SUBSURFACE SIMULATION USING STOCHASTIC MODELING TECHNIQUES
FOR RELIABILITY BASED DESIGN OF GEO-STRUCTURES
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Zhao Li
August, 2016 SUBSURFACE SIMULATION USING STOCHASTIC MODELING TECHNIQUES
FOR RELIABILITY BASED DESIGN OF GEO-STRUCTURES
Zhao Li
Dissertation
Approved: Accepted:
______Advisor Department Chair Dr. Robert Liang Dr. Wieslaw Binienda
______Committee Member Interim Dean of College Dr. Malena Espanol Dr. Eric Amis
______Committee Member Dean of the Graduate School Dr. Chang Ye Dr. Chand Midha
______Committee Member Date Dr. Junliang Tao
______Committee Member Dr. Zhe Luo
ii ABSTRACT
Obtaining adequate and accurate subsurface lithological stratification is an essential and the first task in solving many geotechnical engineering problems. However, due to limited field observations constraint by geotechnical investigation techniques and project budget, inference of subsurface stratigraphic structure unavoidably involves various degree of uncertainty. To obtain better understanding of the uncertain subsurface stratigraphic structure, there is a need to describe stratigraphic structure in a probabilistic manner, and to estimate the stratigraphic uncertainty with a quantitative measure.
For this end, a stochastic geological modeling framework is proposed in this study to generate possible stratigraphic configurations conditional on available site investigation data, and further develop a compatible uncertainty quantification procedure for estimating the stratigraphic uncertainty. The developed stochastic geological modeling framework, by employing Markov random field with a specific spatial correlation, is intended to describe the inherent heterogeneous, anisotropic and non-stationary characteristics of stratigraphic configurations. In particular, a potential function by means of a local neighborhood system was introduced to account for spatial correlations of lithological units and strata extensions. On the basis of the proposed stochastic geological modeling framework, an uncertainty quantification procedure is established to provide quantitative estimation of the stratigraphic uncertainty. The sensitivity analysis of the
iii proposed geological model is conducted to reveal the influence of mesh density and the model parameter on the simulation results. Bayesian inferential framework is introduced to allow for the estimation of the posterior distribution of model parameter, when additional or subsequent borehole information becomes available.
Furthermore, the uncertainties associated with the interpretation of lithological profiles and the spatial variation of soil properties for each identified lithological unit may be significant and should be considered in the geotechnical design. An integrated approach is proposed for probabilistic analysis and design of geotechnical structures that consider both sources of uncertainty by utilizing a Markov random field (MRF) for stochastic modeling of the stratigraphic profile and a Gaussian random field (GRF) for modeling the spatially varying soil properties within each lithological unit. The detailed simulation procedure in the framework of MRF and GRF are described.
Finally, since various sources of uncertainties exist during the design of geotechnical structures, such as the uncertainties from stratigraphic structure, soil properties, external loading, design methodology, and performance requirements, how to handle these uncertainties becomes a challenge for geotechnical engineers. An integrated approach is also proposed to handle all these uncertainties comprehensively and systematically and incorporates these uncertainties into reliability analysis of deep foundation.
iv ACKNOWLEDGEMENTS
I would like to thank my academic advisor, Prof. Robert Liang, for his support and kind encouragements in the past three years. Indeed, there are many people at The
University of Akron that provides a lot of support when needed. Among them, I would like to thank Dr. Malena Espanol, Dr. Chang Ye, Dr. Junliang Tao, Dr. Zhe Luo and Dr.
Wieslaw Binienda for their help in the PhD program. Also, I would also like to thank my group members: Hui Wang, Xiangrong Wang, Lin Li, Hanjian Fan, Ayako Yajima, and
Zhicheng Gao for helping me in both research and life. In addition, I would like to thank our department secretary Kimberly Stone, who provides me a lot of help during these years. Finally, I would like to thank my parents and my fiancé, Steven Mankoci. They were always supporting me and encouraging me with their best wishes. I would never have been able to finish my dissertation without the help from my committee members, other professors, friends and my family.
v TABLE OF CONTENTS
Page
LIST OF TABLES ...... x
LIST OF FIGURES ...... xi
CHAPTER
I. INTRODUCTION ...... 1
1.1 Overview ...... 1
1.2 Statement of the Problem ...... 4
1.3 Objectives and Scope of Work ...... 7
1.4 Dissertation Outline ...... 11
II. LITERATURE REVIEW ...... 14
2.1 Research Path on Geological Modeling ...... 14
2.1.1 Interpolation Methods...... 15
2.1.2 Stochastic Methods ...... 17
2.2 Markov Random Field ...... 19
2.3 Soil Variability Model ...... 22
2.4 Random Fields Generator ...... 24
2.5 Reliability Analysis ...... 27
III. STOCHASTIC GEOLOGICAL MODELING FRAMEWORK BASED ON MARKOV RANDOM FIELD ...... 30
3.1 Introduction ...... 30
3.2 Neighborhood System, Markov Random Field and Gibbs Distribution ...... 32
vi 3.2.1 Neighborhood System ...... 33
3.2.2 Markov Random Field and Gibbs distribution ...... 33
3.3 Local Transition Sampler and Potential Function ...... 35
3.4 Stratigraphic Uncertainty Quantification ...... 41
3.5 Process for Generating Stratigraphic Realization ...... 42
3.6 Effect of Model Parameters on Model Behavior ...... 44
3.7 Case Study ...... 47
3.7.1 Geology and Observations of the Projects ...... 47
3.7.2 Interpreting the Interface between Soil and Bed Rock ...... 51
3.7.3 Interpreting the Stratigraphic Structure ...... 57
3.8 Summary and Conclusions ...... 60
IV. QUANTIFYING STRATIGRAPHIC UNCERTAINTIES BY STOCHASTIC SIMULATION TECHNIQUES BASED ON MARKOV RANDOM FIELD...... 62
4.1 Introduction ...... 62
4.2 Two Modeling Approaches and Sensitivity Analysis ...... 65
4.2.1 Two Modeling Approaches ...... 65
4.2.2 The Algorithm of Proposed Stochastic Geological Model ...... 72
4.2.3 Sensitivity Analysis ...... 74
4.3 Parameter Estimation...... 80
4.3.1 Bayesian Inferential Framework ...... 80
4.3.2 Illustrative Example ...... 82
4.4 Summary and Conclusions ...... 90
V. PROBABILISTIC EVALUATION OF FOUNDATION PERFORMANCE CONSIDERING STRATIGRAPHIC UNCERTAINTY AND SPATIALLY VARYING SOIL PROPERTIES ...... 92
5.1 Introduction ...... 92
vii 5.2 Framework of the Proposed Method ...... 94
5.3 Stochastic Geological Model based on Markov Random Field ...... 97
5.3.1 Markov Random Fields ...... 98
5.3.2 Prior Energy, Likelihood Energy and Posterior Energy ...... 101
5.3.3 Stratigraphic Uncertainty Quantification Using Information Entropy ... 105
5.3.4 Simulation Procedure ...... 106
5.4 Characterization of Material Properties based on Gaussian Random Field ..... 109
5.5 Numerical Example of a Shallow Foundation Settlement Evaluation ...... 111
5.5.1 Stratigraphy Simulation and Uncertainty Quantification Using MRF ... 112
5.5.2 Realizations for Geomaterial Properties Generated Using GRF ...... 117
5.5.3 Settlement Analysis of Shallow Foundation ...... 119
5.6 Summary and Conclusions ...... 128
VI. QUANTIFY GEOTECHNICAL UNCERTAINTIES FOR PERFORMANCE- BASED RELIABILITY ANALYSIS OF DEEP FOUNDATIONS CONSIDERING MULTIPLE FAILURE MODES ...... 130
6.1 Introduction ...... 130
6.2 Stochastic Geological Model ...... 133
6.3. Uncertainties Characterization ...... 134
6.3.1 Characterization of Stratigraphic Uncertainties ...... 134
6.3.2 Characterization of Material Properties ...... 134
6.3.3 Characterization of Model Errors ...... 136
6.3.4 Characterization of External Loads ...... 137
6.3.5 Characterization of Structure Performance Requirements ...... 137
6.4 Framework of Performance-based Reliability Analysis ...... 138
6.4.1 Reliability Assessment ...... 138
6.4.2 Computational Procedure ...... 139
viii 6.5 Illustrative Example...... 141
6.5.1 Uncertainties Quantification ...... 144
6.5.2 Reliability Analysis of Deep Foundation Considering Multiple Failure Modes ...... 148
6.6 Summary and Conclusions ...... 152
VII. SUMMARY AND CONCLUSIONS ...... 154
7.1 Summary of Work Accomplished ...... 154
7.2 Conclusions ...... 158
7.3 Recommendations for Future Research...... 160
REFERENCES ...... 162
ix LIST OF TABLES
Table Page
4.1 The locations and magnitude of known orientation vectors ...... 83
5.1 Material properties for two lithologic units ...... 126
5.2 The summarization of FEA computational results ...... 126
6.1 Material properties for two lithologic units ...... 143
6.2 Soil properties of borehole samples ...... 143
6.3 The locations and magnitude of known orientation data ...... 144
6.4 Statistical properties of random variables ...... 148
6.5 Feasible designs of shaft at three different locations ...... 151
x LIST OF FIGURES
Figure Page
2.1 An example of a Markov random field ...... 20
3.1 Local neighborhood system ...... 33
3.2 Standard Geometric condition and corresponding spatial correlation model ...... 38
3.3 Neighborhood system (a) neighborhood system of uniform cubic mesh (b) geometric condition of neighboring element pairs ...... 39
3.4 Ellipsoid model for spatial correlation ...... 40
3.5 Sensitivity analysis of model parameter (a) changing a in local neighborhood system ...... 44 (b) corresponding interested probability under different temperature ...... 45
3.6 Sensitivity analysis of polar angle (a) changing in local neighborhood system ...... 46 (b) corresponding interested probability under different temperature ...... 47
3.7 Two-dimensional case study (a) available boreholes in the two-dimensional project (b) borehole logs of the two-dimensional project ...... 48 (c) mesh plot for the two-dimensional project ...... 49
3.8 Three-dimensional case study (a) available boreholes in the three-dimensional project (b) borehole logs of the three-dimensional project (c) mesh plot for the three-dimensional project ...... 50
3.9 Comparison of confident ratios and total information entropy values of different modeling cases for interpreting the soil-rock interface (a) two-dimensional project ...... 52 (b) three-dimensional project ...... 53
3.10 Interpreting the soil-rock interface for two-dimensional project (a) information entropy map (b) map of confident assignments ...... 54
xi 3.11 Map of confident assignments for interpreting the soil-rock interface of the three- dimensional project (a) 4 boreholes (b) 8 boreholes ...... 55 (c) 9 boreholes...... 56
3.12 Interpreting the stratigraphic structure of the two-dimensional project (a) information entropy map (b) map of confident assignments ...... 58
3.13 Map of confident assignments for interpreting the stratigraphic structure of the three-dimensional project ...... 59
4.1 Known information (ground surface soil type, borehole soil types, and strata’s orientation =0°)...... 67
4.2 Simulation for two modeling approaches (a) the evolution of total energy ...... 68 (b) histogram of total number of iterations for all 1000 chains...... 69
4.3 Total information entropy for two modeling approaches along the number of realization s (a) changing of total information entropy (b) measured COV of total information entropy ...... 70
4.4 Realization and information entropy plot for two modeling approaches (a) one possible realization for ICM modeling approach (b) one possible realization for MCMC modeling approach ...... 70 (c) information entropy plot for ICM modeling approach (d) information entropy plot for MCMC modeling approach ...... 71
4.5 Diagram of sampling in MCMC technique ...... 74
4.6 Information entropy plots with different mesh density: from top to bottom 400 elements, 1600 elements, and 6400 elements, respectively, with the left column for the ICM approach and the right column for the MCMC approach ...... 76
4.7 Information entropy plots with different parameter a : from top to bottom =1.5, =3, and =5, respectively, with the left column for ICM approach and the right column for MCMC approach ...... 78
4.8 Total entropy for two modeling approaches...... 79
4.9 Illustrative example-I showing known information (ground surface soil type, borehole soil types, and strata’s orientation) ...... 83
4.10 Joint frequency diagram of the pair of depth ( y1 , y 2 ) ...... 84
4.11 Parameter estimation results for three cases using ICM approach
xii (a) samples and histogram of a for Case 1 (b) samples and histogram of for Case 2 (c) samples and histogram of for Case 3 ...... 86
4.12 Estimation generated by Borehole #2+ground surface soil type +strata’s orientation when =3.1459 (a) a possible realization (b) Information Entropy plot (c) comparison of soil classification information between original boreholes and estimated boreholes from one realization, and information entropy plot at Borehole #1 #3 #4 ...... 87
4.13 Illustrative example-II showing known information (ground surface soil type and borehole soil types) ...... 88
4.14 Estimation using MCMC approach (a) Parameter estimation results (b) Information Entropy plot using Borehole#5+#7 when =2.4970...... 89
5.1 Framework of the proposed method...... 96
5.2 Local neighborhood system...... 99
5.3 Ellipse model for local spatial correlation...... 103
5.4 Scan order and local initial configurations (a) diagram of neighbor element classification (different colors are used to distinguish different element classes) (b) local initial configurations using three different random scan orders (different colors represent different lithological units) ...... 107
5.5 Known information (ground surface soil/rock type and borehole soil types) and the strip footing...... 112
5.6 The evolution of total energy (4 realizations)...... 113
5.7 A possible realization (a) initial subsurface configuration (b) the corresponding stratification estimate after the stochastic energy relaxation...... 114
5.8 Total information entropy (a) the changing of total information entropy along the number of realizations (b) the measured COV of total information entropy along the number of realizations ...... 115
5.9 Information entropy plot and probability plot (a) information entropy plot based on 1000 MAP estimates ...... 116 (b) probability plot for the lithologic unit 1 (c) probability plot for the lithologic unit 2 ...... 117
5.10 A realization of coupled random fields ...... 118
xiii 5.11 Eequ results under the influence of both lithological uncertainty and spatially varied soil properties (a) PDF of (b) CDF of ...... 121
5.12 Simulation results (a) a realization of couple random fields (b) the corresponding settlement diagram of FEA model...... 122
5.13 Compared simulation results (a) another realization of couple random fields (b) the corresponding settlement diagram of FEA model...... 123
5.14 results under the influence of lithological uncertainty (a) PDF of ...... 124 (b) CDF of ...... 125
5.15 Inferred failure probability evolution of two selected units (a) without considering imperfect detection (b) considering missed defects. ... 127
6.1 Flow chart of proposed approach for performance-based reliability design...... 141
6.2 Know information (ground surface soil type, borehole soil types, borehole samples, and strata’s orientation)...... 143
6.3 Two possible stratigraphic configurations...... 145
6.4 The color map of normalized information entropy...... 146
6.5 Realizations of soil properties (a) medium clay (c) sand (e) stiff clay corresponding to stratigraphic configuration Figure 6.4a (b) medium clay (d) sand (f) stiff clay corresponding to stratigraphic configuration Figure 6.4b...... 147
6.6 Example of drilled shaft...... 149
6.7 Convergence of the estimates...... 150
6.8 Probability of failure for different designs (a) location 1 (b) location 2 (c) location 3...... 151
xiv
CHAPTER I
INTRODUCTION
1.1 Overview
Design and analysis of geotechnical structures requires site investigation to obtain sufficient data of site conditions. However, due to limitations of current site investigation technologies and project budget constraints, soil boreholes are typically scarce and sparsely distributed within the project site. Expenditures must be commensurate with both the scope of the project and with the potential consequences of using limited information to make decisions. Generally, several aspects of site geology are of interest in the process of site characterization: (1) the geological nature and classification of deposits and formations (lithological units), (2) the location, distribution, thickness, and material composition of the formations, and (3) the pertinent material properties of formations (Baecher and Christian, 2005).
The observations from site characterization (e.g. borehole logs) are classified with respect to soil types. The classifications can be made by visual inspection, lab measurements, and statistical classification analysis of descriptive properties of soils, such as color, odor, density, water content, and grain size distribution (Baecher and
Christian, 2005). The classified data are subsequently used in the interpretation of
1 lithological (stratigraphic) profile showing the locations, depths, and continuities of soil strata. Geological modeling is commonly used for quantifying geological process, interpreting soil stratification, and identifying natural hazards. A geological model is the numerical equivalent of a three-dimensional (3D) or two-dimensional (2D) geological map complemented by a description of physical quantities in the domain of interest.
Statistical theories are useful in geological modeling based on these observations made on and below the ground surface.
However, almost all types of geological data are subject to several sources of uncertainty due to limited knowledge of underground, including insufficient sample numbers, measurement inaccuracies, imperfect concepts and hypotheses for geological structural simplifications, intrinsic randomness, heterogeneity, and among others
(Wellmann et al., 2010). These types of uncertainties can be broadly classified into three categories: (1) imprecision and measurement error, uncertainty in all types of raw data
(e.g., uncertain geological formation boundary and structure’s orientation); (2) inherent randomness and heterogeneity: uncertainty in interpretation of lithological profile between and extrapolation away from known data points (stratigraphic uncertainty); (3) imprecise knowledge: incomplete and imprecise knowledge of structural existence (e.g. a fault exists or not).The first type of uncertainty has been dealt with geostatistical methods for simple structures. The second type of uncertainty has been handled with a variety of statistical and geostatistical methods (Chilès et al., 2004). The last type of uncertainty is often difficult and impossible to evaluate.
Meanwhile, soil is a complicated material that is formed through a combination of physical and chemical processes, and thus its components vary significantly from one
2 point to another. The variability in soil properties is a complex attribute that results from different sources of uncertainties. As recognized in Phoon and Kulhawy (1999), the primary sources of uncertainties for soil properties include the inherent variability of soil properties, measurement errors, and transformation errors. The first type of uncertainty is due to the natural geological process that produced and continually modified the soil mass in situ. The second one is caused by equipment, testing procedures, testing effects and statistical uncertainty or sampling error that result from limited amounts of known information. The third one is introduced when the raw data from subsurface investigations and field or lab measurements, such as the standard penetration test (SPT) or cone penetration test (CPT), is converted to design soil properties, such as undrained shear strength for cohesive soils or effective friction angle for cohesionless soils, using empirical or other computational models. As a result of the formulation process of soils and the sampling and measuring process, soil properties used in the geo-structure design are always uncertain to some degree. The uncertainties of soil properties should be taken into account properly in the design process in order to guarantee the safety and the serviceability.
In order to tackle the uncertainties in design of geotechnical structures, in the past century, Allowable Stress Design (ASD) attempt to ensure that external loads acting upon a structure or foundation do not exceed some allowable limit. ASD combines uncertainties in loads and resistance into a single factor of safety (FOS), which is the ratio of a set of predicted resistances to a set of design loads, stresses or other demands.
Engineering judgments, along with ASD in selecting appropriate FOS, have been used to tackle the influences of various uncertainties on the reliability analysis of geotechnical
3 structures. In recent years, reliability-based design (RBD) is proposed to deal with the influences of uncertainties on the reliability of geotechnical structures. Federal Highway
Administration (FHWA) has incorporated Load and Resistance Factor Design (LRFD) in the FHWA Design Specifications. In contrast to ASD with its single safety criterion
(FOS), LRFD takes into consideration the variability in loads and resistances separately by defining separate factors on each. A load factor is assigned to uncertainty in loads and a resistance factor is assigned to uncertainty in resistances. This represents a small step toward a better handling of uncertainties associated with loads and resistances. However, during the implementation of LRFD, there are a number of implementation issues and problems confronting state departments of transportation. For instance, one of the outstanding problems is the calibration of the resistance factors (Lai, 2009). Under this situation, based on randomized input, Monte-Carlo Simulation (MCS) is considered as the most robust methodology dealing with reliability analysis, especially when the calculation process is strongly nonlinear or involves many uncertain inputs. Various uncertainties are qualified by a sequence of randomly generated samples. Each set of samples is used as input in MCS approach, then the response of a complex system is calculated repeatedly until a desired or prescribed sample size is achieved. Occordingly, it is possible to study the output of MCS statistically.
1.2 Statement of the Problem
One of the most elementary tasks of site characterization is to interpret or model
(map) local and regional lithological profile and infer the continuity of geological strata or formations. Specifically, it is to identify, characterize and interpolate geological
4 structural features that form the boundaries (contact surfaces) among formations and strata. The common practice to interpret lithological profile is based on the observations of the geological features of a finite number of locations (e.g., borehole logs) which are sparsely distributed. For the purpose of geological mapping, several approaches based on geo-statistical methods or interpolation methods have been established (Auerbach and
Schaeben, 1990; Blanchin and Chilès, 1993; Chilès, Aug, Guillen and Lees, 2004).
Recently, probabilistic approaches have been developed to determine underground soil stratification based on cone penetration test (CPT) data (Cao and Wang, 2012; Wang et al., 2013; Ching et al., 2015) and to identify soil strata in London Clay formation based on water content data (Wang et al., 2014). Among these studies, the contact points of the subsurface stratigraphy are determined probabilistically. However, most of these methods focus on estimate of the subsurface structure (lithological profile) with the Maximum
Probability Estimate (MPE) conditional on local experiences, e.g. kriging, which is point estimation, and hence the uncertainty involved with geological modeling (mapping) cannot be quantified. Moreover, in engineering practice, the interpretation of lithological profile is dealt with mainly by using engineering judgment based on local experience
(Elkateb et al., 2003). The abovementioned probabilistic methods and statistics have been relatively little used in rationalizing the interpretation of soil stratigraphy.
On the other hand, soil properties exhibit spatial variability and spatial dependence as noted by Fenton and Griffiths (2008). After the lithological profile has been estimated, soil properties for each lithological unit are commonly modeled as random fields in existing literature (Fenton and Griffiths, 2008). However, no work has been reported in the literature which takes considerations of the uncertainties associated with
5 interpretation of lithological profile and spatially variation of soil properties for each identified lithological unit. Due to lack of such a geological model which is capable of dealing with uncertainties from geological heterogeneity comprehensively, subsequent analysis, design and maintenance of geotechnical structure are also uncertain. We simply cannot achieve, or cannot afford to achieve absolute safety or guarantee the serviceability for any geo-structure and there is always the existence of underlying geotechnical variability.
In the implementation of ASD, the uncertainties in loads and resistance are limped into a single factor of safety (FOS). The influences of the uncertainties involved in the reliability of geotechnical structures are tackled by the reliability-based design (RBD) approach. In the LRFD reliability analysis, practicing engineers must accept all the invisible predefined assumptions or simplifications (e.g. probability distributions of inputs in the design model as well as the overall design method) made during the calibration of the load and resistance factors. The calibrated load and resistance factors are only applicable to a specific design model and fail to take account for the uncertainty associated with the interpretation of the configuration of lithological profile
(stratigraphic/lithological uncertainty), the degree of uncertainties and variability of soil properties, unforeseen loading conditions or computational model errors, etc.
Furthermore, the system reliability of geo-structure is usually ignored in the reliability analysis. However, in the current LRFD approach, different modes of limit states are considered separately without considering the effects of their interaction. Consequently, there is a potential that the overall failure probability may be underestimated. Therefore,
6 there still remains a critical need for developing a comprehensive and unified reliability analysis approach to consider all uncertainties in a systematic manner.
1.3 Objectives and Scope of Work
The purpose of the research is to develop a novel geological model which is able to interpret subsurface lithological profile and quantify stratigraphic uncertainty, so that the underlying risks from subsurface in reliability-based design of geotechnical structures can be properly taken into account. The objectives of this research are:
1) Propose a fundamentally sound stochastic geological model to estimate the
lithological profile and quantify stratigraphic uncertainty using limited known
information based on Markov random field (MRF).
2) Develop an integrated approach to quantify geotechnical uncertainties in a systematic
manner and incorporate these uncertainties into a probabilistic framework to evaluate
the performance of geo-structures.
To solve the problems and achieve the objectives mentioned above, the specific scope of work is enumerated as follows:
A stochastic geological model is developed for the interpretation of configurations of lithological profile based on MRF using site characterization data, such as the ground surface soil/rock types, the soil/rock types and their boundaries at the borehole locations, and the strata orientation information from physical tests. In order to consider the spatial dependencies or correlated features of geological structure, a neighborhood system and an ellipse model are proposed. The potential functions used to specify the MRF are
7 carefully designed to reflect the spatial correlation of the geological structure by means of a local neighborhood system and an ellipse model. A step-by-step modeling procedure for subsurface lithological profile simulation is developed. Following the modeling procedure, the stratigraphic uncertainties can be quantified and visualized by using the concept of information entropy (Wellmann, Horowitz, Schill and Regenauer-Lieb, 2010).
The proposed geological modeling and uncertainty quantification framework is applied to real borehole data to demonstrate the favorable role of the developed methodology for site investigation process of geotechnical practice. The proposed geological model is applied to a two-dimensional real construction project to demonstrate its capability in modeling two-dimensional site. The proposed geological model is applied to a three-dimensional real construction project to demonstrate its applicability in modeling three-dimensional geological body. The proposed modeling framework is used to as a tool which can inform whether the site characterization data is enough and provide suggestion for the locations of the further borehole drilling.
Two modeling approaches are introduced to simulate subsurface geological structures to accommodate different confidence levels on geological structure type (i.e., layered vs. others). The first approach is based on the assumption that geological body has the layered structure. The confidence on the layered system is based on prior knowledge, such as orientation measurements (e.g., seismic surveys), interpretation of geological information and data, and engineering judgment. Under this condition, the iterated conditional modes (ICM) algorithm (Besag, 1986) is adopted. If one concerns about the uncertainty in field investigation data, then Markov chain Monte Carlo (MCMC) method is adopted to allow introducing more randomness into the initial simulation
8 process. Consequently, it is reasonable to anticipate greater degree of uncertainty in the simulated subsurface profile generated via MCMC technique.
The sensitivity analysis of the proposed stochastic geological model is conducted to evaluate the performance of the proposed model under different parameter configurations for two modeling approaches. The sensitivity analysis includes evaluation of the influence of the mesh density on the performance of the proposed model and the influence of the model parameter on the behavior of the proposed model. The Bayesian inferential framework is developed to estimate the model parameter when additional borehole information becomes available. The likelihood for a specific model parameter can be obtained by comparing the generated stratigraphic realizations with additional borehole information. The samples of posterior distribution of model parameter are obtained through the adaptive Metropolis (AM) algorithm—which is based on the classical Metropolis–Hasting algorithm (Metropolis et al., 1953; Hastings, 1970), and the model parameter is selected as the mean value of the samples.
In order to consider both stratigraphic uncertainties and spatially varied material properties, an integrated approach is proposed. The proposed stochastic geological modelling method for simulation of stratigraphic profile is based on MRF with a novel neighborhood system that can reflect the inherent heterogeneous (multiple lithological units) and anisotropic (direction-dependent spatial correlation) characteristics of a geological body in generating lithological profiles. Based on the stratigraphic configurations generated by the geological model, the spatial variability of geo-material properties of each lithological unit is handled by two-dimensional (2D) Gaussian random fields (GRF) with incorporating the correlation function. The proposed integrated
9 approach is applied to the probabilistic analysis of shallow foundation considering both stratigraphic uncertainties and spatially varied material properties through the finite element analysis/Monte Carlo simulation (MCS) framework. To be more specifically, each generated combination of stratigraphic profile and soil properties is used in finite element analysis (FEA) of geo-structure (such as a strip footing) by using Monte Carlo simulations. Then the performance of geo-structure can be assessed through interpretation of the FEA analysis output. Moreover, the effects of simulated uncertain subsurface profiles together with two-dimensional spatial random elastic modulus on the computed settlement of a strip footing were elucidated by the proposed probabilistic analysis approach. At last, the probability distributions of equivalent elastic modulus can be obtained for further analysis of the strip footing underlain by layers of different lithological units.
An integrated approach to quantify geotechnical uncertainties in a systematic manner and incorporate these uncertainties into a probabilistic framework is proposed to evaluate the serviceability performance of the drilled shaft. The uncertainties that affect the reliability of the designed drilled shaft are identified as: subsurface soil stratigraphy, soil properties, external loads, model errors relative to the soil-pile interaction model (p-y model and t-z model), and performance criteria. The stratigraphic uncertainty is quantified by the proposed stochastic geological modelling method. The spatial variability of soil property of each identified soil layer is modeled conditional on the relative known borehole samples by using 2D GRF, in which each soil property is statistically characterized by the mean, variance, and correlation length. The uncertainties from loading conditions, computational models, and performance criteria are handled by
10 their respective prescribed probability distributions. Based on the relative probabilistic models, the generated random samples are input into the soil-pile interaction model to evaluate the performance of the drilled shaft by using MCS. Three failure modes are considered in the performance-based reliability analysis of a drilled shaft, including the vertical movement, lateral deflection, and angular distortion at the top of the pile. The system failure is considered as any of the induced displacements exceeds the corresponding allowable movement. The target reliability index is used to measure whether the system performance is satisfactory, which is defined based on the desired performance level. The details of the proposed reliability analysis framework are provided by a step-by-step procedure description, and a computer program is developed to implement the developed methodology.
1.4 Dissertation Outline
A total of seven chapters are included in this dissertation. The remaining chapters are organized as follows:
Chapter 2 presents a summary of the literature review. In the literature review, the research history of geological modeling is introduced first. The development and applications of Markov random fields are briefly described. The assumptions and the uncertainties of the methods are discussed in detail, and the random field–based modeling of soil properties is explained. The reliability methods (such as Monte Carlo simulation) are discussed, and recent developments in reliability-based design methodology are reviewed.
11 Chapter 3 introduces the basic assumption of the proposed geological model. The mathematical theories of Markov random fields and Gibbs distribution are elucidated in details. The proposed neighborhood system and the ellipse model are introduced to allow for taking into account of spatial correlations of lithological unit and strata extension.
Also, details are provided regarding the prior energy, likelihood energy, posterior energy and their potential functions. Rather than giving a deterministic estimate of the formation boundary, the proposed framework is designed to reflect the inherent heterogeneous and anisotropic characteristics of geological structure by using Markov random field theory.
Then the concept of information entropy is introduced in the post-process as a measure indicator. In addition, the step-by-step simulation procedure is presented. At last, the proposed geological modeling framework is applied to two real highway construction projects to demonstrate its ability in subsurface simulation and uncertainty quantification.
Chapter 4 presents two geological modeling approaches involving the ICM and
MCMC techniques for constructing stochastic heterogeneous geological model using three types of site investigation data, including ground surface soil/rock type, borehole soil/rock classification information and strata’ orientation information. The comparison between using these two modeling techniques is also carried out. Through mesh density and model parameter sensitivity analysis, insights were provided on the performance of the proposed model. A method based Bayesian inference framework is developed for determining model parameter when additional borehole log becomes available. Specific conclusions based on numerical examples are included.
Chapter 5 presents an integrated approach for probabilistic analysis and design of geotechnical structures considering both sources of uncertainties by utilizing Markov
12 random field (MRF) for stochastic modeling of stratigraphic profile and Gaussian random field (GRF) for spatially varied soil properties within each lithological unit. The detailed simulation procedure in the framework of MRF and GRF are described. A simple strip footing is used as an example to illustrate the application of the proposed approach. The numerical example clearly indicates that both sources of uncertainties should be considered in a probabilistic approach for foundation design.
Chapter 6 presents an integrated approach to handle various geotechnical uncertainties comprehensively and systematically and incorporates these uncertainties into reliability analysis of deep foundation. The uncertainties that affect the reliability of the designed drilled shaft are identified as: subsurface soil stratigraphy, soil properties, external loads, model errors relative to the soil-pile interaction model (p-y model and t-z model), and performance criteria. The stratigraphic uncertainty is quantified by the proposed stochastic geological modelling method. Then the spatial variability of soil property of each identified soil layer is modeled conditional on the relative known borehole samples by using 2D GRF. The uncertainties from loading conditions, computational models, and performance criteria are handled by their respective prescribed probability distributions. Three failure modes are considered in the performance-based reliability analysis of a drilled shaft, including the vertical movement, lateral deflection, and angular distortion at the top of the pile. The system failure is considered as any of the induced displacements exceeds the corresponding allowable movement and is evaluated via the proposed reliability analysis framework.
Chapter 7 presents the summary of the research and the associated conclusions. This chapter also provides recommendations for future research.
13
CHAPTER II
LITERATURE REVIEW
2.1 Research Path on Geological Modeling
Geological bodies usually have complicated geometries derived from a long history of sedimentation, structural deformations, and weathering. However, the geological data or site characterization data are always sparse and limited, and nothing is known between over-sampled locations such as geological maps, cross-sections or boreholes (Calcagno et al., 2008). Classic measurements made in the field or by interpreting boreholes give the contact locations of geological formations and the dip of the formations, which are only contact points. Geological modeling consists in inferring a configuration of subsurface lithological profile based on the limited available geological data (e.g., contact locations and orientation data). Two-dimensional (2D) methods are used to construct horizons cross-section of geological body (Galera et al., 2003). Three-dimensional (3D) geological models are useful in better understanding geological formation and in quantifying physical process (Calcagno, Chilès, Courrioux and Guillen, 2008). A series of geometric models have been established (Wijns et al., 2003; Wu et al., 2005; Houlding, 2012) in interpretation of a representation of the geological body.
14 Probabilistic models and statistic methods have been developed to aid the interpretation process of lithological profile. There are at least two main reasons why probabilistic models are useful in this process. First, they can be used to make inferences about the underground stratification of unobserved area of a site. The inferences based on probabilistic models are more powerful than those based on intuition. Second, they can be used to inform the adequacy of site characterization data and optimize the allocation of site exploration, if the probabilistic models are capable to quantify the stratigraphic uncertainty (Baecher and Christian, 2005).
2.1.1 Interpolation Methods
The commonly used interpolation methods include Kriging, Cokriging, and Potential field. Kriging method was formalized as a statistical approach and generally used in geostatistics, which is based on the empirical work of Danie G Krige in evaluating mineral resources. Kriging is basically the optimal prediction or the best linear unbiased estimation with the additional ability to estimate certain aspects of the mean trend (Stein,
2012). Kriging is a form of weighted averaging in which the weights are chosen such that the error is less than for any other linear sum (Cressie, 1990). The Kriging method depends on two types of information: knowledge of how the mean value varies functionally with position and the covariance structure of the field. The covariance structure is related to the selected correlation function, which will be discussed in Section
2.3. The basic idea of Kriging method is to estimate the unknown point using a weighted linear combination of the values of each observation point. The weights are determined based on the covariance structure, which reflects not only the distance between points and
15 the effects of differing lithological units. If a point is close to one of the observed points, then the corresponding weight would be high. On the other hand, if two points are located in different soil layers, the corresponding weight should be small. Kriging method has been applied in mining, meteorology, statistics and other disciplines. More details of
Kriging method can be found in (Cressie, 1990). The common cokriging methods are multivariate extensions of the kriging system of equations, and use two or more additional attributes (Knotters et al., 1995). Cokriging methods are used to take advantage of the covariance between two or more regionalized variables that are related, and are appropriate when the main attribute of interest (well data) is sparse, but related secondary information (seismic) is abundant. The cross-covariance function is needed so that many different types and scales of data can be integrated. Cokriging requires the same conditions to be satisfied as kriging does, but demands more variography, modeling, and computation time (Knotters, Brus and Voshaar, 1995).
The potential field method was designed to build three-dimensional (3D) geological models from available data from geology and mining exploration, such as borehole data, structural data related to geological interfaces, geological map, and interpretations of geologists. The potential field considers a geological interface as a particular isopotential surface of a scalar field defined in the 3D space. In the interpolation of the potential field, two main types of data are coded: interface points (e.g., 3D points discretizing geological contours and formation intersections of boreholes) and strutual data (e.g., unit vectors normal to stratification which is measured on outcrops or in boreholes). Due to the difficulty in inferring the covariance of the potential field, the covariance can be identified from the structural data, which enables it to associate sensible cokriging
16 standard deviations to potential field estimates. More details of the potential field method can be found in (Chilès, Aug, Guillen and Lees, 2004).
However, most of these methods focus on estimate of the subsurface structure
(geological model) with the Maximum Probability Estimate (MPE) conditional on local experiences, e.g. kriging, which is point estimation, and hence the lithological uncertainty cannot be quantified. Actually, the ability to know the uncertainty degree of the estimation is as important as the estimate itself. The uncertainty on the formation boundaries and volumes of various lithological units is often a major part of geotechnical uncertainty.
2.1.2 Stochastic Methods
Some stochastic methods have also been developed to characterize heterogeneity of geological formations, such as Markov chain model, multi-point geostatistics model, and etc. Markov chain model is applied to the geological field to model discrete variables such as lithological units or facies. Instead of using variograms or autocovariance functions to quantify the spatial structures, the Markov chain model is based on the conditional probabilities. The conditional probabilities have the advantage that they are interpreted geologically easier than variogram or autocovariance functions (Elfeki and
Dekking, 2001). The Markov chain probabilistic model possess a property that is usually characterized as "memorylessness" or called Markov property: the probability distribution of the next state depends only on the current state and not on the sequence of events that preceded it (Ang and Tang, 2007). In one-dimensional problems, a Markov chain is described by a single transition probability matrix (Matsuo et al., 1994).
17 Transition probabilities correspond to relative frequencies of transitions from a certain state to another state. However, Markov chains are frequently assumed to be time- homogeneous, so that the process is described by a single, time-independent transition probability matrix. Thus, the Markov chain model is subjected to a strong assumption: the transition probability matrix is stationary and ergodic.
Multiple-point geostatistics was used for modeling subsurface heterogeneity (Caers,
2002; Hu and Chugunova, 2008). Multiple-point geostatistics model does not rely on variogram models, instead, it allows capturing spatial structure from so-called “training images”. Training images are essentially a database of geological patterns (even can be
3D geological patterns). Multiple-point geostatistics model could learn and borrow these features obtained from the training images (Tang et al., 2013). Once the required patterns are extracted from the training images, they need to be anchored to subsurface data, such as well-log, seismic, and production data. The training image works in the multiple-point geostatistics as a statistical device of geological patterns or measure for geological heterogeneity, which can contain multiple-point information and is more intuitive than the variogram (Caers and Zhang, 2004). The idea of using multiple-point statistics from training images in geological modeling was proposed by (Guardiano and Srivastava,
1993). However, the construction and implement of training images also suffers the principles of stationary and ergodicity. In other words, the patterns extracted from the training image will cover the entire simulation domain. Also, the size of training images could have great impacts on the simulations. A small training image would lead to large fluctuations of large range correlations.
18 2.2 Markov Random Field
In this study, geological modeling is based on Markov random field (MRF) theory, which is also a geostatistical method. MRF theory, as one of the most sophisticated spatial statistical models, provides a convenient and consistent way for modeling context dependent entities such as correlated features and for analyzing the spatial dependencies of physical phenomena (Zhang et al., 2001; Li, 2009). The theory of MRF has been studied for a long time (Besag, 1974; Cross and Jain, 1983; Besag, 1986; Tjelmeland and
Besag, 1998; Tolpekin and Stein, 2009) and MRF has been widely applied to various fields, such as computer graphics and image restoration (Besag, 1974; Norberg et al.,
2002; Serpico and Moser, 2006; Li, 2009). In the domain of physics and probability, a random field is to be a Markov random field if it satisfies Markov properties. A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A Markov random field is a set of random variables having a Markov property described by an undirected graph (Figure 2.1) and a MRF extends this property to two or more dimensions or to random variables defined for an interconnected network of items. In
Figure 2.1, A depends on B and C. B depends on A, C and E. C depends on A, B and D.
D depends on C and E. E depends on B and D.
19 A B
C E
D
Figure 2.1 An example of a Markov random field
A MRF is a graphical model of a joint probability distribution. A MRF is constructed on an graph G (V, E) where V is the set of vertices {i | i 1,2,3,..., N} in the graph and E is the set of edges {i, j} satisfying i, j V and i j . Since the graph is a discrete mathematical structure generated from physical space, a mesh scheme must be chosen to map a physical space to a graph. Each element is considered as a vertex in the mapped graph. If two elements i, j share a common node in the meshed plot, their corresponding vertices are defined as neighbors in the graph with an edge {i, j} E . The neighbors of vertex i is defined as all vertices j V having an edge to vertex :
N (i) {j |{i, j}E, j V} (2-1)
Let L {1, 2, 3, ... , l} be the set of states or labels, then F {Fi}, iV, Fi L denotes a configuration of all vertices (i.e. a realization of possible geological model). Let
Ω {ω {Fi}| iV, Fi L} be a set of all possible subsurface configurations.
F {Fi li}, iV,li L is a MRF with respect to {N (i)},i V if:
20 P(F ω) 0 for all ωΩ (2-2)
P(Fs ls | Fr lr ,r s) P(Fs ls | Fr lr ,r N (s)) (2-3)
Equation (2-3) is called “two-sided” Markov property. It also reflects the local characteristics of MRF.
In this study, the naturally occurring geological body is assumed to be in a “stable state” and the intrinsic spatial correlation of geological structure can be modeled by contextual constraint using MRF theory. In other words, MRF is introduced as the prior distribution to interpret site exploration data. The root of such assumption is based on its successful application in geostatistics, i.e., MRF has been used to model discrete geological structures (Norberg, Rosén, Baran and Baran, 2002), and to consider geological realism and connectivity (Daly, 2005) as well as in geological mapping
(Tolpekin and Stein, 2009). Meanwhile, the practical use of MRF models is largely ascribed to the Hammersley–Clifford theorem, a MRF process is equivalent to a given
Gibbs random field, which has been proven by (Hammersley and Clifford, 1971; Besag,
1974). Such MRF-Gibbs equivalence makes it possible to represent the joint probability distribution of MRF in an explicit formula of energy function (Geman and Geman, 1984), thereby provides a feasible and powerful mechanism for modeling spatial continuity and aggregation of the stratigraphic profile. From the computational perspective, the local property of MRFs leads to the implementation of MRFs in a local and massively parallel manner.
MRF is often used in conjunction with statistical decision and estimation theories in order to formulate objective function because MRF favors the patterns encoded in itself by assigning them with larger probabilities. Maximum a posteriori (MAP) has been used
21 commonly in MRF modeling as the statistical criteria for optimality, which is called
MAP-MRF framework (Geman and Geman, 1984). In the MAP-MRF framework, the objective is to maximize the joint posterior probability of MRF. Two main parts of the framework is to generate samples of the posterior probability and to determine the parameters in it (Li, 2009).
2.3 Soil Variability Model
Apart from stratigraphic uncertainty, the uncertainty from the variability of soil properties exerts significant effects on the reliability-based design of geo-structures. Soils are geological materials which are formed by weathering process and transported by physical means as residual soils. They have been subjected to various stresses, physical and chemical process. Thus, it is well known that soil properties are spatially varied and correlated. The uncertainty observed in soil data comes both from this spatial variability and from testing errors (Baecher and Christian, 2005). The COV for soil density is in the range of 1% to 10% and the COV for undrained shear strength of clay is in the range of
10% to 50% (Phoon, 1995), respectively. Depending on type of test and the equipment, the corresponding COV vary significantly from one soil property to another. The typical
COVs for various soil properties could be found in Phoon and Kulhawy (1999).
The stochastic nature of the soil properties plays an important role on the reliability of geotechnical structure. An accurate variability model for soil properties is essential in the reliability analysis of geo-structure, such as deep foundation. A mathematically sound model for statistically charactering the variability of soil properties is needed (Fenton,
1999). In the past time, mean, standard deviation and probability distribution were used
22 to describe the variations of the soil properties. Soil properties have been modeled by normal distribution, lognormal distribution, or by more flexible distributions, such as
Beta distribution. It is common practice to model soil properties, such as unit weight of soil, elastic modulus, effective friction angle and undrained shear strength, as lognormal distribution (Fenton and Griffiths, 2008; Fan and Liang, 2013; Fan et al., 2014). There are several reasons why a lognormal distribution is used. The first is lognormal variables are always nonnegative, which is appropriate to model positive soil parameter. The second reason is the lognormal is closely related to normal. The logarithm of lognormal distribution is normal distribution. Suppose a soil property of interest (e.g., elastic modulus) is denoted as Y , which is assumed to follow a lognormal distribution. Thus the
natural logarithm of material property Y is a Gaussian random field with mean lnY and
standard deviation lnY , which can be obtained from the relations:
2 2 2 lnY ln(1Y / Y ) (2-4)
1 ln( ) 2 (2-5) lnY Y 2 lnY
where Y and Y are mean and standard deviation of material property . With the distribution parameters, the probability density function of the lognormal distribution is written as
2 1 1 ln Y lnY f (Y) exp 2 (2-6) Y lnY 2 2 lnY
However, this modeling method can only describe the soil properties at the point level. Additional tools are needed to model the variation of soil properties in space. In addition to the mean and the variance, a third parameter called scale of fluctuation
23 (correlation length) θ was suggested by Vanmarcke (1977) to characterize the spatial variability of a random variable. The scale of fluctuation is a measure of the distance within which points in the physical domain are significantly correlated. When two points are farther apart than the scale of fluctuation, it shows little correlation (Fenton and
Griffiths, 2008). The correlation length is used to define a correlation function, which describes how random variables are correlated at different separation distances. Three common correlation functions are Markov correlation function, Gaussian correlation function, and Spherical correlation function, which are expressed as follows:
exp 2 / Exponential (2-7)
exp / 2 Gaussian (2-8)
11.5 /0.5 /3, Spherical (2-9) where ρ is the correlation coefficient at the separation distance of τ, and θ is the correlation length. As a measure of the spatial correlation, the correlation length is essential in the definition of correlation function. A longer correlation length implies that the underlying random field is more uniform. If the correlation length is short, the underlying random field varies more rapidly. The above mentioned correlation functions are used to generated one-dimensional random fields. For higher dimensions, the corresponding correlation functions can be found in (Fenton and Griffiths, 2008).
2.4 Random Fields Generator
Random field models have been applied to model spatially variable properties. In the geotechnical engineering system, soil properties are appropriately represented by random
24 fields. The most common random field generator algorithms include: moving-average methods, covariance matrix decomposition, discrete fourier transform method, and local average subdivision method (Fenton and Griffiths, 2008). Herein only the covariance matrix decomposition and local average subdivision method are described in detail.
Covariance matrix decomposition is a direct method of producing a random file with prescribed covariance structure. The main advantages of this method are its simplicity and accuracy, but it would be time consuming and prone to considerable round-off error.
This method is simple to implement and allows for random fields of irregular geometry
(Zhu and Zhang, 2013). The steps to generate random fields are described as follows:
Step 1: Construct the nn ( n is the total number of elements) covariance matrix
ρ n by computing the correlation coefficient of two points (i.e., centroids of any two elements) spaced some distance apart in any direction in physical domain using one of the correlation functions.
Step 2: The covariance matrix is decomposed into the product of a lower triangular matrix X and its transpose XT by Cholesky decomposition:
T XX ρn (2-10)
Step 3: Generate the random field which possesses the prescribed correlation function by multiplying n uncorrelated Gaussian random variables by the lower triangular matrix X .
Step 4: Transform the correlated Gaussian random field to the lognormally distributed random field by the following equation:
Yi explnY lnY Zi (2-11)
25 where Yi and Z i denote the soil parameter and correlated Gaussian variable at element i , respectively.
The local average subdivision method is a fast and generally accurate method of producing realizations of a discrete “local average” random process. The concept of local average subdivision was originated from the fact that quite a number of engineering measurements are the local averages of the properties under consideration. The LAS algorithm to generate one-dimensional Gaussian random field is briefly described as follows:
Step 1: Generate a normally distributed global average with mean zero and variance obtained through local averaging theory.
Step 2: Subdivide the field into two equal parts and generate two normally distributed values that can preserve the local average of the parent cell. The two values are properly correlated with one another and show the variance according to local average theory.
Step 3: Subdivide each cell into two equal parts and generate two random variables to represent the local average of each cell, which should satisfy the criteria described in
Step 2.
Step 4: Repeat Step 3 until the cell at the desired resolution is obtained.
Simply speaking, random fields in LAS are constructed recursively by subdividing the parent cell into equal parts. More technical details can be found in (Fenton and Griffiths,
2008).
26 2.5 Reliability Analysis
Reliability analysis for geotechnical structures becomes increasingly popular. It deals with the relation with the loads and the resistance of a system. Due to the uncertainties existing in loads and resistance, the result of their interaction is also uncertain. The reliability of a system is commonly described in the form of a reliability index, which can be probabilistically related to a probability of failure. The goal of reliability analysis is to estimate the probability of failure, which accounts for the unacceptable performance. Several main steps for reliability analysis are: firstly, establish analytical model. It can be a simple equation or elaborate computation procedure. Then define distribution function and estimate statistical descriptors of parameters. Appropriate probabilistic distributions are assigned to describe the properties of geotechnical materials, loads, computation model, performance criterion, construction materials, and among others. The statistical descriptors usually include mean, variance, and covariance, the spatial correlation parameters (e.g., scale of fluctuation) can be included as well.
After that, calculate the performance of a system. The performance is determined in terms of the concerned functionality of the system, for example, the settlement of shallow foundation, the lateral deflection of drilled shaft. The last step is to calculate the reliability index and compute the probability of failure (Baecher and Christian, 2005).
Several approaches have been developed for the calculation of the probability of failure. The commonly used ones include First-Order-Second-Moment (FOSM), First-
Order-Reliability-Method (FORM), and Monte-Carlo Simulation (MCS). FOSM is based on the first order Taylor series approximation of the performance function linearized at the mean values of the random variables. FOSM uses only second moment statistics
27 (mean and variance) of the random variables. One of the major advantages of FOSM is that it reveals the relative contribution of each variable to the overall uncertainty in a clear manner, which helps to decide what factors need more investigation (Baecher and
Christian, 2005). The major drawback of FOSM is that the calculated probability of failure can be different for the same probable when stated in different ways (Fenton and
Griffiths, 2008). FORM was proposed by (Hasofer et al., 1973), which is known as geometric reliability analysis method due to the different definition of the reliability index that leads to a geometric interpretation. The reliability index is defined as the minimum distance from the origin of the axes in the reduced co-ordinate system to the limit state surface. The minimum distance point on the limit state surface is called the design point or checking point. The algorithm constructs a linear approximation to the limit state at every search point and finds the distance from the origin to the limit state.
Monte-Carlo Simulation (MCS) is a broad class of computational algorithms that relies on repeated random sampling to obtain the statistically unbiased probability of failure.
Various uncertainties are qualified by a sequence of randomly generated samples. Each set of samples is used as input in Monte Carlo approach, then the response of a complex system is calculated repeatedly until a desired or prescribed sample size is achieved.
MCS approach has a few advantages: 1) it enables designers to make adjustments on the target reliability index to accommodate specific needs for particular projects; 2) the analysis process is transparent, enabling designers to gain insights on how the performance level changes as the design parameter configuration changes; and 3) the estimate of probability of failure is accurate as long as the number of samples is large enough. MCS is commonly considered as the most robust methodology dealing with
28 reliability analysis (Fenton and Griffiths, 2008), which needs a large number of random samples to calculate the probability of failure. In terms of how many trials are necessary to ensure a desired level of accuracy, the relative study can be found in Morgan et al.
(1992), Rubinstein and Kroese (2011) and (Fishman, 2013).
29
CHAPTER III
STOCHASTIC GEOLOGICAL MODELING FRAMEWORK BASED ON MARKOV
RANDOM FIELD
3.1 Introduction
Obtaining adequate and accurate subsurface lithological stratification is an essential and the first task in solving many geotechnical engineering problems. However, it is commonly recognized that there is a tradeoff between the cost invested in geotechnical site investigation and the benefits deriving from the obtained geological information for the subsequent phases of geotechnical design and construction. Consequently, in most practice, design of the geotechnical site investigation, including determining the number and location of soil boring (or other in-situ test), highly relies on local experience with considerable constraint from project budget. On the other hand, regardless of the extent of field geotechnical investigation, in forms of borehole logs, in-situ testing, and geophysical survey, it is impossible to make direct and continuous observations in the entire subsurface domain. Therefore, the inference of the subsurface lithological stratification unavoidably involves various degree of uncertainty. Under these conditions, designers and constructors always face an urgent challenge in obtaining the most credible/confident lithological stratification with the minimum costs. To address this
30 challenge, one requisite and elementary step is to build a quantitative measure to estimate the uncertainty of lithological stratification conditional on a set of observations.
To interpret soil stratification of subsurface geological domain with sparse located observed data, substantial approaches (Auerbach and Schaeben, 1990; Blanchin and
Chilès, 1993; Chilès, Aug, Guillen and Lees, 2004; Guillen et al., 2008; Zhu and Zhang,
2013) have been developed, based on geo-statistical methods or interpolation methods.
However, most of these methods can only give point estimation as the Maximum
Probability Estimate (MPE), without taking uncertainty into consideration. In other words, although a most possible deterministic stratigraphic configuration is obtained, its probability and variability are not measured; thereby the credibility/confidence are not provided. To quantify stratigraphic uncertainty, stochastic modeling methods, such as
Markov chain model (Elfeki and Dekking, 2001) and multiple-point geostatistics (Caers and Zhang, 2004; Hu and Chugunova, 2008; Toftaker and Tjelmeland, 2013), have been established. However, these techniques may rely on certainty hypothesis, such as stationary transition probability matrixes or data templates. To further relax these hypotheses, i.e., to reflect the inherent heterogeneous, anisotropic and non-stationary nature of stratigraphic structure, we proposed an innovative geological modeling framework based on Markov random field (MRF) to describe subsurface stratigraphic structure in a probabilistic manner. On the basis of the obtained probabilistic description of stratigraphic structure, we further develop an uncertainty quantification procedure to provide quantitative measurement of stratigraphic uncertainty conditional on available field observations.
31 The geological body is considered as a spatially correlated system with a certain configuration of different lithological units (i.e., geo-material layers and layers’ orientation information). Markov random field theory, as one of the most sophisticated spatial statistical models, provides a convenient and consistent way for modeling context dependent entities such as correlated features and for analyzing the spatial dependencies of physical phenomena (Zhang, Brady and Smith, 2001; Li, 2009). In this study, the naturally occurring geological body is assumed to be in a “stable state” and the intrinsic spatial correlation of geological structure can be modeled by contextual constraint using
MRF theory. The root of such assumption is based on its successful application in geostatistics, i.e., MRF has been used to model discrete geological structures (Norberg,
Rosén, Baran and Baran, 2002), and to consider geological realism and connectivity
(Daly, 2005) as well as in geological mapping (Tolpekin and Stein, 2009). Meanwhile, as proven by the Hammersley–Clifford theorem (Hammersley and Clifford, 1971; Besag,
1974), a MRF process is equivalent to a given Gibbs random field. Such MRF-Gibbs equivalence makes it possible to represent the joint probability distribution of MRF in an explicit formula of energy function (Geman and Geman, 1984), thereby provides a feasible and powerful mechanism for modeling spatial continuity and aggregation of the stratigraphic profile.
3.2 Neighborhood System, Markov Random Field and Gibbs Distribution
The proposed geological model is constructed by discretizing the geological body of interest into small square elements. The neighborhood system, Markov Random Field, and Gibbs Distribution are first introduced.
32 3.2.1 Neighborhood System
A neighborhood system is developed to represent the spatial correlation. Let
S i | i 1,2,...,n be the set of elements in which i is an element index. In an MRF, the elements in S are related to one another via a neighborhood system, which is defined as
N {Ni | i S} . Ni is the set of all elements which share common node(s) with element
in the meshed plot. Figure 3.1 shows an example of a local neighborhood system.
Element has a local neighborhood system Ni containing 8 neighbors j1,..., j8 but not including itself, and the neighboring relationship is mutual. Note that boundary element has fewer neighbors.
(b)
j j j 8 7 6 j j 7 j 6 8 6
j i j 1 5 j i j 1 5
j j j 4 2 3 j j j 5 2 3 4
3.5 4 4.5 5 Figure 3.1 Local neighborhood system
3.2.2 Markov Random Field and Gibbs Distribution
Let R Ri ,i S be a set of random variables indexed by S , in which each random
variable Ri takes a label ri (i.e., lithological unit label, such as sand, clay, shale, etc.) in
33 its state space L 1,2,...,m,...l of all lithological units (or labels). The event R r
indicates the joint event ( Ri ri ,i S ), where r r1,...,rn denotes a subsurface configuration of R, corresponding to a realization of this random field R. Let
r (r1,...,rn ) | ri L,i S be all possible subsurface configurations. R is said to be a MRF on S with respect to a neighborhood system N , if:
P(R r) 0,r (3-1)
P(R r | r ) P(R r | r ) (3-2) i i S{i} i i Ni where S i is the set of elements not including i . Eq.(1) and Eq.(2) state the positivity and the Markovianity (local characteristics) of MRF, respectively. According to the
Hammersley–Clifford theorem, R is a MRF on S with respect to N if and only if the configurations of obey a Gibbs distribution on with respect to . Many proofs of the theorem can be found, e.g., in Besag (1974) and Li (2009).
Based on the explicit form for the joint distribution P(R r) in terms of the energy function U (r) (Geman and Geman, 1984), the Gibbs distribution relative to the neighborhood system has a probability measure (r) on with the following expression:
(r) Z 1 exp(U(r) /T) (3-3) where Z is a normalizing constant called the partition function of the form
Z exp(U(r)/T) (3-4) r
U (ri ) is the energy function in the form
U(r) Vc (r) (3-5) cC
34 which is a sum of clique potentials Vc (r) over all cliques C . denotes the set of clique c . A clique c is defined as a subset of elements in S in which each pair of elements are
neighbors. The value of Vc (r) depends on the labels of elements belonging to and represents several components of total energy. T stands for “temperature”, which has been discussed in detail in Geman and Geman (1984) and has the common form:
N T(k) , 1 k K (3-6) log(1 k) where k represent the k th iteration, the constant N should be appropriate according to the total number of iterations K . For example, given K=100, N should be 5 so that the temperature approaches to 1(or slightly higher) at the end of the simulation process.
3.3 Local Transition Sampler and Potential Function
1) Local transition sampler
The evaluation of the partition function Z is computationally prohibitive because of a large number of elements involved. In this case, Markov Chain Monte Carlo (MCMC) method is employed to sample realizations from the configuration space to obtain the maximum a posteriori (MAP) estimate. Based on the Gibbs distribution (Geman and
Geman, 1984), there exists an explicit form of the local characteristics to calculate the conditional probability given a local neighborhood system:
P(R r | r ) P(R r | r ) Z 1 exp( V (r ,r )/T),r Λ (3-7) i i S{i} i i Ni i c i j i jNi
Zi exp( Vc (ri ,rj )/T) (3-8) riΛ jNi
35 where Λ L denotes the set of lithological units that appear in the local neighborhood system.
The potential function Vc (ri , rj ) reflects the local spatial correlation of neighboring elements and represents the energy of a Gibbs distribution. In this chapter, the pair-site clique potentials are considered because of the simple form and low computational cost
(Li, 2009). In addition, it is assumed that ri rj can contribute to the potential reduction, since nearing elements tend to have the same states. Therefore, the potential function is defined as:
(i, j) , if j Ni and ri rj Vc (ri ,rj ) (3-9) 0, else where (i, j) is a measure of the local spatial correlation of two neighboring elements i and j . Compared to the total number of elements, it is computationally feasible and fast to enable the development of a sampling method that can be used repeatedly in MCMC based on the local neighborhood system. This is called the local transition sampler which can generate initial configuration and continuously update the configuration considering the heterogeneity and anisotropy of the inherent spatial correlation in a real geological body.
2) Model Parameters
The spatial correlation in the local neighborhood system is decomposed into two components named normal and tangential correlation regarding to plane orientation of geological formation (e.g., sedimentary plane, foliation, and cleavage plane). Figure 3-2a illustrates a standard geometric condition of element and one of its neighbor elements
36 j . is the intersection angle between the line of the centroid of element i to that of element and the X axis of Cartesian coordinate system. Under this condition, the radius length ( ) of an ellipse centered at the centroid of element i with a rotation with axis of a Cartesian coordinate system (see Figure 3.2b) is adopted to represent the local spatial correlation between element and element (i.e., (i, j) = ). The larger
is, the stronger influence (correlation strength) is from the neighbor element on element in terms of having the same geo-material type or lithology unit. The ellipse has a major axis a and a unit miner axis, indicating tangential correlation and normal correlation, respectively. The radius length () , referred to as -function, can be calculated using the following equation:
() a/ cos2 ( ) a2 sin 2 ( ) (3-10)
The rotation is closely related to the orientation information of geological formations, and also reflects the non-stationary property of the local correlation. The parameter a indicates the ratio of strength of tangential correlation and normal correlation, which represents the degree of local anisotropy. Thus, it is possible to represent different types of spatial correlations in a geological model by choosing different combinations of two parameters and a in -function. It is worth noting that the focused spatial correlation here is different from the commonly used definition of correlation length. The former reflects the correlation strength from neighbors in terms of having the same geo-material type (discrete states) based on Markov random field, while the latter indicates the scale of fluctuation of soil properties (continuous states) in a
Gaussian random field.
37 (a) (b) 3
2 Element j ( ) j 1 1 a 0 0 Element i Element i -1
-2
-3 0 -3 -2 -1 0 1 2 3
Figure 3.2 Standard Geometric condition and corresponding spatial correlation model
If it is a three-dimensional (3D) geological body, in the proposed model, the 3D
geological body is discretized into a finite set of elements, where each element is
represented by a uniform cubic lattice. A given element has 26 neighbor elements, and all
27 elements form a local neighborhood system, as illustrated in Figure 3.3a. The spatial
correlation between an unlabeled element i and its neighbor element j is designed to be
related to two features: 1) the neighbor element’s lithological label; and 2) the relative
direction of element j to element , represented by polar angles and from the
centroid of element to the centroid of element in the global spherical coordinate
system O-XYZ, as shown in Figure 3.3b.
38 (a) (b)
Figure 3.3 Neighborhood system (a) neighborhood system of uniform cubic mesh; (b)
geometric condition of neighboring element pairs
First, the interaction of a pair of neighboring elements i, j is considered in the local neighborhood system of element i . The spatial correlation in the three-dimensional geological model is decomposed into three components along the directions of the three axes of a local orthogonal Cartesian coordinate system O-xyz, in which the centroid of element i is the origin. To simulate the different local extension directions of the strata structure, the axis in a local Cartesian coordinate system O-xyz has polar angles and
in the global spherical coordinate frame O-XYZ, as shown in Figure 3.4. Under this condition, the radius length i, j of an ellipsoid centered at the centroid of element i and aligned with the axes of O-xyz is adopted to represent the influence from element j on the changing of the label of element i . The length of the axes of the ellipsoid along the x, y and z directions are defined as , and , respectively. Herein, is set to 1.0, and and are usually set to a higher value than to represent major extension
39 directions of strata in the O-XY plane. Hence, the i, j can be calculated using the following equation:
2 2 2 2 2 2 2 2 i, j rx ry / rx sin sin ry sin cos rx ry cos (3-11) in which , .
Therefore, the potential function for the neighboring element pair i, j is defined as:
(i, j) , if j Ni and ri rj Vc (ri ,rj ) (3-12) 0, else
Figure 3.4 Ellipsoid model for spatial correlation
40 3.4 Stratigraphic Uncertainty Quantification
Given a series of stratigraphic realizations for describing the real uncertain stratigraphic structure, the possible lithological labels for each element and their
corresponding probabilities of being assigned, denoted as Pl (i) ( ), can be calculated.
According to these probabilities, the subsurface domain of interest can be divided into two parts: the confident part assigned with lithological labels which possess probabilities larger than certain confident level, and the uncertain part assigned with multiple possible lithological labels.
For the confident part, the concerned issues are its proportion, location and corresponding lithological labels. To address these concerns, the lithological label of element is said to be confident as , if the corresponding probability is larger than a predefined confidence level. Under this condition, the proportion of the elements with confident lithological labels to all the elements in the entire modeled domain can be computed as an indicator, named as confident ratio in this chapter. Then, the locations of elements with confident lithological assignments are represented by a map of corresponding lithological assignments.
For the uncertain part, there needs to be a quantitative measure to quantify the degree of uncertainty, i.e., the fuzziness of the uncertain stratigraphic structure. For this purpose, we employ the concept of information entropy and compute its value for each element in the modeled geological domain according to the following form:
H(i) Pl (i)log Pl (i) (3-13) lL
41 It is noted that the information entropy of one element is dependent on both the number of possible lithological labels and their corresponding probabilities, a detailed introduction of applying information entropy for measuring geological uncertainty can be found in Wellmann and Regenauer-Lieb (2012). The map of information entropy can provide us a clear view of the uncertainties associated with each element in the MRF.
As an extension of the information entropy, total information entropy for the entire geological body can be calculated as:
1 TH H(i) (3-14) S iS where S denotes the cardinality of the set S . The total information entropy is used to quantify the fuzziness of the stratigraphic structure in the entire modeled geological domain with a single number.
Under the condition that all the model parameters are fixed, both the confident ratio and the fuzziness of a modeled stratigraphic structure should be constants which are exclusively determined by the amount of observation information. Therefore, the fluctuation of confident ratio values and total information entropy values can be examined to demonstrate that whether the adopted number of realizations are sufficient for providing a well description of the real stratigraphic structure, conditional on the given observation data.
3.5 Process for Generating Stratigraphic Realization
Given an initial estimate of real stratigraphic structure, the stratigraphic configuration possessing the minimum posterior energy is referred as the maximum a
42 posteriori (MAP) estimate. In the proposed modeling framework, a series of initial estimates are generated and their corresponding MAP estimates are regarded as possible stratigraphic realizations of the true stratigraphic structure. The MAP estimates are obtained through MCMC sampling by means of the Gibbs sampler with the simulated annealing scheme. Detailed introductions of the employed Gibbs sampler can be found in
Geman and Geman (1984) and Casella and George (1992). The process for generating stratigraphic realizations conditional on observation information is as follows:
Step 1: Discretize the geological domain of interest using proper lattices to form an
MRF.
Step 2: Assign the elements at the borehole locations and the surface are with lithological labels according to the stratification information from borehole logs and surface information. Calculate orientation parameters and for each element i according to available orientation information with the aids of the kriging method.
Step 3: Create a scanning order for the Gibbs sampler on basis of classification of the neighboring elements according to their distance to the borehole locations.
Step 4: Following the created scanning order, generate an initial stratigraphic configuration using Gibbs sampler according to the defined energy functions.
Step 5: Implement further MCMC sampling by means of Gibbs sampler to obtain the
MAP estimate of the generated initial configuration.
43 3.6 Effect of Model Parameters on Model Behavior
In this section, on the basis of 2D geological model, the detailed influence of model parameter a and on the probability of possible lithological units for the unknown element is presented. For simplicity, the parameter and are discussed separately in local neighborhood system with a specific assignment of two lithological units (see
Figure 3.5a) under three different temperature T=4 (initial temperature T=2 (processing temperature) and T=1 (final temperature).
(a)
44 (b)
Figure 3.5 Sensitivity analysis of model parameter (a) changing a in local neighborhood
system; (b) corresponding interested probability under different temperature
First, the parameter is set to be zero, and parameter is increased from 1.0 to 5.0 by 0.1 each time, which represents an increasing strength of correlation in the tangential direction. Along the changing of parameter a , the probability of assigning the unknown element with the blue lithological label, calculated based on Equations (3-7)-(3-10), is plotted in Figure 3.5b. Since the local neighborhood system of the unknown element has a centrosymmetric structure, the parameter a of 1.0, indicating equal normal and tangential correlation, the probability for two lithological units is both 0.5. Due to the centrosymmetric structure, the influence from the four corner neighbors is equal. Along the changing of parameter a , the probability of assigning the blue label increases under all three temperature conditions. The higher probability of blue label indicates that the larger parameter a can lead to stronger tangential correlation. It can be noticed that for three temperature conditions, the parameter a of 5.0 leads to a probability very close to
45 1.0, especially under low temperature T=1. Parameter a larger than 5.0 will lead to strong dominance of tangential correlation, hence the scope of parameter a is recommended as
1.0-5.0.
Then, the parameter a is fixed as 3.0, and the parameter is increased from 0° to
45° by 3° each time (see Figure 3.6a), indicating a rotation of the elliptical model.
Similarly, the probability of the blue lithological label for unknown element is shown in
Figure 3.6b. With a parameter equal to 0°, indicating a horizontal tangential direction, the blue unit has stronger effect and higher probability. While along the changing of parameter from 0° to 45°, the tangential direction (major axis) of ellipse rotates from the horizontal blue pairs to the red pairs in the 45° direction. As a result, the interested probability decreases smoothly, continuously and symmetrically from the higher probability to the lower one. Hence it is noted that the parameter is able to reflect the extension direction of strata through adjusting the orientation of the elliptical model.
(a)
46 (b)
Figure 3.6 Sensitivity analysis of polar angle (a) changing in local neighborhood
system; (b) corresponding interested probability under different temperature
3.7 Case Study
In this section, the proposed geological modeling and uncertainty quantification framework is applied to two real highway construction projects to demonstrate the favorable role of the developed methodology for site investigation process of geotechnical practice.
3.7.1 Geology and Observations of the Projects
1) Two-dimensional project:
The project is to replace an existing 4-span bridge over Interstate-90 in Sheffield,
Lorain County, Ohio. The construction area is located within the Erie Lake plain which is characterized by low relief terrain. Drilling report of 5 boreholes near the site
47 demonstrates the component of overburden soils as a mixture of lacustrine sand, silt, and clay, and the overburden soils are underlain by shale and sandstone. The locations of the available boreholes is marked in Figure 3.7a. Corresponding borehole logs can be found in Figure 3.7b. The stratigraphic profile connecting the available boreholes is converted to a two-dimensional domain of interest, as shown in Figure 3.7c.
(a)
(b)
48 (c)
Figure 3.7 Two-dimensional case study (a) available boreholes in the two-dimensional
project;(b) borehole logs of the two-dimensional project; (c) mesh plot for the two-
dimensional project
2) Three-dimensional project:
The project is to reconstruct Interstate-77/Interstate-76 interchange in Akron,
Summit County, Ohio. Soil samplers obtained from the drilling operation shows that
subsurface material consists of about 8.5 to 28 feet of overburden material over shale.
The overburden material is variable; it contains sand, clay, silt, gravel and mixtures of
them. A three-dimensional subsurface region of the project contains 9 boreholes are
studied using the proposed modeling methodology. The locations of the boreholes,
corresponding borehole logs and the created three-dimensional modeling domain can be
found in Figure 3.8.
49 (a)
(b)
(c)
Figure 3.8 Three-dimensional case study (a) available boreholes in the three -dimensional
project; (b) borehole logs of the three-dimensional project; (c) mesh plot for the three-
dimensional project
50 Since the depths of the investigated regions in both the two-dimensional project and the three-dimensional shallow are shallow and the relief are low or moderate, the orientation parameters for these two modeling examples are set to zeros to represent a horizontal sedimentary condition. Meantime, the correlation strength parameters are set to 3.0 for representing a moderate layered aggregation style under conditions of normal soil deposition.
3.7.2 Interpreting the Interface between Soil and Bed Rock
This section focuses on modeling and estimating the location of the interface between the soil and rock for the investigated construction regions, since such interpretation is crucial and essential for the safety and serviceability of the foundation system of the designed bridge and highway interchange. Under this condition, all the borehole samples for the two modeling examples can be divided into two lithological formations – overburden soil and bed rock, and the lithological unit on the ground surface is set to soil, since no outcrop is observed in both projects.
For the two-dimensional modeling example, different number of borehole logs are adopted as input data to form 3 modeling cases:
Case 1: Borehole #1 + Borehole #5
Case 2: Borehole #1 + Borehole #2 + Borehole #4 + Borehole #5
Case 3: Borehole #1 + Borehole #2 + Borehole #3 + Borehole #4 + Borehole #5.
Following the developed modeling and uncertainty quantification procedure, the confident ratio and the total information entropy values of these three modeling cases can
51 be calculated, as plotted in Figure 3.9a. It can be noted that with the increase of the number of the adopted borehole logs, the confident ratio values exhibit a significant upward trend, and the total information entropy values show a down trend. The confident ratio with confident level of 90% and total information entropy for Case 1 using two borehole logs as input data are 46.16% and 0.3423, while the corresponding values for
Case 3 using information of five boreholes are 89.19% and 0.0809, respectively. Such changes indicate that uncertainty of the location of the soil – rock interface in the modeled domain decreases with the increase of available borehole information.
(a)
52 (b)
Figure 3.9 Confident ratios and total information entropy values (a) different modeling cases for interpreting the soil-rock interface of the two-dimensional project; (b) different
modeling cases for interpreting the soil-rock interface of the three-dimensional project
Same inference can be drawn by comparing the confident lithological assignments and the information entropy maps of these three modeling cases shown in Figure 3.10. It can be noted from the result of Case 1 that the area of the uncertain region as well as the area associated with high information entropy values are the largest. This is because two borehole logs far away from each other cannot provide enough constraint to fix the location of the interface in the region between them. However, the uncertain region in
Case 3 is limited in a much smaller area. This result is easy to understand, since additional boreholes drilled in the highly uncertain regions can eliminate the uncertainty of soil – rock interface near the borehole locations, and the distance between boreholes
53 are closer so that stronger constraint can be exerted to prevent lithological labels between them varying greatly.
(a)
(b)
Figure 3.10 Interpreting the soil-rock interface for two-dimensional project (a)
information entropy map (b) map of confident assignments
54 In the modeling of the three-dimensional example, we also use different groups of borehole logs to establish 3 modeling cases:
Case 1: Borehole #6 – Borehole #9 (4 boreholes)
Case 2: Borehole #6 – Borehole #13 (8 boreholes)
Case 3: Borehole #6 – Borehole #14 (9 boreholes)
From the analysis result, the uncertainty of the location of the soil – rock interface also exhibits down trend with the increasing number of borehole information, manifesting as rise of the confident ratio from 42.16% to 69.84% and drop of the total information entropy from 0.3643 to 0.2004, as shown in Figure 3.9b. Meantime, the confident lithological assignments with confident level of 90% for each case, as illustrated in
Figure 3.11, also exhibits the same variation trend of the stratigraphic uncertainty.
(a) (b)
55 (c)
Figure 3.11 Map of confident assignments for interpreting the soil-rock interface of the
three-dimensional project (a) 4 boreholes; (b) 8 boreholes (c) 9 boreholes
The examples of estimating the location of interface between overburden soils and bed rock are intended to reproduce the geotechnical site investigation process by means of drilling boreholes. Given existing borehole logs, the proposed methodology is able to provide quantitative measurements of the stratigraphic uncertainty (uncertainty of the location of the soil – rock interface). Such quantitative measurements can be used as convincing indicators to evaluate the performance of the geotechnical site investigation, i.e., to decide whether the current obtained knowledge of geological condition is sufficient. Under the condition that additional boreholes are needed for obtaining more credible/confident estimate, the map of confident lithological assignments and the information entropy map can provide suggestion for the locations of the further borehole
56 drilling, since the borehole drilled at the location with thickest uncertain region or highest information entropy value is believed being able to reduce the stratigraphic uncertainty to the most extent. With the aids of the proposed uncertainty quantification procedure, the newly obtained borehole logs can be input for updating the modeling, and the circulation of estimation and suggestion can be continued until certain confident ratio or total information entropy value are reached. No doubt that it is hard to make perfect prediction of the borehole logs before the drilling samples are examined. However, the evolution of the confident ratio and the total information entropy values provide the general trends of the change of the quantified stratigraphic uncertainty, so that it is allowed to make estimations of the effect of further borehole drillings on enhancing the credible of the modeled subsurface stratigraphic structure to some extent.
3.7.3 Interpreting the Stratigraphic Structure
In this section, the soil samples obtained from boreholes are classified according to the AASHTO protocol, detailed classifications are shown in the borehole logs in Figure
3.7b and Figure 3.8b. Each of the classifications in the overburden soil is regarded as one lithological formation for modeling the whole stratigraphic structures.
The stratigraphic structure of the two dimensional project is modeled using all 5 available borehole logs. The map of confident assignments with confident level of 90% and the information entropy map are shown in Figure 3.12a and Figure 3.12b. The confident ratio and the total information entropy value for the modeled case are 72.23% and 0.2132, respectively. Similarly, all 9 available borehole logs are used to model the stratigraphic structure of the three-dimensional project, corresponding map of confident
57 assignments with confident level of 90% is illustrated in Figure 3.13. The confident ratio
and the total information entropy value for the three-dimensional case are 58.16% and
0.3005, respectively.
(a)
(b)
Figure 3.12 Interpreting the stratigraphic structure of the two-dimensional project (a)
Information entropy map (b) Map of confident assignments
58
Figure 3.13 Map of confident assignments for interpreting the stratigraphic structure of
the three-dimensional project
Comparing with the location of the soil-rock interface, given same observation information, the stratigraphic structure considering multiple soil types are much more uncertain, manifesting as lower confident ratio and higher total information entropy.
However, this does not affect the application of the proposed methodology. The proposed
59 stochastic geological modeling framework can still be adopt to generate reasonable stratigraphic realizations conditional on the available borehole logs. On the basis of these stratigraphic realizations, the developed uncertainty quantification procedure also can provide measurement of the stratigraphic uncertainty, as well as guidance for further borehole drillings.
3.8 Summary and Conclusions
The tradeoff between the heavy cost for geotechnical site investigation and the acquired geological knowledge for implementing design and construction of geotechnical projects is an unavoidable problem that almost every designer and constructor have to face. In the past, seeking balance between the costs and the benefits of geotechnical site investigation highly depends on local experience. To optimize the geotechnical site investigation process, we proposed a stratigraphic uncertainty quantification procedure on the basis of a novel stochastic geological modeling framework, which is capable to provide quantitative and objective measurement of the subsurface stratigraphic uncertainty.
The proposed stochastic geological modeling framework is based on MRF with specific spatial correlation in terms of potential function. The designed spatial correlation in terms of energy functions is intended to reflect the heterogeneous, anisotropy and non- stationary characteristics of the subsurface stratigraphic structure. Hence, as the modeling cases shown in Section 3.7, the proposed modeling framework is able to model subsurface stratigraphic structure under multiple common geological conditions
60 conditional on various types of observation data, and to present possible stratigraphic realizations in a probabilistic manner.
On the basis of the proposed stochastic modeling framework, we further develop an analysis procedure for quantifying stratigraphic uncertainty. Confident ratio and total information entropy are adopted as quantitative indicators to reflect the stratigraphic uncertainty of the modeled stratigraphic structure. Confident assignment map and information entropy is able to provide intuitive and visualized descriptions of regions with lithological assignment of certain confident level and stratigraphic uncertainty associated with those uncertain regions.
Studies of the real geotechnical projects demonstrate that the proposed methodology is capable to provide quantitative measurements of stratigraphic uncertainty for both two- dimensional and three-dimensional subsurface domains conditional on real borehole logs.
Moreover, based on the modeling results conditional on existing borehole logs, the proposed methodology can be implemented to estimate the benefits obtained from additional boreholes, and to optimize the borehole locations. Therefore, the proposed stratigraphic uncertainty quantification procedure plays favorable role for obtaining necessary knowledge of stratigraphic structure with the minimized cost of geotechnical site investigation.
61
CHAPTER IV
QUANTIFYING STRATIGRAPHIC UNCERTAINTIES BY STOCHASTIC
SIMULATION TECHNIQUES BASED ON MARKOV RANDOM FIELD
4.1 Introduction
In recent years, the uncertainty arising from characterizing inherently heterogeneous soil median has received increasing attention in geotechnical engineering. Soil heterogeneity can be attributed to two main sources (Elkateb et al., 2003). The first source of heterogeneity, which can be called the inherent spatial variation of soil properties, is that within a single formation layer, the soil properties are different from one point to another in space due to the difference of geological deposition history and human activities. The second source of heterogeneity is the stratigraphic or lithological uncertainty, which can be interpreted as the uncertainty of interfaces (boundaries) between different soil layers or lithological units due to limited subsurface investigation data. Substantial research work has been performed on the former type of soil heterogeneity within one nominally homogeneous layer by using either geo-statistics or random field theory. The common practice is to apply Gaussian random field equipped with specific correlation structure to simulate the spatial variability of soil properties with consideration of inherent spatial correlation (Fenton, 1999; Griffiths and Fenton, 2004;
62 Zhu and Zhang, 2013). For this modeling framework, recent studies have been reported to characterize the soil parameters by using site exploration data (Cao and Wang, 2012;
Wang and Cao, 2013; Gong et al., 2014). However, the treatment of this type of uncertainties resulting from lithological heterogeneity has been dealt with mainly by using engineering judgment based on local experience (Elkateb, Chalaturnyk and
Robertson, 2003). In contrast, the work presented in this chapter focuses on stochastic modeling techniques for quantifying uncertainties of geological structure due to limited site exploration data.
Modeling soil profiles for a project site is commonly done by interpolation using a set of observations from borehole logs spaced some distance apart coupled with local geological experience (Nobre and Sykes, 1992). For such purpose, several approaches based on geo-statistical methods or interpolation methods have been established
(Auerbach and Schaeben, 1990; Blanchin and Chilès, 1993; Chilès, Aug, Guillen and
Lees, 2004). However, most of these methods focus on estimate of the subsurface structure (geological model) with the Maximum Probability Estimate (MPE) conditional on local experiences, e.g. kriging, which is point estimation, and hence the lithological uncertainty cannot be quantified. Recently, probabilistic approaches have been developed to determine underground soil stratification based on cone penetration test (CPT) data
(Cao and Wang, 2012; Wang, Huang and Cao, 2013; Ching, Wang, Juang and Ku, 2015) and to identify soil strata in London Clay formation based on water content data (Wang,
Huang and Cao, 2014). Among these studies, the contact points of the subsurface stratigraphy are determined probabilistically. In the work presented herein, the spatial correlation of lithological units is defined by a graphical model (neighborhood system).
63 Multiple stochastic realizations of subsurface configuration are generated. The statistical analysis of the realizations is used subsequently for stratigraphic uncertainty quantification.
In the context of stochastic simulation of subsurface structure, the Markov chain modeling method (Elfeki and Dekking, 2001) and multiple-point geo-statistics (Caers and
Zhang, 2004) are capable of generating multiple realizations. However, these techniques may suffer certain limitations, such as stationary assumption, or that only a single type of observation data is used (i.e. borehole logs). To further relax these potential limitations, we have developed an innovative simulation method based on Markov random field. This method aims to reflect the inherent heterogeneous (multiple geo-material types), anisotropic (directionally dependent local correlation) and non-stationary (local correlation differs among different points in space) characteristics of geological body, as well as taking into consideration of the intrinsic local correlation of geological structure.
Three types of site investigation data could be used as input in this model, including ground surface soil types, boundaries of different soil layers at each borehole log location, and strata orientation information (e.g., from ground penetration radar test and/or seismic survey data). Two stochastic modeling techniques are developed to generate the corresponding subsurface lithological unit configurations with the purpose of accommodating different confidence levels on geological structure type (i.e., how much confidence do we have regarding prior information showing layered structure or not).
The concept of “information entropy” originally suggested by Wellmann and Regenauer-
Lieb (2012) for a quantitative measure of uncertainty in geological modeling, is adopted herein to quantify stratigraphic uncertainty in the post-processing stage.
64 This chapter is organized as follows. In Section 4.2, two modeling approaches, together with the corresponding numerical example results and model parameter sensitivity analysis are presented. To illustrate the performance of these two modeling approaches, the sensitivity analysis is carried out. In particular, the influence of discretizing mesh density and model parameter on the simulation results is systematically studied. In Section 4.3, the Bayesian inferential framework is introduced to allow for estimating model parameter for several scenarios when additional borehole data may subsequently become available for updating purpose. Finally, summary and conclusions of this chapter are presented in Section 4.4.
4.2 Two Modeling Approaches and Sensitivity Analysis
The mathematical theory and algorithm of the proposed geological model have been introduced in Section 3.2-3.5. In this section, the applicability of two modeling approaches of the proposed geological model and the behavior of the proposed model using different model parameters is discussed.
4.2.1 Two Modeling Approaches
In the present work, two modeling approaches are proposed regarding the uncertainty of field data. The first approach is based on the assumption that geological body has the layered structure. The confidence on the layered system is based on prior knowledge, such as orientation measurements (e.g., seismic surveys), interpretation of geological information and data, and engineering judgment. Under this condition, the
65 iterated conditional modes (ICM) algorithm (Besag, 1986) is adopted. In implementation of the ICM algorithm, the geo-material type with the maximum conditional probability is assigned to each element to generate a MPE as a most likely “guess” of initial configuration. With sufficient field investigation information, this modeling technique is considered as a better approach due to its ability using all available input data.
Nevertheless, there is uncertainty in all types of site investigation data that are used for modeling. If one concerns about the uncertainty in field investigation data, then
MCMC method is adopted to create an initial configuration. Compared to the former
ICM modeling technique, the latter MCMC technique allows introducing more randomness into the initial simulation process. Consequently, it is reasonable to anticipate greater degree of uncertainty in the simulated subsurface profile generated via
MCMC technique.
An example is presented to illustrate the applicability of these two modeling techniques. The physical domain is a 10x10 unit length area and is discretized into 2500 square elements, and the known information is assigned to corresponding elements (see
Figure 4.1). Three formations are assumed in this area. In () -function, the two parameters a =1.5 and =0° are taken as constant for this whole area.
66 Ground surface soil type Borehole 10
9
8
7
6
5
4
3
2
1
0 0 2 4 6 8 10 Formation 1 Formation 2 Formation 3 Figure 4.1 Known information (ground surface soil type, borehole soil types, and strata’s
orientation =0°)
The MAP estimation of subsurface configuration is analogous to consider that the geological system is in a relatively stable state with its lithological units and orientations.
The stochastic energy relaxation sampling scheme (also named as Gibbs sampler)
(Geman and Geman, 1984) is implemented in the simulation. The evolution of the total energy of configuration is adopted as the energy convergence indicator. Figure 4.2a shows the changes of the total energy for both modeling techniques for 100 iterations, which includes the initial configuration generation and the subsequent configuration updating process. As can be seen, the evolution of total energy for both modeling techniques exhibits similar trend. At the beginning, there is a fast decreasing rate of the total energy for both modeling techniques. After approximately 30 iterations, the decreasing rate slows down without significant changes and the total energy becomes
67 much more stable, decreasing may continue but the rate is quite low. Such small scale
fluctuation of the total energy is a result of label changing limited in the region near the
contact surfaces of strata. However, if the total energy keeps decreasing, the completely
stable configuration with the shortest contact surfaces (i.e., highly linear boundaries of
strata) will be generated, which may not be the appropriate simulation realization for real
stratigraphic profile. Therefore, a criterion through measuring the decreasing rate of the
total energy by examining the coefficient of variation (COV) of total energy values from
subsequent sampling is proposed. If the COV of total energy values from subsequent 10
times of sampling is less than 0.1%, the decreasing rate of total energy is regarded as
unobvious and the stochastic energy relaxation is regarded as completed to obtain a MAP
estimate (one chain). Based on the convergence criterion, the number of iterations needed
for all 1000 chains have been checked for two modeling approaches, which is shown in
Figure 4.2b.
4 (a) x 10 -1.09 ICM modeling approach MCMC modeling approach
-1.1
-1.11
-1.12
Total Energy -1.13
-1.14
-1.15 0 10 20 30 40 50 60 70 80 90 100 Number of iterations
68
(b)
Figure 4.2 Simulation for two modeling approaches (a) the evolution of total energy (b)
histogram of total number of iterations for all 1000 chains
Moreover, enough numbers of realizations are needed to represent all the possible
subsurface profiles constraint by investigation data. The total information entropy, which
has been used to measure the fuzziness of the geological body (Wellmann and
Regenauer-Lieb, 2012), can be considered as an indicator of convergence. According to
Equations (3-13)-(3-14), the stable total information entropy indicates the number of
uncertain elements is steady, as well as their probabilities for all possible lithological
units. Specifically, the variation total information entropy is evaluated. Once the COV of
subsequent 500 newly added realizations is less than 0.5%, the total number of
realizations is regarded large enough to represent all the possible stratigraphic profiles.
The changing of total information entropy and the measured COV are plotted in Figure
4.3a and Figure 4.3b along the increasing numbers of realizations. It can be noted that
after about 900 realizations, the convergence criterion has been met. In implementation,
we employ 1000 chains to quantify the uncertainty of the geological model.
69 (a) (b) 0.35 0.08 ICM approach ICM approach 0.3 MCMC approach MCMC approach
0.25 0.06
0.2 0.04 0.15
0.1
0.02 Total information entropy information Total
0.05 entropy information total of COV
0 0 0 200 400 600 800 1000 500 600 700 800 900 1000 Number of realizations Number of realizations
Figure 4.3 Total information entropy for two modeling approaches along the number of realizations (a) the changing of total information entropy; (b) the measured COV of total
information entropy
Figure 4.4a and 4.4b show one possible realization of subsurface lithological profile for each modeling technique, respectively. The ICM simulation yields three formation layers in Figure 4.4a, while the MCMC simulation affords high probability to generate the isolated soil zone (e.g. lens) in Figure 4.4b.
Borehole Borehole (a) (b) 10 10
9 9
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0 0 2 4 6 8 10 0 2 4 6 8 10 Formation 1 Formation 2 Formation 3 Formation 1 Formation 2 Formation 3 Formation 2 Formation 3
70 (c) (d) 10 10 1
9 9 0.9
8 8 0.8
7 7 0.7
6 6 0.6
5 5 0.5
4 4 0.4
3 3 0.3
2 2 0.2
1 1 0.1
0 0 0 0 2 4 6 8 10 0 2 4 6 8 10
Figure 4.4 Realizaion and information entropy plot for two modeling approaches (a) One possible realization for the ICM modeling approach; (b) One possible realization for the
MCMC modeling approach (c) Information entropy plot for the ICM modeling approach;
(d) Information entropy plot for the MCMC modeling approach
Figure 4.4c and 4.4d give the visualization of uncertainties using information entropy for each modeling technique, respectively. Note, the information entropy for elements is normalized in the range of 0-1 to be plotted by using continuous colors. As one can notice, the map of information entropy can provide us a clear view of the uncertainties associated with each element in the discretized geological model. We are sure about the geo-material types of the elements with dark blue color (interior region of each formation and regions around boreholes and ground surface), since given known information exerts strong constraint on the geo-material type of the nearby elements. For the ICM simulation technique, with the assumption that the lithological unit has layered structure, the uncertain area is just located at the contact surfaces of formations (Figure
4.4c), which can be referred as “uncertain band”. On the other hand, the MCMC
71 simulation technique (Figure 4.4d), generates the uncertain area, which is divergent from the formation interface at the borehole location, meaning that more uncertainties exist for these elements further away from the borehole location and more randomness allowed in the MCMC modeling technique.
Based on the above example, we learn that ICM simulation technique is applicable to a case where we are confident that the geo-strata are layered structure. On the other hand, the MCMC simulation technique is preferred for a situation where site exploration data is insufficient and more randomness involved in subsurface structure.
4.2.2 The Algorithm of Proposed Stochastic Geological Model
Step 0: 1) Pre-process the geometric information of the meshed plot (calculate ji and construct neighboring system), assign all known data to the corresponding elements;
and 2) choose the global a , and calculate i at the centroid of element i according to given orientation by using the kriging method;
Step 1: Generate a random scan order based on neighbor element classification;
Step 2: Create an initial configuration following the random scan order:
Evaluate the probability of different lithological units at element based on
Equations (3-7)-(3-10) (note that only neighbor elements which have already been assigned influence the geo-material type of element , the temperature T is computed according to Equation (3-6) as N=4).
Approach I-ICM technique:
Assign the lithological unit with the maximum P(R r | r ) to element ; i i Ni
72 Approach II-MCMC technique:
A random number u is generated from a uniform distribution with a range from zero
ri m1 ri m to one. If P(R r | r ) u P(R r | r ) , assign the geo-material type m i i Ni i i Ni ri 1 ri 1
( m L , L is the set of lithological units) to element i (Figure 4.5).
Step 3: Following the same scan order, update the configuration by using MCMC:
For element i at k th iteration, based on the assignment in the local neighborhood
system, generate the candidate lithological unit mk according to the sampling technique
P(R m | r ) described in Figure 4.5, calculate the ratio i k1 Ni . If 1, accept the P(R m | r ) i k Ni
candidate lithological unit (set Ri mk mk ). If 1 then with probability accept the candidate lithological unit, else reject it.
Repeat Step 3 until the total energy of the configuration (TU U i ) converges, iS
COV(TU k9,...,TU k ) 0.1%,
Save the k th iteration configuration as the MAP estimate of subsurface stratigraphy
(one realization);
Step 4: Repeat Step 2-3 to generate a series of realizations, and calculate the total
1 information entropy: TH Pl (i)log Pl (i) , S iS lL
Until the total information entropy of 500 newly generated realizations converges,
th COV(TH t499,...,TH t ) 0.5% (t is the t realization);
73 Step 5: Quantify the stratigraphic uncertainty based on all the generated realizations by computing the information entropy at each element: H(i) Pl (i)log Pl (i) and lL give the information entropy plot.
Figure 4.5 Diagram of sampling in MCMC technique
4.2.3 Sensitivity Analysis
The sensitivity analysis is conducted to exam the performance of the proposed model under different parameter configurations. In this section, we concentrate our attention to the influence of the mesh density and model parameter a . In the following examples, the geological condition is simplified into two formations for the purpose of further comparison and discussion, so that the uncertain zone of possible boundary
74 between two formations can be well quantified. The interested physical domain is a
10x10 unit length area.
1) Effect of mesh density
Three mesh densities with different element sizes are adopted in the two modeling approaches, i.e., the element length is 0.0125, 0.025, and 0.05. Thus, the simulated physical domain is discretized into 6400, 1600, and 400 elements, respectively. Both a
( =3) and ( =0°) are taken as constant in this entire area.
The visualization of uncertainties using information entropy for three mesh densities for the ICM modeling approach is shown in Figure 4.6a, 4.6c, and 4.6e, respectively. As can be seen, the uncertain area of the lithological unit boundary depends on the mesh density significantly. The larger the element size is, the wider the “uncertain band” is.
Figure 4.6b, 4.6d, and 4.6f show the information entropy plots for the MCMC modeling approach for different mesh densities, respectively. Different from the behavior observed from the ICM modeling approach, the “divergent zone” is shown not very sensitive to the mesh density. The angles of divergence of uncertain area for three mesh densities are almost the same.
75 (a) 10 (b)1 10 9 0.9 9 8 0.8 8
7 0.77
6 0.66
5 0.55
4 0.44
3 0.33
2 0.22
1 0.11
0 0 0 0 2 4 6 8 10 0 2 4 6 8 10
(c) (d) 10 10
9 9
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0 0 2 4 6 8 10 0 2 4 6 8 10
(e) (f) 10 10 1
9 9 0.9
8 8 0.8
7 7 0.7
6 6 0.6
5 5 0.5
4 4 0.4
3 3 0.3
2 2 0.2
1 1 0.1
0 0 0 0 2 4 6 8 10 0 2 4 6 8 10
Figure 4.6 Information entropy plots with different mesh density: from top to bottom 400
elements, 1600 elements, and 6400 elements, respectively, with the left column for the
ICM approach and the right column for the MCMC approach
76 Difference in sensitivity of mesh density between the two approaches can be due to their difference in the inferring process of generating initial configuration. The ICM modeling technique assigns the geo-material type with maximum probability for each element. The probability of each geo-material at a specific element depends on the local neighborhood system and local spatial correlation. Since =0°, the known lithological formation location will be horizontally extended with the uncertain area like a band between two lithological formations. On the other hand, the MCMC modeling technique assigns the geo-material type for a specific element randomly conditional on local neighborhood systems. Thus, the uncertain area diverges from the lithological formation interface at the borehole location.
2) Effect of model parameter a
To exam the sensitivity of model performance due to model parameter a , a single mesh density of 1600 elements is adopted in simulations. Three values of model parameter are selected as follows: =1.5, =3, and =5, respectively. Again, =0° is used for the entire region of interest.
Figure 4.7 shows uncertainty plots for three values of model parameter for the
ICM approach and the MCMC approach. As can be seen in Figure 4.7a, 4.7c, and 4.7e, the “uncertain band” is very sensitive to parameter . When larger value of is used, the uncertain zone becomes narrower. On the other hand, as can be seen in Figure 4.7b, 4.7d, and 4.7f, the “divergent zone” is inversely related to the value of parameter a . It becomes narrower with larger value of the model parameter .
77 (a) 10 (b) 1 10
9 0.9 9
8 0.8 8
7 0.77
6 0.66
5 0.55
4 0.44
3 0.33
2 0.22
1 0.11
0 00 0 2 4 6 8 10 0 2 4 6 8 10
(c) (d) 10 10
9 9
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0 0 2 4 6 8 10 0 2 4 6 8 10
(e) (f) 10 10 1
9 9 0.9
8 8 0.8
7 7 0.7
6 6 0.6
5 5 0.5
4 4 0.4
3 3 0.3
2 2 0.2
1 1 0.1
0 0 0 0 2 4 6 8 10 0 2 4 6 8 10
Figure 4.7 Information entropy plots with different parameter a : from top to bottom
=1.5, =3, and =5, respectively, with the left column for ICM approach and the right
column for MCMC approach
78 Using total entropy as a single number to indicate uncertainty of the entire region of interest, we plot the relationship between total entropy and parameter a in Figure 4.8 for the ICM and MCMC approach. As can be seen, there are higher total entropy for the
MCMC approach due to more randomness involved. For both simulation approaches, the total entropy is decreased as parameter a is increased.
The trend observed in Figure 4.8 can be explained by the physical meaning of parameter , which is used to reflect the degree of anisotropy. Since the normal correlation is a unit, the value of parameter reflects the ratio of tangential correlation to normal correlation. The larger means greater tangential correlation than normal correlation. Based on the simulation experience accumulated until the current research stage, cannot be too large, which can lead to dominance of tangential correlation. The local neighborhood system shown in this chapter has 8 neighbors for each interior element. Therefore, the value of parameter should not be greater than 5 to avoid strong dominance of tangential correlation.
0.2 ICM modeling approach Case 1: a=1.5 MCMC modeling approach 0.15 Case 2: a=3 Case 3: a=5
0.1 Case 1: a=1.5 Case 2: a=3
0.05 Case 3: a=5 Total entropy
0 1 2 3 4 5 a
Figure 4.8 Total entropy for two modeling approaches
79 4.3 Parameter Estimation
To apply the developed model in practice, the parameter a needs to be determined in advance. In this section, for model simplicity and computational feasibility, is taken to be constant for the geological model. However, it is possible to consider parameter as a field, which may vary from one element to another. This is beyond the current scope due to heavy computational efforts required for high dimensional inverse problem solving.
4.3.1 Bayesian Inferential Framework
To estimate the posterior distribution of a , the Bayesian inferential framework can be employed, which is expressed as:
P(x | a) (a) P(a | x) P(x | a) (a) (4-1) P(x) where P(a | x) is the posterior distribution, (a) is the prior distribution, P(x | a) is the likelihood distribution, and P(x)is termed as the marginal likelihood or "model evidence".
Regarding the prior distribution , if no prior information is available, a non- informative prior can be adopted. One possible way to define a non-informative prior is assuming that (a) is a Gaussian distribution with mean of 0 and a relatively large standard deviation (say, 100) to assign almost equal probability to all possible values.
Given any value of parameter a, which is input to the geological model, a series of MAP estimates of subsurface lithological unit configuration can be generated. Thus, the
80 likelihood of this geological model for the specific value of parameter a can be obtained by incorporating additional borehole information.
To obtain samples of P(a | x), the adaptive Metropolis (AM) algorithm is adopted.
The AM algorithm is based on the classical Metropolis-Hasting (MH) algorithm
(Metropolis et al., 1953; Hastings, 1970), which has been successfully used for probabilistic characterization of soil property from limited site investigation data (Wang and Cao, 2013). More details and engineering example of the MH algorithm can be found in Wang and Cao (2013). The basic idea of the AM algorithm is that it uses all the information cumulated from the beginning of the simulation to tune the Gaussian proposal distribution suitably. Otherwise the definition of the AM algorithm is identical to the usual Metropolis process (Haario et al., 2001). To be more specific, at time t, the
sampled consecutive states are a0 ,a1,...,at , where a0 is generated from an arbitrary prior
(initial value). Then a candidate point a is sampled at time t+1 from the Gaussian proposal distribution centered on the current state, and the covariance matrix is calculated from the whole historical samples ( ). The candidate is accepted with acceptance probability:
P(x | a ) P(a ) (4-2) A(at ,a ) min1, P(x | at ) P(at )
We will accept the candidate point with probability A , else reject it and set at1 at .
To begin with, we get 100 samples of a , the accept rate (i.e., percentage of acceptance) is calculated. The commonly used accept rate range of 0.40-0.60 (Wang et al., 2015) is adopted here, which could guarantee the chain mix well. If the accept rate is not within the range 0.40-0.60, the last sample (the current state) is used to lead another 100 samples
81 of a and the acceptance rate is re-evaluated until it is within the range 0.40-0.60.
Following a sufficient burn-in period, the chain approaches its stationary stage and generates samples from P(a | x). For each chain, we generate 1000 samples of after the burn-in period. However, due to the local correlation within a Markov chain, we take every 5th sample to reduce the auto-correlation effect (Gilks, 2005; Wang, Yajima and
Castaneda, 2015) and get 200 approximately independent samples, which we believe could properly reflect the distribution .
4.3.2 Illustrative Example
Example-I: ICM approach
The interested physical region in this example is discretized into 1653 rectangular elements. The geometry (a 20x10 unit length area) and mesh are shown in the Figure 4.9
Three types of lithological units (formations) are assigned in this example. The available data consists of the ground surface soil type information, four boreholes, and 15 orientation vectors (see Figure 4.9). Table 4.1 lists the locations and the magnitude of 15 orientation vectors. In Table 4.1, “0°” indicates the strata orientation is parallel to X axis of Cartesian coordinate system; “+” indicates the strata orientation is counterclockwise from X axis; “-” indicates the strata orientation is clockwise from X axis. In this case, it is assumed that the formation is layered-like, and then the ICM approach seems applicable.
Based on the orientation vectors shown, it is clear that the parameter is not a constant.
To obtain at each element centroid, we employ Gaussian correlation structure in the ordinary kriging interpolation process(Cressie, 1990).
82 Borehole #1 Borehole #2 Borehole #3 Borehole #4 10
9
8
7
6
5
4
3
2
1
0 0 2 4 6 8 10 12 14 16 18 20 Formation 1 Formation 2 Formation 3 Orientation
Figure 4.9 Illustrative example-I showing known information (ground surface soil
type, borehole soil types, and strata’s orientation)
Table 4.1 The locations and magnitude of known orientation vectors
Orientation Orientation # X Y angle(°) # X Y angle(°) 1 3.0 8.0 -37.5 9 13.0 4.0 25.0 2 1.5 3.0 -37.5 10 13.0 9.0 16.7 3 4.0 6.5 -30.0 11 15.0 6.0 30.0 4 7.0 4.0 -8.3 12 14.5 3.0 25.0 5 5.0 5.0 -37.5 13 19.0 8.0 37.5 6 8.0 8.0 -8.3 14 18.0 5.0 37.5 7 9. 3.0 0.0 15 18.0 2.0 37.5 8 10.0 4.0 0.0
Based on the proposed geological model and Bayesian inferential framework with an arbitrary prior value of parameter a , #2 borehole and other geological information are used to generate stratigraphic realizations. The convergence check described in Section 3
83 is used to determine how many subsurface realizations need to be generated (say, 1000).
When additional borehole data is available, the likelihood P(x | a) is calculated based on simulated realizations for each candidate a . In this study, the joint frequency of the depth of contact surfaces at the additional borehole location is regarded as the likelihood for specific . For example, #3 borehole is used as the additional data with the depth of
contact surfaces y1 2.7 and y 2 7 (y1 is the contact surface between formation 1 and
formation 2, y 2 is the contact surface between formation 2 and formation 3). Assume
=3.0, the frequency diagram of a pair of depth ( , ) is shown in Figure 4.10, thus the likelihood can be obtained.
Figure 4.10 Joint frequency diagram of the pair of depth ( y1 , y 2 )
84 Three cases of simulations are designed. In the first one, only #4 borehole data is employed to estimate parameter a . In the second case, two boreholes data (#4+#1) is used to calibrate parameter . Finally, in the third case, all three additional boreholes data
(#4+#1+#3) is used to determine parameter . Figure 4.11 shows the simulation results of P(a | x) for three cases.
1) Case 1: Borehole #2 + ground surface soil type + orientation vectors + Borehole #4
(additional data)
Figure 4.11a shows 200 samples of after burn-in period and the histogram of the results. The mean and variance of samples of are 3.1481 and 0.0976, respectively.
2) Case 2: Borehole #2 + ground surface soil type + orientation vectors + Borehole #4 and
#1(additional data)
The simulation results for when Borehole #4 and #1 are incorporated are plotted in Figure 4.11b. The mean of samples is 3.1147 and the variance of samples has been reduced to 0.0537.
3) Case 3: Borehole #2 + ground surface soil type + orientation vectors + Borehole #4 and
#1and #3 (additional data)
The simulation results incorporating these three additional boreholes as additional data are illustrated in Figure 4.11c. As can be seen, the chain (Figure 4.11c left) converges well and can be called well mixing. We can observe from the histogram
(Figure 4.11c right) that most samples are around 3.1. The mean of samples is 3.1459 and the variance is reduced to 0.0455.
85 (a) 5 1.5 Samples of a
4 1 a
3 Density 0.5
2 0 0 20 40 60 80 100 120 140 160 180 200 2.5 3 3.5 4 (b) Number of samples 5 2 Samples of a 4 1.5
a 3 1 Density 2 0.5
1 0 0 20 40 60 80 100 120 140 160 180 200 2.6 2.8 3 3.2 3.4 3.6 3.8 (c) Number of samples
5 2 4 Samples of a 1.5
a 3 1
2 Density 0.5
1 0 0 20 40 60 80 100 120 140 160 180 200 2.6 2.8 3 3.2 3.4 3.6 3.8 Number of samples
Figure 4.11 Parameter estimation results for three cases using ICM approach (a) samples
and histogram of a for Case 1; (b) samples and histogram of a for Case 2; (c) samples
and histogram of for Case 3
According to the simulated results, the estimated mean value of does not change too much for the three cases. However, the standard deviation of samples has been reduced when additional information is considered. Using the estimated mean ( =3.1459) of samples from Case 3, a MAP configuration based on Borehole#2 and other available information is shown in Figure 4.12a. The uncertainty plot using information entropy is illustrated in Figure 4.12b. Also, the comparison of the soil classification information between the original borehole and the estimated borehole from one realization, together with the uncertainty plot at the location of borehole, are shown in Figure 4.12c. As expected, the simulated formations at the locations of Borehole#1#3#4 (Figure 4.12c) correspond to the original formations well.
86 (a) Borehole #2 10
9
8
7
6
5
4
3
2
1
0 0 2 4 6 8 10 12 14 16 18 20
Formation 1 Formation 2 Formation 3 (b) 10 1
9 0.9
8 0.8
7 0.7
6 0.6
5 0.5
4 0.4
3 0.3
2 0.2
1 0.1
0 0 0 2 4 6 8 10 12 14 16 18 20 (c) 1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 Original Estimated Entropy Original Estimated Entropy Original Estimated Entropy Borehole #1 Borehole #3 Borehole #4
Figure 4.12 Estimation generated by Borehole #2+ground surface soil type +strata’s orientation when a =3.1459 (a) a possbile realization; (b) Information Entropy plot; (c)
87 comparison of soil classification information between original boreholes and estimated
boreholes from one realization, and information entropy plot at Borehole #1 #3 #4
Example-II: MCMC approach
Borehole #5 Borehole #6 Borehole #7 10
9
8
7
6
5
4
3
2
1
0 0 2 4 6 8 10 12 14 16 18 20 Formation 1 Formation 2 Formation 3 Figure 4.13 Illustrative example-II showing known information (ground surface soil type
and borehole soil types)
The same parameter estimation algorithm works for MCMC approach. For simplicity, only one case is designed in this example. Borehole#5+#7, and the ground surface soil type information are used to generate subsurface realizations. Borehole #6 is used as the additional data to estimate parameter a . The geometry of physical domain and known data are shown in the Figure 4.13. The orientation =0° is set in this example.
88 Figure 4.14a shows 200 samples of P(a | x) and the histogram of the results. The mean and variance of samples of P(a | x) are 2.4970 and 0.7141, respectively. Since more uncertainty involved in the MCMC simulation process, realizations can differ from each other. Hence, only the uncertainty plot using information entropy is illustrated in
Figure 4.14b using the estimated mean ( a =2.4970) of samples, based on Borehole#5+#7, and ground surface soil type.
(a) 5 0.6 samples of a 4 0.4
a 3
2 Density 0.2
1 0 0 20 40 60 80 100 120 140 160 180 200 2 3 4 5 Number of samples
(b) 10 1
9 0.9
8 0.8
7 0.7
6 0.6
5 0.5
4 0.4
3 0.3
2 0.2
1 0.1
0 0 0 2 4 6 8 10 12 14 16 18 20 Figure 4.14 Estimation using MCMC approach (a)Parameter estimation results;(b)
Information Entropy plot using Borehole#5+#7 when a=2.4970
89 4.4 Summary and Conclusions
In this chapter, two geological modeling approaches involving the ICM and MCMC techniques were developed for constructing stochastic heterogeneous geological model using three types of site investigation data, including ground surface soil/rock type, borehole soil/rock classification information and strata’ orientation information. Rather than giving a deterministic estimate of the formation boundary, the proposed framework is designed to reflect the inherent heterogeneous and anisotropic characteristics of geological structure by using Markov random field theory. In the context of stochastic simulation, a series of possible inferred results (MAPs) can be generated which enables the quantification of uncertainty of the subsurface formation configuration using information entropy. Through mesh density and model parameter sensitivity analysis, insights were provided on the performance of the proposed model. A method based
Bayesian inference framework was suggested for determining model parameter. Specific conclusions based on numerical examples can be summarized as follows:
The modeling approach using ICM technique to generate initial configuration is applicable to the layered lithological structure. The “uncertain band” is generally located at the possible lithological unit boundary. The modeling approach using MCMC technique can introduce more uncertainties into the initial configuration. The uncertain area is a “divergent zone” from the known formation boundary at borehole location. The farther the site is from the borehole location, the more uncertainty of interpretation of geo-material type is.
When the ICM simulation approach is used, the parameter a exerts strong influence on the extent of uncertain elements. Regarding mesh density effect, with the same , the
90 “uncertain band” changes with different mesh densities; the larger the element size is, the wider the “uncertain band” is. When the MCMC simulation approach is used, parameter a exerts influence on the degree of the divergent uncertain zone. The greater is, the smaller the degree of the divergent zone is. However, the “divergent zone” does not change with mesh density for the same . Both the “uncertain band” and “divergent zone” tend to become narrower and the total information entropy tends to decrease when model parameter is increased.
The proposed Bayesian inferential framework for model parameter estimation proves to be effective to generate samples of . The estimation error (sample variance) decreases as more additional boreholes are incorporated as model evidence.
91
CHAPTER V
PROBABILISTIC EVALUATION OF FOUNDATION PERFORMANCE
CONSIDERING STRATIGRAPHIC UNCERTAINTY AND SPATIALLY VARYING
SOIL PROPERTIES
5.1 Introduction
The purpose of geotechnical site investigation is to obtain information about the site subsurface soil stratification and the corresponding geo-material properties of each soil layer for engineering analysis and design of geostructures. However, the information obtained from a site investigation is limited due to constraints from geotechnical site exploration techniques and project budgets. Therefore, there is a need to develop techniques for modeling the site subsurface soil stratification and the corresponding soil/rock properties based on the limited site investigation data. Two main sources of uncertainty are involved in generating a subsurface soil profile for analysis and design: the uncertainty of the spatial distributions of soil/rock layers (stratigraphic or lithological uncertainties) and the spatial variability of the geomaterial properties for each soil layer.
Developing methods for quantifying these uncertainties is of great importance in the probabilistic analysis of geotechnical problems, as these uncertainties have a significant
92 influence on the computed response of geotechnical structures such as foundations, embankments, and underground excavations.
The spatial variability of geo-material properties for each soil layer has commonly been modeled through the use of random field theory with consideration of the spatial correlations of pertinent soil properties (Phoon and Kulhawy, 1999; Fenton and Griffiths,
2008; Zhu and Zhang, 2013). Recent studies have proposed methods for obtaining statistical descriptors (e.g. mean, variation and correlation length) of soil/rock properties by using site exploration data (Cao and Wang, 2012; Wang and Cao, 2013; Lloret-Cabot et al., 2014). Moreover, the influence of spatially varying properties on geotechnical engineering problems have been evaluated for shallow foundations (Fenton and Griffiths,
2002; Fenton and Griffiths, 2005), deep foundations (Fan and Liang, 2013; Fan and
Liang, 2013), and slope stability (Zhang et al., 2010; Li and Liang, 2014).
However, perhaps due to the complexity associated with the interpretation of stratigraphic profiles, little attention has been paid to incorporating the lithological uncertainties into a probabilistic analysis of geotechnical problems. Most existing subsurface modeling methods, such as geostatistical methods or interpolation methods
(Auerbach and Schaeben, 1990; Blanchin and Chilès, 1993; Chilès, Aug, Guillen and
Lees, 2004), focus on obtaining the optimal estimation of the contact boundary between lithological formations (i.e., the most probable lithological profile) given the limited amount of known information, and hence have difficulty in quantifying stratigraphic uncertainties. Several probabilistic approaches have been developed for soil stratification using single borehole or cone penetration test (CPT) data (Matsuo, Sugai and Yamada,
1994; Cao and Wang, 2012; Wang, Huang and Cao, 2013). Recently, Li et al. (2015)
93 presented a method for stochastic modeling of subsurface stratigraphic profiles using a limited number of sparsely distributed borehole log data. However, in their work, spatial variation of soil properties within each identified strata was not considered.
The main objective of this chapter is to present an integrated approach for considering both stratigraphic uncertainties and spatially varying soil properties of each identified lithological unit in the context of evaluating a two-dimensional (2D) strip footing settlement. The commonly used Gaussian random field (GRF) equipped with the exponential correlation function (Fenton, 1999; Fenton and Griffiths, 2008; Zhu and
Zhang, 2013) is adopted to describe random material properties within each lithological unit. A stochastic geological model (Li, Wang, Wang and Liang, 2015) based on a
Markov random field (MRF) is further improved and used for stochastic modeling of the subsurface stratigraphic profile. By integrating these two approaches, together with a finite element method, we have developed a novel probabilistic analysis procedure for geotechnical problems that considers both sources of uncertainty. To demonstrate the application of the proposed probabilistic analysis method, a 2D shallow foundation example is studied. The influence of both sources of uncertainty on the computed outcome of foundation performance is elucidated in this example.
5.2 Framework of the Proposed Method
The framework of the proposed method contains three main parts: stratigraphy simulation, modeling of the geomaterial properties, and subsequent probabilistic analysis of the foundation. A flow chart that depicts the computational procedure is shown in
94 Figure 5.1. The pre-process includes assigning all known information from the site investigation to the corresponding discretization elements, constructing the neighborhood system used for spatial correlation, and processing the geometric information of the mesh plot.
In the first part of the framework, the known subsurface soil stratification information is input into the stochastic geological model. The stratigraphy modeling process can be briefly described by the following procedure. Based on the MRF, a number of initial stratigraphic configurations are first generated by employing random scan orders and are correspondingly updated K times to obtain possible subsurface lithological profiles based on a Gibbs sampler (Geman and Geman, 1984). The coefficient of variation (COV) of total information entropy (Wellmann, Horowitz, Schill and Regenauer-Lieb, 2010) for all generated realizations is used as the convergence indicator. A more detailed description of the stochastic stratigraphic simulation technique is presented in Section 5.3.
The second part of the framework involves modeling the spatially varying soil properties for each identified lithological unit. A 2D GRF is utilized for generating relevant soil properties. In other words, conditional on each stratigraphic profile generated from the first part of the framework, elements are categorized into the corresponding lithological units, and then realizations of material properties for each lithological unit are generated by corresponding GRF.
95 Assign all known information to the corresponding discretization elements, construct the neighborhood system, and preprocess geometric information of the mesh plot
Stratigraphy simulation based on MRF Generate a random scan order based on element classification and create an initial stratigraphic configuration using known information
Update the initial stratigraphic configuration by using Gibbs sampler and repeat this step K times
Save the kth realization as a new stratigraphic configuration and calculate total information entropy with all previously generated configurations
Evaluate the COV of total No information entropy converged?
Yes Stratigraphic profile modeling is completed and save all t stratigraphic configurations
Material properties modeling based on GRF Categorize elements into the corresponding lithological units for each estimated stratigraphic configuration
Generate a number of realizations of relevant soil properties conditional on the inferred lithological configurations
Save all generated realizations
Probabilistic analysis Input all generated realizations into a FEA program by MCS
Evaluate the influences of both sources of uncertainties on the foundation performance
Figure 5.1 Framework of the proposed method
96 Finally, an FEA program is used to analyze foundation settlement for each combination of stratigraphic profile and soil properties by using Monte Carlo simulations.
The performance of the foundation can be assessed through interpretation of the FEA analysis output.
5.3 Stochastic Geological Model based on Markov Random Field
A stochastic geological model based on an MRF is adopted because it is capable of providing global simulations of subsurface stratigraphy and quantifying lithological uncertainty in cases where site exploration information is sparse. The comprehensive introduction of the stochastic geological model can be found in our previous work (Li,
Wang, Wang and Liang, 2015), where the mathematical details and the estimation of model parameters using a Bayesian inferential framework were presented. In this section, only the basic concepts are introduced to aid in understanding the improved model.
The three distinctive features of the geological model (Li, Wang, Wang and Liang,
2015) can be summarized as follows: (1) An MRF was introduced as the prior distribution to interpret the site exploration data; (2) the potential functions used to specify the MRF were carefully designed to reflect the spatial correlation of the geological structure by means of a local neighborhood system (see below); and (3) the simulation algorithm was appropriately developed to allow for quantification of stratigraphic uncertainty.
97 5.3.1 Markov Random Fields
Markov random fields have been widely applied to texture analysis, due to their ability to model contextual dependent patterns (e.g., correlated features) and analyze the spatial dependencies of physical phenomena (Zhang, Brady and Smith, 2001; Li, 2009).
Within the scope of geoscience, MRFs have been successfully applied to capture the intrinsic spatial correlation of geological structures (Daly, 2005) and to deal with geological mapping (Norberg, Rosén, Baran and Baran, 2002) and land-cover classification problems (Tolpekin and Stein, 2009). In this geological model (Li, Wang,
Wang and Liang, 2015), the subsurface stratigraphic profile can be considered as a stochastic physical structure with stable states (i.e. geomaterial types) in a discretized lattice, and an MRF is employed as the prior to describe the stratigraphic profile. A general introduction to MRFs can be found in Besag (1974), Li (2009), and Tjelmeland and Besag (1998).
In an MRF, the spatial dependencies or correlated features can be considered via a neighborhood system. For example, in a discretized geological region, let
S i | i 1,2,...,n be the set of elements in which i is an element index. A local
neighborhood system Ni for element consists of all elements that share common node(s) with element in the lattice plot, as illustrated in Figure 5.2. Hence, the neighborhood
system is defined as N {Ni | i S}.
98
Figure 5.2 Local neighborhood system
In addition to the capability of considering spatial correlation, the MRF–Gibbs equivalence, which was proven and stated as the Hammersley–Clifford theorem (Besag,
1974; Li, 2009), has rendered computational efficiency through the use of an energy function. In the MRF built on a geological region, we can assign each discretization element with a lithological label from a finite set L of material labels (representing all the lithological units in the domain) to form a possible subsurface stratigraphic
configuration r ri | ri L,i S. The MRF–Gibbs equivalence provides a feasible way to characterize the probability of the configuration r by a Gibbs distribution in terms of the energy fucntion (Geman and Geman, 1984):
P(r) Z 1 exp(U(r)/T) (5-1) where U (r) is termed the energy function of the configuration ; Z is a normalizing constant, called the partition function; T stands for “temperature”, which is decreased to
1.0 along the iteration times in the simulated annealing scheme (Geman and Geman,
99 1984). Hence, we can see that the probability P(r) is proportional to eU (r) ,
P(r) eU(r) .
Another advantage of adopting an MRF is that the posterior distribution of an MRF can be effectively obtained through Markov chain Monte Carlo (MCMC) sampling methods. In the geological model, the MRF is employed as the prior to describe the stratigraphic configuration r as P(r) . Given the true stratigraphic profile , the conditional probability that initial configuration r 0 is estimated as P(r0 | r) , and given the initial estimated configuration , the posterior probability for stratigraphic profile is P(r | r0 ). According to Bayes theorem, P(r | r0 ) P(r0 | r)P(r). Alternately, we can rewrite this expression by means of energy functions: U(r | r0 ) U(r0 | r) U(r), where
U(r | r0 ) , U(r0 | r) , and U (r) are the posterior energy, the likelihood energy, and the prior energy, respectively. By adopting a Gibbs sampler (Geman and Geman, 1984) in
MCMC, a maximum a posteriori (MAP) estimate can be obtained via the maximization of the posteriori distribution or the minimization of the posterior energy. The Gibbs sampler generates a configuration based on the conditional probability of a given MRF rather than by marginalizing its joint distribution . According to the local characteristics of an MRF, P(r | r ) P(r | r ) , where S i is the set of elements not i S{i} i Ni including i , the conditional probability is only related to the material type assignment in local neighborhood system. A detailed argorithm of the Gibbs sampler can be found in Li
(2009).
100 5.3.2 Prior Energy, Likelihood Energy and Posterior Energy
The MRF also presents an advantage in that it is convenient to define certain potential functions, which are used to calculate the energy for specifying the MRF. By means of the potential functions, the desired correlated features to describe geological structure can be accounted for in the MRF.
1) Prior energy
The prior energy is a sum of the clique potentials Vc (r) over all cliques C ,
C U(r) Vc (r) . A clique c denotes a pair of elements that are neighbors, and is the cC
set of clique c .The value of Vc (r) depends on the local configuration of clique , which reflects the local spatial correlation of the neighboring elements. The concept and utilization of clique potential can be found in Serpico and Moser (2006), Tolpekin and
Stein (2009) and Zhang, Brady and Smith (2001). In the geological model, the prior potential function is appropriately designed to incorporate the prior knowledge of the local spatial correlation into the MRF to describe the stratigraphic structure.
The strength of the local spatial correlation between a pair of neighboring elements
(ri ,rj ) is measured by an ellipse model, as illustrated in Figure 5.3. The ellipse’s center is at the centroid of element i . Two features of the ellipse model are associated with the main characteristics of the stratigraphic structures. First, the major axis of the ellipse is oriented in the local major extension direction of a geological stratum, referred to as plane orientation (e.g., sedimentary plane, foliation, and cleavage plane), and is
represented by a polar angle i in the global polar coordinate system – (Figure 5.3).
101 Since the major extension direction of the strata could differ from one location to another, the orientation of the ellipse’s major axis is non-stationary and can be interpolated conditional on known orientation information (e.g., from geophysical tests). Second, the ellipse has a major axis a and a unit minor axis, indicating tangential correlation and normal correlation in the local neighborhood system, respectively. The tangential direction (major axis) is oriented in the plane orientation of a stratum, while the normal direction (minor) is perpendicular to the tangential direction. Generally, the length of major axis a is usually set as larger than 1.0 to represent a stronger correlation along the tangential direction. Thus, the parameter a indicates the ratio of strength of the tangential correlation to that of the normal correlation, representing the degree of anisotropy of a stratigraphic structure. Since each interior element i has eight neighbors (Figure 5.2), we have found in a parametric study that the range of parameter a is between 1.0 and 5.0.
After the orientation and shape of the ellipse model is defined, the strength of the local spatial correlation between element i and its neighboring element j can be represented by the corresponding ellipse radius length q based on the relative positions in the mesh plot. The radius length is calculated using the following equation:
2 2 2 q( ji ) a / cos ( ji i ) a sin ( ji i ) (5-2)
where ji is the relative direction from the centroid of element to the centroid of element in the global polar coordinate system – , as illustrated in Figure 5.3.
102
Figure 5.3 Ellipse Model for Local Spatial Correlation
The larger q is, the stronger the influence (correlation strength) is from the neighbor element j on element i in terms of affecting the geomaterial type (lithological unit) assignment in the simulation. The local correlation strength is incorporated into the prior potential function as follows:
q( ji ) , if jNi and ri rj Vc (ri ,rj ) (5-3) 0, else
This is based on the notion that ri rj can contribute to the reduction of prior potential, since the two neighboring elements tend to possess the same geomaterial type.
103 2) Likelihood energy
The likelihood energy U(r0 | r) is a sum of the likelihood potential function
0 0 0 Vi (ri ,ri ) , i.e., U(r | r) Vi (ri ,ri ) . We propose the likelihood potential function for iS each element in the following form:
0 0 0 1 , if i S and ri ri Vi (ri , ri ) (ri , ri ) (5-4) 0, else
0 0 where (ri ,ri ) is the Kronecker delta function, ri and ri are the material label assignment for element i in the current stratigraphic profile r and the initially estimated
0 0 configuration r , respectively. If ri ri , the likelihood potential (or likelihood energy) will be reduced.
3) Posterior energy and posterior conditional probability
The posterior energy is the sum of the likelihood energy and the prior energy. The
Gibbs sampler (Geman and Geman, 1984) generates the next stratigraphic configuration based on the posterior conditional probability, which is given as:
P(r | r ) P(r | r ) Z 1 exp (V (r , r 0 ) V (r , r )) / T , r , r 0 L ; i S{i} i Ni i i i i c i j i i jNi
Z exp (V (r ,r 0 ) V (r ,r )) /T (5-5) i i i i c i j riL jNi where parameter denotes a smoothing factor that controls the contributions from the prior and the likelihood parts in the posterior energy. A more detailed discussion regarding the choice of parameter can be found in Tolpekin and Stein (2009).
104 5.3.3 Stratigraphic Uncertainty Quantification Using Information Entropy
The concept of information entropy, which has been applied in uncertainty quantification in the context of geology modeling by Wellmann, Horowitz, Schill and
Regenauer-Lieb (2010) and Wellmann and Regenauer-Lieb (2012), is adopted herein to provide a measure of the uncertainty of the stochastic geological simulations. The information entropy for element i is computed as:
H(i) Pl (i)log Pl (i) (5-6) lL
where Pl (i) is the probability of each material label for element i based on the simulation results. Thus, the entropy is 0 when no uncertainty exists (i.e., one lithological unit at element has a probability of 1), and the entropy is highest when all lithological units are equally probable for element .
As an extension of the information entropy, the total information entropy for the entire geological body can be calculated as:
1 TH H(i) (5-7) S iS where S denotes the cardinality of the set S .
The color representation of normalized information entropy superimposed on the mesh plot can give a direct visualization of the spatial distribution of uncertainties associated with each element in the geological model. The total information entropy, on the other hand, can provide a quantitative measure of the uncertainty of the stochastic model for the entire geological body. In fact, the total information entropy is used as a convergence criterion for determining the total number of simulations. For example, we
105 can choose the coefficient of variation (COV) of the total information entropy for the consecutive 200 realizations to be less than 1% as a criterion for the cutoff.
5.3.4 Simulation Procedure
In the proposed stochastic geological model, three types of site investigation data can be used as inputs: 1) ground surface and outcrop information, which provides soil and rock types at the ground surface; 2) borehole information, which indicates the soil formation types and the specific locations of formation boundaries (contact surfaces); and
3) orientation vectors, which gives the orientation information of strata at some locations
(e.g., from geophysical tests). It is noted that only the borehole information is required.
Any other information, if available, can be used as additional inputs to improve subsurface simulations.
To create an initial configuration using the abovementioned known data, the neighbor element classification, illustrated in Figure 5.4a, is established according to the physical distance of the unlabeled elements to the borehole locations. A scan order starts from the nearest element class to the element class that is farthest away. A number of initial configurations are generated by incorporating randomness into the scan order. The randomness is introduced in the scan order of elements within each class to improve the efficiency of scanning as well as to avoid unsymmetrical information intensity caused by using a predefined scan order (e.g., top-down). Figure 5.4b gives examples of three local initial configurations generated by using different random scan orders.
106
(a) (b)
Figure 5.4 Scan order and local initial configurations (a) the diagram of neighbor element
classification (different colors are used to distinguish different element classes); (b) local
initial configurations using three different random scan orders (different colors represent
different lithological units)
Subsequently, a number of MCMC iterations are performed to update each initial
configuration to obtain the corresponding MAP estimate (one chain). For each chain, the
initial stratigraphic profile derived from the input data and the subsequent updating
process follow the same scan order. Repeating this stochastic simulation process, the
generated MAP estimates can be considered as samples of subsurface stratigraphy, from
which the lithological uncertainty can be quantified. The required number of samples is
determined by the convergence criterion in terms of total information entropy described
previously in Section 5.3.3.
107 The step-by-step simulation procedure is described in the following:
Step 0: 1) Pre-process the geometric information of the meshed plot (calculate ji and construct the neighborhood system), assign all known data (i.e., lithological labels) to the corresponding discretization elements; and 2) choose the model parameter a , and
calculate i at the centroid of each element i according to the known strata orientation information by using the kriging method (Cressie, 1990).
Step 1: Generate a random scan order based on the established neighborhood system and the previously described element classification system.
Step 2: Derive an initial configuration from the input data following the random scan order.
Evaluate the probability of different lithological units for each element based on
Eqs. (2) through (5). Since this step is the process for generating the initial configuration,
0 the likelihood energy Vi (ri ,ri ) is set as 0.
A random number u is generated from a uniform distribution with a range from zero
ri m1 ri m to one. If P(R r | r ) u P(R r | r ) , assign the geo-material type m i i Ni i i Ni ri 1 ri 1
( m L , L is the set of all lithological units in the domain) to element i .
Step 3: Following the same scan order, update the entire initial configuration using a
Gibbs sampler.
For element i at the kth iteration, based on the assignment in the local neighborhood
system, update the lithological unit mk according to Eqs. (5.2) through (5.5).
108 Repeat Step 3 until the convergence condition is reached (i.e., when K=1000 iterations have been performed in this work). Save the Kth iteration configuration as the
MAP estimate of the subsurface stratification (one realization).
Step 4: Repeat Steps 1 to 3 to generate a series of realizations and calculate the total
1 information entropy TH Pl (i)log Pl (i) , until the COV of the total S iS lL information entropy of the 200 newly generated realizations converges, i.e.,
th COV(TH t199,...,TH t ) 1%(where t is the t realization);
Step 5: Quantify the stratigraphic uncertainty based on all generated realizations by computing the information entropy at each element H(i) Pl (i)log Pl (i) ; present the lL normalized information entropy in a color map.
5.4 Characterization of Material Properties based on Gaussian Random Field
Characterization of the material properties for a lithological unit has been studied extensively in the past. For example, Fenton and Griffiths (2008) demonstrated the use of a lognormal distribution for soil properties Y, such as strength parameters and elastic modulus. Thus, the natural logarithm of the material property ln(Y) is a Gaussian
random variable with mean lnY and standard deviation lnY .
By incorporating the correlation function into a 2D Gaussian random field, the spatial material variability of material properties for each lithological unit can then be described by the COV and the scale of fluctuation (correlation length). The scale of fluctuation is a measure of the distance within which soil properties in the physical
109 domain are significantly correlated. Zhu and Zhang (2013) have shown that the exponential correlation function, given in Eq. (8), can be used effectively to describe the spatial dependence of Gaussian-distributed variables ln(Y) between two physical points:
exp 2 2 2 (5-8) h h h v
where h and v denote the horizontal and the vertical scales of fluctuation (correlation
length) in a Gaussian random field, respectively, and where h and v are the centroidal separation distance along the horizontal and vertical directions, respectively. Other alternative correlation functions, such as Gaussian and spherical correlation functions, are also possible.
The random fields of material properties are constructed on the stratigraphic configurations generated from the stochastic geological model described in the previous section. In other words, for each stratigraphic profile generated by the stochastic geological model, the material properties of the discretization elements of the same lithological unit are characterized by the statistical descriptors (mean, COV, and correlation length) of the corresponding lithological unit. Due to the irregular geometry of soil layers, the matrix decomposition (MD) method (Myers, 1989) is adopted herein to generate random fields of material properties. In essence, the MD method begins with the construction of the covariance matrix by computing the correlation coefficient of two physical points (i.e., centroids of any two elements) spaced some distance apart in any direction in the physical domain by using Eq. (8). Next, the covariance matrix is decomposed into an upper and lower triangular matrix using the Cholesky decomposition.
Next, a random field which possesses the prescribed covariance function is produced by
110 multiplying uncorrelated Gaussian variables by the lower triangular matrix. Finally, the correlated Gaussian random field is transformed to the lognormally distributed random field by using the following equation:
Yi explnY lnY Zi (5-9)
where Yi and Z i denote the soil parameters and the standard correlated Gaussian variables at element i , respectively. Considering that the MD method is applied at the point level, the local averaging over the square element size is performed by multiplying a variance reduction factor to the variance of a normal variable. Similar techniques for such an application has been reported by Fenton and Griffiths (2008) and Luo et al.
(2012).
5.5 Numerical Example of a Shallow Foundation Settlement Evaluation
In this example, we assume that information from two borehole logs is available for the geological site. Two lithological units are identified from the borehole log (i.e., sand and weak shale). In addition, ground surface lithological unit information is available.
The locations of the boreholes, together with lithological information, are shown in
Figure 5.5. A rough rigid strip footing with a width (B) of 4 m is modeled, as shown in
Figure 5.5. The domain of the geological model is 24 12 m, which is discretized into
4,608 square elements. The relevant soil properties of the two lithological units are summarized in Table 1. In this example, for the sake of simplicity, we adopt the elastic constitutive model in FEA for computing footing settlement, although a more advanced elastic-plastic constitutive model can be used in the FEA computations as well.
111 Borehole #1 Footing Borehole #2 B=4 m 12
10
8
6
4
2
0 0 5 10 15 20 24 Formation 1 Formation 2
Figure 5.5 Known information (ground surface soil/rock type and borehole soil types)
and the strip footing
5.5.1 Stratigraphy Simulation and Uncertainty Quantification Using MRF
In the preprocessing stage (Step 0 in Section 5.3.4), the geometric information of the meshed domain is processed. In this case, the model parameter a is assumed to be a constant at 3.0 to represent moderate tangential correlation. Due to lacking of relevant
orientation data, the polar angle of the ellipse for each element is set to bei = 0°.
In the stochastic simulation process (Steps 1 to 3), the smoothing factor is set as
0.2, which allows the boundary of lithologic units in the MAP estimate to be adjusted within a distance of about 5 to 10 element lengths from the initially estimated boundary in the initial configuration. The initial temperature T is set at 4.0. The change in energy with 1,000 MCMC iterations, including the initial configuration generation and the
112 subsequent configuration updating process, is shown in Figure 5.6 (four chains are shown). As can be noticed from this figure, the posterior energy becomes stable after 100
MCMC iterations for each chain. Figure 5.7a shows one example of a realization of an initially estimated stratigraphic profile, and Figure 5.7b gives the corresponding MAP estimate after the stochastic energy relaxation. It can be seen that the boundary between the lithological units appears to be smoothened.
1500
1400
1300
1200
1100 Energy 1000
900
800
700 0 100 200 300 400 500 600 700 800 900 1000 Number of iterations
Figure 5.6 The evolution of total energy (4 realizations)
113 (a) 12
10
8
6
4
2
0 0 5 10 15 20 24
(b) 12
10
8
6
4
2
0 0 5 10 15 20 24
Figure 5.7 A possible realizaion (a) initial subsurface configuration; (b) the
corresponding stratification estimate after the stochastic energy relaxation
In the post-processing stage (Step 4), the total information entropy and corresponding COV versus the number of realizations are plotted in Figure 5.8a and 5.8b, respectively. The convergence criterion on the basis of the COV of total information
114 entropy is described in Section 5.3.3. It can be seen that after about 300 realizations, the convergence criterion has been met. In this example, we employ 500 chains to quantify the stratigraphic uncertainty.
(a) 0.08
0.06
0.04
0.02
Total information entropy information Total 0 0 100 200 300 400 500 Number of realizations
(b) 0.2
0.15
0.1
0.05
COV of total informationCOV total ofentropy 0 200 250 300 350 400 450 500 Number of realizations
Figure 5.8 Total information entropy (a) the changing of total information entropy along the number of realizations; (b) The measured COV of total information entropy along the
number of realizations
115 Based on the 500 simulated configurations (Step 5), a color map of information entropy for each discretization element can be generated (Figure 5.9a). Note that the information entropy for the elements in Figure 5.9a is normalized to the range of 0 to 1, so that the information entropy can be plotted by using different colors. Thus, the color of information entropy can provide a clear visualization of the uncertainty associated with each element in the discretized geological model. In this example, the dark blue elements indicate the region where the lithological unit assignment is quite certain. Similar information can be plotted in terms of probability for each lithologic unit at the element location. Figure 5.9b and 5.9c show the probability plots for lithologic unit 1 and lithologic unit 2 in the domain, respectively.
(a) 12 1
0.9 10 0.8
0.7 8 0.6
6 0.5
0.4 4 0.3
0.2 2 0.1
0 0 0 5 10 15 20 24
116 (b) 12
10
8
6
4
2
0 0 5 10 15 20 24 Formation 1
(c) 12 1
10 0.8
8 0.6
6
0.4 4
0.2 2
0 0 0 5 10 15 20 24 Formation 2
Figure 5.9 Information entropy plot and probability plot (a) Information entropy plot
based on 1000 MAP estimates; (b) Probability plot for lithologic unit 1 (c) Probability
plot for lithologic unit 2
5.5.2 Realizations for Geomaterial Properties Generated Using GRF
In the shallow foundation example, for the sake of simplicity, we adopt elastic theory in the FEA computation for foundation settlement. The elastic theory used in the
117 foundation settlement can also be found in Fenton and Griffiths (2002). Two properties of geomaterials are generally of interest: the elastic modulus E and the Poisson ratio . In our computation, the Poisson ratio is considered to have a relatively smaller spatial variability compared to the elastic modulus. Therefore, only the elastic modulus is considered as a spatially correlated random field in the settlement analysis. A constant
= 0.3 is used for the two lithological units in the example. The scale of fluctuation is site- dependent, which can be determined by collecting elastic modulus data from different locations at the site and estimating the correlation between points in space as a function of the separation distance (Fenton, 1999). In this example, we adopt 3.0 m and 1.0 m for the horizontal and the vertical scales of fluctuation, respectively, for both lithological units.
In the FEA computation of footing settlement for each generated stratigraphic profile, we generate 100 realizations of the elastic modulus fields in order to characterize the spatial variability. Figure 5.10 gives a sample representation of one realization of the elastic modulus fields for the subsurface lithological configuration shown in Figure 5.7b.
8 x 10 12 5
4.5 10 4
8 3.5 3
6 2.5
2 4 1.5
2 1
0.5
5 10 15 20 24
Figure 5.10 A realization of coupled random fields
118 5.5.3 Settlement Analysis of Shallow Foundation
The FEA model adopts the same mesh as the geological discretization mesh scheme.
It is treated as a plain strain problem with the bottom boundary fixed and the side boundary fixed in the horizontal direction. An FEM computation is performed using an incremental footing displacement. A uniform vertical displacement increment is applied at the nodes immediately under the location of the footing to simulate the rough and rigid footing without rotation.
The settlement under the footing is related to the equivalent effect of random elastic
modulus fields. The concept of the equivalent elastic modulus ( Eequ ) is used as given in
Equation (5-10):
pB S (5-10) Eequ where S is the footing settlement, p is the average loading pressure, and B is the width of the footing. For a given displacement on the footing in the FEA computation, the average loading pressure is computed as the summation of the vertical reaction forces at the corresponding nodes divided by the footing area.
In the following subsections, three cases are considered for comparing the difference in the settlement response of a strip footing: Case I considers the influence of both sources of uncertainties, Case II considers the influence of only stratigraphic uncertainty, and Case III considers the influence of only spatially varying material properties for a given stratigraphy.
119 1) Case I: Effect of both Stratigraphic Uncertainty and Material Spatial Variability
In Case I, the effect of both lithological uncertainty and material spatial variability on the footing settlement is analyzed. The generated subsurface lithological configurations integrated with the corresponding realizations of the elastic modulus fields
( realizations in total) are used in the FEA computations.
The probability density plot and the cumulative distribution plot of the equivalent elastic modulus for Case I are shown in Figure 5.11a and Figure 5.11b, respectively. The
mean and the COV of Eequ are 1.6367e8 Pa and 11.68%, respectively, with a minimum of
1.0046e8 Pa and a maximum of 2.3957e8 Pa. The 95% confidence limit of , which can be obtained through Figure 5.11b, is 1.2547e8 Pa. Assuming that the tolerable footing settlement is 0.05 m, the allowable bearing pressure for this case is 1.5680e6 Pa.
Two realizations of elastic modulus fields are presented in Figure 5.12a and Fig
5.13a for illustrating the combined effects of lithologic stratification and spatial variation of modulus on the computed vertical displacement fields shown in Figure 5.12b and
Figure 5.13b, respectively. Thus, this case illustrates the importance of considering the uncertainty from both the lithologic stratigraphy and the spatial variation of the material properties within each lithologic unit.
120 (a) -8 x 10
Mean: 1.6367e+08 2 Variance: 3.6560e+14
1.5
1 Density
0.5
0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 8 Equvalent elastic modulus (Pa) x 10
(b) 1
0.8
0.6
0.4
Cumulativeprobability 0.2
0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 8 Equvalent elastic modulus (Pa) x 10
Figure 5.11 Eequ results under the influence of both lithological uncertainty and spatially
varied soil properties: (a) PDF of ; (b) CDF of
121 (a) B=4 m 8 x 10 12 5.5
5 10 4.5
4 8 3.5
6 3
2.5
4 2
1.5 2 1
0.5 5 10 15 20 24 (b)
Figure 5.12 Simulation results (a) a realization of couple random fields; (b) the
corresponding settlement diagram of FEA model
122 (a) B=4 m 8 x 10 12 5
4.5 10 4
3.5 8
3
6 2.5
2 4 1.5
2 1
0.5
5 10 15 20 24 (b)
Figure 5.13 Compared simulation results (a) another realization of couple random fields;
(b) The corresponding settlement diagram of FEA model
2) Case II: Effect of Stratigraphic Uncertainty Only
In Case II, a total of 500 subsurface lithological configurations generated from the stochastic geological model are used as cases in the FEA computations. In this case, the elastic modulus of each lithological unit is considered as deterministic using the mean
123 value. Figure 5.14a shows the histogram of the equivalent elastic modulus resulting from the 500 computations with a constant elastic modulus for each formation. As can be seen
from this figure, there is a great variation in Eequ , even if we only consider the uncertainty of the boundary between the two lithological units. The mean and COV of
Eequ are 1.6753e8 Pa and 8.74%, respectively, with a minimum of 1.3886e8 Pa and a maximum of 2.1076e8 Pa. From Figure 5.14b, the 95% confidence value of is found to be 1.4657e8 Pa. If we use the same allowable vertical settlement of 0.05 m as in Case I, the allowable bearing pressure is 1.8321e6 Pa.
-8 (a) x 10
Mean: 1.6753e+08 3 Variance: 2.1455e+14
2.5
2
1.5 Density 1
0.5
0 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 8 Equvalent elastic modulus (Pa) x 10
124 (b) (b)1
0.8
0.6
0.4
0.2 Cumulative probability Cumulative
0 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 8 Equvalent elastic modulus (Pa) x 10
Figure 5.14 Eequ results under the influence of lithological uncertainty: (a) PDF of ;
(b) CDF of
3) Case III: Effect of Material Spatial Variability Only
In Case III, we use the commonly adopted method to draw a straight line between
the two borehole locations to generate one stratigraphy. Using the statistical values for
the elastic modulus listed in Table 5.1, we input 1000 realizations of random elastic
modulus fields into the FEA model to compute the footing settlement. Figure 5.15a
shows the histogram of for 1000 FEM simulations. The minimum and the
maximum are 1.0874e8 Pa and 1.7481e8 Pa, respectively; the mean and the COV of
Eequ are 1.4225e8 Pa and 7.65% , respectively. From Figure 5.15b, the 95% confidence
value of is found to be 1.2669e8 Pa. Based on the same criteria of allowable vertical
settlement of 0.05 m, we can obtain the allowable bearing pressure as 1.5836e6 Pa.
125 From the abovementioned computational results, which are summarized in Table 5.2,
several observations can be made. First, the mean values of Eequ for Case I and Case II are similar, while the mean value of for Case III is much smaller. Second, the COV of for Case I is largest, followed by that of Case II; Case III has the smallest COV of
. Third, the range of for Case I is greater than the ranges for the other two cases.
Fourth, by using the same service limit of 0.05 m, it is found that Case I has the minimum bearing pressure at the 95% confidence value of .
Table 5.1 Material properties for two lithologic units
Mean of COV of Horizontal Vertical
elastic modulus elastic modulus correlation length Correlation length
Sand 1.0e8 Pa 30% 3.0 m 1.0 m
Weak shale 3.0e8 Pa 20% 3.0 m 1.0 m
Table 5.2 The summarization of FEA computational results (MPa)
Case Equivalent elastic modulus 95% confidence Allowable bearing pressure
# Mean COV Min Max value of Eequ for 0.05 m settlement
I 163.67 11.68% 100.46 239.57 125.47 1.5680
II 167.53 8.74% 138.86 210.76 146.57 1.8321
III 142.25 7.65% 108.74 174.81 126.69 1.5836
126 -8 x 10
3.5 Mean: 1.4225e+08 Variance: 1.1829e+14 3
2.5
2
Density 1.5
1
0.5
0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Equvalent elastic modulus (Pa) 8 x 10
1
0.8
0.6
0.4
Cumulative probability Cumulative 0.2
0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 8 Equvalent elastic modulus (Pa) x 10
Figure 5.15 Eequ results under the influence of spatially varied soil properties: (a) PDF of
; (b) CDF of
127 5.6 Summary and Conclusions
In this chapter, an integrated approach was proposed for considering both stratigraphic uncertainties and spatially varied material properties in a probabilistic analysis of strip footing performance. The stochastic geological modeling method for simulation of the stratigraphic profile is based on an MRF with a novel neighborhood system that can reflect the inherent heterogeneous (multiple lithological units) and anisotropic (direction-dependent spatial correlation) characteristics of a geological body in generating lithological profiles. The stochastic geological model uses sparsely distributed borehole log data as inputs for subsequent simulations. The details of modeling techniques were provided in a step-by-step procedure. The spatial variability of the geomaterial properties for each lithological unit, on the other hand, was handled by a
2D GRF. An FEA analysis method employing a discretization scheme compatible with that used by the stochastic geological model is used for subsequent analysis of the performance of geotechnical structures, such as a strip footing.
The effects of simulated uncertain subsurface profiles along with a two-dimensional spatial random elastic modulus on the computed settlement of a strip footing were elucidated through the FEA/MCS framework. Compared with the deterministic analysis method, the probability distributions of equivalent elastic modulus were obtained for further analysis of a strip footing underlain by layers of different lithological units. Based on the criterion of the service limit state, an allowable bearing pressure with a certain level of confidence was obtained.
Three cases of settlement analysis of the strip footing were carried out for comparison of the effects of various sources of uncertainty. It was shown that the
128 uncertainty associated with the subsurface lithologic profile can exert a great influence on the computational results of the strip footing performance. When both types of uncertainty (stratigraphy uncertainty and uncertainty in material properties for each lithologic unit) are considered, a greater variation in the computed footing response and a smaller allowable bearing pressure will result. In engineering practice, it is important to take into account both sources of uncertainty.
129
CHAPTER VI
QUANTIFY GEOTECHNICAL UNCERTAINTIES FOR PERFORMANCE-BASED
RELIABILITY ANALYSIS OF DEEP FOUNDATIONS CONSIDERING MULTIPLE
FAILURE MODES
6.1 Introduction
Geotechnical engineers have always been challenged by various sources of uncertainties during their design of geotechnical structures. Starting with the geotechnical site exploration, engineers need to establish geological lithological profiles (soil stratification) for the project site using a limited number of sparsely distributed soil boring logs. The stratigraphic or lithological uncertainties will be introduced in the interpretation of the configuration of lithological profile. Next, pertinent soil properties and their spatial variations within each lithological unit need to be determined by statistical analysis of lab testing data of a small number of soil samples and/or by a few in-situ testing methods. Then, design methodology itself may introduce uncertainties due to lack of sufficient calibration with actual performance data. Assumptions in loading conditions and the selection of appropriate structure performance requirements also bring in another aspect of uncertainties.
130 In the past, engineering judgments, along with the best practice approach in selecting appropriate factor of safety, have been used to account for uncertainties mentioned. In recent years, Federal Highway Administration (FHWA) has incorporated
Load and Resistance Factor Design (LRFD) in the FHWA Design Specifications. This represents a small step toward a better handling of uncertainties associated with loads and resistances. However, there still remains a critical need for developing a comprehensive and unified approach to consider all uncertainties in a systematic manner.
Probabilistic models and/or statistical methods are a common means to quantify uncertainties in the reliability-based design methodology for geotechnical structures. In terms of the establishment of lithological profile, several approaches based on geo- statistical methods or interpolation methods (Auerbach and Schaeben, 1990; Blanchin and Chilès, 1993; Chilès, Aug, Guillen and Lees, 2004) could provide the optimal estimate of underground stratification. Other probabilistic approaches can interpret subsurface stratigraphy based on cone penetration test (CPT) data (Cao and Wang, 2012;
Wang, Huang and Cao, 2013; Ching, Wang, Juang and Ku, 2015). Moreover, some stochastic modeling techniques have been developed for subsurface stratigraphy simulation using sparsely distributed borehole data, such as Markov chain model (Matsuo,
Sugai and Yamada, 1994; Elfeki and Dekking, 2001) and stochastic geological model based on Markov random field (MRF) (Li, Wang, Wang and Liang, 2015). Among these modeling techniques, the stochastic geological model based on MRF (Li, Wang, Wang and Liang, 2015) could effectively quantify and visualize the lithological uncertainties.
Concerning the spatial variability and spatial correlations of soil properties, it is common practice to apply Gaussian random field to model these properties, such as strength
131 parameter and modulus (Fenton, 1999; Zhu and Zhang, 2013). The uncertainties from loading conditions, computational models, and performance criteria are usually tackled by assuming that each input follows their respective prescribed probability distribution
(Ellingwood, 1980; Phoon and Kulhawy, 1999; Zhang and Ng, 2005; Fan and Liang,
2013; Fan, Huang and Liang, 2014).
The objective of this chapter is to propose an innovative approach to quantify the aforementioned uncertainties in geotechnical design in a systematical manner. The adopted stochastic geological modeling method based on MRF (Li, Wang, Wang and
Liang, 2015), to simulate and quantify stratigraphic uncertainties has been elucidated in
Section 3.2-3.5. In section 6.3, the description of the characterization of various sources of uncertainties via different probabilistic models is presented. More specifically, for each stratigraphic configuration generated from the stochastic geological model, the commonly used two-dimensional (2-D) Gaussian random field (GRF) equipped with the exponential correlation function (Fenton, 1999; Zhu and Zhang, 2013) is adopted to describe uncertain material properties within each lithological unit. The design loads, computational model, and performance requirements are treated as random variables and modeled by their respective prescribed probability distributions. Section 6.4 presents the performance-based reliability analysis framework for deep foundations, including the description of reliability assessment and the computational procedure. The performance of a deep foundation (pile or drilled shaft) is defined in terms of the displacements induced by external loads. Under the combined axial and lateral loading, a pile typically has three distinctive displacements (i.e., axial movement w , lateral deflection and angular distortion ). Failure is defined as the event in which the displacements exceed
132 the corresponding allowable displacements. The system reliability is assessed by considering multiple failure modes. If the target system reliability is met, the design parameters of a pile (i.e., pile diameter D and pile length L) can be determined. Section
6.5 gives an illustrative example on the application of the proposed approach in the performance-based reliability analysis of a deep foundation, including the uncertainties quantification and the subsequent probabilistic analysis. Finally, the summary and conclusions are included in Section 6.6.
6.2 Stochastic Geological Model
The stochastic geological model was firstly introduced in our previous work (Li,
Wang, Wang and Liang, 2015), where the mathematical details, the well-designed spatial correlation functions in terms of local neighborhood system and the estimation of model parameters using Bayesian inferential framework are included.
Three types of site investigation data are used as input in the geological model, including information of soil/rock types at the ground surface, information of soil/rock types and the specific locations of soil/rock boundaries (contact surfaces) provided by borehole logs, and orientation information of geological strata at certain locations (e.g., from geophysical tests). There are two main advantages of the geological model in the interpretation of stratigraphic profile. First, the site-based geological characteristics (e.g. the local spatial correlation and the extending trend of strata) can be encoded into MRF prior via defining the potential function (see below). Secondly, a series of stratigraphic configuration samples, not just one, are generated to quantify lithological uncertainty.
The relevant content can also be found in Section 3.2-3.5.
133 6.3. Uncertainties Characterization
In this section, the aforementioned uncertainties are considered as random variables and are modeled through different probabilistic models.
6.3.1 Characterization of Stratigraphic Uncertainties
The stochastic geological model described previously is adopted herein because this model is able to simulate subsurface stratigraphy in case of limited site exploration information, and at the same time, it allows quantitative estimation of stratigraphic uncertainty. Also, this geological modeling method is able to reflect the inherent heterogeneous (multiple geo-material types), anisotropic (directionally dependent local spatial correlation) and non-stationary (local correlation differs among different points in physical domain) characteristics of geological structure.
6.3.2 Characterization of Material Properties
Characterization of material properties based on known borehole soil samples for each lithologic unit follows the work of Lloret-Cabot, Fenton and Hicks (2014). A lognormal distribution with its statistics (mean and COV), suggested by Fenton and
Griffiths (2008), are employed to characterize material properties Y , such as strength parameters. Thus the natural logarithm of material property ln(Y) is a Gaussian random
variable with mean lnY and standard deviation lnY . The relationship between soil property values is modeled by the correlation model, which is incorporated into a 2D
GRF. The exponential correlation model (Luo, Atamturktur, Cai and Juang, 2012; Zhu
134 and Zhang, 2013) can be effectively used to describe the spatial dependence of Gaussian- distributed variables ln(Y) , given as:
exp 2 2 2 (6-1) h h h v
where h and v denote the horizontal and the vertical scales of fluctuation (correlation
length) in GRF, respectively; h and v are the centroidal separation distances along the horizontal and vertical directions, respectively. Other alternative correlation models, such as Gaussian and Spherical correlation models, are also possible.
In simulation, for each stratigraphic configuration generated by the stochastic geological model described in the previous section, the soil properties of discretization elements of the same lithological unit is modeled by 2D GRF conditional on the measurement data of the corresponding borehole soil samples. Due to the irregular geometry of soil layers, the matrix decomposition (MD) method (Myers, 1989; Zhu and
Zhang, 2013) is adopted to generate random fields of material properties. The local averaging technique (Fenton and Griffiths, 2008; Luo, Atamturktur, Cai and Juang, 2012) is performed by multiplying a variance reduction factor to the variance of normal variable to eliminate the effect of element size. The procedure to generate conditional random field is performed in the following way:
Step 1: Generate a 2-D (unconditional) correlated Gaussian random field Z R using statistical descriptors (mean, COV and correlation length) of the corresponding
lithological unit by the MD method, and extract the values of Z R at the locations s m of
th known soil samples ( s m is the set of elements with known soil property for the m lithological unit).
135 Step 2: Generate an initial interpolated field Z0 by Kriging, using the known
measured soil property at the locations s m .
Step 3: Generate Z R by Kriging using the extracted values of Z R at the locations s m calculated in step 1.
Step 4: Calculate the conditional random field Z CR as:
ZCR (ZR Z R ) Z0 (6-2)
Step 5: Transform the generated conditional Gaussian random field to the lognormally distributed random field by the following equation:
Yi explnY lnY ZCRi (6-3)
where Yi and Z CRi denote the soil parameter and conditional Gaussian variable at element i , respectively.
6.3.3 Characterization of Model Errors
Since any computational model is an idealization or simplification of the real world, there always exist uncertainties (errors) in computational model. To take the model errors into account, a factor is introduced as follows:
Y e t (6-4) Yp where Yt is the true value, Yp is the predicted value based on the computational model, and e is a model error or bias factor to capture the model error. It is noted that can be treated as a random variable which may follow normal or lognormal distribution (Phoon and Kulhawy, 2005). The mean of is a measure of the model accuracy, and the
136 variance of e is a measure of the model uncertainty. In most cases, the mean of is assumed to be 1.0, implying there are no potential under- or over-estimates (Fan and
Liang, 2013; Fan and Liang, 2013).
6.3.4 Characterization of External Loads
The design external loads themselves are uncertain to some degree. Particularly, live loads are a much contributor to the uncertainty. Extensive research has been conducted to develop probabilistic models for dead and occupancy live loads (Ellingwood and Tekie,
1999; Bulleit, 2008), which can be statistically characterized by normal, lognormal, gamma, Type I, and Type 2 Gumbel distributions. According to Nowak and Collins
(2012), coefficients of variation for dead load could range from 0.08 to 0.10 for buildings and bridges, while those for live load could range from 0.18 to 0.89.
6.3.5 Characterization of Structure Performance Requirements
To meet the serviceability performace requirements, the movement induced by external loads should not be greater than the allowable displacement. The allowable displacements are traditionally treated as deterministic values (Roberts and Misra, 2010).
However, the specified allowable displacements are expected to contain uncertainty to some extent. Zhang and Ng (2005) collected and analyzed available performance data for building and bridge foundations. They studied the variability of allowable displacements of structures and fitted them to a lognormal distribution. Due to various designed functions of structures, the tolerable movement criteria should be established by structural analyses, by empirical procedures, or by taking both into consideration.
137 6.4 Framework of Performance-based Reliability Analysis
A performance-based reliability analysis framework is developed to assess the reliability of a deep foundation regarding various sources of uncertainties.
6.4.1 Reliability Assessment
In the probabilistic analysis approach, the reliability of a structure (or the safety of a system) is evaluated by the probability of failure. In this chapter, the performance criteria in terms of the drilled shaft head movements (i.e., axial movement w , lateral deflection
and angular distortion ) are adopted. The failure (the unsatisfactory performance) event is said to occur when the induced displacements at the top of drilled shaft exceed the corresponding allowable displacements. For the Kth serviceability failure mode (i.e.,
K = w , and ), the probability of failure Pf ,K is defined as:
Pf ,K P(yt,K ya,K ) (6-5)
in which yt,K is the load-induced displacement and ya,K is tolerable movement for the
Kth service limit state. Thus, the probability of system failure Pf ,S , defined as the event in which any individual failure mode occurs, can be expressed as:
Pf ,S P(yt,K ya,K ) (6-6) K
Given a target reliability index T , the target probability of failure PT can be calculated using the standard normal cumulative distribution function:
PT 1 (T ) (T ) (6-7)
138 If the probability of system failure Pf ,S is smaller than the target probability of failure PT , the corresponding reliability (or safety) of structure is met. Phoon et al. (2003) and Wang et al. (2011) tabulated the scale of the reliability index (β) corresponding to different performance levels.
Monte Carlo simulation (MCS) provides a statistically unbiased way for evaluating the probability of failure of the Kth limit state and/or the system. Various uncertainties are qualified by a sequence of randomly generated samples. Each set of samples is used as input in Monte Carlo approach, then the response of a complex system is calculated repeatedly until a desired or prescribed sample size is achieved. With the performance criteria defined, whether failure occurs or not can be determined accordingly.
6.4.2 Computational Procedure
In order to implement the performance-based reliability method, a computational procedure is developed, shown in Figure 6.1. The framework comprises two main parts: the assessment of the pile performance to external loading with considering various types of uncertainties and the evaluation of the pile reliability for each limit state and the system through the statistical analysis.
The total sample size for MCS is set as n. Based on Markov random field, n lithological configurations are first generated via the proposed stochastic geological model, and thus the stratigraphic uncertainties can be quantified. Then for the stratification of each lithological configuration, the soil properties of lithological units are modeled conditional on the known borehole soil samples through 2-D GRFs.
139 Furthermore, extract the soil stratification together with the corresponding soil properties at the location of the shaft center from the total n realizations. Each set of soil extraction (totally n sets of soil extraction) is used as input to the soil-pile interaction model. In this study, the commonly used load transfer methods (t-z model and p-y model) are applied to model the non-linear soil-pile interaction. The t-z model is used to model axial soil-pile interaction through t-z curves and q-w curves (t and q represent side shear on the shaft and the tip resistance at the toe, and z and w represent the settlement of the shaft and the toe), while the p-y model is used to model lateral soil-pile interaction through p-y curves (p represents the soil reaction and y represents the lateral deflection).
The vertical movement in response to axial loading, and the lateral deflection and angular distortion in response to lateral loading are solved iteratively using the t-z model and p-y model, respectively, through a finite difference method. In the finite difference method, the pile also needs to be discretized into a finite number of segments. Herein, the pile segment length is taken to be equal to the element length in the stochastic model. Hence, the discretized soil extraction can be easily integrated into the pile analysis. Other sources of uncertainties from the model errors of the load transfer methods, the external loads and the performance requirements are simulated directly according to their prescribed probability distributions.
At last, for each Monte Carlo simulation, the displacements ( w , and ) at the pile head are evaluated according to each set of random variables input. With the total n resulting realizations of displacements and the corresponding allowable movements, not only the probability of failure for each limit state, but also the probability of failure for the system can be evaluated.
140 Specify a geological site and process the site investigation data
Quantify uncertainties Stratigraphic uncertainties: stochastic geological model based on MRF Soil properties: mean, variance and correlation structure based on GRF Loads: probability distributions Performance criteria: probability distributions Model errors: probability distributions
Assume a combination of pile design parameters (D and L)
Assume a sample size n for Monte Carlo simulation and generate n sets of random samples according to prescribed probabilistic models
Evaluate the pile response using one set of random samples as input and determine whether failure occurs considering multiple failure modes
Adjust Repeat n times
No Complete n times of repetitive parametersdesign executions of the previous step?
Yes Perform statistical analysis of resulting n sets of output and
evaluate the probability of failure Pf (design parameters)
No P (design parameters) P ? f T
Yes Obtain a feasible design
Figure 6.1 Flow chart of proposed approach for performance-based reliability design
6.5 Illustrative Example
In this section, the probabilistic analysis of deep foundation is given as an example to illustrate the proposed approach for uncertainties quantification and the developed
141 performance-based reliability analysis framework. The physical domain of interest is
16 10 m and is discretized into square lattices with 4000 elements. It is assumed that two borehole logs are available and three lithologic units (medium clay, sand, and stiff clay) are identified from the borehole logs. The borehole samples are measured for each lithological unit. In addition, the lithologic unit information at the ground surface and the orientation of strata at certain locations are available. The known information is assigned to the corresponding elements, as shown in Figure 6.2. In Figure 6.2, sample A-B indicates the Bth borehole sample in Ath formation, for example, sample 2-6 is the 6th borehole sample in formation 2. The relevant soil properties of three lithological units are listed in Table 6.1 and the measured soil properties of borehole samples are listed in
Table 6.2, respectively. The orientation information of strata is illustrated in Table 6.3. In
Table 6.3, “0°” indicates the strata orientation is parallel to X axis of Cartesian coordinate system; “+” indicates the strata orientation is counterclockwise from X axis; “-” indicates the strata orientation is clockwise from X axis. The three locations in the domain where the drilled shaft might be placed are marked in Figure 6.2.
142 Borehole #1 Location 1 Location 2 Location 3 Borehole #2 10 Sample 1-2 9 Sample 1-1 8 Sample 1-3 Sample 2-1 7 Sample 1-4 Sample 2-2 6
5 Sample 2-3 Sample 2-6
Sample 2-7 4 Sample 2-4
3 Sample 2-5 2 Sample 3-2
1 Sample 3-1 Sample 3-3
0 0 2 4 6 8 10 12 14 16 Formation 1 Formation 3 Borehole sample Formation 2 Orientation
Figure 6.2 Know information (ground surface soil type, borehole soil types, borehole
samples, and strata’s orientation)
Table 6.1 Material properties for two lithologic units
Horizontal Vertical Mean COV correlation length (m)Correlation length (m) 1 Medium clay 100 kPa 25% 3.0 1.0 ( Su ) 2 Sand ( ) 30° 20% 3.0 1.0
3 Stiff clay ( Su ) 240 kPa 20% 3.0 1.0
Table 6.2 Soil properties of borehole samples
Formation #1 #2 #3 #4 #5 #6 #7 /Sample ID
1 Medium Clay ( S u 96 103 97 98 - - - 2 Sand/kPa) ( /°) 28 27 29 31 30 28 30
3Stiff Clay ( /kPa) 290 241 270 - - - -
143 Table 6.3 The locations and magnitude of known orientation data
Orientation Orientation # X Y angle(°) # X Y angle(°) 1 1.0 6.0 45.0 11 1.0 1.0 45.0 2 1.5 7.0 45.0 12 1.5 2.0 45.0 3 3.0 8.0 45.0 13 3.0 3.0 45.0 4 4.5 9.0 0.0 14 4.5 4.0 30.0 5 6.0 8.0 -45.0 15 6.0 3.0 17.0 6 7.5 7.5 -45.0 16 7.5 2.5 6.0 7 9.0 6.5 -30.0 17 9.0 1.5 0.0 8 10.0 5.5 -30.0 18 10.0 0.5 0.0 9 12.0 5.5 0.0 19 12.0 0.5 0.0 10 15.0 7.0 30.0 20 14.0 1.0 0.0 21 15.0 2.0 30.0
6.5.1 Uncertainties Quantification
1) Stratigraphy simulation and stratigraphic uncertainty quantification based on
MRF
In the stratigraphy simulation, the model parameter a is set as 3.0 to represent moderate tangential correlation in the main extension direction of strata. The smoothing factor is assumed to be a constant at 0.2 to allow reasonable adjustment for initially estimated boundaries of lithologic units. Following the simulation procedure in Section
6.2, two possible realizations of subsurface stratigraphy estimated by the stochastic geological model based on the limited known information are given in Figure 6.3. As can be seen, there is difference existing in the distribution of stratigraphic structure between the two realizations.
144 (a) 10
9
8
7
6
5
4
3
2
1
0 0 2 4 6 8 10 12 14 16
(b) 10
9
8
7
6
5
4
3
2
1
0 0 2 4 6 8 10 12 14 16
Figure 6.3 Two possible stratigraphic configurations
In this study, the total sample size for MCS is set as n=50000. Based on the 50000 simulated stratigraphic realizations, the color map of normalized information entropy of each discretization element is shown in Figure 6.4. The region with bright colored elements is the region where there are uncertainties in lithologic unit assignment. The uncertain region is located at the possible boundaries of lithologic units.
145 10 1
9 0.9
8 0.8
7 0.7
6 0.6
5 0.5
4 0.4
3 0.3
2 0.2
1 0.1
0 0 0 2 4 6 8 10 12 14 16
Figure 6.4 The color map of normalized information entropy
2) Realizations of soil properties generation based on GRF
In this example, the strength parameters of each lithologic unit, i.e., undrained shear
strength Su for clay and internal friction angle for sand, are modeled as random fields to characterize the soil spatial variability. We adopt 3.0 m and 1.0 m for the horizontal and the vertical scale of fluctuation for all three lithological units, respectively. For each estimated stratigraphic profile by the geological model, one realization of the random fields of soil properties is generated conditional on the corresponding borehole samples.
The realizations of the random fields of soil strength parameters for the subsurface lithological configurations shown in Figure 6.3 are illustrated in Figure 6.5a, 6.5c, and
6.5e are realizations for medium clay, sand, and stiff clay corresponding to Figure 6.5a, respectively; while Figure 6.5b, 6.5d, and 6.5f are realizations for medium clay, sand, and
146 stiff clay corresponding to Figure 6.4b, respectively. As can be noticed, the range and the distribution of soil properties are different for two realizations.
(a) 10 130 (b) 10 160 9 120 9 150 8 8 140 110 7 7 130 6 100 6 120 5 5 90 110 4 4 80 100 3 3 90 2 70 2 80 1 1 60
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Undrained shear strength (kPa) Undrained shear strength (kPa)
10 34 (c) 10 40 (d) 9 9 38 32 36 8 8 30 34 7 7 32 28 6 6 30 5 26 5 28 4 4 26 24 3 24 3 22 2 22 2 20 20 1 1 18
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Angle of friction () Angle of friction ()
(e) 10 360 (f) 10 340
9 9 340 320 8 8 320 300 7 7 300 6 6 280 280 5 5 260
4 260 4 240 3 3 240 220 2 2 220 1 1 200
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Undrained shear strength (kPa) Undrained shear strength (kPa)
Figure 6.5 Realizations of soil properties (a) medium clay (c) sand (e) stiff clay
corresponding to stratigraphic configuration Figure 6.4a; (b) medium clay (d) sand (f)
stiff clay corresponding to stratigraphic configuration Figure 6.4b
147 3) Random variables modeling
Apart from uncertainties from stratigraphy and material properties spatial variability,
there are 8 other random variables, including model errors epy , etz , and eqw for p-y curves,
t-z curves and q-w curves, respectively, three allowable displacements ( Aw for vertical
movement, A for lateral deflection, and A for angular distortion), lateral load V and axial load Q. The corresponding distributions and statistical properties of these random variables are summarized in Table 6.4. The typical variability levels of those random variables can be found in the literature (Phoon and Kulhawy, 1999; Zhang and Ng, 2005;
Fan, Huang and Liang, 2014).
Table 6.4 Statistical properties of random variables
Variable Distribution Mean COV (%)
epy Lognormal 1.0 5
etz Lognormal 1.0 5
eqw Lognormal 1.0 5
Aw Gumbel 25 mm 10
A Gumbel 20 mm 10
A Gumbel 0.01 10 V Gamma 800 kN 20 Q Gamma 150 kN 15
6.5.2 Reliability Analysis of Deep Foundation Considering Multiple Failure Modes
In the domain, a drilled shaft can be placed at three different locations. The configuration of a drilled shaft is shown in Figure 6.6. To simulate the soil-pile interaction, the t-z curves and the q-w curves for axial soil reaction from Manual (2010) and the p-y curves for stiff clay without free water and sand from Reese et al. (2000) are adopted in this example. The material properties of pile are of typical values, e.g. the
148 elastic modulus of concrete is taken as 26.60 GPa and that of steel is 200 GPa. The load eccentricity (h) of the lateral load V is assumed to be a constant at h=2.5m. Hence the bending moment at the top of the shaft is computed as M V h .
Figure 6.6 Example of drilled shaft
For each combination of generated stratigraphic profile and soil properties, the soil profile at the discretization elements where the shaft is located is extracted, together with other sources of uncertainties, are further input into the p-y model and t-z model iteratively to calculate the response of shaft. It is assumed that the dimension of diameter is neglected in the lithologic profile. The estimates of the failure probabilities for vertical
movement Pf ,w , lateral deflection Pf , , angular distortion Pf , , and system failure Pf ,S are plotted in Figure 6.7. As can be seen, all the failure probabilities converge as the number of simulations increases. As expected, the system failure probability is greater than any of other individual failure probabilities.
149 0.007
0.006
0.005 P 0.004 f,S P 0.003 f,w
Probability of failure of Probability 0.002 P f, 0.001 P f,
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 4 Number of simulations x 10
Figure 6.7 Convergence of the estimates
Figure 6.8 shows the variation of the system failure probability Pf ,S as a function of pile diameter D and pile length L for three different pile locations. The horizontal axis in
Fig.11 represents the variation of L, and the values of for three different values of D
(D=1.1m, 1.2m, and 1.3m) are included in the figure. For different pile locations, given a value of D, decreases as L increases. In this example, the target reliability index of serviceability limit state is set as =2.7, thus the target probability of system failure is
PT ,S =0.0035. The relationship between reliability index and probability of failure can be found in CORPS (1997) and Wang, Au and Kulhawy (2011). To satisfy the target reliability, the feasible designs for shaft at three different locations are selected, which are those below the dashed line of target failure probability. The feasible designs are also listed in Table 6.5. As can be seen, the minimum feasible design parameters vary for
150 these three locations. For a given D, the location 3 has the smallest value of the minimum feasible L, followed by the location 2, and the location 1 has the largest value. The difference is due to the different distributions of soil stratigraphy and the associated spatially varied soil properties at three locations. Among the feasible designs, one can choose the optimal design with consideration of the economic cost (Wang, Au and
Kulhawy, 2011).
(a) 0 (b) 0 (c) 0 10 10 10 D=1.0m D=1.0m D=1.1m D=1.0m D=1.1m D=1.2m D=1.1m D=1.2m D=1.2m -1 -1 -1 10 10 10
-2 -2 -2 10 10 10
-3 -3 -3
10 10 10
Probability of failure of Probability failure of Probability Probability of failure of Probability
-4 -4 -4 10 10 10
-5 -5 -5 10 10 10 6.5 7 7.5 8 8.5 9 6.5 7 7.5 8 8.5 9 6.5 7 7.5 8 8.5 9 Pile length Pile length Pile length
Figure 6.8 Probability of failure for different designs (a) location 1 (b) location 2 (c)
location 3
Table 6.5 Feasible designs of shaft at three different locations
Location 1 Location 2 Location 3
D 1.0m L 9.0m L 8.0m L 7.5m
D 1.1m L 8.5m 7.8m L 7.3m
D 1.2m L 8.0m L 7.5m L 7.0m
151 6.6 Summary and Conclusions
This chapter proposed an integrated approach to quantify geotechnical uncertainties in a systematic manner and incorporate these uncertainties into a probabilistic framework to evaluate the serviceability performance of the drilled shaft. The uncertainties that affect the reliability of the designed drilled shaft are identified as: subsurface soil stratigraphy, soil properties, external loads, model errors relative to the soil-pile interaction model (p-y model and t-z model), and performance criteria.
First, the stratigraphic uncertainty is quantified by the novel stochastic geological modelling method. Based on MRF with specific spatial correlation in terms of potential function, the adopted stochastic geological model can reflect the inherent heterogeneous
(multiple lithological units) and anisotropic (direction-dependent spatial correlation) characteristics of a geological body for generating lithological profiles. The results of stratigraphy simulation using real borehole logs demonstrate the capability of the geological model in the measurement and visualization of lithological uncertainty. Then the spatial variability of soil property of each identified soil layer is modeled conditional on the relative known borehole samples by using 2D GRF, in which each soil property is statistically characterized by the mean, variance, and correlation length. Furthermore, the uncertainties from loading conditions, computational models, and performance criteria are handled by their respective prescribed probability distributions. At last, based on the relative probabilistic models, the generated random samples are input into the soil-pile interaction model to evaluate the performance of the drilled shaft by using MCS.
Three failure modes are considered in the performance-based reliability analysis of a drilled shaft, including the vertical movement, lateral deflection, and angular distortion at
152 the top of the pile. The system failure is considered as any of the induced displacements exceeds the corresponding allowable movement. The target reliability index is used to measure whether the system performance is satisfactory, which is defined based on the desired performance level. The details of the proposed reliability analysis framework are provided by a step-by-step procedure description. The example of the reliability analysis of the drilled shaft, which can be placed at three different locations of the domain, was given to illustrate the application of the proposed integrated approach. The feasible design parameters of the shaft are different from location to location. This difference is attributed to the different soil stratigraphy and soil properties at the three locations.
The proposed integrated approach provides a logic step to taking into consideration the uncertainties in geotechnical engineering comprehensively and systematically, especially allowing dealing with stratigraphic uncertainties which has been paid little attention in engineering practice. Also, the developed performance-based reliability analysis framework can be applied to other geotechnical structures, such as embankments, slope, and underground space.
153
CHAPTER VII
SUMMARY AND CONCLUSIONS
7.1 Summary of Work Accomplished
In this study, a novel geological modeling approach using stochastic simulation techniques has been proposed for reliability-based analysis of geo-structures. The proposed approach provides a logic step to taking into consideration the uncertainties in geotechnical engineering comprehensively and systematically, especially allowing dealing with stratigraphic uncertainties quantitatively which has been paid little attention in engineering practice. This research encompassed two main parts: subsurface stratigraphy simulation and subsequent reliability-based analysis of geo-structures. The subsurface stratigraphy simulation is handled by an innovative stochastic geological model using limited field observation data, which was originally developed in this research. The stochastic geological modeling method is applied to real cases (project sites in Sheffield, Lorain County, Ohio and in Akron, Summit County, Ohio) to evaluate its ability of stratigraphic uncertainty quantification. Through subsurface simulation, the stratigraphic uncertainties can be quantified and incorporated, along with other types of uncertainties (e.g. soil variability), into a probabilistic analysis framework to evaluate the performance of geo-structures.
154 The state-of-the-art literature review clearly supported the need for conducting this research, which revealed that there was a lack of geological modeling techniques.
Stratigraphic profile has been established mainly by using engineering judgment based on local experience. Also, the existing geological modeling methods have limitations in quantifying stratigraphic uncertainty, thus stratigraphic uncertainty has always been ignored in engineering practice. In addition, the uncertainties from geological body haven’t been considered comprehensively in probability-based design/analysis for geotechnical structures.
In order to address the aforementioned problems in geological modeling and reliability analysis, two objectives of the research are: 1) propose a fundamentally sound stochastic geological model to estimate the lithological profile and quantify stratigraphic uncertainty using limited known information based on Markov random field (MRF); 2) develop an integrated approach to quantify geotechnical uncertainties in a systematic manner and incorporate these uncertainties into a probabilistic framework to evaluate the performance of geo-structures. More specific contributions of the research include the following:
An effective stochastic geological modeling framework is proposed based on
Markov random field theory with a novel neighborhood system, which is
conditional on site investigation data, such as observations of soil types from
ground surface, borehole logs, and strata orientation from geophysical tests. The
proposed modeling method is capable of accounting for the inherent
heterogeneous and anisotropic characteristics of geological structure. The
representation of real stratigraphic profile is modeled by means of stochastic
155 simulation, and its uncertainty is quantified by evaluating information entropy of
the possible inferred realizations. Map of information entropy is able to provide
intuitive and visualized descriptions of regions with lithological assignment of
certain confident level and stratigraphic uncertainty associated with those
uncertain regions.
Two modeling approaches (ICM and MCMC) have been introduced to simulate
subsurface geological structures to accommodate different confidence levels on
geological structure type (i.e., layered vs. others). The sensitivity analysis for two
modeling approaches was conducted to reveal the influence of mesh density and
the model parameter on the simulation results. Illustrative examples using
borehole data were presented to elucidate the ability to quantify the geological
structure uncertainty. Furthermore, the applicability of two modeling approaches
and the behavior of the proposed model under different model parameters have
also been discussed in detail.
Bayesian inferential framework has been introduced to allow for the estimation of
the posterior distribution of model parameter, when additional or subsequent
borehole information becomes available. Practical guidance of using the proposed
stochastic geological modeling technique for engineering practice is given.
An integrated approach for probabilistic analysis and design of geotechnical
structures that consider both stratigraphic uncertainty and soil variability by
utilizing a Markov random field (MRF) for stochastic modeling of the
156 stratigraphic profile and a Gaussian random field (GRF) for modeling the
spatially varying soil properties within each lithological unit.
An FEA analysis method employing a discretization scheme compatible with that
used by the stochastic geological model was used for subsequent analysis of the
performance of geotechnical structures, such as a strip footing, through Monte
Carlo simulation. The effects of simulated uncertain subsurface profiles along
with a two-dimensional spatial random elastic modulus on the computed
settlement of a strip footing have been elucidated through the FEA/MCS
framework. Compared with the deterministic analysis method, the probability
distributions of equivalent elastic modulus were obtained for further analysis of a
strip footing underlain by layers of different lithological units. Based on the
criterion of the service limit state, an allowable bearing pressure with a certain
level of confidence was obtained.
An integrated approach to handle all these uncertainties comprehensively and
systematically and incorporates these uncertainties into reliability analysis of deep
foundation. The uncertainties that affect the reliability of the designed drilled
shaft are identified as: subsurface soil stratigraphy, soil properties, external loads,
model errors relative to the soil-pile interaction model (p-y model and t-z model),
and performance criteria. The stratigraphic uncertainty was quantified by the
proposed stochastic geological modeling method. The spatial variability of the
geo-material properties for each lithological unit, on the other hand, was handled
by a 2D Gaussian random field. The uncertainties from loading conditions,
157 computational models, and performance criteria are tackled by their respective
prescribed probability distributions.
The performance-based reliability analysis framework has been developed
to assess the serviceability performance of a drilled shaft which can be placed at
different locations in the simulated geological domain. Three failure modes were
considered in the system reliability analysis, including the vertical settlement,
lateral deflection, and angular distortion at the top of shaft. Failure was defined as
the event that displacements exceed the corresponding allowable movements. The
system failure was considered as any of the induced displacements exceeds the
corresponding allowable movement. The target reliability index was used to
measure whether the system performance is satisfactory, which is defined based
on the desired performance level.
7.2 Conclusions
The main conclusions of this study are summarized as follows.
The proposed geological modeling framework is able to model subsurface
stratigraphic structure under multiple common geological conditions conditional
on various types of observation data, and to present possible stratigraphic
realizations in a probabilistic manner.
Studies of the real geotechnical projects demonstrate that the proposed geological
model is capable to provide quantitative measurements of stratigraphic
158 uncertainty for both two-dimensional and three-dimensional subsurface domains
conditional on real borehole logs. Moreover, based on the modeling results
conditional on existing borehole logs, the proposed geological model can be
implemented to estimate the benefits obtained from additional boreholes, and to
optimize the borehole locations. Therefore, the proposed stratigraphic uncertainty
quantification procedure plays favorable role for obtaining necessary knowledge
of stratigraphic structure with the minimized cost of geotechnical site
investigation.
The modeling approach using ICM technique to generate initial configuration is
applicable to the layered lithological structure. The “uncertain band” is generally
located at the possible lithological unit boundary. The modeling approach using
MCMC technique can introduce more uncertainties into the initial configuration.
The uncertain area is a “divergent zone” from the known formation boundary at
borehole location. The farther the site is from the borehole location, the more
uncertainty of interpretation of geo-material type is.
The proposed Bayesian inferential framework for model parameter estimation
proves to be effective to generate samples of model parameter a. The estimation
error (sample variance) decreases as more additional boreholes are incorporated
as model evidence.
Three cases of settlement analysis of the strip footing were carried out for
comparison of the effects of various sources of uncertainty. It was shown that the
uncertainty associated with the subsurface lithologic profile can exert a great
159 influence on the computational results of the strip footing performance. When
both types of uncertainty (stratigraphy uncertainty and uncertainty in material
properties for each lithologic unit) are considered, a greater variation in the
computed footing response and a smaller allowable bearing pressure will result. In
engineering practice, it is important to take into account both sources of
uncertainty.
The example of the reliability analysis of the drilled shaft, which can be placed at
three different locations of the domain, was given to illustrate the application of
the proposed integrated approach. The feasible design parameters of the shaft
were different from location to location. This difference was attributed to the
different soil stratigraphy and soil properties at the three locations. The developed
performance-based reliability analysis framework can be applied to other
geotechnical structures, such as embankments, slope, and underground space.
7.3 Recommendations for Future Research
The major development efforts of this study were concentrated on the development of mathematically sound geological modeling method and the subsequent application of the geological model in reliability analysis of geo-structures. The following recommendations for future research are suggested:
Consider more reasonable neighborhood system and potential function to reflect
the real world subsurface situations.
160 Develop methodology for geological modeling by using other types of
discretization schemes, such as triangular mesh.
Develop methodology for estimating model parameter a by using global field
investigation data, such as seismic data.
Develop a user-friendly software interface of the geological model for actual
engineering practice.
The cost of conducting field drilling to establish borehole logs can be high;
therefore, a systematic study on the relationship between the uncertainty of the
subsurface stratigraphic profile and the number of borehole logs as well as their
optimum locations could be the subject of extensive research in the future.
Apply the proposed reliability-based design framework to other geotechnical
problems, such as slope, tunnel, embankment, etc.
161
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