## 2 HSR-NMR System Equation and Issues of NMR Inversion  
In this section, we will provide a mathematical description of the relationship between the fluid contents in a core and the NMR measurements. Starting with this equation and the deviation between this theoretical relation and the actual setting up of NMR measurements, we will discuss the issues behind NMR inversion. These discussions provide physical justifications for the use of \(L_{p}\) norm inversion that will be discussed and implemented in the next sections. The acquisition of NMR data is illustrated in Figure 1a-c. The whole core is characterized by fluid content function \(a(x)\), and the coil is characterized by the response map \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Optimization Function** & & **Pros/Cons** & **Applications** \\ \hline \multirow{3}{*}{\(L_{2}\) norm} & \multirow{3}{*}{\(C(\mathbf{m})=\frac{1}{2}\|\mathbf{A}\mathbf{m}-\mathbf{\tilde{d}}\|^{2}\)} & \multirow{3}{*}{*} & Straight forward formulation and & \multirow{3}{*}{*} & Travel time and waveform inversion \\  & & & implementation. Most widely used. & & in geophysics with careful \\ \cline{3-5}  & & & Usually the first choice of inversion. & & pre-processing of data. \\ \cline{3-5}  & & & & sensitive to large data noises. & & Medical imaging and tomography. \\ \hline \multirow{3}{*}{\(L_{p}\) norm} & \multirow{3}{*}{\(C(\mathbf{m})=\frac{1}{p}\|\mathbf{A}\mathbf{m}-\mathbf{\tilde{d}}\|^{p}\)} & \multirow{3}{*}{*} & Implementation more complicated. & \multirow{3}{*}{*} & \multirow{3}{*}{*} \\  & & & Iterations are needed. & & & \\ \cline{3-5}  & & & Can handle non-Gaussian noise and & & & \\ \cline{3-5}  & & & & mitigate impacts of large data noises. & & \\ \cline{3-5}  & & & Usually gives robust and reliable results. & & & \\ \hline \multirow{3}{*}{Mixed norms} & \multirow{3}{*}{\(C(\mathbf{m})=\frac{1}{p}\|\mathbf{A}\mathbf{m}-\mathbf{\tilde{d}}\|^{p_{1}}\)} & \multirow{3}{*}{*} & Special implementations are needed. & \multirow{3}{*}{*} & Waveform inversion in geophysics \\  & & & & & with a prior information for \\ \cline{3-5}  & & & Can handle non-Gaussian noise and & & constraints. \\ \cline{3-5}  & & & & \(+\frac{1}{p_{2}}\|\mathbf{m}-\mathbf{m}_{0}\|^{p_{2}}\) & & Usually gives robust and reliable results. & \\ \cline{3-5}  & & & & Can include more a prior information. & & other sources. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of different inversion schemes. function \(r(x)\). \(x\) is the coordinate along the axis of the core and the coil, respectively. During NMR measuring (Figure 2), the core is moving to the right while the coil is at the fixed position. At each location of the core, an NMR signal is measured. Assume the fluid content in the core is digitized into \(k\) samples, and the response function is digitized into \(n\) samples. The relationship between the \(i^{th}\) sample of NMR signal \(s_{i}\) and the fluid contents \(a_{j}\) can be connected by the following discrete linear equations  
\[s_{1}=a_{1}r_{1} \tag{1}\]  
\[s_{2}=a_{1}r_{2}+a_{2}r_{1} \tag{2}\]  
\[s_{n}=\sum_{i=1}^{n}a_{m-n+i}r_{n-i+1} \tag{3}\]  
\[s_{k+n-1}=a_{k}r_{n} \tag{4}\]  
Equation (1) is when the core is at the leftmost position, and Equation (4) is when the core is at the rightmost position. Equation (3) is for general cases. In terms of \(m<n\) or \(m>k\), we assume \(a_{i}=0\) for \(i\leq 0\) and \(i>n\). To facilitate the implementation of the numerical inversion scheme, the NMR equation can be written in matrix-vector form  
\[\mathbf{Ra}=\mathbf{s} \tag{5}\]  
where \(\mathbf{R}\) is a function of response map and in the form of a variant Toeplitz matrix [17]. The dimension of \(\mathbf{R}\) is \((k+n-1)\times k\). \(\mathbf{a}\) and \(\mathbf{s}\) are vectors composed of \(a_{i},i=1,\cdots,k\) and \(s_{i},i=1,\cdots,k+n-1\), respectively. For example, in the case where \(k=9\) and \(n=5\), matrix \(\mathbf{R}\) and Equation (5) can be written as  
\[\mathbf{Ra}=\left(