Mathematical models for earthquake magnitude dispersion derived from first principles in Mathematical Geoenergy (GEM), Chapter 13.
Note: Chapter 13 also covers the Chandler Wobble. That material is hosted in a separate repository: pukpr/ChandlerWobble. The present subdirectory focuses exclusively on the earthquake magnitude dispersion content of GEM Ch. 13.
Earthquake magnitude dispersion is an archetypal example of the Principle of Maximum Entropy (MaxEnt) applied to geophysical hazard. The derivation in GEM Ch. 13 starts from two elementary observations:
A single-step MaxEnt argument thus provides a first-principles derivation of the most important empirical law in seismology, with $b \approx 1$ (the global average) predicting a mean magnitude excess of $1/\beta = 1/\ln 10 \approx 0.43$ magnitude units above completeness.
When the exponential rate $\beta$ (i.e.\ the $b$-value) is itself uncertain and assigned a second MaxEnt (exponential) prior — super-statistics — the marginal distribution becomes the Lomax (Pareto Type II) distribution. This heavier tail accounts for global $b$-value variability and the occasional extreme event (M > 8) that the single-$\beta$ model underestimates.
Eq. 13-1 — MaxEnt PDF of earthquake magnitude excess $\Delta m = m - m_c$
\[p(\Delta m) = \beta\cdot e^{-\beta \Delta m}, \quad \Delta m \geq 0\]Applying the Principle of Maximum Entropy to the excess magnitude $\Delta m = m - m_c \geq 0$ with only the mean excess $\bar{\Delta m} = 1/\beta$ known yields this exponential distribution. It has coefficient of variation CV = 1.
Eq. 13-2 — Gutenberg-Richter law (empirical, 1944)
\[\log_{10} N(>M) = a - b(M - m_c)\]Number of earthquakes exceeding magnitude $M$ follows a log-linear relation. The constants $a$ (productivity) and $b$ ($\approx 1.0$ globally) characterise the seismogenic zone.
Eq. 13-3 — Equivalence of $b$-value and exponential rate $\beta$
\[\beta = b \ln 10\]With this substitution the G-R exceedance $10^{-b \Delta m}$ equals $\exp(-\beta \Delta m)$, proving that the empirical Gutenberg-Richter law is identical to the MaxEnt exponential distribution.
Eq. 13-4 — CDF and CCDF of magnitude
\[F(\Delta m) = 1 - e^{-\beta \Delta m}\] \[\bar{P}(\Delta m) = e^{-\beta \Delta m} = 10^{-b \Delta m}\]The complementary CDF $\bar{P}$ is the Gutenberg-Richter exceedance probability. Median: $\Delta m_{1/2} = \ln 2 / \beta$.
Eq. 13-5 — Seismic energy from magnitude
\[E = E_0 \cdot 10^{1.5 M}\]Empirical relation between seismic energy $E$ and magnitude $M$. Each unit increase in $M$ corresponds to a $10^{1.5} \approx 31.6$-fold increase in energy; each 2-unit increase gives $\approx 1000$-fold.
Eq. 13-6 — Magnitude from seismic energy (change of variables)
\[M = \tfrac{2}{3} \log_{10}\left(\frac{E}{E_0}\right)\]Inverted form of Eq. 13-5. The factor $2/3$ connects energy (force times distance) to the seismic moment used in the Hanks-Kanamori scale.
Eq. 13-7 — MaxEnt seismic energy distribution
\[p(E) = \frac{1}{\bar{E}}\cdot e^{-E/\bar{E}}, \quad E \geq 0\]Applying MaxEnt to seismic energy with only the mean energy $\bar{E}$ known yields an exponential (Gibbs-Boltzmann) distribution. The transformation Eq. 13-6 then produces a magnitude PDF consistent with the Gutenberg-Richter relation.
Eq. 13-8 — Hanks-Kanamori moment-magnitude formula
\[M_w = \tfrac{2}{3} \log_{10}\left(\frac{M_0}{M_{0,\text{ref}}}\right) + C_0\]Seismic moment $M_0 = \mu A d$ (shear modulus $\times$ rupture area $\times$ average slip) is related to $M_w$ by this empirical formula with $C_0 = 10.7$ when $M_0$ is measured in dyne$\cdot$cm.
When the rate $\beta$ (and hence the $b$-value) varies from region to region, assign it a second MaxEnt prior:
Eq. 13-9 — Exponential prior for regional $\beta$
\[p(\beta) = \frac{1}{\bar{\beta}}\cdot e^{-\beta/\bar{\beta}}\]Eq. 13-10 — Super-statistics marginal integral
\[p(\Delta m) = \int_0^\infty p(\Delta m \mid \beta)\cdot p(\beta)\\,d\beta\]Eq. 13-11 — Lomax / Pareto-II magnitude PDF
\[p(\Delta m) = \frac{\bar{\beta}}{(1 + \bar{\beta}\cdot\Delta m)^2}\]Closed-form result of the super-statistics integral. The power-law tail ($\sim 1/\Delta m^2$ for large $\Delta m$) is heavier than the exponential G-R tail and accounts for the globally elevated frequency of extreme earthquakes. The identical distribution governs lake sizes (GEM Ch. 15) and other geophysical dispersive-aggregation processes.
Eq. 13-12 — Lomax CDF (super-statistics)
\[F(\Delta m) = \frac{\Delta m}{\Delta m + 1/\bar{\beta}}\]Median: $\Delta m_{1/2} = 1/\bar{\beta}$. For $\Delta m \gg 1/\bar{\beta}$ the CCDF $\bar{P} \approx 1/(\bar{\beta}\cdot\Delta m)$ — a pure power law (Pareto tail, exponent 1).
| File | Purpose |
|---|---|
earthquake_symbolic.py |
Symbolic derivation of all equations using SymPy |
earthquake_numerical.py |
Numerical implementation, validation, and composite figure |
earthquake_model_output.png |
Output figure (6 panels) generated by earthquake_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python earthquake_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python earthquake_numerical.py
The Gutenberg-Richter law is MaxEnt: the empirical $\log_{10} N \propto -bM$ relation follows directly from applying the Principle of Maximum Entropy to earthquake magnitude with only the mean magnitude constrained. There is no free parameter beyond $b$ (or equivalently $\beta = b \ln 10$).
$b \approx 1$ is the global attractor: for purely tectonic seismicity, the mean magnitude excess above completeness is $1/\beta = 1/\ln 10 \approx 0.43$ magnitude units. Deviations ($b < 1$ in highly stressed zones; $b > 1$ in volcanic or induced seismicity) reflect non-uniform stress states.
Each 2-magnitude step multiplies energy by $\approx 1000$: the $10^{1.5M}$ energy-magnitude scaling (Eq. 13-5) means that an M 8 earthquake releases about $10^3$ times more energy than an M 6. This large dynamic range is captured naturally by the exponential magnitude distribution.
Super-statistics extends the tail: allowing the $b$-value to vary regionally with a MaxEnt (exponential) prior produces the Lomax distribution (Eq. 13-11), which has a power-law tail $\propto 1/\Delta m^2$. This heavier tail better matches the observed global frequency of M > 8 earthquakes.
Universal structure: the Lomax distribution (Eq. 13-11/13-12) is identical to the lake-size distribution (GEM Ch. 15, Eq. 15-11/15-2). The same two-level MaxEnt argument — an exponential conditional distribution nested inside an exponential prior — governs both lake accumulation and seismic hazard, demonstrating the universal character of entropic dispersion.
Single-parameter models: the exponential (G-R) model has one free parameter ($b$, or equivalently $\beta$); the Lomax (super-statistics) model has one free parameter ($\bar{\beta}$). Neither requires fitting a multi-parameter functional form.