An Inversion of NMR Echo Data Based on a Normalized Iterative Hard Thresholding Algorithm

The inversion of nuclear magnetic resonance (NMR) echo data requires solving the discrete Fredholm integral equation of the first kind, which is an ill-posed problem. In this letter, a surrogate objective function of an NMR inversion without an explicit regularization term based on least-squares fit...

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Bibliographic Details
Published in:IEEE geoscience and remote sensing letters Vol. 15; no. 9; pp. 1332 - 1336
Main Authors: Guo, Jiangfeng, Xie, Ranhong
Format: Journal Article
Language:English
Published: Piscataway IEEE 01.09.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
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Algorithms
Computer simulation
Data
Decomposition
Echoes
Fredholm equations
Histograms
Ill posed problems
Integral equations
Inversion
Iterative algorithms
Iterative methods
Least squares
Linear programming
Magnetic resonance
Mathematical models
NMR
normalized iterative hard thresholding algorithm
Nuclear magnetic resonance
nuclear magnetic resonance (NMR)
Numerical simulations
Objective function
Regularization
Relaxation time
Reservoirs
Sandstone
Sedimentary rocks
Signal to noise ratio
Singular value decomposition
Spectra
ISSN:1545-598X, 1558-0571
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Summary:The inversion of nuclear magnetic resonance (NMR) echo data requires solving the discrete Fredholm integral equation of the first kind, which is an ill-posed problem. In this letter, a surrogate objective function of an NMR inversion without an explicit regularization term based on least-squares fitting is introduced to avoid the process of choosing a regularization parameter, and a normalized iterative hard thresholding algorithm is proposed to solve the surrogate objective function. Furthermore, the inverted <inline-formula> <tex-math notation="LaTeX">T_{2} </tex-math></inline-formula> spectra of the proposed method, the truncated singular value decomposition method, the Butler-Reeds-Dawson method, and the least-squares QR decomposition method are compared using numerical simulation examples. The results show that the proposed method is superior to the other methods because the peaks of the inverted <inline-formula> <tex-math notation="LaTeX">T_{2} </tex-math></inline-formula> spectra with a shorter relaxation time are the most similar to the model at a low signal-to-noise ratio and the root-mean-square errors of the inverted <inline-formula> <tex-math notation="LaTeX">T_{2} </tex-math></inline-formula> spectra are the lowest. Finally, we process the NMR experimental data of tight sandstone using the four methods and verify the effectiveness of the proposed method for solving the NMR echo data inversion problem.
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ISSN:1545-598X
1558-0571
DOI:10.1109/LGRS.2018.2844411