Central Italy · Central Apennines, Italy
amatrice 2016
- mainshock
- M6.0 mw · ingv:7073641
- origin (UTC)
- 2016-08-24 01:36:32
- epicentre
- 42.698°, 13.234° depth 8.1 km
- catalog
- INGV (Italy) FDSN event service network ingv
- zone radius
- 53.2 km Gardner–Knopoff window
- completeness Mc
- 1.8 protocol v3, max over time bins
- aftershocks ≥ Mc
- 3,912 largest M5.4
- window
- 0.5–63.7 daystruncated: in-zone M5.9 (ingv:8669321) at day 63.74
rate decay
Observed rate in log-spaced bins against Omori–Utsu fits made with the first 3, 30 and 120 days only, drawn across the whole window. Where a curve leaves the dots, the early fit extrapolated badly.
- observed rate
- Omori fitted on first 3 d (p = 0.77)
- Omori fitted on first 30 d (p = 0.85)
Table view
| days since mainshock | observed rate | Omori fitted on first 3 d (p = 0.77) | Omori fitted on first 30 d (p = 0.85) |
|---|---|---|---|
| 0.5 | — | 1,068.68 | 1,117.24 |
| 0.55 | 1,036.86 | — | — |
| 0.57 | — | 971.05 | 1,005.69 |
| 0.64 | — | 882.34 | 905.29 |
| 0.65 | 987.7 | — | — |
| 0.73 | — | 801.73 | 814.91 |
| 0.77 | 883.06 | — | — |
| 0.82 | — | 728.48 | 733.55 |
| 0.92 | 610.5 | — | — |
| 0.93 | — | 661.93 | 660.31 |
| 1.05 | — | 601.45 | 594.39 |
| 1.09 | 513.44 | — | — |
| 1.19 | — | 546.5 | 535.05 |
| 1.3 | 369.49 | — | — |
| 1.35 | — | 496.58 | 481.63 |
| 1.53 | — | 451.21 | 433.54 |
| 1.54 | 441.79 | — | — |
| 1.73 | — | 409.99 | 390.26 |
| 1.83 | 393.59 | — | — |
| 1.96 | — | 372.53 | 351.3 |
| 2.18 | 389.28 | — | — |
| 2.22 | — | 338.5 | 316.22 |
| 2.52 | — | 307.57 | 284.65 |
| 2.59 | 320.71 | — | — |
| 2.85 | — | 279.47 | 256.23 |
| 3.08 | 249.12 | — | — |
| 3.23 | — | 253.94 | 230.65 |
| 3.65 | — | 230.74 | 207.62 |
| 3.66 | 173.28 | — | — |
| 4.14 | — | 209.66 | 186.89 |
| 4.35 | 158.98 | — | — |
| 4.68 | — | 190.5 | 168.24 |
| 5.18 | 135.94 | — | — |
| 5.31 | — | 173.1 | 151.44 |
| 6.01 | — | 157.29 | 136.32 |
| 6.16 | 114.33 | — | — |
| 6.8 | — | 142.92 | 122.71 |
| 7.32 | 96.94 | — | — |
| 7.7 | — | 129.86 | 110.46 |
| 8.7 | 86.17 | — | — |
| 8.72 | — | 117.99 | 99.43 |
| 9.88 | — | 107.21 | 89.5 |
| 10.35 | 115.39 | — | — |
| 11.18 | — | 97.42 | 80.57 |
| 12.3 | 87.67 | — | — |
| 12.66 | — | 88.52 | 72.52 |
| 14.34 | — | 80.43 | 65.28 |
| 14.63 | 88.32 | — | — |
| 16.24 | — | 73.08 | 58.77 |
| 17.4 | 51.4 | — | — |
| 18.39 | — | 66.41 | 52.9 |
| 20.68 | 43.51 | — | — |
| 20.82 | — | 60.34 | 47.62 |
| 23.58 | — | 54.83 | 42.86 |
| 24.59 | 40.34 | — | — |
| 26.7 | — | 49.82 | 38.58 |
| 29.24 | 24.66 | — | — |
| 30.23 | — | 45.27 | 34.73 |
| 34.23 | — | 41.13 | 31.26 |
| 34.77 | 16.42 | — | — |
| 38.77 | — | 37.37 | 28.14 |
| 41.34 | 20.93 | — | — |
| 43.9 | — | 33.96 | 25.33 |
| 49.16 | 28.05 | — | — |
| 49.71 | — | 30.86 | 22.8 |
| 56.29 | — | 28.04 | 20.53 |
| 58.45 | 19.74 | — | — |
| 63.74 | — | 25.48 | 18.48 |
7-day forecasts
Each point is a forecast issued at that origin for the next 7 days, with its 90% predictive interval, against the count that followed.
- observed
- Omori–Utsu
- ETAS (fixed μ, subcritical)
- GBM-hybrid · cross/pooled
Table view
| forecast origin (days) | observed | Omori–Utsu | ETAS (fixed μ, subcritical) | GBM-hybrid · cross/pooled |
|---|---|---|---|---|
| 1 | 1,444 | 588.79 (549–629) | 1,592.25 (804–2,596) | 762.73 (718–808) |
| 2 | 1,111 | 1,024.85 (972–1,078) | 1,139.16 (531–1,933) | 1,421.31 (1,360–1,484) |
| 3 | 855 | 1,117.27 (1,063–1,173) | 1,159.14 (531–1,983) | 1,183.4 (1,127–1,240) |
| 5 | 723 | 648.75 (607–691) | 794.63 (276–1,530) | 817.46 (771–865) |
| 7 | 681 | 513.63 (477–551) | 630.32 (203–1,247) | 596.19 (556–637) |
| 10 | 644 | 396.27 (364–429) | 522.67 (151–1,075) | 444.81 (410–480) |
| 14 | 405 | 416.39 (383–450) | 474.21 (234–782) | 400.9 (368–434) |
| 21 | 267 | 309.12 (280–338) | 390.72 (238–574) | 301.7 (273–331) |
| 30 | 126 | 223.54 (199–248) | 303.13 (170–466) | 262.14 (236–289) |
| 45 | 194 | 132.57 (114–152) | 198.96 (96–331) | 154.71 (135–175) |
30-day forecasts
Each point is a forecast issued at that origin for the next 30 days, with its 90% predictive interval, against the count that followed.
- observed
- Omori–Utsu
- ETAS (fixed μ, subcritical)
- GBM-hybrid · cross/pooled
Table view
| forecast origin (days) | observed | Omori–Utsu | ETAS (fixed μ, subcritical) | GBM-hybrid · cross/pooled |
|---|---|---|---|---|
| 1 | 2,778 | 599.06 (559–640) | 2,972.97 (1,332–5,141) | 752.47 (708–798) |
| 2 | 2,378 | 1,984.29 (1,911–2,058) | 2,278.05 (1,055–3,877) | 2,990.14 (2,900–3,080) |
| 3 | 2,056 | 2,575.67 (2,492–2,659) | 2,866.3 (1,245–5,027) | 2,730.44 (2,645–2,817) |
| 5 | 1,747 | 1,455.89 (1,393–1,519) | 2,014.11 (687–3,906) | 1,825.88 (1,756–1,896) |
| 7 | 1,536 | 1,239.64 (1,182–1,298) | 1,769.11 (571–3,501) | 1,483.14 (1,420–1,547) |
| 10 | 1,319 | 1,040.66 (988–1,094) | 1,546.97 (516–3,024) | 1,164.86 (1,109–1,221) |
| 14 | 985 | 1,261.64 (1,204–1,320) | 1,629.75 (821–2,661) | 1,197.46 (1,141–1,255) |
| 21 | 768 | 1,006.41 (955–1,059) | 1,681.15 (1,107–2,353) | 968.31 (917–1,020) |
| 30 | 621 | 767.15 (722–813) | 1,350.48 (856–1,937) | 897.09 (848–947) |
information gain
Per-event log-score difference from the Omori–Utsu law. Above zero: the model assigned more probability to what actually happened.
every window, classical models
modelestimate [95% interval]
ETAS (fixed μ, subcritical)+0.133 [+0.020, +0.226]
ETAS (free)+0.131 [+0.017, +0.225]
Omori–Utsu + NB spread+0.012 [+0.003, +0.031]
windows where learned models forecast too
modelestimate [95% interval]
GBM-hybrid · cross/other_region + NB+0.040 [+0.024, +0.048]
GBM-hybrid · cross/pooled + NB+0.040 [+0.024, +0.048]
Omori–Utsu + NB spread+0.040 [+0.024, +0.049]
GBM-hybrid · cross/pooled+within + NB+0.040 [+0.024, +0.048]
GBM-hybrid · cross/chrono + NB+0.039 [+0.023, +0.049]
GBM · cross/other_region + NB+0.039 [+0.024, +0.047]
ETAS (fixed μ, subcritical)+0.038 [+0.023, +0.046]
Poisson GLM · cross/other_region + NB+0.038 [+0.023, +0.047]
Poisson MLP · cross/other_region + NB+0.038 [+0.022, +0.047]
GBM · cross/chrono + NB+0.038 [+0.023, +0.047]
Poisson GLM · cross/chrono + NB+0.038 [+0.024, +0.047]
Poisson GLM · cross/pooled + NB+0.037 [+0.024, +0.046]
Poisson MLP · cross/pooled + NB+0.037 [+0.021, +0.046]
GBM · cross/pooled + NB+0.036 [+0.023, +0.046]
Poisson MLP · cross/chrono + NB+0.035 [+0.021, +0.044]
ETAS (free)+0.033 [+0.013, +0.045]
GBM · within + NB+0.012 [−0.011, +0.042]
GBM-hybrid · cross/pooled+0.006 [−0.029, +0.017]
GBM-hybrid · cross/pooled+within+0.004 [−0.030, +0.014]
GBM-hybrid · cross/other_region+0.000 [−0.047, +0.017]
GBM-hybrid · cross/chrono+0.000 [−0.037, +0.014]
GBM · within−0.005 [−0.018, +0.021]
Poisson GLM · cross/pooled−0.007 [−0.034, +0.032]
GBM · cross/other_region−0.013 [−0.036, +0.017]
GBM · cross/pooled−0.020 [−0.047, +0.018]
Poisson GLM · cross/chrono−0.027 [−0.060, +0.021]
GBM-hybrid · within + NB−0.037 [−0.123, +0.042]
GBM · cross/chrono−0.044 [−0.071, −0.006]
Poisson GLM · within + NB−0.044 [−0.086, +0.042]
Poisson GLM · cross/other_region−0.062 [−0.108, +0.005]
GBM-hybrid · within−0.083 [−0.165, −0.027]
Poisson GLM · within−0.101 [−0.131, −0.070]
Poisson MLP · cross/other_region−0.187 [−0.297, −0.011]
Poisson MLP · cross/chrono−0.203 [−0.276, −0.089]
Poisson MLP · cross/pooled−0.254 [−0.316, −0.138]
calibration
Randomised probability integral transform over all windows. Flat at 1 is calibrated; a U means intervals too narrow, a hump means too wide.
Omori–Utsu
90% coverage 17% · 30 windows
Omori–Utsu + NB spread
90% coverage 57% · 30 windows
ETAS (fixed μ, subcritical)
90% coverage 87% · 30 windows
GBM-hybrid · within
90% coverage 13% · 15 windows
GBM-hybrid · cross/pooled
90% coverage 20% · 30 windows
where
Forecast density at day 30 for the next 7 days on the 2 km benchmark grid; dots are the 126 epicentres that followed.
KDE (adaptive)
ETAS kernel
Cell GBM
modelestimate [95% interval]
Cell GBM+0.046 [−0.011, +0.100]
KDE (fixed bandwidth)−0.053 [−0.082, −0.023]
ETAS kernel−0.332 [−0.562, +0.146]
fitted parameters
Refitted at every origin on data up to that origin. A parameter on its search bound is flagged (Omori with an asterisk, ETAS by name): early in a sequence the catalog is still incomplete, the rate looks flat, and neither likelihood identifies its decay parameters — the optimum sits on a bound.
| origin (d) | events | Omori p | Omori c (d) | ETAS α | ETAS p | ETAS n₃₆₅ | on bound | free ETAS μ/day | free n₃₆₅ |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 425 | 5.00* | 2.986 | 2.98 | 1.02 | 0.48 | — | 0.24 | 0.48 |
| 2 | 847 | 0.92 | 0.000 | 3.04 | 1.00 | 0.39 | — | 0.24 | 0.39 |
| 3 | 1,184 | 0.77 | 0.000 | 2.66 | 0.96 | 0.69 | — | 201.94 | 0.22 |
| 5 | 1,530 | 1.00 | 0.187 | 2.76 | 1.02 | 0.58 | — | 0.00 | 0.59 |
| 7 | 1,776 | 1.00 | 0.195 | 2.78 | 0.98 | 0.57 | — | 18.50 | 0.48 |
| 10 | 2,039 | 1.00 | 0.183 | 2.90 | 0.97 | 0.53 | — | 13.45 | 0.47 |
| 14 | 2,457 | 0.81 | 0.000 | 2.51 | 0.93 | 0.72 | — | 55.52 | 0.31 |
| 21 | 2,862 | 0.82 | 0.000 | 1.28 | 1.30 | 0.99 | — | 15.72 | 0.84 |
| 30 | 3,186 | 0.85 | 0.000 | 1.30 | 1.28 | 0.98 | — | 0.00 | 1.00 |
| 45 | 3,472 | 0.96 | 0.265 | 1.45 | 1.27 | 0.95 | — | 0.00 | 0.97 |
| 60 | 3,807 | 0.91 | 0.120 | 1.43 | 1.27 | 0.94 | — | 0.00 | 0.96 |
completeness
Mc per log-spaced time bin (bootstrap median of 200 resamples). The sequence Mc is the largest bin value.
| from (d) | to (d) | events, all M | Mc | bootstrap sd | estimator |
|---|---|---|---|---|---|
| 0.50 | 0.9 | 666 | 1.80 | 0.13 | gft95 |
| 0.92 | 1.7 | 935 | 1.60 | 0.10 | gft95 |
| 1.68 | 3.1 | 1,595 | 1.50 | 0.25 | gft95 |
| 3.08 | 5.6 | 1,109 | 1.80 | 0.17 | gft95 |
| 5.65 | 10.3 | 1,917 | 1.45 | 0.10 | gft95 |
| 10.35 | 19.0 | 3,310 | 1.40 | 0.26 | gft90 |
| 18.97 | 34.8 | 3,625 | 1.30 | 0.06 | gft95 |
| 34.77 | 63.7 | 4,618 | 1.30 | 0.14 | gft95 |