ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July S. Casotto, F. Panzetta Università di Padova, Italy and GOCE Italy Consortium Sponsored by ASI Tidal Field Refinement from GOCE and GRACE – A sensitivity study

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Tidal Field Refinement from GOCE ? S. Casotto, F. Panzetta Università di Padova, Italy and GOCE Italy Consortium Sponsored by ASI S. Casotto, F. Panzetta Università di Padova, Italy and GOCE Italy Consortium

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Outline  Tide field representation  Sidebands and sensitivity of satellite orbits to ocean tides  Rationale for ocean tide parameter estimation from GOCE  Roadmap to using GOCE + other missions for OT extraction

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Why study ocean tides?  Tides as “noise” ●Remove ocean tide and load tide from satellite gravity records (e.g., GOCE, GRACE) ●Remove tidal currents from Acoustic Doppler Current Profiler (ADCP) records  Tides as “signal” ●Oceanographic applications (tidal currents in ocean mixing, mean flows, ice formation rates, etc.) ●Geodetic applications (satellite perturbations, tidal loading and station displacements, etc.)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July

6 Ocean Tide Representations  Harmonic constituents ●Doodson (1921) ●FES2004 OT model  Response method ●Originally due to Munk & Cartwright (1966) ●Orthotides variant due to Groves & Reynolds (1975) ●Orthotides are orthogonal over time ●CSR3.0, etc.  Proudman functions ●Orthogonal over space  MASCONS (Mass Concentrations) ●Usually for localized sensitivity (Ray et al.)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July k = Doodson number of the tide constituent = tide amplitude = tide phase = Doodson & Warburg phase correction = Doodson argument Ocean Tide Constituent  = mean lunar time s = mean longitude of the Moon h = mean longitude of the Sun p = mean longitude of the lunar perigee N’ = negative mean longitude of the lunar node p s =mean longitude of the solar perigee

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Spherical Harmonic Representation Amplitude and phase from FES2004 OT model 15 constituents (M 2, S 2, K 1, O 1, … ) Harmonic analysis provides harmonic constants a,b,c,d’s

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Tidal mass displacement → Stokes coefficients variation Ocean Tide Potential Can compute functionals of gravity accelerations gravity gradients

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July The Response Method (1/2) Tide height field as a weighted sum of past, present and future values of the Tide Generating Potential (TGP) TGP coefficients c nm (t) due to Sun, Moon, Planets

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Define admittance G as FT of impulse response M&C credo of smoothness → Linear in each tidal band m = k 1 Basis for extrapolation to minor constituents’ frequencies The Response Method (2/2)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July frequency A, B, C, D Extrapolation to minor constituents

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Tide height as a linear combination of orthotides Tide height as a linear combination of orthotides Orthotide method (1/4) Orthotides result from a convolution with TGP coefficients orthotide order orthotide constants (Groves & Reynolds, 1975) orthoweights CSR3.0

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July harmonic analysis of the convolution weights for each tidal band Total tide height as convolution with the TGP coefficients Total tide height as convolution with the TGP coefficients Orthotide method (2/4)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July SH coefficients of tide height CONVOLUTION SH coefficients of convolution weightsof TGP Orthotide method (3/4)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Ocean Tide potential Orthotide method (4/4) Obtain variations of the Stokes OT coefficients

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Constituents Orthotides ●FES2004 into orthotides representation ●Extract any constituent from CSR4.0  Constituents suitable for frequency analysis ●variant due to Groves & Reynolds (1975)  Orthotides allow efficient computation of gravitational perturbations on satellite orbits – economy of representation So far …

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Ocean tide model improvement from space missions ●Altimetry (TPX/Poseidon, Jason, …) ●Orbit perturbation analysis – very classical, goes back to 1970’s  Sensitivity study ●Use constituents over entire tide spectrum to identify OT coefficients (solution set) ●Beware of aliasing, resonances (orbit is sun-synchronous) and other perturbations  Parameter estimation ●Based on constituents ●Based on orthotides – some caveats ●Based on mascons Now …

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Sensitivity analysis – GOCE Transverse perturbations – Constituent RMS

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Sensitivity analysis – GOCE Spectrum of transverse perturbations

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Rationale ●Exceptionally low orbit of GOCE is highly sensitive to tidal perturbations ●Tidal perturbation power distributed across OT spectrum, not fully intercepted by the 15 constituents of FES2004 ●Official GOCE orbits do not account for admittance tides ●Official orbit accuracy at the 1-3 cm level may leave residual power containing OT signal ●More power, constraints, complementarity from other high accuracy missions (GRACE, …) OT parameter estimation

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Input data ●GOCE GPS phase measurements orbit fit residuals ●GOCE GRADIO measurement residuals – not enough sensitivity ●GRACE GPS residuals + KBRR residuals  Model dynamics ●Orbit perturbations due to OT only ●OT field representation  Measurement models ●SST h-l range ●SST l-l range rate OT parameter estimation

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Kaula-type linear theory ●Available in Orbit Elements or RTN Cartesian ●Limited by use of reference secularly precessing Keplerian orbit ●Need for multi-arc approach  Integral equations ●Also linearized orbit perturbations (Xu, 2008; Schneider, 1968) ●Can use any reference trajectory  Relative orbit methods ●Can use any reference trajectory – no multi-arc needed  Brute force numerical integration ●Need entire force field Orbital Dynamics due to OT

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Can refer to any reference trajectory as the intermediary orbit to evaluate the perturbations ●Single integration arc over 180-day nominal GOCE mission  No need for the partials w.r.t. reference orbit ●Not officially available from the project  Still need the orbit fit residuals ●We learned yesterday that the residuals are being made availlable ●Tracking observations are available, but not equivalent ●Otherwise entire OD process is to be redone Relative Orbital Dynamics Approach

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Classical constituents ●Provides the best identification of relevant parameters in this “selective” application ●Use of “response” background model still possible and more efficient, also in view of decoupling from “sensitive” constituents  MASCONS ●Well-posed inverse problem due to applicable constraints ●Already applied to GRACE (Ray & al.) for localized sensitivity  Response/Orthotides ●Critical if used in parameter inversion – tuned to specific satellite, not sensitive to entire spectrum (better suited to altimeter-based inversion) OT Representation

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Possible misidentification of relevant OT coefficients ●Use of SVD techniques for inversion of normal equations can help solve the singularity  Deep resonances – ●adopt Colombo’s model (essentially ODE solution with multiple eigenvalues)  Sideband constituents associated with longer periods than mission length ●Possibly not a problem due to foreseen total mission length OT parameter inversion (1/2)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July  Sideband constituents not used in official products ●Sideband constituents were considered in preliminary studies, but are not in current official GOCE processing standards  If official GOCE orbits have absorbed residual tidal signal ●OT inversion incomplete, try new POD estimates ●Hopefully not necessary  Inclusion of data from other missions, like GRACE ●Apply the same “reference orbit” philosophy ●Model “instrumental” (RR) measurements (Cheng, 2002) ●Build on current experience, e.g. within “Darota” OT parameter inversion (2/2)

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July ●Tools were developed for handling several ocean tides representations and transforming between them ●Interpolation/extrapolation to minor constituents available ●Linear perturbation analysis using numerical integration underway as verification of analytical approach for identification of sensitive parameters ●System dynamics representation identified ●Input data identified ●Economy of representations is based on excellent quality of reference official GOCE (as well as GRACE) orbits Conclusions

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July ●Need to study all details of GOCE orbit processing standards ●Refine interpolation/extrapolation to sidebands – nonlinearity corrections ●Develop integral equation solution capability ●Develop hybrid response method/mascons model to represent ocean tides ●Verify ideas by running numerical simulations ●Build on experience within GOCE-Italy ●Use data to squeeze out residual power Future work

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Sensitivity analysis – GOCE Radial perturbations – Constituent RMS

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Sensitivity analysis – GOCE Normal perturbations – Constituent RMS

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Sensitivity analysis – GOCE Spectrum of radial perturbations

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July Sensitivity analysis – GOCE Spectrum of normal perturbations