The idea

Mechanical earthquake-cycle models track stress and friction on a fault. Those quantities are hard to measure at the scales that matter, and there may be no unique mechanical model of a given fault. kes takes a macroscopic view instead, in the spirit of Carnot’s engine that works “independently of any mechanism” and Jaynes’ maximum entropy reasoning. It tracks one kinematic quantity, the geometric moment $m$: fault area times average slip, in m³. It then asks what the least-committal earthquake sequence is that respects

• an approximate long-term balance between moment accumulated by tectonic loading and moment released by earthquakes and afterslip;
• three well-established empirical laws: Gutenberg–Richter magnitudes, magnitude–area scaling and Omori aftershock decay.

Where nothing else is known, choices are made by maximizing entropy (uncertainty).

The model has three parts:

1. an earthquake rate set by the moment budget;
2. maximum entropy earthquake locations;
3. maximum entropy afterslip.

All three act on a fault discretized into small patches. In the examples here the fault is a vertical strike-slip fault 200 km long and 25 km deep.

Earthquake rate from the moment budget

Each patch $i$ stores a geometric moment deficit. It grows with tectonic loading and shrinks when the patch slips in an earthquake or in afterslip. Summed over the fault, the deficit is

the moment accumulated minus that released coseismically ($m_\mathrm{c}$) and by afterslip ($\dot m_\mathrm{s}$). The earthquake rate is proportional to this deficit, plus an Omori aftershock term:

The aftershock productivity grows with mainshock magnitude, $K(M) \propto 10^{0.8(M - 6)}$ (Reasenberg & Jones, 1989). The constant $c$ is the value that gives moment balance on average:

Here $\bar{m}$ is the mean moment per earthquake under the Gutenberg–Richter distribution, and $m(M_\mathrm{max})/\dot m_\mathrm{a}$ is the recurrence time of the largest event. A slow feedback nudges $c$ so that release keeps pace with loading.

The rate is accumulated rather than sampled. The running count $n(t) = \int_0^t \rho\,dt'$ produces an earthquake every time it passes a whole number, so the rate climbs steadily while the fault is quiet and drops after large releases. Each new earthquake is built in four steps:

1. Draw a magnitude from a truncated Gutenberg–Richter distribution with slope $b$.
2. Convert it to a rupture area with $\log_{10} a = M - 3.99$, with $a$ in km² (Allen & Hayes, 2017).
3. Grow a circular rupture around the centroid (described next), truncated at the fault edges.
4. Assign slip that decays exponentially away from the centroid, with small random perturbations. Slip is capped so that no patch releases more moment than it has stored.

Runs start from a “spun-up” deficit, as if the fault had already been active for a long time.

Where earthquakes happen: maximum entropy

Where should an earthquake of magnitude $M$ be centred? With no information at all, the maximum entropy answer is “anywhere”: $p_i = 1/N$. kes adds three constraints:

• the probabilities sum to one;
• the expected logarithm of the stored moment at the centroid, $\langle \log m \rangle$, is fixed;
• that expectation can depend on magnitude.

Maximizing the entropy $-\sum_i p_i \ln p_i$ subject to these constraints gives a power law in the stored moment:

The exponent $\gamma$ is the Lagrange multiplier of the moment constraint, written $\lambda_\mathrm{c}^{\{2\}}$ in the paper. It sets how selective an event is:

• With $\gamma_\mathrm{min} = 0$, the smallest earthquakes ignore the moment field and can occur anywhere.
• As $M$ grows, $\gamma$ approaches $\gamma_\mathrm{max}$ (1.5 in the examples), and large earthquakes increasingly favour regions that have stored the most moment.

After a large earthquake, the same distance kernel used for afterslip (below) also raises the probability of nucleating near the rupture. These are aftershocks in space as well as in time.

Afterslip: maximum entropy again

After a large earthquake ($M \ge 7$ in the examples), patches that still hold residual moment $m_i^\mathrm{r}$ can slip slowly. kes again seeks the least-committal distribution, this time for afterslip speed $\sigma$, constrained only by a finite mean speed. Maximizing the relative entropy gives an exponential distribution. Taking the residual moment as the driving potential gives afterslip that decays exponentially in time and releases exactly what is left.

Where afterslip happens is set by a spatial kernel that decays with the distance $r_i$ from the rupture, $\phi(r_i) = e^{-r_i/l}$ with $l \approx 10$ km. The initial velocity is the product of three factors:

The second expression is the time evolution, which keeps $\int_0^\infty v_i\,dt = m_i^\mathrm{r}$. Here $\beta = 1/3$, because moment scales as length cubed (see Eq. 8 of the paper). Inside the rupture the kernel is large but most of the moment is already spent. Far away, moment is available but the kernel is small. So afterslip peaks in a halo just outside the rupture.

What a sequence looks like

The paper’s example is a 1,000-year run on 0.5 km × 0.5 km patches:

• loading of 10 mm/yr everywhere, plus a 20 mm/yr Gaussian “pulse” centred at $x = 100$ km;
• $b = 1$ and $M = 5$–8;
• Omori $p = 1$ with a one-day offset;
• afterslip triggered by $M \ge 7$ events.

It produced 1,272 earthquakes. Accumulation came to 97% of release, and afterslip accounted for 18% of the release.

The fault-plane snapshots show the same run before, during and after an $M = 7.15$ earthquake. In each pair, the upper panel is the moment released that year and the lower panel is the accumulated deficit.

Randomness versus loading

Because runs are cheap, kes can generate ensembles. Two experiments separate the effects of chance and of loading:

• Changing only the random seed produces qualitatively different histories. Event counts vary by about 35%, and long quiescent periods appear in some runs but not others.
• Shifting the loading pulse with the seed fixed changes the timing and number of events by less than 1%. The largest earthquakes move with the pulse.

Nothing in kes adjusts earthquake size, so the catalogs recover the prescribed $b$-value. The Omori exponent comes out lower than prescribed, because the moment-budget rate briefly drops after large releases and then recovers as loading resumes.

How kes differs from ETAS

kes uses the same Omori aftershock term as the widely used ETAS model, but differs from it in several ways:

• Driver. It tracks a moment field that is loaded tectonically and depleted by slip, rather than a self-exciting point process.
• Finite ruptures. Earthquakes are spatially extended, not point sources.
• Locations. Centroids are chosen from the current stored moment, not from kernels around past epicentres.
• Afterslip. Afterslip spends stored moment and so lowers local earthquake likelihood. It has no direct analogue in ETAS.

Together these produce behaviours a constant-background ETAS model does not: quasi-periodicity, semi-clustered large events and quiescent epochs.

The algorithm

1. Compute the rate constant $c$ from the moment balance.
2. For each time step:
   1. Release afterslip from earlier large earthquakes.
   2. Add tectonic loading.
   3. Compute the rate $\rho(t)$ from the moment deficit and the Omori term.
   4. Accumulate $\rho\,\Delta t$, and emit one earthquake for each whole number passed.
   5. For each earthquake:
      1. Draw a magnitude.
      2. Choose a centroid from $p(i\,|\,M)$.
      3. Generate slip.
      4. Update the moment budget.
      5. Start afterslip if the earthquake is large enough.