## 1 Introduction

[2] Properties of volcanic systems are commonly inferred by using geodetic observations to constrain simple kinematic models. These models do not explicitly include magmatic processes or the ways that such processes give rise to observable fields, and as a result are not well-suited for constraint by diverse time-evolving data sets and cannot be used to directly estimate most properties of the magma in the system. These limitations reduce our ability to leverage observations in the understanding of complex volcanic systems. For instance, while it is possible to use geodetic data with a kinematic model of a spherical magma chamber to constrain the chamber's location and volume change, its total volume and pressure change cannot in general be independently estimated [*McTigue*, 1987], and little insight into properties of the magma and the forces that actually give rise to displacements may be obtained. Such constraint requires additional information and a model that includes at least some physical-chemical properties of the magma [e.g., *Mastin et al*., 2008].

[3] Physics-based forward models can link processes occurring in the magmatic system with diverse observations including ground deformation, extrusion rate, gas emissions, and gravity changes, and therefore allow all observations to be used simultaneously to constrain properties of the volcanic system. Such models are also able in principle to predict the full evolution of an eruption given only initial conditions and material properties, which results in a more relevant model parameterization that is well-suited to inverse techniques and allows us to use the full temporal evolution of the data in the inverse procedure.

[4] Models of volcanic eruptions that include the physics governing magmatic processes [e.g., *Jaupart*, 1996; *Melnik and Sparks*, 1999; *Mastin and Ghiorso*, 2000; *Massol et al*., 2001; *Barmin et al*., 2002; *Costa et al*., 2007] provide valuable insight into various types of eruptive behavior, including cyclic activity. In order to use these models in an inversion, they must relate changes in pressures and tractions in magma to stresses and strains in the host rock in order to predict observations such as ground deformation. *Anderson and Segall* [2011] (hereafter referred to as Part 1) developed a relatively simple numerical, physics-based model of an effusively erupting silicic volcano and showed that, with coupling to the host rock, it could be used to predict ground displacement and lava extrusion time series.

[5] To the extent that a model is a reasonable representation of a volcanic system, comparing model predictions with data allows us to infer properties of the volcanic system. Many commonly used inverse techniques are cast as optimization problems, in which the goal is to determine a “best-fitting” set of model parameters which minimizes the misfit between observations and model predictions (usually calculated in a least-squares sense). Such approaches do not attempt to fully characterize the uncertainty associated with model parameters, and as a result, conclusions drawn from them may be inadequate or even misleading (consider, for instance, that very different sets of model parameters may yield very similar fits to the data). This is particularly true when the forward model is highly nonlinear, as is the case for most physics-based models (due in part to complex constitutive laws) or when inverting using diverse data sets.

[6] When inverting using a physics-based model, and when computationally feasible, quantifying as well as possible the full uncertainties associated with estimated model parameters may be a better choice than optimization [e.g., *Sambridge and Mosegaard*, 2002; *Melnik and Sparks*, 2005]. In a Bayesian approach to the inverse problem, probability distributions are used to characterize estimated model parameters as well as any available independent a priori information. (Physics-based models are well-suited for constraint by a priori information; for example, a petrological observation could be used to constrain a model's melt water content.) By comparing model predictions with data and combining with prior information, it is possible to formulate a posterior probability density function (PDF) which characterizes uncertainty in the estimated model parameters, taking into account uncertainty in both the data and a priori information (in this work we do not attempt to account for uncertainty associated with the fact that the forward model must be an imperfect representation of the earth). The posterior PDF may be efficiently characterized using the Markov Chain Monte Carlo (MCMC) algorithm.

[7] In this work we develop a novel technique for using a physics-based forward model with diverse volcanological data sets in a probabilistic inverse procedure. We apply our technique to the 2004–2008 eruption of Mount St. Helens, using wide prior bounds on model parameters (weakly informative prior information) to test the ability of the data to constrain these parameters. We compare results to independent information from petrology, geodesy, and other techniques. We also compare inversions using simple analytical models with those using physics-based models, and inverting different types of data (net GPS displacements, time-dependent geodetic data, and time-dependent geodetic data plus observations of lava dome growth). We are able to demonstrate that the approach described in this work offers a number of important advantages over more traditional techniques, including the remarkable ability to infer melt volatile content using observations of ground displacements and lava dome extrusion, and the ability to estimate magma chamber recharge from a deeper source. The inclusion of additional data sets, such as gravity changes or gas emissions, should in the future provide additional constraints on model parameters, and because the principles outlined in this work are general, with changes to the forward model the technique should be applicable to a wide variety of volcanoes.