Recently, math professor George Papanicolaou delivered a seminar to our Geophysics Department in which he presented to us an amazing proposition: Random scatterers can give us superresolution because they can enlarge the effective aperture.
I will propose it this way: Scatterers give rise to seismic coda. Maybe we can use it effectively even though we may not know the location of the scatterers.
In this paper I sketch how this could happen in the realistic case of near-surface scattering and I indicate how it could be tested and demonstrated.
Except for one essential feature, the earth model that we examine in this initial exploratory phase is a two-dimensional horizontally layered earth. The essential departure is that the top roughness that acts as point scatterers. We might think of it as fine scale surface topography. Alternately, we might think of it as a thin water layer with boulders strewn around, all acting as point scatterers of random amplitude, polarity and location. The one-dimensional earth model itself has arbitrary velocity v(z), multiple reflections, shear waves, anisotropy, etc.
Visualize this geometry:
. r1 r2 r3 r4 r5 r6 r7 r8 r9
We are interested in ray paths like the one from the shot s to the reflector, to the rock r9, to the reflector, to the geophone g. Both paths to and from the rock r9 include all arrivals, both direct and reflected. We convolve the 1-D simple earth's s to r9 response with its r9 to g response. Whatever the response is to one rock, the response to all the rocks will be the one-rock response convolved horizontally by random numbers.
There will be several conjectures. The simplest is that from a zero-offset section, spectral factorization Kolmogoroff (1939) via the helix Claerbout (1998) manufactures a common midpoint gather from which we can do velocity analysis.