Towards a Physical Understanding of the Earthquake Frequency Distribution

Summary In this paper the cumulative frequency‐magnitude relationship is replaced by the cumulative frequency‐moment equation in which the B ‐value takes the place of the b ‐valuewhere the constants α and B can be observed or derived from the magnitude‐moment and frequency‐magnitude relationships. The average and maximal moments of a given set of earthquakes are found to be directly related to B and α respectively:The total cumulative moment of an earthquake sequence can be expressed aswhich is approximately equal to twice M 0 (max) for most observed sets of earthquakes. Using the definition of the moment we then derive the general area frequency relationshipwhich shows that B gives the distribution of the product of average displacement D and fault area A . We may reformulate this in terms of the product of stress‐drop Δσ and fault areaIf we make the assumption that the stress‐brop is a known function of the source dimension,which can be verified for a given sequence, we may express the frequency of earthquake occurrence as a function of one source parameterfrom which we can obtain the mean rupture area of a set as a function of B , γ and the smallest area in the setAlternatively we may eliminate the area and obtainFrom this we may calculate the ratio of the average stress‐drops of two sets with different B . From the different b ‐values of Denver earthquakes during low and high injection pressure the stress‐drop is computed as 30 per cent higher at low‐pore pressure. This difference is in good agreement with the difference of the respectively necessary failure stresses, which is 14 per cent. In addition high apparent stresses and high stress‐drop Δσ were found to correlate with low b ‐ and low B ‐values, as they do in microfracture experiments. The combination of high average , high mean Δσ, large mean A and low b ‐ or low B ‐value can be explained by high regional shear stress.