Central Italy · Central Apennines, Italy

l’aquila 2009

mainshock: M6.1 mw · ingv:1895389
origin (UTC): 2009-04-06 01:32:40
epicentre: 42.342°, 13.380° depth 8.3 km
catalog: INGV (Italy) FDSN event service network ingv
zone radius: 54.7 km Gardner–Knopoff window
completeness Mc: 2.1 protocol v3, max over time bins
aftershocks ≥ Mc: 2,183 largest M5.4
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 = 1.08)
Omori fitted on first 30 d (p = 1.22)
Omori fitted on first 120 d (p = 1.18)

Table view

days since mainshock	observed rate	Omori fitted on first 3 d (p = 1.08)	Omori fitted on first 30 d (p = 1.22)	Omori fitted on first 120 d (p = 1.18)
0.5	—	766.84	598.98	633.38
0.56	647.84	—	—	—
0.59	—	638.48	544.73	568.79
0.7	—	531.6	491.11	506.58
0.71	535.73	—	—	—
0.83	—	442.62	438.95	447.55
0.9	465.67	—	—	—
0.98	—	368.53	388.99	392.36
1.14	234.17	—	—	—
1.16	—	306.84	341.86	341.46
1.38	—	255.48	298.07	295.14
1.44	234.97	—	—	—
1.63	—	212.71	257.94	253.49
1.83	211.2	—	—	—
1.93	—	177.11	221.65	216.46
2.29	—	147.46	189.24	183.87
2.31	146.72	—	—	—
2.71	—	122.78	160.61	155.46
2.92	181.15	—	—	—
3.21	—	102.23	135.58	130.88
3.7	140.86	—	—	—
3.8	—	85.11	113.91	109.78
4.5	—	70.87	95.29	91.78
4.68	90.49	—	—	—
5.33	—	59	79.41	76.52
5.93	63.64	—	—	—
6.31	—	49.13	65.96	63.63
7.48	—	40.9	54.63	52.81
7.5	63.85	—	—	—
8.85	—	34.06	45.13	43.74
9.49	54.47	—	—	—
10.48	—	28.36	37.2	36.17
12.01	29.99	—	—	—
12.41	—	23.61	30.61	29.87
14.7	—	19.66	25.14	24.64
15.2	18.4	—	—	—
17.41	—	16.37	20.63	20.31
19.23	18.73	—	—	—
20.61	—	13.63	16.9	16.72
24.34	14.28	—	—	—
24.41	—	11.35	13.83	13.76
28.91	—	9.45	11.31	11.31
30.8	7.84	—	—	—
34.23	—	7.87	9.24	9.3
38.98	5.33	—	—	—
40.54	—	6.55	7.55	7.64
48	—	5.45	6.16	6.27
49.33	3.69	—	—	—
56.84	—	4.54	5.03	5.15
62.42	2.58	—	—	—
67.31	—	3.78	4.1	4.22
78.99	5.52	—	—	—
79.71	—	3.15	3.34	3.47
94.39	—	2.62	2.73	2.84
99.97	3.86	—	—	—
111.78	—	2.18	2.22	2.33
126.5	2.24	—	—	—
132.37	—	1.82	1.81	1.91
156.75	—	1.51	1.47	1.57
160.1	1.14	—	—	—
185.62	—	1.26	1.2	1.29
202.6	0.79	—	—	—
219.8	—	1.05	0.98	1.05
256.4	0.48	—	—	—
260.29	—	0.87	0.8	0.86
308.23	—	0.73	0.65	0.71
324.5	0.6	—	—	—
365	—	0.61	0.53	0.58

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	841	445.07 (411–480)	816.91 (746–890)	692 (649–736)
2	686	461.46 (426–497)	596.13 (245–1,071)	455.13 (420–490)
3	603	378.04 (346–410)	669.49 (290–1,175)	454.61 (420–490)
5	407	426.31 (393–461)	232.47 (69–472)	412.44 (379–446)
7	318	310 (281–339)	252.64 (90–482)	361.25 (330–393)
10	171	281.66 (254–310)	316.81 (154–527)	323.55 (294–353)
14	129	183.91 (162–206)	168.66 (61–318)	206.47 (183–230)
21	100	101.78 (85–119)	120.9 (42–233)	120.34 (103–139)
30	57	66.72 (54–80)	80.06 (26–157)	80.09 (66–95)
45	20	33.68 (24–43)	39.28 (7–91)	40.42 (30–51)
60	22	22.1 (15–30)	34.59 (5–84)	26.23 (18–35)
90	30	17.65 (11–25)	33.84 (7–77)	19.98 (13–28)
120	14	14.53 (9–21)	26.33 (3–66)	12.96 (7–19)
180	3	8.84 (4–14)	13.51 (0.5–45)	11.51 (6–17)
240	4	5.9 (2–10)	10.16 (0.5–32)	7 (3–12)

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	1,335	455.95 (421–491)	1,237.84 (1,082–1,402)	1,166.17 (1,110–1,223)
2	1,111	782.59 (737–829)	961.95 (358–1,802)	810.05 (764–857)
3	988	717.15 (673–761)	1,251.86 (474–2,330)	877.15 (829–926)
5	729	1,020.84 (969–1,074)	392.34 (132–764)	1,011.87 (960–1,064)
7	620	774.35 (729–820)	543.32 (186–1,051)	905.81 (857–956)
10	429	784.03 (738–830)	939.19 (458–1,558)	901.69 (853–951)
14	331	527.98 (490–566)	541.39 (224–970)	592.74 (553–633)
21	220	300.28 (272–329)	414.24 (176–733)	355.02 (324–386)
30	143	210.32 (187–234)	313.54 (133–555)	252.43 (227–279)
45	92	111.42 (94–129)	180.91 (62–350)	133.73 (115–153)
60	132	76.59 (62–91)	147.89 (47–294)	90.9 (76–107)
90	103	66.08 (53–80)	139.11 (39–289)	74.81 (61–89)
120	62	56.36 (44–69)	110.51 (24–249)	50.27 (39–62)
180	27	35.32 (26–45)	65.54 (9–164)	45.99 (35–57)
240	15	23.92 (16–32)	52.97 (2–158)	28.39 (20–37)

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)+0.111 [+0.038, +0.156]

ETAS (free)+0.056 [+0.034, +0.079]

Omori–Utsu + NB spread+0.021 [+0.003, +0.087]

45 windows · 10,531 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]

Omori–Utsu + NB spread+0.095 [+0.032, +0.128]

Poisson GLM · cross/other_region + NB+0.094 [+0.030, +0.128]

GBM-hybrid · cross/chrono + NB+0.093 [+0.030, +0.126]

GBM-hybrid · cross/pooled+within + NB+0.092 [+0.028, +0.126]

Poisson GLM · cross/pooled + NB+0.091 [+0.021, +0.125]

GBM-hybrid · cross/other_region + NB+0.091 [+0.026, +0.125]

GBM-hybrid · cross/pooled + NB+0.091 [+0.027, +0.124]

Poisson MLP · cross/pooled + NB+0.091 [+0.021, +0.126]

ETAS (fixed μ, subcritical)+0.091 [+0.024, +0.125]

Poisson MLP · cross/other_region + NB+0.090 [+0.022, +0.125]

GBM · cross/chrono + NB+0.088 [+0.012, +0.123]

Poisson MLP · cross/chrono + NB+0.088 [+0.017, +0.123]

GBM · cross/other_region + NB+0.085 [−0.001, +0.120]

GBM · cross/pooled + NB+0.084 [−0.001, +0.118]

Poisson GLM · cross/chrono + NB+0.083 [+0.014, +0.119]

ETAS (free)+0.080 [+0.016, +0.112]

Poisson GLM · cross/other_region+0.077 [−0.026, +0.123]

GBM · cross/chrono+0.058 [−0.066, +0.102]

Poisson MLP · cross/other_region+0.050 [−0.077, +0.100]

Poisson MLP · cross/pooled+0.046 [−0.037, +0.084]

GBM · within + NB+0.030 [+0.003, +0.047]

Poisson GLM · cross/pooled+0.026 [−0.037, +0.056]

GBM · cross/other_region+0.020 [−0.147, +0.075]

Poisson MLP · cross/chrono+0.004 [−0.076, +0.038]

GBM · within−0.010 [−0.092, +0.021]

GBM · cross/pooled−0.024 [−0.154, +0.017]

GBM-hybrid · cross/chrono−0.035 [−0.046, −0.001]

GBM-hybrid · within + NB−0.040 [−0.094, +0.051]

GBM-hybrid · cross/pooled−0.072 [−0.094, −0.006]

Poisson GLM · within + NB−0.075 [−0.133, +0.050]

GBM-hybrid · cross/pooled+within−0.076 [−0.098, −0.011]

GBM-hybrid · cross/other_region−0.090 [−0.120, −0.000]

GBM-hybrid · within−0.091 [−0.125, −0.004]

Poisson GLM · within−0.099 [−0.149, −0.004]

Poisson GLM · cross/chrono−0.156 [−0.280, −0.105]

30 windows · 2,197 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 82% · 45 windows

ETAS (fixed μ, subcritical)

90% coverage 96% · 45 windows

GBM-hybrid · within

90% coverage 47% · 30 windows

GBM-hybrid · cross/pooled

90% coverage 33% · 45 windows

where

Forecast density at day 30 for the next 7 days on the 2 km benchmark grid; dots are the 57 epicentres that followed.

KDE (adaptive)

ETAS kernel

Cell GBM

modelestimate [95% interval]

ETAS kernel+0.217 [+0.098, +0.495]

Cell GBM+0.155 [+0.025, +0.524]

KDE (fixed bandwidth)−0.167 [−0.373, −0.078]

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	270	5.00*	3.556	6.18	1.23	0.01	—	0.05	0.00
2	502	1.15	0.049	0.66	1.87	0.94	—	0.10	0.94
3	638	1.08	0.000	0.85	1.71	0.95	—	0.00	0.95
5	908	0.87	0.000	1.05	2.04	0.89	—	0.00	0.89
7	1,036	0.92	0.000	1.07	1.80	0.91	—	5.65	0.87
10	1,241	0.86	0.000	1.02	1.51	0.95	—	6.20	0.89
14	1,354	0.96	0.076	1.09	1.55	0.93	—	0.97	0.92
21	1,483	1.17	0.511	1.13	1.54	0.92	—	1.97	0.89
30	1,599	1.22	0.636	1.21	1.49	0.90	—	2.41	0.85
45	1,685	1.39	1.161	1.27	1.53	0.88	—	0.00	0.89
60	1,742	1.43	1.280	1.33	1.51	0.87	—	0.40	0.86
90	1,874	1.25	0.671	1.37	1.52	0.87	—	0.29	0.87
120	1,977	1.18	0.463	1.44	1.49	0.85	—	0.47	0.85
180	2,070	1.19	0.505	1.56	1.48	0.82	—	0.40	0.81
240	2,117	1.22	0.591	1.57	1.49	0.81	—	0.22	0.81

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	529	2.05	0.08	gft95
1.14	2.6	892	1.80	0.24	gft90
2.60	5.9	1,741	1.60	0.10	gft90
5.93	13.5	2,753	1.60	0.07	gft95
13.51	30.8	2,728	1.50	0.09	gft95
30.80	70.2	2,679	1.50	0.14	gft95
70.22	160.1	3,806	1.30	0.06	gft95
160.09	365.0	1,888	1.65	0.16	gft95