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本文讨论地球物理资料解释中三类典型的不适定问题,主要内容如下: 1.综述与三类典型不适定问题有关的问题及研究概况; 2.建立一个更接近物理实际的不适定问题的新提法; 3.给出三类典型不适定问题解的Hlder连续依赖性的估计式。 相似文献
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由于地球物理场的等效性、观测数据集的有限性和数据观测存在误差等问题,地球物理反演通常为不适定问题.Tikhonov正则化是应用最广泛的减少反演不适定性的方法之一,稳定函数的引入一方面提高了反演稳定性;另一方面将先验信息融入反演,提高了反演的效果.稳定函数的构建是正则化反演的关键之一,本文提出一种可以融入任意数量先验物性信息的对称多项式模型约束稳定函数.通过引入该函数,达到充分利用岩石先验物性信息、提高反演效果的目的.重力模型及实测数据反演表明,基于对称多项式的模型约束反演,可以实现任意数量岩石统计物性的利用,并获得理想的异常体边界和密度范围约束效果. 相似文献
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结合反问题技巧对一维海温模式变分资料同化的理论分析及数值试验 总被引：19，自引：0，他引：19
以一维海温模式为例, 采用变分资料同化技术及处理数学物理反问题的技巧, 对初始条件、边界条件和模式参数等进行最优估计和确定. 为了克服反问题中不适定性带来的困难, 例如对于依赖于时间和空间的模式参数的估计, 引进了反问题中的正则化思想, 在目标泛函中增加了稳定泛函, 其目的在于克服不适定和计算不稳定. 数值试验结果表明, 与采用通常的变分同化技术相比, 引进正则化思想以后的变分同化技术, 无论目标泛函的下降速度、迭代收敛速度, 还是解的精度都有较明显的改进. 相似文献
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地球表面的各向异性特性可以用地表二向反射函数（BRDF）恰当地描述.BRDF的核心是利用线性核驱动模型,数学上表述为各向同性核、体散射核和几何光学核的线性组合.随着多角度遥感领域的发展,BRDF模型越来越被看作是可以反演重要的有关地表生物的或气候的参数,比如说叶面积指数和地表反照率.一个线性逼近的核驱动BRDF模型通常可以写成下述形式（Roujean等,1992）：fiso＋kvol（ti,tv,φ）fvol＋kgeo（ti,tv,φ）fgeo=r（ti,tv,φ）,其中r表示地表的二向反射;kvol和kgeo为通常所说的核函数,即为已知的入射和观测几何特性的函数,分别描述了体散射和几何散射（包括折射和反射）;ti是太阳方向天顶角,tv是观测方向天顶角;φ表示太阳-观测方向的相对方位角;fiso,fvol和fgeo为未知的待反演参数,可以用来拟合观测.计算过程的稳定性是由核矩阵的代数算子特征谱和观测噪音/误差来刻画的.因此为了计算地表反照率,成功反演模型参数是至关重要的环节.我们首先考虑了为计算BRDF模型反演的光滑解方法.业已知道,这是一个不适定的反问题.不适定性是由线性核驱动BRDF模型的欠定性表征的,比如说观测严重不足或观测方向范围有限,或者是观测数据高度线性相关以及噪音的污染等.例如,一次单角度观测可以导致一个欠定的系统（核算子的零空间含有非零向量）或者系统无解（系数矩阵的秩不等于增广矩阵的秩）.因此,光滑性或正则化技巧应当加以利用来压制不适定性.Li等（2001）应用先验知识把原始模型转换为一个超定的模型并求得最小二乘解.Pokrovsky等（2002）应用QR分解反演BRDF模型.Wang等（2007）考虑到了反演的正则化策略并提出了不适定地表参数反演的一个完整的正则化理论.在文中,强调从不同的空间添加先验信息于反演模型中.首先从数学物理的观点,第一次提出了? 相似文献
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反问题中的最大熵方法 总被引：1，自引：0，他引：1
一、引言 1940年以前物理学家只对适定问题感兴趣。所谓适定问题是指问题的解存在、唯一,并且关于数据是稳定的。但是在对统计力学中却出现了不恰当的解释和处理。E. T. Jaynes(1957)指出统计力学问题应该看作一个不适定问题,它的解可用确定广义逆的信息论方法得出,这个见解在物理学的很多领域以及工程、医学、统计学、经济学和搜索理论中发展为处理不适定问题的普遍方法、投影重建图象是一个反问题,由于测量的不精确性,数据的不完备以及噪声等问题,而是不适定的。为了处理这些不适定问题,信息论提供了最大熵原理与最小交叉熵原理。 相似文献
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作为全局非线性优化的新方法之一的遗传算法,近年来已从生物工程流行到大地电磁测深资料解释中.然而,大地电磁反演问题具有不适定性,解的非唯一性.通过结合求解不适定问题的Tikhonov正则化方法,本文采用实数编码遗传算法求解大地电磁二维反演问题.此算法在构建目标函数时引入正则化的思想,利用遗传算法求解最优化问题.常规的基于局部线性化的最优化反演方法易使解陷入局部极小值,而且严重的依赖初始模型的选择.与传统线性化的迭代反演方法相比,实数编码遗传算法能够克服传统方法的不足且能获得更好的反演结果.通过对大地电磁测深理论模型进行计算,结果表明:该算法具有收敛速度快、解的精度高和避免出现早熟等优点,可用于大地电磁资料解释. 相似文献
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地球物理中一类构造成像问题可以用双曲型偏微分方程不适定问题来描述，这时表征地质构造的地下反射波场由这类不适定问题的解来表示。作者已撰文讨论了这类不适定问题的提法并提出了数值求解这类基本问题的逼近问题。 相似文献
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Gennady A. Kivman 《Surveys in Geophysics》1997,18(6):621-643
The problem of deriving tidal fields from observations by reason of incompleteness and imperfectness of every data set practically available has an infinitely large number of allowable solutions fitting the data within measurement errors and hence can be treated as ill-posed. Therefore, interpolating the data always relies on some a priori assumptions concerning the tides, which provide a rule of sampling or, in other words, a regularization of the ill-posed problem. Data assimilation procedures used in large scale tide modeling are viewed in a common mathematical framework as such regularizations. It is shown that they all (basis functions expansion, parameter estimation, nudging, objective analysis, general inversion, and extended general inversion), including those (objective analysis and general inversion) originally formulated in stochastic terms, may be considered as utilizations of one of the three general methods suggested by the theory of ill-posed problems. The problem of grid refinement critical for inverse methods and nudging is discussed. 相似文献
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A. S. Leonov 《Izvestiya Physics of the Solid Earth》2011,47(6):531-540
A new scheme is proposed for a posteriori estimation of the accuracy of the approximate solutions to the linear ill-posed
problems of the continuation of potential geophysical fields. When special methods are applied for the solution of the ill-posed
problems of interest, namely, the algorithms of extraoptimal regularization, these a posteriori estimates have the optimal
(in the order of magnitude) accuracy. The proposed theory is illustrated by numerical experiments. 相似文献
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Regularization methods are used to recover a unique and stable solution in ill-posed geophysical inverse problems. Due to the connection of homogeneous operators that arise in many geophysical inverse problems to the Fourier basis, for these operators classical regularization methods possess some limitations that one may try to circumvent by wavelet techniques.
In this paper, we introduce a two-step wavelet-based regularization method that combines classical regularization methods with wavelet transform to solve ill-posed linear inverse problems in geophysics. The power of the two-step wavelet-based regularization for linear inversion is twofold. First, regularization parameter choice is straightforward; it is obtained from a priori estimate of data variance. Second, in two-step wavelet-based regularization the basis can simultaneously diagonalize both the operator and the prior information about the model to be recovered. The latter is performed by wavelet-vaguelette decomposition using orthogonal symmetric fractional B-spline wavelets.
In the two-step wavelet-based regularization method, at the first step where fully classical tools are used, data is inverted for the Moore-Penrose solution of the problem, which is subsequently used as a preliminary input model for the second step. Also in this step, a model-independent estimate of data variance is made using nonparametric estimation and L-curve analysis. At the second step, wavelet-based regularization is used to partially recover the smoothness properties of the exact model from the oscillatory preliminary model.
We illustrated the efficiency of the method by applying on a synthetic vertical seismic profiling data. The results indicate that a simple non-linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods. 相似文献
In this paper, we introduce a two-step wavelet-based regularization method that combines classical regularization methods with wavelet transform to solve ill-posed linear inverse problems in geophysics. The power of the two-step wavelet-based regularization for linear inversion is twofold. First, regularization parameter choice is straightforward; it is obtained from a priori estimate of data variance. Second, in two-step wavelet-based regularization the basis can simultaneously diagonalize both the operator and the prior information about the model to be recovered. The latter is performed by wavelet-vaguelette decomposition using orthogonal symmetric fractional B-spline wavelets.
In the two-step wavelet-based regularization method, at the first step where fully classical tools are used, data is inverted for the Moore-Penrose solution of the problem, which is subsequently used as a preliminary input model for the second step. Also in this step, a model-independent estimate of data variance is made using nonparametric estimation and L-curve analysis. At the second step, wavelet-based regularization is used to partially recover the smoothness properties of the exact model from the oscillatory preliminary model.
We illustrated the efficiency of the method by applying on a synthetic vertical seismic profiling data. The results indicate that a simple non-linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods. 相似文献
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本文讨论地球内部构造反演问题的某些新途径.其内容如下:地球构造反问题与固有值反问题;反散射问题中介质间断性的成象与因果广义Radon变换,包含地球构造反问题的新提法,反散射问题的线性化,古典Radon变换与广义Radon变换,线性化反问题的渐近解,渐近解和偏移格式. 相似文献
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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. 相似文献
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Michael S. Zhdanov 《Geophysical Prospecting》2009,57(4):463-478
The interpretation of potential and electromagnetic fields observed over 3D geological structures remains one of the most challenging problems of exploration geophysics. In this paper I present an overview of novel methods of inversion and imaging of gravity and electromagnetic data, which are based on new advances in the regularization theory related to the application of special stabilizing functionals, which allow the reconstruction of both smooth images of the underground geological structures and models with sharp geological boundaries. I demonstrate that sharp-boundary geophysical inversion can improve the efficiency and resolution of the inverse problem solution. The methods are illustrated with synthetic and practical examples of the 3D inversion of potential and electromagnetic field data. 相似文献
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The anisotropy of the land surface can be best described by the bidirectional reflectance distribution function (BRDF). As
the field of multiangular remote sensing advances, it is increasingly probable that BRDF models can be inverted to estimate
the important biological or climatological parameters of the earth surface such as leaf area index and albedo. The state-of-the-art
of BRDF is the use of the linear kernel-driven models, mathematically described as the linear combination of the isotropic
kernel, volume scattering kernel and geometric optics kernel. The computational stability is characterized by the algebraic
operator spectrum of the kernel-matrix and the observation errors. Therefore, the retrieval of the model coefficients is of
great importance for computation of the land surface albedos. We first consider the smoothing solution method of the kernel-driven
BRDF models for retrieval of land surface albedos. This is known as an ill-posed inverse problem. The ill-posedness arises
from that the linear kernel driven BRDF model is usually underdetermined if there are too few looks or poor directional ranges,
or the observations are highly dependent. For example, a single angular observation may lead to an under-determined system
whose solution is infinite (the null space of the kernel operator contains nonzero vectors) or no solution (the rank of the
coefficient matrix is not equal to the augmented matrix). Therefore, some smoothing or regularization technique should be
applied to suppress the ill-posedness. So far, least squares error methods with a priori knowledge, QR decomposition method for inversion of the BRDF model and regularization theories for ill-posed inversion were
developed. In this paper, we emphasize on imposing a priori information in different spaces. We first propose a general a priori imposed regularization model problem, and then address two forms of regularization scheme. The first one is a regularized
singular value decomposition method, and then we propose a retrieval method in I
1 space. We show that the proposed method is suitable for solving land surface parameter retrieval problem if the sampling
data are poor. Numerical experiments are also given to show the efficiency of the proposed methods.
Supported by National Natural Science Foundation of China (Grant Nos. 10501051, 10871191), and Key Project of Chinese National
Programs for Fundamental Research and Development (Grant Nos. 2007CB714400, 2005CB422104) 相似文献