THE COMPLEX STRAIN APPROXIMATION IN SPACE AND TIME APPLIED TO THE KINEMATICAL ANALYSIS OF RELATIVE HORIZONTAL CRUSTAL MOVEMENTS

The analysis of repeated geodetic observations has become an important tool for the investigation of the kinematics of tectonic plates. 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 discontinuities in space and time. A mathematical model of simultaneous network adjustment and strain approximation is elaborated. It enables 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. 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-homogeneous strain-fields in space. The method is applied to the 1970-80 Hollister network, which had been observed by the U.S. Geological Survey. An approximation model is evaluated which incorporates third-degree complex algebraic polynomials with four block translation terms in space and fifth-degree 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.