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The definition of a geophysically meaningful International Terrestrial Reference System Problems and prospects The definition of a geophysically meaningful International Terrestrial Reference System Problems and prospects Athanasios Dermanis Department of Geodesy and Surveying Aristotle University of Thessaloniki X Y Z EGU General Assembly 2006 April 2-7, Vienna

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Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS) Objective (practice):Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions x i (t), y i (t), z i (t) of points P i in a global geodetic network

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Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS) Objective (practice):Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions x i (t), y i (t), z i (t) of points P i in a global geodetic network Available data:Coordinates x i (t k ), y i (t k ), z i (t k ) of points in subnetworks at discrete epochs t k (e.g. daily or weekly solutions) with respect to a different reference systems for each subnetwork

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Objective (theory): Express deformation of the earth by the temporal variation of earth point coordinates with respect to an optimally selected International Terrestrial Reference System (ITRS) Objective (practice):Provide a realization of the ITRS by an International Terrestrial Reference Frame (ITRF), consisting of the coordinate functions x i (t), y i (t), z i (t) of points P i in a global geodetic network Available data:Coordinates x i (t k ), y i (t k ), z i (t k ) of points in subnetworks at discrete epochs t k (e.g. daily or weekly solutions) with respect to a different reference systems for each subnetwork Problem (theory):Introduce a principle leading to an optimal choice of the reference frame Problem (practice):Combine the known shapes of the overlapping subnetworks into the shape of a global network expressed by coordinates with respect to the already defined optimal reference system

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The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal For the whole earth dm x Tisserand principle: Vanishing relative angular momentum Minimal relative kinetic energy

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The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal For the whole earth For a network of discrete points dm x Tisserand principle: Vanishing relative angular momentum Minimal relative kinetic energy Discrete Tisserand principle: Vanishing relative angular momentum Minimal relative kinetic energy xixi i

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The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal The optimal choice of reference system General characteristic: Apparent motion (coordinate variation) of mass points should be minimal For the whole earth For a network of discrete points dm x Tisserand principle:Discrete Tisserand principle: Define orientation of the reference system. For the position: xixi i Preservation of the center of mass: Trivial choice: m i =1

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The “linear” model Is there a linear model for deformation? No ! It is not possible to have all shape related parameters which are coordinate invariants as linear functions of time!

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The “linear” model Is there a linear model for deformation? No ! A planar counterexample: s1s1 s2s2 s4s4 s5s5 s3s3 s i = a i t + b i,i = 1,2,3,4,5 The 5 shape defining sides are linear functions of time: It is not possible to have all shape related parameters which are coordinate invariants as linear functions of time!

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The “linear” model Is there a linear model for deformation? No ! A planar counterexample: s1s1 s2s2 s4s4 s5s5 s3s3 d s i = a i t + b i,i = 1,2,3,4,5 d = nonlinear in t The 5 shape defining sides are linear functions of time: The 2nd diagonal of the quadrilateral is not a linear function of time! It is not possible to have all shape related parameters which are coordinate invariants as linear functions of time!

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The “linear” model What is the meaning of linear model for coordinate functions? Linear coordinates in one reference system x(t) = x 0 t + v x, y(t) = y 0 t + v y, z(t) = z 0 t + v z, x(t) = x 0 t + vx(t) = R(t) x 0 t + R(t) v + c(t) x 0 (t) t + v x´(t) = R(t) x(t) + c(t) Non-linear coordinates in another CHANGE OF REFERENCE SYSTEM

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The “linear” model x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand R(t) = R = const. c(t) = c = const. R(t) = R = const. c(t) = c = const. What is the meaning of linear model for coordinate functions? x(t) = x 0 t + v x, y(t) = y 0 t + v y, z(t) = z 0 t + v z,

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The “linear” model Ad hoc definition of “linear deformation” model: Coordinates in one (and hence all) Tisserand system are linear functions of time x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand x(t) x´(t) = R(t) x(t) + c(t) are both Tisserand Note:Tisserand reference system here means: Orientation by Tisserand principle (zero relative angular momentum) Position by center of mass preservation principle R(t) = R = const. c(t) = c = const. R(t) = R = const. c(t) = c = const. What is the meaning of linear model for coordinate functions? x(t) = x 0 + t v x, y(t) = y 0 + t v y, z(t) = z 0 + t v z,

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The Tisserand reference system conditions for the linear model Vanishing relative angular momentum: Constant (zero) center of mass coordinates: Tisserand conditions: Usual choice: Tisserand conditions:

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The Tisserand reference system conditions for the linear model Vanishing relative angular momentum: Constant (zero) center of mass coordinates: Tisserand conditions: Usual choice: Tisserand conditions: Used at the stage of data analysis as additional constraints + Usual inner constraints for the definition of reference frame at epoch t 0 = Selection of a single Tisserand frame out of a set of dynamically equivalent ones Used at the stage of data analysis as additional constraints + Usual inner constraints for the definition of reference frame at epoch t 0 = Selection of a single Tisserand frame out of a set of dynamically equivalent ones

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Accessing a geophysically meaningful reference system x(t) (earth) Tisserand x(t) (earth) Tisserand ~ x(t) = R(t) x(t) ~ x(t) (discrete) Tisserand x(t) (discrete) Tisserand dm To access R(t) transforming h R (Earth) to h R (Earth) = 0 we need an approximation of h R (Earth) ~

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Accessing a geophysically meaningful reference system x(t) (earth) Tisserand x(t) (earth) Tisserand ~ x(t) = R(t) x(t) ~ x(t) (discrete) Tisserand x(t) (discrete) Tisserand dm To access R(t) transforming h R (Earth) to h R (Earth) = 0 we need an approximation of h R (Earth) ~ Earth E = P 1 + … + P K + … + P L + P 0 Network D = D 1 + … + D K + … + D L PlatesInterior Subnetwork on plate P K To access R(t) transforming h R (Earth) to h R (Earth) = 0 we need an approximation of h R (Earth) 0 Use D K to access h P K (t) !

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Estimation of the rigid motion of a plate Plate P K Subnetwork D K x = RK xx = RK x Transformation such that original velocities X Y Z X Y Z

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Estimation of the rigid motion of a plate Plate P K Subnetwork D K RKRK KK x = RK xx = RK x Transformation such that original velocities transformed velocities determine rotation R K to reference system best fitted to D K X Y Z X Y Z

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Estimation of the rigid motion of a plate Plate P K Subnetwork D K RKRK KK x = RK xx = RK x Transformation such that original velocities transformed velocities velocities due to rigid plate rotation only determine rotation R K to reference system best fitted to D K assume rigid motion of plate P K due only to rotation R K X Y Z X Y Z

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Estimation of the rigid motion of a plate Plate P K Subnetwork D K RKRK KK x = RK xx = RK x Transformation such that original velocities transformed velocities velocities due to rigid plate rotation only determine rotation R K to reference system best fitted to D K assume rigid motion of plate P K due only to rotation R K compute relative angular momentum of plate P K due only to rotation R K X Y Z X Y Z

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Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated

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Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i

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Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix Subnetwork D K 1

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Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix rotation vector inertia matrix Subnetwork D K Plate P K 1 2

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Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix rotation vector inertia matrix Subnetwork D K Plate P K total rotation vector 1 2 3

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Final transformation to an approximate Earth-Tisserand reference system x = R x ~ Previously estimated Approximate (numerically sufficient) solution for the linear model: x i (t) = x 0i + t v i relative angular momentum inertia matrix rotation vector inertia matrix Subnetwork D K Plate P K total rotation vector 1 2 3 4 new linear model only velocities updated !

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A copy of this presentation can be downloaded from: http://der.topo.auth.gr A copy of this presentation can be downloaded from: http://der.topo.auth.gr Thanks for your attention !

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VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy May 29 - June 2, 2006, Wuhan, China VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy: Challenge and Role of Modern Geodesy May 29 - June 2, 2006, Wuhan, China the Intercommission Committee on Theory (ICCT) of the International Association of Geodesy (IAG)

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