Linear least-squares inversion

Tarantola (1987) formalizes the geophysical inverse problem by providing a theoretical approach to compensate for experimental deficiencies (e.g., acquisition geometry, complex overburden), while being consistent with the acquired data. His approach can be summarized as follows: given a linear modeling operator ${\bf L}$, compute synthetic data d using ${\bf d}={\bf L}{\bf m}$ where m is a reflectivity model. Given the recorded data ${\bf d}_{obs}$, a quadratic cost function,

$$S({\bf m})=\Vert {\bf d} - {\bf d}_{obs} \Vert^2 =\Vert {\bf L}{\bf m} - {\bf d}_{obs} \Vert^2,$$ (1)

is formed. The reflectivity model $\hat{{\bf m}}$ that minimizes $S({\bf m})$ is given by the following:  

$$\hat{{\bf m}}=({\bf L}'{\bf L})^{-1}{\bf L}' {\bf d}_{obs} = {\bf H}^{-1} {\bf m}_{mig},$$ (2)

where ${\bf L}'$ (migration operator) is the adjoint of the linear modeling operator ${\bf L}$, ${\bf m}_{mig}$ is the migration image, and ${\bf H}={\bf L}'{\bf L}$ is the Hessian of $S({\bf m})$.

The main difficulty with this approach is the explicit calculation of the inverse Hessian. In practice, it is more feasible to compute the least-squares inverse image as the solution of the linear system,  

$${\bf H} \hat{{\bf m}}={\bf m}_{mig},$$ (3)

by using an iterative inversion algorithm.

Equation 3 states that if we convolve the Hessian matrix ${\bf H}$ with a perfect model we should obtain the migration result ("Hessian impulse response"). In the next section we will study the approximations involved in the computation of the Hessian matrix, and will try to understand how far the "Hessian impulse response" computed using the approximated Hessian (from equation 3) is from the real migration result.

 

• Subsurface-offset Hessian
• Data fitting goal