共查询到20条相似文献，搜索用时 31 毫秒
1.
2.
本文讨论地球物理中的某些反问题与不适定问题。其内容如下:场的延拓与偏微分方程的不适定Cauchy问题;不适定问题稳定性的性态;线性不适定问题的正则化算法。 相似文献
3.
V. N. Koneshov V. B. Nepoklonov R. A. Sermyagin E. A. Lidovskaya 《Izvestiya Physics of the Solid Earth》2014,50(1):127-136
The methods and techniques for estimating the accuracy of global models of the Earth’s gravity field in the form of spherical harmonic expansion of the geopotential are analyzed. Various methods for obtaining the a priori and a posteriori estimates for the accuracy are considered and classified. The application of different approaches is illustrated by numerical examples for nine models, including those recently developed using the modern methods of space geodesy. The basic requirements for the database and software for estimating the accuracy are formulated. 相似文献
4.
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
5.
利用伴随算子L*,直接的偏移方法通常导致一个低分辨率或模糊的地震成像.线性化偏移反演方法需求解一个最小二乘问题.但直接的最小二乘方法的数值不稳定,为目视解译带来困难.本文建立约束正则化数学模型,研究了地震偏移反演成像问题的迭代正则化求解方法.首先对最小二乘问题施加正则化约束,接着利用梯度迭代法求解反演成像问题,特别是提出了共轭梯度方法的混合实现技巧.为了表征该方法的可实际利用性,分别对一维,二维和三维地震模型进行了数值模拟.结果表明该正则偏移反演成像方法是有效的,对于实际的地震成像问题有着良好的应用前景. 相似文献
6.
Axelle Viré Jiansheng Xiang Frank Milthaler Patrick Emmet Farrell Matthew David Piggott John-Paul Latham Dimitrios Pavlidis Christopher Charles Pain 《Ocean Dynamics》2012,62(10-12):1487-1501
Fluid–structure interactions are modelled by coupling the finite element fluid/ocean model ‘Fluidity-ICOM’ with a combined finite–discrete element solid model ‘Y3D’. Because separate meshes are used for the fluids and solids, the present method is flexible in terms of discretisation schemes used for each material. Also, it can tackle multiple solids impacting on one another, without having ill-posed problems in the resolution of the fluid’s equations. Importantly, the proposed approach ensures that Newton’s third law is satisfied at the discrete level. This is done by first computing the action–reaction force on a supermesh, i.e. a function superspace of the fluid and solid meshes, and then projecting it to both meshes to use it as a source term in the fluid and solid equations. This paper demonstrates the properties of spatial conservation and accuracy of the method for a sphere immersed in a fluid, with prescribed fluid and solid velocities. While spatial conservation is shown to be independent of the mesh resolutions, accuracy requires fine resolutions in both fluid and solid meshes. It is further highlighted that unstructured meshes adapted to the solid concentration field reduce the numerical errors, in comparison with uniformly structured meshes with the same number of elements. The method is verified on flow past a falling sphere. Its potential for ocean applications is further shown through the simulation of vortex-induced vibrations of two cylinders and the flow past two flexible fibres. 相似文献
7.
The maximum entropy principle of the information theory gives rise to a general regularization strategy for ill-posed inverse problems. The methods based on this principle have become standard in various branches of engineering sciences. Of course, ill-posed problems frequently appear in Earth sciences, too. Nonetheless, the concept of maximum entropy is not very popular here. Therefore, we review the basic approaches employing the principle of maximum entropy in one way or another. We can distinguish at least three different approaches, partly yielding coincident results.
One possible area of application is the determination of Earth and planetary models, although the paper cannot treat this in its practical complexity. Most of the discussion is restricted to the determination of the Earth's mass density function from various sources of data. Three sample problems are solved using the principle of maximum entropy: a spherical and an ellipsoidal problem related to the Earth and an ellipsoidal problem related to Mars. This illustrates the numerical procedure, which is non-trivial in many cases. It also shows some results, partly compared to standard solutions. The pros and cons of the approaches are discussed. 相似文献
8.
高频天波返回散射探测作为重要的电离层探测手段,能够实现遥远区域电离层空间上的连续监测,探测获取的返回散射扫频电离图显示了探测频率-群路径-回波能量三者之间的关系.由于电离图包含了探测路径上的电离层状态信息,通过对其反演可以实时获取大面积范围的电离层参数.本文提出了一种基于解空间约束的返回散射前沿反演算法,能够重构电离层水平不均匀结构.针对反演非线性问题,采用Newton-Kontorovich方法进行求解,同时又引入了求解不适定问题的Tikhonov正则化方法,有益于解的稳定性和唯一性.利用模拟数据和实测数据分别对本文建立的算法进行了验证,并与Fridman和Fridman于1994年提出的反演方法进行了对比.结果表明,本文算法反演结果稳定,对返回散射前沿判读误差不敏感,与Fridman和Fridman 1994年方法相比,本文方法对电离层局部精细结构反演更加准确,具有较高的反演精度.本文提出的算法不但能够反演白天和夜间这种电离层较平稳时期的电离层状态,而且对于日出/日落时段等电子浓度分布变化较快情形下的电离层,也有很好的反演效果,表明了该算法在处理复杂多变的实际探测的返回散射电离图中的应用价值. 相似文献
9.
煤矿井下微震震源准确定位,对于动力灾害监测预警具有重要意义.由于微震震源需要通过井下传感器接收信息反演确定,传感器的安装位置限制于煤矿井下巷道周围,传感器沿巷道近平面的不合理布置将大大降低震源定位精度.针对由传感器信息反演震源位置引起的病态问题,本文提出了基于微震监测测点优化布置的震源高精度定位算法.首先通过计算系数矩阵条件数,判定病态问题;然后利用中心化法和行平衡法联合进行病态矩阵预处理.对预处理后的矩阵A、b利用L曲线法计算正则参数,结合Tikhonov正则化算法计算得到震源坐标正则解.研究结果表明,中心化法有效降低了矩阵数量级,行平衡预处理降低了病态条件数,预处理后Tikhonov正则解的震源坐标误差最小可以达到3.09m,与预处理前的高斯消去解相比误差大大降低.通过上述优化处理,实现了井下受限空间微震监测震源高精度定位. 相似文献
10.
Vir&#; Axelle Xiang Jiansheng Milthaler Frank Farrell Patrick Emmet Piggott Matthew David Latham John-Paul Pavlidis Dimitrios Pain Christopher Charles 《Ocean Dynamics》2012,62(10):1487-1501
Fluid–structure interactions are modelled by coupling the finite element fluid/ocean model ‘Fluidity-ICOM’ with a combined finite–discrete element solid model ‘Y3D’. Because separate meshes are used for the fluids and solids, the present method is flexible in terms of discretisation schemes used for each material. Also, it can tackle multiple solids impacting on one another, without having ill-posed problems in the resolution of the fluid’s equations. Importantly, the proposed approach ensures that Newton’s third law is satisfied at the discrete level. This is done by first computing the action–reaction force on a supermesh, i.e. a function superspace of the fluid and solid meshes, and then projecting it to both meshes to use it as a source term in the fluid and solid equations. This paper demonstrates the properties of spatial conservation and accuracy of the method for a sphere immersed in a fluid, with prescribed fluid and solid velocities. While spatial conservation is shown to be independent of the mesh resolutions, accuracy requires fine resolutions in both fluid and solid meshes. It is further highlighted that unstructured meshes adapted to the solid concentration field reduce the numerical errors, in comparison with uniformly structured meshes with the same number of elements. The method is verified on flow past a falling sphere. Its potential for ocean applications is further shown through the simulation of vortex-induced vibrations of two cylinders and the flow past two flexible fibres.
相似文献11.
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
12.
地球物理反演是获取地球信息的重要手段,其求解具有严重的不适定性.为获得稳定的反问题结果,通常需要在目标泛函中加入正则化约束项.正确地估计正则化参数一直是地球物理反问题中的难点.目前存在的选取方法需要根据大量的试验来确定正则化参数,工作量十分巨大,并且存在很大的经验性,很难得到最优的正则化参数.针对这个问题,本文提出了一种基于广义Stein无偏风险估计的正则化参数求取方法.该方法的具体思路是通过求解模型参数均方误差的广义Stein无偏风险估计函数,在反问题求解过程中自动求取正则化参数.本文模型测试结果表明,相比于目前常用的方法,通过该方法得到的正则化参数是最优的. 相似文献
13.
常规AVO三参数反演是通过Zoeppritz方程的近似公式来建立AVO正演模拟的过程,然而在P波入射角过临界角和弹性参数在纵向上变化剧烈的情况下,Zoeppritz方程近似公式精度有限.针对这种情况,可以使用精确的Zoeppritz方程来构建反演目标函数,由于精确Zoeppritz方程中P波反射系数和弹性参数之间是一种复杂的非线性关系,通常解决途径是利用非线性的优化算法来进行数值计算,但是非线性优化算法的缺点是计算量过大;另外一种途径是利用广义线性反演的方法,通过泰勒一阶展开式将P波反射振幅展开后,用线性关系近似表达非线性关系,经过几次迭代后,在理论上可以达到很高的精度,但是广义线性反演算法的核心部分--Jacobian矩阵由于矩阵条件数过大,往往会造成反演算法的不稳定,其应用范围得到了限制.贝叶斯反演方法是通过引入模型参数的先验分布结合噪声的似然函数,生成模型参数的后验分布,通过求取模型参数的最大后验概率分布来得到模型参数的反演解,由于引入模型参数的先验分布信息,可以有效的降低反演的不适定问题.本文将两种反演算法的思想相结合,利用广义线性反演算法的思想,构建AVO正演模拟的过程来提高大角度地震数据反演的精度,同时结合贝叶斯理论,通过引入模型参数的先验分布信息构建反演目标函数的正则化项,可以有效降低由于Jacob矩阵条件数过大带来的反演不适定问题,该算法假设模型参数服从三变量柯西分布. 相似文献
14.
C. R. DIETRICH 《水文研究》1996,10(6):893-902
The convolution assumption between excess rainfall and runoff provides a framework in which catchment runoff can be predicted with reasonable accuracy and moderate computational cost. Associated with it, the deconvolution problem of estimating unitgraph ordinates from rainfall–runoff events involves a matrix with a particularly simple structure. This matrix structure is used here as a basis on which the ill-posed nature of deconvolution is analysed. As a result, based on a simple transform of the excess rainfall data, a very simple criterion is derived to test the degree to which deconvolution may yield a unit hydrograph estimate displaying spurious oscillations of large magnitude. This has practical implications as the solution to an ill-posed problem can be very sensitive to errors in the model and the data and therefore may need to be stabilized. Illustration of these issues is provided using published rainfall–runoff data. 相似文献
15.
各向异性介质中利用跨孔qP波和qSV波走时的统计法成像 总被引：2，自引：1，他引：1
跨孔层析成像通常在各向同性条件下进行,地壳中存在着广泛的各向异性,考虑各向异性介质的成像具有重要的理论意义和实际价值,本文以各向异性介质的走时扰动理论为基础,进行了qP波和qSV波跨孔走时的联合成像,与各向同性情形相比,各向异性介质的成像是一个更为病态的问题。本文首次将正则化方法与统计检验理论相结合,提出了统计法跨孔成像。数值模拟结果表明,此法具有精度良好、计算稳定和较少受初始模型限制的优点,并可进一步推广到其他地球物理反问题求解。 相似文献
16.
17.
18.
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
19.
A pseudo-seismic interpretation method is an alternative way to process and explain transient electromagnetic (TEM) data, and has become a popular research field in recent years. TEM signals which satisfy the diffusion equation can be converted by means of a mathematical transformation into ones which obey the wave equation. For an ill-posed problem of this kind of transformation, a sub-regularization algorithm is developed in this paper to extract a virtual wavelet of the TEM field. According to the conventional designation of TEM recordings, the entire integration period is divided into seven time intervals. In order to avoid low accuracy in the calculations, high-density wavefield data has been calculated based on the former sub-division. Therefore, the virtual wavelet can be extracted successfully by using an optimized algorithm to obtain high-density integral coefficients for all time windows, and a satisfactory condition number of the coefficient matrix while taking a different channel number in each time period. The Tikhonov regularization inversion scheme is used to determine the optimal parameters based on minimizing a least squares misfit, and the Newton iterative formula is used to obtain optimal regularization parameters. Both synthetic model simulations and a real data interpretation example indicate that the proposed pseudo-seismic wavefield method is a suitable alternative way to interpret TEM data. 相似文献
20.
本文讨论了线性广义反演方法对位场延拓问题的应用。如果考虑能量有限的约束,应用拉格朗日乘子法便可得到与随机逆相同的反演公式（Franklin）,而不必假定模型为高斯白噪的随机过程。对反演算子进行谱分解之后,拉格朗日乘子起到折衷因子的作用,因此改变拉格朗日乘子的值便可找到在分辨率和误差之间取最佳折衷的反问题数值解。将这种广义反演技术用于位场从任意曲面向下延拓到源顶面,给出了较好的结果。与BG方法相比,位场向下延拓结果的精度相近,但计算速度可以提高。 相似文献