Using surface displacement and strain observations to determine deformation at depth, with an application to Long Valley Caldera, California

We present a method for the inversion of surface displacement and strain to determine six independent combinations of the strain components at depth. These six combinations represent three mutually perpendicular double forces without moment and three pairs of double forces with moment. Furthermore, they may be directly related to a moment tensor density distribution for static strain. The method is linear and, unlike other methods, does not depend on an a priori source model. Due to the linearity of the formulation, methods such as generalized inversion or l 1 residual norm minimization may be used to determine models of deformation at depth. The methods presented are applicable to a variety of available geodetic data sets. The technique is applied to repeat leveling data taken between 1982 and 1985 in Long Valley caldera. We determined a model of the magma intrusion suspected to have occurred beneath the caldera. The model derived minimized the l 2 norm of the difference between the predicted and observed data subject to the constraint that only intrusion occur. The model indicates that most volume expansion took place in the caldera region above 8 km±2 km. In the 0–4 km depth range the expansion is widely distributed with a maximum value of 0.005 km 3 . Deeper, 4–8 km, there is a concentration of intrusion in the central portion of the caldera. The maximum volume increase here is 0.030 km 3 .