Applications of the POCS Inversion Method

*Abstract.* We apply the method of Projection
Onto Convex Sets (POCS) to the problem of solving geophysical
inverse problems. The advantage of this iterative method is its
flexibility in handling non-linear equality and inequality
contraints, including constraints on the spectrum of unknown
functions. We give examples of using POCS to interpolate topographic
profiles, topographic maps, and the physical properties of the
earth between well logs.

*Fig 1.*(A) Every point in function space corresponds
to a different function. These functions are grouped into sets sharing
a common property, as indicated by the closed curves. A point of
intersection of the sets (in the shaded region) is found by starting
with an initial guess of the function, and then sequentially projecting
it onto the surfaces of the sets. (PostScript
Version).

*Fig 2.*(A) Synthetic field with anisotropic
spectrum. (B) Well logs through A. (C) POCS interpolation based
on well logs in B. (D) Synthetic topography with a fractal character.
(E) Smooth spline interpolation of 32x32 point, subsampled version
of D. (F) Rough POCS inversion acheived with constraint of fractal
spectrum, where the fractal dimension is estimated from the subsampled
data. (G) Smooth POCS interpolation acheived with constraint of
exponentially decaying spectrum, is similar to E. (H) Smooth spline
interpolation of ETOPO5 Rocky Mountain data acheived; (I) Rough POCS
interpolation of ETOPO5 Rocky Mountain data acheived with constraint
of fractal spectrum, where the fractal dimension is estimated from the
data. Images H and I are 1200 km across.
(Enlargements
A,
B,
C,
D,
E,
F,
G,
H,
I).

Menke, W., Applications of the POCS inversion method to interpolating topography and other geophysical fields, Geophys. Res. Lett. 18, 435-438, 1991.

POCS fractal interpolation program pocs_surface.tar.Z (WARNING. NO WARRANTY IS PROVIDED WITH THIS SOFTWARE. USE AT YOUR OWN RISK!)