General Theory
The determination of deformation parameters is based on the finite elements theory.
The general equation for a field of displacement can be written as
where
is
the vector of displacement vectors,
is
the vector of coordinates,
is the vector of translation
elements, and
is
the gradient of displacements, composed by symmetrical tensor of
deformation
and
anti-symmetrical tensor of rotation
.
After
rearrangements we can finally express the displacements
in
the direction of coordinate axes
by
following equations
where
and
are the components of tensor
and
are
the components of tensor
.
The
components of both tensors as well as translation
in
equation are unknown and can be determined by solving the system,
introducing parameters
for each point of displacement
.
Finally,
the parameters of deformation,as principal axes of compression or extension, total dilatation,shear deformations and their orientation
can be determined using the known formulae.
It should be stressed that the parameters of deformation do not depend on the coordinate system used, and can define the general trends of compression or extension at given area under study.
With the aim to express the deformation parameters in the plane, the approximation of field of displacement vectors was applied, using polynomials. After solving necessary equations of the following type
the
values
were determined as the derivatives of values
for respective coordinates. After full processing, the vectors of
displacement are computed and displayed in rectangular grid of an
arbitrary scale. Within the same grid, the deformation parameters for
each intersection can be determined.
From:
Jan Kostelecký, Milan Talich and Pavel Vyskočil: Crustal Deformation Analysis in the International Center on Recent Crustal Movements [Journal of the Geodetic society of Japan, Vol. 40, No. 4, (1994). pp, 301-308].
Full solution:
[1] Talich Milan: Using GPS Data for Deformations Analysis. 1994 IERS and IGS Workshop Paris and Saint Mandé, 1994, March 21-25, p.6.
[2] Kostelecký Jan: Výpočetní programy pro interpretaci horizontálních posunů. Technická zpráva. Zdiby: VÚGTK, 1986, 21s.
[3] Talich Milan, Kostelecký Jan, Vyskočil Pavel: Modelling of Horizontal Movements and Computing Method of Deformation Fields at the International Center On Recent Crustal Movements. In: Zpráva o řeąení vědeckého projektu 4-01-60 Geodézie a dynamika Země v roce 1993, edice VÚGTK, 1993, s.104-115.
[4] Talich, Milan: Geodynamika s GPS. Sborník prací VÚGTK 1996. - Roč.41. - Zdiby: VÚGTK, 1997. - S.31-40. http://www.vugtk.cz/odis/sborniky/sb96/talich.htm
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