Global Terrestrial Reference Systems and Frames Zuheir ALTAMIMI Laboratoire de Recherche en Géodésie Institut National de l’Information Géographique et Forestière France E-mail: [email protected] Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 1 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 OUTLINE • What is a Terrestrial Reference System (TRS), why is it needed and how is it realized ? • Concept and Definition • TRS Realization by a Frame (TRF) • International Terrestrial Reference System (ITRS) and its realization: the International Terrestrial Reference Frame (ITRF) • ITRF2008 Geodetic & Geophysical Results • How to access the ITRF ? • GNSS associated reference systems and their relationship to ITRF: – World Geodetic System (WGS84) – Galileo Terrestrial Reference Frame (GTRF) 2 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Defining a Reference System & Frame: Three main conceptual levels : Ideal Terrestrial Reference System (TRS): Ideal, mathematical, theoretical system Terrestrial Reference Frame (TRF) : Numerical realization of the TRS to which users have access Coordinate System : cartesian (X,Y,Z), geographic ( λ, φ, h), ......... – The TRF is a materialization of the TRS inheriting the mathematical properties of the TRS – As the TRS, the TRF has an origin, scale & orientation – TRF is constructed using space geodesy observations Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Ideal Terrestrial Reference System A tridimensional reference frame (mathematical sense) Defined in an Euclidian affine space of dimension 3: Affine Frame (O,E) where: O: point in space ( Origin ) E: vector base: orthogonal with the same length: - vectors co-linear to the base ( Orientation ) - unit of length ( Scale ) r r r 2 δ = = λ = E E .E = λ δ ( ij 1, i j) i i= 3,2,1 i j ij 4 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Terrestrial Reference Frame in the context of space geodesy • Origin : Z – Center of mass of the Earth System P • Scale (unit of length) : SI unit k • Orientation : j i – Equatorial (Z axis is approximately o Y the direction of the Earth pole) X 5 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Transformation between TRS (1/2) 6 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Transformation between TRS (2/2) 7 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 From one RF to another ? Z2 Z1 Rz Y2 T O2 O1 X2 Y1 Ry Rx X1 X X T D −−− R R X x z y === +++ +++ −−− Y Y Ty R z D R x Y Z Z −−− R R D Z 2 1 Tz y x 1 8 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Coordinate Systems Z – Cartesian: X, Y, Z P – Ellipsoidal: λλλ, ϕϕϕ, h R – Mapping: E, N, h z – Spherical: R, θθθ, λλλ θθθ – Cylindrical: l, λλλ, Z o Y λλλ l X l cos λλλ Rcos θθθ cos λλλ OP l sin λλλ OP Rcos θθθ sin λλλ z Rsin θθθ Cylindrical Spherical 9 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Ellipsoidal and Cartesian Coordinates: Ellipsoid definition a: semi major axis b b: semi minor axis f: flattening a e: eccentricity a 2 −−− b2 a −−− b e 2 === , f === a 2 a (a,b), (a,f ), or (a,e 2) define entirely and geometrically the ellipsoid 10 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 === +++ λλλ ϕϕϕ Ellipsoidal and X (N h)cos cos Cartesian Coordinates Y === (N +++ h)sin λλλ cos ϕϕϕ Z === [N 1( −−− e 2 ) +++ h]sin ϕϕϕ Z Zero meridian Local meridian a N === 2 2 h 1 −−− e sin ϕϕϕ b Prime Vertical Radius of Curvature a o ϕ λλλ Y equator X 11 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 (X, Y, Z) ==> ( λλλ,ϕϕϕ,h) Y f =1− 1− e2 R = X 2 +Y 2 + Z 2 λ = arctg X Z e2a µ = arctg ()1− f + X 2 +Y 2 R − + 2 3 µ ϕ = Z 1( f ) e asin arctg 2 2 2 3 1( − f )[ X +Y − e acos µ] h = X 2 +Y 2 [cos ϕ + Z sin ϕ]− a 1− e2 sin 2 ϕ 12 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Map Projection Mathematical function from an ellipsoid to a plane (map) E = f( λλλ,ϕϕϕ) N = g( λλλ,ϕϕϕ) P Z h f & g N p P0 ϕ E Y o λ Mapping coordinates: (E,N,h) X 13 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Why a Reference System/Frame is needed? • Precise Orbit Determination for: – GNSS: Global Navigation Satellite Systems – Other satellite missions: Altimetry, Oceanography, Gravity • Earth Sciences Applications – Tectonic motion and crustal deformation – Mean sea level variations – Earth rotation – … • Geo-referencing applications – Navigation: Aviation, Terrestrial, Maritime – National geodetic systems – Cartography & Positioning 14 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 What is a Reference Frame? • Earth fixed/centred RF: allows determination of station location/position as a function of time • It seems so simple, but … we have to deal with: – Relativity theory – Forces acting on the satellite – The atmosphere Origin, Scale & Orientation – Earth rotation – Solid Earth and ocean tides – Tectonic motion – … • Station positions and velocities are now determined with mm and mm/yr precision 15 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 "Motions" of the deformable Earth • Nearly linear motion: – Tectonic motion: horizontal – Post-Galcial Rebound: Vertical & Horizontal • Non-Linear motion: – Seasonal: Annual, Semi & Inter-Annual caused by loading effects – Rupture, transient: uneven motion caused by EQ, Volcano Eruptions, etc. 16 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Crust-based TRF The instantaneous position of a point on Earth Crust at epoch t could be written as : === +++ & −−− +++ ∆∆∆ X (t) X 0 X .( t t0 ) ∑∑∑ X i (t) i X0 : point position at a reference epoch t0 X& : point linear velocity ∆∆∆ : high frequency time variations: X i (t) - Solid Earth, Ocean & Pole tides - Loading effects: atmosphere, ocean, hydrology, Post-glacial-Rebound - ... Earthquakes 17 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Reference Frame Representations • "Quasi-Instanteneous" Frame: mean station positions at "short" interval: – One hour, 6-h, 12-h, one day, one week ==> Non-linear motion embedded in time series of instanteneous frames • Long-Term Secular Frame: mean station positions at a reference epoch (t 0) and station velocities: X(t) = X 0 + V*(t – t0) 18 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Implementation of a TRF • Definition at a chosen epoch, by selecting 7 parameters, tending to satisfy the theoretical definition of the corresponding TRS • A law of time evolution, by selecting 7 rates of the 7 parameters, assuming linear station motion! • ==> 14 parameters are needed to define a TRF 19 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 How to define the 14 parameters ? « TRF definition » • Origin & rate: CoM (Satellite Techniques) • Scale & rate: depends on physical parameters • Orientation: conventional • Orient. Rate: conventional: Geophysical meaning (Tectonic Plate Motion) • ==> Lack of information for some parameters: – Orientation & rate (all techniques) – Origin & rate in case of VLBI – ==> Rank Deficiency in terms of Normal Eq. System 20 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 Implmentation of a TRF in practice The normal equation constructed upon observations of space techniques is written in the form of: (1) where are the linearized unknowns Eq. (1) is a singular system: has a rank deficiency equal to the number of TRF parameters not given by the observations. Additional constraints are needed: • Tight constraints ( σσσ≤≤≤ 10 -10 ) m Applied over station • Removable constraints ( σσσ≅≅≅ 10 -5 ) m coordinates • Loose constraints ( σσσ≥≥≥ 1) m • Minimum constraints (applied over the TRF parameters, see next) 21 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 TRF definition using minimum constraints (1/3) The standard relation linking two TRFs 1 and 2 is: θθθ is the vector of the 7 (14) transformation parameters Least squares adjustment gives for θθθ : A : desigin matrix of partial derivatives given in the next slide 22 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 The Design matrix A 14 parameters 7 parameters Note: A could be reduced to specific parameters. E.g. if only rotations and rotation rates are needed, then the first 4 columns of the two parts of A are deleted 23 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 TRF definition using minimum constraints (2/3) • The equation of minimum constraints is written as: It nullifies the 7 (14) transformation parameters between TRF 1 and TRF 2 at the ΣΣΣθθθ level • The normal equation form is written as: ΣΣΣθθθ is a diagonal matrix containing small variances of the 7(14) parameters, usually at the level of 0.1 mm 24 Technical Seminar on Reference Frame in Practice Rome - Italy, 4th- 5th May 2012 TRF definition using minimum constraints (3/3) Considering the normal equation of space geodesy: (1) where are the linearized unknowns Selecting a reference solution XR ,the equation of minimal constraints is given by: (2) AccumulatingPar cumul de (1) (1) and et (2),(2), weon