Consider the five layer model in Figure . Each layer has unit traveltime thickness (so integration is simply summation). Let the squared interval velocities be (a,b,c,d,e) with strong reliable reflections at the base of layer c and layer e, and weak, incoherent, ``bad'' reflections at bases of (a,b,d). Thus we measure Vc2 the RMS velocity squared of the top three layers and Ve2 that for all five layers. Since we have no reflection from at the base of the fourth layer, the velocity in the fourth layer is not measured but a matter for choice. In a smooth linear fit we would want d=(c+e)/2. In a blocky fit we would want d=e.
Figure 4 A layered earth model. The layer interfaces cause reflections. Each layer has a constant velocity in its interior.
Our screen for good reflections looks like (0,0,1,0,1) and our screen for bad ones looks like the complement (1,1,0,1,0). We put these screens on the diagonals of diagonal matrices and .Our fitting goals are: