## 1 Introduction  
Source rock cores in oil and gas reservoirs are routinely extracted for a better understanding of the conditions of the reservoirs. Subsurface properties, such as porosity, permeability, lithology, and other fluid-related properties, are measured from cores in the lab. Accurate measurements of these parameters are critical for the plausible estimation of reservoir productivity. However, accuracy is just one aspect of the whole story. Often, the measured data contain various types of errors from different sources such as equipment, spatial or angular coverages, and noises in the measuring environments. Some of these errors are random, and some are due to systematic reasons. In these situations, reliable and robust estimation of the fluid-related properties is the foremost important objective. Many different methods have been proposed to measure the fluid contents from the core. A thorough discussion of different measurements is beyond the focus of this paper. In this paper, we will focus on the inversion of fluid contents from a non-destructive method, high spatial resolution NMR (HSR-NMR) measurements. Low field NMR has been widely used in the oil and gas industry to study porosity, permeability, fluid saturation, and behaviors in the rock pores [1; 2; 3; 4; 5]. For readers who are interested in different approaches to measuring the fluid saturation from cores, they are referred to the paper [6]. The major elements and main procedures of HSR-NMR measurements are illustrated in Figure 1. Figure 1a shows the coil and the digitized response function \(r\), and Figure 1b shows a long rock sample and the digitized fluid content function \(a\). Figure 1c illustrates the relative position between the coil and the rock sample. The discrete \(r_{i}\) and \(a_{i}\) are defined inside the cells. Note that due to the limited size of the coil, most part of the sample is outside of the coil. The magnetic field produced by the coil shows a significant difference at the middle and on the two ends of the coil. The main features of HSR-NMR compared with the conventional approach can be summarized as follows: (1) HSR-NMR is a non-destructive technology, which means the same cores are still available for other measurements after NMR. (2) HSR-NMR allows for the same spatial resolution between digitized coils and digitized fluid contents. (3) HSR-NMR measurement can achieve the desired spatial resolution. There are many similarities between inverting fluid content from NMR data and solving inverse problems in exploration geophysics, for example, in ray-based travel time inversion for macro velocity [7, 8], velocity inversion using finite-frequency travel time [9, 10, 11], post-migration imaging velocity tomography [12], and waveform inversion [13, 14]. Extensive research has been performed, and highly effective algorithms have been developed to solve complicated model-building problems in practice [8, 13, 15, 16]. These inversion problems share similar features with the inversion of fluid content from NMR data in terms of the form \(\alpha_{1}\), \(\alpha_{2}\)  
of noise identification, mitigation of noise impacts, and inversion robustness and accuracy. Table 1 below compares some advantages and disadvantages of several most often used inversion schemes with mainly focusing on geophysical tomography and similar problems. For a comprehensive review, the readers are referred to [14]. The major contributions of this paper are outlined as follows. First, we use both numerical geophysical experiments and NMR field data to confirm that \(L_{p}\)-norm inversion can suppress the impacts of the data with large deviations. Second, quantitative values are suggested for parameter \(p\) for typical practical problems with non-Gaussian noise distribution. Third, we demonstrate that \(L_{p}\)-norm inversion can provide accurate and robust results for fluid contents using MMR data measured from the whole core, which are substantially larger than the size of the coil.