[1] Operator linearizeation in non-linear inversion

A linear operator L depicts a physics process. It can be written as  

$$L\textbf{m}=\textbf{d},$$ (24)

where $\textbf{d}$ is the synthetic wavefield and $\textbf{m}$ is a medium model. The linear operator L can be seen as a function, which can be expanded into a Taylor series near a known model $\textbf{m}_{0}$ as follows:  

$$L\textbf{m}=L\textbf{m}_{0}+\frac{\partial L}{\partial \text... ... \textbf{m}^{n}}\left( \triangle \textbf{m}\right) ^{n}+\cdots.$$ (25)

Omitting all the terms that are higher than second-order yields a linearized equation:  

$$\frac{\partial L}{\partial \textbf{m}}\triangle \textbf{m} \approx L\textbf{m} -L\textbf{m}_{0}.$$ (26)

If $L\textbf{m}$ stands for the observed data, and if $L\textbf{m}_{0}$ synthesizes a wavefield with a known background model and a given operator, equation (26) can be rewritten as  

$$\frac{\partial L}{\partial \textbf{m}}\triangle \textbf{m} \approx \textbf{d}^{obs}- \textbf{d}^{cal}.$$ (27)

Equation (27) can be regarded as a matrix equation, which may be ill-conditioned. The model disturbance can be solved by many linear algebraic algorithms. If the background model $\textbf{m}_{0}$ is very close to the true model, the true model can be approached by some iterative algorithms. This idea is meaningful, but impractical. In fact, equation (27) can be simplified to  

$$A\delta \textbf{m}=\delta \textbf{d},$$ (28)

where $A=\partial L/\partial \textbf{m}$ and $\delta \textbf{d}= \textbf{d}^{obs}-\textbf{d}^{cal}$. Least-squares methods are then used to solve the inverse problem. Equations (9) and (15) can also be expressed in the form of equation (28). Comparing equation (28) with equations (9) and (15), it is clearly seen that the main difference between linearized inversion and non-linear waveform inversion consists in the forward modeling operator. The operator after Born approximation models only the primaries; however, the Frechèt derivative $A=\partial L/\partial \textbf{m}$ models all the wave phenomena. The Born approximation should be replaced by the De Wolf approximation. The non-linear inversion incurs much higher calculation costs.