Brownian Passage-Time Model

A particularly interesting renewal model is the Brownian passage-time Model. It was originally introduced by Matthews et al (2002) and Ellsworth et al (1999) to provide a physically-motivated renewal model for earthquake recurrence. It is based on the properties of the Brownian relaxation oscillator (BRO). A Brownian passage-time model considers an event (earthquakes in Matthews' model or renewed eruptive activity in our case) as a realization of a point process in which new eruptive activity will occur when a state variable (or a set of them) reaches a threshold ($X_f$) and at which time the state variable returns to a base ground level ($X_0$). Adding Brownian perturbations to steady loading of the state variable $X$ produces a stochastic load-state process. An eruption relaxes the load state to the characteristic ground level and begins a new cycle. The load-state process is a BRO, while intervals between events have a distribution known as Brownian passage-time distribution. Note that this is the name used in physics literature; in statistics literature it is often known as Inverse Gaussian or Wald distribution (Matthews et al, 2002).

In the conceptual Model of Matthews et al (2002), the loading of the system has two components: (1) a constant-rate loading component, $\lambda t$, and (2) a random component, $\epsilon (t) = \sigma W(t)$, that is defined as a Brownian motion (where $W$ is a standard Brownian motion and $\sigma$ is a nonnegative scale parameter). Standard Brownian motion is simply integrated stationary increments where the distribution of the increments is Gaussian (which might be motivated by central-limit arguments if we consider perturbations as the sum of many small, independent contributions), with zero mean and constant variance. The Brownian perturbation process for the state variable $X(t)$ (see figure 2 in Matthews et al (2002)) is defined as:

$$X(t) = \lambda t + \sigma W(t)$$ (2)

An event will occur when $X(t) \geq X_f$; event times are seen as ``first passage'' or ``hitting'' times of Brownian motion with drift (Matthews et al, 2002). The BRO are a family of stochastic renewal processes defined by four parameters: the drift or mean loading ($\lambda$), the perturbation rate ($\sigma^2$), the ground state ($X_0$), and the failure state ($X_f$). On the other hand, the recurrence properties of the BRO (repose times) are described by a Brownian passage-time distribution which is characterized by two parameters: (1) the mean time or period between events, ($\mu$), and (2) the aperiodicity of the mean time, $\alpha$, which is equivalent to the familiar coefficient of variation (defined in equation 1). The probability density for the Brownian passage-time model is given by:

$$f(t; \mu; \alpha) = \left ( \frac{\mu}{2 \pi \alpha^2 t^3} ... ...bf{e}^{ \left\{-\frac{(t-\mu)^2}{2 \alpha^2 \mu t} \right\} }$$ (3)

The state variable $X(t)$ is a formal parameter of a point process model and represents a constant-rate mean path that embodies a macroscopic view of a uniform loading of the volcanic system. It may summarize the macro-mechanics of the volcanic system controlled by one or more physical variables. An explicit definition of the driving physical parameters may be unrealistic and impossible to demonstrate from our analysis. Independently of its physical nature, the state variable should be a parameter that accumulates with time during repose episodes, up to a critical value beyond which the system becomes perturbed enough and a new eruptive process may be triggered. Then, the eruptive process relaxes the system and the state variable returns to a ground level and a new cycle starts. The perturbation factor $\epsilon(t)$ represents the total sum of all other factors which may play a role in the recurrent eruptive process considered and/or that may randomly disturb the state variable producing the aperiodicity of the mean time between eruptions (e.g. effects from tectonic environment, changes in the magma rate supply, compositional changes, etc.).