occasionally receiving distinctly erroneous results; therefore, we suggest carrying out the tests independently twice, using samples [1.25] and [1.26], respectively. The tests results should be comparable.
When testing H02 we use
– Step 7. Calculating the test p-value. To do this we count how many samples of R, drawn in step 6, result in the kernel estimates of PDF [1.3], which only have one mode (or only one bump when testing H02). This can be achieved by studying the first derivative of the estimate obtained from [1.23] (the second derivative obtained from [1.24] when testing H02) in which [1.25] or [1.26] replaces The tests p-values are:
and
for testing H01 and H02, respectively.
– Step 8. Calibrating p-values. Lasocki and Papadimitriou (2006) checked the agreement between the p-values obtained from [1.27] and [1.28], and the expected p-values from the p-value definition. To do this they applied the testing procedure to a large number of Monte Carlo samples drawn from the exponential distribution [1.19]. Their tests showed that the p-values estimates, [1.27] and [1.28], could significantly deviate from their actual values, probably because of the skewness of the exponential distribution. Because this can impact the final test result, the cited authors suggested calibrating the p-values estimates, [1.27] and [1.28]. For this purpose, we draw a large number of samples from the exponential distribution [1.19] with the parameter β obtained in step 3. The calibrated p-value estimate, pm_cat, is the proportion of these samples, whose p-values obtained after performing steps 5–7 are less than or equal to pm. The calibration of pm is done in the same way.
The comparison of the calibrated p-value with the assumed test significance level α finalizes the test(s).
In a considerable proportion of studied cases, the above-presented tests questioned the exponential model for magnitude distribution. Such results were obtained for both worldwide and regional catalogs of natural earthquakes (Lasocki and Papadimitriou 2006; Lasocki 2007), as well as for anthropogenic seismicity data (e.g. Lasocki 2001; Lasocki and Orlecka-Sikora 2008, Urban et al. 2016; Leptokaropoulos 2020). Furthermore, the additional modes or bumps located in the ranges of larger magnitudes, as shown above, can yield significant errors of seismic hazard estimates. In this connection, new approaches for estimating the magnitude distributions are needed.
The smooth bootstrap test for multimodality of magnitude distribution has been implemented on the IS-EPOS Platform (tcs.ah-epos.eu). The platform integrates and provides free access to the research infrastructure of anthropogenic seismicity and the related hazards (Orlecka-Sikora et al. 2020).
1.3. Kernel estimation of magnitude distribution
The kernel estimation of magnitude distribution follows the general kernel estimation methods presented in Silverman (1986), with some adaptations to the specific features of magnitude (Kijko et al. 2001; Orlecka-Sikora and Lasocki 2005; Lasocki and Orlecka-Sikora 2008).
As already mentioned, magnitude datasets contain many repetitions. The kernel estimation of distribution functions is applicable for continuous random variables; hence, firstly, we should randomize magnitudes according to [1.21].
The estimation is based on the sample being representative of a population. The definition of the catalog completeness level, Mc, implicates the use of the magnitudes M ≥ Mc only.
The magnitude PDF is a steeply, exponentially like, the decreasing function. The larger the magnitudes are, the more sparse they are in data samples. In the seismic hazard studies, we are mainly interested in larger magnitudes. To ensure a better estimation of the distribution functions in the sparse data range, we use the estimators with the adaptive kernel [1.5] and [1.6]. For the left-hand side limited distribution of magnitude they take the form of:
[1.29]
The magnitude PDF has the global maximum at the catalog completeness level, Mc, and is zero for M < Mc. For this reason the data sample is mirrored symmetrically around Mc, and the estimation is carried out using the estimator [1.12] in the way described in section 1.1.
When the existence of a strict, single value upper bound to the magnitude range, Mmax, is assumed, the estimators of magnitude distribution functions are:
[1.31]
Kijko et al. (2001) compared the performances of the kernel estimation of magnitude distribution functions and the estimation based on the exponential distribution model [1.20]. For this purpose, they estimated the distribution functions, using Monte Carlo samples drawn from two distributions, mimicking real instances of magnitude distribution. The considered starting distributions were the exponential distribution
