| A method is presented which uses observations of surface displacement to place constraints on the components of the strain tensor at depth. Using a volume integral form of Volterra's integral and a series expansion method, the problem is discretized. An external inverse approach is described allowing the calculation of bounds on model parameters from bounds on the data. Methods of placing bounds on generalized moments of the perturbing body are developed, and techniques of handling errors in the data are discussed.; A special case, the use of uplift measurements to bound fractional volume change in the subsurface, is developed. The connection between extremal bound techniques and the method of ideal bodies is explored. As an application, uplift data from 1982, 1983 and 1985, are used to constrain the depth and horizontal extent of any possible magma intrusion at depth. It was found that, in order to satisfy the 1982, 1983 and 1985 uplift data, volume expansion was required above depths of 11, 9 and 8 kilometers respectively. Furthermore, bounds on fractional volume change of individual blocks were calculated.; Finally, a method is developed which determines multiple solutions to non-linear inverse problems. An extension of Newton's method for non-linear optimization, the algorithm projects through solutions and searches for additional ones. The technique is applied to finding best fitting polyhedral volume source for a given displacement anomaly. This method has applications in many areas of geophysics. |
