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 • 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 12 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.
INTRODUCTIONMany methods have been proposed to determine subsurface deformation, such as fault shear, from surface displacement observations. Unfortunately, most methods rely on local linearization about some assumed source geometry. Because of this, these methods are dependent on the type of model assumed. If the form of this model is not general enough to include all possible source types or multiple sources, then the results may be misleading. Dieterich and Decker [1975] have illustrated the ambiguity caused by variations in source shape. Considering a variety of volume sources fitting the same vertical uplift data, these authors note that source depths varied by a factor of 3.5.In this paper we present a stable, systematic method for the inversion of surface displacement and strain to obtain a measure of deformation at depth without the need of assuming an initial source model. To do this, we examine the problem in terms of certain combinations of the tensor components of strain at depth. The deformation is characterized by the threedimensional distribution of six linear combinations of strain components. The distribution is discretized using a series expansion method, and a linear inverse problem is presented without local linearization. The linearity is an important feature of the method because it allows one to compute the data resolution matrix, the parameter resolution matrix, and the parameter covariance matrix [Menke, 1984]. The basis functions for the series can be general enough to include all source types. This inverse problem may then be solved by minimizing the difference between the predicted data and the...
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