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Statistical Relations Among Earthquake Magnitude, Surface Rupture Length, and Surface Fault Displacement

By M.G. Bonilla1, R.K. Mark1, and J.J. Lienkaemper1

U.S. Geological Survey Open-File Report 84-256
Online Version 1.0


Prepared in cooperation with
U.S. Nuclear Regulatory Commission


Dallas L. Peck, Director

This report is preliminary and has not been reviewed for conformity with U.S. Geological Survey (USGS) editorial standards or with the North American Stratigraphic Code. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

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In order to refine correlations of surface-wave magnitude, fault rupture length at the ground surface, and fault displacement at the surface by including the uncertainties in these variables, the existing data were critically reviewed and a new data base was compiled. Earthquake magnitudes were redetermined as necessary to make them as consistent as possible with the Gutenberg methods and results, which necessarily make up much of the data base. Measurement errors were estimated for the three variables for 58 moderate to large shallow-focus earthquakes. Regression analyses were then made utilizing the estimated measurement errors.

The regression analysis demonstrates that the relations among the variables magnitude, length, and displacement are stochastic in nature. The stochastic variance, introduced in part by incomplete surface expression of seismogenic faulting, variation in shear modulus, and regional factors, dominates the estimated measurement errors. Thus, it is appropriate to use ordinary least squares for the regression models, rather than regression models based upon an underlying deterministic relation with the variance resulting from measurement errors.

Significant differences exist in correlations of certain combinations of length, displacement, and magnitude when events are qrouped by fault type or by region, including attenuation regions delineated by Evernden and others. Subdivision of the data results in too few data for some fault types and regions, and for these only regressions using all of the data as a group are reported.

Estimates of the magnitude and the standard deviation of the magnitude of a prehistoric or future earthquake associated with a fault can be made by correlating M with the logarithms of rupture length, fault displacement, or the product of length and displacement. Fault rupture area could be reliably estimated for about 20 of the events in the data set. Regression of MS on rupture area did not result in a marked improvement over regressions that did not involve rupture area. Because no subduction-zone earthquakes are included in this study, the reported results do not apply to such zones.

Diagram showing classification of fault types
Relation of surface rupture length to earthquake magnitude for all types of faults. Error bars shown for each faulting event (from figure 2A in the following report)


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Statistical Relations Among Earthquake Magnitude, Surface Rupture Length, and Surface Fault Displacement

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Maintained by:Michael Diggles
Date created: October 22, 2000
Last modified: May 27, 2005