Target-oriented Hessian

In general, the synthetic data for one frequency, a shot positioned at ${\bf x}_s=(0,x_s,y_s)$ and a receiver positioned at ${\bf x}_r=(0,x_r,y_r)$ can be given by a linear operator ${\bf L}$ acting on the full model space ${\bf m}({\bf x})$ with ${\bf x}=(z,x,y)$ (${\bf x}=(z,x)$ in 2D) as  

$${\bf d}({\bf x}_s,{\bf x}_r;\omega) = {\bf L}{\bf m}({\bf x}... ... \, {\bf G}({\bf x},{\bf x}_r;\omega) \, {\bf m}({\bf x}),$$ (6)

where ${\bf G}({\bf x},{\bf x}_s;\omega)$ and ${\bf G}({\bf x},{\bf x}_r;\omega)$ are the Green functions from the shot position ${\bf x}_s$ and the receiver position ${\bf x}_r$ to a point in the model space ${\bf x}$, respectively.

In equation (6), we use two important properties Ehinger et al. (1996): first, the Green functions are computed by means of the one-way wave equation, and second, the extrapolation is performed by using the adequate paraxial wave equations (flux conservation) Bamberger et al. (1988).

The quadratic cost function is

$$S({\bf m}) = \sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x}_r} \Vert {\bf d} - {\bf d}_{obs} \Vert^2,$$ (7)

and its second derivative with respect to the model parameters ${\bf m}({\bf x})$ and ${\bf m}({\bf y})$ is the Hessian

$$\begin{eqnarray} {\bf H}({\bf x},{\bf y})&=& \frac{\partial^2{S({\bf m})}}{\par... ...'}({\bf x},{\bf x}_r;\omega) {\bf G}({\bf y},{\bf x}_r;\omega), \end{eqnarray}$$ (8)

where ${\bf G'}({\bf x},{\bf x}_r;\omega)$ is the adjoint of ${\bf G}({\bf x},{\bf x}_r;\omega)$.

Notice that computing ${\bf H}({\bf x},{\bf y})$ in equation (8) needs only the precomputed Green functions at model points ${\bf x}$ and ${\bf y}$. Thus, the size of the problem can be considerably reduced by computing the Green functions only at the target location ${\bf x}_T$, reducing equation (8) to  

$${\bf H}({\bf x}_T,{\bf y}_T)=\sum_{\omega} \sum_{{\bf x}_s}... ... x}_T,{\bf x}_r;\omega) {\bf G}({\bf y}_T,{\bf x}_r;\omega).$$ (9)