Thisstudytestedfivemethodswidelyusedinestimatingthecompletemagnitudes(MC)ofearthquakecatalogs.Usingcatalogsofobservedearthquakeproperties,wetesttheperformanceofthesefivealgorithmsunderseveralchallengingconditions,suchassmallvolumeofeventsandspatial-temporalheterogeneity,inordertoseewhetherthealgorithmsarestableandinagreementwithknowndata.Wefindthatthemaximumcurvaturemethod(MAXC)hasperfectstability,butwillsignificantlyunderestimateMCunlessheterogeneityisabsent.MCestimatedbytheb-valuestabilitymethod(MBS)requiresmanyeventstoreachastableresult.Resultsfromthegoodnessoffitmethod(GFT)wereunstablewhenheterogeneityloweredthefitnesslevel.Theentiremagnituderangemethod(EMR)isrelativelystableinmostconditions,andcanreflectthechangeinMCwhenheterogeneityexists,butwhentheincompletepartoftheearthquakecatalogisdismissed,thismethodfails.Themedian-basedanalysisofthesegmentslopemethod(MBASS)cantoleratesmallsamplesize,butisincapableofreflectingthemissingdegreeofsmalleventsinaftershocksequences.InconditionswhereMCchangesrapidly,suchasinaftershocksequences,observingthetimesequencedirectlycangiveapreciseestimationofthecompletesub-catalog,butonlywhenthenumberofeventsavailableforstudyislargeenoughcantheMAXC,GFT,andMBSmethodsgiveasimilarlyreliableestimation.
