Testing aftershock models on a time-scale of decades

Abstract

Aftershocks are the smaller earthquakes that normally follow a larger earthquake, the so- called mainshock. The frequency of aftershock decays with time from the mainshock. We use aftershock models for earthquake forecasting that mathematically describe this decay. According to these models, aftershock activity can continue for years, even thousands of years. Earthquakes that occur outside a mainshock-aftershock sequence are called background seismicity.

We investigate three questions regarding aftershock occurrence:

1. How long can we detect aftershocks before they merge with the background seismicity?;
2. Can a single set of model parameters in our aftershock models describe all aftershock sequences well?; and
3. For how long following a mainshock can we forecast aftershocks accurately?

We found that the duration of an aftershock sequence, i.e. the time before it aftershocks merge with the background seismicity, is difficult to determine. It depends on both the background seismicity and the mainshock magnitude, and can vary from a few days to many years. In New Zealand, the longest duration was found for the Canterbury sequence because the background seismicity was very low prior to the 2010 Darfield earthquake.

In response to the second question, we found that a single set of aftershock parameters did not describe all aftershock sequences well. However, the alternative approach of fitting individual sequences had many uncertainties, and unfortunately did not necessarily provide better results than using a uniform set of model parameters.

To address the third question, we conducted a numerical experiment where we created synthetic earthquake catalogues based on one of our aftershock models. We found that the time in which aftershock models can effectively forecast earthquakes of magnitude 4 and larger was only about 100 or 1000 days after a mainshock of magnitude 6 or 7, respectively.

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Technical Abstract

In New Zealand we use two aftershock models for earthquake forecasting: The Short-Term Earthquake Probability (STEP) model and variations of the Epidemic Type Aftershock Sequence (ETAS) model. These aftershock models are based on the Omori-Utsu model, a power law that describes the decay of aftershock rate with time. The models imply that aftershock activity continues for thousands if not million of years. Since homogeneous earthquake catalogues are generally available for time periods of 30 – 50 years, it is difficult to assess the validity of the aftershock models for very long time periods. This project set out to test aftershock models on a time-scale of decades. In particular, we address the following three questions:

1. For how long following a mainshock is it possible to detect aftershocks in earthquake catalogues and, in particular, how does aftershock detectability depend on the background seismicity rate?
2. How well can a universal set of ETAS parameters (constrained by physical models) forecast triggered seismicity within the observed uncertainties/variability?
3. How does the forecast ability of the ETAS model vary with an increasing time horizon¹ for individual earthquake sequences?

We use a mix of ETAS simulations and analyses of real earthquake catalogues to address the questions. For our analyses, we distinguish three different time scales of aftershock activity: (1) the triggering time T, which is the duration of the physical triggering process of a single event; (2) the apparent aftershock duration Ta which is the time period in which aftershocks dominate the seismicity; and (3) the effective forecasting period Tf within which earthquake rate estimates are significantly improved by time-dependent seismicity models after a large earthquake.

A finite value of T is expected from a physical point of view, but has not been incorporated in standard ETAS model applications so far. During this project we introduce and estimate for the first time finite T-values in the modified ETAS model.

Although estimates of T were only weakly constrained and potentially subject to biases due to limited catalogue length and cluster selection, our comparative analysis of synthetic sequences gave some robust results: We found that T has an impact on the estimates of the other ETAS-parameters and reduces the mismatch between the power law decay parameter in the ETAS model and predictions of physics-based models. Furthermore, the predicted inverse proportionality between T and the background rate is in agreement with the observed trend in the estimated values of T for empirical earthquake sequences.

We estimated Ta for all earthquake sequences with at least 50 earthquakes in our three different earthquake catalogues, as well as for simulated sequences. We found that many sequences had durations Ta of less than one year and only few lasted longer than 10 years. This finding contradicts our current aftershock models. We have suggested two ways of changing the models but pursuing these is outside the scope of this EQC project.

Our forecast experiment with universal ETAS parameters confirmed earlier work that universal ETAS parameters do not fit all sequence well. However, estimating parameters for an on-going sequence has too many uncertainties and does not lead to stable results. It was outside the scope of this project to investigate what universal set of parameters might be best.

The effective forecasting period TF depends on several factors, including (1) the number and quality of data available; (2) the quality of the model, i.e. how well the model describes the observed seismicity and (3) the magnitude difference between mainshock and cut-off magnitude. We conducted a numerical experiment of ETAS simulations and found that after approximately 100 days for M = 6 and 1000 days for M = 7, the forecast of the time-invariant Poisson model becomes equal to or better than that of the modified ETAS model.

In summary, our project on “Testing aftershock models on time-scale of decades” answered the three questions posed above. We found that many aftershock sequences cannot be detected above the background seismicity for more than 1 year, and only few sequences last longer than 10 years. A universal set of ETAS parameters does not fit all earthquake sequences well, but fitting the parameters to individual sequences introduces many uncertainties. Finally, the effective forecasting time of the ETAS model is only in the order of 100 and 1000 days for mainshocks of M6 and M7, respectively.

¹ The time horizon here mean the time into the future for which an earthquake forecast applies. We later refer to this as “effective forecasting period Tf“ within which earthquake rate estimates are significantly improved by time-dependent seismicity models after a large earthquake.

 

Published paper:
S Hainnzl, A Christopersen, D Rhoades, D Harte, Statistical estimation of the duration of aftershock sequences. Geophysical Journal International (2016) 205, 1180-1189 - Advance Access publication 2016 February 24, GJI Seismology - doi:10.1093/gji/ggw075