Institute of Geophysics, University of TehranJournal of the Earth and Space Physics2538-371X35220090723A proposal for deformation analysis via direct computation of strain tensor elements from the time-wise changes in the distances and angles in a geodetic network
Case study: Deformation computation of the geodynamic network of IranA proposal for deformation analysis via direct computation of strain tensor elements from the time-wise changes in the distances and angles in a geodetic network
Case study: Deformation computation of the geodynamic network of Iran21421FAAlirezaArdalan0000-0001-5549-3189MehdiRaoofianJournal Article19700101A method for deformation computation based on strain tensor elements, as an alternative to the usual way of application of gradient of displacement vector, is proposed. The method computes directly the strain tensor elements from the computed/observed changes in distances and angles between the stations of a geodetic network in two epochs of observations. Displacement vector which is determined from the coordinate differences with respect to “reference” and “current” states depends on the definition of coordinate system and as such can not be considered as suitable measure of deformation. On the contrary from strain tensor invariant parameters like “dilatation” and “maximum shear” can be computed which allow correct interpretation of deformation. The strain tensor can be derived from the difference between line elements of a massive body in the reference and current states as follows:
(1)
Where and are the coordinates of points in the current and reference states of the body, respectively. For computation of strain tensor directly from changes in distances and angles between stations of a geodetic network in the two states, let us start with the presentation of strain tensor as:
(2)
Therefore by substitution of equation (2) in equation (1) we have:
(3)
In the equation (3) and are the distances between geodetic network stations in the reference and current states, respectively, and , are defined as follows:
(4)
The above relations are taken from continuum mechanics, which assumes continuity in the massive body, however, in practice for the numerical computation of strain we need to discretize the body into finite element of, for example, triangular shapes in 2-D space. The triangular elements can be generated by Delaunay triangulation. Then, for each triangle three equations of the type equation (3) can be written, and via the solution of the system of equations unknown parameters can be estimated.
For the angular observation from the definition of the inner product the following equation can be developed:
(5)
Equation (5) can equivalently be written as:
(6)
where the differential elements dx and dy are defined as below:
(7)
Alternatively, the finite difference method can also be used for computation of the strain tensor in a point-wise manner as below:
(8)
where in equation (8) is the elongation in the azimuth defined as:
(9)
In this paper we numerically tested the above mentioned method for strain tensor computation by simulated examples and then applied the method to the geodynamic network of Iran.A method for deformation computation based on strain tensor elements, as an alternative to the usual way of application of gradient of displacement vector, is proposed. The method computes directly the strain tensor elements from the computed/observed changes in distances and angles between the stations of a geodetic network in two epochs of observations. Displacement vector which is determined from the coordinate differences with respect to “reference” and “current” states depends on the definition of coordinate system and as such can not be considered as suitable measure of deformation. On the contrary from strain tensor invariant parameters like “dilatation” and “maximum shear” can be computed which allow correct interpretation of deformation. The strain tensor can be derived from the difference between line elements of a massive body in the reference and current states as follows:
(1)
Where and are the coordinates of points in the current and reference states of the body, respectively. For computation of strain tensor directly from changes in distances and angles between stations of a geodetic network in the two states, let us start with the presentation of strain tensor as:
(2)
Therefore by substitution of equation (2) in equation (1) we have:
(3)
In the equation (3) and are the distances between geodetic network stations in the reference and current states, respectively, and , are defined as follows:
(4)
The above relations are taken from continuum mechanics, which assumes continuity in the massive body, however, in practice for the numerical computation of strain we need to discretize the body into finite element of, for example, triangular shapes in 2-D space. The triangular elements can be generated by Delaunay triangulation. Then, for each triangle three equations of the type equation (3) can be written, and via the solution of the system of equations unknown parameters can be estimated.
For the angular observation from the definition of the inner product the following equation can be developed:
(5)
Equation (5) can equivalently be written as:
(6)
where the differential elements dx and dy are defined as below:
(7)
Alternatively, the finite difference method can also be used for computation of the strain tensor in a point-wise manner as below:
(8)
where in equation (8) is the elongation in the azimuth defined as:
(9)
In this paper we numerically tested the above mentioned method for strain tensor computation by simulated examples and then applied the method to the geodynamic network of Iran.https://jesphys.ut.ac.ir/article_21421_588d3823b12c9f43af59e53cd2fa8d10.pdf