We have derived a convenient approximation to both the data-space and the model-space inverses. It is natural to ask when we should use one and when the other. An important factor to take into account is the characteristics of LTL and LLT, and, in particular, how difficult it is to compute their approximate inverses.
The computation of each element of LTL requires the evaluation of an inner product in the data space. This is an AMO transformation from a data trace di to a data trace dj. The computation of each element of LLT is an AMO transformation between two model elements, mi and mj. The costs of the inversion is therefore determined by the relative size of the model space compared to the input data.
To suppress noise and estimate velocities, we tend to collect highly redundant data; Therefore the problem can be often considered overdetermined. The model-space inverse is usually recommended for use in overdetermined problems. However, the seismic inversion problem is often locally underdetermined where irregularities in acquisition geometry create gaps in the spatial coverage of the midpoint plane. Further, in many interesting problems, the size of the model space is not fixed, but arbitrarily determined according to the desired resolution of the resulting image. Therefore, the problem is never genuinely overdetermined, as often perceived.
Later in the discussion we show the results of two different inversions when solving a globally overdetermined, locally underdetermind problem.