Central Italy · Central Apennines, Italy

norcia 2016

mainshock: M6.5 mw · ingv:8863681
origin (UTC): 2016-10-30 06:40:17
epicentre: 42.830°, 13.109° depth 10 km
catalog: INGV (Italy) FDSN event service network ingv
zone radius: 61.3 km Gardner–Knopoff window
completeness Mc: 2.3 protocol v3, max over time bins
aftershocks ≥ Mc: 3,778 largest M5.5
window: 0.5–365 daysfull year

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 = 2.75)
Omori fitted on first 30 d (p = 0.94)
Omori fitted on first 120 d (p = 0.79)

Table view

days since mainshock	observed rate	Omori fitted on first 3 d (p = 2.75)	Omori fitted on first 30 d (p = 0.94)	Omori fitted on first 120 d (p = 0.79)
0.5	—	823.45	838.23	634.63
0.56	813.56	—	—	—
0.59	—	741.51	714.84	555.14
0.7	—	658.21	609.6	485.6
0.71	619.07	—	—	—
0.83	—	575.23	519.86	424.77
0.9	512.71	—	—	—
0.98	—	494.38	443.33	371.56
1.14	420.01	—	—	—
1.16	—	417.41	378.07	325.02
1.38	—	345.95	322.41	284.31
1.44	317.21	—	—	—
1.63	—	281.27	274.95	248.69
1.83	255.3	—	—	—
1.93	—	224.25	234.48	217.54
2.29	—	175.32	199.96	190.29
2.31	196.24	—	—	—
2.71	—	134.43	170.52	166.46
2.92	107.24	—	—	—
3.21	—	101.13	145.42	145.61
3.7	114.52	—	—	—
3.8	—	74.71	124.01	127.37
4.5	—	54.24	105.76	111.41
4.68	110.4	—	—	—
5.33	—	38.74	90.19	97.46
5.93	82.95	—	—	—
6.31	—	27.25	76.91	85.25
7.48	—	18.91	65.59	74.57
7.5	60.46	—	—	—
8.85	—	12.96	55.93	65.23
9.49	44.65	—	—	—
10.48	—	8.78	47.7	57.06
12.01	40.93	—	—	—
12.41	—	5.89	40.68	49.91
14.7	—	3.91	34.69	43.66
15.2	35.41	—	—	—
17.41	—	2.58	29.58	38.19
19.23	25.12	—	—	—
20.61	—	1.69	25.23	33.41
24.34	18.98	—	—	—
24.41	—	1.1	21.51	29.22
28.91	—	0.71	18.35	25.56
30.8	23.25	—	—	—
34.23	—	0.46	15.65	22.36
38.98	15.33	—	—	—
40.54	—	0.29	13.34	19.56
48	—	0.19	11.38	17.11
49.33	12.63	—	—	—
56.84	—	0.12	9.7	14.97
62.42	7.74	—	—	—
67.31	—	0.077	8.28	13.09
78.99	33.36	—	—	—
79.71	—	0.049	7.06	11.45
94.39	—	0.031	6.02	10.02
99.97	7.38	—	—	—
111.78	—	0.02	5.13	8.76
126.5	4.32	—	—	—
132.37	—	0.012	4.38	7.66
156.75	—	0.0078	3.73	6.7
160.1	3.65	—	—	—
185.62	—	0.0049	3.18	5.86
202.6	2.61	—	—	—
219.8	—	0.0031	2.71	5.13
256.4	1.59	—	—	—
260.29	—	0.002	2.31	4.49
308.23	—	0.0012	1.97	3.93
324.5	1.03	—	—	—
365	—	0.00078	1.68	3.43

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	942	769.87 (725–816)	838.78 (608–1,099)	1,003.09 (951–1,055)
2	659	731.6 (687–776)	795.96 (452–1,218)	869.12 (821–918)
3	550	249.61 (224–276)	502 (145–1,032)	280.79 (254–309)
5	397	375.95 (344–408)	415.14 (93–925)	440.12 (406–475)
7	327	309.01 (280–338)	331.73 (147–575)	313.71 (285–343)
10	271	237.65 (213–263)	255.74 (125–423)	246.26 (221–272)
14	204	189.47 (167–212)	217.77 (57–462)	210.77 (187–235)
21	136	139.15 (120–159)	153.79 (17–405)	150.88 (131–171)
30	114	112.15 (95–130)	167.4 (3–568)	116.16 (99–134)
45	97	81.88 (67–97)	108.87 (1–396)	85.14 (70–101)
60	44	64.11 (51–78)	72.96 (23–146)	71.17 (58–85)
90	68	77.17 (63–92)	106.11 (39–200)	59.69 (47–73)
120	28	56.69 (45–69)	58.78 (0.5–265)	44.57 (34–56)
180	18	35.38 (26–45)	41.37 (0.5–215)	39.62 (30–50)
240	14	24.93 (17–33)	21.48 (0.5–75)	27.65 (19–37)

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	1,647	1,110.7 (1,056–1,166)	1,251.54 (886–1,668)	1,562.9 (1,498–1,628)
2	1,333	1,379.31 (1,319–1,441)	1,530.14 (984–2,174)	1,891.66 (1,820–1,963)
3	1,187	305.95 (277–335)	961.13 (279–1,975)	338.16 (308–369)
5	993	824.31 (777–872)	907.23 (273–1,840)	984.65 (933–1,037)
7	863	740.18 (696–785)	822.84 (366–1,426)	745.88 (701–791)
10	745	620.05 (579–661)	705.21 (241–1,366)	639.04 (598–681)
14	640	538.03 (500–576)	617.76 (263–1,093)	595.27 (555–636)
21	542	432.45 (399–467)	494.48 (160–977)	468.91 (434–505)
30	408	375.93 (344–408)	566.3 (81–1,414)	389.35 (357–422)
45	268	293.01 (265–321)	419.08 (12–1,344)	304.69 (276–334)
60	705	238.06 (213–264)	302.62 (102–588)	264.29 (238–291)
90	198	304 (276–333)	380.45 (157–683)	235.15 (210–261)
120	107	227.04 (203–252)	235.31 (6–763)	178.51 (157–201)
180	80	144.32 (125–164)	152.08 (2–519)	161.64 (141–183)
240	60	102.75 (86–120)	106.66 (5–317)	113.97 (97–132)

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)+0.095 [+0.017, +0.197]

ETAS (free)+0.095 [+0.017, +0.197]

Omori–Utsu + NB spread+0.028 [+0.003, +0.110]

45 windows · 14,586 events · moving-block bootstrap. Naive references (constant rate, persistence, uniform density) score far lower and are left out of the plot; they appear in the tables.

windows where learned models forecast too

modelestimate [95% interval]

GBM-hybrid · cross/pooled + NB+0.083 [+0.012, +0.231]

GBM-hybrid · cross/pooled+within + NB+0.083 [+0.012, +0.231]

GBM-hybrid · cross/chrono + NB+0.083 [+0.012, +0.231]

Omori–Utsu + NB spread+0.083 [+0.012, +0.231]

GBM-hybrid · cross/other_region + NB+0.082 [+0.012, +0.230]

Poisson MLP · cross/chrono + NB+0.081 [+0.012, +0.223]

Poisson MLP · cross/pooled + NB+0.081 [+0.012, +0.225]

Poisson MLP · cross/other_region + NB+0.080 [+0.010, +0.228]

ETAS (free)+0.080 [+0.009, +0.231]

GBM · cross/other_region + NB+0.080 [+0.011, +0.224]

Poisson GLM · cross/other_region + NB+0.080 [+0.010, +0.227]

ETAS (fixed μ, subcritical)+0.079 [+0.007, +0.230]

Poisson GLM · cross/pooled + NB+0.076 [+0.013, +0.208]

GBM · cross/pooled + NB+0.076 [+0.011, +0.212]

Poisson GLM · cross/chrono + NB+0.076 [+0.013, +0.208]

GBM · cross/chrono + NB+0.076 [+0.008, +0.217]

GBM-hybrid · within + NB+0.074 [+0.008, +0.211]

Poisson GLM · within + NB+0.074 [+0.011, +0.204]

GBM · within + NB+0.052 [−0.031, +0.223]

Poisson MLP · cross/chrono+0.021 [−0.003, +0.064]

GBM-hybrid · cross/pooled+0.019 [+0.004, +0.051]

GBM-hybrid · cross/pooled+within+0.017 [−0.000, +0.049]

GBM-hybrid · cross/chrono+0.014 [−0.003, +0.050]

Poisson MLP · cross/pooled+0.001 [−0.011, +0.009]

GBM-hybrid · cross/other_region−0.006 [−0.077, +0.043]

GBM · cross/pooled−0.009 [−0.021, +0.002]

Poisson GLM · cross/pooled−0.018 [−0.052, −0.004]

GBM · cross/chrono−0.020 [−0.034, −0.001]

GBM · cross/other_region−0.028 [−0.055, −0.015]

Poisson GLM · cross/chrono−0.031 [−0.085, −0.007]

Poisson GLM · within−0.062 [−0.245, +0.023]

Poisson GLM · cross/other_region−0.064 [−0.095, −0.017]

GBM · within−0.077 [−0.189, −0.033]

GBM-hybrid · within−0.086 [−0.362, +0.008]

Poisson MLP · cross/other_region−0.100 [−0.188, −0.045]

30 windows · 4,940 events. Within-sequence models abstain until 24 training windows have closed.

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 44% · 45 windows

Omori–Utsu + NB spread

90% coverage 76% · 45 windows

ETAS (fixed μ, subcritical)

90% coverage 98% · 45 windows

GBM-hybrid · within

90% coverage 33% · 30 windows

GBM-hybrid · cross/pooled

90% coverage 47% · 45 windows

where

Forecast density at day 30 for the next 7 days on the 2 km benchmark grid; dots are the 114 epicentres that followed.

KDE (adaptive)

ETAS kernel

Cell GBM

modelestimate [95% interval]

Cell GBM+0.187 [−0.256, +0.466]

KDE (fixed bandwidth)+0.042 [−0.049, +0.129]

ETAS kernel−0.294 [−0.570, +0.002]

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	314	1.25	0.126	3.50	1.27	0.10	—	2.25	0.09
2	641	0.96	0.000	3.36	1.00	0.14	—	95.84	0.13
3	799	2.75	1.872	3.46	1.17	0.16	—	0.00	0.16
5	1,030	1.03	0.000	3.30	1.10	0.19	—	35.23	0.16
7	1,188	1.01	0.000	3.27	1.07	0.20	—	28.24	0.16
10	1,349	1.00	0.000	3.23	1.06	0.20	—	15.92	0.17
14	1,515	0.99	0.000	3.04	1.05	0.27	—	13.66	0.21
21	1,719	0.97	0.000	3.04	1.03	0.26	—	7.54	0.22
30	1,937	0.94	0.000	2.44	1.08	0.63	—	5.18	0.53
45	2,176	0.93	0.000	2.56	1.04	0.54	—	4.45	0.44
60	2,345	0.93	0.000	2.48	1.05	0.58	—	2.44	0.52
90	3,050	0.76	0.000	2.37	1.10	0.63	—	2.01	0.56
120	3,248	0.79	0.000	2.42	1.12	0.61	—	0.10	0.61
180	3,478	0.83	0.000	2.31	1.15	0.68	—	0.00	0.69
240	3,617	0.86	0.000	2.05	1.20	0.79	—	0.00	0.75

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	1.1	947	2.30	0.20	gft95
1.14	2.6	2,578	2.00	0.14	gft95
2.60	5.9	2,016	1.90	0.03	gft95
5.93	13.5	3,623	1.80	0.16	gft95
13.51	30.8	5,594	1.70	0.05	gft95
30.80	70.2	11,466	1.50	0.02	gft95
70.22	160.1	16,465	1.40	0.02	gft95
160.09	365.0	11,992	1.20	0.04	gft95