Comparing the Gradual Deformation with the Probability Perturbation Method for Solving Inverse Problems

Inverse problems are ubiquitous in the Earth Sciences. Many such problems are ill-posed in the sense that multiple solutions can be found that match the data to be inverted. To impose restrictions on these solutions, a prior distribution of the model parameters is required. In a spatial context this prior model can be as simple as a Multi-Gaussian law with prior covariance matrix, or could come in the form of a complex training image describing the prior statistics of the model parameters. In this paper, two methods for generating inverse solutions constrained to such prior model are compared. The gradual deformation method treats the problem of finding inverse solution as an optimization problem. Using a perturbation mechanism, the gradual deformation method searches (optimizes) in the prior model space for those solutions that match the data to be inverted. The perturbation mechanism guarantees that the prior model statistics are honored. However, it is shown with a simple example that this perturbation method does not necessarily draw accurately samples from a given posterior distribution when the inverse problem is framed within a Bayesian context. On the other hand, the probability perturbation method approaches the inverse problem as a data integration problem. This method explicitly deals with the problem of combining prior probabilities with pre-posterior probabilities derived from the data. It is shown that the sampling properties of the probability perturbation method approach the accuracy of well-known Markov chain Monte Carlo samplers such as the rejection sampler. The paper uses simple examples to illustrate the clear differences between these two methods     

粘弹性波动方程的系数反演

波动方程的系数反演是一种用于识别地下介质物理力学参数的重要方法。过去这方面的研究一般都建立在弹性模型的基础上,本文则提出了粘弹性反演模型,这种模型更能真实地反映波在地下传播的实际情形。在粘弹性反演模型的基础上,我们还提出了一整套行之有效的数值反演方法,并在频域中完成了反演计算。最后的数值计算结果是令人满意的,证明了本文的模型和方法的合理性。     

非线性抛物型方程反问题的一种新算法

利用反问题新算法—时间域正演反演法研究非线性抛物型方程的逆时反问题。该方法处理反问题的主要思路是先求解相对应的正问题,获得解在特定时间网格处的近似值;然后构造一个合适的全纯映射,在象空间中获得解在时间网格对应点处的近似值;最后利用解析函数的唯一延拓性质实现反问题的逆时间反演。数值模拟例子说明了方法的有效性。     

On Mineralogical Studies and the Use of Mineralogical Information in Solving Petro- and Ore Genesis Problems

Geology of Ore Deposits - The deep penetration of crystal chemistry and physics of minerals into mineralogy, as well as the introduction of modern physical methods, has advanced mineralogical...     

水环境逆边界逆动态混合控制精确算法

根据逆边界逆动态控制理论,将河流水污染动态控制问题提为逆边界逆动态混合控制问题。针对多个或单个污染源排放浓度和排放总量计算,提出了一维对流-扩散方程逆控制的精确算法。该方法与现行最优控制方法相比,其优点是充分考虑了河流沿程的稀释混合容量,并能充分考虑水质动态标准和社会经济变化等因素,可获得动态控制精确解的近似解。     

Study on Precipitation Image Downscaling Based on the Method of Ill-posed Problems Solving

Downscaling of remote sensing precipitation products and the forecasting of circulation model are always the intense interests in hydrology and meteorology. The essence of downscaling is primarily to enhance resolution of observation or simulated rainfall field, and to appropriately increase its details or high frequency characteristics. Precipitation, as the main driving factors of the earth’s hydrologic cycle, not only affects the moisture and heat condition of a certain river basin, but also affects the global water and heat circulation. Based on the properties of rainfall self-similarity structure, the mathematically ill-posed precipitation problem solving method was used in low resolution downscaling precipitation for high resolution reconstruction. When solving the downscaling ill-posed problem, the greedy method of orthogonal matching pursuit was introduced so as to get the best high-resolution estimation in an optimal sense. It is hard to imagine that we might be able to find very similar (in mathematical norms) precipitation patterns over relatively large storm-scales. However, finding similar features over sufficiently small sub-storm scales seems more feasible. Based on the characteristics that small scale organized precipitation features tend to recur across different storm environments, the precipitation of both high and low resolution was obtained by training, which could be used to reconstruct the desired high-resolution precipitation field. Multi-source merged precipitation products were used in this experiment. Given the consideration of incompleteness of merged precipitation dataset, it was firstly interpolated based on the method of Fields of Experts (FoEs), which could solve the problem that common interpolation method could hardly work on the interpolation for dataset where consecutive missing data exists. Secondly, ideal experiments of precipitation products downscaling were carried out, where smooth coupling sampling and resampling operator were adopted respectively. Assessment based on the metrics of fidelity and spatial structural similarity demonstrates that the method used in this paper is feasible.     

A Projection Method for Solving Systems of Linear Equations: Gravimetry Applications

Doklady Earth Sciences - Existing methods for solving inverse problems, such as the regularization method, look mostly for a quasi-solution that may not be a solution to the original problem, but...     

The Probability Perturbation Method: A New Look at Bayesian Inverse Modeling

Building of models in the Earth Sciences often requires the solution of an inverse problem: some unknown model parameters need to be calibrated with actual measurements. In most cases, the set of measurements cannot completely and uniquely determine the model parameters; hence multiple models can describe the same data set. Bayesian inverse theory provides a framework for solving this problem. Bayesian methods rely on the fact that the conditional probability of the model parameters given the data (the posterior) is proportional to the likelihood of observing the data and a prior belief expressed as a prior distribution of the model parameters. In case the prior distribution is not Gaussian and the relation between data and parameters (forward model) is strongly non-linear, one has to resort to iterative samplers, often Markov chain Monte Carlo methods, for generating samples that fit the data likelihood and reflect the prior model statistics. While theoretically sound, such methods can be slow to converge, and are often impractical when the forward model is CPU demanding. In this paper, we propose a new sampling method that allows to sample from a variety of priors and condition model parameters to a variety of data types. The method does not rely on the traditional Bayesian decomposition of posterior into likelihood and prior, instead it uses so-called pre-posterior distributions, i.e. the probability of the model parameters given some subset of the data. The use of pre-posterior allows to decompose the data into so-called, “easy data” (or linear data) and “difficult data” (or nonlinear data). The method relies on fast non-iterative sequential simulation to generate model realizations. The difficult data is matched by perturbing an initial realization using a perturbation mechanism termed “probability perturbation.” The probability perturbation method moves the initial guess closer to matching the difficult data, while maintaining the prior model statistics and the conditioning to the linear data. Several examples are used to illustrate the properties of this method.     

Solving the Seismic Inverse Problem Using an Ensemble of Models: A Case Study of Inversion of Receiver Functions

Doklady Earth Sciences - This paper proposes a new method for determining the seismic parameters of the subsurface below a seismic station using the waveforms of converted waves. The subsurface...     

基于随机加权先验的P-Ⅲ分布参数贝叶斯估计

研究随机加权先验法进行P-Ⅲ分布参数贝叶斯估计。应用随机加权法确定分布参数的先验分布,MCMC自适应采样算法(AM)进行参数的后验分布采样,并与矩法、极大似然法和概率权重矩法等传统水文频率分析方法进行比较。实例表明,AM方法估算参数下,实测样本与对应频率设计值离差平方和最小,是一种可行的水文频率分析途径。     

用信息量法寻找地下水

信息量法,在国外于七十年代已在找矿上应用,于此,笔者仅对在找水工作上怎样应用此法作一阐述。     

求解岩体裂隙产状三维概率分布的数值方法

测线法是一种广泛使用的裂隙几何特征野外观测技术,但它获得的一维产状观测数据不能代表三维空间内的概率分布.在实测裂隙倾向和倾角之间相互独立的假设基础上,借用概率论和微积分建立了一维数据和三维分布的数值解关系式,进而提出一种由一维观测数据求解三维概率分布的方法.该方法的实现步骤是:(1)通过关系式数值求解产状的三维累积概率...     

用惯性力求解结构的自振频率

基于多自由度体系的惯性力方程,把最大惯性力加到体系质点位置,以此受力为依据,求出质点处的位移。根据各质点处位移变形协调条件,得出各质点处位移的相互关系,进而求出结构中质量的自振频率。此法对求解多自由度体系的自由振动,与传统方法相比具有方法简便、便于掌握等特点。     

揭示地下隐伏信息的专题抽取法

根据遥感图像多波段成像的特性,讨论了利用典型地物波谱曲线分离地下隐伏信息或揭示地下异常信息的专题抽取法。     

一种新的遥感蚀变信息编图方法

利用遥感蚀变信息进行成矿预测已经得到普遍应用, 但是并没有系统的编图方法, 提出遥感蚀变信息等密度制图法以进行改进.它借鉴了地球化学场的编图思路, 通过统计单位面积内各蚀变信息所占的面积百分比, 进而生成面密度等值线图.利用该方法并结合线、环构造信息在内蒙古东乌旗地区圈定了8个成矿远景区.这种新的编图方法, 既能有效增强多种蚀变同时出现时小面积蚀变信息, 又能更直观的表达蚀变信息的总体空间分布特征和浓集中心, 对圈定成矿远景区和优选野外验证点十分有利.      

频域波动方程的多重网格法

针对差分法求解过程中细网格计算所需时间和空间要求高以及在粗网格上计算又达不到所要求的精度等问题，利用粗网格上的残差校正特性消除迭代误差的低频分量，同时利用细网格上的松弛光滑特性消除迭代误差的高频分量，采用不同疏密的网格距消除不同频率范围内的误差分量，将多重网格方法成功应用于频域波动方程的求解，并给出了实际算例。算例中，分别用Gauss-Seidel迭代法和多重网格法求其数值解，并绘出效果图；比较得出多重网格方法具有精度高、收敛速度快和易于实现的特性，更适合于高维波动问题的求解。     

高频燃烧—红外碳硫仪测定地质样品中的碳和硫

应用HIR-944B型高频-红外碳硫分析仪,对不同地质样品中碳、硫的测定进行了研究,称样30～60mg,加入0.4g纯铁屑及1.7g钨粒助熔剂,高温燃烧分解试样,红外检测,可定量地质样品中质量为0～O.9mg的硫及质量为0～15mg的碳.用该仪器测定地质标样中碳、硫的结果与标准值符合,碳和硫11次测定的RSD分别是＜2.6%和＜3.0%.     

An Example of Realistic Modelling of Rock Dynamics Problems: FEM/DEM Simulation of Dynamic Brazilian Test on Barre Granite

The scope of this study is to numerically simulate the behaviour of Brazilian disc specimens as observed in laboratory during dynamic, high-strain rate, indirect tensile tests using an innovative combined finite-discrete element method (FEM/DEM) research code. Laboratory experiments using a split Hopkinson pressure bar (SHPB) apparatus were conducted by the authors and the measured indirect tensile strength values were used to verify the FEM/DEM models. In the models the applied boundary conditions, related to the loading rate of the specimen, were matched with the experimental observations. The results of the numerical simulations, including tensile strength and failure time, are in agreement with the laboratory findings. The main failure mechanisms, i.e. tensile splitting along loading axis and shear failure close to loading platens are captured by the numerical model. A linear relationship between tensile strength and loading rate is found for the range of dynamic strain rates tested and simulated. The simulation results are in good agreement with laboratory observations and demonstrate the potential for using FEM/DEM to realistically model dynamic response of rocks.