Download Diapositive 1 - Michel KASSER

EISSA Leila1,3,4, KASSER Michel²
Email : [email protected], [email protected]
(1)LAREG (Laboratoire de recherche En Géodésie), (2)ENSG, 6 – 8 av Blaise Pascal, 77455 Marne la Vallée cedex 2, France,
(3) université Paris-Est, Cité Descartes 5, bd Descartes, Champs sur Marne 77454 MARNE LA VALLEE, (4)Tishreen University, Lattaquia Syria.
Introduction:
Computation and representation of errors:
Tensor calculation:
The space geodesy measurements quantifies the deformation of the earth crust with an accuracy of level of mm
per year.
Our aim is :
• to assess the deformations for a geologically active site : tectonic zone, seismic zone, landslide etc …
• to represent the strain rate by tensors evenly spread (independent of reference frame and of location of
measurement sites).
• to represent the degree of significance of the obtained deformations.
To make the link between the geodetic measures and the geophysical interpretation.
Data sources:
Data sources to assess and represent the deformation of the crust are:
The variance analysis is particularly complex, given the unstability of the representation of the tensor form around the
isotropic case, we therefore adopt a method of Monte-Carlo.
From displacement vectors on the grid, we calculate the strain
rate tensor inside each square K. For each grid we compute the
variation in length for the four sides and the two diagonals, we
write the following equations :
We have two sets of available measurements on a given geodetic network with the resulting variances of the two
compensations. We then synthesize for the two periods random sets of coordinates for each measuring point. Thus
we calculate the corresponding tensors. We repeat this operation 100 times. Therefore as we have 100 datasets we
compute 100 times the interpolation grid. At the end we will have 100 tensors components.
From these datasets we estimate uncertainties of normal and shear components as well as uncertainty of maximum
shear direction.
The calculation of tensor components uses the finite element
method (Pagarete and al, 1990), for infinitesimal
deformations the relative variations of each side between two
epochs is written (Kasser and Thom, 1995; Welsch, 1983):
Now we can represent the strain rate tensor components and their uncertainties on the same map.
• Displacement fields derived from space geodetic techniques such as (GPS, SLR,VLBI, DORIS).
• Displacement fields from correlation between two ortho-images
Where:
is tensor component in X,
is tensor component in Y and
is shearing component
Respectively are the azimut, the length and the variation of length of the side k
In this equation we have three unknowns
The determination of these three unknown requires at least three equations, but we have six equations (one
for each side) so we solve by least square adjustment.
We build the following matrix system:
Where:
H is matrix coefficients following the equation (2)
e is components vector
U vector of the sex relative variation lengths of each segment of the mesh K
Example :
We apply this methodology on the well known fault area of San Andreas fault in California.
GPS EUREF
Ortho-images correlation
Interpolation:
Conclusion:
In order to represent the rate tensors of region of interest, we must interpolate a grid covering the study area.
The tool in development will make the “connection” between the expertise of geodesists and that of
geophysicists in order to acquire all available data, without any omission or artificially added ones.
This will greatly facilitate the interpretation of phenomena, directly from a velocity field measured by space
geodesy or any other techniques of measurement.
Many interpolation methods have been tested such as: linear, bicubic, nearest neighbours and spline
interpolations.
This software can be integrated with GMT or at least will have an output GMT.
We have conducted a test on a fictive site.
Geographic situation
Interpolation grid
Tensor classical representation
We note that it is difficult to interpret this kind of representation, furthermore we have no idea about the uncertainties of
obtained tensors. So the heart of our work begins here :
Aim :
To represent the isotropic deformation and shearing part on the same map in an intuitive mapping.
We want to represent these deformations with colour themes, and to combine them so as to ease the interpretation.
Choice :
Bicubic interpolation
Spline interpolation
Combined solution with nearest
neighbours
and spline interpolations
A colour scale to represent the isotropic part (applied over the entire mesh) with maximum saturation and a segment of
variable width to represent the shearing part, its value and its direction. The variation of the saturation will be used for the
error analysis display.
References
bicubic interpolation may give an unexpected discontinuity
spline interpolation works well but it diverges when one moves away from sites of measurements
J.Pagarete, m.Kasser, J.-C. Ruegg Évaluation et représentation des erreurs sur les déformations d’un réseau
géodésique: utilisation de la méthode de Monte Carlo. Bull. Géod. 64 (1990) pp. 63-72
Compromise: use spline to interpolate (inside the convex hull of measurement sites) and nearest neighbours
Hans-Gert Kahle, Marc Cocard, Yannick Peter, Alain Geiger, Robert Reilinger, Aykut Barka, and George Veis
GPS-derived strain rate field within the boundary zones of the Eurasian, African, and Arabian Plates. J Geophys
Res , VOL.105, NO.B10, PAGES 23,353-23,370, oct 10, 2000
to extrapolate (outside this convex hull)
What is a strain rate tensor?
Chung-Pai Chang, Tsui-Yu Chang, Jaques Angelier, Honn Kao, Jian-cheng Lee, Shui-Beih Yu strain and stress
field in Taiwan oblique convergent system: constraints from GPS observation and tectonic data. Earth and
planetary Science Letters 214 (2003) 115-127
Any deformation deforms an infinitesimal circle into an
ellipse. Its axes are greater or smaller than the radius of
initial circle. This can characterize the deformation.
So a strain tensor is generally represented either by
axes of the ellipse either by the triplet isotropic part,
shearing part and direction of maximum shear.
A.Caporali, S.Martin, M.Massironi and S.Baccini state of strain in the Italian crust from geodetic data.
Sébastien Leprince, Sylvain Barbot, François Ayoub, and Jean-Philippe Avouac
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING,
VOL. 45, NO. 6, JUNE 2007
a classical strain rate tensor
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