A parallel evolutionary strategy based simulation–optimization approach for solving groundwater source identification problems

Groundwater characterization involves the resolution of unknown system characteristics from observation data, and is often classified as an inverse problem. Inverse problems are difficult to solve due to natural ill-posedness and computational intractability. Here we adopt the use of a simulation–optimization approach that couples a numerical pollutant-transport simulation model with evolutionary search algorithms for solution of the inverse problem. In this approach, the numerical transport model is solved iteratively during the evolutionary search. This process can be computationally intensive since several hundreds to thousands of forward model evaluations are typically required for solution. Given the potential computational intractability of such a simulation–optimization approach, parallel computation is employed to ease and enable the solution of such problems. In this paper, several variations of a groundwater source identification problem is examined in terms of solution quality and computational performance. The computational experiments were performed on the TeraGrid cluster available at the National Center for Supercomputing Applications. The results demonstrate the performance of the parallel simulation–optimization approach in terms of solution quality and computational performance.     

基于球面边值问题的点质量调和分析方法

对全球扰动点质量模型而言,可以假定虚拟扰动质点系位于地球内部同一Bjerhamar球面上,同时把边值界面视为球面.本文针对这一假设下所形成的线性方程组的系数阵,运用快速傅里叶变换的方法,得到了点质量模型解算中利用分块循环矩阵分解大型线性方程组的新方法.全球30′×30′扰动点质量模型259200阶方程组的解算分解为720个360阶方程组的解算,解决了点质量模型构建中大型线性方程组的稳定解算问题.推导了全球点质量模型与球谐位系数模型的转换关系,得到了一种基于球面边值问题的点质量调和分析方法.数值模拟试验表明,在适当选取点质量埋深度的情况下,本文的点质量调和分析方法较传统的调和分析方法精度更高.     

Solutions of the inherent problem of the equivalence in direct current resistivity and electromagnetic methods through global optimization and joint inversion by successive refinement of model space

The problem of equivalence in direct current (DC) resistivity and electromagnetic methods for a thin resistive and conducting layer is well‐known. Attempts have been made in the past to resolve this problem through joint inversion. However, equivalence still remains an unresolved problem. In the present study, an effort is made to reduce non‐uniqueness due to equivalence using global optimization and joint inversion by successive refinement of the model space. A number of solutions derived for DC resistivity data using very fast simulated annealing global inversion that fits the observations equally well, follow the equivalence principle and show a definite trend. For a thin conductive layer, the quotient between resistivity and thickness is constant, while for a resistive one, the product between these magnitudes is constant. Three approaches to obtain very fast simulated annealing solutions are tested. In the first one, layer resistivities and thicknesses are optimized in a linear domain. In the second, layer resistivities are optimized in the logarithmic domain and thicknesses in the linear domain. Lastly, both layer resistivities and thicknesses are optimized in the logarithmic domain. Only model data from the mean models, corresponding to very fast simulated annealing solutions obtained for approach three, always fit the observations. The mean model defined by multiple very fast simulated annealing solutions shows extremely large uncertainty (almost 100%) in the final solution after inversion of individual DC resistivity or electromagnetic (EM) data sets. Uncertainty associated with the intermediate resistive and conducting layers after global optimization and joint inversion is still large. In order to reduce the large uncertainty associated with the intermediate layer, global optimization is performed over several iterations by reducing and redefining the search limits of model parameters according to the uncertainty in the solution. The new minimum and maximum limits are obtained from the uncertainty in the previous iteration. Though the misfit error reduces in the solution after successive refinement of the model space in individual inversion, it is observed that the mean model drifts away from the actual model. However, successive refinement of the model space using global optimization and joint inversion reduces uncertainty to a very low level in 4–5 iterations. This approach works very well in resolving the problem of equivalence for resistive as well as for conducting layers. The efficacy of the approach has been demonstrated using DC resistivity and EM data, however, it can be applied to any geophysical data to solve the inherent ambiguities in the interpretations.     

From Bayes to Tarantola: New insights to understand uncertainty in inverse problems

Anyone working on inverse problems is aware of their ill-posed character. In the case of inverse problems, this concept (ill-posed) proposed by J. Hadamard in 1902, admits revision since it is somehow related to their ill-conditioning and the use of local optimization methods to find their solution. A more general and interesting approach regarding risk analysis and epistemological decision making would consist in analyzing the existence of families of equivalent model parameters that are compatible with the prior information and predict the observed data within the same error bounds. Otherwise said, the ill-posed character of discrete inverse problems (ill-conditioning) originates that their solution is uncertain. Traditionally nonlinear inverse problems in discrete form have been solved via local optimization methods with regularization, but linear analysis techniques failed to account for the uncertainty in the solution that it is adopted. As a result of this fact uncertainty analysis in nonlinear inverse problems has been approached in a probabilistic framework (Bayesian approach), but these methods are hindered by the curse of dimensionality and by the high computational cost needed to solve the corresponding forward problems. Global optimization techniques are very attractive, but most of the times are heuristic and have the same limitations than Monte Carlo methods. New research is needed to provide uncertainty estimates, especially in the case of high dimensional nonlinear inverse problems with very costly forward problems. After the discredit of deterministic methods and some initial years of Bayesian fever, now the pendulum seems to return back, because practitioners are aware that the uncertainty analysis in high dimensional nonlinear inverse problems cannot (and should not be) solved via random sampling methodologies. The main reason is that the uncertainty “space” of nonlinear inverse problems has a mathematical structure that is embedded in the forward physics and also in the observed data. Thus, problems with structure should be approached via linear algebra and optimization techniques. This paper provides new insights to understand uncertainty from a deterministic point of view, which is a necessary step to design more efficient methods to sample the uncertainty region(s) of equivalent solutions.     

Interpretation of marine magnetic anomalies. Part II. Analysis of the new method and algorithm based on the least squares method

In this work, the problem of interpreting linear marine magnetic anomalies in the context of the spreading model is considered. Based on the least squares method, a new algorithm for determining the position of direct and reverse polarity blocks is proposed. The global minimum of the residual is found by a combination of the multistart and Monte Carlo methods. Using an example of a field of three blocks perturbed by a normal noise, it is shown that the method yields an error close to the minimum possible. Using a model constructed by the inversion scale, it is revealed that the proposed algorithm allows one to find positions of block boundaries of the field without noise with a higher accuracy determined in fact by the computation time.     

Pareto Joint Inversion of 2D Magnetotelluric and Gravity Data — Towards Practical Applications

In this paper, a Pareto inversion based global optimization approach, to obtain results of joint inversion of two types of geophysical data sets, is formulated. 2D magnetotelluric and gravity data were used for tests, but presented solution is flexible enough to be used for combination of any kind of two or more target functions, as long as misfits can be calculated and forward problems solved. To minimize dimensionality of the solution, space and introduce straightforward regularization Sharp Boundary Interface (SBI) method was applied. As a main optimization engine, Particle Swarm Optimization (PSO) was used. Synthetic examples based on a real geological model were used to test proposed approach and show its usefulness in practical applications.     

基于多目标优化的有限角度图像重建算法及实现

本文综合考虑了图像局部和整体的平滑性,以及最大熵准则,重新建立新的多目标优化模型,并在求解过程中引入同伦参数加快收敛速度。通过仿真实验,验证了所提算法能较好地改善有限角度下图像重建的质量及提高重建速度。     

DIRECT CURRENT ELECTRIC POTENTIAL FIELD ASSOCIATED WITH TWO SPHERICAL CONDUCTORS IN A WHOLE-SPACE1

Bispherical coordinates are used to derive an exact mathematical solution for the potential field generated by direct current electric conduction in an earth model consisting of two spherical inclusions in a uniform whole-space. The solution takes the form of a spherical harmonic expansion in bispherical coordinates; coefficients in the expansion are obtained by solving sets of linear equations. Rapid forward modelling of numerous interesting situations in d.c. resistivity prospecting is facilitated by the generality and computational efficiency inherent to this new solution. For example, the accuracy of image (or superposition) methods for calculating potential solutions can be quantified. Similarly, the ability of d.c. conduction methods to resolve two distinct bounded bodies in three-dimensional space can be examined by repeatedly calculating the secondary potential or apparent resistivity response of an earth model as a selected parameter is varied. Synthetic mise à la masse, crosshole, or areal potential data sets can be generated for subsequent use in inversion studies. Improvements in solution technique derived here also apply to a simpler model consisting of a single sphere buried in a half-space.     

基于高阶累积量ARMA模型线性非线性结合的地震子波提取方法研究

在地震子波非因果、混合相位的假设下,本文应用自回归滑动平均(ARMA)模型对地震子波进行参数化建模,并提出利用线性(矩阵方程法)和非线性(ARMA拟合方法)相结合的参数估计方式对该模型进行参数估计.在利用矩阵方程法确定模型参数范围的基础上,利用累积量拟合法精确估计参数.理论分析和仿真结果表明,该方式有较好的适应性：一方面提高了子波估计精度,避免单独使用矩阵方程法在短数据地震记录情况下可能带来的估计误差;另一方面提高了子波提取运算效率,降低了ARMA模型拟合方法参数范围确定的复杂性,避免了单纯使用滑动平均（MA）模型拟合法估计过多参数所导致的运算规模过大问题.初步应用结果表明该方法是有效可行的.     

An efficiency-improved genetic algorithm and its application on multimodal functions and a 2D common reflection surface stacking problem

Although Genetic Algorithms have found many successful applications in the field of exploration geophysics, the convergence speed remains a big challenge as Genetic Algorithms usually require a huge amount of fitness function evaluations. In this paper, we propose an efficiency-improved Genetic Algorithm, which has both a good global search capability and a good local search capability, and is also capable of robustly handling the premature convergence challenge commonly seen in linear and directed non-linear optimization methods. In our new genetic algorithm, the global search capability is performed via a modified island model, while the local search capability is provided by a novel self-adaptive differential evolution fine tuning scheme. Premature convergence is dealt with via a local exhaustive search method. We first demonstrate the much improved convergence speed of this efficiency-improved Genetic Algorithm over that of our previously proposed advanced Genetic Algorithm on several multimodal functions. We further demonstrate the effectiveness of our efficiency-improved Genetic Algorithm by applying it to a two-dimensional common reflection surface stacking problem, which is a highly nonlinear geophysical optimization problem, to obtain very encouraging results.     

地震层析成像反演中解的定量评价及其应用

对地震层析成像非线性问题线性化处理之后,各种反演算法归纳成为对不适定方 程的求解．地震层析成像反演算法的解的物理意义是给出地质结构,因此对于解的可靠性及 分辨率研究非常重要．然而许多反演算法不能给出解的评价方法,因而对解的可信度产生怀 疑．本研究根据解估计的分辨率矩阵的原理,提出ＬＳＱＲ（Ｌｅａｓｔ Ｓｑｕａｒｅ ＱＲ）算法解协方差矩 阵的评价算法,用相关分析可以为那些在求解过程中得不到分辨率矩阵的反演方法提供解的 定量评价．并用本文提出的解的定量评价方法试评了一个实际地壳模型的地震层析成像的 速度重建结果．     

A hybrid inversion method of damped least squares with simulated annealing used for Rayleigh wave dispersion curve inversion

Surface wave methods are becoming increasingly popular in many geotechnical applications and in earthquake seismology due to their noninvasive characteristics.Inverse surface wave dispersion curves are a crucial step in most surface wave methods.Many inversion methods have been applied to surface wave dispersion curve inversion,including linearized inversion and nonlinearized inversion methods.In this study,a hybrid inversion method of Damped Least Squares(DLS) with Very Fast Simulated Annealing(VFSA) is developed for multi-mode Rayleigh wave dispersion curve inversion.Both synthetic and in situ fi eld data were used to verify the validity of the proposed method.The results show that the proposed method is superior to the conventional VFSA method in aiming at global minimum,especially when parameter searching space is adjacent to real values of the parameters.The advantage of the new method is that it retains both the merits of VFSA for global search and DLS for local search.At high temperatures,the global search dominates the runs,while at a low temperatures,the local search dominates the runs.Thus,at low temperatures,the proposed method can almost directly approach the actual model.     

一种新的地球物理反演方法——模拟原子跃迁反演法

详细研究了一般地球物理反问题的迭代优化求解过程与物理学中原子跃迁过程的对应关系,建立了反演问题中模型空间、初始模型、局部极值模型、最优化模型等与原子的态空间、定态、激发态、基态等的对应关系. 在此基础上,模拟了物理学中原子从激发态向基态跃迁的物理过程,建立了一种与原子跃迁过程相对应的非线性随机跃迁数学模型和模型解跃迁搜索准则,导出了适用于一般地球物理资料的模拟原子跃迁的非线性反演算法. 用理论测试函数对这种新的反演方法进行了数值试验,结果表明该方法具有解不依赖于初始模型、收敛速度快等优点.     

基于粒子群优化的理论变异函数拟合方法研究

变异函数是地统计学中区域化变量空间结构分析和空间局部插值的主要分析工具.理论变异函数模型的获取是地质统计学中的基础性工作,它是了解区域化变量的变异特征、进一步对地质统计学计算的必要环节.针对现有的理论变异函数的拟合方法,如人工拟合法、线性规划拟合法、加权多项式拟合法、目标规划拟合法等的不足之处,充分利用粒子群优化算法在求解非线性优化问题时具有的全局寻优的特点,提出基于粒子群优化的理论变异函数拟合方法.在实例应用中,分别利用粒子群优化算法和加权多项式拟合方法进行理论变异函数拟合,交叉验证结果表明粒子群优化算法预测精度较高,具有较强的稳健性.     

Numerical modeling of the propagation and morphological changes of turbidity currents using a cost-saving strategy of solution updating

Existing layer-averaged numerical models for turbidity currents have mostly adopted the global minimum time step (GMiTS) for solution updating, which confines their computational efficiency and limits their attractiveness for field applications. This paper presents a highly efficient layer-averaged numerical model for turbidity currents by implementing the combined approach of the local graded-time-step (LGTS) and the global maximum-time-step (GMaTS). The governing equations are solved for unstructured triangular meshes by the shock-capturing finite volume method along with a set of well-balanced evaluations of the numerical flux and geometrical slope source terms. The quantitative accuracy of the model, given reasonably estimated empirical and model parameters (e.g., bed friction, water entrainment, sediment deposition and erosion coefficients), is demonstrated by comparing the numerical solutions against laboratory data of the current front positions and deposition profiles, as well as field data of the current front positions. The improved computational efficiency is demonstrated by comparing the computational cost of the present model against that of a traditional model that uses a GMiTS. For the present simulated cases, the maximum reduction of the computational cost is approximately 80% (e.g., a simulation that cost 1 h before will only require 12 min with the new model).     

An algorithmic approach to the total least-squares problem with linear and quadratic constraints

Proper incorporation of linear and quadratic constraints is critical in estimating parameters from a system of equations. These constraints may be used to avoid a trivial solution, to mitigate biases, to guarantee the stability of the estimation, to impose a certain “natural” structure on the system involved, and to incorporate prior knowledge about the system. The Total Least-Squares (TLS) approach as applied to the Errors-In-Variables (EIV) model is the proper method to treat problems where all the data are affected by random errors. A set of efficient algorithms has been developed previously to solve the TLS problem, and a few procedures have been proposed to treat TLS problems with linear constraints and TLS problems with a quadratic constraint. In this contribution, a new algorithm is presented to solve TLS problems with both linear and quadratic constraints. The new algorithm is developed using the Euler-Lagrange theorem while following an optimization process that minimizes a target function. Two numerical examples are employed to demonstrate the use of the new approach in a geodetic setting.     

Applications of hybrid genetic algorithms in seismic tomography

Almost all earth sciences inverse problems are nonlinear and involve a large number of unknown parameters, making the application of analytical inversion methods quite restrictive. In practice, most analytical methods are local in nature and rely on a linearized form of the problem equations, adopting an iterative procedure which typically employs partial derivatives in order to optimize the starting (initial) model by minimizing a misfit (penalty) function. Unfortunately, especially for highly non-linear cases, the final model strongly depends on the initial model, hence it is prone to solution-entrapment in local minima of the misfit function, while the derivative calculation is often computationally inefficient and creates instabilities when numerical approximations are used. An alternative is to employ global techniques which do not rely on partial derivatives, are independent of the misfit form and are computationally robust. Such methods employ pseudo-randomly generated models (sampling an appropriately selected section of the model space) which are assessed in terms of their data-fit. A typical example is the class of methods known as genetic algorithms (GA), which achieves the aforementioned approximation through model representation and manipulations, and has attracted the attention of the earth sciences community during the last decade, with several applications already presented for several geophysical problems.In this paper, we examine the efficiency of the combination of the typical regularized least-squares and genetic methods for a typical seismic tomography problem. The proposed approach combines a local (LOM) and a global (GOM) optimization method, in an attempt to overcome the limitations of each individual approach, such as local minima and slow convergence, respectively. The potential of both optimization methods is tested and compared, both independently and jointly, using the several test models and synthetic refraction travel-time date sets that employ the same experimental geometry, wavelength and geometrical characteristics of the model anomalies. Moreover, real data from a crosswell tomographic project for the subsurface mapping of an ancient wall foundation are used for testing the efficiency of the proposed algorithm. The results show that the combined use of both methods can exploit the benefits of each approach, leading to improved final models and producing realistic velocity models, without significantly increasing the required computation time.     

Impact of simulation model solver performance on ground water management problems

Ground water management models require the repeated solution of a simulation model to identify an optimal solution to the management problem. Limited precision in simulation model calculations can cause optimization algorithms to produce erroneous solutions. Experiments are conducted on a transient field application with a streamflow depletion control management formulation solved with a response matrix approach. The experiment consists of solving the management model with different levels of simulation model solution precision and comparing the differences in optimal solutions obtained. The precision of simulation model solutions is controlled by choice of solver and convergence parameter and is monitored by observing reported budget discrepancy. The difference in management model solutions results from errors in computation of response coefficients. Error in the largest response coefficients is found to have the most significant impact on the optimal solution. Methods for diagnosing the adequacy of precision when simulation models are used in a management model framework are proposed.     

基于全局弱式无单元法直流电阻率正演模拟

全局弱式无单元法是在有限单元法基础上发展起来的一种数值模拟方法,它采用局部支持域内的节点信息来构造形函数实现局部精确逼近,摆脱了单元,仅依赖于节点信息,具有预处理简单、模拟精度高、灵活性强的特点,适用于复杂地电条件下直流电阻率正演模拟.本文采用RPIM构造直流电阻率全局弱式无单元法形函数,利用RPIM形函数推导了直流电阻率全局弱式无单元法方程.然后,编制了直流电阻率全局弱式无单元法正演模拟Fortran程序,利用该程序对典型的地电模型进行了正演模拟,并将正演结果与基于线性插值的FEM正演结果及解析解进行对比,结果表明采用RPIM形函数的全局弱式无单元法用于直流电阻率正演模拟的正确性及有效性,且在同等条件下,全局弱式无单元法模拟精度高于矩形剖分的FEM,更有利于指导电法勘探的数据解译;利用该程序对复杂地电模型进行了正演模拟,结果表明全局弱式无单元法对复杂地电模型模拟效果良好,适应性强,灵活性高,可任意加密节点提高模拟精度.