Enhancing the ETAS model: incorporating rate-dependent incompleteness, constructing a representative data set and reducing bias in inversions

SUMMARY The development of reliable operational earthquake forecasts is dependent upon managing uncertainty and bias in the parameter estimations obtained from models like the Epidemic-Type Aftershock Sequence (ETAS) model. Given the intrinsic complexity of the ETAS model, this paper is motivated by the questions: ‘What constitutes a representative sample for fitting the ETAS model?’ and ‘What biases should we be aware of during survey design?’. In this regard, our primary focus is on enhancing the ETAS model’s performance when dealing with short-term temporally transient incompleteness, a common phenomenon observed within early aftershock sequences due to waveform overlaps following significant earthquakes. We introduce a methodological modification to the inversion algorithm of the ETAS model, enabling the model to effectively operate on incomplete data and produce accurate estimates of the ETAS parameters. We build on a Bayesian approach known as inlabru, which is based on the Integrated Nested Laplace Approximation method. This approach provides posterior distributions of model parameters instead of point estimates, thereby incorporating uncertainties. Through a series of synthetic experiments, we compare the performance of our modified version of the ETAS model with the original (standard) version when applied to incomplete data sets. We demonstrate that the modified ETAS model effectively retrieves posterior distributions across a wide range of mainshock magnitudes and can adapt to various forms of data incompleteness, whereas the original model exhibits bias. In order to put the scale of bias into context, we compare and contrast further biases arising from different scenarios using simulated data sets. We consider: (1) sensitivity analysis of the modified ETAS model to a time binning strategy; (2) the impact of including and conditioning on the historic run-in period; (3) the impact of combination of magnitudes and trade-off between the two productivity parameters K and $\alpha$; and (4) the sensitivity to incompleteness parameter choices. Finally, we explore the utility of our modified approach on three real earthquake sequences including the 2016 Amatrice earthquake in Italy, the 2017 Kermanshah earthquake in Iran, and the 2019 Ridgecrest earthquake in the US. The outcomes suggest a significant reduction in biases, underlining a marked improvement in parameter estimation accuracy for the modified ETAS model, substantiating its potential as a robust tool in seismicity analysis.