Target-oriented Hessian

Since accurate imaging of reflectors is more important in the neighborhood of the reservoir, it makes sense to apply a target-oriented strategy to reduce the number of depth steps. A way to achieve this objective is to write the modeling operator ${\bf L}$ in a target-oriented fashion and explicitly compute the 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  

$${\bf d}({\bf x}_s,{\bf x}_r;\omega) = {\bf L}{\bf m}({\bf x}... ... \, {\bf G}({\bf x},{\bf x}_r;\omega) \, {\bf m}({\bf x}),$$ (4)

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 ${\bf x}$.

In equation (4), two important properties have been used 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

$$S({\bf m}) = \sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x}_r} ... ...ft[ {\bf d}({\bf x}_s,{\bf x}_r;\omega)-{\bf d}_{obs} \right],$$ (5)

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

$$\begin{eqnarray} {\bf H}({\bf x},{\bf y})&=& \frac{\partial^2{S({\bf m})}}{\par... ...'}({\bf x},{\bf x}_r;\omega) {\bf G}({\bf y},{\bf x}_r;\omega). \end{eqnarray}$$ (6) (7)

Notice that to compute ${\bf H}({\bf x},{\bf y})$ in equation (7), only the precomputed Green functions at model points ${\bf x}$ and ${\bf y}$ are needed. Thus, the size of the problem can be considerably reduced by storing the Green functions only at the target location ${\bf x}_T$. Then equation (7) reduces to

$${\bf H}({\bf x}_T,{\bf y}_T)=\sum_{\omega} \sum_{{\bf x}_s}... ...bf x}_T,{\bf x}_r;\omega) {\bf G}({\bf y}_T,{\bf x}_r;\omega),$$ (8)

where the Hessian is computed only at the target location.

In the next section we show three numerical examples of Hessians estimated with the proposed target-oriented approach.