Forecasting lava flows with a logic tree
A logic tree can be used to weight different models of specific volcanic phenomena, like probability of lava flows inundating a specific location. Rather than the nodes (red circles) being events, they are models, with inputs and outputs. A given branch of the logic tree comprises all the models used to estimate the probability of the phenomenon occurring. Nodes with the same parent and in the same column represent alternative models for the same volcanic process. Alternative models are assigned weights by changing the values along lines connecting nodes. The weights emanating from a single parent must sum to 1. The probability based on the model output multiplied by their weight is the weighted probability. The sum of the weighted probabilities is the expected value of the ensemble models (our best guess of the probability of lava flow inundation considering alternative models).
This probability model of lava flow inundation of Flagstaff (AZ) depends on a location model (where will an eruptive vent form, given an eruption?), a recurrence rate model (how frequently are new vents expected to form?), and a lava flow model (will lava reach Flagstaff, given the vent location?).
Key Concepts
The ensemble probability (or ensemble model) is the sum of the weighted probabilities. This is a way to combine models (or different opinions about models). Judgment is used to weight models. In expert elicitation, individuals complete trees and then ensemble probabilites from different individuals are weighted and combined.
Logic trees are a great way to test the significance of alternative models in terms of their impact on hazard (or other outcomes). That is, by changing a branch and its weight, how much does the ensemble probability change? Such experiments might be informative about how to spend time and resources.
Logic trees provide a framework for data assimilation. Consider the alternative location models in this example. Given a seismic swarm or magnetotelluric anomaly, one might alter the location models or their weights. Data assimilation almost always must be done through implementation of a probability model that is sensitive to the data, rather than using the data directly as a logic tree node.
Logic trees provide a means of using databases of observations, including libraries of numerical model results in hazard forecasts. For example, a library of lava flow model output (areas inundated) can be developed, and this library sampled using alternative weights for lava volume distributions, spatial density models of probable vent location, or likelihood of eruptions.
Some References
Hincks et al., 2014 discuss logic trees applied to eruption forecasting and their relation to Bayes' theorem and Bayesian networks.
Coppersmith et al., 2009 discuss logic trees applied to expert elicitation for possible volcanic eruptions through Yucca Mountain (NV).
Logic trees are alluded to, mixed with event trees, in Marzocchi et al., 2012
There is much more literature on logic trees in the probabilistic seismic hazard assessment literature, Starting with Cornell (1968) and summarized in Field et al. on principles of open software design for PSHA.