Geodetic Methods for Monitoring Crustal Motion and Deformation

Abstract

The use of geodetic data for crustal deformation studies is studied for the two possible cases: (a) The comparison of shape at two epochs for which station coordinates are available in order to compute, at any desired point, invariant planar deformation parameters (strain parameters), such as principal strains, principal elongations, dilatation and shear. (b) The utilization of coordinates and velocities at a particular epoch for the computation of the time derivatives of the deformation parameters (strain rate parameters). The classical approximate “infinitesimal” theory is presented as well as the widely used finite element method with triangular elements for the interpolation of station displacements and velocities. In addition, a new completely rigorous planar deformation theory, based on the singular value decomposition of the deformation gradient matrix, is presented for both strain and strain rate parameter computation. The invariance characteristics of all the above deformation parameters, under changes of the involved reference systems, are studied, from a purely geodetic point of view different from that in classical mechanics. Emphasis is given to the separation of rigid motion of independent tectonic regions from their internal deformation, utilizing the concept of a discrete Tisserand reference system that best fits the geodetic subnetwork covering the relevant region. Interpolation of displacements or velocities using stochastic minimum mean square error prediction (known as collocation or kriging) is also examined with emphasis on how it can become statistically relevant and rigorous based on sample covariance and cross-covariance functions. It is also shown how the planar deformation can be adapted to the study of surface deformation, with applications to the study of shell-like constructions in geodetic engineering and the deformation of the physical surface of the earth. The most important application presented, is the study of horizontal deformation on the surface of the reference ellipsoid, utilizing either differences of geodetic coordinates between two epochs, or the horizontal components of station velocities. Finally, it is also shown how the rigorous theory of planar deformation can be extended to the three-dimensional case.

Zusammenfassung

Die Verwendung geodätischer Daten für Studien der Krustendeformation führt zu zwei möglichen Fällen: (a) dem Vergleich der Ausprägung zweier Epochen, für die Stationskoordinaten verfügbar sind, um in jeder gewünschten Genauigkeit invariante planare Deformationsparameter zu berechnen, (b) der Nutzanwendung von Koordinaten und Geschwindigkeiten, um während einer bestimmten Epoche Zeitableitungen der Deformationsparameter zu bestimmen. Der vorliegende Beitrag widmet sich der klassischen approximativen ,,infinitesimalen“ Theorie ebenso wie der weit verbreiteten Finite-Element-Methode mittels triangulärer Elemente zur Interpolation von Stationsverschiebungen und Geschwindigkeiten. Eine strenge Theorie der Erweiterung planarer Deformation auf den dreidimensionalen Fall wird aufgewiesen.

This chapter is part of the series Handbuch der Geodsie, volume “Mathematische Geodsie/ Mathematical Geodesy”, edited by Willi Freeden, Kaiserslautern.

Appendix A: Unambiguous Diagonalization of a 2 × 2 Symmetric Matrix

From the well-known fact that symmetric matrices have real positive eigenvalues and orthonormal eigenvalues we may set

$$\displaystyle \begin{aligned} & \mathbf{M} = \left[ {{\begin{array}{c@{\quad }c} a & b \\ b & d \\ \end{array} }} \right] = \left[ {{\begin{array}{c@{\quad }c} {\cos \alpha } & { - \sin \alpha } \\ {\sin \alpha } & {\cos \alpha } \\ \end{array} }} \right]\left[ {{\begin{array}{c@{\quad }c} {l_1^2 } & 0 \\ 0 & {l_2^2 } \\ \end{array} }} \right]\left[ {{\begin{array}{c@{\quad }c} {\cos \alpha } & {\sin \alpha } \\ { - \sin \alpha } & {\cos \alpha } \\ \end{array} }} \right] = \mathbf{R}( - \alpha ){\mathbf{L}}^2\mathbf{R}(\alpha ), \end{aligned} $$

(A1)

$$\displaystyle \begin{aligned} & l_1^2 = A + B,\qquad l_2^2 = A - B,\qquad l_1^2 \ge l_2^2,{} \end{aligned} $$

(A2)

where

$$\displaystyle \begin{aligned} A = \dfrac{a + d}{2},\quad B = \dfrac{1}{2}\sqrt{(a - d)^2 + 4b^2}, \end{aligned} $$

(A3)

while the angle α is determined in the correct quadrilateral from

$$\displaystyle \begin{aligned} \sin \alpha = \mathrm{sgn}(b)\sqrt{\dfrac{1 - P}{2}}, \cos \alpha = \sqrt{\dfrac{1 + P}{2}}, \end{aligned} $$

(A4)

where

$$\displaystyle \begin{aligned} P = \dfrac{a - d}{2B}. \end{aligned} $$

(A5)

The above formulas for the angle α avoid the ambiguity in the usual relation $\tan 2\alpha = \dfrac {2b}{a - d}$.