**1. Introduction**

The Gutenberg-Richter law [1] is the most widely applied magnitude frequency distribution for earthquakes. If we look only to the distribution of the magnitudes, independently from the rate of events, this law corresponds to an exponential distribution [2]. In this case, it depends on only one parameter (the so-called b-value), controlling the slope of the distribution, and does not have an upper bound for the magnitude. In order to have a more physical behavior for the right tail of the magnitude distribution, two other formulations of this law are usually applied: the truncated and the tapered Gutenberg-Richter distributions [3]. The truncated version applies a hard bound to the tail, i.e., a maximum magnitude (Mmax). Instead, the tapered version applies a soft bound, i.e., a corner magnitude (*CM*): the probability of an earthquake bigger than the corner magnitude decreases very rapidly asymptotically reaching zero (see Figure 1).

**Citation:** Taroni, M.; Selva, J.; Zhuang, J. Estimation of the Tapered Gutenberg-Richter Distribution Parameters for Catalogs with Variable Completeness: An Application to the Atlantic Ridge Seismicity. *Appl. Sci.* **2021**, *11*, 12166. https://doi.org/ 10.3390/app112412166

Academic Editors: Stefania Gentili, Rita Di Giovambattista, Robert Shcherbakov, Filippos Vallianatos and Fernando M.S.F. Marques

Received: 9 July 2021 Accepted: 3 December 2021 Published: 20 December 2021

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**Figure 1.** Probability density functions of the tapered (**a**) and truncated (**b**) Gutenberg-Richter distributions, in a log10 Y-axis scale.

For these two formulations of the Gutenberg-Richter distribution, we need an extra parameter to be estimated: the maximum and the corner magnitude for the truncated and the tapered distributions, respectively. Regarding the estimation of the maximum magnitude,Zöller and Holschneider summarized very well the state of the art: "*the earthquake history in a fault zone tells us almost nothing about Mmax*" [4]. This and other papers [5–7] clearly show that a maximum likelihood estimation (MLE) of Mmax is not applicable, as the MLE is equal to the maximum observed magnitude and this may be problematic, considering the relatively short observation time as compared with mean recurrence times of large magnitude events. Conversely, the corner magnitude can be properly estimated if a sufficiently large amount of data is available [8]. The tapered Gutenberg-Richter distribution, also called tapered Pareto distribution or "Kagan distribution" by some statistical seismologists,was deeply investigated primarily by Kagan and Schoenberg [8], and then by Kagan [3], Schoenberg and Patel [9], and Geist and Parsons [7]. All these works use seismic catalogs with a single magnitude of completeness. These methods do not need any assumption on the distribution of the number of events. However, the size of the catalog can largely be expanded by adopting multiple levels of completeness, with a completeness magnitude that decreases in time, as the quality and quantity of the available instrumentation improve (Figure 2). This allows including in the estimation both the large number of relatively small events recorded by modern monitoring networks, and the larger events that occurred in the past, possibly also from pre-instrumental times [10].

Existing methods [10–12] that deal with this problem need an assumption regarding the distribution through the time of the events. The distribution usually assumed is the Poisson distribution. This assumption is not always correct for the events in seismic catalogs, in particular if the magnitude of completeness of the catalog is lower than Mw 6.5 [13], forcing the application of declustering algorithms. On the other hand, declustering decreases the number of usable data and may introduce important biases in parameter estimations [14], which may even depend on the declustering algorithm selected.

This paper aims to develop a method to perform the parameters' estimation for catalog with a variable magnitude of completeness (see Figure 2), without making any assumption on the distribution of the number of events. Thus, such a method can take the pros of both the previously described approaches, avoiding the cons relative to the single level of completeness and the Poisson assumption, allowing to use more data in the data estimation.

In the following, we first introduce the method and then we apply it to the Atlantic ridge seismicity. This region is characterized by shallow seismicity with a prevalence of normal/strike-slip mechanisms. The statistics of seismicity for oceanic spreading ridges was already studied in Bird et al. [15] and Bird and Kagan [16], which estimated for these zones a corner magnitude lower than other parts of the world (*CM* ≈ 5.8). Here, focusing on the Atlantic ridge with a longer catalog and our newly developed methodology, we improve the estimation of the parameters of the tapered Gutenberg-Richter distribution exploiting the potentiality of the newly developed method.

**Figure 2.** Time vs. magnitude plot for a catalog with a variable magnitude of completeness (*M*(*i*) *min*). The grey line represents the completeness, black dots the seismic events.
