A stochastic model reduction method for nonlinear unconfined flow with multiple random input fields

In this paper we present a stochastic model reduction method for efficiently solving nonlinear unconfined flow problems in heterogeneous random porous media. The input random fields of flow model are parameterized in a stochastic space for simulation. This often results in high stochastic dimensionality due to small correlation length of the covariance functions of the input fields. To efficiently treat the high-dimensional stochastic problem, we extend a recently proposed hybrid high-dimensional model representation (HDMR) technique to high-dimensional problems with multiple random input fields and integrate it with a sparse grid stochastic collocation method (SGSCM). Hybrid HDMR can decompose the high-dimensional model into a moderate M-dimensional model and a few one-dimensional models. The moderate dimensional model only depends on the most M important random dimensions, which are identified from the full stochastic space by sensitivity analysis. To extend the hybrid HDMR, we consider two different criteria for sensitivity test. Each of the derived low-dimensional stochastic models is solved by the SGSCM. This leads to a set of uncoupled deterministic problems at the collocation points, which can be solved by a deterministic solver. To demonstrate the efficiency and accuracy of the proposed method, a few numerical experiments are carried out for the unconfined flow problems in heterogeneous porous media with different correlation lengths. The results show that a good trade-off between computational complexity and approximation accuracy can be achieved for stochastic unconfined flow problems by selecting a suitable number of the most important dimensions in the M-dimensional model of hybrid HDMR.     

Wavelet-vaguelette decomposition of spatiotemporal random fields

A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodifferential equations of parabolic type is derived in the cases of random initial conditions and random forcing. The family of spatiotemporal models considered can represent anomalous diffusion processes when the spatial operator involved is a fractional or multifractional pseudodifferential operator. The results obtained are applied to the generation of the sample paths of Gaussian spatiotemporal random fields in the family studied.     

楼板振动概率分析的微分方程方法

本文针对作用于楼板上的多台机器扰和的复杂特性。给出了一个较接近于实际的随机扰力模型，从理论上建立了楼板振地国频率分析的微分方程方法，对于给定的随机扰和模型，给出了结构响应统计参数的计算公式，使得对楼板的随机振动分析得到深入，在计算机上得到简化。     

Streamflow simulation and skewness preservation based on the bootstrapped stochastic models

The stochastic model has been widely used for the simulation study. However, there was a difficulty in the reproduction of the skewness of observed series and so the stochastic model for the skewness preservation was appeared. While the skewness in the residuals of the stochastic model has been considered for the skewness preservation this study uses a random resampling technique of residuals from the stochastic models for the simulation study and for the investigation of the skewness coefficient. The main advantage of this resampling scheme, called the bootstrap method is that it does not rely on the assumption of population distribution and this study uses the combined model of the stochastic and bootstrapped models. The stochastic and bootstrapped stochastic (or combined) models are used for the investigations of skewness preservation and of the reproduction of probability density function between the simulated series. The models are applied to the annual and monthly streamflows of Yongdam site in Korea and Yakima river, Washington, USA for the streamflow simulation study then the statistics and probability density functions for the observed and simulated streamflows are compared. As the results the bootstrapped stochastic model reproduces the skewness and probability density function much better than the stochastic model. This evidences suggest that the bootstrapped stochastic model might be more appropriate than the stochastic model for the preservation of skewness and for simulation purposes of the series.     

Conditional simulations of water–oil flow in heterogeneous porous media

This study is an extension of the stochastic analysis of transient two-phase flow in randomly heterogeneous porous media (Chen et al. in Water Resour Res 42:W03425, 2006), by incorporating direct measurements of the random soil properties. The log-transformed intrinsic permeability, soil pore size distribution parameter, and van Genuchten fitting parameter are treated as stochastic variables that are normally distributed with a separable exponential covariance model. These three random variables conditioned on given measurements are decomposed via Karhunen–Loève decomposition. Combined with the conditional eigenvalues and eigenfunctions of random variables, we conduct a series of numerical simulations using stochastic transient water–oil flow model (Chen et al. in Water Resour Res 42:W03425, 2006) based on the KLME approach to investigate how the number and location of measurement points, different random soil properties, as well as the correlation length of the random soil properties, affect the stochastic behavior of water and oil flow in heterogeneous porous media.     

Quantifying the sources of simulation uncertainty in natural catastrophe models

The risk from natural catastrophes is typically estimated using complex simulation models involving multiple stochastic components in a nested structure. This risk is principally assessed via the mean annual loss, and selected quantiles of the annual loss. Determining an appropriate simulation strategy is important in order to achieve satisfactory convergence of these statistics, without excessive computation time and data storage requirements. This necessitates an understanding of the relative contribution of each of the stochastic components to the total variance of the statistics. A simple framework using random effects models and analysis of variance is used to partition the variance of the annual loss, which permits calculation of the variance of the mean annual loss with varying numbers of samples of each of the components. An extension to quantiles is developed using the empirical distribution function in combination with bootstrapping. The methods are applied to a European flood model, where the primary stochastic component relates to the frequency and severity of flood events, and three secondary components relate to defence levels, exposure locations and building vulnerability. As expected, it is found that the uncertainty due to the secondary components increases as the size of the portfolio of exposures decreases, and is higher for industrial and commercial business, compared with residential for all statistics of interest. In addition, interesting insights are gained as to the impact of flood defences on convergence.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved     

Influence of ground-motion correlation on probabilistic assessments of seismic hazard and loss: sensitivity analysis

Recent studies have shown that the proper treatment of ground-motion variability and, particularly, the correlation of ground motion are essential for the estimation of the seismic hazard, damage and loss for distributed portfolios. In this work we compared the effects of variations in the between-earthquake correlation and in the site-to-site correlation on probabilistic estimations of seismic damage and loss for the extended objects (hypothetical portfolio) and critical elements (e.g. bridges) of a network. Taiwan Island has been chosen as a test case for this study because of relatively high seismicity and previous experience in earthquake hazard modelling. The hazard and loss estimations were performed using Monte Carlo approach on the basis of stochastic catalogues and random ground-motion fields. We showed that the influence of correlation on parameters of seismic hazard, characteristics of loss distribution and the probability of damage depend, on one hand, on level of hazard and probability level of interest (return period) and, on the other hand, the relative influence of each type of correlation is not equal.     

Stochastic dynamic analysis of a layered half-space

A stochastic thin-layer method is developed for the analysis of wave propagation in a layered half-space. A random field of shear moduli in the layered system is considered in terms of multiple correlated random variables. Expanding the random moduli and uncertain responses by means of Hermite polynomial chaos expansions and applying the Galerkin method in the spatial as well as stochastic domains, stochastic versions of thin-layer methods for a layered half-space in plane strain and antiplane shear are obtained. In order to represent the infinite half-space, continued-fraction absorbing boundary conditions are included in the thin-layer models of the half-space. Using these stochastic methods, dynamic responses of a layered half-space subjected to line loads are examined. Means, coefficients of variance, and probability density functions of the half-space responses with a varying correlation coefficient of the shear moduli are computed and verified by comparison with Monte Carlo simulations. It is demonstrated that accurate probabilistic dynamic analysis is possible using the developed stochastic thin-layer methods for a layered half-space.     

Stochastic Lagrangian trajectory model for drifting objects in the ocean

The prediction of drifting object trajectories in the ocean is a complex problem plagued with uncertainties. This problem is usually solved simulating the possible trajectories based on wind and advective numerical and/or instrumental data in real time, which are incorporated into Lagrangian trajectory models. However, both data and Lagrangian models are approximations of reality and when comparing trajectory data collected from drifter exercises with respect to Lagrangian models results, they differ considerably. This paper introduces a stochastic Lagrangian trajectory model that allows quantifying the uncertainties related to: (i) the wind and currents numerical and/or instrumental data, and (ii) the Lagrangian trajectory model. These uncertainties are accounted for within the model through random model parameters. The quantification of these uncertainties consists in an estimation problem, where the parameters of the probability distribution functions of the random variables are estimated based on drifter exercise data. Particularly, it is assumed that estimated parameters maximize the likelihood of our model to reproduce the trajectories from the exercise. Once the probability distribution parameters are estimated, they can be used to simulate different trajectories, obtaining location probability density functions at different times. The advantage of this method is that it allows: (i) site specific calibration, and (ii) comparing uncertainties related to different wind and currents predictive tools. The proposed method is applied to data collected during the DRIFTER Project (eranet AMPERA, VI Programa Marco), showing very good predictive skills.     

基于Metropolis优化的叠前全局迭代地质统计学反演方法

叠前地质统计学反演将随机模拟与叠前反演相结合,不仅可以反演各种储层弹性参数,还提高了反演结果的分辨率.基于联合概率分布的直接序贯协模拟方法可以在原始数据域对数据进行模拟,不需要对数据进行高斯变换,拓展了地质统计学反演的应用范围;而联合概率分布的应用确保了反演参数之间相关性,提高了反演的精度.本文将基于联合概率分布的直接序贯协模拟方法与蒙特卡洛抽样算法相结合,参考全局随机反演策略,提出了基于蒙特卡洛优化算法的全局迭代地质统计学反演方法.为了提高反演的稳定性,我们修改了局部相关系数的计算公式,提出了一种新的基于目标函数的优化局部相关系数计算公式并应用到协模拟之中.模型测试及实际数据应用表明,该方法可以很好的应用于叠前反演之中.     

Reduction of reflections from above surface objects in GPR data

During a ground-penetrating radar (GPR) survey, special attention must be paid to objects located above the earth's surface. Due to the low-loss character of electromagnetic propagation in air and high velocity, above-surface reflections or diffractions can overwhelm subsurface events, making the interpretation a difficult task. The relative sensitivity of reflections and diffractions originating from above-surface objects is a function of the antenna radiation characteristics, the lateral and vertical dimensions of the objects and their position with respect to the antennas. The largest amplitude reflections and diffractions are expected when the polarization of the electric field is parallel to the long-axis of the object. Near the surface in the E-plane, the electric field is vertically polarized and has a larger amplitude than the horizontally polarized electric field in the H-plane. Numerical modeling of reflections from three above surface objects (a vertical plane and elongated horizontal and vertical objects) demonstrate that the largest amplitude difference occurs when an elongated vertical object is present in the E- or H-plane. The calculated reflection from the elongated vertical object present in the E-plane was 21 times larger than when it was present in the H-plane. In 60-m long field data sets, reflections from interfering trees present in the E-plane were at several positions >15 times larger and on average 6 times larger than when the trees were present in the H-plane. These large amplitude differences indicate that appropriate orientation of the antennas can be used to minimize the effects of above-surface reflections and diffractions.     

Continuous time random walks for non-local radial solute transport

This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests.