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Satellite geodesy. Basic concepts. I1.1a Z b a r Meridian plane h φ X-Y = geocentric latitude φ = geodetic latitude r = radial distance, h = ellipsoidal height a = semi-major axis, b = semi-minor axis z = axis of rotation, 1900. flattening = (a-b)/a. C.C.Tscherning, University of Copenhagen, 2007-10-25 1 Coordinate-systems Example: Frederiksværk φ=560, λ=120, h= 50 m C.C.Tscherning, 2007-10-25. Geoid and mean sea level H N Earth surface Geoid: gravity potential constant Ellipsoid h=H+N=Orthometric height + geoid height along plumb-line =HN+ζ=Normal height + height anomaly, along plumb-line of gravity normal field C.C.Tscherning, 2007-10-25. Gravity potential, Kaula Chap. 1. • Attraction (force): mM F k 2 r • Direction from gravity center of m to M. • With m = 1 (unitless), then acceleration kM a 2 2 r r C.C.Tscherning, 2007-10-25. Gradient of scalar potential, V, V x V kM a V , V ( x, y, z ) point - mass r y V z C.C.Tscherning, 2007-10-25. Volume distribution, ρ(x,y,z) V ( x' , y ' , z ' ) ( x, y , z ) k ( x x' ) 2 ( y y ' ) 2 ( z z ' ) 2 Earth dx dy dz k d r Earth • V fulfills Laplace equation 2 2 2 V 2 V 2 V 2 V 0, outside masses x y z C.C.Tscherning, 2007-10-25. Spherical coordinates • Geocentric latitude • Longitude, λ, r = distance to origin. x r cos cos y r cos sin z r sin Volume - measure : d dxdydz cos r 2 d ddr C.C.Tscherning, 2007-10-25. Laplace in spherical coordinates V 1 2 V ) (cos r r r cos 2 1 V cos 2 2 Solution : V R(r ) ( ) ( ), 1 V 2 r or r , ( ) cos m or sin m R(r ) r ( ) Pnm (sin ), n degree, m order. n 1 n C.C.Tscherning, 2007-10-25. Spherical harmonics • Define: cos m , m 0 an Vnm (r , , ) n 1 Pn|m| (sin ) r sin | m | , m 0 then V (r , , ) kM n 0 n C m n V (r , , ) nm nm C.C.Tscherning, 2007-10-25. Orthogonal basis functions • Generalizes Fourier-series from the plane 90 180 V nm 90 (r , , )Vij (r , , ) cos d d 0 kMCnm for n i, m j for n i or m j 0 Functions may be (fully)nor malized, then C nm Cnm and Pnm Pnm C.C.Tscherning, 2007-10-25. Centrifugal potential • On the surface of the Earth we also measure the centrifugál acceleration, 1 2 2 2 W V r cos r 2 1 2 2 2 V (x y ) 2 rotational velocity (in inertial space). C.C.Tscherning, 2007-10-25. Normal potential, U • • • • Good approximation to potential of ideal Earth Reference ellipsoid is equipotential surface, U=U0, ideal geoid. It has correct total mass, M. It has correct centrifugal potential 2 a 2 a 4 2 U 1 J 2 ( ) P20 (sin ) J 4 ( ) P40 (sin ) .... r cos 2 r r 2 2 3 2 J 2 f (1 f / 2) m / 3(1 m f ) ...... 3 2 7 m 2 a / g e , (gravity at Equator) • Knowledge of the series development of the gravity potential can be used to derive the flattening of the Earth ! C.C.Tscherning, 2007-10-25. Satellite Geodesy: distances or ranges • If we know the orbit of the satellite and measure distances (or distance differences) to the satellite, then we may determine our own position: GPS, Doppler. S2 S1 Us C.C.Tscherning, 2007-10-25. S3 Satellite Geodesy: ranging to the Earth • Known orbit. One or two way ranging. Envisat • Satellite altimetry Ocean or ice ERS1 • PRARE • Synthetic Aperture Radar C.C.Tscherning, 2007-10-25. Satellite Geodesy: from ground to satelite • Laser ranging • Optical directions • Enables orbit determination • Enables gravity potential estimation C.C.Tscherning, 2007-10-25. LAGEOS Satellite to satellite tracking • We measure the position of a satellite from other satellites: SST • Enables calculation of position and velocity • Enables calculation of Kinetic Energy S2 S1 S4 C.C.Tscherning, 2007-10-25. S3 Satellite to satellite tracking, Low-Low • We measure range rates between two satellites • Gives (approximately) difference in Kinetic energy S1 GRACE S2 S4 Range-rate C.C.Tscherning, 2007-10-25. S3 S4 SST and gradiometry • Measurement of acceleration differences inside satellite (GOCE). • Determines second-order derivatives of potential. Mass1 Mass2 C.C.Tscherning, 2007-10-25. Task of Geodesy: to determine • Points on the surface of the Earth (and Planets) and changes (geodynamics, climate changes, sea-level) • Geoid/quasi-geoid determines state of sea-level if no currents – tides – salinity or temperature variations • Determines tides, (geostrophic) currents etc. If geoid known • Position of the Earth in Inertial system, Earth rotation, pole-position, Earth elastic parameters. • Changes of gravity (due to crustal uplift or hydrosphere changes) C.C.Tscherning, 2007-10-25.