The data-space inverse

To estimate the data-space inverse we need to approximate an inverse for the cross product matrix LL^T. This inverse acts as a filter for the data space. Each element $\bf D_{ij}$ of $(\bf L \bf L^T)^{-1}$measures the correlation between a data element $\bf d_i$ and another data element $\bf d_j$. The computation of each element $\bf D_{ij}$ requires the evaluation of an inner product in the model space. Since the model space is regularly sampled, the inner products can then be computed analytically (pending available representation of the chained operator, $\bf L \bf L^T$).

Considering an irregularly sampled input of n seismic traces, the cross-product operator LL^T can be expressed in matrix notations as:

$$\bf L \bf L^T = \left[ \begin{array} {cccc} \left[ L_{(m,d... ..._n)}L^T_{(m,d_n)}\right] \end{array} \right] \EQNLABEL{equ6}$$ (49)

Each inner product $\left[ L_{(m,d_i)}L^T_{(m,d_j)}\right]$ is a reconstruction of a data trace with offset $\bf h_i$ to a new trace with offset $\bf h_j$. Therefore the mapping is an AMO transformation, and ${\bf D}$ can be written as

$${\bf D}= \left[ \begin{array} {ccccc} I & D_{(h_1,h_2)} & ... ...D_{(h_n,h_3)} &...... & I \end{array} \right] \EQNLABEL{equ7}$$ (50)

where $\bf D_{(h_i,h_j)}$ is AMO from input offset $\bf h_i$ to output offset $\bf h_j$and, $\bf I$ is the identity operator (mapping from $\bf h_i$ to $\bf h_i$). Conforming to the definition of AMO, $\bf D_{(h_i,h_j)}$ is the adjoint of $\bf D_{(h_j,h_i)}$; therefore, the filter ${\bf D}$ is Hermitian with diagonal elements being the identity and off-diagonal elements being trace to trace AMO transforms. This is a fundamental definition of D that will allow a fast and efficient numerical approximation of its inverse.

The data-space inverse can then be expressed as a two-step solution where the data is first filtered with the inverse of the operator D then the adjoint is applied to the filtered data to solve for a model. The solution for ${\bf m}$ from equation equ3 can be written in terms of D as:

$$\bf m= \bf L^T {\bf D}^{-1} \bf d \EQNLABEL{equ8}$$ (51)

Now, changing the problem formulation variable ${\bf d}$ to $\hat{ \bf d}$, where

$$\hat{ \bf d}={\bf D}^{-1} \bf d \EQNLABEL{equ9}$$ (52)

and recasting the problem as

$$\bf m= \bf L^T \hat{ \bf d} \EQNLABEL{equ8}$$ (53)

we need now to solve for $\hat{ \bf d}$ by computing the inverse of ${\bf D}$ from the system of equations:

$$\bf d={\bf D} \hat{ \bf d} \EQNLABEL{equ10}$$ (54)

Once the inverse of ${\bf D}$ is estimated to yield the filtered data $\hat{ \bf d}$, we merely solve for the initial model $\bf m= \bf L^T \hat{ \bf d}$.

Notice that after filtering, we can apply any imaging operator $\bf L^T$ to invert for ${\bf m}$.