## 3 Robust Inversion for Solving Geophysical Problems
For \(p<2\), we can convert (11) into a similar form as (13),  
\[\frac{\partial C(\mathbf{m})}{\partial m_{k}}=\sum_{i=1}^{M}\left\{\left|\sum _{j=1}^{N}A_{ij}m_{j}-\widetilde{d}_{i}\right|^{p-2}\left(\sum_{j=1}^{N}A_{ ij}m_{j}-\widetilde{d}_{i}\right)A_{ik}\right\} \tag{14}\]  
Equation (14) can be written as  
\[(\mathbf{D}\mathbf{A})^{T}(\mathbf{D}\mathbf{A})\mathbf{m}=(\mathbf{D} \mathbf{A})^{T}\widetilde{\mathbf{d}} \tag{15}\]  
where \(\mathbf{D}\) is a diagonal matrix,  
\[D_{ii}=\sqrt{W_{ii}} \tag{16}\]  
and  
\[\widetilde{d}_{i}=\sum_{j=1}^{N}A_{ij}m_{j}+\widetilde{n}_{i}=\sum_{j=1}^{N}A _{ij}\widetilde{m}_{j},i=1,\cdots,M \tag{8}\]0  
\[\widetilde{d}_{i}=\sum_{j=1}^{N}A_{ij}m_{j}+\widetilde{n}_{i}=\sum_{j=1}^{N}A _{ij}\widetilde{m}_{j},i=1,\cdots,M \tag{8}\]1  
Comparing Equation (15) with Equation (13), we can see that the optimal solution of the \(L_{p}\)-norm optimization is the solution to the \(L_{2}\) optimization, with the original matrix being weighted by the inverse of fitting errors. Therefore, the \(L_{p}\) norm is a nonlinear inversion with the operator changed with each iteration when the model is updated. This is a very desired property of the \(L_{p}\)-norm inversion, which can be seen from the following empirical analysis. We consider one scenario in which the error of an input data point, for example \(r_{k}\), is exceptionally large. Intuitively, this means \(\vec{d}_{k}\) deviates farther away from the trend of most other measurements. For the case of \(p=2\), all the data point residuals are weighted equally. Therefore, all input data \(d_{k}\) are treated equally. For, \(p<2\), the \(k^{th}\) row of **DA** and the \(k^{th}\) point of \(\widetilde{\textbf{Dd}}\) are scaled by a factor of \(\frac{1}{\left|r_{i}\right|^{2-p}}\) from **A** and \(\widetilde{\textbf{d}}\), respectively. In the extreme case of \(\left|r_{i}\right|\rightarrow\infty\), the \(k^{th}\) row of **DA** and the \(k^{th}\) point of \(\widetilde{\textbf{Dd}}\) are zeroes. Therefore, the impact of the \(k^{th}\) data is completely removed. It can be proven that the residual due to the final solution of \(L_{1}\) norm optimization is the median value of the residuals of all input measurements. Therefore, as long as more than half the values of the data inputs are normal, the \(L_{1}\) norm optimization can still provide reasonable estimations for the model. In order to focus on the physical aspect of inversion problems, the details of workflows and numerical procedures are given in the next section. Figure 3 shows the cost functions, their first-order derivatives, and the weights for different values of \(p\). The first-order derivative, which plays the same role as the influence function [25], needs to increase not so rapidly for large errors for the sake of robustness. From Figure 3c, we can see that as \(p\) decreases, Equation (15) puts more and more weight on the small value of residuals, the residuals of the data measurements with smaller deviations. As the errors of data measurements increase, for small values of \(p\), the weight decreases quickly, therefore reducing their impact on the inversion. This is contrary to the case of \(p=2\), where the weight is one. \[\widetilde{d}_{i}=\sum_{j=1}^{N}A_{ij}m_{j}+\widetilde{n}_{i}=\sum_{j=1}^{N}A _{ij}\widetilde{m}_{j},i=1,\cdots,M \tag{8}\]2 _Cont._  
As an example for demonstrating the effectiveness of \(L_{p}\) norm (\(p<2\))-based optimization for suppressing the impact of input data with markedly large errors, we show below a typical geophysical problem, the inversion of seismic wave velocity from travel time picks. Figure 4a below shows a simple model and travel time acquisition patterns. The seismic velocity in this model is homogeneous with the exact value of \(v=1\) m/s. The source points are located on a horizon in the model. The receivers are distributed along the surface at \(z=0\) m. The solid black lines from a source to a receiver point are rays. Since the velocity model is homogeneous, the rays are straight lines.