Upsampling of the PCube+ Posterior Markov Chain

PCube+ represents posterior lithology-fluid class probabilities as a discrete-time
Markov chain at the sampling resolution of the input seismic data. This note presents
a method for constructing a finer posterior chain without upsampling the seismic input
or rerunning the inversion at higher resolution. Posterior marginals are interpolated
in logit space using Catmull-Rom splines. Fine-step transition matrices are approximated
from the coarse transitions while preserving legal transitions, and are subsequently
corrected to make the interpolated marginals reachable. Derived classification, thickness,
and elastic-property outputs are then calculated from the upsampled posterior.

Examples demonstrate smoother classification and elastic-property maps, together with
reduced sample-interval bias in a synthetic thickness estimate. Synthetic seismic and
residual RMS values remain comparable to those obtained from the original results,
indicating that the procedure does not significantly alter the seismic response. The
method introduces no new seismic information and should not be interpreted as equivalent
to performing PCube+ inversion at a finer resolution. It provides a computationally
inexpensive means of improving visualization and the sampling of posterior-derived
quantities.