1. 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

2. 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

3. 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

4. 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

5. Linear Least-Squares Regression BASIC 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.

6. 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

7. Which regression relation?
Standard Linear Regression
Other Regressions

8. Here we focus on the performance of
Standard Regression
Orthogonal Regression
= s2y / s2x

9. s2x  0, s2y > 0 s2x > 0, s2y  0 s2x > 0, s2y > 0
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 > 0
Y  b X + a
s2x > 0, s2y  0
Y  b X + a
s2x > 0, s2y > 0
Y = b X + a

10. 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

11. GOR SR
ISR
Example:
(X, Y) exponentially distributed,
Exponentially distributed errors added to X and Y,
True slope (b) = 1.

12. 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.

13. Ratios h between variances Absolute values of errors sx, sy
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

14. 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 < btrue < 2
0.05 < sx, sy < 0.50
0.25 < h < 0.3

15. Generation of the datasets
1) 103 couples of magnitudes (Mx, My) with
3.5 < Mx, My < 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.

20. Results GOR is always the best fitting procedure
However, if h is unknown…

22. 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

24. If nothing is known about the variable variances, compare your case to the whole set of figures posted in to get some examples for chosing the best regression procedure

25. A typical Italian dataset
Data no. ms Mw
109 mL
121 mb
204 s ms
0.28 mL
0.22 mb
0.37 Mw
0.18
Italian earthquakes in
s computed for each earthquake each time it was recorded by at least 3 stations.

26. Mw-ms

27. Mw-mL

28. Mw-mb

29. 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…

30. BSSA, (2007)
While the most realistic result should be…

31. The use of SR without any discussion on the applicability of the model is unfortunately still too common

32. The problem of the magnitude conversion
can of course be approached also through other techniques
IN EUROPE
Panza et al., 2003; Cavallini and Rebez, 1996; Kaverina et al., 1996; Gutdeutsch et al., 2002; Grünthal and Wahlström, 2003; Stromeyer et al., 2004:
Apply the OR for magnitude-intensity relations
but h = 1
Gutdeutsch et al., 2002 finds the mL-ms relation through OR (h = 1) for the Kàrnik (1996) catalouge of central-southern Europe
Grünthal and Wahlström, 2003 applied the c2 regression to central Europe
Gutdeutsch and Keiser are studing the c2 regressions for magnitudes

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