The reality of the scaling law of earthquake-source spectra?

Attempts to build a “constant-stress-drop” scaling of an earthquake-source spectrum have invariably met with difficulties. Physically, such a scaling would mean that the low-frequency content of the spectrum would control the high-frequency one, reducing the number of the parameters governing the time history of a shear dislocation to one. This is technically achieved through relationships of the corner frequency of the spectrum to the fault size, inevitably introduced in an arbitrary manner using a constant termed “stress drop”. Throughout decades of observations, this quantity has never proved to be constant. This fact has fundamental physical reasons. The dislocation motion is controlled by two independent parameters: the final static offset and the speed at which it is reached. The former controls the low-frequency asymptote of the spectrum while the latter its high-frequency content. There is no physical reason to believe that the static displacement should predetermine the slip rate, which would be implied if the “stress drop” were constant. Reducing the two parameters to just one (the seismic moment or magnitude) in a “scaling law” has no strict justification; this would necessarily involve arbitrary assumptions about the relationship of one parameter to the other. This explains why the “constant-stress-drop” scaling in seismology has been believed in but never reconciled with the data.