Frequency-magnitude distributions, and their associated uncertainties, are of key importance in statistical seismology. When fitting these distributions, the assumption of Gaussian residuals is invalid since event numbers are both discrete and of unequal variance. In general, the observed number in any given magnitude range is described by a binomial distribution which, given a large total number of events of all magnitudes, approximates to a Poisson distribution for a sufficiently small probability associated with that range. In this paper, we examine four earthquake catalogues: New Zealand (Institute of Geological and Nuclear Sciences), Southern California (Southern California Earthquake Center), the Preliminary Determination of Epicentres and the Harvard Centroid Moment Tensor (both held by the United States Geological Survey). Using independent Poisson distributions to model the observations, we demonstrate a simple way of estimating the uncertainty on the total number of events occurring in a fixed time period.
It is well documented that typical catalogues containing large numbers of earthquake magnitudes are closely approximated by power-law or gamma frequency distributions [1,2,3,4]. This paper addresses the characterisation of counting errors (that is, the uncertainties in histogram frequencies) required when fitting such a distribution via the maximum likelihood method, rather than the choice of model itself (for which see [5]). We follow this with an empirical demonstration of the Poisson approximation for total event-rate uncertainty [used in 5]. Our analysis provides evidence to support the assumption in seismic hazard assessment that earthquakes are Poisson processes [6,7,8,9], which is routinely stated yet seldom tested or used as a constraint when fitting frequencymagnitude distributions. Use is made of the Statistical Seismology Library [10], specifically the data downloaded from the New Zealand Institute of Geological and Nuclear Sciences (GNS, http://www.gns.cri.nz), the Southern California Earthquake Center (SCEC, http://www.scec.org) and the United States Geological Survey (USGS, http://www.usgs.gov), along with associated R functions for extracting the data.
Consider a large sample of N earthquakes. In order to estimate the underlying proportions of different magnitudes, which reflect physical properties of the system, the data are binned into m magnitude ranges containing n events such that m i=1 n i = N . Since n are discrete, a Gaussian model for each n i is inappropriate and may introduce significant biases in parameter estimations [11,12,13]. Hence when fitting some relationship with magnitudes M, n f it = f (M), linear regression must take the generalised, rather than least-squares, form [14]. Weighted least squares is an alternative approach which we do not consider here. The set n is described as a multinomial distribution; should we wish to test whether two different samples n and n ′ are significantly different given a fixed N “trials”, confidence intervals that reflect the simultaneous occurrence of all n must be constructed using a Bayesian approach [15]. However, in the case of earthquake catalogues, it is the temporal duration rather than the number of events that is fixed. Observational variability is not, therefore, constrained to balance a higher n i at some magnitude with a lower n j elsewhere, and n are well approximated by independent binomial distributions [16].
Each incremental magnitude range (M i -δM/2,M i + δM/2) contains a proportion of the total number of events and hence a probability p i with which any event will fall in that range. Providing the overall duration of the catalogue is greater than that of any significant correlations between either magnitudes or inter-event times, n i can be modelled as a binomial experiment with N independent trials each having a probability of “success” p i [16]. The binomial distribution converges towards the Poisson distribution as N → ∞ while N p 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 earthquake catalogues are consistent with a Poisson hypothesis, while in Sect. 3 we derive the resulting Poisson distributions of the total numbers of events, which provide simple measures of uncertainty in event rates. Tensor (CMT, Jan 1977 -June 1999, <100 km focal depth). While we impose no additional temporal or spatial filters on the raw data, magnitude limits are chosen to minimise the effects of incompleteness at lower magnitudes and undersampling of higher magnitudes. Following [5], who demonstrate the use of an objective Bayesian information criterion for choosing between functions, we seek to fit each catalogue with either a single power-law distribution
M being already on a log scale, or a gamma distribution
where a, b, c and k are constants. The gamma distribution consists of a power law of seismic moment or energy at the lower magnitudes followed by an exponential roll-off. Unlike pure power laws, its integration is finite and so it represents a physical generalisation of the Gutenberg-Richter law; for examples see [19] and references therein. For internal consistency, the Poisson assumption in [5] is indeed valid as we now demonstrate.
As explained in Sect. 1, generalised linear regression is required since we have 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% confidence intervals and plotted in Fig. 1. In all four catalogues, the binomial and Poisson residuals are almost indistinguishable, and show no significant deviation from the expected 1 in 20 exceedance rate when counting those points that lie outside the 95% confidence limits. Equal bin widths ∆M = 0.1 are used as is co
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