Observed Magnitude Frequency Distributions
K. R. Felzer
Seismicity averaged over the state of California or over any large region therein has
been found to robustly follow the Gutenberg-Richter magnitude frequency
distribution (Gutenberg and Richter, 1944) with b=1.0 (Felzer, 2006; Hutton, 2010).
Assuming this magnitude-frequency distribution and assuming that it ends abruptly
at some maximum magnitude Mmax, we calculate average annual seismicity rates
for the time periods 1850-9/30/2011 (full duration of useable catalog) and
1984/1/1 – 9/30/2011 (modern instrumental catalog), for the entire UCERF3 study
area and for the northern and southern portions. For catalog completeness
estimates we use the thresholds calculated by Felzer (2008) for the state as a whole,
with the exception that we dropped the completeness threshold from 1850-1865
from 8.0 to 7.4, based on re-assessment that an M >7.4 earthquake would probably
have been felt over a sufficiently wide area of the state to have been noted, despite
the sparse contemporary population (Table 1). We calculate both magnitudeaveraged rates, which use the assumption of a Gutenberg-Richter distribution, b
value, and Mmax, and direct count rates, which do not use any of these assumptions.
These rate calculations are explained in detail below:
Calculation of magnitude-averaged rates:




The catalog was corrected for magnitude rounding and magnitude errors
using the methods given in Felzer, 2008.
The number of earthquakes ≥ the completeness magnitude, Mc, was counted
for each completeness era.
Each count was converted into a rate of 5≤M≤Mmax earthquakes/year by
using the Gutenberg-Richter distribution with b=1.
The 5≤M≤Mmax rates for each era were averaged together with each era
weighted linearly by its duration. This contrasts with the Weichert (1980)
method used in National Seismic Hazard Maps in which each era is weighted
by the number of earthquakes it contains. If the underlying rate is the same
in each era the Weichert (1980) method produces a more accurate result;
however if the seismicity rate might change with time, which is plausible,
then the Weichert (1980) method produces an inaccurate result that is
heavily skewed towards the instrumental era.
Calculation of direct count rates:


The largest magnitude completeness threshold is chosen for each data set.
For example, if we are looking at a particular region from 1850-2011, and
that region was only complete to M 6.5 before 1880, then M 6.5 will be
chosen as the completeness threshold for the entire data set.
The total number of earthquakes occurring in each half magnitude unit above
the data set’s completeness threshold are counted.

No magnitude rounding or error corrections are performed, as these require
assumptions about the magnitude-frequency distribution. Likewise the
counts given here are incremental only, not cumulative, so a value of Mmax
does not need to be assumed.
The stationary Poissonian function is used to calculate uncertainties for both sets of
rates. For the direct count rates this calculation is straightforward; for the
magnitude-averaged rates separate uncertainties are calculated for each
completeness era, producing a lowest and highest possible rate (at 95% confidence,
two-tailed) for each time period. These lowest and highest rates are then averaged
together with weighting by the length of the time period that each represents, in the
same way that the rates from each completeness era are averaged together. The
resulting confidence intervals represent the range of true underlying seismicity
rates for each catalog that is being analyzed. For example, let’s say we were looking
at a single year and we observed 20 earthquakes during that year; the Poissonian
function tells us that a true underlying rate of 12.5 to 30 earthquakes/year during
could have produced our observation (at 95% confidence). The confidence
intervals that we calculate do not constrain what may occur in other time periods, as
the time scales over which seismicity rates in California may show systematic
variations is not known. As a test we looked at decadal time periods in southern
California and the full state from 1932-2011, and found that more from decade to
decade than would be expected from a pure stationary Poissonian.
Finally for the calculation of the magnitude-averaged rates we need a value for
Mmax. The maximum magnitude for the state can be estimated given a seismicity
rate at small magnitudes and average long term seismic moment release rate. We
estimated a minimum seismicity rate for 1850-2011 by assuming infinite maximum
magnitude; this gives 8 M≥5 earthquakes/year; and then used the estimates of Ward
(1998) for statewide seismic moment rate. Given the uncertainty on the moment
rate we recovered a maximum magnitude between 8.0 – 8.25, which is in agreement
with the estimate of Bird and Kagan (2004) that continental transform boundaries
have a corner magnitude of 8.01 -0.21 +0.47. The larger magnitude is in better
agreement with the grand inversion, which has possible magnitudes of 8.5 or larger.
Therefore the preferred rates below use Mmax = 8.25 but the uncertainties include
uncertainties calculated with Mmax = 8.0.
Note 1: All confidence limits given in the tables below are two-tailed 95%
confidence estimates.
Note 2: Earthquake rates represent where earthquakes nucleated, not where they
are propagating. The 1857 Ft. Tejon earthquake, for example, is recorded as a
northern California event because its epicenter in the source catalog is just north of
the northern/southern California boundary. This increases the pre-instrumental
seismicity rate in Northern California.
Table 1: Completeness magnitudes for the entire UCERF3 catalog*
Starting year
Catalog completeness threshold
1850
7.4
1870
7.2
1885
7.1
1910
6.9
1932
6.0
1942
5.6
1957
5.1
1997
4.0
*Completeness thresholds may not apply in the immediate aftermath of large
earthquakes
Table 2: Magnitude averaged seismicity rates, Entire UCERF3 region, 18509.30.2011. Rates are given in earthquakes/year
Magnitude
M≥4
M≥4.5
M≥5
M≥5.5
M≥6.0
M≥6.5
M≥7
M≥7.5
Preferred rate
85.4
27.0
8.5
2.7
0.85
0.27
0.085
0.027
Lower bound
29.7
9.4
3.0
0.94
0.30
0.09
0.03
0.009
Upper bound
321.6
101.7
32.16
10.17
3.22
1.02
0.32
0.10
Table 3: Magnitude averaged seismicity rates, Entire UCERF3 region, 19849.30.2011. Rates are given in earthquakes/year
Magnitude
M≥4
M≥4.5
M≥5
M≥5.5
M≥6.0
M≥6.5
M≥7
M≥7.5
Preferred rate
75.4
23.9
7.5
2.4
0.75
0.24
0.075
0.024
Lower bound
62.7
19.8
6.3
2.0
0.63
0.20
0.06
0.02
Upper bound
88
27.8
8.8
2.8
0.88
0.28
0.09
0.028
Table 4: Magnitude averaged seismicity rates, Southern California, 1850-9.30.2011.
Rates are given in earthquakes/year
Magnitude
M≥4
M≥4.5
M≥5
M≥5.5
M≥6.0
M≥6.5
M≥7
M≥7.5
Preferred rate
41.5
13.1
4.15
1.31
0.415
0.131
0.0415
0.0131
Lower bound
14.0
4.4
1.4
0.44
0.14
0.04
0.014
0.004
Upper bound
168
53.1
16.8
5.3
1.68
0.53
0.168
0.053
Table 4: Magnitude averaged seismicity rates, Southern California, 1984-9.30.2011.
Rates are given in earthquakes/year
Magnitude
M≥4
M≥4.5
M≥5
M≥5.5
M≥6.0
M≥6.5
M≥7
M≥7.5
Preferred rate
49.2
15.6
4.92
1.55
0.492
0.156
0.049
0.015
Lower bound
39.5
12.5
3.95
1.25
0.40
0.12
0.04
0.013
Upper bound
60.9
19.3
6.1
1.93
0.61
0.19
0.06
0.02
Table 5: Magnitude averaged seismicity rates, Northern California, 1850-9.30.2011.
Rates are given in earthquakes/year
Magnitude
M≥4
M≥4.5
M≥5
M≥5.5
M≥6.0
M≥6.5
M≥7
M≥7.5
Preferred rate
43.9
13.9
4.4
1.4
0.44
0.139
0.044
0.014
Lower bound
10.6
3.4
1.1
0.34
0.11
0.03
0.01
0.0034
Upper bound
214
67.7
21.4
6.8
2.1
0.7
0.21
0.07
Table 6: Magnitude averaged seismicity rates, Northern California, 1984-9.30.2011.
Rates are given in earthquakes/year
Magnitude
M≥4
M≥4.5
M≥5
Preferred rate
26.4
8.35
2.64
Lower bound
19.6
6.2
1.96
Upper bound
36.0
11.4
3.6
M≥5.5
M≥6.0
M≥6.5
M≥7
M≥7.5
0.84
0.27
0.084
0.026
0.0083
0.62
0.20
0.06
0.02
0.006
1.1
0.36
0.11
0.04
0.011
Table 7: Direct count earthquake rates, whole state, 1850-9.30.2011. Rates are
incremental and are given in earthquakes/year.
Magnitude
M 7.5--8
Preferred rate
0.0247
Lower bound
0.0074
Upper bound
0.0618
Table 8: Direct count earthquake rates, whole state, 1984-9.30.2011. Rates are
incremental and are given in earthquakes/year.
Magnitude
M 5.5—6.0
M 6.0—6.5
M 6.5—7.0
M 7.0—7.5
Preferred rate
1.37
0.40
0.25
0.144
Lower bound
1.10
0.20
0.13
0.043
Upper bound
1.78
0.67
0.51
0.360
Table 9: Direct count earthquake rates, southern California, 1850-9.30.2011. Rates
are incremental and are given in earthquakes/year.
Magnitude
M 7.5—8.0
Preferred rate
0.0124
Lower bound
0.0025
Upper bound
0.0445
Table 10: Direct count earthquake rates, southern California, 1984-9.30.2011.
Rates are incremental and are given in earthquakes/year.
Magnitude
M 5.5—6.0
M 6.0—6.5
M 6.5—7.0
M 7.0—7.5
Preferred rate
0.94
0.25
0.07
0.1081
Lower bound
0.66
0.13
0.014
0.032
Upper bound
1.3
0.50
0.26
0.32
Table 11: Direct count earthquake rates, northern California, 1850-9.30.2011.
Rates are incremental and are given in earthquakes/year.
Magnitude
M 7.5—8.0
Preferred rate
0.0124
Lower bound
0.0025
Upper bound
0.0445
Table 12: Direct count earthquake rates, northern California, 1984-9.30.2011.
Rates are incremental and are given in earthquakes/year.
Magnitude
M 5.5—6.0
M 6.0—6.5
M 6.5—7.0
M 7.0—7.5
Preferred rate
0.43
0.14
0.18
0.036
Lower bound
0.26
0.04
0.07
0.004
Upper bound
0.74
0.36
0.42
0.20
References
Bird, P. and Y. Y. Kagan (2004), Plate-tectonic analysis of shallow seismicity: apparent
boundary width, beta, corner magnitude, coupled lithosphere thickness, and coupling in
seven tectonic settings, Bull. Seis. Soc. Am., 94, 2380-2399.
Felzer, Karen R. (2006), Calculating the Gutenberg-Richter b value, Eos Trans. AGU 87(52),
Fall Meet. Suppl., Abstract S42C-08.
Felzer, Karen R. (2008), Calculating California seismicity rates, Appendix I, the Uniform
California Earthquake Rupture Forecast (UCERF 2), U. S. Geological Survey Open File
Report 2007-1437I.
Gutenberg, B. and Charles F. Richter (1944), Frequency of earthquakes in California, Bull.
Seis. Soc. Am, 4, 185-188.
Hutton, K. L. and J. Woessner and E. Hauksson (2010), Earthquake monitoring in southern
California for seventy-seven years, Bull. Seis. Soc. Am, 100, 423-446.
Weichert, D. H. (1980). Estimation of the earthquake recurrence parameters for unequal
observation periods for different magnitudes. Bull. Seis. Soc. Am. 70, 1337–1346.