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Correcting exponentiality test for binned earthquake magnitudes

This paper demonstrates that the standard practice of adding uniform noise to binned earthquake magnitudes fails to recover the underlying continuous exponential distribution, leading to systematic overestimation of the magnitude of completeness, and proposes a corrected truncated exponential noise distribution that accurately restores exponentiality for statistical testing.

Original authors: Angela Stallone, Ilaria Spassiani

Published 2026-03-03

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CC BY 4.0

Original authors: Angela Stallone, Ilaria Spassiani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Counting Earthquakes with a Ruler

Imagine you are trying to measure the height of a forest of trees. You know that tree heights generally follow a smooth, predictable curve (tall trees are rare, short trees are common). This is similar to how earthquakes work: small ones happen all the time, and big ones are rare, following a rule called the Gutenberg-Richter law.

However, there's a catch. The "ruler" we use to measure earthquakes (seismographs) isn't infinitely precise. It can't tell the difference between a magnitude 4.00 and a 4.01. It only sees them as 4.0.

This turns our smooth, continuous curve of earthquake sizes into a staircase. Instead of a smooth slide, the data looks like steps. In statistics, this is called "binning" or "discretizing."

The Problem: The "Magic Dust" Mistake

Scientists need to find the "Magnitude of Completeness" ( $M_c$ ). Think of this as the "noise floor." Below this level, our instruments are too weak to hear the tiny earthquakes, so the data is messy and incomplete. Above this level, the data is clean and follows the smooth curve.

To find this level, scientists use a statistical test called the Lilliefors test. But this test is picky: it only works on smooth, continuous data, not staircases.

The Old Fix (Uniform Dithering):
To trick the test into thinking the data is smooth, scientists used to sprinkle "magic dust" on the steps. They would add a tiny bit of random noise to every earthquake magnitude.

The Analogy: Imagine you have a staircase made of wooden blocks. To make it look like a smooth ramp, you sprinkle a handful of sand (uniform noise) over the edges. You hope the sand fills the gaps perfectly so the ramp looks smooth.
The Flaw: The paper shows that this "sand" is the wrong shape. It's just random, flat sand. When you add it to the steps, you don't get a smooth ramp; you get a jagged, wobbly ramp that looks a bit like a staircase even after the sand is added.

The Consequence: Missing the Small Earthquakes

Because this "wobbly ramp" doesn't look perfectly smooth, the Lilliefors test gets confused. It thinks, "Hey, this data isn't following the perfect curve I expected!"

So, the test starts rejecting the data. It says, "Okay, the data is too messy to be trusted, so we must throw away the smaller earthquakes and only look at the big ones."

The Result: Scientists end up thinking the "noise floor" is much higher than it actually is. They ignore thousands of tiny, real earthquakes because they think their instruments are too noisy to hear them. In large, high-tech catalogs with millions of events, this error can make them miss earthquakes that are one full magnitude unit smaller than they should be. That's a huge difference in energy!

The Solution: The "Perfect Filler"

The authors of this paper asked: "If we want to turn a staircase back into a smooth ramp, what kind of 'sand' do we actually need?"

They did the math and realized the "sand" shouldn't be random and flat. It needs to be shaped like a truncated exponential curve.

The Analogy: Imagine the gaps between the wooden blocks aren't empty; they are filled with a special, squishy gel that naturally tapers off. If you pour this specific gel into the gaps, it perfectly fills the shape of the missing ramp.
The New Method: Instead of adding random noise, the authors propose adding this specific "truncated exponential" noise.

The Result: A Smooth Ride

When they tested this new method:

The "wobbly ramp" disappeared.
The data looked perfectly smooth again.
The Lilliefors test stopped complaining.
Scientists could finally trust the data all the way down to the very smallest earthquakes, even in massive, high-resolution catalogs.

Summary in One Sentence

The paper proves that the old way of "smoothing out" earthquake data (adding random noise) was actually hiding the smallest earthquakes by making the data look messy, and they have found the exact mathematical "glue" needed to fix the data so we can see the full picture of seismic activity.

1. Problem Statement

Earthquake magnitudes above the magnitude of completeness ( $M_c$ ) are theoretically modeled by the Gutenberg-Richter (GR) law, which follows a continuous exponential distribution. However, real-world seismic catalogs report magnitudes with finite resolution (e.g., one decimal place), resulting in discrete (binned) data that follows a geometric distribution.

To apply standard statistical tests for exponentiality (specifically the Lilliefors test, which assumes continuous data), the standard practice is to add uniform noise (dithering) to the binned magnitudes to "smooth" them into a continuous variable. The authors identify a critical flaw in this methodology:

The Bias: Adding uniform noise to a geometric distribution does not recover the underlying continuous exponential distribution. Instead, it creates a piecewise-constant (staircase) residual lifetime distribution.
The Consequence: This systematic deviation from the true exponential model causes the Lilliefors test to reject the null hypothesis (exponentiality) more frequently as catalog size increases or bin width widens. This leads to a systematic overestimation of the magnitude of completeness ( $M_c$ ), with biases exceeding one magnitude unit in large, high-resolution catalogs.

2. Methodology

The authors employed a combination of analytical derivation and numerical simulation to investigate the issue and propose a solution.

A. Analytical Derivation

Modeling Binned Data: They mathematically demonstrated that discretizing a continuous exponential variable via a floor function (binning) results in a geometric distribution.
Analyzing Uniform Dithering: They derived the probability density function (PDF) of a binned magnitude plus uniform noise. They proved that the resulting distribution is a sum of rectangular bumps with exponentially decaying heights (a staircase function), rather than a smooth exponential curve.
Deriving the Correct Noise: To mathematically restore the continuous exponential distribution from the discrete geometric form, they derived the exact distribution required for the added noise variable ( $Y$ ). They showed that $Y$ must follow a truncated exponential distribution within each bin $[0, \Delta m)$ , rather than a uniform distribution.

B. Numerical Experiments

The authors conducted two sets of numerical tests using synthetic catalogs generated from the geometric distribution (simulating binned data):

Uniform Noise Test: They applied standard uniform dithering to synthetic catalogs of varying sizes (1,000 to 1,000,000 events) and bin widths. They measured the estimated $M_c$ and the Lilliefors test rejection probability.
Truncated Exponential Noise Test: They repeated the experiments using the newly derived truncated exponential noise.
Comparison: They compared the rejection probabilities and $M_c$ estimates against the known ground truth ( $M_c = M_{min}$ ) to assess bias and statistical consistency.

3. Key Contributions

Identification of Methodological Flaw: The paper provides analytical proof that uniform dithering fails to recover the exponential distribution, introducing a systematic bias that is often masked in small catalogs but becomes severe in large datasets.
Derivation of Exact Correction: The authors derived the precise mathematical form of the noise required to transform discrete geometric data back into a continuous exponential distribution: a truncated exponential distribution defined within each magnitude bin.
Quantification of Bias: They quantified the magnitude of the overestimation error, showing that for high-resolution catalogs (e.g., $10^6$ events), uniform dithering can lead to $M_c$ overestimations of $>1.0$ magnitude units.

4. Results

Uniform Dithering Failure:
- In synthetic catalogs with $10^6$ events and a standard bin width ( $\Delta m = 0.1$ ), the Lilliefors test rejected the exponential hypothesis 100% of the time when uniform noise was used.
- The estimated $M_c$ shifted from the true value of 1.0 to 2.65, a massive overestimation.
- The rejection probability increased with both catalog size and bin width, indicating that the test correctly detected the "staircase" artifact introduced by uniform noise.
Truncated Exponential Success:
- When using the proposed truncated exponential noise, the estimated $M_c$ remained stable and close to the true value (e.g., 1.03 for $10^6$ events) across all catalog sizes.
- The Lilliefors rejection probabilities remained close to the significance level ( $\alpha = 0.1$ ) and did not systematically increase with catalog size, except for negligible deviations at extreme sizes likely due to numerical precision limits.
- The method successfully restored the continuous exponential distribution, eliminating the systematic bias.

5. Significance

Impact on Seismology: Accurate estimation of $M_c$ is fundamental for calculating the $b$ -value and assessing seismic hazard. The systematic overestimation caused by uniform dithering leads to the erroneous exclusion of valid data (events below the inflated $M_c$ ), skewing statistical analyses.
Relevance to High-Resolution Catalogs: The problem is most acute for modern, high-resolution catalogs generated via template matching or machine learning, which contain orders of magnitude more events than routine catalogs. These large datasets have high statistical power, making them highly sensitive to the subtle biases introduced by uniform dithering.
General Applicability: While illustrated using the Lilliefors test, the authors note that the identified bias and the proposed correction apply generally to any statistical test of exponentiality applied to discretized magnitude data.
Implementation: The authors provide a Python implementation of the truncated exponential dithering method, offering a simple, drop-in replacement for current practices to ensure statistically consistent results.

In conclusion, the paper argues that the standard practice of adding uniform noise to binned earthquake magnitudes is mathematically flawed. It proposes a rigorous, analytically derived correction (truncated exponential noise) that ensures the validity of exponentiality tests and the accuracy of $M_c$ estimation in modern seismology.