Gao Yang, Xiao Lizhi, Zhang Yi, Xie Qingming
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China; College of Science, China University of Petroleum, Beijing 102249, China.
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China.
J Magn Reson. 2016 Oct;271:1-6. doi: 10.1016/j.jmr.2016.07.018. Epub 2016 Jul 28.
The inversion of NMR relaxation time involves the Fredholm integral equation of the first kind. Due to its ill-posedness, numerical solutions to this type of equations are often found much less accurate and bear little resemblance to the true solution. There has been a strong interest in finding a well-posed method for this ill-posed problem since 1950s. In this paper, we prove the existence, the uniqueness, the stability and the convergence of the generalized Phillips-Twomey regularization method for solving this type of equations. Numerical simulations and core analyses arising from NMR transverse relaxation time inversion are conducted to show the effectiveness of the generalized Phillips-Twomey method. Both the simulation results and the core analyses agree well with the model and the realities.
核磁共振弛豫时间的反演涉及第一类弗雷德霍姆积分方程。由于其不适定性，这类方程的数值解往往精度低得多，与真实解几乎没有相似之处。自20世纪50年代以来，人们一直热衷于为这个不适定问题找到一种适定方法。在本文中，我们证明了广义菲利普斯 - 特沃梅正则化方法用于求解这类方程时的存在性、唯一性、稳定性和收敛性。进行了核磁共振横向弛豫时间反演的数值模拟和岩心分析，以表明广义菲利普斯 - 特沃梅方法的有效性。模拟结果和岩心分析都与模型和实际情况吻合得很好。
