The model-space inverse

Similarly to the data-space filter, computing the model-space inverse reduces to estimating an inverse for a cross-product matrix $\bf L^T \bf L$. This inverse acts as a filter for the model space. Each element $\bf M_{ij}$ of $( \bf L^T \bf L)^{-1}$ measures the correlation between a model element $\bf m_i$ and another model element $\bf m_j$. The computation of each element $\bf M_{ij}$ requires the evaluation of an inner product in the data space. Since that the data-space is irregularly sampled, the computation must be carried numerically.

The size of $\bf L^T \bf L$ is the square of the size of the model. Given that the adjoint operator, $\bf L^T$, is AMO-Stacking then the size of the model is generally much smaller than the data. This leads to a more affordable computation of $\bf M=( \bf L^T \bf L)^{-1}$ compared to the costs of computing the data-space filter.