Earthquake Occurrence: Short- and Long-term Models and their Validation

By Rodolfo Console, Maura Murru and Giuseppe Falcone

Earthquake Occurrence provides the reader with a review of algorithms applicable for modeling seismicity, such as short-term earthquake clustering and pseudo-periodic long-term behavior of major earthquakes. The concept of the likelihood ratio of a set of observations under different hypotheses is applied for comparison among various models.

In short-term models, known by the term ETAS, the occurrence space and time rate density of earthquakes is modeled as the sum of two terms, one representing the independent or spontaneous events, and the other representing the activity triggered by previous earthquakes. Examples of the application of such algorithms in real cases are also reported.

Dealing with long-term recurrence models, renewal time-dependent models, implying a pseudo-periodicity of earthquake occurrence, are compared with the simple time-independent Poisson model, in which every event occurs regardless of what has occurred in the past.

The book also introduces a number of computer codes developed by the authors over decades of seismological research.

• Language: English
• Publisher: Wiley
• Release date: Jul 17, 2017
• ISBN: 9781119372226

Book preview

Table of Contents

Cover

Title

Copyright

Foreword

Preface

Introduction

1 Seismicity and Earthquake Catalogues Described as Point Processes

1.1. The Gutenberg–Richter law

1.2. The time-independent Poisson model

1.3. Occurrence rate density as a space–time continuous variable

1.4. Time-independent spatial distribution

1.5. Clustered seismicity

1.6. Epidemic models

2 The Likelihood of a Hypothesis

2.1. The Bayes theorem

2.2. Likelihood function

2.3. Alternative formulations

2.4. Likelihood ratio

3 The Likelihood for a Model of Continuous Rate Density Distribution

3.1. The limit case of regions of infinitesimal dimensions

3.2. The case of discrete regions

3.3. The case of time independence

3.4. The likelihood of an epidemic model in a 4-D space of parameters

4 Forecast Verification Procedures

4.1. Scoring procedures

4.2. The binary diagrams

4.3. Statistical tests implemented within CSEP

5 Applications of Epidemic Models

5.1. Declustering a catalogue through an epidemic model

5.2. Earthquake forecasting

5.3. Seismic hazard maps for short-term forecast

6 Long-term Earthquake Occurrence Models

6.1. The empirical Gutenberg–Richter law and the time-independent model under the Poisson hypothesis

6.2. Statistics of inter-event times

6.3. The truncated magnitude distribution

6.4. Earthquake rate assessment under a renewal time-dependent model

6.5. Validation and comparison of renewal time-dependent models

6.6. The Cornell method for time-independent seismic hazard assessment

6.7. Acknowledgments

7 Computer Programs and Examples of their Use

7.1. PDE2REC, ZMAP2REC

7.2. REC2PDE

7.3. SMOOTH

7.4. LIKELAQP

7.5. LIKSTAQP

7.6. BPT

Bibliography

Index

End User License Agreement

List of Tables

4 Forecast Verification Procedures

Table 4.1.Contingency table

Table 4.2.Contingency tables for the ETAS model applied to the Italian catalogue from 1 January 2006 to 11 October 2007 as a function of a threshold occurrence rate value (r), expressed as events/day/100 km²

5 Applications of Epidemic Models

Table 5.1.Probability of an earthquake of magnitude ≥ 5 in 24 h in the area of L’Aquila. The red arrows point to the occurrence of the main observed earthquakes

6 Long-term Earthquake Occurrence Models

Table 6.1.Statistical parameters of the inter-event times distribution obtained from a Monte Carlo procedure on 19 earthquake datasets for each of six renewal models (from [CON 12])

Table 6.2.Confidence levels by which the Poisson distribution can be rejected, with their uncertainties. These values refer to different renewal models and to any of the 19 sites considered in this study (from [CON 12])

Table 6.3.Estimated effects on the city of Messina for the nine earthquakes of the first 100 years of a 100 ky synthetic catalogue (from [CON 17a])

List of Illustrations

1 Seismicity and Earthquake Catalogues Described as Point Processes

Figure 1.1.An example of cumulative number of earthquakes versus the magnitude threshold

Figure 1.2.Probability density distribution of the largest magnitude in a sample with a minimum magnitude M0=4.0. The different colors represent samples with different numbers of events. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 1.3.a) Epicentral distribution of the earthquakes with magnitude M ≥ 2.0 reported by the Southern California Earthquake Data Center in the time period 1984–2002. b) Smoothed distribution of the Southern California (1984–2002) seismicity obtained by the smoothing algorithm of equation [1.26] with a correlation distance c = 5.0 km. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

2 The Likelihood of a Hypothesis

Figure 2.1.Intersection of two events

Figure 2.2.Intersection of various events. Event A is totally contained in B, and B is divided into various mutually exclusive sub-events Bi (i = 1,…,N)

4 Forecast Verification Procedures

Figure 4.1.An example of the log-performance factor achieved by a model named ETASb [ZHU 04] and a model called ERS (epidemic rate-state), respectively (by [CON 06a]), against a plain time-independent Poisson model, plotted versus time for the whole test period provided by the JMA (1 January 1970–31 December 2003). The origin time is 1 January 1994 at 00:00 UTC. The occurrence time of the earthquakes with magnitude M≥6.5 are indicated by black triangles. The sharp positive steps correspond to the events that occurred in a space–time point of large expected rate density with respect to the Poisson model. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 4.2.Relative operating characteristic diagrams (ROC). Plots of the hit rate H (fraction of earthquakes that occur on alarm cells), versus the false alarm rate F (ratio between the false alarms and the total number of alarms). In a), the scale of the F parameter covers the whole range between 0 and 1. In b), the scale of the F parameter covers only the range between 0 and 1·10−3. The trend of the H parameter for random forecasts is also shown (from [MUR 09]). For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 4.3.Molchan’s error diagram for a) the ETAS model and b) a spatially heterogeneous Poisson model (by [CON 10b]). In both figures are also shown, as diagonal lines, the results expected for a spatially homogeneous time-independent Poisson model. These plots show the fraction of failures to predict, ν, versus the fraction of alarm targets, τ, for different magnitude thresholds. It is important to note that ν and τ can be connected to the Aki probability gain by the relation: G = (1−ν)/τ

Figure 4.4.Probability gain versus false alarm rate (F). In a), the scale of the F parameter covers the whole range between 0 and 1. In b), the scale of the F parameter covers only the range between 0 and 1·10−2. The magnitude values are also reported above each line besides given in the legend. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

5 Applications of Epidemic Models

Figure 5.1.a) Cumulative distribution of the number of events in the southern California (1984–2002) earthquake catalogue. b) Cumulative distribution of the weights pi over the time spanned by the southern California (1984–2002) earthquake catalogue that is normalized to the total number of events; see step 3 in the text of this section for the definition of pi. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 5.2.Smoothed distribution of the southern California (1984–2002) seismicity obtained by the algorithm of equation [1.26] with a correlation distance c=5.0 km and applied to the catalogue weighted by the algorithm described in this section. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 5.3.Distribution of the probability of independence for the 60,480 events of the southern California earthquake catalogue

Figure 5.4.Comparison between the initial and final smoothed estimate of the spatial occurrence rate density for the learning period (17 April 2005–15 March 2009) (events/day/deg2). For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 5.5.Distribution of probability for an event of the learning period to be spontaneous

Figure 5.6.Examples of the expected daily seismicity rate forecast at 00:00 UTC by the ETAS model on the following days: a) 30 March 2009 (after the M4.0 foreshock occurred at 13:38 UTC), b) 6 April 2009 before the mainshock occurred at 01:32 UTC), c) 7 April 2009 (before the M5.7 shock occurred at 17:47 UTC) and d) 30 April 2009. The units of the color scale are the number of events (M≥2.0)/day/deg². The black dots show the epicenters of the earthquakes recorded in the 24 h following the forecast. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 5.7.Comparison between forecast and observed rates during the test period (16 March 2009–30 June 2009) for events M≥2.0 per day. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 5.8.Enlargements of the expected rate density maps generated for all the Italian territory, corresponding to a zone of 100×100 km centered on the point of the maximum expected rate during various time periods proceeding and following the Montefeltro sequence that occurred on 2006 among the Emilia Romagna, Marche and Toscana regions. (a) The map represents the modeled occurrence rate density, ML ≥ 3.8 (events/day/km²), for the Montefeltro zone (100×100 km) on 29 August 2006 at 12:00 UTC, which is 22 h before the mainshock (3.8 ML) of the Montefeltro sequence that occurred on 30 August 2006 at 10:01 UTC. (b) As in (a) on 30 August 2006 at 00:00 UTC, which is 10 h before the mainshock of the sequence. Gray dots represent the only seven foreshocks of the sequence. (c) As in (a) and (b) on 30 August 2006 at 12:00 UTC, which is 2 h after the mainshock. Black dots represent the aftershocks and the foreshocks of the sequence. The red star is the hypocenter of the mainshock. (d) As in (a), (b) and (c) on 6 September 2006 at 00:00 UTC, which is 1 week after the mainshock (from [MUR 09]). For a color version of this figure, see www.iste.co.uk/console/earthquake.zip.

Figure 5.9.Real-time hazard maps automatically calculated by the stochastic model of earthquake clustering (ETAS). These areas are centered on the points of maximum rate density, respectively shown in Figure 5.8(b) and (c). Colors show the probability of exceeding PGA 0.01 g in 24 h in an area of 100×100 km centered on the point of the maximum expected rate. (a) On 30 August 2006 at 00:00 UTC, which is 10 h before the mainshock of the Montefeltro sequence that occurred among the Emilia Romagna, Marche and Toscana regions. (b) As in (a), on 30 August 2006 at 12:00 UTC, which is 2 h after the mainshock of this sequence (from [MUR 09]). For a color version of this figure, see www.iste.co.uk/console/earthquake.zipuake.zip

6 Long-term Earthquake Occurrence Models

Figure 6.1. Frequeency magnitudde distribution of earthquakees observed inn New Zealannd (1960–20055) and its bestt fit according to the Gutenbberg–Richter laaw (red line). FFor a color verrsion of this figure, see www.iste.co.uk/coonsole/earthquuake.zip

Figure 6.2.pdf distribution and hazard function of five different renewal models compared with the Poisson model for a recurrence time Tr = 1 and a coefficient of variation Cv = 0.5. For a color version of this figure, see www.iste.co.uk/console/earthquake.zip

Figure 6.3.Cumulative distributions of dlnLs for 1,000 synthetic sequences obtained from the Poisson distribution compared with the observed dlnLs computed under a variable-σ log-normal distribution, with its uncertainty, for each site considered in this study. The ordinate of the real dlnL in the synthetic distribution gives the probability that the observed dlnL comes by chance from a random distribution. The vertical lines show the observed dlnLs, and the horizontal lines show the respective probabilities. The standard deviations and their respective probabilities are shown by the dotted lines (from [CON 12])

Figure 6.4.Mean values of the log-likelihood ratio for