Yan Kagan grew up and was educated in Moscow, Russia. In 1974 he came to UCLA, and working with Leon Knopoff, David Jackson, Peter Bird, and Frederick Schoenberg applied his mathematical/statistical model to seismicity analysis. Since 1999 these results have been used to produce daily earthquake forecasts for several seismically active regions and currently for the whole Earth. The performance and predictive skill of these forecasts is now being tested by several research groups.

• Preface xiiiAcknowledgments xviiList of Abbreviations xixList of Mathematical Symbols xxiPART I MODELS 11 Motivation: Earthquake science challenges 32 Seismological background 62.1 Earthquakes 62.2 Earthquake catalogs 82.3 Description of modern earthquake catalogs 112.4 Earthquake temporal occurrence: quasi-periodic, Poisson, or clustered? 142.5 Earthquake faults: one fault, several faults, or an infinite number of faults? 162.6 Statistical and physical models of seismicity 182.7 Laboratory and theoretical studies of fracture 193 Stochastic processes and earthquake occurrence models 213.1 Earthquake clustering and branching processes 213.2 Several problems and challenges 243.3 Critical continuum-state branching model of earthquake rupture 26PART II STATISTICS 314 Statistical distributions of earthquake numbers: Consequence of branching process 334.1 Theoretical considerations 344.2 Observed earthquake numbers distribution 435 Earthquake size distribution 545.1 Magnitude versus seismic moment 545.2 Seismic moment distribution 565.3 Is 𝛽 ≡ 1¨M2? 605.4 Seismic moment sum distribution 80 5.5 Length of aftershock zone (earthquake spatial scaling) 865.6 Maximum or corner magnitude: 2004 Sumatra and 2011 Tohoku mega-earthquakes 906 Temporal earthquake distribution 966.1 Omori’s law 966.2 Seismic moment release in earthquakes and aftershocks 976.3 Random shear stress and Omori’s law 1076.4 Aftershock temporal distribution, theoretical analysis 1106.5 Temporal distribution of aftershocks: Observations 1166.6 Example: The New Madrid earthquake sequence of 1811–12 1216.7 Conclusion 1237 Earthquake location distribution 1257.1 Multipoint spatial statistical moments 1257.2 Sources of error and bias in estimating the correlation dimension 1277.3 Correlation dimension for earthquake catalogs 1417.4 Conclusion 1458 Focal mechanism orientation and source complexity 1468.1 Random stress tensor and seismic moment tensor 1478.2 Geometric complexity of earthquake focal zone and fault systems 1508.3 Rotation of double-couple (DC) earthquake moment tensor and quaternions 1548.4 Focal mechanism symmetry 1598.5 Earthquake focal mechanism and crystallographic texture statistics 1638.6 Rotation angle distributions 1678.7 Focal mechanisms statistics 1708.8 Models for complex earthquake sources 177PART III TESTABLE FORECASTS 1839 Global earthquake patterns 1859.1 Earthquake time-space patterns 1859.2 Defining global tectonic zones 1879.3 Corner magnitudes in the tectonic zones 1889.4 Critical branching model (CBM) of earthquake occurrence 1909.5 Likelihood analysis of catalogs 1979.6 Results of the catalogs’ statistical analysis 20410 Long- and short-term earthquake forecasting 20610.1 Phenomenological branching models and earthquake occurrence estimation 20610.2 Long-term rate density estimates 20710.3 Short-term forecasts 21510.4 Example: earthquake forecasts during the Tohoku sequence 21810.5 Forecast results and their discussion 22410.6 Earthquake fault propagation modeling and earthquake rate estimation 22611 Testing long-term earthquake forecasts: Likelihood methods and error diagrams 22911.1 Preamble 22911.2 Log-likelihood and information score 23011.3 Error diagram (ED) 23511.4 Tests and optimization for global high-resolution forecasts 24711.5 Summary of testing results 25012 Future prospects and problems 25312.1 Community efforts for statistical seismicity analysis and earthquake forecast testing 25312.2 Results and challenges 25412.3 Future developments 256References 260Index 281

Yan Kagan grew up and was educated in Moscow, Russia. In 1974 he came to UCLA, and working with Leon Knopoff, David Jackson, Peter Bird, and Frederick Schoenberg applied his mathematical/statistical model to seismicity analysis. Since 1999 these results have been used to produce daily earthquake forecasts for several seismically active regions and currently for the whole Earth. The performance and predictive skill of these forecasts is now being tested by several research groups.

• Preface xiiiAcknowledgments xviiList of Abbreviations xixList of Mathematical Symbols xxiPART I MODELS 11 Motivation: Earthquake science challenges 32 Seismological background 62.1 Earthquakes 62.2 Earthquake catalogs 82.3 Description of modern earthquake catalogs 112.4 Earthquake temporal occurrence: quasi-periodic, Poisson, or clustered? 142.5 Earthquake faults: one fault, several faults, or an infinite number of faults? 162.6 Statistical and physical models of seismicity 182.7 Laboratory and theoretical studies of fracture 193 Stochastic processes and earthquake occurrence models 213.1 Earthquake clustering and branching processes 213.2 Several problems and challenges 243.3 Critical continuum-state branching model of earthquake rupture 26PART II STATISTICS 314 Statistical distributions of earthquake numbers: Consequence of branching process 334.1 Theoretical considerations 344.2 Observed earthquake numbers distribution 435 Earthquake size distribution 545.1 Magnitude versus seismic moment 545.2 Seismic moment distribution 565.3 Is 𝛽 ≡ 1¨M2? 605.4 Seismic moment sum distribution 80 5.5 Length of aftershock zone (earthquake spatial scaling) 865.6 Maximum or corner magnitude: 2004 Sumatra and 2011 Tohoku mega-earthquakes 906 Temporal earthquake distribution 966.1 Omori’s law 966.2 Seismic moment release in earthquakes and aftershocks 976.3 Random shear stress and Omori’s law 1076.4 Aftershock temporal distribution, theoretical analysis 1106.5 Temporal distribution of aftershocks: Observations 1166.6 Example: The New Madrid earthquake sequence of 1811–12 1216.7 Conclusion 1237 Earthquake location distribution 1257.1 Multipoint spatial statistical moments 1257.2 Sources of error and bias in estimating the correlation dimension 1277.3 Correlation dimension for earthquake catalogs 1417.4 Conclusion 1458 Focal mechanism orientation and source complexity 1468.1 Random stress tensor and seismic moment tensor 1478.2 Geometric complexity of earthquake focal zone and fault systems 1508.3 Rotation of double-couple (DC) earthquake moment tensor and quaternions 1548.4 Focal mechanism symmetry 1598.5 Earthquake focal mechanism and crystallographic texture statistics 1638.6 Rotation angle distributions 1678.7 Focal mechanisms statistics 1708.8 Models for complex earthquake sources 177PART III TESTABLE FORECASTS 1839 Global earthquake patterns 1859.1 Earthquake time-space patterns 1859.2 Defining global tectonic zones 1879.3 Corner magnitudes in the tectonic zones 1889.4 Critical branching model (CBM) of earthquake occurrence 1909.5 Likelihood analysis of catalogs 1979.6 Results of the catalogs’ statistical analysis 20410 Long- and short-term earthquake forecasting 20610.1 Phenomenological branching models and earthquake occurrence estimation 20610.2 Long-term rate density estimates 20710.3 Short-term forecasts 21510.4 Example: earthquake forecasts during the Tohoku sequence 21810.5 Forecast results and their discussion 22410.6 Earthquake fault propagation modeling and earthquake rate estimation 22611 Testing long-term earthquake forecasts: Likelihood methods and error diagrams 22911.1 Preamble 22911.2 Log-likelihood and information score 23011.3 Error diagram (ED) 23511.4 Tests and optimization for global high-resolution forecasts 24711.5 Summary of testing results 25012 Future prospects and problems 25312.1 Community efforts for statistical seismicity analysis and earthquake forecast testing 25312.2 Results and challenges 25412.3 Future developments 256References 260Index 281

Yan Kagan grew up and was educated in Moscow, Russia. In 1974 he came to UCLA, and working with Leon Knopoff, David Jackson, Peter Bird, and Frederick Schoenberg applied his mathematical/statistical model to seismicity analysis. Since 1999 these results have been used to produce daily earthquake forecasts for several seismically active regions and currently for the whole Earth. The performance and predictive skill of these forecasts is now being tested by several research groups.

• Preface xiiiAcknowledgments xviiList of Abbreviations xixList of Mathematical Symbols xxiPART I MODELS 11 Motivation: Earthquake science challenges 32 Seismological background 62.1 Earthquakes 62.2 Earthquake catalogs 82.3 Description of modern earthquake catalogs 112.4 Earthquake temporal occurrence: quasi-periodic, Poisson, or clustered? 142.5 Earthquake faults: one fault, several faults, or an infinite number of faults? 162.6 Statistical and physical models of seismicity 182.7 Laboratory and theoretical studies of fracture 193 Stochastic processes and earthquake occurrence models 213.1 Earthquake clustering and branching processes 213.2 Several problems and challenges 243.3 Critical continuum-state branching model of earthquake rupture 26PART II STATISTICS 314 Statistical distributions of earthquake numbers: Consequence of branching process 334.1 Theoretical considerations 344.2 Observed earthquake numbers distribution 435 Earthquake size distribution 545.1 Magnitude versus seismic moment 545.2 Seismic moment distribution 565.3 Is 𝛽 ≡ 1¨M2? 605.4 Seismic moment sum distribution 80 5.5 Length of aftershock zone (earthquake spatial scaling) 865.6 Maximum or corner magnitude: 2004 Sumatra and 2011 Tohoku mega-earthquakes 906 Temporal earthquake distribution 966.1 Omori’s law 966.2 Seismic moment release in earthquakes and aftershocks 976.3 Random shear stress and Omori’s law 1076.4 Aftershock temporal distribution, theoretical analysis 1106.5 Temporal distribution of aftershocks: Observations 1166.6 Example: The New Madrid earthquake sequence of 1811–12 1216.7 Conclusion 1237 Earthquake location distribution 1257.1 Multipoint spatial statistical moments 1257.2 Sources of error and bias in estimating the correlation dimension 1277.3 Correlation dimension for earthquake catalogs 1417.4 Conclusion 1458 Focal mechanism orientation and source complexity 1468.1 Random stress tensor and seismic moment tensor 1478.2 Geometric complexity of earthquake focal zone and fault systems 1508.3 Rotation of double-couple (DC) earthquake moment tensor and quaternions 1548.4 Focal mechanism symmetry 1598.5 Earthquake focal mechanism and crystallographic texture statistics 1638.6 Rotation angle distributions 1678.7 Focal mechanisms statistics 1708.8 Models for complex earthquake sources 177PART III TESTABLE FORECASTS 1839 Global earthquake patterns 1859.1 Earthquake time-space patterns 1859.2 Defining global tectonic zones 1879.3 Corner magnitudes in the tectonic zones 1889.4 Critical branching model (CBM) of earthquake occurrence 1909.5 Likelihood analysis of catalogs 1979.6 Results of the catalogs’ statistical analysis 20410 Long- and short-term earthquake forecasting 20610.1 Phenomenological branching models and earthquake occurrence estimation 20610.2 Long-term rate density estimates 20710.3 Short-term forecasts 21510.4 Example: earthquake forecasts during the Tohoku sequence 21810.5 Forecast results and their discussion 22410.6 Earthquake fault propagation modeling and earthquake rate estimation 22611 Testing long-term earthquake forecasts: Likelihood methods and error diagrams 22911.1 Preamble 22911.2 Log-likelihood and information score 23011.3 Error diagram (ED) 23511.4 Tests and optimization for global high-resolution forecasts 24711.5 Summary of testing results 25012 Future prospects and problems 25312.1 Community efforts for statistical seismicity analysis and earthquake forecast testing 25312.2 Results and challenges 25412.3 Future developments 256References 260Index 281