## 3 Robust Inversion for Solving Geophysical Problems  
Similar to the NMR Equations (5) and (6), in geophysics, many inverse problems, such as travel time inversion [7,8], global full-waveform inversion [14], and amplitude inversion for absorption coefficients [18] have been investigated. These inverse problems can be formulated as the following linear relations  
\[d_{i}=\sum_{j=1}^{N}A_{ij}m_{j},i=1,\cdots,M \tag{7}\]  
where \(m_{j}\) are the discretized model variables or model perturbations, \(d_{i}\) are the discretized observations. \(A_{ij}\) is the sensitivity kernel of the \(i^{th}\) data measurement \(d_{i}\) with respect to the \(j^{th}\) model \(m_{j}\). If the measurements \(d_{i}\) are free of noises and errors, and the linear theoretical relation (7) correctly reflects the true relationship between the model and the data, Equation (7) can be solved using any standard least square-based methods [8,14,19]. In practice, the measured data may contain various types of errors from different sources such as equipment, spatial or angular coverages, and noise in the measuring environments. Some of these errors are random and can be described by Gaussian distribution, and others cannot be described as any systematic distributions. In the case where noise is present, if we assume the same linear relationship between the model and the data, the relationship between data and model can be written symbolically as  
\[\widetilde{d}_{i}=\sum_{j=1}^{N}A_{ij}m_{j}+\widetilde{n}_{i}=\sum_{j=1}^{N}A _{ij}\widetilde{m}_{j},i=1,\cdots,M \tag{8}\]  
where \(\widetilde{d}_{i}\) are the actual measurements, and the unknown \(\widetilde{n}_{i}\) are the errors of the measurements. \(\widetilde{m}_{j}\) are model parameters that can be inverted using the aforementioned standard methods when the noise is considered part of the input. However, this should be avoided since the information contained in \(\widetilde{n}_{i}\) about the model may not be useful, and quite often, indiscriminate use of \(\widetilde{n}_{i}\) as input of inversion may lead to incorrect estimation of the model. Therefore, a robust inversion, which can remove or mitigate the impact of \(\widetilde{n}_{i}\) and still recover the true \(m_{j}\) rather \(\widetilde{m}_{j}\), is expected. To mitigate the impact of the noise on the inversion, we need a cost function that is more tolerant of larger errors [20,21]. This type of inversion is called robust estimation [22]. One such option is to cast the inversion of \(m_{j}\) from (8) into the following \(L_{p}\)-norm optimization problem  
\[C(\mathbf{m})=\frac{1}{p}\sum_{i=1}^{M}\left|\sum_{j=1}^{N}A_{ij}m_{j}- \widetilde{d}_{i}\right|^{p} \tag{9}\]  
The optimal solution is to find the minimum point of (9). This is given by  
\[\frac{\partial C(\mathbf{m})}{\partial m_{k}}=0,k=1,\cdots,N \tag{10}\]  
Using (10), we have  
\[\frac{\partial C(\mathbf{m})}{\partial m_{k}}=\sum_{i=1}^{M}\left\{\left|\sum _{j=1}^{N}A_{ij}m_{j}-\widetilde{d}_{i}\right|^{p-1}\text{sgn}\left(\sum_{j=1 }^{N}A_{ij}m_{j}-\widetilde{d}_{i}\right)A_{ik}\right\} \tag{11}\]  
For \(p=2\), Equation (11) is reduced to  
\[\frac{\partial C(\mathbf{m})}{\partial m_{k}}=\sum_{i=1}^{M}\left\{\left|\sum _{j=1}^{N}A_{ij}m_{j}-\widetilde{d}_{i}\right|\text{sgn}\left(\sum_{j=1}^{N}A _{ij}m_{j}-\widetilde{d}_{i}\right)A_{ik}\right\}=\sum_{i=1}^{M}\left\{\left( \sum_{j=1}^{N}A_{ij}m_{j}-\widetilde{d}_{i}\right)A_{ik}\right\} \tag{12}\]  
In matrix-vector form, Equation (12) is [23,24]  
\[\mathbf{A}^{T}\mathbf{A}\mathbf{m}=\mathbf{A}^{T}\widetilde{\mathbf{d}} \tag{13}\]  
which is equivalent to assuming the noise distribution in Equation (8) is Gaussian [14,19].