Ocea ad atospheric tides stadards (used for EIGEN gravity fied odeig Richard Biacae (CNES/GRGS IERS Workshop o Covetios: 0- Septeber 007
Ocea tides odeig The height of ocea tides is expressed by a su over waves : Z is the apitude of the wave ( ϕt Z ( ϕ cos ( (t ψ ( ϕ ξ ψ is the phase is the Doodso arguet which is expressed i iear cobiatio of 6 variabes : ( 5 s ( 5 h ( 5 p ( 5 N ( p 3 4 5 6 s ( t τ 5 These 6 variabes with decreasig frequecies represet the fudaeta arguets accordig to the Su ad Moo otios : τ : age of the ea uar day (.03505 d s : age of the ea tropic oth (7.358 d h : age of the ea tropic year (365.4 d p : age of the ea uar perigee (8.8473 y N : age of the ea uar ode (8.69 y p s : age of the periheio (0940.8 y ( 0 3... defies the specie (og period diura sei-diura ter-diura... the group (i geera : 9 ad 3 the costituet ( 3 9.
The apitude (Z ad phase (ψ of the differet waves of tides represeted by cotida aps ca be expaded i spherica haroic fuctios of Z cosψ ad Z siψ : The we have : ξ ( ϕ t ( Z cosψ cos Z siψ si Writig : C a d Z Z C cosψ siψ a a a c c ( a cos b si P ( siϕ ( c cos d si P ( siϕ d d d b b ( cos cos si si ( cos cos si si ( cos si si cos ( cos si si cos ad S c b P S ( siϕ the height of tide becoes : ξ ( ϕ t P ( siϕ C cos( χ S si( χ c b [ ]
Procedure # readig the poit grid by ec_grie(_quart_degre # covertig by gpg; poit grid C ea grid C # covertig by aahs; ea grid C a b haroics # covertig by gpg; poit grid S ea grid S # covertig by aahs; ea grid S c d haroics # covertig by covers_hs; (a d /(b c haroics C / S ocea tides haroics # appyig by cor_eips the eipsoida correctio
FES004 Ocea tide ode: FES004 oraized ode (fev. 004 up to (0000 i c (og period fro FES00 up to (5050 equiibriu O/O atospheric tide NOT icuded Doodso Csi Ccos Csi- Ccos- C eps C- eps- 55.565 O 0-0.540594 0.000000 0.000000 0.000000 0.5406 70.000 0.0000 0.000 55.575 O 0-0.0058 0.000000 0.000000 0.000000 0.005 70.000 0.0000 0.000 56.554 Sa 0-0.046604-0.000903 0.000000 0.000000 0.0466 68.890 0.0000 0.000 57.555 Ssa 0-0.96385-0.00794 0.000000 0.000000 0.966 67.94 0.0000 0.000 65.455 M 0-0.47940-0.084083 0.000000 0.000000 0.4865 60.047 0.0000 0.000 75.555 Mf 0-0.805539-0.363 0.000000 0.000000 0.8394 53.66 0.0000 0.000 85.455 Mt 0-0.3908-0.04948 0.000000 0.000000 0.476 50.439 0.0000 0.000 93.555 Msq 0-0.0939-0.006674 0.000000 0.000000 0.005 5.008 0.0000 0.000 35.655 Q -0.990 0.30834 0.049563-0.0395 0.440 36.60 0.099 66.339 45.555 O -.480706.38307 0.455-0.79480.06 33.047 0.939 50.3 63.555 P -0.505544 0.55083 0.795-0.5944 0.7478 37.468 0.380 3.87 65.555 K -.530097.66093 0.8450-0.7850.583 37.348.535 3.889 35.755 N -0.0444 0.3483 0.00987-0.0067 0.49 34.834 0.0 94.55 45.655 N -0.577967 0.958489 0.6389 0.009450.93 38.90 0.67 86.670 55.555 M -3.3375 3.84073 0.78634 0.430793 5.005 39.908 0.8966 6.83 73.555 S -.8860.93664 0.04090 0.4978.859 35. 0.48 5.565 75.555 K -0.337696 0.3588-0.00066 0.4679 0.497 36.737 0.47 359.669 455.555 M4 4 4-0.00055-0.00344 0.0055-0.000 0.003 89.949 0.005 5.33
LAGEOS ocea tides fit over 9 y (oraized coefficiets : LAGEOS FES-004 differece doodso darw C (c ε (deg C (c ε (deg ΔC (% Δε (% 55.565 Ω 0 0.4387 3.7 0.5406 70.00 9. 6. 55.575 Ω 0 0.306 5.05 0.005 70.00 6044. 80. 56.554 Sa 0 0.5800 6.6 0.0466 68.89 44. 65. 57.555 Ssa 0 0.7390 77.7 0.966 67.9 49. 5. Ω Sa ad Ssa resuts are ot to be cosidered i ters of tides but ore probaby i ters of ass dispaceet
Apitude (c 000 00 0 0 00 M S K O N Sa Ssa K P Q Nu Mu M N L Mf T J M MSf R Eta Psi Phi OO Eps M3 Lada Pi Tau M4 Q siga MS4 Teta Ro MS6 M6 MS Odes du spectre Décopositio haroique du spectre de arée (pour a base de doées péagiques ST95 Ode Pourcetage d iportace M 335% S 6% K 0% O 70% N 68% K 33% Q 5% N 0% Tota 759% Pourcetage d iportace des pricipaes odes du spectre (source : thèse F. Lefèvre No de Nobre de Arguet de Doodso Coef. har. Fréquece ( /h Fréquece (rad/s Période (jours Orig ie Darwi Doodso α i M 0 055.555-050458 0 - L S 0 055.555-034 0 - S S a 056.554 h p 0076 0040667 000000099 365594 S S sa 057.555 h 00787 008373 0000000398 86 S S ta 058.554 3h p 00047 03040 00000005973 7493 S M s 063.655 s h p 00578 0475 000000860 389 L M 065.455 s p 00854 05443747 000000639 75546 L M sf 073.555 s h 00370 058958 000000495 47653 L M f 075.555 s 0564 098033 0000005334 36608 L M st 083.655 3 s h p 000569 5695548 00000076094 95569 L M t 085.455 3 s p 00995 644078 0000007966 939 L M sq 093.555 4s h 000478 3988 000000486 70958 L Q 5.755 τ 3 s p 000955 84486 000006709 678 S σ 7.555 τ 3 s h 0053 97398 000006675 603 L Q 35.655 τ s p 0076 33986609 00000649585 95 S ρ 37.455 τ s h p 0037 347545 000006537 35 L O 45.555 τ s 037689 39430356 00000675977 0758 L τ 47.555 τ s h 00049 40579 00000679960 0695 M 55.655 τ p 00964 44966939 000007080 0347 L M 55.655 τ p 00040 4487403 0000070369 095 L χ 57.455 τ h p 000566 45695476 0000070635 095 L π 6.556 τ s 3h p 0009 4978647 000007338 0055 S P 63.555 τ s h 07554 4958934 00000759 007 S L K 65.555 τ s 03633 5040686 0000079 09973 L S K 65.555 τ s 0687 5040686 0000079 09973 S ψ 66.554 τ s h p 00043 508353 000007303 09946 S ϕ 67.555 τ s h 000756 53059 0000073394 0999 S 73.655 τ s h p 000566 555897 000007507 09670 L J 75.455 τ s p 00954 55854433 00000755604 0964 L SO 83.455 τ 3s h 00049 60569644 00000778464 0934 L OO 85.655 τ 3s N' 0063 63907 0000078446 0994 L ν 95.455 τ 4s p 0003 66834764 00000808838 0899 L ε 7.655 τ s p N' 00067 7346964 000035563 05486 L N 35.755 τ s p 0030 78953548 000035405 05363 L μ 37.555 τ 4s 4h 00777 7968084 0000355937 05363 L N 45.655 τ s p 07387 8439795 0000378797 0574 L ν 47.455 τ s h p 003303 85583 00003839 056 L M 55.555 τ 0908 898404 000040589 0575 L 63.655 τ s h p 000670 9455653 000048049 0509 L L 65.455 τ s p 00567 9584700 000043580 05078 L T 7.556 τ s 3h p 00479 95589333 0000433058 05075 S S 73.555 τ s h 0486 300000000 000045444 05000 S R 74.554 τ s h p 000354 30040667 000045643 04993 S S K 75.555 τ s 003648 3008373 000045843 04986 S 75.555 τ s 007858 3008373 000045843 04986 L K L Pricipaes coposates extraites du déveoppeet de Doodso (source : thèse F. Lefèvre
The adittace fuctio is defied for each wave as the ratio betwee the potetia geeratig tides ad the ocea respose : ( ( ( t U t G ϕ ϕ ξ ϕ The behavior of the adittace fuctio is sooth i frequecy so that it ca be iterpoated ieary betwee ai waves of cose frequecies : ( ad ( ( ( G G G Adittace fuctio ( ( ( [ ] ( ( ( [ ] ( ( ( [ ] ( ( ( [ ] χ χ ϕ χ χ ϕ χ χ ϕ χ χ ϕ ξ si C cos S (si P si si S cos C (si P cos H H. si C cos S (si P si si S cos C (si P cos H H. ( i eve odd e i Y gh U ϕ with ( ( U G U G ξ ξ ( U U U U U G ξ ξ ξ with O. Coobo thesis of S Casotto (989
Copariso of CHAMP orbits over 0 days with the oe coputed with P fro FES004 Orbit differeces ( 3D Radia Nora Tagetia without P 44 0 47 36 P with adittace 3 4 adittace (65 waves 90 6 74 35
ELLIPSOIDAL CORRECTIONS TO SPHERICAL HARMONICS OF SURFACE PHENOMENA GRAVITATIONAL EFFECTS Georges Baio CNES-GRGS 8 Av. Edouard Bei 340 TOULOUSE Cedex 9 Frace georges.baio@ces.fr Subitted Septeber 9 003 To Techica Uiversity of Graz Istitut für Geodäsie Specia pubicatio i hoour of H. Moritz
(E O RE ϕ H Φ H P H ' H ( η / with η Φ ϕ Φ H 4 ϕ e si Φ cosφ e R0 4 ( ϕ' R ( ε si ϕ' ε si ϕ' R E 3 0 4 H R 0... H R 0 si 3 Φ cos Φ ( δ o K C i S δk { } r( ϕ ' ' * r ' dr' Y ( ϕ' ' dσ ' α ρ with α [( δ ( MR ] σ R E ( ϕ ' 0 with r' RE( ϕ ' H' ( ϕ' ' ad R E ( ϕ' 4 R0 ( ε si ϕ' ε 4 si ϕ'... 3 r' 3 { r ' /( 3 } R ( ϕ' { R E E [ H '/ R ( / [ H '/ R E ( ϕ'] E ( ϕ']... R 3 E ( ϕ'} δ K δ K α β ρ 4 * R ( ε si ϕ' ε 4 si ϕ' hkyk ( ϕ' '.Y( ϕ' ' dσ' o σ h h h ( ε a b c R o R 0 k R 0
Eipsoida correctio Mode : Ocea tide ode: FES004 oraized ode (i c Mode : Ocea tide ode: FES004 oraized ode with eipsoida correctio (i c ode ode - doodso darw c(c eps(deg c(c eps(deg. differece / reative(% 55.555 M 0 0.5888 40.804 0.5885 39.990-0.0003-0.804 0.0-0.5 55.555 M 0.8 80.4369.766 80.00-0.0046-0.459-0.4-0. 55.555 M 3 0.4460 5.06.44 5.40-0.0038 0.04-0.3 0.0 55.555 M 4 0.6966 95.4730.695 95.3580-0.004-0.50-0. -0. 55.555 M 5 0.36 38.68. 38.050-0.040-0.566 -. -0.3 55.555 M 0.399 40.86 0.3979 4.40 0.0050 0.3798.3 0. 55.555 M.050 50.48.055 50.090 0.004-0.463 0.4-0.3 55.555 M 3.578 345.465.555 346.4350-0.003 0.9699-0. 0.5 55.555 M 4.344 30.4333.397 30.3530-0.045-0.0803-0.6 0.0 55.555 M 5 3.5506 37.774 3.59 37.780-0.034 0.0536-0.9 0.0 55.555 M 5.05 39.9495 5.005 39.9080 0.0090-0.045 0. 0.0 55.555 M 3 0.8508 7.80 0.8549 7.0060 0.004 0.959 0.5 0. 55.555 M 4 4.7345 8.885 4.74 9.000-0.004 0.58-0. 0. 55.555 M 5.7639 9.930.7569 9.9690-0.0070 0.0560-0.4 0.0 55.555 M 3 3 3.388 36.596 3.347 36.7060-0.004 0.764-0. 0. 55.555 M 4 3 3.76 78.5463 3.746 78.4950-0.050-0.053-0.4 0.0 55.555 M 5 3.86 95.8639.8309 95.7940-0.030-0.0699 -. 0.0 55.555 M 4 4 3.7494 30.6906 3.7439 30.640-0.0055-0.0766-0. 0.0 55.555 M 5 4 4.070 49.3 4.0430 49.0-0.07-0.00-0.7 0.0 55.555 M 5 5 0.8065 70.980 0.80 70.6670-0.0054-0.33-0.7-0.
GRGS-GRACE tide ode Mode : Ocea tide ode: FES004 oraized ode (fev. 004 up to (0000 i c (og period fro FES00 up to (5050 equiibriu O/O atospheric tide Mode : OCEAN TIDES fro GRACE soutio doodso ode ode - c(c eps(deg c(c eps(deg. differece / reative(% 35.655 Q 0.440 36.6096 0.4603 33.7090 0.0363 -.9006 8.6 -.6 45.555 O.06 33.0474.9586 35.550-0.0676.076-3.3. 63.555 P 0.7478 37.4678 0.677 36.990-0.0707 -.688-9.5-0.6 65.555 K.583 37.3477.4783 33.8800 0.00 6.533 9.7 3.6 35.755 N 0.49 34.8343 0.484 338.00 0.0065-3.633 4.6 -.0 45.655 N.93 38.90.00 330.640-0.0093.7039-0.8 0.9 55.555 M 5.005 39.9080 4.835 3.60-0.970.540-3.9 0.7 73.555 S.859 35..894 309.970 0.068-5.95 3.7 -.9 75.555 K 0.497 36.737 0.573 3.00-0.354-5.75-47.8-3.
Tepora evoutio of the geoid differeces derived fro the ea atospheric tide odes based o 6h ECMWF pressure fied data (985-00 for August each 3h. 0h 3h 6h 9h h 5h 8h h
S ECMWF derived atospheric tides (Bode Biacae003 oraized ode (i bar Doodso Darw Csi Ccos C eps 64.556 S 0 0-0.0047 0.004887 0.05 95. 64.556 S 0-0.057 0.0075 0.036 30.74 64.556 S 0-0.0999-0.09367 0.093 8.64 64.556 S 3 0-0.09447-0.00090 0.095 66.79 64.556 S 0.548 0.063876 0.0 73.99 64.556 S 0.043 0.0486 0.087 37.633 64.556 S 3-0.536-0.030 0.59 64.849 64.556 S 4 0.006730-0.05007 0.064 55.846 64.556 S 0.04045 0.0584 0.066 37.78 64.556 S 3-0.000839 0.0486 0.049 358.067 64.556 S 4-0.0700-0.04535 0.0307 4.706 64.556 S 5 0.0070-0.0067 0.003 47.38 64.556 S 3 3-0.063-0.00 0.097 35.579 64.556 S 4 3-0.0887 0.0558 0.03 98.34 64.556 S 5 3-0.0853-0.003835 0.089 58.97 64.556 S 6 3-0.009-0.030 0.0 86.30 64.556 S 4 4-0.083 0.0696 0.094 336.339 64.556 S 5 4 0.0365-0.00437 0.039 00. 64.556 S 6 4 0.005458 0.00786 0.0096 34.769 64.556 S 7 4-0.00848 0.00407 0.008 344.695 64.556 S 5 5-0.09909 0.03089 0.0369 37.365 64.556 S 6 5 0.03443-0.00804 0.0343 93.06 64.556 S 7 5 0.093 0.0075 0.00 83.896 64.556 S 8 5-0.00684-0.00046 0.0069 66.36 S ECMWF derived atospheric tides (Bode Biacae003 oraized ode (i bar Doodso Darw Csi Ccos C eps 73.555 S 0 0-0.0067 0.00479 0.0049 345.00 73.555 S 0 0.00307-0.008 0.0033.985 73.555 S 0 0.04690-0.0495 0.00 56.997 73.555 S 3 0 0.040 0.00365 0.08 7.000 73.555 S 0.0533-0.003030 0.053 96.874 73.555 S 0.067-0.075 0.056 5.84 73.555 S 3-0.09507 0.009509 0.07 95.988 73.555 S 4-0.00488 0.0054 0.0043 85.406 73.555 S 0.83369-0.46898 0.5474 48.8 73.555 S 3 0.0098 0.06639 0.084 0.0 73.555 S 4-0.03633 0.070506 0.0793 33.750 73.555 S 5-0.00440-0.0087 0.0053 36.878 73.555 S 3 3 0.06795 0.00556 0.077 7.88 73.555 S 4 3 0.00383-0.000086 0.0038 9.9 73.555 S 5 3-0.004878 0.006430 0.008 3.85 73.555 S 6 3 0.00879 0.00580 0.005 49.940 73.555 S 4 4-0.05098 0.00088 0.05 73.39 73.555 S 5 4 0.0048 0.0089 0.0085 6.630 73.555 S 6 4-0.00503 0.00043 0.006 85.365 73.555 S 7 4-0.004306-0.00009 0.0044 56.8 73.555 S 5 5 0.00934-0.000794 0.0094 94.867 73.555 S 6 5 0.000304 0.00054 0.0006 30.0 73.555 S 7 5 0.003957-0.000054 0.0040 90.78 73.555 S 8 5 0.00583 0.00430 0.0055 74.854
Sa ECMWF derived atospheric tides (Bode Biacae003 oraized ode (i bar Doodso Darw Csi Ccos C eps 56.554 Sa 0 0-0.68856-0.049683 0.760 53.604 56.554 Sa 0 0.0507-0.057879 0.00 8.848 56.554 Sa 0 0.0795 0.65379 0.6577 6.8 56.554 Sa 3 0 -.5490-0.37665.806 53.9 56.554 Sa 0.0478 0.0890 0.59.08 56.554 Sa 0.3067 0.548859 0.685 9.54 56.554 Sa 3-0.04330 0.33606 0.3388 35.8 56.554 Sa 4-0.705-0.7063 0.04 3.50 56.554 Sa -0.898 0.589 0.706 305.968 56.554 Sa 3-0.5685-0.04370 0.603 60.633 56.554 Sa 4-0.6484 0.00388 0.643 70.69 56.554 Sa 5-0.9783-0.0067 0.973 68.94 56.554 Sa 3 3 0.33799-0.037 0.375 03.338 56.554 Sa 4 3 0.09600 0.4783 0.60.660 56.554 Sa 5 3 0.885 0.700 0.34 58.435 56.554 Sa 6 3 0.07368 0.56099 0.57 6.349 56.554 Sa 4 4 0.0390-0.3590 0.33 73.969 56.554 Sa 5 4-0.03664-0.0340 0.09 5.830 56.554 Sa 6 4 0.0634-0.03474 0.070 9.37 56.554 Sa 7 4-0.053530 0.04686 0.0555 85.34 56.554 Sa 5 5-0.0389-0.035348 0.0380 0.455 56.554 Sa 6 5 0.093559-0.805 0.466 40.38 56.554 Sa 7 5 0.0565-0.08943 0.056 47.876 56.554 Sa 8 5 0.0478-0.03463 0.0584 5.99 Ssa ECMWF derived atospheric tides (Bode Biacae003 oraized ode (i bar Doodso Darw Csi Ccos C eps 57.555 Ssa 0 0 0.0648-0.043967 0.053 49.000 57.555 Ssa 0 0.3356 0.36954 0.465 30.09 57.555 Ssa 0-0.03797 0.093489 0.007 338.5 57.555 Ssa 3 0 0.440373 0.98389 0.4830 65.748 57.555 Ssa -0.8784-0.054635 0.956 53.783 57.555 Ssa -0.0079 0.0846 0.04 85.678 57.555 Ssa 3 0.00 0.068 0.0666 9.479 57.555 Ssa 4 0.0533 0.04395 0.0485 3.365 57.555 Ssa -0.03308 0.0868 0.0354 9.86 57.555 Ssa 3-0.07309 0.0088 0.074 7.34 57.555 Ssa 4 0.08970 0.00777 0.0300 74.984 57.555 Ssa 5-0.08689-0.05389 0.036 4.79 57.555 Ssa 3 3-0.07506-0.066689 0.004 8.363 57.555 Ssa 4 3 0.00785-0.0564 0.0567 7.065 57.555 Ssa 5 3-0.399-0.06846 0.306 38.507 57.555 Ssa 6 3-0.069-0.077537 0.0808 96.33 57.555 Ssa 4 4-0.00336-0.039 0.035 93.856 57.555 Ssa 5 4 0.04857-0.037387 0.063 7.68 57.555 Ssa 6 4-0.0390 0.04806 0.003 36.804 57.555 Ssa 7 4 0.0359 0.0330 0.0383 69.655 57.555 Ssa 5 5 0.004073-0.05976 0.063 7.089 57.555 Ssa 6 5 0.000599-0.0694 0.069 78.76 57.555 Ssa 7 5-0.037086-0.009590 0.0383 55.50 57.555 Ssa 8 5-0.0847-0.007664 0.094 54.9
Copariso to the Haurwitz ad Cowey ode Mode : H W atospheric tides (Haurwitz Cowey 973 oraized ode up to (85 (i bar Mode : ECMWF derived atospheric tides (Bode Biacae 003 oraized ode up to (85 (i bar doodso ode ode - c(b eps(deg c(b eps(deg. differece / reative(% 64.556 S 0.885.000 0.0 73.990-0.0675 6.989-3.4 34.0 64.556 S 0.0365 330.9997 0.087 37.6330-0.078 66.6333-48.8 37.0 64.556 S 3 0.0860 97.000 0.59 64.8490 0.0399 67.8489 46.4 37.7 73.555 S 0.550 59.0000 0.5474 48.80-0.0036-0.780-0.7-5.7 73.555 S 3 0.060 8.0007 0.084 0.00 0.004-60.7807 9. -33.8 73.555 S 4 0.0540 330.9995 0.0793 33.7500 0.053.7505 46.9.0