Complex crustal strain approximation
The analysis of repeated geodetic observations has become an important tool for the investigation of the kinematics of tectonic plate boundary zones. The most appropriate analytical method for such investigations of contemporary crustal deformation is the strain analysis, a method of differential geometry. In attempting to find an elegant mathematical formulation to describe plane strain, the use of complex analysis proves to be very advantageous. The analytical modeling of spatially and temporally continuous and discontinuous displacement fields is developed using least-squares approximation of generalized polynomials. Algebraic polynomials are proposed for the continuous approximation, whereas specifically designed step functions are used to model the discontinuities in space and time. A mathematical model of simultaneous network adjustment and strain approximation is elaborated. It yields a general analytical method which enables strain-rates, or accumulated strain and fault-slip, to be determined from various types of geodetic measurements. In contrast to the widely used observation method (Frank’s method), this approach does not rely on repeated observations of the same observables. Repeatedly observed networks of non-identical design can be analyzed. The constraints incorporated by the approximation model allow strain estimation even when the network of some observation epochs suffer from formulation or configuration defects with respect to positions. Experiments with various graphical representations of strain are carried out. Strain pedal-curves and shear-rosettes expressing extension and shear in a given direction, plotted at equally spaced grid points, provide a comprehensive display of non-homogenous strain-fields in space. Confidence regions associated with extension and shear in a given direction are plotted together with these strain figures. A software package ‘CRUSTRAIN’ is developed for the simultaneous adjustment and strain approximation and for the display of the estimated strain parameters. The method is first tested with synthetic data and then with a real kinematic network. The method is applied to the 1970-80 Hollister network, which had been observed by the U.S. Geological Survey. This applications reveals the strength as well as the limitations of the proposed technique. An approximation model is evaluated which incorporates third-degree complex algebraic polynomials with three episodic terms in time. This approximation estimates co-seismic fault-slip and strain release associated with three moderate earthquakes which occurred in the Hollister area within the time interval in question.