A simple spatial ETAS model was fitted to NZ earthquake events with magnitude ≥ 4, occurring between 1965 and 2011 (inclusive). The spatial region is: 164°E – 182°E, 35°S – 48°S, and depth ≤ 40km. The particular model used here is described in 10.1093/gji/ggs026 and 10.1093/gji/ggu129.
The conditional intensity function (like an instantaneous event rate) of a spatial ETAS model fitted to NZ earthquake events, with magnitude ≥ 4 and depth ≤ 40km, is shown as a movie in nz.avi (MPEG-4, 15.1 Mb). The movie runs from 1970 until mid 2013. Only events with magnitude ≥ 4 are included in the NZ movie because the catalogue is incomplete for events of a lower magnitude in this space-time region. The movie plays at 5 frames per second. Each frame represents 20 days of real time.
In fact, the movie is an approximation to the intensity function. For each 20 day interval, the spatial region is divided up into spatial cells of 0.2° by 0.2°. The intensity function is integrated over each space-time volume, i.e. each cell by each 20 day interval. Hence, it is the expected number of events within each space-time volume, given the model and history of the process; and is therefore a smoothed version of the intensity function.
Similarly, the conditional intensity function of the model during the Darfield–Christchurch sequence (1 Sept 2010 until 30 June 2012) is also represented as a movie in chch.avi (MPEG-4, 12.7 Mb). It includes events with magnitude ≥ 3. This movie plays at 6 frames per second. Each frame represents 15 minutes of real time. For each 15 minute interval, the spatial region is divided up into cells of 0.02° by 0.02°. Using the same model at a different magnitude cut-off is not strictly correct (10.1093/gji/ggv524), as the model was calibrated for events with magnitude ≥ 4. This will cause the values of the plotted log-rate to be incorrect, though the visual dynamics are accurate under the assumption that these smaller events have a triggering effect.
Currently, during periods of benign seismicity, forecasts are made every 2nd Tuesday at 00:00hr (UTC) for 28 days. Over a shorter time interval there are too fewer events to usefully evaluate the efficacy of the model. It should also be noted that the forecast intervals are overlapping, and hence the results are not independent. The NZ catalogue (ssNZ package) versions used to make and assess each forecast are listed in details.txt; and are archived on the GNS ftp website until that of 2019-09-22 12:00 (UTC), and on Google Drive (uses tinyurl.com) since then.
The first group of plots in summary.pdf are modified boxplots of the ETAS forecast count distributions for events of magnitude ≥ 4 derived from pg 1 of counts.pdf (blue histogram) of the given forecast. The added triangles on the boxplot extend down to the 2.5 percentile and up to the 97.5 percentile, and hence cover 95% of the simulated number of events. If the real data conforms to such a model, then there is only a 5% chance that the number of ultimately observed events is not within this range. Further, over many such forecasts, we would expect 25% to be in the lower part of the box, 25% in the upper part, 22.5% in the lower triangle, and 22.5% in the upper triangle. This is clearly not happening; the red point represents the number of real events. The model is under-fitting for a time after major events (see forecasts during Kaikoura 2016), and consequently over-fitting during times of more benign seismicity. This has occurred in nearly all major NZ event sequences in the last 50 years (see Tables 6 & 7 in 10.1093/gji/ggs026). One can also see the lack of independence, as expected, in that the results tend to occur in pairs; however, over many such forecasts, we would expect the results to conform to the above percentages.
Do the real observed events occur in space–time locations that are consistent with the model? This is assessed by calculating the log-likelihood (10.1093/gji/ggu442) in the forecasted space–time volume for all simulated sequences, and comparing with that of the real events (2nd half of summary.pdf). However, the distribution of the likelihood is dependent on the number of events. This can be seen on pg 5 of counts.pdf. A rather large number of simulations are required to get a good empirical estimate of each distribution on pg 5, and to satisfactorily describe the heavy tail behaviour on pg 1. At the end of the forecasting period, that boxplot from pg 5 coinciding with the number of observed events is added to summary.pdf along with the log-likelihood of the real data (red point). These boxplots of the log-likelihoods are assessed in the same way as for the count data above.
Note that the ETAS forecast count distribution for magnitude ≥ 4 (pg 1 of counts.pdf, blue distribution) is a highly contagious distribution (see Eq. 18 in 10.1093/gji/ggx146 for the probability generating function), with a much longer tail than the corresponding Poisson distribution with the same mean (grey distribution). This long tail represents the small probability of getting a large event and the potentially large number of triggered events. Note that as the magnitude threshold is increased (pp 2–4) the difference between the two distributions diminishes.
The forecasting procedure and associated problems are discussed in more depth here: 10.1093/gji/ggz088.