Regression problems Regression problems for magnitudefor 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

The need for a unified magnitudeThe need for a unified magnitude

A large variety of earthquake size A large variety of earthquake size indicator exists (mindicator exists (ms,s, m mb,b, m md, d, mmL,L, M M0, 0,

MMe,e, M Mw ,w ,etc.)etc.)

Each one with a different meaningEach one with a different meaning

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

MMww

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

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.

Fail to satisfy the basic assumptions may:

1) Lead to wrong magnitude conversions,

2) Have severe consequences on the b-value of the Gutenberg-Richter magnitude-frequency distribution, which is the basis for probabilistic seismic hazard estimates

Which regression relation?Which regression relation?

Standard Linear RegressionStandard Linear Regression Other RegressionsOther Regressions

Here we focus on the performance ofHere we focus on the performance of

Standard RegressionStandard Regression Orthogonal RegressionOrthogonal Regression

2y / 2

x

SR Standard least-squares Regression

ISR Inverse Standard least-squares Regression

GOR General Orthogonal Regression

OR Orthogonal Regression. Special case of GOR with

2x ~ 2

y

=1

2x 0, 2

y > 0 Y b X + a

2x > 0, 2

y 0 Y b X + a

2x > 0, 2

y > 0 Y = b X + a

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

GOR

SR

ISR

Example:

(X, Y) exponentially distributed,

Exponentially distributed errors added to X and Y,

True slope () = 1.

However the point with GOR is

That the error ratio between the y and the x variables (2

y / 2x) needs to be

known.

In practice is mostly ignored since the seismological data centers do not publish standard deviations for their average event magnitudes.

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

Slopes

Ratios between variances

Absolute values of errors x, y

which may be enocuntered when converting magnitudes

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 <true < 2 0.05 <x, y < 0.50 0.25 < < 0.3

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 x and y

Steps 1) to 3) are repeated 103 times and 103 SR, ISR, GOR and OR regressions are performed to obtain the average SR, ISR, GOR, OR and their deviations.

Generation of the datasets

Results

GOR is always the best fitting procedure

However, if is unknown…

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

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

A typical Italian datasetA typical Italian dataset

Data no.

ms Mw 109

mL Mw 121

mb Mw 204

ms 0.28

mL 0.22

mb 0.37

Mw 0.18

Italian earthquakes in 1981-1996.

computed for each earthquake each time it was recorded by at least 3 stations.

MMww-m-mss

MMww-m-mLL

MMww-m-mbb

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…

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

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

The problem of the magnitude conversion can of course be approached also through other techniques

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:

1. Apply the OR for magnitude-intensity relations

2. but = 1

2. Gutdeutsch et al., 2002 finds the mL-ms relation through OR ( = 1) for the Kàrnik (1996) catalouge of central-southern Europe

3. Grünthal and Wahlström, 2003 applied the 2 regression to central Europe

4. Gutdeutsch and Keiser are studing the 2 regressions for magnitudes

IN EUROPEIN EUROPE

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