位场向下延拓的数值计算方法
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摘要
位场向下延拓的研究不仅仅是在重磁资料的解译中,而且在导航方面也有着重要的作用。位场的向下延拓归结于求解一个第一类Fredholm型的积分方程,具有不适定性,如何获得稳定的、精度高、抑制噪声能力强、计算效率高的下延方法是本文的研究重点。
位场的向下延拓问题是典型的第一类褶积型线性反演问题,无论直接求解还是迭代求解实质上都是求解该问题的广义逆。而每一种方法都有自己的优缺点,比如常用的有Tikhonov正则化和Landweber迭代方法。Tikhonov方法适合比较容易求得广义逆的反问题,但存在阻尼因子敏感的问题,而Landweber方法适合难以求得广义逆的反问题,即Landweber方法通过迭代来获得反问题求解所需要的广义逆,该方法需要迭代计算效率相对较低。为了系统的研究位场的向下延拓方法,本文首先介绍了求解广义逆的直接正则化方法——奇异值分解法,由于该算法是在空间域进行计算,只能适合较少量的数据计算,当数据量较大时,必须采用频率域方法、空间域迭代算法或空间域和频率域结合的迭代算法,因此迭代法的研究是本文的重中之重。对于迭代算法的“半收敛”现象,本章结尾处通过奇异值分解的原理进行阐释,这也是本文囊括奇异值分解法的意义。
本文的迭代法包括了徐世浙院士提出的积分迭代法和已有的Tikhonov正则化迭代法,同时,也包括了本文提出的位场向下延拓的相关系数法、采用Barzilai-Borwein法求解下延方程组的快速向下延拓方法,加速的Landweber迭代法—位场向下延拓的υ半迭代法,以及包含了本文首次将krylov子空间方法(CGNR,LSQR,GMRES, MINRES, Lanczos)应用到位场的向下延拓中,实现了位场向下延拓的krylov子空间方法。另外,在位场向下延拓的krvlov子空间方法的算法实现上,快速傅里叶变换算法的引入是极为重要的一步,否则会受到数据量大的限制。
The downward continuation of potential fields plays an important role not only in gravity and magnetic data interpretation, but also in geomagnetic navigation. The downward continuation problem is essentially the solution of the Fredholem integral equation of the first kind. And the equation is ill-posed. The point of this research is how to get stability, high precision, strong anti-noise ability and efficient calculation for downward continuation of potential fields.
The field downward continuation problem is typical of the first class of convolution type linear inversion problem, whether direct solution or the iterate solution is actually to solve the problem through the generalized inverse, but every method has its advantages and disadvanta-ges.Like we commonly used Tikhonov normalization and Landweber iteration method. Tikhonov method is suitable for getting the generalized inverse problem easier, but damping factor is sensitive, Landweber method is suitable for getting the generalized inverse problem harder, in other words Landweber methods get the needed generalized inverse which is used for solving inverse problem through iteration, this method need the efficiency of iterative calculation is relatively low. In order to research potential field of downward continuation method systemat-icially, this article firstly introduces solving the direct normalization method for generalized inv-erse-the Singular Value Decompostion method, when the amount of data calculation is small, this algorithm can be used to calculate in spatial domain, while the large number data exists, we must adopt frequency domain method, the iteration algorithm for spatial domain or frequency domain and frequency domain combined with the iterative algorithm, so the iterative method research in this paper is priority among priorities. For iterative algorithm usemi-convergent "phenomenon, we interpret this phenomenon at the end of this chapter through the singular value decomposition principle, this is reason why this paper includes singular value decomposition method.
The iterative method talk about in this article included integral iteration method proposed by Shizhe Xu academician and existing Tikhonov normalization iterative method, also included field downward continuation of the correlation coefficient method proposed by this article. We realize the potential field downward continuation of krylov subspace method through using the Barzilai-Borwein method to solve the downward equations of rapid downward continuation method, accelerated Landweber iterative method-potential field downward continuation v semis-iteration method, and contains the this article firstly applied krylov subspace method (CGNR, LSQR, GMRES, MINRES, Lanczos)to field downward continuation. In addition, in application of the potential field downward continuation of krylov subspace method, introducing fast Fourier transform algorithm is a very important step,or it will be limited by the large number of data.
位场的向下延拓问题是典型的第一类褶积型线性反演问题,无论直接求解还是迭代求解实质上都是求解该问题的广义逆。而每一种方法都有自己的优缺点,比如常用的有Tikhonov正则化和Landweber迭代方法。Tikhonov方法适合比较容易求得广义逆的反问题,但存在阻尼因子敏感的问题,而Landweber方法适合难以求得广义逆的反问题,即Landweber方法通过迭代来获得反问题求解所需要的广义逆,该方法需要迭代计算效率相对较低。为了系统的研究位场的向下延拓方法,本文首先介绍了求解广义逆的直接正则化方法——奇异值分解法,由于该算法是在空间域进行计算,只能适合较少量的数据计算,当数据量较大时,必须采用频率域方法、空间域迭代算法或空间域和频率域结合的迭代算法,因此迭代法的研究是本文的重中之重。对于迭代算法的“半收敛”现象,本章结尾处通过奇异值分解的原理进行阐释,这也是本文囊括奇异值分解法的意义。
本文的迭代法包括了徐世浙院士提出的积分迭代法和已有的Tikhonov正则化迭代法,同时,也包括了本文提出的位场向下延拓的相关系数法、采用Barzilai-Borwein法求解下延方程组的快速向下延拓方法,加速的Landweber迭代法—位场向下延拓的υ半迭代法,以及包含了本文首次将krylov子空间方法(CGNR,LSQR,GMRES, MINRES, Lanczos)应用到位场的向下延拓中,实现了位场向下延拓的krylov子空间方法。另外,在位场向下延拓的krvlov子空间方法的算法实现上,快速傅里叶变换算法的引入是极为重要的一步,否则会受到数据量大的限制。
The downward continuation of potential fields plays an important role not only in gravity and magnetic data interpretation, but also in geomagnetic navigation. The downward continuation problem is essentially the solution of the Fredholem integral equation of the first kind. And the equation is ill-posed. The point of this research is how to get stability, high precision, strong anti-noise ability and efficient calculation for downward continuation of potential fields.
The field downward continuation problem is typical of the first class of convolution type linear inversion problem, whether direct solution or the iterate solution is actually to solve the problem through the generalized inverse, but every method has its advantages and disadvanta-ges.Like we commonly used Tikhonov normalization and Landweber iteration method. Tikhonov method is suitable for getting the generalized inverse problem easier, but damping factor is sensitive, Landweber method is suitable for getting the generalized inverse problem harder, in other words Landweber methods get the needed generalized inverse which is used for solving inverse problem through iteration, this method need the efficiency of iterative calculation is relatively low. In order to research potential field of downward continuation method systemat-icially, this article firstly introduces solving the direct normalization method for generalized inv-erse-the Singular Value Decompostion method, when the amount of data calculation is small, this algorithm can be used to calculate in spatial domain, while the large number data exists, we must adopt frequency domain method, the iteration algorithm for spatial domain or frequency domain and frequency domain combined with the iterative algorithm, so the iterative method research in this paper is priority among priorities. For iterative algorithm usemi-convergent "phenomenon, we interpret this phenomenon at the end of this chapter through the singular value decomposition principle, this is reason why this paper includes singular value decomposition method.
The iterative method talk about in this article included integral iteration method proposed by Shizhe Xu academician and existing Tikhonov normalization iterative method, also included field downward continuation of the correlation coefficient method proposed by this article. We realize the potential field downward continuation of krylov subspace method through using the Barzilai-Borwein method to solve the downward equations of rapid downward continuation method, accelerated Landweber iterative method-potential field downward continuation v semis-iteration method, and contains the this article firstly applied krylov subspace method (CGNR, LSQR, GMRES, MINRES, Lanczos)to field downward continuation. In addition, in application of the potential field downward continuation of krylov subspace method, introducing fast Fourier transform algorithm is a very important step,or it will be limited by the large number of data.
引文
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