# Post Seismic Catalog Incompleteness and Aftershock Forecasting

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Catalog Incompleteness

## 3. The Influence of STAI on Model Parameters

#### 3.1. The Influence of STAI on the ETAS Parameters

#### 3.2. Is STAI Related to the Static ${m}_{c}$?

## 4. The Origin of STAI and the Envelope Function

- For $t-{t}_{0}<\tau $, ${\mu}_{e}\left(t\right)$ increases to a maximum value ${\mu}_{M}$.
- For $\tau <t-{t}_{0}<{t}_{M}$, ${\mu}_{e}\left(t\right)$ follows a logarithmic decay as$${\mu}_{e}\left(t\right)\simeq {\mu}_{M}-qlog(t-{t}_{0}).$$
- For $t-{t}_{0}>{t}_{M}$, the average value of the envelope $\langle {\mu}_{e}\left(t\right)\rangle $ is still logarithmic but with different coefficients:$$\langle {\mu}_{e}\left(t\right)\rangle ={\mu}_{M}-\varphi log(t-{t}_{0})-\Delta \mu ,$$

#### Numerical Generation of the Envelope Function

## 5. The ETASI Model

#### 5.1. ETASI2

#### 5.2. Dynamical Scaling ETAS Model

## 6. Automatic Procedures for Short-Term Aftershock Forecasting

#### 6.1. The Omi Method

#### 6.2. The Lippiello Method

#### Test of the Procedure

#### 6.3. Comparison between the Omi and the Lippiello Methods

- (i)
- It is faster. Indeed, aftershock localization is a non-trivial routine involving the elaboration of at least the seismic signal from three different seismic stations.
- (ii)
- It works when only few events are identified by the automatic detection routine whereas the Omi et al method needs that at least ∼30 aftershocks must be identified [15].
- (iii)
- It provides the in-situ occurrence probability by simply installing a seismic station in the site of interest. This could be particularly useful in areas with a very low dense seismic network and where automatic detection routines are not efficient.
- (iv)
- It provides directly in output the probability of peaks of the local ground velocity and therefore it overcomes the large amount of uncertainty [56], which is present in the attenuation relations necessary to convert aftershock occurrence probability to the local ground motion intensity.

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Omori, F. On the after-shocks of earthquakes. J. Coll. Sci. Imp. Univ. Tokyo
**1894**, 7, 111–200. [Google Scholar] - Utsu, T.; Ogata, Y.; Ritsuko, S.; Matsu’ura. The Centenary of the Omori Formula for a Decay Law of Aftershock Activity. J. Phys. Earth
**1995**, 43, 1–33. [Google Scholar] [CrossRef] - Ogata, Y. Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes. J. Am. Stat. Assoc.
**1988**, 83, 9–27. [Google Scholar] [CrossRef] - Helmstetter, A.; Kagan, Y.Y.; Jackson, D.D. Comparison of Short-Term and Time-Independent Earthquake Forecast Models for Southern California. Bull. Seismol. Soc. Am.
**2006**, 96, 90–106. [Google Scholar] [CrossRef][Green Version] - Ogata, Y.; Katsura, K. Immediate and updated forecasting of aftershock hazard. Geophys. Res. Lett.
**2006**, 33. [Google Scholar] [CrossRef] - Werner, M.J.; Helmstetter, A.; Jackson, D.D.; Kagan, Y.Y. High-Resolution Long-Term and Short-Term Earthquake Forecasts for California. Bull. Seismol. Soc. Am.
**2011**, 101, 1630–1648. [Google Scholar] [CrossRef][Green Version] - Omi, T.; Ogata, Y.; Hirata, Y.; Aihara, K. Forecasting large aftershocks within one day after the main shock. Sci. Rep.
**2013**, 3, 2218. [Google Scholar] [CrossRef] - Omi, T.; Ogata, Y.; Hirata, Y.; Aihara, K. Estimating the ETAS model from an early aftershock sequence. Geophys. Res. Lett.
**2014**, 41, 850–857. [Google Scholar] [CrossRef] - Omi, T.; Ogata, Y.; Hirata, Y.; Aihara, K. Intermediate-term forecasting of aftershocks from an early aftershock sequence: Bayesian and ensemble forecasting approaches. J. Geophys. Res. Solid Earth
**2015**, 120, 2561–2578. [Google Scholar] [CrossRef] - Omi, T.; Ogata, Y.; Shiomi, K.; Enescu, B.; Sawazaki, K.; Aihara, K. Automatic Aftershock Forecasting: A Test Using Real-Time Seismicity Data in Japan. Bull. Seismol. Soc. Am.
**2016**, 106, 2450. [Google Scholar] [CrossRef] - Hainzl, S. Rate-Dependent Incompleteness of Earthquake Catalogs. Seismol. Res. Lett.
**2016**, 87, 337–344. [Google Scholar] [CrossRef] - Hainzl, S. Apparent triggering function of aftershocks resulting from rate-dependent incompleteness of earthquake catalogs. J. Geophys. Res. Solid Earth
**2016**, 121, 6499–6509. [Google Scholar] [CrossRef][Green Version] - Zhuang, J.; Ogata, Y.; Wang, T. Data completeness of the Kumamoto earthquake sequence in the JMA catalog and its influence on the estimation of the ETAS parameters. Earth Planets Space
**2017**, 69, 36. [Google Scholar] [CrossRef][Green Version] - Seif, S.; Mignan, A.; Zechar, J.D.; Werner, M.J.; Wiemer, S. Estimating ETAS: The effects of truncation, missing data, and model assumptions. J. Geophys. Res. Solid Earth
**2017**, 122, 449–469. [Google Scholar] [CrossRef] - Omi, T.; Ogata, Y.; Shiomi, K.; Enescu, B.; Sawazaki, K.; Aihara, K. Implementation of a Real-Time System for Automatic Aftershock Forecasting in Japan. Seismol. Res. Lett.
**2018**, 90, 242. [Google Scholar] [CrossRef] - Lippiello, E.; Cirillo, A.; Godano, G.; Papadimitriou, E.; Karakostas, V. Real-time forecast of aftershocks from a single seismic station signal. Geophys. Res. Lett.
**2016**, 43, 6252–6258. [Google Scholar] [CrossRef] - Lippiello, E.; Petrillo, C.; Godano, C.; Tramelli, A.; Papadimitriou, E.; Karakostas, V. Forecasting of the first hour aftershocks by means of the perceived magnitude. Nat. Commun.
**2019**, 10, 2953. [Google Scholar] [CrossRef] - Mignan, A.; Werner, M.J.; Wiemer, S.; Chen, C.C.; Wu, Y.M. Bayesian Estimation of the Spatially Varying Completeness Magnitude of Earthquake Catalogs. Bull. Seismol. Soc. Am.
**2011**, 101, 1371–1385. [Google Scholar] [CrossRef] - Lippiello, E.; Godano, C.; de Arcangelis, L. The earthquake magnitude is influenced by previous seismicity. Geophys. Res. Lett.
**2012**, 39, L05309. [Google Scholar] [CrossRef] - Amorése, D. Applying a Change-Point Detection Method on Frequency-Magnitude Distributions. Bull. Seismol. Soc. Am.
**2007**, 97, 1742–1749. [Google Scholar] [CrossRef] - Schorlemmer, D.; Woessner, J. Probability of Detecting an Earthquake. Bull. Seismol. Soc. Am.
**2008**, 98, 2103–2117. [Google Scholar] [CrossRef] - Kagan, Y.Y. Short-Term Properties of Earthquake Catalogs and Models of Earthquake Source. Bull. Seismol. Soc. Am.
**2004**, 94, 1207–1228. [Google Scholar] [CrossRef][Green Version] - Helmstetter, A.; Kagan, Y.Y.; Jackson, D.D. Importance of small earthquakes for stress transfers and earthquake triggering. J. Geophys. Res. Solid Earth
**2005**, 110, B05S08. [Google Scholar] [CrossRef] - Shcherbakov, R.; Turcotte, D.L.; Rundle, J.B. A generalized Omori’s law for earthquake aftershock decay. Geophys. Res. Lett.
**2004**, 31, L11613. [Google Scholar] [CrossRef] - Reasenberg, P.A.; Jones, L.M. Earthquake Hazard After a Mainshock in California. Science
**1989**, 243, 1173–1176. [Google Scholar] [CrossRef] - Peng, Z.; Vidale, J.E.; Ishii, M.; Helmstetter, A. Seismicity rate immediately before and after main shock rupture from high-frequency waveforms in Japan. J. Geophys. Res. Solid Earth
**2007**, 112, B03306. [Google Scholar] [CrossRef] - Lennartz, S.; Bunde, A.; Turcotte, D.L. Missing data in aftershock sequences: Explaining the deviations from scaling laws. Phys. Rev. E
**2008**, 78, 041115. [Google Scholar] [CrossRef] - Helmstetter, A.; Kagan, Y.Y.; Jackson, D.D. High-resolution Time-independent Grid-based Forecast for M≥5 Earthquakes in California. Seismol. Res. Lett.
**2007**, 78, 78–86. [Google Scholar] [CrossRef] - Marzocchi, W.; Lombardi, A.M. Real-time forecasting following a damaging earthquake. Geophys. Res. Lett.
**2009**, 36, L21302. [Google Scholar] [CrossRef] - Zhuang, J.; Ogata, Y.; Vere-Jones, D. Stochastic declustering of space-time earthquake occurrences. J. Am. Stat. Assoc.
**2002**, 97, 369–380. [Google Scholar] [CrossRef] - De Arcangelis, L.; Godano, C.; Lippiello, E. The overlap of aftershock coda-waves and short-term post seismic forecasting. J. Geophys. Res. Solid Earth
**2018**, 123, 5661–5674. [Google Scholar] [CrossRef] - Baiesi, M.; Paczuski, M. Complex networks of earthquakes and aftershocks. Nonlinear Process. Geophys.
**2005**, 12, 1–11. [Google Scholar] [CrossRef] - Zaliapin, I.; Gabrielov, A.; Keilis-Borok, V.; Wong, H. Clustering Analysis of Seismicity and Aftershock Identification. Phys. Rev. Lett.
**2008**, 101, 018501. [Google Scholar] [CrossRef][Green Version] - Zaliapin, I.; Ben-Zion, Y. Earthquake clusters in southern California I: Identification and stability. J. Geophys. Res. Solid Earth
**2013**, 118, 2847–2864. [Google Scholar] [CrossRef] - Moradpour, J.; Hainzl, S.; Davidsen, J. Nontrivial decay of aftershock density with distance in Southern California. J. Geophys. Res. Solid Earth
**2014**, 119, 5518–5535. [Google Scholar] [CrossRef][Green Version] - Lippiello, E.; de Arcangelis, L.; Godano, C. Influence of Time and Space Correlations on Earthquake Magnitude. Phys. Rev. Lett.
**2008**, 100, 038501. [Google Scholar] [CrossRef][Green Version] - Lippiello, E.; de Arcangelis, L.; Godano, C. Time, Space and Magnitude Correlations in Earthquake Occurrence. Int. J. Mod. Phys. B
**2009**, 23, 5583–5596. [Google Scholar] [CrossRef] - Lippiello, E.; Godano, C.; de Arcangelis, L. Magnitude correlations in the Olami-Feder-Christensen model. Europhys. Lett.
**2013**, 102, 59002. [Google Scholar] [CrossRef] - Peng, Z.; Zhao, P. Migration of early aftershocks following the 2004 Parkfield earthquake. Nat. Geosci.
**2009**, 2, 877–881. [Google Scholar] [CrossRef] - Enescu, B.; Mori, J.; Miyazawa, M.; Kano, Y. Omori-Utsu Law c-Values Associated with Recent Moderate Earthquakes in JapanShort Note. Bull. Seismol. Soc. Am.
**2009**, 99, 884–891. [Google Scholar] [CrossRef] - Odaka, T.; Ashiya, K.; Tsukada, S.; Sato, S.; Ohtake, K.; Nozaka, D. A New Method of Quickly Estimating Epicentral Distance and Magnitude from a Single Seismic Record. Bull. Seismol. Soc. Am.
**2003**, 93, 526–532. [Google Scholar] [CrossRef] - Aki, K.; Chouet, B. Origin of coda waves: Source, attenuation, and scattering effects. J. Geophys. Res.
**1975**, 80, 3322–3342. [Google Scholar] [CrossRef] - Lee, W.H.K.; Bennett, R.E.; Meagher, K.L. A Method of Estimating Magnitude of Local Earthquakes from Signal Duration; US Department of the Interior, Geological Survey: Washington, DC, USA, 1972; Volume 28.
- Sawazaki, K.; Enescu, B. Imaging the high-frequency energy radiation process of a main shock and its early aftershock sequence: The case of the 2008 Iwate - Miyagi Nairiku earthquake, Japan. J. Geophys. Res. Solid Earth
**2014**, 119, 4729–4746. [Google Scholar] [CrossRef] - De Arcangelis, L.; Godano, C.; Grasso, J.R.; Lippiello, E. Statistical physics approach to earthquake occurrence and forecasting. Phys. Rep.
**2016**, 628, 1–91. [Google Scholar] [CrossRef] - Lippiello, E.; Bottiglieri, M.; Godano, C.; de Arcangelis, L. Dynamical scaling and generalized Omori law. Geophys. Res. Lett.
**2007**, 34, L23301. [Google Scholar] [CrossRef] - Lippiello, E.; Godano, C.; de Arcangelis, L. Dynamical Scaling in Branching Models for Seismicity. Phys. Rev. Lett.
**2007**, 98, 098501. [Google Scholar] [CrossRef][Green Version] - Bottiglieri, M.; Lippiello, E.; Godano, C.; de Arcangelis, L. Identification and spatiotemporal organization of aftershocks. J. Geophys. Res. Solid Earth
**2009**, 114, B03303. [Google Scholar] [CrossRef] - Lippiello, E.; de Arcangelis, L.; Godano, C. Role of Static Stress Diffusion in the Spatiotemporal Organization of Aftershocks. Phys. Rev. Lett.
**2009**, 103, 038501. [Google Scholar] [CrossRef][Green Version] - Bottiglieri, M.; de Arcangelis, L.; Godano, C.; Lippiello, E. Multiple-Time Scaling and Universal Behavior of the Earthquake Interevent Time Distribution. Phys. Rev. Lett.
**2010**, 104, 158501. [Google Scholar] [CrossRef] - Bottiglieri, M.; Lippiello, E.; Godano, C.; de Arcangelis, L. Comparison of branching models for seismicity and likelihood maximization through simulated annealing. J. Geophys. Res. Solid Earth
**2011**, 116, B02303. [Google Scholar] [CrossRef] - Lippiello, E.; Godano, C.; de Arcangelis, L. The Relevance of Foreshocks in Earthquake Triggering: A Statistical Study. Entropy
**2019**, 21, 173. [Google Scholar] [CrossRef] - Ogata, Y. Space-time Point-process Models for Earthquake Occurrences. Ann. Inst. Math. Stat.
**1988**, 50, 379–402. [Google Scholar] [CrossRef] - Ogata, Y. A Monte Carlo method for high dimensional integration. Numer. Math.
**1989**, 55, 137–157. [Google Scholar] [CrossRef] - Reasenberg, P.A.; Jones, L.M. Earthquake Aftershocks: Update. Science
**1994**, 265, 1251–1252. [Google Scholar] [CrossRef][Green Version] - Cornell, C.A. Engineering seismic risk analysis. Bull. Seismol. Soc. Am.
**1968**, 58, 1583–1606. [Google Scholar]

**Figure 1.**Magnitude completeness in Southern California. The value of ${m}_{c}$ can be obtained by the color bar and triangles identify the location of seismic stations. Green dashed lines define Region 1. Region 2 is the complement to Region 1 with respect to the entire Southern California (From [19]).

**Figure 2.**The number of aftershocks with magnitude larger than ${m}_{th}$ for the ${m}_{M}=7.3$ Landers earthquake in Southern California, evaluated in different temporal windows $\delta t$ from the main shock. The green dashed line is the exponential behavior expected according to the GR law with $b=1$.

**Figure 3.**Examples of the estimated time-varying $50\%$ detection rate $\mu \left(t\right)$ (solid curve) of magnitudes and estimated various time-varying completeness magnitudes (dotted curve) as indicated in the inset, superimposed on the magnitude-time plot of the observed aftershocks during the first day of the main shock. From [9].

**Figure 4.**Different panels correspond to the ETAS parameters $\mu ,K,A=K{\int}_{0}^{\infty}{(t+c)}^{-p}dt,c,\alpha ,p$ (see axis labels) estimated from the Kumamoto aftershock sequence with different magnitude thresholds. The red and black dots are the estimates based on the original and the replenished datasets, respectively. Unit of measures are $da{y}^{-1},da{y}^{p-1},day$ for $\mu ,K,c$ respectively and the other quantities are adimensional except $A=K{\int}_{0}^{\infty}{(t+c)}^{-p}dt$ which represents the productivity from an event of magnitude ${m}_{c}$. From [13].

**Figure 5.**The ETAS parameters are plotted against ${m}_{th}$ for synthetic catalogs simulated with parameters from Southern California (gray) and compared with the parameter for the incomplete synthetic catalog (orange). The “true” parameter values are plotted with black dashed lines the grey shadowed region represents the $95\%$ quantiles of 30 synthetic ETAS catalogs. The orange shadowed region represents the $95\%$ quantiles of 30 synthetic ETAS incomplete catalogs. From [13].

**Figure 6.**The number of events identified as aftershocks by the BP declustering procedure with magnitude larger than ${m}_{th}$, which occurred at a temporal distance t from events identified as mainshocks with magnitude $m\in [{m}_{M},{m}_{M}+1)$, is divided by the number of identified mainshocks and plotted versus t. Different panels correspond to different values of the mainshock magnitude class $m\in [{m}_{M},{m}_{M}+1)$. Different colors correspond to results for different geographic regions: Region 1 (open green symbols) and Region 2 (filled magenta symbols). Different symbols indicate different values of the lower threshold: ${m}_{th}=1.5$ (circles), ${m}_{th}=2.5$ (squares) and ${m}_{th}=3.5$ (diamonds).

**Figure 7.**(Color online) (

**a**) The same data in Figure 6 are plotted as function of $t/\tau $, with $\tau ={10}^{d({m}_{M}-{m}_{th})}$ proportional to ${c}_{meas}$ (Equation (8)) with $d=1$, for different values of ${m}_{M}$ and ${m}_{th}$. Filled (empty) colored symbols are used for data of Region 1 (Region 2). The magenta continuous line is the scaling function $F\left(x\right)=Alog\left(1+B{x}^{-p}\right)$ with $A=0.35$, $B=70$ and $p=1.1$, whereas the dashed green line is the scaling function $F\left(x\right)=A{(x/B+1)}^{-p}$ with $A=300$, $B=7$ and $p=1.1$. (

**b**) The aftershock density $\rho (t,{m}_{M},{m}_{th})$ in the ETASI1 catalog, with a blind time $\Delta t=1$ min, is plotted as a function of $t/\tau $. Different values of ${m}_{M}$ and ${m}_{th}$ are plotted with different symbols: stars for ${m}_{M}-{m}_{th}=2.5$, crosses for ${m}_{M}-{m}_{th}=1.5$ and plus for ${m}_{M}-{m}_{th}=0.5$. Different colors correspond to different values of ${K}_{0}$ and of the average background rate ${r}_{B}$: ${K}_{0}=0.035$ and ${r}_{B}=4.38$ days${}^{-1}$ (black), ${K}_{0}=0.035$ and ${r}_{B}=8.3$ days${}^{-1}$ (green) and ${K}_{0}=0.068$ and ${r}_{B}=4.38$ days${}^{-1}$ (red). Magenta continuous and green dashed lines are the same scaling functions $F\left(x\right)$ plotted in (

**a**). (Inset) The value of $\Delta m$ (Equation (8)) as function of ${\mathrm{log}}_{10}\left({K}_{o}\right)$ for the ETASI model with a blind time $\Delta t=1$ min (black crosses). The cyan line is the theoretical prediction (Equation (18)). From [31].

**Figure 8.**The quantity ${\mu}_{e}\left(t\right)$ (green circles) after the the Hector Mine earthquake in California recorded at the station CIGSC located at a distance of 92 km from the main shock epicenter. The magenta crosses indicate the (logarithmically binned with bin value $0.1$) average value of $\mu \left(t\right)$, the red continuous lines represent the results of the logarithmic fit (Equation (13)) for $t-{t}_{0}>{t}_{M}$. The dashed blue lines represent the quantity ${\mu}_{min}\left(t\right)$ and orange lines are used for results of numerical simulations for the theoretical envelope ${\mu}_{th}\left(t\right)$, defined in Equation (15). The values of the best-fitting parameters in Equation (6) are $K=0.95,c=0.18$ days and $\tau =8$ s.

**Figure 9.**Information gain per aftershock of the forecasts based on the Hi-net and JMA catalogs, respectively, relative to the generic model for the cases with: (

**a**) ${M}_{t}={M}_{c}$; and (

**b**) ${M}_{t}=3.95$. If the lower bound of the error bar is greater than 0, the forecast is significantly better than the generic model with a probability larger than the $95\%$. From Omi et al. [10].

**Figure 10.**(

**Left**) Black dotted-lines represent the envelope function ${\mu}_{e}\left(t\right)$ of the Lixouri earthquake in Greece recorded at the station LKD2 located at 70 km from the mainshock epicenter. Colored dot-dashed lines are used for ${\mu}_{th}\left(t\right)$ with $\tau =11$ s, $K=1.15$ and different values of c ranging in the interval $[0.01,4.5]$ days. Black circles represent the logarithmic fit (Equation (15)) in the interval $[10,160]$ min of the experimental envelope function, whereas continuous lines are used for the best fit of the numerical ${\mu}_{th}\left(t\right)$, with c increasing from $0.01$ to $4.5$ days from top to bottom. (

**Right**) The same as in the left panel but plotting numerical data ${\mu}_{th}\left(t\right)$ with $\tau =11$ s, $c=1.15$ days and different values of $K\in [0.75,2.95]$. Continuous lines are the the logarithmic fits of numerical data, in $[10,160]$ min, with K increasing from bottom to top. From Reference [16].

**Figure 11.**The quantities ${N}_{3}\left(t\right)$ (

**top**) and ${N}_{2}\left(t\right)$ (

**bottom**) are plotted for $t-{t}_{0}>{T}_{2}=160$ min as green circles for the three main-aftershock sequences of Figure 1: the 26 January 2014 $m=6.1$ Lixouri earthquake (

**left**), the 16 October 1999 $m=7.1$ Hector Mine earthquake (

**middle**) and the 6 April 2009 $m=5.9$ L’Aquila earthquake (

**right**). Red squares are the expected values according to Equation (6) using the average values obtained in Reference [55]. The orange curves are the expected values using in Equation (6) the best parameters K and c inverted from the experimental fit of $\tau $, $\varphi $ and $\Delta \mu $. The error bars in each plot incorporate both the uncertainty in the estimate of $(\Delta \mu ,\varphi )$ and fluctuations in the aftershock number for given values of $\overline{K}$ and $\overline{c}$. From Reference [16].

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lippiello, E.; Cirillo, A.; Godano, C.; Papadimitriou, E.; Karakostas, V. Post Seismic Catalog Incompleteness and Aftershock Forecasting. *Geosciences* **2019**, *9*, 355.
https://doi.org/10.3390/geosciences9080355

**AMA Style**

Lippiello E, Cirillo A, Godano C, Papadimitriou E, Karakostas V. Post Seismic Catalog Incompleteness and Aftershock Forecasting. *Geosciences*. 2019; 9(8):355.
https://doi.org/10.3390/geosciences9080355

**Chicago/Turabian Style**

Lippiello, Eugenio, Alessandra Cirillo, Cataldo Godano, Elefetheria Papadimitriou, and Vassilis Karakostas. 2019. "Post Seismic Catalog Incompleteness and Aftershock Forecasting" *Geosciences* 9, no. 8: 355.
https://doi.org/10.3390/geosciences9080355