Magma Depletion: An alternative to time-homogeneity for forecasting vent distribution in volcanic fields

For small volume eruptions, such as those common for volcanic fields, the location of an eruptive vent controls the hazards, their intensities, and ultimately the impact of the eruption. An eruption through water can result in a highly explosive event, and an eruption beneath a hospital or critical infrastructure can cause significant long-term impacts. We look here at long-term probabilistic assessments, the outputs of which inform evacuation plans, the (re)location of vital infrastructure, and inform the placement of early-warning monitoring equipment.

Current estimates of future vent locations are based on point-process methods with probability surfaces built from patterns, clusters, and/or lineaments identified from previous vent locations. These all assume that locations with more past-vents are more likely to produce future-vents, or in other words a null hypothesis

H_o: The likelihood of an eruption at the location of an existing vent is a local maximum of the spatial density surface.

Critically, under this model the occurrence of an eruption does not change the likelihood of further eruptions at that locality. We investigate here an alternative (but not necessarily better) hypothesis of magma depletion, i.e., that after an eruption, the magma source at depth is depleted by the volume of the eruption in this area, lessening the likelihood by creating a local depression in the probability surface. More formally we consider the alternative hypothesis

H_a: The likelihood of an eruption at the location of an existing vent is a local minimum of the spatial density surface.

We present the mathematics and code for various alternatives to current kernel density estimates, and then set out to try and disprove our null hypothesis by examining goodness of fit to data, all using the exemplar of the Auckland Volcanic Field, New Zealand