD-BCS), which can directly generate high-resolution 3D random field samples (RFSs) from sparse measurements. The evaluation results show that ASSD-BCS provides accurate estimates with quantified uncertainty from sparse measurements. In addition, ASSD-BCS exhibits remarkably high computational efficiency and performs robustly under different benchmarking cases.

Data-driven simulation of two-dimensional cross-correlated random fields from limited measurements using joint sparse representation

Cross-correlated random fields are an essential tool for simultaneously modeling both auto- and cross-correlation structures of spatial or temporal quantities in stochastic analysis of structures or systems. Existing cross-correlated random field simulation methods often require explicit information about random field parameters as inputs. However, in engineering practice, site-specific measurements of different quantities are often limited, non-co-located and irregularly distributed within a given site because of time, budget, or space constraints as well as missing data. It is notoriously difficult to properly estimate reliable random field parameters from limited non-co-located measurements with an irregular spatial pattern, particularly the auto-correlation and cross-correlation structures of a two-dimensional (2D) cross-correlated random field. To deal with this issue, this study proposes a novel 2D cross-correlated random field generator for simulating 2D cross-correlated random field samples (RFSs) directly from sparsely measured non-co-located data points with unequal measurement intervals. Using a joint sparse representation, auto- and cross-correlation structures of different spatial/temporal quantities are exploited simultaneously from sparse measurements, followed by the generation of cross-correlated RFSs using Bayesian compressive sampling (BCS) and Markov chain Monte Carlo (MCMC) simulation in a data-driven manner. The proposed generator is demonstrated using 2D data of two correlated geotechnical properties. The results indicate that the RFSs generated using the proposed method from sparse measurements can properly characterize the spatial auto- and cross-correlation structures of different geotechnical properties.

Data-driven and physics-informed Bayesian learning of spatiotemporally varying consolidation settlement from sparse site investigation and settlement monitoring data

A digital twin of a geotechnical project (e.g., a reclamation or ground improvement project) is a virtual model that aims to continuously learn from actual observations (e.g., site investigation and monitoring data) and improve model prediction (e.g., spatiotemporally varying consolidation settlement). However, real geotechnical observation data obtained from a site are often spatially sparse (e.g., site investigation data) and spatiotemporally varying (e.g., settlement monitoring data). The sparse and spatiotemporally varying data pose great challenges for continuous learning of data and improvement in model prediction. To address these challenges, this study proposes a novel data-driven and physics-informed Bayesian learning framework that automatically develops ground models from spatially sparse site investigation data, performs geotechnical analysis, and integrates geotechnical analysis results with limited, but spatiotemporally varying, settlement monitoring data to improve model prediction in a systematic and quantitative manner. The proposed method contains three key components, (1) data-driven ground modeling by Bayesian compressive sampling (BCS) using sparse site investigation data as input, (2) finite element modeling (FEM) of consolidation settlement that incorporates domain knowledge, and (3) Bayesian sparse dictionary learning of settlement monitoring data together with FEM results. The proposed method is illustrated using a real ground improvement project, and the results show that the proposed approach performs well.

"Conical" Frustum Multi-Beam Phased Arrays for Air Traffic Control Radars.

The design of conical frustum phased array antennas for air traffic control (ATC) radar systems is addressed. The array architecture, which is controlled by a fully digital beam-forming (DBF) network, is composed by a set of equal vertical modules. Each module consists of a linear sparse array that generates on receive multiple instantaneous beams pointing along different directions in elevation. To reach the best trade-off between the antenna complexity (i.e., minimum number of array elements and/or radio frequency components) and radiation performance (i.e., matching a set of reference patterns), the synthesis problem is formulated in the Compressive Sampling (CS) framework. Then, the positions of the array elements and the complex excitations for generating each single beam are jointly determined through a customized version of the Bayesian CS (BCS) tool. Representative numerical results, concerned with ideal as well as real antenna models, are reported both to validate the proposed design strategy and to assess the effectiveness of the synthesized modular sparse array architecture also in comparison with conventional arrays with uniformly-spaced elements.

Adaptive sampling strategy for characterizing spatial distribution of soil liquefaction potential using cone penetration test

Characterizing spatial distribution of soil liquefaction potential is critical for assessing liquefaction-related hazards (e.g. building damages caused by liquefaction-induced differential settlement). However, in engineering practice, soil liquefaction potential is usually measured at limited locations in a specific site using in situ tests, e.g. cone penetration tests (CPTs), due to the restrictions of time, cost and access to subsurface space. In these cases, liquefaction potential of soil at untested locations requires to be interpreted from limited measured data points using proper interpolation method, leading to remarkable statistical uncertainty in liquefaction assessment. This underlines an important question of how to optimize the locations of CPT soundings and determine the minimum number of CPTs for achieving a target reliability level of liquefaction assessment. To tackle this issue, this study proposes a smart sampling strategy for determining the minimum number of CPTs and their optimal locations in a self-adaptive and data-driven manner. The proposed sampling strategy leverages on information entropy and Bayesian compressive sampling (BCS). Both simulated and real CPT data are used to demonstrate the proposed method. Illustrative examples indicate that the proposed method can adaptively and sequentially select the required number and optimal locations of CPTs.

Fusion of geotechnical and geophysical data for 2D subsurface site characterization using multi-source Bayesian compressive sampling

Subsurface site characterization is essential for geotechnical engineering applications (e.g., slope stability analysis and deep excavation design), which is usually achieved through geotechnical site investigation and might be supplemented by geophysical survey. Geotechnical and geophysical investigations are complementary in many aspects. Geotechnical investigation provides direct measurement data with high accuracy but only at limited locations. On the other hand, geophysical survey provides abundant two-dimensional (2D) or three-dimensional measurements, but the data are often indirect. In addition, geotechnical and geophysical data are usually correlated. Therefore, fusion of geotechnical and geophysical data during site characterization is beneficial. This paper proposed a novel data fusion method, called multi-source Bayesian compressive sampling, for fusion of geotechnical and geophysical data and statistical characterization of 2D subsurface profiles. The proposed method is data-driven and non-parametric, without the need for an empirical parametric function between geotechnical and geophysical data. The proposed method was illustrated and validated using both numerical and real-life examples. The results show that the proposed method not only properly characterizes 2D subsurface profiles but also explicitly quantifies the statistical uncertainty associated with the site characterization.

Uncertainty quantification of nonlinear system stochastic response estimates based on the Wiener path integral technique: A Bayesian compressive sampling treatment

The Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems is enhanced herein based on a Bayesian compressive sampling (CS) treatment. Specifically, first, sparse expansions for the system response joint probability density function (PDF) are employed. Next, exploiting the localization capabilities of the WPI technique for direct evaluation of specific PDF points leads to an underdetermined linear system of equations for the expansion coefficients. Further, relying on a Bayesian CS solution formulation yields a posterior distribution for the expansion coefficient vector. In this regard, a significant advantage of the herein developed methodology relates to the fact that the uncertainty of the response PDF estimates obtained by the WPI technique is quantified. Furthermore, an adaptive scheme is proposed based on the quantified uncertainty of the estimates for optimal selection of PDF sample points. This yields considerably fewer boundary value problems to be solved as part of the WPI technique, and thus, the associated computational cost is significantly reduced. Two indicative numerical examples pertaining to a Duffing nonlinear oscillator and to an oscillator with asymmetric nonlinearities are considered for demonstrating the capabilities of the developed technique. Comparisons with pertinent Monte Carlo simulation data are included as well.

CPT-based probabilistic liquefaction assessment considering soil spatial variability, interpolation uncertainty and model uncertainty

In engineering practice, simplified procedure based on cone penetration test (CPT) results is widely used for evaluating soil liquefaction potential. Since the CPT-based simplified procedure was developed from observations during past earthquakes, it is semi-empirical and involves significant uncertainty (e.g., model uncertainty). In addition, due to time, budget and access constraints, CPTs are often sparsely conducted at a specific site, leading to a significant uncertainty associated with interpolation of the limited CPT soundings, particularly along horizontal direction. Furthermore, it is well-recognized that spatial variability of soil properties has a remarkable effect on soil liquefaction. All these variability and uncertainties greatly affect the seismic liquefaction assessment results, particularly spatial distribution of liquefiable soils in a site, and liquefaction-induced damage. This underscores a question of how to properly incorporate these variability and uncertainties in liquefaction assessment, e.g., how to characterize spatial distribution of soil liquefaction potential in a site with quantitative consideration of the abovementioned variability and uncertainty. To address this issue, this paper develops a novel probabilistic method for characterizing spatial distribution of soil liquefaction potential through factor of safety, FS against liquefaction in a vertical cross-section using Bayesian compressive sampling and Monte Carlo simulation. Using the proposed method, many random field samples of FS cross-section are obtained directly from limited CPT measurements. The proposed method is illustrated using both a simulated data example and a set of real CPT data from Christchurch, New Zealand. It is shown that the proposed method performs well and provides reasonable liquefaction assessment results.

Interpolation of extremely sparse geo-data by data fusion and collaborative Bayesian compressive sampling

In geotechnical or geological engineering, geo-data interpolation based on measurements is often needed for engineering design and analysis. However, measurements are sometimes extremely sparse (e.g., several, or even just a few, data points) because of limited access to the subsurface and the cost of tests. It is, therefore, difficult to properly interpolate the measurements. On the other hand, multiple data sources (e.g., standard penetration tests, SPT, and cone penetration tests, CPT) often exist in engineering practice, and data fusion methods (e.g., cokriging) have been developed to leverage the correlation among multiple data sources for interpolation of sparse geo-data. Performance of cokriging depends on proper modeling of spatial variability using variogram models. However, the construction of proper variogram models requires many measurement data points. Therefore, it is very challenging to properly interpolate extremely sparse geo-data due to the difficulty in obtaining suitable variogram models. In this study, a novel data fusion method, called collaborative Bayesian compressive sampling (Co-BCS), is proposed to tackle this problem. Equations of the proposed Co-BCS method are derived, and the method is illustrated using real data. The results show that the proposed method not only properly interprets extremely sparse geo-data by integrating correlated secondary data sources but also quantifies the associated interpolation uncertainty simultaneously.

Non-parametric construction of site-specific non-Gaussian multivariate joint probability distribution from sparse measurements

Construction of a joint probability distribution for correlated geotechnical properties is often needed in geotechnical reliability-based analysis and design. Geotechnical properties vary across sites and follow site-specific and non-Gaussian probability distribution, because of different geological processes that soils and rocks in different sites have undergone. In addition, site-specific measurements on geotechnical properties are usually sparse and correlated in geotechnical practice, leading to the difficulty in estimation of meaningful joint probability distribution when using conventional parametric statistical methods. To address this issue, this paper develops a non-parametric and data-driven approach for characterizing the site-specific non-Gaussian multivariate joint probability distribution without pre-selection of the marginal probability distribution type. Two key components for constructing a multivariate joint probability distribution (i.e., marginal probability distribution for each variable and correlation matrix for all variables) are estimated separately using Bayesian Gaussian mixture and Bayesian compressive sampling (BCS) and Karhunen-Loève (KL) expansion. The proposed method is illustrated using both simulated and real data in geotechnical site investigation. The results show that the proposed method performs well for both examples. The proposed method does not model spatial variability explicitly but is based on the assumption of independent measurement data along depth, and thus measurement records are required to be sufficiently spaced apart in depth (e.g., more than 1 scale of fluctuation) to satisfy this assumption.

Selection of standard penetration test number for geotechnical investigation of a vertical cross section considering spatial variability and correlation in soil properties

The standard penetration test (SPT) is a widely used in situ test for characterizing variation of subsurface soil properties, and results of site investigation are usually simplified as a two-dimensional (2D) vertical cross section for subsequent geotechnical design and construction. Current geotechnical design codes and guidelines only provide general recommendations for selection of an appropriate number of in situ tests (e.g., SPT) (e.g., the greater variability of subsurface conditions, the larger number of SPTs required to obtain sufficient underground information). No quantitative or rational method is available for selecting the appropriate number of SPTs considering spatial variability and correlation in subsurface conditions. A comprehensive parametric study is carried out in this study to investigate the influence of spatial variability in subsurface conditions on the minimum SPT number needed for satisfying an accuracy requirement of site investigation. Random field is adopted to model spatial variation and correlation in soil properties in typical site conditions, and 2D Bayesian compressive sampling is used to interpolate sparse SPT data. Based on the parametric study results, a statistical chart is developed for geotechnical engineers to conveniently select an appropriate number of SPTs in a vertical cross section. Real SPT data from New Zealand are used to illustrate and validate the proposed method.

Rational determination of cone penetration test quantity in a two-dimensional vertical cross-section for site investigation

Site investigation plays a critical role in underground space development and geotechnical engineering practice, and it aims at collecting data for characterization of subsurface soil properties and stratification, which are often simplified as a two-dimensional (2D) vertical cross-section for subsequent geotechnical design and analysis in practice. In-situ tests, such as cone penetration tests (CPT), are often performed during site investigation. Existing tunneling or geotechnical guidelines or design codes only provide conceptual principle for determination of CPT quantity (e.g., accuracy of site investigation improves as the CPT quantity increases) and recommend some rule-of-thumb minimal quantities (e.g., at least one CPT every 100 m). No rational method is available to determine the minimal CPT quantity for achieving a target level of accuracy in an objective and quantitative manner. This paper aims to develop a rational method for addressing this issue. The method developed is based on an extensive parametric study that uses random field simulation to cover typical CPT data encountered in geotechnical engineering practice and Bayesian compressive sampling to interpolate the spatially varying, but correlated, CPT data along both depth and horizontal directions simultaneously. The proposed method is illustrated and validated using real CPT data, and it is shown to perform well.

Data-driven determination of sample number and efficient sampling locations for geotechnical site investigation of a cross-section using Voronoi diagram and Bayesian compressive sampling

Geotechnical analyses and designs in practice are often performed using a two-dimensional (2D) cross-section, information of which is obtained from site investigation. The quality of site investigation results depends greatly on the number and locations of sampling during site investigation. However, increasing the sample number requires additional expenditure, human resources, and time. In addition, geotechnical site investigation is a multi-stage process, and the measurements at preliminary stage are often sparse and limited, hence additional samples might be needed in later stages. This study develops a smart sampling strategy for planning of multistage geotechnical site investigation of a cross-section using Voronoi diagram, Bayesian compressive sampling (BCS), and information entropy. The proposed method is non-parametric and data-driven, and it can determine both the necessary sample number and their corresponding optimal sampling locations. The proposed smart sampling strategy applies Voronoi diagram to determine the efficient sampling locations of measurements at preliminary stage of site investigation, and uses BCS and information entropy to automatically decide whether or not additional samples are needed and their efficient sampling locations in a self-adaptive and data-driven manner. The proposed method is illustrated using real soil data and showed to perform well and robustly.

Smart sampling strategy for investigating spatial distribution of subsurface shallow gas pressure in Hangzhou Bay area of China

Subsurface shallow gas has been widely discovered in coastal areas across continents at buried depths of approximately 10–100 m. Its presence poses a significant hazard and risk during underground construction, e.g., risk of explosion and fire when using tunnel boring machines for subway construction. Therefore, it is important to determine the spatial distribution of the shallow gas pressure during site investigation for subsequent hazard analysis and mitigation. In engineering practice, the gas pressure is often sparsely measured at a limited number of locations using modified cone penetration tests (CPTs) owing to time, budget, resource, and technical constraints. There are no rational or quantitative methods available for the planning of site investigation of gas pressure. Therefore, it is important to determine ways to properly and efficiently plan a site investigation of the shallow gas pressure during geotechnical site investigation (e.g., how to determine the required number of modified CPTs (i.e., the sample size) and their corresponding optimal locations). A smart sampling strategy is developed for planning of site investigation in this study that uses innovative data analytic methods (e.g., Bayesian compressive sampling, BCS, and information entropy) to automatically determine sample size and optimal sampling location for gas pressure characterization. The proposed smart sampling strategy is illustrated using numerical examples. The results show that the proposed smart sampling strategy performs reasonably well.

Differentiation of discrete data with unequal measurement intervals and quantification of uncertainty in differentiation using Bayesian compressive sampling

Abstract Calculation of derivatives on discrete measurement data with unequal intervals is often required in geotechnical engineering, such as interpretation of stiffness reduction curve of soil from pressuremeter test data, pile lateral responses from inclinometer data. Such a task is however tricky and challenging, because a small error or noise in the measurements may amplify and lead to huge fluctuations in the derivatives obtained. The amplification becomes increasingly significant as the order of derivative increases. It is therefore of great importance to evaluate reliability of the derivatives obtained and quantify the uncertainty associated with the derivative calculation. A Bayesian compressive sampling-based method is proposed in this paper to address this problem. It not only provides high-order derivatives on discrete measurement data, even at un-sampled locations, but also quantifies the uncertainty associated with the derivatives obtained and offers an index to evaluate reliability of the derivatives obtained. The proposed approach is illustrated using both real-life pressuremeter data and numerical example of pile lateral responses. A comparison is also made between the proposed method and several existing methods in geotechnical literature. It shows that the proposed method performs better than existing methods and it is applicable to problems with both elastic and plastic soil responses.

Determination of efficient sampling locations in geotechnical site characterization using information entropy and Bayesian compressive sampling

Site characterization is indispensable in geotechnical engineering practice, and measurements on soil properties are performed through in situ tests, laboratory tests or other methods. However, due to time or budget limit, technical or access constraints, etc., the measurements are usually taken at a limited number of locations. This leads to a question of how to select the efficient locations for measurements or sampling such that as much information as possible on the spatial variability of soil properties can be obtained from a given number of measurements. In addition, geotechnical site characterization is a multi-stage process, and additional measurements might be required at a later stage of site characterization. In such a case, how the additional sampling locations can be selected efficiently such that the pre-existing measurements obtained from the preliminary stages of site characterization can be best used and how can as much information as possible on soil properties be further obtained are problems engineers face. This paper aims to address these two problems using information entropy and Bayesian compressive sampling. Real cone penetration test data along both vertical and horizontal directions are used to illustrate and validate the proposed methods. Results show that the proposed methods are very effective and robust in selecting efficient sampling locations for geotechnical site characterization.

Direct simulation of two-dimensional isotropic or anisotropic random field from sparse measurement using Bayesian compressive sampling

Random field theory has been increasingly adopted to simulate spatially varying environmental properties and hydrogeological data in recent years. In a two-dimensional (2D) stochastic analysis, variation of the environmental properties or hydrogeological data along different directions can be similar (i.e., isotropic) or quite different (i.e., anisotropic). To model the spatially isotropic or anisotropic variability in a stochastic analysis, conventional random field generators generally require a vast amount of measurement data to identify the random field parameters (e.g., mean, variance, and correlation structure and correlation length in different directions). However, measurement data available in practice are usually sparse and limited. The random field parameters estimated from sparse measurements might be unreliable, and the subsequent random field modeling or stochastic analysis might be misleading. This underscores the significance and challenge of generating 2D isotropic or anisotropic random fields from sparse measurements. This paper develops a novel 2D random field generator, which does not require a parametric form of correlation function or estimation of correlation length and other random field parameters, and directly generates 2D isotropic or anisotropic random field samples from sparse measurements. The proposed generator is highly efficient because simulation of a 2D random field is achieved by generation of a short 1D random vector. The effectiveness and applicability of the proposed generator are illustrated using isotropic and anisotropic numerical examples.