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Volcano geodesy often involves the use of models to explain observed surface deformation. A variety of forward models are used, from analytical point sources to numerical simulations that consider complex magma system geometries, topography, and material properties. Various inversion methods can then be used to relate observed volcano data to models. Ideally, forward models should be verified through intercomparison, to check for implementation errors and quantify the error induced by any approximations used. Additionally, forward models and inversion methods should be validated through tests with synthetic and/or real data, to determine their ability to match data and estimate parameter values within uncertainty. However, to date, there have not been comprehensive verification and validation efforts in volcano geodesy. Here, we report on the first phase of the Drivers of Volcano Deformation (DVD) exercises, which were designed to build community involvement through web-based exercises involving calculations of static elastic displacement around pressurized magma reservoirs. The forward model exercises begin with a spherical reservoir in a homogeneous half space, then introduce topography, heterogeneous elastic properties, and spheroidal geometries. The inversion exercises provide synthetic noisy surface displacement data for a spherical reservoir in a homogeneous half space and assess consistency in estimates of reservoir location and volume/pressure change. There is variability in the results from both forward modeling and inversions, which highlights the strengths and limitations of different forward models, as well as the importance of inversion method choice and uncertainty quantification. This first phase of the DVD exercises serves as a community resource and will facilitate further efforts to develop standards of reproducibility. [ABSTRACT FROM AUTHOR]

This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported. [ABSTRACT FROM AUTHOR]