ETASCatGen: An ETAS Earthquake Catalog Generator

The etascatgen Python module and C++ library generates space-agnostic earthquake catalogs according to the Epidemic-Type Aftershock Sequence (ETAS) model of temporal earthquake clustering. Specifically, ETASCatGen is based on the model described by Ogata (1988). This model is a marked linear stationary Hawkes process (Hawkes, 1971) with time-dependent intensity

$$\lambda(t) = \mu + \sum\limits_{t_i < t} f(M_i) g(t-t_i)\,,$$

where $t_i$ are the occurrence time of past earthquakes (the events), and $M_i$ are the magnitudes of the earthquakes (the markers).

In the epidemic-type model of aftershock excitement proposed by Ogata (1988), the rate of aftershock generation depends on the magnitude of generated earthquakes,

$$f(M) = \exp\big(\alpha(M - M_r)\big)\,,$$

where $M_r$ is the ‘reference magnitude’ (Ogata, 1988; $f(M)$ is labeled $c(M)$ therein) and $\alpha$ is the effect of the magnitude (marker) onto the offspring rate. Furthermore, the time decay of the aftershock generation rate is given by the ‘[...] the modified Omori formula (Utsu 1961) [...]’ (Ogata, 1988):

$$g(t) = \frac{K}{(t+c)^p}\,,$$

where $K$ scales the rate, $p$ is the exponent of the time decay, and $c$ can be understood as a time lag. Having a power law time dependency of the Hawkes process can lead to memory effects that do not occur in the frequently used exponential kernel (e.g., Kwan, 2023).

Once an earthquake occurs, the magnitude is drawn (independently) from a Gutenberg-Richter distribution with an adjustable $b$-value confined to the magnitude interval $M_\mathrm{min} \leq M \leq M_\mathrm{max}$,

$$\phi(M) = \frac{\beta \exp\big(-\beta (M - M_\mathrm{min})\big)} {1 - \exp\big(-\beta(M_\mathrm{max} - M_\mathrm{min})\big)}$$

with $\beta = \ln(10) b$ (Ogata, 1988).

For the Hawkes process to be stationary, the expected number of offspring triggered by an earthquake needs to be less than one. This is expressed in the condition

$$\left\langle f(M) \int\limits_0^\infty \frac{K}{(t+c)^p \mathrm{d}t} \right\rangle_M < 1$$

which leads to an upper bound on $K$:

$$K < K^* = \left\lbrace \begin{array}{ll} \frac{1}{\tau(M_\mathrm{max} - M_\mathrm{min})} &: \quad \alpha = \beta \\\ \frac{\alpha - \beta} {\tau\left(\exp\left((\alpha-\beta)M_\mathrm{max}\right) - \exp\left((\alpha-\beta)M_\mathrm{min}\right) \right)} &: \quad \alpha \neq \beta \end{array}\right.$$

with

$$\tau = \frac{c^{1-p}\exp\big((\beta - \alpha) M_\mathrm{min}\big)} {(p-1)\left(1 - \exp\big(-\beta(M_\mathrm{max} - M_\mathrm{min}\big)\right)}$$

The ETASCatGen module takes a number of choices with regards to the parameters:

• Without loss of generality, $M_r$ is chosen to be identical to $M_\mathrm{min}$.
• The scale parameter $K$ is expressed as a fraction of $K^*$, the critical $K$ at which the Hawkes process catastrophically self-resonates (that is, an expected offspring number of 1).
• Stationarity requires that $p &gt; 1$.

Usage

ETASCatGen comes with one function: generate_catalog_M_t. This generates a catalog of earthquakes with occurrence times and magnitudes.

Python

ETASCatGen is built on NumPy and Cyantities (for physical unit support). It comprises of one function that can be used like so:

# Import necessary packages:
from etascatgen import generate_catalog_M_t
from cyantities import Quantity
import numpy as np # to use np.log

# Set a time scale:
yr = Quantity(60 * 60 * 24 * 365.25, 's')

# Set the parameters of the ETAS model and the
# Gutenberg-Richter distribution:
mu_0 = 1e-3 / yr
Mmin = 4.0
Mmax = 9.0
beta = np.log(10) * 1.0
alpha = beta
p = 1.2  # significant memory
c = yr / 16
offspring_fraction = 0.95

# Build a large catalog.
N = 10000000

# The Hawkes process might take some time
# to approach a steady state since we
# start at t=0 and not t=-inf (in particular
# when we are at p < 1.5).
# `N_skip` starts the process by discarding
# the first N_skip events.
# NOTE: this example is not necessarily in
#       steady state!
#       What is more, whether the ETAS
#       model is mean-square ergodic
#       in this parameter range
#       (1.0 < p < 1.5) is unclear to the
#       author at the time of writing.
N_skip = 10*N


Mi, ti = generate_catalog_M_t(
    N,
    mu_0,
    Mmin,
    Mmax,
    beta,
    alpha,
    p,
    c,
    offspring_fraction,
    N_skip
)

Install

You can build and install ETASCatGen from the project's root directory using Pip:

pip install --user .