Calculation of the Rate of M≥6.5 Earthquakes for 

California and Adjacent Portions of Nevada and Mexico 

By Arthur Frankel and Charles Mueller 

Open-File Report 2008–1112 

U.S. Department of the Interior 

U.S. Geological Survey 

U.S. Department of the Interior 

DIRK KEMPTHORNE, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

U.S. Geological Survey, Reston, Virginia 2008 

For product and ordering information: 

World Wide Web: http://www.usgs.gov/pubprod 

Telephone: 1-888-ASK-USGS 

For more information on the USGS—the Federal source for science about the Earth, 

its natural and living resources, natural hazards, and the environment: 

World Wide Web: http://www.usgs.gov 

Telephone: 1-888-ASK-USGS 

Suggested citation: 

Frankel, Arthur, and Mueller, Charles, 2008, Calculation of the rate of M≥6.5 earthquakes for California and 

adjacent portions of Nevada and Mexico, U.S. Geological Survey Open-File Report 2008-1112, 14 p. 

Any use of trade, product, or firm names is for descriptive purposes only and does not imply 

endorsement by the U.S. Government. 

Although this report is in the public domain, permission must be secured from the individual 

copyright owners to reproduce any copyrighted material contained within this report. 

ii
iii

Contents 

Introduction .........................................................................................................................................................................1

Procedure ............................................................................................................................................................................1

Results ..................................................................................................................................................................................3

References Cited ................................................................................................................................................................5

Figures 

1. M≥6.0 earthquakes since 1850 in the Working Group area of interest, with zones selected for this study..7 

2. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in extensional zone plotted against time. ..........................8 

3. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in San Francisco Bay zone plotted against time ..............9 

4. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in Los Angeles zone plotted against time. .......................10 

5. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in Mendocino zone plotted against time..........................11 

6. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in remainder area plotted against time...........................12 

7. Cumulative number of earthquakes M≥6.5 in remainder area plotted against time........................................13 

8. Cumulative number of earthquakes M≥7.0 in remainder area plotted against time........................................14 

Tables 

1. Tests of procedure using synthetic catalogs............................................................................................................6 

2. Results of Bayesian procedure for CGS/WG catalog ..............................................................................................6 
Calculation of the Rate of M≥6.5 Earthquakes for 

California and Adjacent Portions of Nevada and 

Mexico 

By Arthur Frankel and Charles Mueller 

Introduction 

One of the key issues in the development of an earthquake recurrence model for California 

and adjacent portions of Nevada and Mexico is the comparison of the predicted rates of 

earthquakes with the observed rates (see, for example, Petersen and others, 2000). Therefore, it is 

important to make an accurate determination of the observed rate of M≥6.5 earthquakes in 

California and the adjacent region. 

As part of the Working Group on California Earthquake Probabilities (WGCEP) effort, 

Felzer (2007) pointed out various issues with the earthquake catalog that must be addressed in 

order to accurately assess the rates of earthquakes. She noted that the uncertainty in the magnitude 

and the rounding procedures significantly affect the apparent rate of earthquakes in the catalog. 

Felzer (2007) assigned magnitude uncertainties and rounding parameters to the catalog compiled by 

the California Geological Survey (CGS). We declustered this catalog (which we denote as 

CGS/WG) using the procedure of Gardner and Knopoff (1974). The resulting catalog was used in 

the analysis described below, including the magnitude uncertainty and rounding determined for 

each earthquake by Felzer (2007). The region of study is identical to that used in WGCEP and 

consists of California with a border region including parts of Nevada, Oregon, Arizona, and 

Mexico. 

Procedure 

We have developed a procedure to calculate observed earthquake rates from an earthquake 

catalog, accounting for magnitude uncertainty and magnitude rounding. We present a Bayesian 

method that corrects for the effect of the magnitude uncertainty in calculating the observed rates. 

Using Bayes theorem, the probability distribution p(M | Mobs) of the true magnitude M of each 

event given an observed (calculated) magnitude Mobs can be determined from 

p(M | Mobs) = p(M) p(Mobs | M) / P(Mobs) . (1) 

Here p(M) is the a priori distribution of M, which is the Gutenberg-Richter recurrence 

relation, so that p(M)= 10-bM . p(Mobs | M) is the probability distribution of the observed magnitude 

1
given the true magnitude M, assuming the magnitude uncertainty is normally distributed with a 

standard deviation of σ, 

p(Mobs | M) = e . − − M M obs ( )/ 2 2 2σ

P(Mobs) in equation (1) is the total probability of having Mobs. Substituting into equation (1), we find 

pMM c e obs

bM M M obs (| ) ( )/ = −− − 10 2 2 2σ

. (2) 

Here c is a constant that is calculated so that the probability density for each event sums to unity. 

In our procedure to find observed rates from a catalog with magnitude rounding and 

magnitude uncertainty, we first account for magnitude rounding. We create a probability 

distribution for each event, which has a magnitude Mreported in the catalog. The distribution ranges 

between Mreported- round/2 and Mreported + round/2, where round is the rounding increment for that 

event (for example, 0.5 or 0.1) and the amplitude of the distribution has the form of 10-bM. The 

distribution is normalized to give unit area. 

Then, for each magnitude bin in the rounding distribution we apply equation (2) to account 

for magnitude uncertainty. Each bin in the rounding distribution represents a possible measured 

magnitude of the earthquake Mobs, before rounding was applied. 

The probability density in each bin is the product of the density from the rounding 

distribution and the density from equation (2) for that magnitude bin. We apply our Bayesian 

procedure for Mobs-M within ±4σ. We found that using even larger limits resulted in no significant 

change to the results. 

The probability densities in each magnitude bin are added for all the events and then 

cumulative counts and rates are calculated above various magnitude thresholds. The total number 

of events in all the magnitude bins equals the original number of earthquakes in the catalog. 

We tested this procedure using synthetic catalogs of earthquakes. We first produced 

synthetic catalogs from a Gutenberg-Richter distribution of earthquakes with respect to magnitude. 

Then, for each earthquake, a Gaussian random-number generator was used to assign an “observed” 

magnitude adjustment, given a specified sigma for the magnitude uncertainty. Finally, the resulting 

magnitude is rounded to the specified precision. 

We used various combinations of magnitude uncertainty and rounding to test our method. 

The selected parameters span the values found in the actual catalog. For each combination, we 

produced five catalogs each starting with 10,000 earthquakes with M≥4.0. The b-value chosen was 

0.8, as is found in our analysis of the actual catalog and by Felzer (2007). Table 1 shows the 

average number of events with M≥6.5 in the catalogs before the magnitude uncertainty and 

rounding were applied (“actual number of earthquakes”). 

2
 Table 1 also shows the average number of events with M≥6.5 calculated from the 

procedure described above. Note the similarity of the actual and calculated numbers of events, 

indicating that the methodology does an excellent job recovering the proper rate of earthquakes. 

Results 

In our estimation of rates in the actual catalog, we divide the region into zones to determine 

the proper observation time to consider. Figure 1 is a map of M≥6.0 earthquakes since 1850, 

showing the zones we selected. Two of these zones (San Francisco Bay and Los Angeles) were 

chosen because they encompass large population centers where one might expect longer periods of 

catalog completeness. The extensional zone contains Reno and towns such as Bishop and 

Independence which were populated in the mid-1800s (Felzer, 2007). We also outlined a zone 

around earthquakes in the Mendocino area, because the catalog there seems to effectively start in 

1870. 

Plots of cumulative number of events with 6.0 ≤ M ≤ 6.5 versus time were made for each 

zone. We do need to include events with catalog magnitudes less than 6.5, because some of those 

earthquakes will contribute to the rate of M≥6.5 earthquakes because of magnitude uncertainty. In 

our rate calculation for M≥6.5 we use earthquakes in the catalog with magnitudes M≥5.0. In 

practice there is no significant difference in the rates found when using a minimum magnitude of 

5.0 and 6.0. Obviously, we do not expect the M5 earthquakes to be complete for the time periods 

we considered. 

For the extensional zone, it is clear that there is a high rate of 6.0 ≤ M ≤ 6.5 earthquakes 

from 1850 to about 1890, with a lower rate from 1890 to 2000 (see fig. 2). In order to include this 

higher rate in our calculations we extended the observation time for this zone back to 1850. This 

zone encompasses the area around Reno, as well as the Mammoth Lakes and Owens Valley areas. 

It is well documented (Ellsworth and others, 1981) that the San Francisco Bay area 

experienced a surge of activity in the 50 years prior to the 1906 M7.8 earthquake. This is apparent 

in the cumulative plot (fig. 3) of events in our San Francisco Bay zone with 6.0 ≤ M ≤ 6.5. This 

area extends from the San Francisco Bay area southeast to the Parkfield area. The slope is much 

steeper from 1850-1900 than from 1900 to 2006. For the rate calculation for this zone, we start at 

1850. 

We also chose a zone that encompassed the greater Los Angeles, San Diego, and Santa 

Barbara areas. The plot of events with 6.0 ≤ M ≤ 6.5 (fig. 4) shows a burst of activity in the 1850s 

with a variable rate after that. We chose an observation time of 1850 for this zone. 

The Mendocino area exhibits a higher rate of 6.0 ≤ M ≤ 6.5 earthquakes from 1870 to 1920 

and a lower rate from 1920 to the present (fig. 5). There are only three earthquakes with M≥4.0 in 

the catalog for this region prior to 1870. These earthquakes were assigned to the same location and 

had a catalog magnitude of 5.5. Therefore, we start our calculation at 1870 for this region. 

A key issue is what year to start the observation period for the events in the rest of 

California. The plot of 6.0 ≤ M ≤ 6.5 events (fig. 6) is not conclusive. The rate from 1850 to 1890 is 

3
relatively low, but similarly low rates are observed during other periods. Figure 7 shows the 

cumulative number of M≥6.5 for the region. From this plot it is clear that the low rate of M≥6.5 

events prior to 1890 is anomalous compared to the period since 1890. Therefore, we chose 1890 as 

the start of our observation period for this zone. Most of the earthquakes in the remainder area 

occur in the Mojave zone specified by Felzer (2007). She derives a completeness time of 1910 for 

M=6.6 earthquakes in this zone. Thus our choice of 1890 as the observation time for M≥6.5 

events in the remainder zone is reasonably consistent with her completeness time for this 

magnitude range. 

We also implemented an alternative approach to calculate the rate of M≥6.5 events in the 

remainder area based on completeness times assigned for each magnitude bin. When determining 

the rate of M≥6.5 events, one can use a longer completeness time for M7 earthquakes than M6.5 

earthquakes. The disadvantage of this method is that it uses different periods of time for different 

magnitude ranges above M6.5. In Figure 8, we show the cumulative number of M≥7.0 earthquakes 

in the remainder area. There are no M≥7.0 events in the remainder area from 1850 to 1891. It is not 

clear if this is due to incompleteness of the catalog or to a real lull of production. There is another 

period of similar duration, 1953-1991, with no M≥7.0 earthquakes in the remainder area. 

We implement this alternative procedure by using the bins in our Bayesian calculation 

(equation 1). For each magnitude bin, the rate is determined from the effective number of events in 

that bin divided by the completeness period for that bin. Note that these bins also include the 

distribution from the rounding. The rates in each magnitude bin are then accumulated to get the 

total rate above a certain magnitude threshold. 

In the alternative procedure, we use the completeness times developed by Felzer (2007) 

based on the distribution of population and newspapers across California as a function of time. All 

but one of the M≥6.5 events in the remainder area are in either Felzer’s Mojave or “rest of state” 

zones (exception is the 1927 Hosgri earthquake). Because most of the events are in the Mojave 

zone, we use Felzer’s completeness times for that zone: M< 6.9 1890, M6.9 to 7.0 1880, M7.0 to 

7.1 1875, M7.1 to 7.3 1870, M7.3 to 7.4 1865, M7.4 to 8.0 1855, M≥8.0 1850. We found that this 

produces an 8 percent decrease in the calculated rate of M≥6.5 events in the remainder area. 

Because there were no M≥7.0 events in the catalog from 1850 to1890, but we are applying 

completeness times prior to 1890 for these magnitudes, this lowers the calculated rate of M≥6.5 

events compared to using an 1890 observation time for all magnitudes. The central question here is 

whether the lack of M≥7.0 events in the catalog for the remainder area from 1850 to1890 is real or 

an artifact of catalog incompleteness. 

Table 2 shows the observation time, number of M≥6.5 events, and rate we calculated for 

each zone. For the remainder zone and total, we cite the values determined from both methods 

described above. 

To estimate uncertainty bounds, we apply the standard estimate of uncertainty for a 

Poissonian process that the standard deviation (sigma) of the observed count of earthquakes equals 

the square root of the number of earthquakes (see Weichert, 1980). We calculated the standard 

deviation in rate in each zone from the square root of the effective number of events with M≥6.5 

4
divided by the observation period in each region (table 2). Then we added the variances of the five 

regions and took the square root of the sum to get the standard deviation for the total rate. We 

found a two-sigma value of 0.086 for the total rate. For the alternative method, the variance was 

determined for each magnitude bin based on the effective count and completeness time for that bin. 

These variances were summed for bins with M≥6.5 events and the square root was taken to find the 

standard deviation for the remainder area. We think this is a minimum estimate of the uncertainty, 

because we have not considered other sources of uncertainty. 

It is reasonable to average the rates determined by the two methods, so our recommended 

determination of the observed rate of M≥6.5 is 0.246 ± 0.085 (for two sigma) per year. We stress 

that this rate is likely an underestimate of the true rate. For example, it is unlikely that the catalog 

for the extensional zone is complete to M6.5 back to 1850 and it is certainly incomplete for M6.0-

6.4 earthquakes back to 1850. So the surge of the events in the catalog from 1850 to 1900, may still 

under-represent the actual increase in events during this time. 

Acknowledgments 

David Perkins suggested summing the variances of the rates in each region to get the total 

variance. We thank David Perkins and Mark Petersen for their reviews and valuable advice. 

References Cited 

Ellsworth, W.L., Lindh, A.G., Prescott, W.H. and Herd, D.G., 1981, The 1906 San Francisco 

earthquake and the seismic cycle, in Earthquake Prediction: D.W. Simpson and P.G. 

Richards, eds. American Geophysical Union, Washington, D.C., p. 126-140. 

Felzer, K. (2007). Appendix I: Calculating California seismicity rates, in The Uniform California 

Earthquake Rupture Forecast, Version 2 (UCERF 2): 2007 Working Group on California 

Earthquake Probabilities and the USGS National Seismic Hazard Mapping Program 

(NSHMP), U.S. Geological Survey Open-File Report 2007-1437-I. 

Gardner, J.K. and Knopoff, L., 1974, Is the sequence of earthquakes in southern California, with 

aftershocks removed, Poissonian? Bulletin of the Seismological Society of America, v. 64, 

p. 1363-1367. 

Petersen, M.D., Cramer, C.H., Reichle, M.S., Frankel, A.D., and Hanks, T.C., 2000, Discrepancy 

between earthquake rates implied by historic earthquakes and a consensus geologic source 

model for California, Bulletin of the Seismological Society of America, v. 90, p. 1117-

1132. 

Weichert, D.H. (1980). Estimation of the earthquake recurrence parameters for unequal observation 

periods for different magnitudes, Bulletin of the Seismological Society of America, v. 70, p. 

1337-1346. 

5
Table 1. Tests of procedure using synthetic catalogs 

Magnitude 

Uncertainty 

(?) 

Rounding 

Actual Number 

of M≥6.5 

Earthquakes 

Calculated Number of 

M≥6.5 Earthquakes from Bayesian 

Procedure 

0.333 0.1 99.6 99.2 

0.4 0.5 100.2 98.4 

0.1 0.1 97.2 96.7 

0.2 0.01 93.0 93.8 

Table 2. Results of Bayesian procedure for CGS/WG catalog 

zone 

observation 

start 

calculated 

number of 

M≥6.5 events 

calculated rate of 

M≥6.5 events per year 

estimated standard 

deviation of rate 

extensional 1850 3.91 0.0251 0.0127 

San 

Francisco 

Bay 

1850 7.09 0.0454 0.0171 

Los Angeles 1850 5.05 0.0324 0.0144 

Mendocino 1870 6.39 0.0470 0.0186 

remainder of 

region 1890 11.55 0.0996*,0.0918** 0.0293*,0.0271** 

total 0.250*, 0.242** 0.043*, 0.042** 

* determined from M≥6.0 events since 1890 

** determined from alternative method using completeness times as a function of magnitude above 

M6.9 

6
Figure 1. M ≥ 6.0 earthquakes since 1850 in the Working Group area of interest, with zones 

selected for this study. Earthquakes from the California Geological Survey/Working Group 

for California Earthquake Probabilities catalog. 

7
Figure 2. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in extensional zone plotted 

against time. 

8
Figure 3. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in San Francisco Bay zone 

plotted against time. 

9
Figure 4. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in Los Angeles zone plotted 

against time. 

10
Figure 5. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in Mendocino zone plotted 

against time. 

11
Figure 6. Cumulative number of earthquakes 6.0 ≤ M ≤ 6.5 in remainder area plotted 

against time. 

12
Figure 7. Cumulative number of earthquakes M ≥ 6.5 in remainder area plotted against 

time. 

13
Figure 8. Cumulative number of earthquakes M ≥ 7.0 in remainder area plotted against 

time. 

14