Signal/Noise Separation and Velocity Estimation
, by William S. Harlan
To interpret noisy data one must recognize the lateral coherence of
geologic events, their statistical predictability. We define "focusing"
as increasing the statistical independence of samples. By the central
limit theorem, focusing signal with some invertible, linear transformation L
makes signal more non-gaussian; the same L must defocus noise and make
it more gaussian. A measure F, defined from cross entropy, measures
non-gaussianity from transformed data and of transformed, artificially
incoherent data provide enough information to estimate the amplitudes
distributions of transformed signal and noise--errors only increase the
estimate of noise. With these distributions one can recognize and extract
samples containing the highest percentage of signal. Iterative estimations
of signal and noise improve the extractions of each.
If we remove bed reflections an dnoise, F will determing the best migration
velocit for the remaining diffractions. Slant stacjs map lines to points,
greatly concentrating continuous reflections. We invert the strongest
extracted events and subtract them from the data, leaving diffractions and
noise. Next, we migrate with man velocities, extract focused events, and
invert. We find the least-squares sum of these events best resembling the
diffractions in the original data. Migration of these diffractions and estimate
velocities for a window of data containing a growth fault. A spatially variable
least-squares supposition allows variable velocities.
Localized slant stacks allow a laterally adaptable extraction of locally linear
events. For a stackes section we successfully extract weak signal with highly
variable coherency from behind strong gaussian noise. An more general local
transformation expresses events as a sum of short second-order curves.
An algorithm similar to that for diffractions will allow wave-equation velocity
analyses of a shot or midpoin tgather. Unlike NMP, migration will image skewed
the hyperbolas of dipping reflectors. The algorithm treats offset truncations as
another removable form of noise and escapes artifacts usually plaguing the wave-
equation method.
One may remove non-gaussian noise from shot gathers by first removing the
strongest signal, then estimating amplitude distributions for noise and remaining
signal. Without the strongest signal present, some samples may be recognized as
contianing a small percentage of signal. Extracting these events and then subtracting
from the original data removes the strongest noise. This procedure successfully removes
ground-roll and other noise from a common-shot gather.
STANFORD