地球物理反演中病态矩阵方程正则化解算方法研究
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摘要
地球物理反演是地球物理探测数据最重要的解释方法技术.求解地球物理反演问题常会涉及到大型病态矩阵方程的求解.理论上讲正则化方法是处理病态问题的有效手段,但在实践上正则化参数的选择却是一个困难的问题.本文在比较系统地研究了正则化方法的基础上,针对实际计算中常会碰到的问题,将其与Active-Set算法、差分进化算法等相结合,发展了一些新的病态矩阵方程正则化解算方法.论文的主要内容包括:
1.研究了在实际反演中遇到参数有非负要求特性时的反演计算方法.将原问题转化为一个带非负约束的阻尼最小二乘问题,并用Active-Set算法求解.通过对理论模型进行数值模拟计算,验证了将Tikhonov正则化方法与Active-Set算法相结合的A-TR算法的有效性.应用到实际双频电导率成像反演,也取得了满意的结果.
2.研究了差分进化算法在地球物理反演中的几种应用.为加速差分进化算法的收敛速度,提出了将种群熵的自适应差分进化(ARDE)算法以及粒子群差分进化混合(PSODE)算法分别与Tikhonov正则化方法结合.在大型反演计算中,这两种方法可以在不影响反演效果的前提下,不同程度地提高收敛速度,降低时间成本.同时结合LSQR和差分进化算法的优点,提出了基于LSQR算法的差分混合(HDE)算法,避开了Tikhonov和TSVD等直接正则化算法在选取正则化参数上的困难,同时具有数值稳定性好、不依赖于初值、不易陷入局部极值和收敛速度快等优点,适宜于在正则化参数选取困难情况时的地球物理反演问题的求解.
3.提出了一种双参数混合正则化方法.引入了带有二阶正则算子的正则化项,并应用L-曲线法、偏差原理和广义交叉校验准则的优化组合确定了最佳正则化参数.数值模拟实验和实际数据处理实验结果表明了该方法的可行性.这是一种将高阶正则化算子应用于实际反演计算的新的尝试.
基于数值模拟实验和实际数据处理实验,认为研究发展的A-TR算法、HDE等算法各有其不同的适用条件,A-TR算法适用于求解反演参数有非负约束的情况,而当正则化参数选取困难时,可采用HDE算法.针对本文所考察的双频电导率反演问题,由于电导率的非负性,采用A-TR方法可得到更加精细可靠的重建图像.
Geophysical inversion is a key technique in geophysical exploration. Geophysical inversion often relates to solving ill-posed, large matrix equations. Theoretically, regularization is an effective method in dealing with ill-posed problems. Its application in practical geophysical inversion problems, however, still has many difficulties in selecting regularization parameters. Based on systematic study of regularization, this dissertation presents a few newly developed regularization methods that apply Active-Set algorithm, Differential Evolution algorithm, and a few others. These regularization methods focus on solving ill-posed matrix equations arising from practical problems. Main results of the dissertation include:
Tikhonov regularization and Active-Set algorithm are applied together to the geophysical inversion problems so that the problems with non-negative parameters are converted into problems of non-negative damped least square algorithm, which can be further solved by the Active-Set algorithm. The improved recursive algorithm is further verified by numerical simulation. Satisfactory results are obtained by applying this algorithm to electrical conductivity imagery inversion.
Furthermore, differential evolution algorithms are also studied. To improve the rate of convergence of Differential Evolution algorithms, two new Tikhonov regularization algorithms are proposed that respectively employ Adaptive Recursive Differential Evolution (ARDE) algorithm based on population entropy and Particle Swarm Optimization and Differential Evolution (PSODE) algorithm. Without any compromise in effectiveness, these two algorithms both improve the convergence speed and thus reduce computation cost. Still further, a new DE algorithm based on LSQS, which inherits the advantages of both LSQR and DE, is designed. This new algorithm avoids the common difficulty of regularization parameter selection in Tikhonov and TSVD algorithms. It also displays superior stability, independence on initial values, unlikelihood of local extrema, and faster convergence. This algorithm is particularly suitable to solve the problem of selection of regularization parameters in the study of geophysical inversion.
Finally, the selection method of regularization term is also studied. A regularization term with a second order regularization operator is introduced to propose a mixed regularization method with two parameters. The L-curve criterion, discrepancy principle, and generalized cross-validation are applied to determine the optimal value of the regularization parameter. The validity and superiority of the proposed method is verified by numerical simulation of the theoretical model.
Based numeric simulations and results from actual data processing, it is found that the newly developed A-TR regularization algorithm, HDE regularization algorithm are effective in certain conditions. The A-TR regularization algorithm is applicable to inversion problems that require non-negative parameters, whereas HDE regularization algorithm is applicable to those inversion problems whose selection of regularization parameters is difficult. Because of non-negative conductivity, A-TR algorithm will get more detailed and reliable imagery for dual-frequency conductivity inversion problems investigated by this paper.
1.研究了在实际反演中遇到参数有非负要求特性时的反演计算方法.将原问题转化为一个带非负约束的阻尼最小二乘问题,并用Active-Set算法求解.通过对理论模型进行数值模拟计算,验证了将Tikhonov正则化方法与Active-Set算法相结合的A-TR算法的有效性.应用到实际双频电导率成像反演,也取得了满意的结果.
2.研究了差分进化算法在地球物理反演中的几种应用.为加速差分进化算法的收敛速度,提出了将种群熵的自适应差分进化(ARDE)算法以及粒子群差分进化混合(PSODE)算法分别与Tikhonov正则化方法结合.在大型反演计算中,这两种方法可以在不影响反演效果的前提下,不同程度地提高收敛速度,降低时间成本.同时结合LSQR和差分进化算法的优点,提出了基于LSQR算法的差分混合(HDE)算法,避开了Tikhonov和TSVD等直接正则化算法在选取正则化参数上的困难,同时具有数值稳定性好、不依赖于初值、不易陷入局部极值和收敛速度快等优点,适宜于在正则化参数选取困难情况时的地球物理反演问题的求解.
3.提出了一种双参数混合正则化方法.引入了带有二阶正则算子的正则化项,并应用L-曲线法、偏差原理和广义交叉校验准则的优化组合确定了最佳正则化参数.数值模拟实验和实际数据处理实验结果表明了该方法的可行性.这是一种将高阶正则化算子应用于实际反演计算的新的尝试.
基于数值模拟实验和实际数据处理实验,认为研究发展的A-TR算法、HDE等算法各有其不同的适用条件,A-TR算法适用于求解反演参数有非负约束的情况,而当正则化参数选取困难时,可采用HDE算法.针对本文所考察的双频电导率反演问题,由于电导率的非负性,采用A-TR方法可得到更加精细可靠的重建图像.
Geophysical inversion is a key technique in geophysical exploration. Geophysical inversion often relates to solving ill-posed, large matrix equations. Theoretically, regularization is an effective method in dealing with ill-posed problems. Its application in practical geophysical inversion problems, however, still has many difficulties in selecting regularization parameters. Based on systematic study of regularization, this dissertation presents a few newly developed regularization methods that apply Active-Set algorithm, Differential Evolution algorithm, and a few others. These regularization methods focus on solving ill-posed matrix equations arising from practical problems. Main results of the dissertation include:
Tikhonov regularization and Active-Set algorithm are applied together to the geophysical inversion problems so that the problems with non-negative parameters are converted into problems of non-negative damped least square algorithm, which can be further solved by the Active-Set algorithm. The improved recursive algorithm is further verified by numerical simulation. Satisfactory results are obtained by applying this algorithm to electrical conductivity imagery inversion.
Furthermore, differential evolution algorithms are also studied. To improve the rate of convergence of Differential Evolution algorithms, two new Tikhonov regularization algorithms are proposed that respectively employ Adaptive Recursive Differential Evolution (ARDE) algorithm based on population entropy and Particle Swarm Optimization and Differential Evolution (PSODE) algorithm. Without any compromise in effectiveness, these two algorithms both improve the convergence speed and thus reduce computation cost. Still further, a new DE algorithm based on LSQS, which inherits the advantages of both LSQR and DE, is designed. This new algorithm avoids the common difficulty of regularization parameter selection in Tikhonov and TSVD algorithms. It also displays superior stability, independence on initial values, unlikelihood of local extrema, and faster convergence. This algorithm is particularly suitable to solve the problem of selection of regularization parameters in the study of geophysical inversion.
Finally, the selection method of regularization term is also studied. A regularization term with a second order regularization operator is introduced to propose a mixed regularization method with two parameters. The L-curve criterion, discrepancy principle, and generalized cross-validation are applied to determine the optimal value of the regularization parameter. The validity and superiority of the proposed method is verified by numerical simulation of the theoretical model.
Based numeric simulations and results from actual data processing, it is found that the newly developed A-TR regularization algorithm, HDE regularization algorithm are effective in certain conditions. The A-TR regularization algorithm is applicable to inversion problems that require non-negative parameters, whereas HDE regularization algorithm is applicable to those inversion problems whose selection of regularization parameters is difficult. Because of non-negative conductivity, A-TR algorithm will get more detailed and reliable imagery for dual-frequency conductivity inversion problems investigated by this paper.
引文
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