## 2 Methods  
### Maximum Likelihood Estimation of the Parameters  
The Gutenberg-Richter distribution and all its derivations were originally developed using magnitudes. If we use seismic moments (_Mom_) instead of magnitudes, the Gutenberg-Richter distributions (unbounded/truncated/tapered) correspond to the Pareto distributions defined in [PERSON] (1966), with slope parameter \(\beta\) equal to 2/3 the b-value. The probability density function of the tapered distribution is (1966):  
\[f(Mom) =\left(\frac{\beta}{Mom}+\frac{1}{C_{Mom}}\right)\left(\frac{Mom_{ min}}{Mom}\right)^{\beta}exp\left(\frac{Mom_{min}-Mom}{C_{Mom}}\right) \tag{1}\] \[for\;Mom_{min}\leq Mom<\infty\]  
where \(Mom_{min}\) is the seismic moment of completeness of the catalog, \(\beta\) is the parameter controlling the slope of the distribution, and \(C_{Mom}\) is the corner moment that controls the tail of the distribution. We stress that it is always possible to pass from the seismic moment to the magnitude distribution (here, we adopt the relationship defined in [PERSON] (2016). In this case the corner moment \(C_{Mom}\) is called \"corner magnitude\" (\(C_{M}\)), and the seismic moment of completeness corresponds to the magnitude of completeness. If we have a seismic moment of completeness that varies with time (\(Mom_{min}^{(i)}\), Figure 2), we can easily rewrite Equation (1) with:  
\[f_{(i)}(Mom) =\left(\frac{\beta}{Mom}+\frac{1}{C_{Mom}}\right)\left(\frac{Mom_ {min}^{(i)}}{Mom}\right)^{\beta}exp\left(\frac{Mom_{min}^{(i)}-Mom}{C_{Mom}}\right)\] \[for\;Mom_{min}^{(i)}\leq Mom<\infty\]  
This relationship allows referring each observation to the completeness that holds at the time of its occurrence: in this time frame, indeed, Equation (2) describes the statistical distribution that holds. Being both the parameters of the distribution (\(C_{Mom}\) and \(\beta\)) in common to all these distributions (\(Mom_{min}^{(i)}\) is a parameter related to the seismic catalog, estimated independently), their likelihood holds for all such distributions. Thus, if we have a seismic catalog with  
Figure 2: Time vs. magnitude plot for a catalog with a variable magnitude of completeness (\(M_{min}^{(i)}\)). The grey line represents the completeness, black dots the seismic events. \(N\) earthquakes with moments \(x_{1},\ldots,x_{i},\ldots,x_{N}\), the log-likelihood of the tapered Pareto distribution becomes:  
\[LL(x_{1},\ldots,x_{N}|\beta,C_{Mom})=\sum_{i=1}^{N}ln\Big{[}f_{(i)}(x_{i})\Big{]} \tag{3}\]  
In Equation (3) the probability density function \(f_{(i)}\) depends on the seismic moment of completeness relative to the i-th earthquake. In Figure 3 we summarize the scheme of our methodology, applied to completeness thresholds relative to Figure 2: in this case the log-likelihood of Equation (3) is obtained by summing up the log-likelihoods relative to the three different thresholds of completeness. Notably, if the seismic moment of completeness is the same for all the events, Equation (3) becomes the classical log-likelihood for the tapered Pareto distribution [3]. To maximize the likelihood of observations, and evaluate the maximum likelihood estimation (MLE) of the parameters \(\beta\) and \(C_{Mom}\), we adopt a brute-force approach, that is, we evaluated the log-likelihood for many potential combinations of the parameters, covering the entire parameter space [12], This allows obtain the complete description of the log-likelihood function: the maximum of the function (\(LL_{\text{max}}\)) is, by definition, the MLE of the parameters. Moreover, this approach allows evaluating also the shape of the log-likelihood function, which is particularly useful to assess the uncertainty associated with the parameters' estimation, as it will be shown in the next section. We stress the simplicity of our approach: to move from Equation (1) to Equation (2) we only need to substitute \(Mom_{min}\) with \(Mom_{min}^{(i)}\), i.e., using the time-variable seismic moment of completeness instead of the fixed one.