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Additional info for Case Studies in Spatial Point Process Modeling (Lecture Notes in Statistics)
Transversality is violated only when rx (xs x) = ;rx (xr x). We claim that this cannot occur when x 2 , under the hypothesis enunciated at the beginning of this section. Indeed, if r (xs x) = ;r (xr x), then the ray from xr to x, traversed backwards, is a continuation of the ray from xs to x, because both are the x-projections of the solutions of Hamilton's equations with data (x r (xs x)). In particular, points x0 on this ray near xr are touched by this \turned" ray, obviously, but also by the straight line to xs, which lies entirely in fz zdg as soon as jx0 ; xrj is small enough.
In fact: Suppose P = p(x D) is pseudodi erential, and u is a distribution, smooth near x0 2 IRn. Then Pu is smooth near x0. This statement replaces the vague \locii of high frequency components": pseudodi erential operators do not create singularities in new locations. Thus any singularities of the processed data set above are amongst the singularities of = = ]. In fact, the converse is also true, as follows from the inversion theory of the next section. Thus the locations of the singularities of = = ] are found by the above \before-stack migration" procedure.
Note the limited aperture and migration artifacts. 63 Plane 1 Trace 300 200 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 200 300 300 400 400 500 500 600 600 700 700 800 800 900 900 1000 1000 1100 1100 1200 1200 1300 1300 1400 1400 Trace 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 Plane 1 Figure 8(d): Relative velocity perturbation - Figure 3(b) as a function of 2 variables, constant in x, depending only on z. Note that the locii of rapid change line up well with those in Figure 8(c).