Earthquake Rate Model 2.2 of the 

2007 Working Group for California 

Earthquake Probabilities, 

Appendix D: Magnitude-Area 

Relationships 

By Ross S. Stein1

Open-File Report 2007-1162 

2007 

U.S. Deprtment of the Interior 

U.S. Geological Survey 

1

U.S. Geological Survey, Menlo Park, Calif. 
U.S. Department of the Interior 

DIRK KEMPTHORNE, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

U.S. Geological Survey, Reston, Virginia 2007 

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: 

Stein. R.S., 2007, Earthquake rate model 2.2 of the 2007 Working Group 

for California Earthquake Probabilities, appendix D; magnitude-area 

relationship: U.S. Geological Survey Open-File Report 2007-1162 

[http://pubs.usgs.gov/of/2007/1162/]. 

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. 
Contents 

Summary............................................................................................................................. 1 

Needs of the Working Group ............................................................................................. 1 

Magnitude-Area Summit Meeting ...................................................................................... 1 

Estimation of Downdip Fault Dimension, W ..................................................................... 2 

Limitations Associated with Estimating W from Microseismicity .................................... 3 

Magnitude-Area Relations Used to Infer Earthquake Size................................................. 5 

Request for Additional Analysis by Donald Wells ............................................................ 7 

Uncertainty on the Mean Magnitude .................................................................................. 7 

Impact on the Magnitude-Frequency Distribution for California....................................... 8 

Conclusions......................................................................................................................... 8 

References........................................................................................................................... 8 

Supplementary Material.................................................................................................... 10 

i
Summary 

To estimate the down-dip coseismic fault dimension, W, the Executive Committee has 

chosen the Nazareth and Hauksson (2004) method, which uses the 99% depth of 

background seismicity to assign W. For the predicted earthquake magnitude-fault area 

scaling used to estimate the maximum magnitude of an earthquake rupture from a fault’s 

length, L, and W, the Committee has assigned equal weight to the Ellsworth B (Working 

Group on California Earthquake Probabilities, 2003) and Hanks and Bakun (2002) (as 

updated in 2007) equations. The former uses a single relation; the latter uses a bilinear 

relation which changes slope at M=6.65 (A=537 km2). 

Needs of the Working Group

The Working Group for California Earthquake Probabilities (WGCEP) must be able to 

assign earthquake magnitudes from inferred fault geometry. The length of past fault 

ruptures and the length of continuous fault traces are the best observed parameters. In 

contrast, the down-dip fault dimension W is poorly resolved, resulting in considerable 

uncertainty in fault area, A. The Working Group thus seeks a proxy for fault width that 

can be applied to all California faults, with appropriate uncertainties. For vertical strikeslip faults, W is closely related to the depth of the brittle-ductile transition; for thrust and 

normal faults, W is less well constrained by the transition, and uncertsainties are greater. 

Empirical relations between fault area and moment-magnitude suggest that the static 

earthquake shear stress drop is roughly constant over a large range of magnitudes; the 

magnitude of the stress drop may transition to fault-length scaling for continental strikeslip earthquakes over Mw=7 (Scholz, 1990). We will thus consider several magnitude-area 

scaling relationships. The frequency-magnitude distribution for California is partly 

dependent on the magnitude-area relation, and so these two efforts must be consistent in 

approach and in parameter definitions. 

Magnitude-Area Summit Meeting 

The Working Group hosted a meeting to solicit advice from the scientific community on 

magnitude-area relations. Participants reviewed the Working Group 02 approaches and 

considered subsequent advances pertinent to this problem. Paul Somerville was also 

contracted by the Working Group to prepare a review paper on the subject. Present at the 

November 1, 2006, Menlo Park videoconference were Bill Ellsworth, Jessica Murray, 

Donald Wells, Tom Hanks, Bill Bakun, Ruth Harris, Paul Somerville, Ned Field, Ken 

Hudnut, Colin Williams, Egill Haukkson and Ross Stein. In addition, Paul Segall, Roland 

Bürgmann, and Jim Savage contributed references or written comments. Paul Somerville 

provided a written review of magnitude-area relations for the Working Group, and Tom 

Hanks wrote a comment on the Somerville report, which is referenced at the end of this 

one. The agenda and presentations given at the meeting by Ellsworth, Field, Somerville, 

Hanks, Murray, and Williams, as well as the Hanks letter are available at: 

http://www.WGCEP.org/activities/meetings/110106

1
Figure 1. Relocated seismicity from Hauksson and Shearer (2005) compared with the 

inferred 300 and 400°C isotherms, suggesting that seismicity extends to about 400°-

425°C across the Los Angeles Basin (Collin Williams, written comm., 

http://pubs.usgs.gov/of/2003/of03-214/WG02_OFR-03-214_AppendixA.pdf) 

Estimation of Downdip Fault Dimension, W

Limited geodetic sampling and large ambiguities in the inferred maximum depth of 

coseismic slip and the interseismic locking depth render geodetic estimates of the downdip fault dimension, W, inadequate for the purposes of WGCEP. The problem is 

compounded when faults are closely spaced, which is frequently the case, and this arises 

regardless of the quality and density of the surface deformation data because the locking 

depth (and slip rates) inferred on such faults are not independent. There is wide 

consensus on these conclusions by tectonic geodesists, including Hudnut, Murray, 

Savage, Bürgmann, and Segall. In contrast, the microseismicity coverage is more 

complete and uniform than is the geodetic coverage. 

2
The higher the observed heat flow, the shallower the maximum depth of seismicity is 

found to extend. Heat flow observations are generally consistent with the assumption that 

the maximum depth of seismicity along California faults corresponds to the 350°-400°C 

isotherm (figure 1). This means that the lower depth of seismicity may trace the brittleductile transition on most faults (see Collin Williams’ Appendix in Working Group on 

California Earthquake Probabilities, 2003). But heat flow coverage is even more spatially 

limited than the geodetic data and the observations are variable in quality. Thus, we 

regard the thermal measurements and models more as a test of other approaches, rather 

than an indpendent data set that can be used to estimate a lower depth of faulting for all 

California faults. 

The WGCEP has therefore used the method of Nazareth and Hauksson (2004) to 

estimate the lower seismogenic fault depth from background seismicity. Nazareth and 

Hauksson demonstrate, albeit with a limited southern California dataset, that the depth 

above which 99.9% of the moment release of background seismicity occurs reasonably 

estimates the maximum depth of rupture in moderate to large earthquakes. 

Limitations Associated with Estimating W from 

Microseismicity 

The approach of Nazareth and Hauksson (2004) suffers where background seismicity is 

sparse, such as along the San Andreas Cholame-Simmler segment, and parts of the 

Mojave segment, along many of the Eastern California shear zone faults, and along the 

San Gregorio fault. In other areas, such as the many thrust faults, seismicity is distributed 

and the lower depth of seismicity may lie beneath the major thrust fault. This may be the 

case for the Coalinga and Kettleman Hills thrusts. But even with these acknowledged 

weaknesses, coverage is still far superior to geodetic, heat flow, and seismic profiling. 

For the thrust faults, seismic inferences can be modified by published interpretations of 

reflection and refraction profiles (for example, Shaw and Suppe, 1996). 

It is also possible that the lower depth extent of M≥7.4 shocks will prove to exceed the 

lower depth of background seismicity, in which case the moment release per unit fault 

length increases for large earthquakes. The Denali earthquake may have ruptured to as 

much as 20 km depth, whereas background seismicity only extended to about 10 km 

depth. However, both the depth of seismicity and the teleseismic inversion of the lower 

depth of faulting along the Denali fault are very poorly constrained, rendering this 

comparison dubious. In onctrast, geodetic estimates of the rupture depth of 10-12 km 

(Wright and others, 2004) are consistent with the aftershock and background seismicity, 

and thus require no coseismic deepening (figure 2). 

3
Figure 2. The aftershocks of the 2002 M=7.9 Denali shock (Eberhart-Phillips and 

others, 2003) compared to geodetic (Wright and others, 2004) and telesesmic (Asano and 

others, 2005; Oglesby and others; 2004, Ozacar and others, 2003) inferences for the 

lower depth of coseismic rupture. 

Rolandone and others (2004) argued that the immediate Landers aftershocks extended 3 

km deeper than the background seismicity, and returned to the background depth over the 

succeeding 3-4 years (figure 3). However, this result could be an artifact of the depth 

scatter, which increases with the number of earthquakes measured. Because the rate of 

aftershocks decays with time, there is a much greater sample during the first posteismic 

year than afterward. In the Landers area, the background seismicity rate was so low that 

the depth extent is also poorly determined. 

Figure 3. Depth of seismicity along the 1992 Landers rupture, showing a coseismic 

deepening of the lower depth of seismicity followed by a gradual return to pre-1992 

depths, from Rolandone and others (2004).

In contrast, the 1999 Izmit aftershocks have a very similar depth distribution to the 

background seismicity measured with the same IZINET seismic network beginning in 

1992 by Ito and others (2002) (figure 4). Here the background rate is much higher than at 

Landers and so is more reliable. Thus we find that the evidence does not suggest that 

4
earthquake ruptures commonly exceed the background seismicity depth by a significant 

amount. 

Figure 4. Depth cross-sections along the 1999 Izmit, Turkey, rupture from Ito and others

(2002) exhibit little, if any, coseismic deepening. 

The Executive Committee of the Working Group thus believes that contemporary 

seismicity provides the most sound and consistent method available to estimate the lower 

depth. The upper depth of faulting is left unresolved by this method, but we will 

arbitrarily set this to be 0 km except in special cases. This might overestimate the 

potential fault area, but few alternative assumptions exist. 

Magnitude-Area Relations Used to Infer Earthquake Size 

Equations relating Mw to rupture area, A, are derived from empirical earthquake datasets, 

most notably Wells and Coppersmith (1994). Of greatest importance to WGCEP are large 

strike-slip earthquakes, for which the Wells and Coppersmith sample is small. Although 

essential, these datasets suffer from uncertainty in the down-dip dimension, W, and for 

many historical earthquakes there are also large uncertainties in Mw and in some cases, 

even the length of the rupture, L. Published equations by Wells and Coppersmith (1994), 

Hanks and Bakun (2002), Ellsworth (Working Group on California Earthquake 

Probabilities, 2003), as well as Somerville (2006), present alternatives. 

5
 Mw = 4.20 + 1.0 log(A) (‘Ellsworth B’ of Working Group, 2003) 

 Mw = 3.98 + 1.0 log(A) (for A<500 km2; Hanks and Bakun, 2007) 

 Mw = 3.08 + 4/3 log(A) (for A>500 km2; Hanks and Bakun, 2007) 

 Mw = 3.87 + 1.05 log(A) (Somerville, 2006) 

 Mw = 3.98 + 1.02 log(A) (Wells and Coppersmith, 1994) 

Figure 5. Revised observations and model fit by Hanks and Bakun (2007). The light 

dashed line is the continuation of the A<501 km2

 curve for all A. The added observations 

are are in red, and appear in the table. 

Hanks and Bakun (2007) is an update of Hanks and Bakun (2002). Somerville, by 

contrast uses a more recent and so more uniform dataset. But Somerville makes extensive 

use of teleseismic and strong motion inversions of coseismic slip, which tend to distribute 

some slip to the lower depth of the surrface over whgich slip is inverted. It is also 

currently unclear that his method to infer W is compatible with how W is treated in the 

WGCEP fault section database and deformation model. For example, the USGS-CGSSCEC California Reference Geologic Fault Parameter Database uses L and W. In 

contrast, Somerville trims the fault area on the basis of indpendent criteria (Somerville 

6
and others, 1999). How broad areas of low or no slip within a rupture zone is treated in 

calculating fault area needs consideration. 

Hanks and Bakun (2002) provide a physical basis for a bilinear scaling in their equation 

(stress drop scaling by fault area for Mw<7 and scaling by fault length for Mw>7). 

However, there are too few observations at Mw≥7.5 to be confident in the departure from 

a single slope. In response to a request from the WGCEP, Hanks and Bakun (2007) 

updated their inventory of large strike-slip earthquakes by adding seven new events. The 

new data are compatible with the nearly identical bilinear relation proposed by Hanks and 

Bakun (2002). 

For uniformity and consistency with the 2002 National Seismic Hazard Mapping Project, 

the Executive Committee of the WGCEP has assigned equal weight to the Working 

Group on California Earthquake Probabilities (2003) ‘Ellsworth B’ relation, and to the 

Hanks and Bakun (2002) bilinear equations as updated by the inclusion of the additional 

data in Hanks and Bakun (2007). The Somerville (2006) and Wells and Coppersmith 

(1994) relations are nearly identical, and it could be argued that they should also assigned 

some weight. However, we judge that Wells and Coppersmith (1994) must be updated 

before it could be included, and that Somerville (2006) treats fault area in a manner 

incompatible with our assignment of earthquake magnitude from fault area. For this 

reason, they are not included. 

Request for Additional Analysis by Donald Wells

The Executive Committee has asked Donald Wells to include the 12 years of earthquakes 

that have struck since his 1994 paper was published. In addition, there are new studies of 

many of the pre-1994 earthquakes that should cause these values to be reassessed. For 

example, the rutpure length of the 1973 Mw=7.5 Luhuo earthquake on the Xian Shiehe 

fault may need modification (Zhou, Allen, and Kanamori, BSSA, 73, 1585-1597, 1983). 

Thus far, we have received no reply to this request. 

Uncertainty on the Mean Magnitude 

Although two magnitude-area relations will be used with equal weight, there remains a 

value of M where the two curves intersect, and so near this interesection at M=7.6, the 

epistemic uncertainty (due to lack of knowledge) on M for a given A will be smaller than 

at other values. In part to ameliorate this localized problem, and in part to reflect 

epistemic uncertainty on all earthquake magnitudes, ±0.1 M unit correction is associated 

with all magnitudes. Some 60% of the weight will be given to the derived magnitude, 

20% at -0.1 units, and 20% at +0.1 units. These values and their associated weights were 

chosen to follow the same procedure as the National Seismic Hazard Mapping Project 

uses for its maps. The aleatory uncertainty (due to random variability) is set to be 0.12 M 

units, consistent with the Working Group (2002) and the National Seismic Hazard 

Mapping Project (2002). 

7
Impact on the Magnitude-Frequency Distribution for 

California 

Although many parameter selections are needed to produce an all-California magnitudefrequency distribution, the WGCEP has found that most realizations of this distribution 

suffer from too high a rate of 6.25<M<6.75 earthquakes in comparison to the inferred 

historical rate, a cause of considerable concern. Global catalogs for the past 30-100 years 

(Bird and Kagan, 2004), as well as local network catalogs with a much greater range of 

magnitude completeness and a longer historical record, such as in the Kanto (greater 

Tokyo) region of Japan (Grunewald and Stein, 2006), exhibit no bulge, and so we regard 

the excess rate of moderate magnitude earthquakes in the California model as an artifact. 

Although the Somerville (2006) or Wells and Coppersmith (1994) magnitude-area 

relations exacerbate this excess earthquake frequency at about M~6.7, none of the 

magnitude-area relations we have considered removes it. So the magnitude-area relation 

is not the largest contributing factor in the disagreement between the model and data. 

Nevertheless, since the bulge is almost certainly a model artifact rather than a ture feature 

of seismicity, the Executive Committee has favored magnitude-area assumptions that 

tend to minimize the excess rate of M~6.7 earthquakes. 

Conclusions 

The WGCEP is charged with delivering a Poisson probability model to the National 

Seismic Hazard Mapping Project. For this, the Executive Committee has used the 

Nazareth and Hauksson (2004) method to estimate W. The few faults with known creep 

are also assigned aseismicity factors. The WGCEP assigns equal weight to the Ellsworth 

B (Working Group on California Earthquake Probabilities, 2003) and Hanks and Bakun

(2007) magnitude-area relations. 

Acknowledgements. We greatly appreciate thoughtful reviews by Robert Wesson, 

Michael Blanpied, Ruth Harris, Ray Weldon and Ned Field. 

References 

Asano, K., Iwata, T., and Irikura, K., 2005, Estimation of source rupture process and 

strong ground motion simulation of the 2002 Denali, Alaska, earthquake, Bull. 

Seism. Soc. Am. 95, 1701-1715. 

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. Seismol. Soc. Amer., 94, 2380-2399; 

DOI: 2310.1785/0120030107. 

Eberhart-Phillips and others, 2003, The 2002 Denali Fault Earthquake, Alaska: A Large 

Magnitude, Slip-Partitioned Event, Science, 300, 1113–1118, DOI: 

10.1126/science.1082703. 

Grunewald, E.D., and R.S. Stein,2006, A new 1649–1884 catalog of destructive 

earthquakes near Tokyo and implications for the long-term seismic process, J. 

Geophys. Res., 111, doi:10.1029/2005JB004059.

8
Hanks, T. C., and W. H. Bakun, 2002, A bilinear source-scaling model for M-log A 

Observations of continental earthquakes, Bull. Seismol. Soc. Amer., 92, 1841-

1846. 

Hanks, T. C., and W. H. Bakun, 2007, M—log A observations for recent large 

earthquakes, submitted to Bull. Seismol. Soc. Amer. 

Hauksson, E., and P. Shearer, 2005, Southern California Hypocenter Relocation with 

Waveform Cross-Correlation, Part 1: Results Using the Double-Difference 

Method, Bull. Seismol. Soc. Amer., 95, 896–903, doi:810.1785/0120040167. 

Ito, A., and others, 2002, Aftershock activity of the 1999 Izmit, Turkey, earthquake 

revleaed from microearthquake observations, Bull. Seismol. Soc. Amer., 92, 418-

427. 

Nazareth, J. J., and E. Hauksson, 2004, The seismogenic thickness od the southern 

California crust, Bull. Seismol. Soc. Amer., 94, 940-960. 

Oglesby, D.D., D.S. Dreger, R.A. Harris, N. Ratchkovski, and R. Hansen, 2004, Inverse 

kinematic and forward dynamic models of the 2002 Denali, Alaska earthquake. 

Bull. Seism. Soc. Am. 94, S214-S233. 

Ozacar, A., S. L. Beck and D. H. Christensen, 2003, Source process of the 3 November 

2002 Denali fault earthquake (central Alaska) from teleseismic observations, 

Geophys. Res. Let. 30, 1638.

Rolandone, F., R. Bürgmann, and R. M. Nadeau, 2004, The evolution of the seismicaseismic transition during the earthquake cycle: Constraints from the timedependent distribution of aftershocks, Geophys. Res. Lets., 31, L23610, 

doi:10.1029/2004GL021379. 

Scholz, C.H., The Mechanics of Earthquake Faulting, Cambridge Univ. Press, 

Cambridge, 439 p. 

Shaw, J.H., and J. Suppe, 1996, Earthquake hazards of active blind-thrust faults under the 

central Los Angeles basin, California, J. Geophys. Res., 101, 8623-8642. 

Somerville, P., 2006, Review of magnitude-area scaling of crustal earthquakes, Rep. to 

WGCEP, 22 pp, URS Corp., Pasadena. 

Somerville, P. G. and others, 1999, Characterizing earthquake slip models for the 

prediction of strong ground motion, Seism. Res. Lett., 70, 59-80. 

Wells, D. L., and K. J. Coppersmith, 1994, New empirical relationships among 

magnitude, rupture length, rupture width, rupture area, and surface displacement, 

Bull. Seismol. Soc. Amer., 84, 974-1002. 

Working Group on California Earthquake Probabilities, 2003, Earthquake Probabilities in 

the San Francisco Bay Region: 2002–2031, U.S. Geol. Surv. Open File Rep. 03-

214. 

Wright, T.J., Z. Lu, and C. Wicks, 2004, Constraining the Slip Distribution and Fault 

Geometry of the Mw 7.9, 3 November 2002, Denali Fault Earthquake with 

Interferometric Synthetic Aperture Radar and Global Positioning System Data, 

Bull. Seismol. Soc. Amer., 94, 175-189. doi:10.1785/0120040623. 

9
Supplementary Material 

Paul Somerville’s report to the Working Group on California Earthquake Probabilities, 

revised after review by the Executive Committee, and a detailed Comment by Tom Hanks 

and a Reply by Paul Somerville, are available online through this link: 

http://www.WGCEP.org/resources/documents/ERM2_1_Report/SomervilleReport_1127

06.pdf 

10