Please download to get full document.

View again

of 9

A Poisson model for earthquake frequency uncertainties in seismic hazard analysis

A Poisson model for earthquake frequency uncertainties in seismic hazard analysis
2 views9 pages
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Documenttranscript
    a  r   X   i  v  :   0   8   0   7 .   2   3   9   6  v   2   [  p   h  y  s   i  c  s .  g  e  o  -  p   h   ]   3   S  e  p   2   0   0   8  A Poisson modelfor earthquake frequency uncertaintiesin seismic hazard analysis J. Greenhough & I. G. MainSchool of Geosciences, University of Edinburgh, UK john.greenhough@ed.ac.uk, ian.main@ed.ac.ukSeptember 3, 2008 Abstract Frequency-magnitudedistributions, and their associated uncertainties,are of key importance in statistical seismology. When fitting these dis-tributions, the assumption of Gaussian residuals is invalid since eventnumbers are both discrete and of unequal variance. In general, the ob-served number in any given magnitude range is described by a binomialdistribution which, given a large total number of events of all magnitudes,approximates to a Poisson distribution for a sufficiently small probabilityassociated with that range. In this paper, we examine four earthquakecatalogues: New Zealand (Institute of Geological and Nuclear Sciences),Southern California (Southern California Earthquake Center), the Pre-liminary Determination of Epicentres and the Harvard Centroid MomentTensor (both held by the United States Geological Survey). Using inde-pendent Poisson distributions to model the observations, we demonstratea simple way of estimating the uncertainty on the total number of eventsoccurring in a fixed time period. 1 Introduction It is well documented that typical catalogues containing large numbers of earth-quake magnitudes are closely approximated by power-law or gamma frequencydistributions [1, 2, 3, 4]. This paper addresses the characterisation of count- ing errors (that is, the uncertainties in histogram frequencies) required whenfitting such a distribution via the maximum likelihood method, rather thanthe choice of model itself (for which see [5]). We follow this with an empiri-cal demonstration of the Poisson approximation for total event-rate uncertainty[used in 5]. Our analysis provides evidence to support the assumption in seismichazard assessment that earthquakes are Poisson processes [6, 7, 8, 9], which is routinely stated yet seldom tested or used as a constraint when fitting frequency-magnitude distributions. Use is made of the Statistical Seismology Library [10],1  specifically the data downloaded from the New Zealand Institute of Geologicaland Nuclear Sciences (GNS, http://www.gns.cri.nz), the Southern CaliforniaEarthquake Center (SCEC, http://www.scec.org) and the United States Geo-logical Survey (USGS, http://www.usgs.gov), along with associated R functionsfor extracting the data.Consider a large sample of   N   earthquakes. In order to estimate the underly-ing proportions of different magnitudes, which reflect physical properties of thesystem, the data are binned into  m  magnitude ranges containing  n  events suchthat  mi =1 n i  =  N  . Since n are discrete, a Gaussian model for each n i  is inappro-priate and may introduce significant biases in parameter estimations [11, 12, 13]. Hence when fitting some relationship with magnitudes  M ,  n fit  =  f  ( M ), lin-ear regression must take the generalised, rather than least-squares, form [14].Weighted least squares is an alternative approach which we do not considerhere. The set  n  is described as a multinomial distribution; should we wish totest whether two different samples  n  and  n ′ are significantly different given afixed  N   “trials”, confidence intervals that reflect the simultaneous occurrence of all  n  must be constructed using a Bayesian approach [15]. However, in the caseof earthquake catalogues, it is the temporal duration rather than the numberof events that is fixed. Observational variability is not, therefore, constrainedto balance a higher  n i  at some magnitude with a lower  n j  elsewhere, and  n  arewell approximated by independent binomial distributions [16].Each incremental magnitude range ( M  i − δM/ 2, M  i + δM/ 2) contains a pro-portion of the total number of events and hence a probability  p i  with which anyevent will fall in that range. Providing the overall duration of the catalogueis greater than that of any significant correlations between either magnitudesor inter-event times,  n i  can be modelled as a binomial experiment with  N   in-dependent trials each having a probability of “success”  p i  [16]. The binomialdistribution converges towards the Poisson distribution as  N   →∞ while  Np i remains fixed. Various rules of thumb are quoted to suggest values of   N   and  p i for which a Poisson approximation may be valid; see for example [17, 18]. Here, we show empirically in Sect. 2 that the frequencies in four natural earthquakecatalogues are consistent with a Poisson hypothesis, while in Sect. 3 we derivethe resulting Poisson distributions of the total numbers of events, which providesimple measures of uncertainty in event rates. 2 Frequency-magnitude Distributions Four earthquake catalogues are analysed: New Zealand (1460 – Mar 2007),Southern California (Jan 1932 – May 2007), the Preliminary Determination of Epicentres (PDE, Jan 1964 – Sep 2006) and the Harvard Centroid MomentTensor (CMT, Jan 1977 – June 1999,  < 100 km focal depth). While we imposeno additional temporal or spatial filters on the raw data, magnitude limits arechosen to minimise the effects of incompleteness at lower magnitudes and un-dersampling of higher magnitudes. Following [5], who demonstrate the use of an objective Bayesian information criterion for choosing between functions, we2  seek to fit each catalogue with either a single power-law distributionlog 10 n  =  a − b M ,  (1) M  being already on a log scale, or a gamma distributionlog 10 n  =  a − b M − c exp( k M ) ,  (2)where a, b, c and k are constants. The gamma distribution consists of a powerlaw of seismic moment or energy at the lower magnitudes followed by an ex-ponential roll-off. Unlike pure power laws, its integration is finite and so itrepresents a physical generalisation of the Gutenberg-Richter law; for examplessee [19] and references therein. For internal consistency, the Poisson assumptionin [5] is indeed valid as we now demonstrate.As explained in Sect. 1, generalised linear regression is required since wehave non-Gaussian counting errors on each bin. To test the consistency of these counting errors with the Gaussian, binomial and Poisson distributions,the residuals (observations minus chosen fit) are normalised to their 95% confi-dence intervals and plotted in Fig. 1. In all four catalogues, the binomial andPoisson residuals are almost indistinguishable, and show no significant deviationfrom the expected 1 in 20 exceedance rate when counting those points that lieoutside the 95% confidence limits. Equal bin widths ∆ M   = 0 . 1 are used as iscommon practice in earthquake hazard analysis; while this underestimates theintrinsic physical uncertainty of earthquake magnitude determination, for thepresent purposes the Poisson model appears to be a good proxy. At least, for thecatalogues considered here and with ∆ M   = 0 . 1, the Poisson model is valid. Byway of a further check, the value  b  of the fitted power-lawslope (Equs. 1, 2) givenbinomial errors is, to two significant figures, equal to that given Poisson errors,for all four catalogues. Constant Gaussian errors systematically overestimatefrequency uncertainties on the smaller magnitudes, leading to differences in  b  of +10% and  − 30% respectively for the Southern California and PDE data (seecaption of Fig. 2). These are caused by over-weighting the exponential compo-nents of the gamma distributions and exemplify worst-case results of incorrecterror structures. In Fig. 2, then, we need only plot the fits and uncertaintiesusing the Poisson model. Let us now describe, in Sect. 3, the usefulness of thisresult for estimating event-rate uncertainties. 3 Event-rate Uncertainties Having established that independent Poisson distributions characterise the mag-nitude frequencies in these four catalogues (importantly, these data span suffi-ciently large times and distances as to minimise dependencies due to clustering),we now ask how this impacts on uncertainties in total numbers of events. Whilewe cannot create equivalent catalogues by re-sampling the same regions underthe same physical conditions, we can simulate  S   = 10 5 samples from each magni-tude range by keeping the fitted mean  λ i  constant (representing the underlying3  4.0 4.5 5.0 5.5 6.0 6.5 7.0  −   2 −   1   0   1   2 magnitude   r  e  s   i   d  u  a   l   /   9   5   %   c  o  n   f   i   d  e  n  c  e   l   i  m   i   t 4.0 4.5 5.0 5.5 6.0 6.5 7.0  −   2 −   1   0   1   2 4.0 4.5 5.0 5.5 6.0 6.5 7.0  −   2 −   1   0   1   2 4.0 4.5 5.0 5.5 6.0 6.5 7.0  −   2 −   1   0   1   2 BinomialPoissonGaussian (a) 3 4 5 6 7  −   2 −   1   0   1   2 magnitude   r  e  s   i   d  u  a   l   /   9   5   %   c  o  n   f   i   d  e  n  c  e   l   i  m   i   t 3 4 5 6 7  −   2 −   1   0   1   2 3 4 5 6 7  −   2 −   1   0   1   2 3 4 5 6 7  −   2 −   1   0   1   2 BinomialPoissonGaussian (b) 5.0 5.5 6.0 6.5  −   2 −   1   0   1   2 magnitude   r  e  s   i   d  u  a   l   /   9   5   %   c  o  n   f   i   d  e  n  c  e   l   i  m   i   t 5.0 5.5 6.0 6.5  −   2 −   1   0   1   2 5.0 5.5 6.0 6.5  −   2 −   1   0   1   2 5.0 5.5 6.0 6.5  −   2 −   1   0   1   2 BinomialPoissonGaussian (c) 5.5 6.0 6.5 7.0 7.5 8.0  −   2 −   1   0   1   2 magnitude   r  e  s   i   d  u  a   l   /   9   5   %   c  o  n   f   i   d  e  n  c  e   l   i  m   i   t 5.5 6.0 6.5 7.0 7.5 8.0  −   2 −   1   0   1   2 5.5 6.0 6.5 7.0 7.5 8.0  −   2 −   1   0   1   2 5.5 6.0 6.5 7.0 7.5 8.0  −   2 −   1   0   1   2 BinomialPoissonGaussian (d) Figure 1: Residuals of fitted frequency-magnitude distributions fromGNS/SCEC/USGS catalogues: (a) New Zealand, (b) Southern California, (c)PDE, (d) CMT. (solid line) Best fit to Eq. 1 or 2; (dashed lines) 95% confidence limits of respective distribution.4  4.0 4.5 5.0 5.5 6.0 6.5 7.0         1  .        0        1  .        5        2  .        0        2  .        5        3  .        0        3  .        5 magnitude         l      o      g         1        0         (        f      r      e      q      u      e      n      c      y        ) (a) 3 4 5 6 7         0  .        0        0  .        5        1  .        0        1  .        5        2  .        0        2  .        5        3  .        0        3  .        5 magnitude         l      o      g         1        0         (        f      r      e      q      u      e      n      c      y        ) (b) 5.0 5.5 6.0 6.5         1  .        0        1  .        5        2  .        0        2  .        5        3  .        0        3  .        5        4  .        0 magnitude         l      o      g         1        0         (        f      r      e      q      u      e      n      c      y        ) (c) 5.5 6.0 6.5 7.0 7.5 8.0         0  .        0        0  .        5        1  .        0        1  .        5        2  .        0        2  .        5        3  .        0 magnitude         l      o      g         1        0         (        f      r      e      q      u      e      n      c      y        ) (d) Figure 2: Frequency-magnitude distributions from GNS/SCEC/USGS cata-logues. (solid line) Best fit to Eq. 1 or 2: (a) New Zealand, power law  b  = 1 . 0;(b) Southern California, gamma  b  = 0 . 91; (c) PDE, gamma  b  = 0 . 91; (d) CMT,gamma  b  = 0 . 85. (dashed lines) 95% Poisson confidence limits. UnweightedGaussian regression leads to  b -value estimates of (a) 0.98, (b) 1.03, (c) 0.66, (d)0.83.5
Advertisement
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks
SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...

Sign Now!

We are very appreciated for your Prompt Action!

x