On estimating frequency-magnitude relations from heterogeneous catalogs

The paper studies the effect of magnitude errors on heterogeneous catalogs, by applying the apparent magnitude theory (seeTinti andMulargia, 1985a), which proves to be the most natural and rigorous approach to the problem. Heterogeneities in seismic catalogs are due to a number of various sources and affect both instrumental as well as noninstrumental earthquake compilations.The most frequent basis of heterogeneity is certainly that the recent instrumental records are to be combined with the historic and prehistoric event listings to secure a time coverage, considerably longer than the recurrence time of the major earthquakes. Therefore the case which attracts the greatest attention in the present analysis is that of a catalog consisting of a subset of higher quality data, generallyS ₁, spanning the interval T ₁ (the instrumental catalog), and of a second subset of more uncertain magnitude determination, generallyS ₂, covering a vastly longer interval T ₂ (the historic and/or the geologic catalog). The magnitude threshold of the subcatalogS ₁ is supposedly smaller than that ofS ₂, which, as we will see, is one of the major causes of discrepancy between the apparent magnitude and the true magnitude distributions. We will further suppose that true magnitude occurrences conform to theGutenberg-Richter (GR) law, because the assumption simplified the analysis without reducing the relevancy of our findings.The main results are: 1) the apparent occurrence rate exceeds the true occurrence rate from a certain magnitude onward, saym _GR; 2) the apparent occurrence rate shows two distinct GR regimes separated by an intermediate transition region. The offset between the two regimes is the essential outcome ofS ₁ being heterogeneous with respect toS ₂. The most important consequences of this study are that: 1) it provides a basis to infer the parameters of the true magnitude distribution, by correcting the bias deriving from heterogeneous magnitude errors; 2) it demonstrates that the double GR decay, that several authors have taken as the incontestable proof of the failure of the GR law and of the experimental evidence of the characteristic earthquake theory, is instead perfectly consistent with a GR-type seismicity.