[1.15]
where λ is the rate of earthquake occurrence.
The seismic hazard is often only characterized by its source component, namely, by the probability that an earthquake of a magnitude greater than or equal to M occurs within D time units (the exceedance probability), R(M, D), or the expected average time between successive occurrences of earthquakes with magnitudes ≥ M (the mean return period), T(M). The exceedance probability reads:
If the seismic occurrence process is considered as Poissonian then:
and
1.2. Complexity of magnitude distribution
The most often used model of magnitude distribution, fM(M), FM(M), results from the empirical Gutenberg–Richter relation, which predicts a linear dependence of the logarithm of the number of earthquakes with magnitudes greater than or equal to M on M. This yields a piecewise distribution of magnitude:
where β is the distribution parameter and MC is the magnitude value beginning from where all earthquakes have been statistically recorded and are in the earthquake catalog. MC is called the catalog completeness level. Earthquake magnitude represents the physical size of an earthquake; hence in an environment of finite dimensions, as seismogenic zones are, it cannot be unlimited. For this reason, among others, we often amend the model [1.19] with an endpoint. The upper-bounded model of magnitude PDF with a hard endpoint is:
where Mmax is the upper limit of magnitude distribution (e.g. Cosentino et al. 1977).
The great popularity of the model [1.19] stems from two facts. First, it is a one-parameter model with a very simple form, and its parameter can be readily estimated, even from poorly populated samples. Second, the linear Gutenberg–Richter relation, at first glance, seems to fit almost all seismicity cases. This caused the Gutenberg–Richter relation to often be considered as a universal law, though there were works indicating breaks in scaling in the frequency-magnitude (early examples: Schwartz and Coppersmith 1984; Davison and Scholz 1985; Pacheco et al. 1992). The fit of the Gutenberg–Richter relation, and thus of the models [1.19] and [1.20] to actual earthquake samples, was rarely verified by rigorous statistical testing, and visible deviations from the model were interpreted as statistical scatter.
For obvious reasons, in seismic hazard problems, the attention is devoted to stronger magnitudes, whose exceedance probability, 1−FM(M), is low. This exceedance probability of magnitude enters the equations characterizing hazards in either the negative exponent (equations [1.16] and [1.17]), or in the denominator (equation [1.18]). When a magnitude distribution model was inappropriate in the range of larger magnitudes, this would result in significant systematic errors of hazard parameters.
There are parametric alternatives to the exponential models [1.19] and [1.20] (see, e.g., Lasocki 1993; Utsu 1999, and the references therein; Jackson and Kagan 1999; Kagan 1999; Pisarenko and Sornette 2003). The drawback in all of them is that their PDFs either decrease monotonically, or have at most one mode, that is, one local maximum at the catalog completeness level, MC. Such distributions cannot correctly model the modes at larger magnitudes and faster than linear drops of the logarithmic number of observations in the range of larger magnitudes, that is, the features observed and expected based on physical models of seismicity. The same drawback holds for the exponential distribution model.
Two approaches can be proposed for investigating how well the exponential models [1.19] and [1.20] fit data. Within the first approach, the null hypothesis is H0 (the magnitude distribution is exponential) against the alternative that it is not exponential. This hypothesis can be readily assessed by means of the Kolmogorov-Smirnov one-sample test, or with better precision by the Anderson- Darling test. These tests can also verify some of the mentioned alternative models. When we estimate model parameters from the data sample, and then this model is tested, the Kolmogorov-Smirnov test statistics should be modified. Both tests are very popular but will not be described here. Examples of when the above null hypothesis was disproved can be found in Urban et al. (2016) and Leptokaropoulos (2020). It is worth noting that both tests are suitable for continuous random variables, whose samples do not contain repeated values. Magnitude is a continuous random variable, but in seismic catalogs magnitudes have a limited number of digits, usually one digit after the decimal point. Therefore, magnitude data samples have many repetitions. To remedy the problem, Lasocki and Papadimitriou (2006) recommended randomizing the data before testing through the following equation:
where M is the magnitude value taken from catalog, δM is the length of the magnitude round off interval, u is the random value drawn from the uniform distribution in the [0,1] interval, F(•) is the CDF of an exponential model fitting the data, F-1(•) denotes its inverse function and M* is the randomized value of magnitude. This randomization procedure will return the original catalog magnitude when M*is rounded to δM, and the randomized values in every interval [M-0.5δM, M+0.5δM] repeat exponential distribution of the whole dataset. We can object that the best fitted exponential distribution is used first to randomize the data, and next, this fit is tested. Indeed, this randomization slightly amplifies the probability that H0 will not be rejected. However, other randomizations, e.g., the one assuming normal distribution in [M−0.5δM, M+0.5δM], is logically incorrect and strongly acts against H0.
Within the second approach, two independent null hypotheses