Linear least squares inversion

Tarantola (1987) formalizes the geophysical inverse problem from a Bayesian point of view. He gives a theoretical background to compensate for the reflection experiment's deficiencies (acquisition geometry, obstacles, etc.), while being consistent with the acquired data.

Under this scheme, data and model (subsurface image) are assumed to have Gaussian distribution (a priori data and model probability density are Gaussian). The forward operator is assumed to be linear (or weakly non-linear). The resulting subsurface image is the mean of a posterior Gaussian probability density in the model space. Its expression for a general linear (forward modeling) operator ${\bf L}$ is  

$$\hat{{\bf m}}=({\bf L}^t {\bf C}_D^{-1}{\bf L}+ {\bf C}_M^{-... ... {\bf C}_D^{-1}{\bf d}_{obs}+{\bf C}_M^{-1}{\bf m}_{prior}),$$ (1)

where $\hat{{\bf m}}$ is the mean of a posterior Gaussian probability density, ${\bf L}^t$ is the transpose of the linear forward operator, ${\bf C}_D$ is the data covariance, ${\bf C}_M$ is the model covariance, ${\bf d}_{obs}$ is the measured data and ${\bf m}_{prior}$ is the prior model.

Equation (1) can be solved by using gradient-based methods like steepest descent. This iterative algorithm can be written in the form $m_{n+1}=m_n+\delta m_n$ Tarantola (1987), where the term $\delta m_n$ depends on the gradient of the function to be optimized, a metric in the model space, and an ad-hoc constant.

 

• Target-oriented least squares inversion strategy