The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The rank deficiency in estimation theory and the definition of reference frames V Hotine-Marussi Symposium on Mathematical Geodesy - June 17-21, Matera, Italy Athanasios Dermanis Department of Geodesy and Surveying - The Aristotle University of Thessaloniki

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Description of local vectors - tensors Description of local vectors - tensors: Description of space-time Description of space-time: Relativity theory Relativity theory: Coordinates Field of local vector bases

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Newtonian physics: Separation of space from time - Euclidean space model Parallel translation of vector basis to any point (axiom of parallelism) Cartesian coordinates = = components of position vector: Position vector concept: Description of local vectors - tensors: Description of space: Reference frame Reference frame: Origin + 3 base vectors (orthonormal for convenience)

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Reference frame Reference frame: A modeling device for the provision of - a field of (parallel) local frames for local tensor representation, - a system of (cartesian) coordinates for the description of “places” (also curvilinear coordinates defined as functions of cartesian). has no physical meaning A reference frame has no physical meaning and cannot be determined from observations. It must be established by introducing arbitrary conventions in the process of data analysis. advantage Its use has the advantage of facilitating mathematical modeling of physical processes and disadvantage the disadvantage of leading to equations with no unique solutions (rank deficient models)

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Physical system Physical system covered by observations: Set of all parameters that can be virtually expressed as functions of observables. Modeling Simplification isolation Modeling : Simplification of natural processes and isolation of a part of nature in relation to a performed set of measurements corresponding to a set of observables. Errors Errors: discrepancies accounting for effects of isolation and simplification. Model parameters Model parameters: observables & additional unknown parameters Model equations Model equations: equations of the form Condition equations Examples:, (no unknowns ): Condition equations Observation equations, (describing set ): Observation equations Parametric rank Parametric rank of physical system: Minimal number of parameters needed to describe the system (all other parameters can be expressed as functions of a describing set of parameters).Modeling

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Use of m parameters x describing a system S' S' System covered by observations S  S' with rank r  m S Geodetic network example: S = shape and size S' = shape, size and position y = angles and distances, x = coordinates Question Question: When does a parameter q(x)  S ? Or: is q(x) determinable from the observables? Answer Answer: When it can be expressed as a function of y : q(x)  d(y)  d(f(x)) ! determinability condition q  d  f determinability condition Problem circumvention Problem circumvention: Arbitrarily fix S'  S by d  m  r conditions c(x)  0 (minimal constraints) arbitrary reference frame Geodetic network example: Use minimal constraints to introduce an arbitrary reference frame Improper modeling(model without full rank) Improper modeling y  f ( x ) (model without full rank)

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames BLUE = BLUE of :Uniformly unbiased ( ), linear ( ) with minimum mean square error ( ) estimability “Stochastic” definition of estimability: is estimable when there exists a uniformly unbiased linear estimate of estimability “Deterministic” condition of estimability: (linear version of determinability condition ) Estimable parameter = it can be expressed as function of the observables: Parameter estimability in rank-deficient linear models

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames where is any solution of the normal equations: BLUE of :No unique solution because has If is estimable: Invariant Invariant estimate! If, are two solutions of the normal equations then: Invariant Invariant estimates of observables! Invariance of observables and estimable quantities

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model Solutions of normal equations: Transformations leaving observables invariant: leave estimable quantities invariant:

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model To each y  R(A) corresponds a solution space S y parallel to the null space N(A)  S 0 As y varies over R(A) the disjoint solution spaces S y cover the whole space X fibering They form a fibering of X

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model Minimum norm solution If is a basis for : Inner constraints

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The geometry of the linear model Minimal constraints Minimal constraints: section The linear subspace is a section of the fibering of the solution spaces, i.e. it intersects each in a single element Solution for BLUE estimates: - Satisfying constraints and normal equations

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The peculiar character of trivial constraints Trivial constraints Trivial constraints: Can be applied a priori: Unique solution: Estimable !? The unknowns are estimable only if the trivial constraints are “true”, i.e. they have a valid physical meaning within the model! The same holds true for any minimal constraints ! formalestimability Note the difference between formal estimability when are arbitrary true estimability and true estimability when are physically valid within the model!

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames These are parameters relating only to the shape and size of the network, estimable independent from its position and orientation and thus estimable parameters. Furthermore: The trivial constraints determine the reference frame in a deterministic way, independently of the uncertainties present in the stochastic estimates of network coordinates. For any point P : z P = distance of P from P 1 P 2 P 3 plane y P = distance of projection of P on P 1 P 2 P 3 plane from line P 1 P 2 x P = distance of projection of P 1 on P 1 P 2 line from point P 1 P 2 Physically meaningful trivial constraints in geodetic networks Angle and distance observations: shape and size determination Trivial constraints: Origin at P 1 x axis along P 1 P 2 line x, y plane at P 1 P 2 P 3 plane

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Space of observationsSpace of coordinates The reference frame problem in geodetic networks: The nonlinear picture Space of observables:Every point y = = a particular network shape Least-squares solutions: Transformations leaving observables invariant: leave estimable quantities invariant: Coordinates corresponding to the same shape (different frames):

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The reference frame problem in geodetic networks: The nonlinear picture General form of transformations leaving observables (shape) invariant: Space of observationsSpace of coordinates Transformation parameters: Similarity transformation. s  0  Rigid transformation, R = orthogonal

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The reference frame problem in geodetic networks: The nonlinear picture Fixing a point the parameters of the transformation Space of observationsSpace of coordinates may serve as a set of coordinates on

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The reference frame problem in geodetic networks: The nonlinear picture Fixing a point the parameters of the transformation Space of observationsSpace of coordinates may serve as a set of coordinates on Basis of tangent space to at : Columns of the matrix of inner constraints :

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The evolution of the reference frame in time The problem: Observations determine the shape at every epoch t Space of coordinates Open question: What is a optimal frame? Parallel Parallel frames RF and RF’: Two parallel frames are dynamically equivalent. Thus the initial epoch reference frame x 0 (t) must be arbitrarily fixed ! Starting from any frame RF 0 an optimal frame RF can be computed by determining the relevant transformation parameters p from x 0 (t) to x(t) : A reference frame RF is a curve in X intersecting the manifolds S y (t) at a single point x(t) for each t At the shape manifold S y (t) select a frame x(t) in a smooth way

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames all equivalent They are all equivalent and lead to a system of nonlinear ordinary differential equations to be satisfied by the parameters p(t) of transformation from a given to the optimal frame. Initial conditions (integration constants): Choice between equivalent parallel frames. The optimal (time evolution) of the reference frame The surveying discrete time practice: Use coordinates estimates of epoch t k  1 as approximate coordinates for the epoch t k. Best fit the frame of t k to the frame of t k  1 by applying inner constraints on  x k  x k  x k  1. Meissl ladder Build a Meissl ladder! Alternatives for continuous time: geodesic A geodesic on the manifold Meissl A generalization of the Meissl ladder concept: Tisserand A discrete Tisserand principle (vanishing relative angular momentum):

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The ITRF approach Introduction of an analytical time evolution model - Deformation linear in time Plus the standard inner constraints for corrections at the initial epoch Discrete Tisserand condition: Linearized form: Usual form for :

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Reference frames in earth rotation theories Inertial frame:Terrestrial frame: =angular momentum, = torqueEquations of rotational motion: In terrestrial frame:Liouville equations Euler geometric equations matrix (tensor) of inertia relative angular momentum rotation (angular velocity) vector where:

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Reference frames in earth rotation theories geocenter Terrestrial frame origin: geocenter Simplified Liouville equations: Terrestrial frame orientation: Simplification of rotation equations - 2 choices diagonal inertia tensor - NOT USED Axes of figure Axes of figure: vanishing relative angular momentum Tisserand axes Tisserand axes: A family of parallel frames: Arbitrary choice of initial epoch orientation !

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Connecting the geodetic network reference frame with the geophysical earth frame of rotation theory The connection between geodetic (ITRF) and Tisserand frame requires: Geodetic frame Geodetic frame:Discrete, its definition depends on the shape of the defining network. Tisserand frame Tisserand frame:Continuous, its definition depends on mass distribution and motion for the whole earth. -Estimation of the position of geocenter with respect to the geodetic frame from satellite geodetic techniques (gravity field also unknown) -Estimation of earth motion and density from surface (network) motions and geophysical data.

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Estimability of geocenter position Problem unknowns:Coordinates and gravity field parameters Both depend on reference frame Induced transformation on function coefficients : Coordinate transformation of a point P due to change of reference frame: Representation of unknown function V(P) (gravitational potential) in terms of basis functions involving coordinates:

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Estimability conditions for geocenter coordinates : The null space basis for function coefficients : columns of from The null space basis for coordinates : columns of from Inner constraints: Estimability of geocenter coordinates The combined null space: columns of Linearized model:

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames We seek only: We need to realize the expansion: The null space for coordinates and unknown function coefficients Expand: For curvilinear coordinates :

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames The null space for spherical harmonic coefficients Expand: where Compute: Required expansions for spherical harmonics: Translation: Rotation: Scale:

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames Estimating the direction of the earth Tisserand axes Transformation from ITRF coordinates x to estimates of Tisserand coordinates: Each plate P K is represented by a subnetwork D K - Assume rotation-only model for plates Angular momentum and inertia matrix for each subnetwork D K : Plate rotation vector:Plate inertia matrix: determines one out of infinite parallel Tisserand frames ! ITRF to Tisserand frame rotation vector and matrix: weighted mean !

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames This powerpoint presentation is available at:

The Aristotle University of Thessaloniki - Department of Geodesy and Surveying A. Dermanis: The rank deficiency in estimation theory and the definition of reference frames See you in Thessaloniki ! 3rd Meeting of the International Gravity and Geoid Commission IAG - Section III August 26-30, 2002 Thessaloniki, Greece