Low-field nuclear magnetic resonance (NMR) is a widely employed technique in geoscience. However, signal-to-noise ratio (SNR) is always an issue in low-field NMR measurement, which should be carefully addressed to ensure the accuracy of relaxation spectrum reconstruction for subsequent petro-physical interpretation and applications. This paper presents a novel denoising method for low-field NMR measurements utilizing double sparsity dictionary learning (DSDL), which is an unsupervised machine learning approach. The elaborate trained dictionary models could be directly implemented to denoise raw spin echoes with different signal-to-noise ratios (SNRs). After denoising, digital phase-sensitive detection (DPSD) and phase rotation are conducted to obtain the multi-exponential decay signals and fundamental noise signals for subsequent spectrum reconstruction. In this study, numerical simulations is mainly conducted. The pre-set T2 spectrum models are built to derive raw spin echoes with different SNRs through forward modeling, and then are used to train the double sparsity dictionary models. The dictionary models are trained on raw echo datasets with different Gaussian distributed noise and same porosity, and tested on raw echo datasets with same Gaussian noise and different porosities. The echo data before and after denosing are all inverted by using Singular Value Decomposition (SVD) method, which is a non-objective inversion algorithm to avoid the parameter selection like commonly used regularization inversion algorithm. All the inverted results are compared with the forwarding spectrum models. It is demonstrated that the DSDL method could effectively improve the quality of low-field NMR measurements, resulting in accurate relaxation spectrum.

To tackle the noisy label problem, sample selection methods have demonstrated particular advantages. An emerging method, named Tripartite, was proposed to divide the data into clean, uncertain, and noisy subsets and applies a low-weight training strategy to the uncertain subset to alleviate the harm of noisy labels. However, we find the uncertain subset selected by the Tripartite contains many noisy samples as well, which considerably lowers the performance of models even when using low-weight training. Therefore, we propose the Quadripartite, which further divides the uncertain subset into an uncertain clean and an uncertain noisy subset. The data visualization shows that these two new subsets have a high sample selection quality. It addresses a tough challenge that most sample selection methods face: how to reliably select the hard clean samples. The Quadripartite employs a weighted learning on the uncertain clean subset to maximize the utilization of clean samples, while dropping the uncertain noisy subset during the training stage to mitigate the negative effects caused by noisy samples. Extensive experiments show Quadripartite outperforms the state-of-the-art approaches on four benchmark datasets.