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Volcanologists increasingly rely on numerical simulations to better understand the dynamics of erupting volcanoes. Mathematical models are often used to explain the geological processes responsible for eruption deposits found in the geological record, and to better characterize possible hazards from future volcanic activity. Examples of models include the finite-element flow and transport codes used to simulate pyroclastic flows, lahars, and volcanic debris avalanches (Iverson 1997; Patra et al. 2005), analytical solutions or finite-difference approximations to the advection-diffusion equation that are used to model tephra dispersion (Suzuki 1983) and gas emissions from quiescentvolcanoes (Delmelle et al. 2001), and cellular automata algorithms that model the advance of lava (Barca et al. 1994). Commonality among these examples involves the fact that the parameters to be estimated are related to the dynamics of volcanic activity derived from field observations. For instance, how well can the magnitude of an eruption be estimated from measurements made of tephra deposits?

One solution to this question lies in coupling numerical simulations to inversion methods that search for an optimal set of parameters that explain the physical observations. For example, volcanologists make observations of tephra thickness and variations in particle size to help estimate the parameters that describe the dynamics of the volcanic eruption that created the deposit. These parameters include eruption volume, eruption column height, and wind velocity as a function of height above the ground. The difficulty in this approach is that volcanologists must deduce multiple parameters that characterize

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