DOI: 10.1785/bssa0720051677. The standard error of the magnitude-frequency<i>b</i>value

The standard error of the magnitude-frequencybvalue

10.1785/bssa0720051677

Crossref journal-article

Seismological Society of America (SSA)

Bulletin of the Seismological Society of America (790)

Abstract

abstractEstimated b values in log N = a − bM are widely used in seismicity comparisons and risk analysis, but uncertainties have been little explored. In this paper, the usual F probability density distribution for b is given and compared with an asymptotic form for temporally varying b. Convenient tables for the standard error of b are given that allow statistical tests to accompany investigations of both temporal and spatial variations of b. With large samples and slow temporal changes in b, the standard error of b isσ ( b ^ ) = 2.30 b 2 σ ( M _ ) ,whereσ 2 ( M _ ) = ∑ i − 1 n ( M i − M _ ) 2 / n ( n − 1 ) .In an example from central California, stable estimates of b require a space-time window containing about 100 earthquakes. From 1952 to 1978, the average b and 90 per cent confidence limits are 0.95 (+0.94, −0.30). Some fluctuations of b are statistically significant but some are not. Within 90 per cent confidence limits, b changes from a low of 0.60 (+0.11, −0.09) in 1955 to a high of 1.39 (+0.25, −0.21) in 1967 and drops to 0.72 (+0.13, −0.10) in 1975. In this example, no correlation between large earthquakes (M &gt; 5) and b variations occurred.

Bibliography

Shi, Y., & Bolt, B. A. (1982). The standard error of the magnitude-frequencybvalue. Bulletin of the Seismological Society of America, 72(5), 1677â1687.

Authors 2

Yaolin Shi (first)
Bruce A. Bolt (additional)

References 14 Referenced 507

{'key': '2021063015295453600_BIB1', 'volume-title': 'Handbook of Mathematical Functions', 'author': 'Abramowitz', 'year': '(1965)'} / Handbook of Mathematical Functions by Abramowitz ((1965))
{'key': '2021063015295453600_BIB2', 'first-page': '237', 'article-title': 'Maximum likelihood estimate of b in the formula log N = a − bM and its confidence limits', 'volume': '43', 'author': 'Aki', 'year': '(1965)', 'journal-title': 'Bull. Earthquake Res. Inst., Tokyo Univ.'} / Bull. Earthquake Res. Inst., Tokyo Univ. / Maximum likelihood estimate of b in the formula log N = a − bM and its confidence limits by Aki ((1965))
{'key': '2021063015295453600_BIB3', 'article-title': 'Catalog of earthquakes in northern California and adjoining areas', 'author': 'Bolt', 'year': '(1975)'} / Catalog of earthquakes in northern California and adjoining areas by Bolt ((1975))
10.1785/BSSA0630020517 / Bull. Seism. Soc. Am. / Variation of the frequency-magnitude relation during earthquake sequences in New Zealand by Gibowicz ((1973))
{'key': '2021063015295453600_BIB5', 'volume-title': 'Seismicity of the Earth', 'author': 'Gutenberg', 'year': '(1954)', 'edition': '2nd ed.'} / Seismicity of the Earth by Gutenberg ((1954))
{'key': '2021063015295453600_BIB6', 'first-page': '101', 'article-title': 'Time and space scanning of the b value—A method for monitoring the development of catastrophic earthquakes', 'volume': '21', 'author': 'Li', 'year': '(1978)', 'journal-title': 'Acta Geophys. Sin.'} / Acta Geophys. Sin. / Time and space scanning of the b value—A method for monitoring the development of catastrophic earthquakes by Li ((1978))
10.1016/0031-9201(79)90115-8 / Phys. Earth Planet. Interiors / Variation of energy release, rate of occurrence and b value of earthquakes in the main seismic region, New Zealand by Robinson ((1979))
10.1785/BSSA0580010399 / Bull. Seism. Soc. Am. / The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes by Scholz ((1968))
10.1038/289136a0 / Nature / The b-value as an earthquake precursor by Smith ((1981))
10.1785/BSSA0560010185 / Bull. Seism. Soc. Am. / Differences between aftershocks and foreshocks in the relationship of magnitude to frequency of occurrence for the great Chilean earthquake of 1960 by Suyehiro ((1966))
{'key': '2021063015295453600_BIB11', 'first-page': '99', 'article-title': 'A method for determining the value of b in formula log N = a - bM showing the magnitude-frequency relation for earthquakes', 'volume': '13', 'author': 'Utsu', 'year': '(1965)', 'journal-title': 'Geophys. Bull. Hokkaido Univ.'} / Geophys. Bull. Hokkaido Univ. / A method for determining the value of b in formula log N = a - bM showing the magnitude-frequency relation for earthquakes by Utsu ((1965))
10.4294/jpe1952.14.37 / J. Phys. Earth / A statistical significance test of the difference in b-value between two earthquake groups by Utsu ((1966))
{'key': '2021063015295453600_BIB13', 'first-page': '292', 'article-title': 'On the method of estimating b value and its standard error', 'volume': '3', 'author': 'Zhang', 'year': '(1981)', 'journal-title': 'Acta Seismologica Sin.'} / Acta Seismologica Sin. / On the method of estimating b value and its standard error by Zhang ((1981))
{'key': '2021063015295453600_BIB14', 'first-page': '24', 'article-title': 'Proc. conf. on tectonic problems on the San Andreas fault system', 'author': 'Wyss', 'year': '(1973)'} / Proc. conf. on tectonic problems on the San Andreas fault system by Wyss ((1973))

Dates

Type	When
Created	4 years, 2 months ago (June 30, 2021, 2:32 p.m.)
Deposited	2 years, 7 months ago (Jan. 1, 2023, 11:59 p.m.)
Indexed	9 hours, 40 minutes ago (Aug. 30, 2025, 12:28 p.m.)
Issued	42 years, 10 months ago (Oct. 1, 1982)
Published	42 years, 10 months ago (Oct. 1, 1982)
Published Online	42 years, 10 months ago (Oct. 1, 1982)
Published Print	42 years, 10 months ago (Oct. 1, 1982)

Funders 0

None

BibTeX

@article{Shi_1982, title={The standard error of the magnitude-frequencybvalue}, volume={72}, ISSN={0037-1106}, url={http://dx.doi.org/10.1785/bssa0720051677}, DOI={10.1785/bssa0720051677}, number={5}, journal={Bulletin of the Seismological Society of America}, publisher={Seismological Society of America (SSA)}, author={Shi, Yaolin and Bolt, Bruce A.}, year={1982}, month=oct, pages={1677–1687} }

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  "abstract": "<jats:title>abstract</jats:title><jats:p>Estimated b values in log N = a \u2212 bM are widely used in seismicity comparisons and risk analysis, but uncertainties have been little explored. In this paper, the usual F probability density distribution for b is given and compared with an asymptotic form for temporally varying b. Convenient tables for the standard error of b are given that allow statistical tests to accompany investigations of both temporal and spatial variations of b. With large samples and slow temporal changes in b, the standard error of b is</jats:p><jats:p>\u03c3 ( b ^ ) = 2.30 b 2 \u03c3 ( M _ ) ,</jats:p><jats:p>where</jats:p><jats:p>\u03c3 2 ( M _ ) = \u2211 i \u2212 1 n ( M i \u2212 M _ ) 2 / n ( n \u2212 1 ) .</jats:p><jats:p>In an example from central California, stable estimates of b require a space-time window containing about 100 earthquakes. From 1952 to 1978, the average b and 90 per cent confidence limits are 0.95 (+0.94, \u22120.30). Some fluctuations of b are statistically significant but some are not. Within 90 per cent confidence limits, b changes from a low of 0.60 (+0.11, \u22120.09) in 1955 to a high of 1.39 (+0.25, \u22120.21) in 1967 and drops to 0.72 (+0.13, \u22120.10) in 1975. In this example, no correlation between large earthquakes (M &amp;gt; 5) and b variations occurred.</jats:p>",
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