  The surface roughness of several stylolites in limestones was measured using
high resolution laser profilometry. The 1D signals obtained were statistically
analyzed to determine the scaling behavior and calculate a roughness exponent,
also called Hurst exponent. Statistical methods based on the characterization
of a single Hurst exponent imply strong assumptions on the mathematical
characteristics of the signal: the derivative of the signal (or local
increments) should be stationary and have finite variance. The analysis of the
measured stylolites show that these properties are not always verified
simultaneously. The stylolite profiles show persistence and jumps and several
stylolites are not regular, with alternating regular and irregular portions. A
new statistical method is proposed here, based on a non-stationary but Gaussian
model, to estimate the roughness of the profiles and quantify the heterogeneity
of stylolites. This statistical method is based on two parameters: the local
roughness (H) which describes the local amplitude of the stylolite, and the
amount of irregularities on the signal (\mu), which can be linked to the
heterogeneities initially present in the rock before the stylolite formed.
Using this technique, a classification of the stylolites in two families is
proposed: those for which the morphology is homogeneous everywhere and those
with alternating regular and irregular portions.
