New Zealand · Canterbury, New Zealand

darfield 2010

mainshock: M7.2 mw · geonet:3366146
origin (UTC): 2010-09-03 16:35:41
epicentre: -43.527°, 172.168° depth 11 km
catalog: GeoNet / GNS Science (New Zealand) FDSN event service network gns
zone radius: 74.9 km Gardner–Knopoff window
completeness Mc: 2.8 protocol v3, max over time bins
aftershocks ≥ Mc: 4,641 largest M6.2
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 = 0.12)
Omori fitted on first 30 d (p = 1.06)
Omori fitted on first 120 d (p = 1.08)

Table view

days since mainshock	observed rate	Omori fitted on first 3 d (p = 0.12)	Omori fitted on first 30 d (p = 1.06)	Omori fitted on first 120 d (p = 1.08)
0.5	—	261.24	342.72	345.31
0.56	248.59	—	—	—
0.59	—	256.08	323.52	325.8
0.7	—	251.01	303.33	305.28
0.71	244.06	—	—	—
0.83	—	246.04	282.37	283.99
0.9	291.63	—	—	—
0.98	—	241.17	260.93	262.23
1.14	197	—	—	—
1.16	—	236.4	239.31	240.29
1.38	—	231.72	217.82	218.5
1.44	223.22	—	—	—
1.63	—	227.13	196.77	197.18
1.83	225.13	—	—	—
1.93	—	222.64	176.46	176.62
2.29	—	218.23	157.12	157.07
2.31	227.42	—	—	—
2.71	—	213.92	138.96	138.72
2.92	171.01	—	—	—
3.21	—	209.68	122.13	121.74
3.7	87.03	—	—	—
3.8	—	205.53	106.7	106.19
4.5	—	201.47	92.71	92.12
4.68	78.73	—	—	—
5.33	—	197.48	80.17	79.52
5.93	61.5	—	—	—
6.31	—	193.57	69.01	68.33
7.48	—	189.74	59.17	58.48
7.5	38.99	—	—	—
8.85	—	185.99	50.56	49.87
9.49	38.4	—	—	—
10.48	—	182.3	43.07	42.4
12.01	42.69	—	—	—
12.41	—	178.7	36.59	35.94
14.7	—	175.16	31.01	30.4
15.2	34.85	—	—	—
17.41	—	171.69	26.22	25.66
19.23	26	—	—	—
20.61	—	168.3	22.14	21.61
24.34	18.28	—	—	—
24.41	—	164.97	18.66	18.18
28.91	—	161.7	15.71	15.27
30.8	14.44	—	—	—
34.23	—	158.5	13.21	12.82
38.98	9.46	—	—	—
40.54	—	155.37	11.1	10.74
48	—	152.29	9.32	9
49.33	8.68	—	—	—
56.84	—	149.28	7.82	7.53
62.42	6.18	—	—	—
67.31	—	146.32	6.55	6.3
78.99	5.26	—	—	—
79.71	—	143.43	5.49	5.27
94.39	—	140.59	4.6	4.4
99.97	3.22	—	—	—
111.78	—	137.81	3.85	3.68
126.5	4.82	—	—	—
132.37	—	135.08	3.22	3.07
156.75	—	132.41	2.7	2.56
160.1	27.5	—	—	—
185.62	—	129.79	2.26	2.14
202.6	8.8	—	—	—
219.8	—	127.22	1.89	1.79
256.4	9.75	—	—	—
260.29	—	124.7	1.58	1.49
308.23	—	122.23	1.32	1.24
324.5	5.38	—	—	—
365	—	119.81	1.1	1.04

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	769	1,872.59 (1,802–1,944)	1,770.4 (1,331–2,260)	1,902.54 (1,831–1,975)
2	596	1,104.74 (1,050–1,160)	940.53 (745–1,155)	1,010.9 (959–1,063)
3	411	1,359.93 (1,300–1,421)	1,274.65 (1,026–1,545)	1,209.09 (1,152–1,267)
5	306	260.8 (235–288)	323.46 (73–718)	282.96 (256–311)
7	283	175.33 (154–197)	276.69 (71–590)	197.82 (175–221)
10	266	103.33 (87–120)	148.29 (25–355)	114.15 (97–132)
14	193	162.86 (142–184)	275.61 (84–556)	168.28 (147–190)
21	138	127.03 (109–146)	183.33 (69–342)	115.56 (98–134)
30	78	94.92 (79–111)	122.12 (42–235)	102.42 (86–119)
45	60	61.77 (49–75)	88.93 (30–171)	69.01 (56–83)
60	48	44.33 (34–56)	57.72 (14–124)	48.97 (38–61)
90	23	29.13 (21–38)	41.42 (9–93)	31.43 (23–41)
120	27	23.15 (16–31)	42.05 (5–109)	24.72 (17–33)
180	127	53.8 (42–66)	181.96 (26–452)	36.74 (27–47)
240	30	47.2 (36–59)	55.42 (0.5–222)	32.23 (23–42)

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	1,398	7,984.71 (7,838–8,132)	6,742.72 (3,780–10.398k)	8,060.14 (7,913–8,208)
2	1,197	3,553.41 (3,456–3,652)	2,182.54 (1,549–2,905)	3,157.4 (3,065–3,250)
3	985	5,238.53 (5,120–5,358)	4,428.88 (2,867–6,267)	4,499 (4,389–4,610)
5	839	352.65 (322–384)	596.55 (25–1,833)	382.62 (351–415)
7	724	249.04 (223–275)	505.5 (49–1,365)	282.48 (255–310)
10	641	160.12 (140–181)	280.99 (10–879)	177.83 (156–200)
14	515	435.8 (402–470)	613.54 (60–1,655)	450.3 (416–485)
21	390	386.22 (354–419)	665.67 (170–1,425)	351.33 (321–382)
30	279	311.24 (283–341)	560.44 (191–1,086)	335.84 (306–366)
45	215	215.52 (192–240)	414.14 (140–804)	240.76 (216–267)
60	164	160.44 (140–182)	306.11 (88–629)	177.24 (156–199)
90	134	110.65 (94–128)	214.98 (57–453)	119.37 (102–138)
120	106	90.6 (75–107)	215.89 (28–548)	96.76 (81–113)
180	306	222.41 (198–247)	688.83 (119–1,644)	151.88 (132–172)
240	131	196.88 (174–220)	276.51 (4–950)	134.45 (116–154)

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)+0.810 [+0.105, +1.150]

ETAS (free)+0.648 [+0.091, +0.909]

Omori–Utsu + NB spread+0.070 [−0.001, +0.214]

45 windows · 12,184 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]

Poisson GLM · cross/other_region + NB+0.137 [−0.002, +0.236]

Poisson MLP · cross/pooled + NB+0.137 [−0.004, +0.236]

Poisson GLM · cross/pooled + NB+0.137 [−0.002, +0.236]

GBM · cross/chrono + NB+0.136 [−0.005, +0.234]

GBM · cross/pooled + NB+0.136 [−0.005, +0.235]

Poisson GLM · cross/chrono + NB+0.136 [−0.004, +0.234]

GBM · cross/other_region + NB+0.136 [−0.005, +0.235]

ETAS (fixed μ, subcritical)+0.134 [−0.008, +0.233]

Poisson MLP · cross/other_region + NB+0.134 [−0.009, +0.233]

GBM-hybrid · cross/other_region + NB+0.133 [−0.007, +0.231]

Omori–Utsu + NB spread+0.133 [−0.008, +0.231]

GBM-hybrid · cross/pooled + NB+0.133 [−0.007, +0.230]

GBM-hybrid · cross/pooled+within + NB+0.133 [−0.008, +0.230]

GBM-hybrid · cross/chrono + NB+0.133 [−0.008, +0.230]

Poisson MLP · cross/chrono + NB+0.129 [−0.007, +0.229]

ETAS (free)+0.122 [−0.028, +0.225]

Poisson GLM · cross/other_region+0.118 [−0.001, +0.217]

GBM · cross/pooled+0.117 [−0.007, +0.218]

GBM · cross/other_region+0.116 [−0.006, +0.218]

Poisson GLM · cross/pooled+0.116 [−0.003, +0.218]

GBM · within + NB+0.112 [−0.019, +0.205]

Poisson MLP · cross/pooled+0.112 [−0.027, +0.210]

GBM · cross/chrono+0.105 [−0.007, +0.196]

Poisson GLM · cross/chrono+0.086 [−0.018, +0.173]

Poisson MLP · cross/other_region+0.067 [−0.130, +0.184]

GBM · within+0.052 [−0.139, +0.173]

Poisson MLP · cross/chrono+0.026 [−0.293, +0.191]

Poisson GLM · within + NB+0.019 [−0.009, +0.041]

GBM-hybrid · cross/other_region+0.005 [−0.047, +0.025]

GBM-hybrid · cross/pooled−0.001 [−0.069, +0.025]

GBM-hybrid · cross/pooled+within−0.014 [−0.071, +0.005]

GBM-hybrid · cross/chrono−0.016 [−0.077, +0.005]

GBM-hybrid · within + NB−0.044 [−0.105, +0.041]

Poisson GLM · within−0.092 [−0.334, +0.020]

GBM-hybrid · within−0.114 [−0.187, −0.032]

30 windows · 4,041 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 33% · 45 windows

Omori–Utsu + NB spread

90% coverage 67% · 45 windows

ETAS (fixed μ, subcritical)

90% coverage 80% · 45 windows

GBM-hybrid · within

90% coverage 37% · 30 windows

GBM-hybrid · cross/pooled

90% coverage 40% · 45 windows

where

Forecast density at day 30 for the next 7 days on the 2 km benchmark grid; dots are the 78 epicentres that followed.

KDE (adaptive)

ETAS kernel

Cell GBM

modelestimate [95% interval]

ETAS kernel+0.053 [−0.096, +0.304]

Cell GBM−0.053 [−0.081, +0.021]

KDE (fixed bandwidth)−0.109 [−0.120, −0.064]

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	134	0.05*	100.000	7.17	1.96	0.99	c	0.10	93.09
2	346	0.31	0.405	5.89	1.19	0.22	—	0.44	1.56
3	571	0.12	0.000	5.69	0.59	0.86	—	195.52	0.05
5	743	5.00*	13.436	0.37	4.50	0.96	p	0.48	0.96
7	873	5.00*	13.986	0.39	4.50	0.95	p	5.96	0.92
10	982	4.48	12.431	0.47	4.50	0.95	p	4.03	0.91
14	1,156	1.23	1.666	0.40	4.50	0.96	p	6.44	0.89
21	1,349	1.09	1.231	0.70	2.03	0.97	—	6.71	0.86
30	1,519	1.06	1.152	0.88	1.67	0.99	—	4.82	0.86
45	1,687	1.09	1.259	0.99	1.54	0.99	—	1.85	0.94
60	1,798	1.12	1.392	1.15	1.44	0.99	—	0.00	1.02
90	1,962	1.12	1.378	1.29	1.34	0.99	—	0.00	1.01
120	2,096	1.08	1.158	1.50	1.22	0.99	—	0.00	1.02
180	3,217	0.61	0.000	1.67	1.21	0.99	—	0.20	1.16
240	3,709	0.60	0.000	1.76	1.17	0.99	—	0.19	1.11

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	190	2.80	0.28	gft90
1.14	2.6	453	2.70	0.32	gft95
2.60	5.9	498	2.70	0.13	gft95
5.93	13.5	579	2.50	0.54	gft95
13.51	30.8	787	2.60	0.23	gft95
30.80	70.2	869	2.50	0.13	gft95
70.22	160.1	880	2.50	0.09	gft95
160.09	365.0	5,250	2.80	0.38	gft95