ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July 2010 1 S. Casotto, F. Panzetta Università di Padova, Italy and GOCE Italy Consortium Sponsored

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July 20101

S. Casotto, F. Panzetta

Università di Padova, Italy

and GOCE Italy Consortium

Sponsored by ASI

Tidal Field Refinement from GOCETidal Field Refinement from GOCEand GRACE – A sensitivity studyand GRACE – A sensitivity study


Tidal Field Refinement from GOCETidal Field Refinement from GOCE

??S. Casotto, F. Panzetta



Sponsored by ASI

S. Casotto, F. Panzetta




OutlineOutline

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


Why study ocean tides?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.)



Ocean Tide RepresentationsOcean 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.)


• k = Doodson number of the tide constituent• = tide amplitude• = tide phase• = Doodson & Warburg phase correction• = Doodson argument

Ocean Tide ConstituentOcean Tide Constituent

( , , ) ( , ) cos[ ( ) ( , )]t Z t k k k k k

1 2 3 4 5 6( ) ( 5) ( 5) ( 5) ( 5) ' ( 5) st k k s k h k p k N k p k

( , )Z k

k

( , ) k

k

• = 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• ps = mean longitude of the solar perigee

k


Spherical Harmonic RepresentationSpherical Harmonic Representation

0 0

0 0

cos [ cos( ) sin( )] (sin )

sin [ cos( ) si

( , ) ( ,

n( )] (

)

( , ) ( , ) sin )

N nmn

n m

N nmn

n m

nm nm

nm nm

Z

Z

m m

mc P

a Pb

dm

k kk

kk

k

k

k

• Amplitude and phase from FES2004 OT model

• 15 constituents (M2, S2, K1, O1 , … )

• Harmonic analysis provides harmonic constants a,b,c,d’s


Tidal mass displacement → Stokes coefficients variation

Ocean Tide PotentialOcean Tide Potential

1( )

21

( )2

nm nm nm

nm

nm

nm nm nm

F a d

F c b

C

S

kk

k

k kk

'4 1

2 1w n

nmnm

G kF

gN n

0(2 1)(2 )( )!

( )!m

nm

n n mN

n m

0 0

cos( ) sin( )] (sin )nN n

meOT nm nm n

n m

aGMU C m S m P

r r

k k k k k k

k

Can compute functionals of gravity• accelerations

• gravity gradients


The Response Method (1/2)The Response Method (1/2)

Tide height field as a weighted sum of past,Tide height field as a weighted sum of past,

present and future values of the present and future values of the Tide Generating Potential Tide Generating Potential (TGP)(TGP)

TGP coefficients TGP coefficients ccnmnm(t) (t) due to Sun, Moon, Planetsdue to Sun, Moon, Planets

,

( , , )( () , ) n

S

n m s Ss mm c tgt s t

0

,

4 2(sin )

2(

1) j

n

imj mm en j

j Sun Mo jn

n jm

o

GMc

aP e

gr n rt


Define admittance Define admittance GG as FT of impulse response as FT of impulse response

M&C credo of smoothness → Linear in each tidal band M&C credo of smoothness → Linear in each tidal band m = km = k11

Basis for extrapolation to minor constituents’ frequenciesBasis for extrapolation to minor constituents’ frequencies

1( , ) ( , )s

k i s ts

s S

G g e

kk

*( , , ) ( , ) ( ),nmt G c t k k k

The Response Method (2/2)The Response Method (2/2)


. . ..

frequency

Extrapolation to minor constituentsExtrapolation to minor constituents


Tide height as a linear combination of Tide height as a linear combination of orthotidesorthotides

Orthotide method (1/4)Orthotide method (1/4)

OrthotidesOrthotides result from a convolution with TGP coefficients result from a convolution with TGP coefficients

orthotide orderorthotide order

orthotide constantsorthotide constants(Groves & Reynolds, 1975)(Groves & Reynolds, 1975)

orthoweightsorthoweights

1( )

0

( , (( , , ) ))I

mi

ii

m

m kzt t

( ,( ), ,) ) (ii iz iu v

) ( )(S

nms

mis

mi

S

c t st W t

CSR3.0CSR3.0


harmonic analysis of the convolution weights for each tidal band harmonic analysis of the convolution weights for each tidal band

Total Total tide height as convolution with the TGP coefficientstide height as convolution with the TGP coefficients

,

(( , , ) (, ) )S

nmn m s S

mswt c t s t

0 0

( , ) [ ( ) cos( ) ( )sin( )] (sin )L l

m m m ps lp lp l

l p

w D s p E s p P


( ( , )), ,( ) m mis

mi

is smw W z g


SH coefficients of tide heightSH coefficients of tide height

CONVOLUTIONCONVOLUTION

SH coefficients SH coefficients

of convolution weights of TGP

( )nmc t

*

*

( ) ( ) ( )

( ) ( ) ( )

m mlp lp nm

s

m mlp lp nm

s

A t D s c t s t

B t E s c t s t


( ), ( )m mlp lpD s E s


Ocean Tide potential

4 1 '( )

2 1

4 1 '( )

2 1

( )

( )

mw llp

m

mw llp

lp

lpm

G kA t

g l

G

C t

Sk

B tg l

t

0 0

( , , ) [ cos( ) sin( )] (sin )lL l

peOT l

l plp lp

aGMU r p

rC p P

rS


Obtain variations of the Stokes OT coefficientsObtain variations of the Stokes OT coefficients


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 …So far …


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 …Now …


Sensitivity analysis – GOCE Sensitivity analysis – GOCE Transverse perturbations – Constituent RMSTransverse perturbations – Constituent RMS


Sensitivity analysis – GOCE Sensitivity analysis – GOCE

Spectrum of transverse perturbationsSpectrum of transverse perturbations


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 estimationOT parameter estimation


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 estimationOT parameter estimation


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 OTOrbital Dynamics due to OT


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 ApproachRelative Orbital Dynamics Approach


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 RepresentationOT Representation


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)OT parameter inversion (1/2)


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)OT parameter inversion (2/2)


● 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

ConclusionsConclusions


● 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 workFuture work

k



Sensitivity analysis – GOCE Sensitivity analysis – GOCE Radial perturbations – Constituent RMSRadial perturbations – Constituent RMS


Sensitivity analysis – GOCE Sensitivity analysis – GOCE Normal perturbations – Constituent RMSNormal perturbations – Constituent RMS



Spectrum of radial perturbationsSpectrum of radial perturbations



Spectrum of normal perturbationsSpectrum of normal perturbations

Documents

ESA Living Planet Symposium, Bergen, Norway, 27 June-2 July 2010 1 S. Casotto, F. Panzetta Università di Padova, Italy and GOCE Italy Consortium Sponsored