Enhancing gradient-based parameter estimation with an evolutionary approach

Traditionally, the calibration of groundwater models has depended on gradient-based local optimization methods. These methods provide a reasonable degree of success only when the objective function is smooth, second-order differentiable, and satisfies the Lipschitz's condition. For complicated and highly nonlinear objective functions it is almost impractical to satisfy these conditions simultaneously. Research in the calibration of conceptual rainfall-runoff models, has shown that global optimization methods are more successful in locating the global optimum in the region of multiple local optima. In this study, a global optimization technique, known as shuffle complex evolution (SCE), is coupled to the gradient-based Lavenberg–Marquardt algorithm (GBLM). The resultant hybrid global optimization algorithm (SCEGB) is then deployed in parallel testing with SCE and GBLM to solve several inverse problems where parameters of a nonlinear numerical groundwater flow model are estimated. Using perfect (i.e. noise-free) observation data, it is shown SCEGB and SCE are successful at identifying the global optimum and predicting all model parameters; whereas, the commonly applied GBLM fails to identify the optimum. In subsequent inverse simulations using observation data corrupted with noise, SCEGB and SCE again outperform GBLM by consistently producing more accurate parameter estimates. Finally, in all simulations the hybrid SCEGB is seen to be equally effective as SCE but computationally more efficient.     

Applying a real‐coded multi‐population genetic algorithm to multi‐reservoir operation

The primary objective of this study is to propose a real‐coded hypercubic distributed genetic algorithm (HDGA) for optimizing reservoir operation system. A conventional genetic algorithm (GA) is often trapped into local optimums during the optimization procedure. To prevent premature convergence and to obtain near‐global optimal solutions, the HDGA is designed to have various subpopulations that are processed using separate and parallel GAs. The hypercubic topology with a small diameter spreads good solutions rapidly throughout all of the subpopulations, and a migration mechanism, which exchanges chromosomes among the subpopulations, exchanges information during the joint optimization to maintain diversity and thus avoid a systematic premature convergence toward a single local optimum. Three genetic operators, i.e. linear ranking selection, blend‐α crossover and Gaussian mutation, are applied to search for the optimal reservoir releases. First, a benchmark problem, the four‐reservoir operation system, is considered to investigate the applicability and effectiveness of the proposed approach. The results show that the known global optimal solution can be effectively and stably achieved by the HDGA. The HDGA is then applied in the planning of a multi‐reservoir system in northern Taiwan, considering a water reservoir development scenario to the year 2021. The results searched by an HDGA minimize the water deficit of this reservoir system and provide much better performance than the conventional GA in terms of obtaining lower values of the objective function and avoiding local optimal solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Water‐Saving Crop Planning Using Multiple Objective Chaos Particle Swarm Optimization for Sustainable Agricultural and Soil Resources Development

Establishing a water‐saving planting structure is necessary for the arid, water‐deficient regions of northern China and of the world. Optimizing and adjusting a water‐saving agricultural planting structure is a typical semi‐structured, multi‐level, multi‐objective group decision‐making problem. Therefore, optimization can be best achieved with a swarm intelligence algorithm. We build an optimization model for a water‐saving planting structure with four target functions: (1) maximum total net output, (2) total grain yield, (3) ecological benefits, and (4) water productivity. The decision variable is the yearly seeded area of different crops, and its restrictions are the farmland area, the agricultural water resources, and the needs of the people and other farming‐related industries. Multiple objective particle swarm optimization (MOPSO) is an efficient optimization method, but its main shortcoming is that it can easily fall into a local optimum. Multiple objective chaos particle swarm optimization (MOCPSO) will greatly improve the searching performance of the algorithm by placing chaos technology with the advantages of ergodicity into MOPSO. When MOCPSO is used to solve the multi‐objective optimization model in the middle portion of the Heihe River basin, the results show that MOCPSO has the advantages of a high convergence speed and a tendency not to fall easily into a local optimum. After adopting a water‐saving agricultural planting structure, irrigation water would be reduced by about 7%, which would provide tangible economic, social, and ecological benefits for sustainable agricultural development.     

The determination of the best fitting and the extreme source models by a global inversion method

In this paper, a global inversion method is developed for seismic moment tensor inversion by using the body wave forms. The algorithm depends on neither the selection of starting model nor the forms of objective function and constraints. When the error function, measure of the difference between the observed and synthetic waveforms, is chosen as the objective function, the best fitting source model is found; when a certain combination in seismic moment tensor elements is selected as the objective function and the values of error function are constrained in a suitable bound, the extreme source models can be produced by minimizing or maximizing this combination. By changing the form of the combination of moment tensor elements, a variety of different source characteristics can be considered. Therefore the extreme solution provides an estimation of the uncertainty in the best fitting source model. The seismic waveform data was used to evaluate the effectiveness of this algorithm. This research was supported by the National Natural Science Foundation of China.     

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.     

基于MCMC的叠前地震反演方法研究

马尔科夫链蒙特卡洛方法(MCMC)是一种启发式的全局寻优算法[1].它在贝叶斯框架下,利用已有资料进行约束,既可使最优解满足参数的统计特性,又通过融入的先验信息,提高解的精度;寻优过程可跳出局部最优,得到全局最优解.利用MCMC方法,可以得到大量来自于后验概率分布的样本,不仅可以得到每个未知参数的估计值,而且可以得到与...     

简化混合域全波形反演多GPU加速策略

全波形反演利用地震记录中的振幅、走时和相位等信息,通过拟合实际地震记录和计算波场来定量提取地下介质的弹性参数,进而为勘探地震成像、速度建模以及大尺度构造演化分析等提供可靠依据.但全波形反演计算量巨大,特别是应用于三维大区块叠前数据时,生产成本仍然很高.本文介绍并比较了时间域和频率域的全波形反演方法,综合两者的优点,最终采用混合域的反演算法,并且在此基础上做了进一步的简化以提高计算效率.针对全波形反演方法应用于大规模叠前数据时易陷入局部极小值的问题,我们提出对模型数据进行分割,同时在数个小模型内进行梯度搜索,然后对比各个局域的梯度,最终找出合适的全局下降方向,以克服局部极小的隐患.该方法能够充分利用GPU的硬件特性.在GPU环境下实现本文所提出的简化混合域全波形反演算法.数值计算实例体现出新方法具有良好的计算效率、反演精度和算法可扩展性.     

Ray tomography: Errors and error functions

Tomography is the inversion of boundary projections to reconstruct the internal characteristics of the medium between the source and detector boreholes. Tomography is used to image the structure of geological formations and localized inhomogenieties. This imaging technique may be applied to either seismic or electromagnetic data, typically recorded as transmission measurements between two or more boreholes. Algebraic algorithms are error-driven solutions where the goal is to minimize the error between measured and predicted projections. The purpose of this study is to assess the effect of the ray propagation model, the measurement errors, and the error functions on the resolving ability of algebraic algorithms. The problem under consideration is the identification of a two-dimensional circular anomaly surveyed using crosshole measurements. The results show that: (1) convergence to the position of the circular anomaly in depth between vertical boreholes is significantly better than for convergence in the horizontal direction; (2) error surfaces may not be convex, even in the absence of measurement and model errors; (3) the distribution of information content significantly affects the convexity of averaging error functions; (4) measurement noise and model inaccuracy manifest in increased residuals and in reduced convergence gradients near optimum convergence; (5) the maximum ray error function increases convergence gradients compared with the average error function, and is unaffected by the distribution of information content; however, it has higher probability of local minima. Therefore, inversions based on the minimization of the maximum ray error may be advantageous in crosshole tomography but it requires smooth projections. These results are applicable to both electromagnetic and seismic data for wavelengths significantly smaller than the size of anomalies.     

Nonlinear Rayleigh wave inversion based on the shuffled frog-leaping algorithm

At present, near-surface shear wave velocities are mainly calculated through Rayleigh wave dispersion-curve inversions in engineering surface investigations, but the required calculations pose a highly nonlinear global optimization problem. In order to alleviate the risk of falling into a local optimal solution, this paper introduces a new global optimization method, the shuffle frog-leaping algorithm (SFLA), into the Rayleigh wave dispersion-curve inversion process. SFLA is a swarm-intelligence-based algorithm that simulates a group of frogs searching for food. It uses a few parameters, achieves rapid convergence, and is capability of effective global searching. In order to test the reliability and calculation performance of SFLA, noise-free and noisy synthetic datasets were inverted. We conducted a comparative analysis with other established algorithms using the noise-free dataset, and then tested the ability of SFLA to cope with data noise. Finally, we inverted a real-world example to examine the applicability of SFLA. Results from both synthetic and field data demonstrated the effectiveness of SFLA in the interpretation of Rayleigh wave dispersion curves. We found that SFLA is superior to the established methods in terms of both reliability and computational efficiency, so it offers great potential to improve our ability to solve geophysical inversion problems.     

Determining the focal mechanisms and depths of relatively small earthquakes using a few stations by full‐waveform modelling

Determining the focal mechanism of earthquakes helps us to better define faults and understand the stress regime. This technique can be helpful in the oil and gas industry where it can be applied to microseismic events. The objective of this paper is to find double couple focal mechanisms, excluding scalar seismic moments, and the depths of small earthquakes using data from relatively few local stations. This objective is met by generating three‐component synthetic seismograms to match the observed normalized velocity seismograms. We first calculate Green's functions given an initial estimate of the earthquake's hypocentre, the locations of the seismic recording stations and a 1D velocity model of the region for a series of depths. Then, we calculate the moment tensor for different combinations of strikes, dips and rakes for each depth. These moment tensors are combined with the Green's functions and then convolved with a source time function to produce synthetic seismograms. We use a grid search to find the synthetic seismogram with the largest objective function that best fits all three components of the observed velocity seismogram. These parameters define the focal mechanism solution of an earthquake. We tested the method using three earthquakes in Southern California with moment magnitudes of 5.0, 5.1 and 4.4 using the frequency range 0.1–2.0 Hz. The source mechanisms of the events were determined independently using data from a multitude of stations. Our results obtained, from as few as three stations, generally match those obtained by the Southern California Earthquake Data Center. The main advantage of this method is that we use relatively high‐frequency full‐waveforms, including those from short‐period instruments, which makes it possible to find the focal mechanism and depth of earthquakes using as few as three stations when the velocity structure is known.     

A hybrid fast algorithm for first arrivals tomography

A hybrid algorithm, combining Monte-Carlo optimization with simultaneous iterative reconstructive technique (SIRT) tomography, is used to invert first arrival traveltimes from seismic data for building a velocity model. Stochastic algorithms may localize a point around the global minimum of the misfit function but are not suitable for identifying the precise solution. On the other hand, a tomographic model reconstruction, based on a local linearization, will only be successful if an initial model already close to the best solution is available. To overcome these problems, in the method proposed here, a first model obtained using a classical Monte Carlo-based optimization is used as a good initial guess for starting the local search with the SIRT tomographic reconstruction. In the forward problem, the first-break times are calculated by solving the eikonal equation through a velocity model with a fast finite-difference method instead of the traditional slow ray-tracing technique. In addition, for the SIRT tomography the seismic energy from sources to receivers is propagated by applying a fast Fresnel volume approach which when combined with turning rays can handle models with both positive and negative velocity gradients. The performance of this two-step optimization scheme has been tested on synthetic and field data for building a geologically plausible velocity model.This is an efficient and fast search mechanism, which permits insertion of geophysical, geological and geodynamic a priori constraints into the grid model and ray path is completed avoided. Extension of the technique to 3D data and also to the solution of 'static correction' problems is easily feasible.     

Inversion of ocean-bottom seismometer (OBS) waveforms for oceanic crust structure: a synthetic study

The waveform inversion method is applied—using synthetic ocean-bottom seismometer(OBS) data—to study oceanic crust structure. A niching genetic algorithm(NGA) is used to implement the inversion for the thickness and P-wave velocity of each layer, and to update the model by minimizing the objective function, which consists of the misfit and cross-correlation of observed and synthetic waveforms. The influence of specific NGA method parameters is discussed, and suitable values are presented.The NGA method works well for various observation systems, such as those with irregular and sparse distribution of receivers as well as single receiver systems. A strategy is proposed to accelerate the convergence rate by a factor of five with no increase in computational complexity; this is achieved using a first inversion with several generations to impose a restriction on the preset range of each parameter and then conducting a second inversion with the new range. Despite the successes of this method,its usage is limited. A shallow water layer is not favored because the direct wave in water will suppress the useful reflection signals from the crust. A more precise calculation of the air-gun source signal should be considered in order to better simulate waveforms generated in realistic situations; further studies are required to investigate this issue.     

大地电磁测深资料的二次函数逼近非线性反演

将二次函数逼近非线性优化首次应用于大地电磁测深反演问题,该反演方法利用二次函数有唯一最小值的特点进行逼近大地电磁反演模型,从而避免了常规的迭代反演过程中陷入局部极小问题,实现了对目标函数求全局极小,较好地解决了非唯一性问题;同时该方法不用求灵敏度矩阵,且对初始模型无任何要求。通过理论模型检验、井旁MT点反演结果与测井曲线的对比及MT测线的反演电阻率深度剖面与地震测线的时间剖面对比均表明,本文方法取得较好的应用效果。     

波动方程反演的全局优化方法研究

复杂介质波动方程反演是地球物理研究中的重要问题,通常表述为特定目标函数最优化,难点是多参数、非线性和不适定性.局部和全局优化方法都不能实现快速全局优化.本文概述了地震波勘探反演问题的理论基础和研究进展,阐述了反演中优化问题的解决方法和面临的困难,并提出了一种确定性全局优化的新方法.通过在优化参数空间识别并划分局部优化解及其附近区域,只需有限次参数空间划分过程就能发现所有局部解（集合）;基于复杂目标函数多尺度结构分析,提出多尺度参数空间分区优化方法的研究方向.该方法收敛速度快,优化结果不依赖初始解的选取,是对非线性全局优化问题的一个新探索.     

基于柯西分布约束和快速近端目标函数优化的三维重力反演方法

优化算法的选取在很大程度上影响着三维重力反演的计算效率,从而制约着三维重力反演的实用性.在复杂地质构造背景下,不同岩性单元之间可能会发生物性突变,产生尖锐边界.为此,本文提出了一种新的基于柯西分布约束和快速近端目标函数（Fast Proximal Objective Function,FPOF）优化的三维重力反演方法.FPOF优化方法的一个突出特点是在每一步迭代过程中逐一计算剖分网格内的未知密度参数,因此,有较低的计算复杂度和较高的计算效率.此外,目标函数中柯西范数（Cauchy norm）的引入会对反演结果施加稀疏性,有助于产生块状效果.理论模型测试表明,本文方法不仅能产生更加聚焦的反演效果,而且反演所需的时间也比传统的共轭梯度优化方法少.最后将本文方法应用于我国西部某地区实际重力数据,反演结果与已知的地质信息有较好的一致性.     

Pseudodynamic testing of strain-softening systems with adaptive time steps

A structure may exhibit a severe strain-softening behaviour when subjected to strong earthquake excitation. Pseudodynamic testing of such structures using an implicit time-integration algorithm may be conceived of as a problem, since the Newton-type iterations, which are often required when structural non-linearity develops, may not converge under these circumstances. An unconditionally stable implicit time-integration algorithm implemented with Newton-type iterations is analysed to provide an insight into this problem. A simple convergence condition is derived to detect possible divergence. The condition is shown to be a sufficient criterion for convergence for general multiple-degree-of-freedom structures, and it is used later on to develop an adaptive time-stepping strategy to avoid divergence under severe strain-softening conditions. The implementation of this technique for pseudodynamic testing is presented. As demonstrated by numerical examples, the algorithm proves to be effective and reliable.     

量子遗传算法在大地电磁反演中的应用

量子遗传算法（QGA）以量子理论为基础,通过利用量子位编码代替经典遗传算法的二进制位编码,利用量子旋转门定向更新种群来代替传统方法中种群的选择、交叉和变异过程,使得算法具有一定的内在并行运算能力和量子的隧道效应,从而加快了搜索速度,改善了收敛速度,并具有更强的全局寻优能力.本文针对地球物理反演问题的非线性、多极值特点提出一套实现方案,通过理论模型和实测数据试验对比研究,表明量子遗传方法在大地电磁反演中的寻优质量和效果明显优于传统遗传算法.     

大地电磁阻尼粒子群优化反演法研究

粒子群优化算法（PSO）是模仿鸟群寻找食物的社会行为的一种全局最优化算法,在多维空间函数寻优、动态目标寻优等方面有着收敛速度快、解质量高且需要设置的参数较少等优点.本文在研究常规粒子群优化算法的基础上,对常规的粒子群算法进行了改进,提出了一种新的惯性权重ω参数振荡递减策略,加快了PSO算法的收敛速度,构造的新算法称为阻尼粒子群优化算法.在MATLAB 6.5 编程环境中对阻尼PSO算法进行了数值实验,并对大地电磁测深的理论模型和实测数据进行了反演试算,结果表明,阻尼PSO算法不依赖于初始模型、能够搜索到全局极值,不易陷入局部极值,是一种快速有效的地球物理反演方法.     

Application of a land surface model for simulating rainfall streamflow hydrograph: 1. Model calibration

Different variants of parameters’ calibration of land surface model SWAP were examined with the aim to maximize the accuracy of reproducing rainfall runoff hydrograph. The optimization of parameter values was automated based on two different algorithms for the search of the global optimum of an objective function: a random search technique and a shuffled complex evolution method SCE-UA. In both cases, two objective functions, based on the mean systematic error and the Nash and Sutcliffe coefficient of efficiency, were used. The number of calibrated parameters varied from 10 to 15, and their values were within the reasonable range so as not to contradict the physical meaning and to ensure the best agreement between the simulated and observed daily river runoff. The streamflow hydrographs for some rivers in USA simulated with the use of different sets of optimized parameters were compared with observation data.     

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.