GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 1, PAGES 21-24, JANUARY 1,2001
Error spectrum for the global M2 ocean tide
R. D. Ray,• R. J.Eanes,
2 G. D. Egbert,
3 andN. K. Pavlis
4
Abstract. The mostaccuratedeterminations
of the global The tidal errorspectrum
is estimated
hereby two comoceantidesarecurrentlybasedon altimetermeasurements
plementary
methods:by computing
differences
in existing
madeby the Topex/Poseidon
satellite.The errorspectrummodelsandby estimating
errorsthroughglobalinversecalcorresponding
to theM2 tidalsolutionis hereestimated,
pri- culationswherereasonably
realistica priori errorcovarimarilyby inverse
methods
andsecondarily
by simplediffer- ancesareemployed.
Because
we areinterested
in theeffect
encingof severalof the besttidal models.The tidal error of tide modelerrorson the gravityfield as determinedby
spectrumis flatterthanthe tidal signalspectrum,and it ex- satellitegeodesy,the erroris expressed
in termsof equivaceeds10% of thesignalat sphericalharmonicdegree15 and lent geoidheighterror,aswasrecentlydoneby Wahret al.
above. The tide errorsalsoexceedthe anticipatedsensitiv- [1998]. Two possiblesystematic
errorsin the conversion
to
ity of the upcomingGRACEsatellitegravitymissionfor all tidal geoidvariationsare alsoexamined:the commonasdegreesbelow 40, andpossiblybelow 50.
sumptions
of a constantseawaterdensityand of a spherical earth. In this note we confine ourselves to the dominant
1. Introduction
semidiurnal lunar tide M2.
Sincethe launchof the Topex/poseidon
satellitein 1992
considerable
progresshasbeenmadein our knowledgeof
the globaloceantides[e.g.,Le Provostet al., 1995]. This
progresshas been quantifiedin severalways: by numerouscomparisons
with in situtide determinations,
primarily
with coastaltide gaugesat small islandsand bottompressurerecorderson the sea floor; by variancereductiontests
with independentsatellitealtimeterdata; and by comparisonswith tidal estimatesfrom satellitetracking [e.g., An-
2. Formalism
dersen et al. , 1995; Shum et al., 1997; Desai et al. , 1997].
(Testsof tidal currentvelocitiesfrom someof the new tide
modelshavealsobeenmadeby Dushawet al. [ 1997] andby
others,althoughtidalcurrentsarenotthetopicof thepresent
paper.)
Anotherapproachto understanding
accuraciesof ocean
tide modelsis to determinetheir errorspectra--thatis, their
erroramplitudes
asa functionof the degreen of a spherical
harmonicexpansion.In additionto characterizing
modelaccuracies,reliableerrorspectrahavemanyapplications[e.g.,
Tsaoussiand Koblinsky,1994]. One application(prompting
the presentstudy)involvesestimatingthe effectof aliased
tide modelerrorson monthlydeterminations
of the Earth's
gravityfield as anticipatedto occurduringthe upcoming
GRACEsatellitemission. The first step of this work is to
determinethe errorspectrumof the bestavailablemodels.
We employthe sphericalharmonicnormalization
commonly usedin physicalgeodesy[Heiskanenand Mortiz,
1967],wheretheLegendre
functions
Pnm(cos
O)satisfy
.;_•
[Pnm(•t)]
2dtt
- 2(2
- Sm,0).
1
The contributionof the oceantidesto the geoidundulation
N is expressedas
•SN(O,
qo,
t) - a • • Pnm(cos0)[•SCnmcOsmqo+•SSnm
n= 1 m=0
(1)
wherea is the earth'smeanradiusand (O,9) are spherical
polarcoordinates.
Thetime-varying
Stokescoefficients
are
computed
fromthetidalelevation
•(O,q0,
t) by [cf Wahret
al., 1998]
•Snm(t)
Me(2n+
1)f/ •Pnm
sinmq0
{•Cnm(t)
}--a2pw(l
+k•n)
{cosmq0
}dg•(2)
wherePwisthemeandensity
of theocean,
Me isthemassof
theearth,andk•nisa loading
Lovenumber
of degree
n which
accounts
for thegeoidperturbation
fromoceantidalloading
of the solid earth.
Define the Stokescoefficients•Cnmand •Snmas the root
overa complete
tidalcycleT (12.42hours)of
1Laboratory
forTerrestrial
Physics,
NASA GoddardSpaceFlight meansquares
thetime-varying
coefficients.
Thenthetidalcontribution
to
Center,Greenbelt,Maryland.
2Center
for SpaceResearch,
University
of Texas,Austin,Texas. thevarianceof thegeoid,takenovertheentireglobe,is
3Collegeof OceanicandAtmospheric
Sciences,
OregonState
University,Corvallis,Oregon.
1 //f•N2dtd•--a2E(•2nmq-•n
4Raytheon
ITSS, Greenbelt,
Maryland.
4r•T
We takethe amplitudespectrumto be the quantity
Copyright
2001 bytheAmericanGeophysical
Union.
1/2
Papernumber2000GL011674.
(3)
•Nnm-a(•2nmq-•2nm)
0094-8276/01/2000GL011674505.00
21
22
RAY ET AL.' ERROR SPECTRUM
The degreeamplitudespectrumis [Wahret al., 1998]
•Nn -- a
n
--2
(Z aXn
•Cnmq-
rn
\m-O
3. Tidal error spectra
(4)
FOR THE M2 TIDE
tions(LTE) and prior statisticalassumptions
aboutthe errorsin theseequations[Egbertet al., 1994]. Posteriorerrors
canthusbe calculatedto characterize
uncertaintyin the estimatedfieldsarisingfrom incompletealtimetercoverageand
errorsin thedata.We usea Monte Carloapproach[Dushaw
et al., 1997], which we summarize here. A series of ran-
dom realizationsof the LTE forcingerror (which accounts
in the equations,bathymetryerrors,and
Figure 1 showsthe spectrumof the oceantide computed for approximations
numerical
grid
truncation)
are generatedusingtheprior cofrom threerecentglobalmodelsof the M2 oceantide. (The
variance
derived
in
Egbert
et
al. [ 1994]. For eachrealization
figure strictly showsthe tide elevationspectrum,not the
i
the
perturbed
LTE
are
solved
numericallyto yield synthetic
geoidspectrum;the two are relatedby the factorpreceding
tidal
elevations
•i,
i
=
1,...,
N.
(Fortheresults
reported
here
the integralin (2).) The threemodelsare all deducedfrom
we
use
N
-20.)
These
elevation
fields
are
each
sampled
Topex/Poseidon
altimetryusingdifferentmethodsof analysis, including(for TPXO.4) formal data assimilationinto a with thespatialandtemporalpatternof thealtimeter,random
of noiseandnon-tidaloceanograhydrodynamicmodel. All are improvements
to earliertide dataerrors(representative
modelsthat Shumet al. [1997] found were amongthe most phy in the T/P data) are added,and the syntheticdata are
accurateof thoseavailablefor the entireglobalocean.Solu- inverted(usingthe sameprocedureappliedto real T/P data
tion GOT99.2is describedby Ray (1999); CSR4.ois similar togenerate
TPXO.4)
forestimates
•i. Thedifferences
•i- •i
in designbut differentin somedetails,includingthe method are representative
of errorsin the assimilationestimatesof
usedfor tidal analysis.AssimilationsolutionTPXO.4 is an tidal elevations.The error fields are found to be largestin
updateto thatdescribedby Egbertet al. [1994] andEgbert thepolarseas(beyondthe satellitedatacoverage)andin the
[1997].BothCSR4.O
andGOT99.2
default
toapurelyhydro- shallowseassurroundingIndonesia.The root meansquare
dynamicsolution[Le Provostet al., 1994] in polarregions amplitudesof sphericalharmoniccoefficientsof theN error
abovethe Topex inclinationof 66ø, while TPXO.4 extends fieldsareusedto estimatethe tidal errorspectrum.
polewardsvia its own linearizedhydrodynamicmodel. The
Parametersof the prior covariancecanbe adjustedsothat
threespectraareclearlyin closeagreement,althoughcareful the amplitudeand spatialstructureof the randomrealizaexaminationrevealsminor differences.The largestvariation tions•i aregrossly
consistent
(in amplitude
andspatialstrucin the spectrais a functionof degreen, notorderm. The cor- ture) with the T/P data, but it mustbe admittedthat signifirespondingdegreeamplitudespectrumis shownas the top cantuncertaintiesremainaboutthe "true" prior covariance.
We have thus performedcalculationsfor severaldifferent
curvein Figure2 (basedon TPXO.4).
The degreeamplitudespectraof the tide model errors, (consistent)prior covariancesto test the sensitivityof our
to poorlyconstrained
parameters.Curves1 and
basedon differentapproaches
to estimatingthe errorsin the conclusions
coefficients•SC,mand•SS,
m,are shownasthe coloredcurves 2 in Figure 2 give tidal error amplitudespectracalculated
labeled1-4 in Figure2. ']['hey
stemfromtwocomplementary usingdecorrelationlength scalesof 300 and 600 km, remethods.
spectively,for the LTE forcingerrors. The two curvesare
in goodagreementfor degreeslessthan 10. For higherde3.1 Error estimates from inverse methods
greesthe shorterdynamicalerrordecorrelationlengthscale
The assimilationsolutionTPXO.4 canbe viewedasan op- results(quitesensibly)in somewhatlargererrors. But diftimal interpolation
or smoothingof the T/P altimeterdata, ferencesbetweenthe two casesare not so great;both cross
usinga covariance
determinedby the Laplacetidal equa- the GRACEsensitivitycurveneardegree50.
O
GOT99.2 •
-,"
TPXO.4a •
Figure1. Spherical
harmonicspectra
of threerecentglobaloceantidemodelsasa functionof degreen andorderm. Vertical
axisrepresents
spectralamplitudein seasurfaceheight.
RAY ET AL.: ERROR SPECTRUM FOR THE M2 TIDE
23
derestimate the real errors since identical methods of anal-
ysis were used, and indeedcurve 4 falls below curves 1-3
everywhereexceptfor the very lowestdegreeswhereit rises
abovecurves1 and 2. Exceptfor thesevery low degrees,
curves3 and4 areconsistently
belowcurves1 and2.
10ø E
3.3 Other systematicerrors
ß
•. 10'•-
.
E
o
10 =,.
.
10• 0
.,
10
20
30
40
50
60
degree
Figure 2. Degreeamplitudegeoidspectraof the globalM2
tide, variousestimatesof the present-dayM2 errors,and
the anticipatedsensitivityfrom 3 monthsof data from the
GRACE satellitegravity mission. Error curvesare shown
in color and are numberedas follows: (1) inverseestimate
corresponding
to TPXO.4 assuminga 300-km decorrelation
lengthfor the dynamicalerrorcovariance;(2) sameexcept
with 600-km decorrelationscale; (3) estimatefrom the differencein modelsTPXO.4 and CSR4.o;(4) estimatefrom the
differencein modelsCSR4.oandCSR3.0;(5) errorcausedby
the neglectof earthflattening;(6) errorcausedby useof a
constantdensityfor seawater,with the largererroremploy-
Curves5 and6 in Figure2 exploretheeffectof twopossible systematic
errorsin theestimates
of thetidalgeoidvariations. For this purpose,both assumethat thereis no error
at all in thetidalelevations
• (whicharetakenfrommodel
TPXO.4).Curve5 showsthe errorcausedby the neglectin
Eq. (2) of the earth'sflattening.Specifically,
it is the spectral differencebetweena harmonicexpansion
basedupona
sphericalearth surface(radius 6371 km) and one basedon
a "serrated"ellipsoidalsurface[Jekeli, 1981]. The error is
seento peakarounddegrees5 to 7.
Curve 6 (actuallycomprisingtwo curves)showsthe error causedby the commonassumptionof constantseawater
density,whichallowedPwin Eq. (2) to be outsidethedouble
integral.Theerrorin suchassumption
hasbeencomputed
by
includingPwwithinthespatialintegration
whereit thenrepresentsthe averagedensitywithin the watercolumnat any
location.This densityhasbeencomputedfrom the Levims
et al. [ 1994] climatologicaldensities.The two errorcurves
correspond
toassuming
Pwiseithera constant
1025kgm-3
or1035kgm-3. Thelatterismorerepresentative
ofthetrue
ingdensity
1025kgm-3 andthesmaller
themorerealistic globalmeanseawaterdensity. The formeris morerepre1035kgm-3.
sentativeof surfacewaters,andit inducesthe largererrors,
actuallycrossing
severalof theothercurvesbelowdegree6.
3.2 Error
estimates
from model differences
A moresimplisticapproachto estimatingthe error in the
tidal coefficientsis by computingthedifferencesamongour
three adoptedmodels. Since all are basedon the samealtimeterdata, this approachlikely underestimates
the errors
4. Discussion
Curves1-3 in Figure2 represent
ourbestestimates
of the
errorspectrum
for presentmodelsof the globalM2 ocean
somewhat, but it is still a useful check on the inverse results.
Figure 3 showsthe maximum elevationdifferencebetween TPXO.4and CSR4.o,computedat every point of the
global ocean. Throughoutmost of the open ocean where
Topex/Poseidon
data are of greatestbenefit,the differences
betweenthese two models are small, generally less than
2 cm. In coastalregionswhere the tidal wavelengthsare
shorterandthetideis lesseasilymappedby relativelywidely
spacedsatelliteobservations,
modeldifferencesare significantlylarger.Someof thesedifferencesare alsoin regions
of extremelylargetidal amplitudeswhereevena smallpercentageerrorcanproducelargediscrepancies.
As expected,
modeldifferencesarealsolargein polarlatitudesoutsidethe
satellitelimits; resultsin theseregionsrely completelyon
hydrodynamicmodelingand (possibly)a sparseset of tide
gaugemeasurements.
Figure 3 certainlysuggests
locations
wherefuturemodelingeffortsshouldbe directed.
Curve3 in Figure'2showsthedegreeamplitudespectrum
computedfrom Figure 3. A similar resultis obtainedfrom
differences
involvingtheGOT99.2model.Curve4 is a spectrum computedfrom the differencebetweenCSR4.oand its
earlierversionCSR3.0.This curveis especiallyproneto un-
30'$
80'S
60'E
120'E
0
180'
i
2
4
120'W
6
8
10
$O'W
20
o'
oll
Figure3. Maximumseasurfaceheightdifference
between
theCSR4.oandTPXO.
4 modelsof theM2 globaloceantide,
in cm. Note the color scale is nonlinear. The maximum
difference(in HudsonStrait) exceeds4 meters.Somebasin
boundary
errors(notin TPXO.4)arediscussed
by Smithand
Andersen[ 1997].
24
RAY ET AL.: ERRORSPECTRUMFORTHE M2 TIDE
tide. Curve 4 is, as expected,an underestimate.The er- Desai,S. D., J. M. Wahr,andY. Chao,Erroranalysis
of empirical
oceantidemodels
estimated
fromTopex/Poseidon
altimetry,
J.
ror spectrumis considerably
flatterthanthesignalspectrum.
Geophys.Res.,102, 25157-25172, 1997.
Abovedegree15 or so,the errorexceeds10% of the signal.
B. D. Cornuelle,
Becausecurve 5 everywherefalls below curves1-4 we Dushaw,B. D., G. D. Egbert,P. F. Worcester,
B. M. Howe,andK. Metzger,A Topex/Poseidon
globaltidal
concludethat errorscausedby neglectingthe earth'sflatmodel(TPXO.2)andbarotropic
tidalcurrents
determined
from
teningare smallerthan the errorsinducedby incomplete long-range
acoustic
transmissions,
Prog.Oceanogr.,
40, 337367, 1997.
knowledgeof the tidal elevations
•. Until furthersignifiEgbert,
G. D., Tidaldatainversion:interpolation
andinference,
cantimprovements
in ( areforthcoming,
thespherical
earth
Prog. Oceanogr.,40, 53-80, 1997.
approximationappearsjustified (althoughthe "serrated"elEgbert,
G. D., A. F.Bennett,
andM. G. G. Foreman,
Topex/Poseilipsoidapproximationis nearlyas simpleto implementas
dontidesestimated
usinga globalinverse
method,
J. Geophys.
the sphericalapproximation).Similarly,errorsinducedby
Res., 99, 24821-24852, 1994.
W. A. andH. Moritz,PhysicalGeodesy,
Freeman
and
usinga constantseawaterdensityin (2) can be ignoredif Heiskanen,
Co., San Francisco, 1967.
one adoptsa sufficientlyrepresentative
densitynear 1035
Jekeli, C., The downwardcontinuationto the Earth's surfaceof
kgm-3;a density
of 1025kgm-3 induces
errors
exceeding
curves 1 and 2 when n < 5.
truncated
spherical
andellipsoidal
harmonic
series
ofthegravity
andheightanomalies,
Rep.323, Dept. Geodetic
Science,
Ohio
In the contextof the upcomingGRACEgravityfield mis-
State Univ., 1981.
C., M. L. Genco,F. Lyard,P.Vincent,andP.Canceil,
sion,weobserve
thattheexpected
GRACE
sensitivity
curve Le Provost,
Spectroscopy
of theworldocean
tidesfroma finiteelement
hyin Figure2 crosses
abovethetidalsignalcurveonlyat dedrodynamic
model,J. Geophys.
Res.,99, 24777-24797,1994.
grees
above
56.Thisimplies
thatocean-tide
forcemodeling LeProvost,
C., A. F.Bennett,
andD. E. Cartwright,
Oceantidesfor
onthesatellite
mustemploymuchhigherdegree
spherical andfromTopex/Poseidon,
Science,
267,639-642, 1995.
harmonic
expansions
thanis traditionally
donein satellite Levitus,S., R. Burgett,andT. Boyer,WormOceanAtlas1994,
U.S. Dept.Commerce,
Washington,
1994.
Ray,
R.
D.,
A
global
ocean
tide
model
fromTopex/Poseidon
alMoreimportantly,
weobserve
thatpresent-day
tidemodel
timetry:
GOT99.2,
NASATech.Memo.209478,58pp.,1999.
errorsexceed
theexpected
GRACEsensitivity
at all spheri- Shum,C. K., et al.,Accuracy
assessments
of recentglobalocean
calharmonic
degrees
belowabout40 or50. It is important tidemodels,J. Geophys.
Res.,102,25173-25194,1997.
geodesy.
to acknowledge,
however,the temporalmismatch
in these Smith, A. J. E. and O. B. Andersen,Errors in recentoceantide
curves.TheGRACEsensitivity
corresponds
to a long-term models:possible
originandcause,Prog.Oceanogr.,
40, 325(3-month)estimate
of thegravityfield;thetidalerrorcurves 336, 1997.
Tsaoussi,
L. S. andC. J. Koblinsky,An errorcovariance
modelfor
correspondto mean errors over 12.4 hours. To someextent
theeffectof thetideerrorswill bemitigated
by temporal
seasurface
topography
andvelocityderivedfromTopex/Poseidonaltimetry,J. Geophys.
Res.,99, 24669-24683, 1994.
averaging. To what extent is still unclear. The tide errors Wahr,J.,M. Molenaar,
andF.Bryan,Timevariability
of theEarth's
willcertainly
aliasintolonger,
non-tidal
periods
(depending gravityfield:hydrological
andoceanic
effectsandtheirpossible
detection
usingGRACE,J. Geophys.
Res.,103, 30205-30229,
onthesatellitesamplingcharacteristics).
Whethertheerrors
1998.
couldseriously
hindertheestimation
of thegravityfieldat
thelevelof theinstrument
precision
orhinder
thegeophys- R. J. Eanes,CSR, Univ. Texas,3925 W. BrakerLane, Austin,
icalinterpretation
of thegravityfieldtemporal
variations
is TX 78759-5321
anobjectof ongoingstudy.
G. D. Egbert,COAS, OregonStateUniversity,Corvallis,OR
97331-5503
References
N. K. Pavlis, RaytheonITSS, 7701 GreenbeltRd., Greenbelt,
MD 20770
R. D. Ray, NASA/GSFC, Code 926, Greenbelt,MD 20771;
Andersen,
O.B.,P.L. Woodworth,
andR.A. Flather,
Intercompar-richard.ray@gsfc.nasa.gov
isonof recentoceantidemodels,
J. Geophys.
Res.,100,2526125282, 1995.
(ReceivedApril 9, 2000; accepted
October4, 2000.)