Abstract:

The authors discussed both the cutoff method and damping method for NMR T₂ spectrum inversion, which are based on the theory of generalized inverse and singular value decomposition. At first, the solution of M-P generalized inverse of the inversion problem is presented. Then M-P generalized inverse of coefficient matrix is obtained by means of singular value decomposition. The stability and resolution of the solution are ensured by discarding some small singular values or adding damping term. Finally, the non-negativity constraint of solution is realized by iterative technique. Numerical simulation and practical application of core demonstrates：As to bimodal spectrum, when the signal to noise ratio(SNR)≥10, T₂ spectra inverted have high resolution and stability. Cutoff factors have a small range of option while damping factors have a large range of option. When SNR<10, the resolution and stability of solution are poor in cutoff method while stability is well though low resolution of short-components, high resolution of long-components in damping method. As to trimodal spectrum, when SNR≥20, T₂ spectra inverted with both methods has trimodal feature with good continuity and smoothness. When SNR<20, both methods cannot invert trimodal spectrum close to the artificial spectrum. As a result, both of the methods can be applied to the T₂ spectrum inversion of bimodal spectrum when SNR≥10 and of trimodal spectrum when SNR≥20. Taking the resolution and stability of solution, ability of smoothing noise, difficulty of selecting cutoff and damping factors into account, the damping method is a little better than cutoff method.

Key words: well logging, nuclear magnetic resonance (NMR), M-P generalized inverse, singular value decomposition, T₂ spectrum, inversion