Order parameter fluctuations in natural time and b-value variation before large earthquakes

Self-similarity may stem from two origins: the process' increments infinite variance and/or process' memory. The $b$-value of the Gutenberg-Richter law comes from the first origin. In the frame of natural time analysis of earthquake data, a fall of the b-value observed before large earthquakes reflects an increase of the order parameter fluctuations upon approaching the critical point (mainshock). The increase of these fluctuations, however, is also influenced from the second origin of self-similarity, i.e., temporal correlations between earthquake magnitudes. This is supported by observations and simulations of an earthquake model.