Correcting exponentiality test for binned earthquake magnitudes

TL;DR

The paper addresses biases in testing exponentiality of earthquake magnitudes due to binning, which affects magnitude-of-completeness estimates. It shows that standard uniform dithering fails to restore the continuous exponential and induces a residual lifetime artifact that becomes detectable with larger catalogs or coarser bins. The authors derive the exact noise distribution— a truncated exponential on $[0,\\Delta m)$—that reconstitutes the exponential distribution when adding noise, and they validate this approach with numerical tests showing Lilliefors rejection rates consistent with the significance level. This correction removes a methodological bias, improving completeness estimation and GR parameter inference, particularly for high-resolution catalogs.

Abstract

In theory, earthquake magnitudes follow an exponential distribution. In practice, however, earthquake catalogs report magnitudes with finite resolution, resulting in a discrete (geometric) distribution. To determine the lowest magnitude above which seismic events are completely recorded, the Lilliefors test is commonly applied. Because this test assumes continuous data, it is standard practice to add uniform noise to binned magnitudes prior to testing exponentiality. This work shows analytically that uniform dithering cannot recover the exponential distribution from its geometric form. It instead returns a piecewise-constant residual lifetime distribution, whose deviation from the exponential model becomes detectable as catalog size or bin width increases. Numerical experiments confirm that this deviation yields an overestimation of the magnitude of completeness in large catalogs. We therefore derive the exact noise distribution - a truncated exponential on the bin interval - that correctly restores the continuous exponential distribution over the whole magnitude range. Numerical tests show that this correction yields Lilliefors rejection rates consistent with the significance level for all bin widths and catalog sizes. The proposed solution eliminates a methodological bias in completeness estimation, which especially impacts high-resolution catalogs.

Figures (4)

• Figure 1: Numerical test showing how $\Delta m$ impacts the approximation quality of the exponential distribution (solid black line). The analytical solutions of the piecewise constant staircase PDF (left) and CDF (right) are calculated for the following values of $\Delta m$: 0.1, 0.2, 0.3, 0.4, 0.5. Since the maximum magnitude is fixed ($M_{max}=5.0$), here $N$ is a dependent variable: $N = (M_{max} - M_0)/\Delta_M$, where $M_0=1.0$ is the minimum magnitude.
• Figure 2: Numerical test showing how $N$ impacts the approximation quality of the exponential distribution. Same as the previous test (see Figure \ref{['fig:testA']}), but now we fix $\Delta m = 0.1$ and explore the following values for $N$: 5, 10, 20, 50.
• Figure 3: Lilliefors test rejection rates (%) for $\alpha$ = 0.1, computed from 1,000 simulations, each incorporating 100 noise realizations, across varying bin widths ($\Delta m$) and catalog sizes. See Table \ref{['Tab:Rejection_Rates']}.
• Figure 4: Same as Figure \ref{['fig:lilliefors']}, but magnitudes are dithered with truncated exponential noise. See also Table \ref{['Tab:Rejection_Rates_Trunc']}.