Abstract
Based on the elastic theory, we derived 2-D analytical solutions for estimating (1) the depth of a magma chamber giving rise to an initial caldera (ring-fault) of a given radius, and (2) the magma chamber volume change required for the ring-fault formation. To facilitate mathematical treatment, we assume, first, that the magma chamber may be modelled either as a finite sphere or a point source and, second, that the ring-fault formation is due to magma chamber contraction (volume reduction) in the radial directions. Using the Coulomb criterion, we show that the relationship between ring-fault radius and chamber depth can be described by a linear function where the proportionality coefficient consists of elastic constants and the ratio (a/d), where a and d are the radius and depth of the finite-sphere magma chamber, respectively. In the point-source model the volume change required for ring-fault formation is proportional to the cube of the magma chamber depth. In the finite-sphere model, the solution consists of two terms. The first reflects the effects of the magma chamber depth; the second reflects the effects of a sphere radius and contributes to the volume change required for ring-fault formation if the a/d ratio is large. However, the effect of the second term is negligible if the a/d ratio is small. Since we consider only surface stresses in deriving the equations, the present results cannot be used to explain the mechanical details of caldera formation. Nevertheless, because our solutions are similar to relationships derived from observational data, they may provide useful first-order estimates of the magma chamber depth and volume change during caldera formation.
