California · Parkfield, California

parkfield 2004

mainshock: M6.0 mw · nc30228270
origin (UTC): 2004-09-28 17:15:24
epicentre: 35.818°, -120.366° depth 8.1 km
catalog: USGS ComCat network nc
zone radius: 53.2 km Gardner–Knopoff window
completeness Mc: 1.2 protocol v3, max over time bins
aftershocks ≥ Mc: 2,666 largest M5.0
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 = 0.70)
Omori fitted on first 30 d (p = 0.79)
Omori fitted on first 120 d (p = 0.66)

Table view

days since mainshock	observed rate	Omori fitted on first 3 d (p = 0.70)	Omori fitted on first 30 d (p = 0.79)	Omori fitted on first 120 d (p = 0.66)
0.5	—	234.92	251.32	195.81
0.56	263.66	—	—	—
0.59	—	208.75	219.79	175.09
0.7	—	185.48	192.22	156.56
0.71	214.29	—	—	—
0.83	—	164.82	168.11	139.99
0.9	145.82	—	—	—
0.98	—	146.45	147.02	125.18
1.14	104.07	—	—	—
1.16	—	130.13	128.58	111.93
1.38	—	115.63	112.45	100.08
1.44	88.11	—	—	—
1.63	—	102.75	98.35	89.49
1.83	67.31	—	—	—
1.93	—	91.3	86.01	80.02
2.29	—	81.12	75.22	71.55
2.31	95.37	—	—	—
2.71	—	72.09	65.78	63.98
2.92	94.2	—	—	—
3.21	—	64.05	57.53	57.21
3.7	42.37	—	—	—
3.8	—	56.92	50.32	51.16
4.5	—	50.57	44	45.74
4.68	34.39	—	—	—
5.33	—	44.94	38.48	40.9
5.93	35.04	—	—	—
6.31	—	39.93	33.66	36.57
7.48	—	35.48	29.43	32.7
7.5	23.17	—	—	—
8.85	—	31.53	25.74	29.24
9.49	27.24	—	—	—
10.48	—	28.01	22.51	26.15
12.01	20.11	—	—	—
12.41	—	24.89	19.69	23.38
14.7	—	22.12	17.22	20.91
15.2	20.07	—	—	—
17.41	—	19.65	15.06	18.69
19.23	14.1	—	—	—
20.61	—	17.46	13.17	16.72
24.34	10.27	—	—	—
24.41	—	15.52	11.52	14.95
28.91	—	13.79	10.07	13.37
30.8	8.67	—	—	—
34.23	—	12.25	8.81	11.95
38.98	13.26	—	—	—
40.54	—	10.89	7.7	10.69
48	—	9.67	6.74	9.56
49.33	10.22	—	—	—
56.84	—	8.6	5.89	8.54
62.42	9.37	—	—	—
67.31	—	7.64	5.15	7.64
78.99	8.37	—	—	—
79.71	—	6.79	4.51	6.83
94.39	—	6.03	3.94	6.11
99.97	5.68	—	—	—
111.78	—	5.36	3.45	5.46
126.5	6.33	—	—	—
132.37	—	4.76	3.01	4.88
156.75	—	4.23	2.64	4.37
160.1	4.79	—	—	—
185.62	—	3.76	2.31	3.91
202.6	3.66	—	—	—
219.8	—	3.34	2.02	3.49
256.4	5.06	—	—	—
260.29	—	2.97	1.76	3.12
308.23	—	2.64	1.54	2.79
324.5	3.9	—	—	—
365	—	2.34	1.35	2.5

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	365	241.3 (216–267)	181.24 (101–280)	308.98 (280–338)
2	302	137.07 (118–157)	99.4 (51–160)	146.89 (127–167)
3	232	292.4 (265–321)	383.11 (271–511)	262.84 (236–290)
5	188	193.11 (171–216)	159.18 (92–241)	178.8 (157–201)
7	160	152.62 (133–173)	135.45 (74–211)	171.76 (150–194)
10	149	120.96 (103–139)	124.04 (68–193)	131.87 (113–151)
14	118	102.79 (86–120)	104.77 (59–161)	92.82 (77–109)
21	73	83.36 (69–99)	88.02 (50–135)	87.01 (72–103)
30	67	62.9 (50–76)	69.63 (24–134)	74.58 (61–89)
45	76	54.65 (43–67)	57.98 (26–100)	53.46 (42–66)
60	64	51.45 (40–64)	71.63 (41–108)	46.93 (36–58)
90	41	43.26 (33–54)	53.98 (29–84)	37.15 (27–47)
120	42	35.8 (26–46)	46.46 (24–73)	38.73 (29–49)
180	20	29.64 (21–39)	41.92 (14–80)	32.64 (24–42)
240	56	24.84 (17–33)	38.3 (19–62)	24.57 (17–33)

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	718	360.39 (329–392)	221.76 (128–335)	441.63 (407–476)
2	642	217.87 (194–242)	139.84 (82–210)	231.11 (206–256)
3	553	708.4 (665–752)	1,171.85 (826–1,567)	614.21 (574–655)
5	477	479.83 (444–516)	356.75 (221–518)	444.28 (410–479)
7	440	400.52 (368–434)	348.26 (217–504)	452.55 (418–488)
10	406	340.59 (311–371)	372.44 (231–540)	371.3 (340–403)
14	372	312.58 (284–342)	331.72 (209–477)	282.27 (255–310)
21	337	275.04 (248–303)	305.18 (176–462)	287.1 (260–315)
30	335	218.73 (195–243)	246.58 (129–394)	259.37 (233–286)
45	282	202.68 (180–226)	223.72 (126–343)	198.27 (175–222)
60	245	198.01 (175–221)	274.28 (185–377)	180.6 (159–203)
90	169	172.34 (151–194)	240.43 (155–340)	148.01 (128–168)
120	174	144.96 (125–165)	221.88 (139–320)	156.82 (137–178)
180	90	122.31 (104–141)	185.4 (96–298)	134.69 (116–154)
240	162	103.43 (87–120)	179.2 (113–257)	102.29 (86–119)

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)+0.071 [+0.005, +0.128]

ETAS (free)+0.051 [−0.037, +0.115]

Omori–Utsu + NB spread+0.009 [+0.000, +0.021]

45 windows · 7,764 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]

ETAS (fixed μ, subcritical)+0.024 [+0.006, +0.038]

Poisson GLM · cross/chrono + NB+0.022 [+0.005, +0.037]

Poisson GLM · cross/pooled + NB+0.022 [+0.005, +0.037]

GBM-hybrid · cross/other_region + NB+0.022 [+0.004, +0.038]

GBM-hybrid · cross/pooled+within + NB+0.022 [+0.005, +0.037]

Omori–Utsu + NB spread+0.022 [+0.005, +0.037]

GBM-hybrid · cross/pooled + NB+0.022 [+0.005, +0.037]

GBM · cross/chrono + NB+0.021 [+0.004, +0.036]

GBM · cross/pooled + NB+0.021 [+0.004, +0.035]

ETAS (free)+0.021 [−0.000, +0.037]

Poisson MLP · cross/chrono + NB+0.020 [+0.003, +0.036]

GBM-hybrid · cross/chrono + NB+0.018 [+0.002, +0.033]

Poisson MLP · cross/other_region + NB+0.018 [−0.000, +0.032]

Poisson GLM · cross/other_region + NB+0.018 [−0.001, +0.034]

GBM · cross/other_region + NB+0.018 [+0.001, +0.030]

GBM-hybrid · cross/other_region+0.017 [+0.004, +0.027]

GBM-hybrid · within + NB+0.012 [−0.014, +0.036]

Poisson GLM · cross/chrono+0.011 [−0.008, +0.026]

Poisson MLP · cross/pooled + NB+0.006 [−0.012, +0.023]

GBM-hybrid · cross/pooled−0.000 [−0.012, +0.012]

GBM-hybrid · cross/pooled+within−0.001 [−0.013, +0.012]

GBM · within + NB−0.008 [−0.047, +0.027]

GBM-hybrid · within−0.009 [−0.027, +0.008]

Poisson GLM · within + NB−0.013 [−0.056, +0.035]

Poisson MLP · cross/chrono−0.021 [−0.046, +0.020]

GBM · cross/chrono−0.022 [−0.040, +0.001]

Poisson GLM · cross/pooled−0.023 [−0.040, −0.004]

Poisson GLM · within−0.026 [−0.060, +0.014]

GBM-hybrid · cross/chrono−0.040 [−0.070, −0.015]

GBM · cross/pooled−0.049 [−0.075, −0.019]

GBM · within−0.050 [−0.091, −0.011]

Poisson MLP · cross/other_region−0.060 [−0.085, −0.048]

GBM · cross/other_region−0.087 [−0.134, −0.054]

Poisson GLM · cross/other_region−0.104 [−0.158, −0.062]

Poisson MLP · cross/pooled−0.349 [−0.406, −0.300]

30 windows · 3,397 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 47% · 45 windows

Omori–Utsu + NB spread

90% coverage 78% · 45 windows

ETAS (fixed μ, subcritical)

90% coverage 82% · 45 windows

GBM-hybrid · within

90% coverage 43% · 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 67 epicentres that followed.

KDE (adaptive)

ETAS kernel

Cell GBM

modelestimate [95% interval]

Cell GBM+0.125 [+0.050, +0.198]

KDE (fixed bandwidth)−0.151 [−0.244, −0.108]

ETAS kernel−0.754 [−1.852, −0.061]

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	ETAS α	ETAS p	ETAS n₃₆₅	on bound	free ETAS μ/day	free n₃₆₅
1	100	1.16	4.65	2.37	0.07	—	0.10	0.11
2	183	1.26	4.98	2.22	0.03	—	34.15	0.02
3	279	0.70	0.61	1.32	0.99	—	77.07	0.05
5	373	0.80	0.00	1.92	0.82	alpha	14.98	0.68
7	437	0.82	0.18	1.78	0.83	—	6.66	0.74
10	511	0.83	0.18	1.61	0.85	—	5.75	0.75
14	597	0.81	0.18	1.67	0.84	—	5.98	0.71
21	715	0.78	4.28	0.80	0.13	—	5.87	0.12
30	810	0.79	4.19	0.81	0.15	—	0.83	0.15
45	981	0.74	4.38	0.76	0.13	—	6.54	0.12
60	1,145	0.69	0.17	1.52	0.86	—	8.06	0.11
90	1,390	0.66	0.53	1.40	0.88	—	6.32	0.10
120	1,559	0.66	1.13	1.15	0.91	—	4.11	0.09
180	1,885	0.64	1.79	0.82	0.79	—	2.92	0.29
240	2,121	0.64	0.83	1.24	0.87	—	0.00	0.99

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	186	1.00	0.13	gft90
1.14	2.6	191	1.10	0.12	gft90
2.60	5.9	246	1.00	0.17	gft90
5.93	13.5	329	1.00	0.37	gft95
13.51	30.8	344	1.10	0.24	gft90
30.80	70.2	683	1.20	0.12	gft95
70.22	160.1	877	1.20	0.20	gft95
160.09	365.0	1,229	1.10	0.18	gft90