California · Central Coast, California

san simeon 2003

mainshock: M6.5 mw · nc21323712
origin (UTC): 2003-12-22 19:15:56
epicentre: 35.700°, -121.100° depth 8.4 km
catalog: USGS ComCat network nc
zone radius: 61.3 km Gardner–Knopoff window
completeness Mc: 2.1 protocol v3, max over time bins
aftershocks ≥ Mc: 2,150 largest M4.7
window: 0.5–280.9 daystruncated: aftershock zone of registered mainshock parkfield_2004 (M6.0) intersects; its origin is at day 280.92

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 = 0.52)
Omori fitted on first 30 d (p = 1.22)
Omori fitted on first 120 d (p = 1.00)

Table view

days since mainshock	observed rate	Omori fitted on first 3 d (p = 0.52)	Omori fitted on first 30 d (p = 1.22)	Omori fitted on first 120 d (p = 1.00)
0.5	—	338.19	314.66	355.84
0.56	394.14	—	—	—
0.59	—	310.89	298.54	329.65
0.69	—	285.79	281.41	303.38
0.7	276.65	—	—	—
0.81	—	262.72	263.43	277.39
0.88	205.62	—	—	—
0.96	—	241.51	244.8	252
1.1	236.01	—	—	—
1.13	—	222.02	225.76	227.51
1.32	—	204.1	206.57	204.17
1.38	210.58	—	—	—
1.56	—	187.62	187.51	182.2
1.73	129.79	—	—	—
1.83	—	172.48	168.86	161.72
2.16	—	158.55	150.85	142.85
2.17	146.15	—	—	—
2.54	—	145.75	133.73	125.61
2.73	171.63	—	—	—
2.98	—	133.99	117.66	110
3.42	89.11	—	—	—
3.51	—	123.17	102.79	95.97
4.13	—	113.23	89.2	83.46
4.28	67.99	—	—	—
4.85	—	104.09	76.92	72.36
5.37	59.16	—	—	—
5.71	—	95.69	65.95	62.58
6.71	—	87.96	56.24	53.99
6.73	70.12	—	—	—
7.9	—	80.86	47.74	46.49
8.44	38.16	—	—	—
9.29	—	74.33	40.35	39.97
10.58	39.19	—	—	—
10.93	—	68.33	33.97	34.3
12.85	—	62.82	28.5	29.4
13.27	35.58	—	—	—
15.12	—	57.75	23.84	25.17
16.64	23.61	—	—	—
17.78	—	53.09	19.89	21.53
20.86	13.33	—	—	—
20.92	—	48.8	16.56	18.4
24.61	—	44.86	13.76	15.72
26.15	11.48	—	—	—
28.94	—	41.24	11.41	13.41
32.78	12.25	—	—	—
34.04	—	37.91	9.45	11.44
40.04	—	34.85	7.81	9.76
41.1	6.12	—	—	—
47.1	—	32.04	6.45	8.32
51.53	8.05	—	—	—
55.41	—	29.45	5.33	7.09
64.61	5.53	—	—	—
65.17	—	27.07	4.39	6.04
76.66	—	24.89	3.62	5.14
81	7.14	—	—	—
90.17	—	22.88	2.98	4.38
101.5	3.82	—	—	—
106.06	—	21.03	2.45	3.73
124.76	—	19.33	2.02	3.17
127.3	2.5	—	—	—
146.75	—	17.77	1.66	2.7
159.6	2.1	—	—	—
172.61	—	16.34	1.36	2.3
200.1	1.83	—	—	—
203.04	—	15.02	1.12	1.95
238.82	—	13.81	0.92	1.66
250.9	1.41	—	—	—
280.92	—	12.69	0.76	1.41

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	681	128.16 (110–147)	155.27 (64–278)	83.07 (68–98)
2	537	481.26 (445–518)	556.11 (323–839)	518.55 (481–556)
3	414	653.73 (612–696)	755.3 (419–1,171)	645.09 (604–687)
5	335	231.18 (206–256)	458.54 (196–810)	256.69 (231–283)
7	270	292.41 (265–321)	395.24 (209–629)	341.73 (312–372)
10	241	193.57 (171–217)	272.59 (117–481)	215.72 (192–240)
14	165	172.12 (151–194)	219.44 (83–409)	185.54 (163–208)
21	78	119.32 (102–138)	176.52 (15–485)	129.32 (111–148)
30	86	67.69 (54–81)	120.91 (2–412)	80.34 (66–95)
45	57	45.29 (35–57)	80.6 (8–217)	53.75 (42–66)
60	39	36.43 (27–47)	62.93 (13–141)	40.2 (30–51)
90	36	30.39 (22–40)	57.82 (1–186)	30.88 (22–40)
120	16	22.42 (15–30)	36.15 (1–117)	24.73 (17–33)
180	12	14.27 (8–21)	33.13 (0.5–193)	17.06 (11–24)
240	7	10.46 (5–16)	18.72 (0.5–74)	12.53 (7–19)

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	1,178	129.2 (111–148)	159.43 (67–283)	55.72 (44–68)
2	1,008	1,035.28 (983–1,088)	1,329.53 (658–2,191)	1,206.01 (1,149–1,263)
3	861	1,808.21 (1,739–1,878)	2,248.7 (1,061–3,794)	1,731.61 (1,663–1,800)
5	730	382.25 (350–415)	1,544.99 (473–3,116)	423.95 (390–458)
7	619	764.84 (720–811)	1,316.93 (615–2,232)	893.86 (845–943)
10	513	497.97 (462–535)	1,015.01 (404–1,849)	554.95 (516–594)
14	394	505.17 (468–542)	842.39 (328–1,549)	544.55 (506–583)
21	283	372.85 (341–405)	747.27 (88–1,943)	404.13 (371–437)
30	247	214.64 (191–239)	549.31 (64–1,430)	254.74 (229–281)
45	196	155.32 (135–176)	399.21 (43–1,057)	184.33 (162–207)
60	199	131.8 (113–151)	330.15 (46–826)	145.42 (126–166)
90	107	117.01 (100–135)	264.3 (10–819)	122.65 (105–141)
120	65	88.3 (73–104)	185.02 (0.5–754)	97.4 (81–114)
180	47	57.61 (45–70)	129.55 (0.5–580)	68.87 (56–83)
240	42	42.8 (32–54)	98.12 (0.5–414)	51.29 (40–63)

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)+0.263 [+0.008, +0.378]

ETAS (free)+0.257 [+0.004, +0.373]

Omori–Utsu + NB spread+0.004 [+0.001, +0.015]

45 windows · 10,129 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/pooled + NB+0.014 [+0.001, +0.027]

Poisson GLM · cross/chrono + NB+0.013 [+0.003, +0.029]

GBM · cross/chrono + NB+0.012 [+0.000, +0.028]

Poisson GLM · cross/other_region + NB+0.012 [−0.005, +0.028]

Poisson MLP · cross/pooled + NB+0.012 [−0.001, +0.028]

Poisson MLP · cross/other_region + NB+0.012 [−0.004, +0.029]

Omori–Utsu + NB spread+0.011 [+0.001, +0.026]

GBM-hybrid · cross/chrono + NB+0.011 [−0.000, +0.026]

GBM · cross/pooled + NB+0.010 [−0.008, +0.023]

GBM · within + NB+0.008 [−0.004, +0.021]

GBM-hybrid · cross/pooled+within + NB+0.008 [−0.004, +0.024]

GBM-hybrid · cross/pooled + NB+0.008 [−0.004, +0.024]

ETAS (fixed μ, subcritical)+0.008 [−0.007, +0.023]

GBM · cross/other_region + NB+0.007 [−0.016, +0.022]

GBM-hybrid · cross/other_region + NB+0.006 [−0.006, +0.022]

ETAS (free)+0.004 [−0.013, +0.020]

Poisson GLM · within + NB+0.003 [−0.004, +0.015]

Poisson MLP · cross/chrono + NB+0.003 [−0.009, +0.021]

Poisson GLM · cross/pooled+0.000 [−0.031, +0.014]

Poisson MLP · cross/other_region+0.000 [−0.026, +0.025]

Poisson GLM · cross/other_region−0.000 [−0.044, +0.022]

GBM-hybrid · cross/chrono−0.003 [−0.013, +0.005]

GBM · cross/pooled−0.004 [−0.040, +0.011]

GBM-hybrid · cross/pooled+within−0.004 [−0.020, +0.010]

GBM · within−0.004 [−0.030, +0.009]

GBM-hybrid · cross/pooled−0.004 [−0.021, +0.011]

GBM · cross/other_region−0.006 [−0.050, +0.010]

GBM-hybrid · within + NB−0.009 [−0.028, +0.016]

Poisson GLM · cross/chrono−0.019 [−0.054, +0.012]

GBM · cross/chrono−0.024 [−0.043, +0.001]

Poisson MLP · cross/pooled−0.034 [−0.118, −0.003]

GBM-hybrid · within−0.035 [−0.069, −0.012]

Poisson GLM · within−0.038 [−0.078, −0.019]

GBM-hybrid · cross/other_region−0.041 [−0.086, −0.006]

Poisson MLP · cross/chrono−0.164 [−0.228, −0.118]

30 windows · 2,956 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 51% · 45 windows

Omori–Utsu + NB spread

90% coverage 78% · 45 windows

ETAS (fixed μ, subcritical)

90% coverage 89% · 45 windows

GBM-hybrid · within

90% coverage 53% · 30 windows

GBM-hybrid · cross/pooled

90% coverage 49% · 45 windows

where

Forecast density at day 30 for the next 7 days on the 2 km benchmark grid; dots are the 86 epicentres that followed.

KDE (adaptive)

ETAS kernel

Cell GBM

modelestimate [95% interval]

Cell GBM−0.021 [−0.067, +0.058]

KDE (fixed bandwidth)−0.036 [−0.063, −0.001]

ETAS kernel−0.217 [−0.401, −0.157]

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	139	5.00*	1.996	2.82	4.50	0.14	p	0.05	0.14
2	320	0.77	0.000	2.40	0.95	0.77	—	0.49	0.76
3	481	0.52	0.000	0.87	1.64	0.99	—	90.97	0.66
5	638	2.15	3.493	1.04	1.62	0.99	—	1.86	1.36
7	770	0.85	0.327	0.81	1.50	0.99	—	24.86	0.92
10	895	1.09	0.908	0.94	1.47	0.99	—	1.34	1.27
14	1,040	0.92	0.455	1.07	1.41	0.99	—	1.99	1.25
21	1,205	0.98	0.649	2.28	0.93	0.99	—	0.00	1.19
30	1,303	1.22	1.496	2.13	1.00	0.99	—	0.00	1.10
45	1,441	1.19	1.413	2.00	1.02	0.99	—	0.00	1.04
60	1,550	1.12	1.115	2.01	1.01	0.98	—	0.00	0.99
90	1,749	0.98	0.542	2.07	0.99	0.94	—	0.00	0.94
120	1,856	1.00	0.607	2.06	1.00	0.90	—	0.00	0.90
180	1,994	1.02	0.711	2.11	1.01	0.86	—	0.00	0.87
240	2,093	1.03	0.754	2.13	1.02	0.84	—	0.00	0.84

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	328	2.00	0.12	gft95
1.10	2.4	452	2.00	0.11	gft95
2.43	5.4	657	1.90	0.29	gft90
5.37	11.9	749	2.00	0.12	gft90
11.85	26.2	905	2.10	0.13	gft90
26.15	57.7	814	2.10	0.07	gft95
57.70	127.3	1,865	2.00	0.30	gft95
127.31	280.9	2,909	1.30	0.18	gft95