Regression problems for magnitude

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Regression problems for magnitude

• Silvia Castellaro1, Peter Bormann2, Francesco
  Mulargia1 and Yan Y. Kagan3
• 1 Sett. Geofisica, Università Bologna (Italy)
• 2 GFZ, Potsdam (Germany)
• 3 UCLA, Los Angeles
• IUGG (Perugia), 11 July 2007

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The need for a unified magnitude

• A large variety of earthquake size indicator
  exists (ms, mb, md, mL, M0, Me, Mw ,etc.)
• Each one with a different meaning

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• Ignoring the fact that a single indicator of size
  may be inadequate in seismic hazard estimates,
• The state of the art is to use
• on account of its better definition in
  seismological terms

Mw
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The magnitude conversion problem

• In converting magnitude,
• It is commonly assumed that the relation Mx My
  is linear
• (this is justified as long as none of them shows
  a much stronger saturation than the other)
• Least-Squares Linear Regression
• is so popular that it is mostly applied without
  checking whether its basic requirements are
  satisfied

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Linear Least-Squares RegressionBASIC ASSUMPTIONS

• The uncertainty in the independent variable is at
  least one order of magnitude smaller than the one
  on the dependent variable,
• Both data and uncertainties are normally
  distributed,
• Residuals are constant.

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• Fail to satisfy the basic assumptions may
• Lead to wrong magnitude conversions,
• Have severe consequences on the b-value of the
  Gutenberg-Richter magnitude-frequency
  distribution, which is the basis for
  probabilistic seismic hazard estimates

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Which regression relation?
Standard Linear Regression Other Regressions

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Here we focus on the performance of
Standard Regression Orthogonal Regression

• s2y / s2x

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• SR Standard least-squares Regression
• ISR Inverse Standard least-squares Regression
• GOR General Orthogonal Regression
• OR Orthogonal Regression. Special case of GOR
  with
• s2x s2y
• h 1

• s2x ? 0, s2y gt 0
• Y ? b X a
• s2x gt 0, s2y ? 0
• Y ? b X a
• s2x gt 0, s2y gt 0
• Y b X a

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It has already been demonstrated that GOR
produces better results than SR/ISR (Castellaro
et al., GJI, 2006)

• On normally, log-normally and exponentially
  distributed variables
• On normally, log-normally and exponentially
  dstributed errors
• On different amount of errors

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GOR
SR
ISR
Example (X, Y) exponentially
distributed, Exponentially distributed errors
added to X and Y, True slope (b) 1.
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However the point with GOR is

• That the error ratio between the y and the x
  variables (h s2y / s2x) needs to be known.
• In practice h is mostly ignored since the
  seismological data centers do not publish
  standard deviations for their average event
  magnitudes.

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To define the performance of the different
procedures we run enough simulations to cover the
ranges of

• Slopes b
• Ratios h between variances
• Absolute values of errors sx, sy
• which may be enocuntered when converting
  magnitudes

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Parameters used in the simulations

• In order to produce realistic simulations,
  parameters are inferred from the study of CENC
  (Chinese Earthquake Network Center), GRSN (German
  Regional Seismic Network) and Italian official
  catalogues
• 0.5 lt btrue lt 2
• 0.05 lt sx, sy lt 0.50
• 0.25 lt ?h lt 0.3

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Generation of the datasets

• 1) 103 couples of magnitudes (Mx, My) with
• 3.5 lt Mx, My lt 9.5
• 2) Sampled from exponential distribution (Utsu,
  1999, Kagan, 2002, 2002b, 2005, Zailiapin et al.
  2005)
• 3) From (Mx, My) to (mx, my) by addition of
  errors sampled from Gaussian distributions with
  deviations sx and sy
• Steps 1) to 3) are repeated 103 times and 103 SR,
  ISR, GOR and OR regressions are performed to
  obtain the average bSR, bISR, bGOR, bOR and their
  deviations.

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Results

• GOR is always the best fitting procedure
• However, if h is unknown

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• Attention should be paid to the mb-MS relation
  which is not linear (due to saturation of the
  short-period mb for strong earthquakes), has an
  error ratio of about 2 and usually a rather large
  absolute scatter in the mb data

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If nothing is known about the variable variances,
compare your case to the whole set of figures
posted in www.terraemoti.net to get some examples
for chosing the best regression procedure
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A typical Italian dataset
Data no.
ms Mw 109
mL Mw 121
mb Mw 204
s
ms 0.28
mL 0.22
mb 0.37
Mw 0.18
Italian earthquakes in 1981-1996. s computed for
each earthquake each time it was recorded by at
least 3 stations.
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Mw-ms
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Mw-mL
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Mw-mb
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• The magnitude conversion problem may appear a
  solved problem while it is not!
• For example, some authors in the BSSA (2007)
  state this work likely represent the final stage
  of calculating local magnitude relation ML-Md by
  regression analysis but they forgot to
  consider the variable errors at all!
• It follows that

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BSSA, (2007)
While the most realistic result should be
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The use of SR without any discussion on the
applicability of the model is unfortunately still
too common
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The problem of the magnitude conversion can of
course be approached also through other techniques
IN EUROPE

1. Panza et al., 2003 Cavallini and Rebez, 1996
   Kaverina et al., 1996 Gutdeutsch et al., 2002
   Grünthal and Wahlström, 2003 Stromeyer et al.,
   2004
2. Apply the OR for magnitude-intensity relations
3. but h 1
4. Gutdeutsch et al., 2002 finds the mL-ms relation
   through OR (h 1) for the Kàrnik (1996)
   catalouge of central-southern Europe
5. Grünthal and Wahlström, 2003 applied the c2
   regression to central Europe
6. Gutdeutsch and Keiser are studing the c2
   regressions for magnitudes

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• The software to run SR, ISR, OR and GOR is
  available on www.terraemoti.net