Tidal Corrections to Direction Observations Catherine Le Cocq SLAC - Metrology Department - Advanced Methods Group

Displacement-Strain Relation Q P´ Q´ Displacement vector u u = PP’ = x’-x Final state Strain tensor e Original state Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Angular Formula r´ q´ Final state r q Original state 2 infinitesimal vectors q and r r q θ r´ q´ θ´ transform into q’ and r’ Final state Introducing =-’  and  the cosine directors of q and r Original state Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Surface Strain Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Mathematical Cartography Original state = sphere or ellipsoid as a model of earth p (,) Final state = plane or map P(X,Y) Fundamental quadratic form Linear alteration = Angular alteration =  = A- = V-- Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Tidal Displacement Vector At point P (r,,), the tidal displacement vector u is given in spherical coordinates by North-South horizontal component East-West horizontal component Vertical component W is the tidal potential, g is the mean gravity at point P h and l are the first and third Love numbers Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Tidal Potential B φ P l d C r Equator North South γ O α E δ B H Zenith Pole φ Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Tidal Strain Tensor Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Celestial Coordinates: , Computational Steps Location r,, Love Numbers l,h Date&Time d,ut Local Sideral Time: s Celestial Coordinates: , Hour Angle: H Strain Tensor: e Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

 = 37°25’  = 122°12’30’’ h=0 on 11-7-2000 at 1AM Azimuth Variations  = 37°25’  = 122°12’30’’ h=0 on 11-7-2000 at 1AM 1.08e-8 degree 4.39e-7 degree Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq

Same location between 11-6-2000 and 11-9-2000 with  = -/2 Time Variations Same location between 11-6-2000 and 11-9-2000 with  = -/2 Ground Motion Workshop, November 6-9, 2000, SLAC Catherine Le Cocq