## 2 Eruption Forecasting Algorithm  
An event tree is a decision making framework that estimates the probability of a set of pre-defined events using a tree like decision making process. Each branch or node of the tree represents a set of possible outcomes for a particular event, which increase in specificity with each step. Progression to the next branch is accomplished when the probability estimate for the preceding node exceeds a pre-defined threshold. Our algorithm is rooted in a general state of unrest and grows to forecast volcanic activity in increasing detail, such as the probability of eruption, intensity, and location. The event tree utilized in this research is shown in Figure 1. Each node estimates the conditional probability for a specific event via Bayes theorem, which is defined as  
\[P(n|n-1)=\frac{P(n,n-1)}{P(n-1)}=\frac{P(n-1|n)P(n)}{P(n-1|n)P(n)+P(n-1|n^{ \prime})P(n^{\prime})} \tag{1}\]  
where \(P(n-1|n)\) is the likelihood, \(P(n)\) is the prior probability, \(P(n-1)\) is the marginal probability, \(P(n|n-1)\) is the posterior probability, the prime symbol represents the complement, and \(n\) is the event tree node. Our event tree implementation is comprised of the following nodes:  
* Node 1. _Unrest: Does the geophysical activity at the selected volcano exceed a predetermined threshold?_ Geophysical activity includes heightened seismicity or surface deformation. If the amount of unrest exceeds a predefined threshold, the decision making process progresses to the  
Figure 1: Schematic representation of the event tree, where the clone label indicates the tree structure at that point is identical to that below. The color code represents the USGS ground-based hazard declarations and are superimposed over their respective event tree branch. next branch, otherwise it terminates and restarts when new data are available. If the predefined threshold is exceeded, the probability that unrest is occurring equals one, which is given by \[P(\Theta_{1})=P(1)= \begin{cases}0&\text{if }\
u_{1}=0\\ 1&\text{if }\
u_{1}>0\end{cases} \] (2) where \(\
u_{1}\) is the summation of several explanatory variables that are defined in Section 2.3. * Node 2. _Magmatic Intrusion: Is the observed unrest the result of a magmatic intrusion?_ Fluid motion can be detected from the presence of heightened levels of seismicity having specific spectral characteristics or continuous or rapidly varying surface deformation. Possible causes of unrest that are not related to fluid motion or not indicative of an impending eruption include tectonic and geothermal activity. If the unrest is related to fluid motion, the process progresses to the next branch, otherwise it terminates and restarts when new data are available. The probability that an intrusive event is occurring is given by \[P(\Theta_{2})=P(2|1)=\frac{P(1|2)P(2)}{P(1|2)P(2)+P(1|2^{\prime})P(2^{\prime})}\] (3) where \(P(2)\) is defined as \[P(2)=P(Intrusion|X_{2,N})\] (4) and \(X_{2,N}\) is a node specific set of \(N\) independent explanatory variables described in Section 2.4. * Node 3. _Eruption: Does the detected fluid motion have the potential to reach the surface and cause an eruption?_ If the source of the unrest is a magmatic intrusion with associated seismicity, the probability that the fluid will reach the surface may be high if that particular volcano has a historically high eruption frequency. If the observed, modeled, and historic information support this hypothesis, the algorithm moves to the next branch, otherwise it terminates and restarts when new data are available. The probability that an eruption will occur is given by \[P(\Theta_{3})=P(3|2)=\frac{P(2|3)P(3)}{P(2|3)P(3)+P(2|3^{\prime})P(3^{\prime})}\] (5) where \(P(3)\) is defined as \[P(3)=P(Eruption|X_{3,N})\] (6) and \(X_{3,N}\) is a node specific set of \(N\) independent explanatory variables described in Section 2.4. * Node 4.