Earthquake Mc

Earthquake Mc

Core Task

Estimate and explain the magnitude of completeness (Mc) for an earthquake catalog by comparing normalized cumulative occurrence curves for a range of threshold magnitudes. Use this skill when a catalog must be filtered to a complete magnitude range before downstream seismicity statistics.

The motivating paper is:

• Correlation of earthquake occurrence among major fault zones in the eastern margin of the Tibetan Plateau through Big Data Analysis, The Innovation Geoscience, 2025.
• It cites Petrillo and Zhuang (2023), Verifying the Magnitude Dependence in Earthquake Occurrence, Phys. Rev. Lett. 131, 154101.
• In the EMQTP case, threshold curves were tested for Mc = 0.5, 1.0, 1.5, 2.0, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.5. Curves were dispersed for Mc <= 2.0 and nearly overlapped for Mc >= 2.5, so the paper adopted Mc = 2.5.

Do not hard-code 2.5 for new catalogs. Treat it as an example from that paper.

Required Inputs

Require or infer:

• Event origin time or an event order column.
• Magnitude column.
• Study region, time window, catalog source, magnitude type, and magnitude precision if available.
• Candidate magnitude thresholds, usually spanning below and above the likely completeness magnitude.

Before estimating Mc, check:

• Time values are parseable and sorted chronologically.
• Magnitudes are numeric and use one magnitude scale or documented mixed scales.
• The catalog time span does not combine strongly incompatible network eras without noting the limitation.
• Each candidate threshold has enough events to support a curve. Flag thresholds with too few events.

Normalized Quantity Method

For each threshold magnitude m_th, select events with M >= m_th. Compute the normalized cumulative count as a function of time:

textCopy code
F(t | m_th) = count(events with time <= t and M >= m_th) / count(events with M >= m_th)

This is the empirical cumulative distribution of occurrence times for events above each threshold.

Interpretation:

• If m_th is below the completeness magnitude, early or low-detectability periods usually miss small events. The normalized curves for lower thresholds separate from higher-threshold curves.
• If m_th is at or above Mc, the curves for all higher thresholds should become visually indistinct or nearly collapse, apart from small noise.
• Choose the lowest threshold at which the curve family is stable/collapsed and event counts remain adequate.

The method is a catalog-completeness diagnostic, not a standalone proof. Combine the plot with event counts, known station/network changes, magnitude precision, and study purpose.

Workflow

1. Load the catalog and identify time and magnitude fields.
2. Define candidate thresholds. If the user does not provide them, use the catalog magnitude precision:
   • 0.1 spacing for high-resolution catalogs.
   • 0.5 spacing for coarse historical or regional catalogs.
   • Include several thresholds below and above the apparent transition.
3. For each threshold, calculate event count and F(t | m_th).
4. Plot all normalized cumulative curves on the same time axis. Overlay catalog events or a magnitude-time scatter when useful.
5. Quantify curve separation using a common time grid:
   • Interpolate each threshold curve on the same grid.
   • Compare the envelope of curves from each candidate threshold upward.
   • Report max and median separation; use this only as a guide, not a substitute for visual inspection.
6. Select the lowest threshold where higher-threshold curves collapse and sample size is still sufficient.
7. Write a short justification that includes candidate thresholds, accepted Mc, rejected lower thresholds, event counts, and known caveats.

Script

Use scripts/estimate_mc_normalized.py when the user provides a CSV catalog or asks for a reproducible calculation.

Example:

bashCopy code
python scripts/estimate_mc_normalized.py catalog.csv \
  --time-column time \
  --mag-column magnitude \
  --thresholds 0.5,1.0,1.5,2.0,2.5,2.6,2.7,2.8,2.9,3.0,3.5 \
  --out-dir mc_results

The script writes:

• normalized_curves.csv
• threshold_summary.csv
• mc_report.md
• normalized_curves.png if matplotlib is installed

Treat the script recommendation as a first pass. Always inspect the plot and the event counts before giving the final Mc.

Reporting Template

When reporting Mc, include:

textCopy code
Catalog: <source, region, time window>
Method: normalized cumulative occurrence curves after Petrillo & Zhuang (2023)
Candidate thresholds: <list>
Accepted Mc: <value>
Reason: curves below <value> are separated; curves from <value> upward nearly overlap/collapse.
Event count at Mc: <n>
Caveats: <network changes, magnitude type, sparse high-threshold curves, mixed catalog sources, etc.>
Downstream use: use events with M >= Mc for statistics requiring catalog completeness.

Relationship To b-Value

Do not claim that the EMQTP paper computed a b-value unless the user provides additional supporting text. In that paper's main text, the relevant section uses the normalized quantity method to choose Mc; it does not present a b-value calculation workflow.

If the user asks to calculate b-value after Mc, use a standard Gutenberg-Richter/Aki-Utsu maximum-likelihood estimate on events with M >= Mc:

textCopy code
b = log10(e) / (mean(M) - (Mc - deltaM / 2))

Use deltaM equal to the catalog magnitude bin width, commonly 0.1 when magnitudes are rounded to 0.1. State whether the b-value is calculated from the raw or declustered catalog.