Applied Mathematics in EM Studies with Special Emphasis on an Uncertainty Quantification and 3-D Integral Equation Modelling

Despite impressive progress in the development and application of electromagnetic (EM) deterministic inverse schemes to map the 3-D distribution of electrical conductivity within the Earth, there is one question which remains poorly addressed—uncertainty quantification of the recovered conductivity models. Apparently, only an inversion based on a statistical approach provides a systematic framework to quantify such uncertainties. The Metropolis–Hastings (M–H) algorithm is the most popular technique for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. However, all statistical inverse schemes require an enormous amount of forward simulations and thus appear to be extremely demanding computationally, if not prohibitive, if a 3-D set up is invoked. This urges development of fast and scalable 3-D modelling codes which can run large-scale 3-D models of practical interest for fractions of a second on high-performance multi-core platforms. But, even with these codes, the challenge for M–H methods is to construct proposal functions that simultaneously provide a good approximation of the target density function while being inexpensive to be sampled. In this paper we address both of these issues. First we introduce a variant of the M–H method which uses information about the local gradient and Hessian of the penalty function. This, in particular, allows us to exploit adjoint-based machinery that has been instrumental for the fast solution of deterministic inverse problems. We explain why this modification of M–H significantly accelerates sampling of the posterior probability distribution. In addition we show how Hessian handling (inverse, square root) can be made practicable by a low-rank approximation using the Lanczos algorithm. Ultimately we discuss uncertainty analysis based on stochastic inversion results. In addition, we demonstrate how this analysis can be performed within a deterministic approach. In the second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.     

Uncertainty in Shear-Wave Velocity Profiles

Journal of Seismology - This paper considers the uncertainty in the shear wave velocity (Vs) of soil and rock profiles for use in earthquake site response calculations. This uncertainty is an...     

AVO反演的不确定性分析

叠前地震数据反演可以得到比常规叠后波阻抗反演更丰富、更有效的岩性信息,但叠前数据体的噪声及其它因素严重影响了AVO反演的稳定性,如何评估AVO反演结果的可靠性显得尤为重要.本文从贝叶斯理论出发,假定均匀先验分布、高斯噪音分布,推出不确定性分析方程,利用协方差矩阵中的方差描述反演问题的不确定性,模型研究显示反演不确定性与叠前信噪比、纵横波速度比、覆盖次数及反演采用的角度范围相关,方法预测的反演误差可定量解释反演结果的可靠性,为解释人员提供有效的质量监控手段.     

Uncertainty Analysis in Atmospheric Dispersion Modeling

The concentration of a pollutant in the atmosphere is a random variable that cannot be predicted accurately, but can be described using quantities such as ensemble mean, variance, and probability distribution. There is growing recognition that the modeled concentrations of hazardous contaminants in the atmosphere should be described in a probabilistic framework. This paper discusses the various types of uncertainties in atmospheric dispersion models, and reviews sensitivity/uncertainty analysis methods to characterize and/or reduce them. Evaluation and quantification of the range of uncertainties in predictions yield a deeper insight into the capabilities and limitations of atmospheric dispersion models, and increase our confidence in decision-making based on models.     

Uncertainty estimates for surface runoff models

A lower bound for variance in surface runoff modelling estimates is advanced. The bound is derived using a linear unit hydrograph approach which utilizes a discretization of the catchment into an arbitrary number of subareas, a linear routing technique for channel flow effects, a variable effective rainfall distribution over the catchment, and calibration parameter distributions developed in correlating rainfall-runoff data by the model. The uncertainty bound reflects the dominating influence of the unknown rainfall distribution over the catchment and is expressed as a distribution function that can be reduced only by supplying additional rainfall-runoff data. It is recommended that this uncertainty distribution in modelling results be included in flood control design studies in order to incorporate a prescribed level of confidence in flood protection facilities.     

起算震级与衰减不确定性校正

分析表明,起算震级Ｍ０对危险性分析结果的表现为当Ｍ０取取值不够低时,场地的危险性将被低估。此种低估影响与场地的地震危险性高低潜源距场地的距离及衰减公式的形式和标准差有关。     

Uncertainty Analysis of Strong-Motion and Seismic Hazard

In this article we present the modelling of uncertainty in strong-motion studies for engineering applications, particularly for the assessment of earthquake hazard. We examine and quantify the sources of uncertainty in the basic variables involved in ground motion estimation equations, including those associated with the seismological parameters, which we derive from a considerable number of strong-motion records. Models derived from regression analysis result in ground motion equations with uncertain parameters, which are directly related to the selected basic variables thus providing an uncertainty measure for the derivative variable. These uncertainties are exemplified and quantified. An alternative approach is presented which is based on theoretical modelling defining a functional relationship on a set of independent basic variables. Uncertainty in the derivative variable is then readily obtained when the uncertainties of the basic variables have been defined. In order to simplify the presentation, only the case of shallow strike-slip earthquakes is presented. We conclude that the uncertainty is approximately the same as given by the residuals typical for regression modelling. This implies that uncertainty in ground motion modelling cannot be reduced below certain limits, which is in accordance with findings reported in the literature. Finally we discuss the implications of the presented methodology in hazard analyses, which is sensitive to the truncation of the internal error term, commonly given as an integral part of ground motion estimation equations. The presented methodology does not suffer from this shortcoming; it does not require truncation of the error term and yields realistic hazard estimates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Quantifying Data Worth Toward Reducing Predictive Uncertainty

The present study demonstrates a methodology for optimization of environmental data acquisition. Based on the premise that the worth of data increases in proportion to its ability to reduce the uncertainty of key model predictions, the methodology can be used to compare the worth of different data types, gathered at different locations within study areas of arbitrary complexity. The method is applied to a hypothetical nonlinear, variable density numerical model of salt and heat transport. The relative utilities of temperature and concentration measurements at different locations within the model domain are assessed in terms of their ability to reduce the uncertainty associated with predictions of movement of the salt water interface in response to a decrease in fresh water recharge. In order to test the sensitivity of the method to nonlinear model behavior, analyses were repeated for multiple realizations of system properties. Rankings of observation worth were similar for all realizations, indicating robust performance of the methodology when employed in conjunction with a highly nonlinear model. The analysis showed that while concentration and temperature measurements can both aid in the prediction of interface movement, concentration measurements, especially when taken in proximity to the interface at locations where the interface is expected to move, are of greater worth than temperature measurements. Nevertheless, it was also demonstrated that pairs of temperature measurements, taken in strategic locations with respect to the interface, can also lead to more precise predictions of interface movement.     

Remediation of Heterogeneous Aquifers Subject to Uncertainty

Optimal cost pump-and-treat ground water remediation designs for containment of a contaminated aquifer are often developed using deterministic ground water models to predict ground water flow. Uncertainty in hydraulic conductivity fields used in these models results in remediation designs that are unreliable. The degree to which uncertainty contributes to the reliability of remediation designs as measured by the characterization of the uncertainty is shown to differ depending upon the geologic environments of the models. This conclusion is drawn from the optimal design costs for multiple deterministic models generated to represent the uncertainty of four distinct models with different geologic environments. A multi scenario approach that includes uncertainty into the remediation design called the deterministic method for optimization subject to uncertainty (DMOU) is applied to these distinct models. It is found that the DMOU is a method for determining a remediation design subject to uncertainty that requires minimal postprocessing efforts. Preprocessing, however, is required for the application of the DMOU to unique problems. In the ground water remediation design problems, the orientation of geologic facies with respect to the orientation of flow patterns, pumping well locations, and constraint locations are shown to affect the preprocessing, the solutions to the DMOU problems, and the computational efficiency of the DMOU approach. The results of the DMOU are compared to the results of a statistical analysis of the effects of the uncertainty on remediation designs. This comparison validates the efficacy of the DMOU and illustrates the computational advantages to using the DMOU over statistical measures.     

军事应用地震学研究

介绍了美国若干军事设施遭受地震破坏的情况，并提出开展军事应用地震学研究的建议。     

Uncertainty estimation of pathlines in ground water models

A method is proposed to estimate the uncertainty of the location of pathlines in two-dimensional, steady-state confined or unconfined flow in aquifers due to the uncertainty of the spatially variable unconditional hydraulic conductivity or transmissivity field. The method is based on concepts of the semianalytical first-order theory given in Stauffer et al. (2002, 2004), which allows estimates of the lateral second moment (variance) of the location of a moving particle. However, this method is reformulated in order to account for nonuniform recharge and nonuniform aquifer thickness. One prominent application is the uncertainty estimation of the catchment of a pumping well by considering the boundary pathlines starting at a stagnation point. In this method, the advective transport of particles is considered, based on the velocity field. In the case of a well catchment, backtracking is applied by using the reversed velocity field. Spatial variability of hydraulic conductivity or transmissivity is considered by taking into account an isotropic exponential covariance function of log-transformed values with parameters describing the variance and correlation length. The method allows postprocessing of results from ground water models with respect to uncertainty estimation. The code PPPath, which was developed for this purpose, provides a postprocessing of pathline computations under PMWIN, which is based on MODFLOW. In order to test the methodology, it was applied to results from Monte Carlo simulations for catchments of pumping wells. The results correspond well. Practical applications illustrate the use of the method in aquifers.