JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B5, PAGES 10,039-10,054, MAY 10, 1997
Marine gravity anomalyfrom Geosatand ERS 1 satellitealtimetry David T. Sandwell Instituteof Geophysics andPlanetaryPhysics, ScrippsInstitution of Oceanography, Universityof California,La Jolla
Walter H. F. Smith Geosciences Laboratory,NOAA, SilverSpring,Maryland
Abstract.CloselyspacedsatellitealtimeterprofilescollectedduringtheGeosatGeodeticMission(-6 km) andthe ERS 1 GeodeticPhase(8 km) areeasilyconvertedto gridsof verticalgravitygradientand gravityanomaly.The long-wavelength radialorbiterroris suppressed belowthenoiselevelof the altimeterby takingthe along-trackderivativeof eachprofile. Ascendinganddescending slopeprofiles aretheninterpolated ontoseparate uniformgrids. Thesefour gridsarecombinedto formcomparable gridsof eastandnorthverticaldeflectionusinganiterationschemethatinterpolates datagapswith minimumcurvature.The verticalgravitygradientis calculateddirectlyfrom the derivativesof the verticaldeflectiongrids,while Fourieranalysisis requiredto constructgravityanomaliesfrom thetwo verticaldeflectiongrids. Thesetechniques areappliedto a combinationof high-density datafrom the densemappingphasesof GeosatandERS 1 alongwith lower-density buthigher-accuracy profilesfrom theirrepeatorbitphases.A comparison with shipboard gravitydatashowstheaccuracyof the satellitederivedgravityanomalyis about4-7 mGalfor randomshiptracks.The accuracy improvesto 3 mGal whentheshiptrackfollowsa GeosatExactRepeatMissiontrackline. Thesedataprovidethefirstview of the oceanfloor structures in manyremoteareasof theEarth. Someapplications includeinertial navigation, predictionof seafloordepth,planningshipboard surveys,platetectonics, isostasy of volcanoesandspreading ridges,andpetroleumexploration. previous work [Heiskanen and Moritz, 1967; Briggs, 1974; Haxby et al., 1983; Sandwell, 1984; Freedmanand Parsons,
Introduction
Radar altimeter measurementsof the marine geoid collected
1986; Haxby and Weissel,1986; Roest,1987; Smithand Wessel, 1990; Rumreeland Haagmans,1990;Haxby and Hayes, 1991; Sandwell,1992;Laxonand McAdoo,1994;Hwang and Parsons, over all the oceanbasins[Haxby et al., 1983]. However,because 1996]. We thenassess the qualityof the griddeddataandsuggest of insufficienttrack density,it has taken 16 years for the full possiblewaysto improvethe resolutionof the gravityfield along potentialof the satellitealtimetermethodto be realized. The stackedprofiles. Finally, we providea tour of the gravityfield of high-densitycoverageobtainedby ERS 1 duringits geodetic the oceansandpointout a few of thenew andinterestingfeatures. mappingphase(April 1994 to March 1995) promptedthe U.S. We do not attemptto review othermethodsfor recoveryof shortNavy to declassifyall of the Geosataltimeterdataon June22, wavelength gravity information from satellite altimetry [e.g., 1995. We are grateful to the European Space Agency for Wakker et al., 1993; Cazenave et al., 1995; Andersen et al., 1995;
duringthe Seasataltimetermissiongavemarinegeodesists and geophysicists a hopeof uncovering thedetailsin thegravityfield
extending theERS 1 mappingphasesothatan equatorialground Hwang and Parsons, 1995] but instead focus on our recipe, trackspacingof 8 km couldbe completed.The combination of discusshow it wasdeveloped,andexplainwhy certainprocesses thesetwo high-densitydatasetsprovidedthe first detailedview are used. Over the next months and years our recipe may of all the ocean basinsat a 10-km resolution. Consideringthe undergorevisions,andperhapsotherswill devisebetterrecipes. sparseshipboard coverageof manyoceanareas[Smith,1993], This researchhastwo main components.The first component, thesenew altimeterdata are arguablythe most importantmarine presentedin this manuscript,is a traditionaljournal article with geologyandgeophysics datasetcollectedoverthepastdecade. appendices.Appendix A coversthe relationshipbetweengeoid The focusof this paperis on the efficientrecoveryof marine height,verticaldeflection,gravity gradient,and gravity anomaly. gravityanomalies andotherderivatives of thepotentialusingdata Appendix B coversthe statisticalmethodto estimatenorth and from satellitealtimetershaving different orbital inclinationsand eastverticaldeflectionsfrom along-trackslopes.AppendixC has differentnoisecharacteristics; no attemptis madeto recoversea the derivationof analytic formulae to calculatethe approximate surface topography(i.e., geoid height plus ocean dynamic satellitepositionand velocity as a functionof time sinceequator topography).After an introductionto radar altimetryand its crossingand latitude. The secondpart of our researchis more inherentlimitations, we discussdata availability and our recipe for constructing gridded gravity anomalies from altimeter
profiles.The method(recipe)presented hereis basedlargelyon Copyfight1997by theAmericanGeophysical Union. Papernumber96JB03223. 0148-0227/97/96JB-03223 $09.00
suitablefor electronicmediaandwill be submittedfor publication to Earth Interactions. It containsmany color images of the marine gravity field (http://topex.ucsd.edu); the images are layered so the reader can zoom in on features of interest. In addition, the companion electronic article contains all of the computercodeusedto constructthe resultsshownin both papers as well as links to files of gridded gravity anomaly, vertical gravity gradient, and high-quality postscriptplots of gravity
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Geosat
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wavelengthgravitysignals(equation(A9)). Consideran anomaly on the ocean floor with a 16-km wavelength and a 15-mGal amplitude(i.e., a typical value for oceans).On the oceansurface this anomaly will be reduced to 3.1 mGal by upward continuation.
The
second
limitation
is due to the short-
wavelengthnoise from oceansurfacewaves (typically > 1 m). The radarpulsereflectsfrom an areaof oceansurface(footprint) that grows with increasing sea state [Stewart, 1985]. The superposition of thereflectionsfromthisareastabilizestheshape of the echo,but it alsosmoothsthe echoso that the timing of its leadingedgeis moreuncertain.By averagingmanyechoes(1000 Hz) over multiple repeatcycles,one can achievea 10 to 20-mm rangeprecision[Noreus 1995; Yaleet al., 1995]. Over a distance of 4 km (i.e., 1/4 wavelength)this corresponds to a sea surface slope error of 4 grad and a gravity error of about 4 mGal (AppendixA). The combinationof thesetwo limitationsmakesit difficult to improvethe resolution.Considertrying to improve the resolutionby a factorof 2, upwardcontinuationwill attenuate
thesignalby an additional factorof e2 = 7.4, soassuming a Gaussian error distribution, the number of measurementsmust be
increased bya factorof e4 = 55. At longer wavelengths orin shallower water the situation is not as bad, but nevertheless the
Figure 1. A pulse-limitedradaraltimeterorbitsat an altitudeof about 800 km and measures the distance to the closest ocean
surfaceby recordingthe traveltime of a pulse. A globaltracking networkalongwith preciseorbitcalculations basedon theJGM-3 gravity model[Neremet al., 1994] is usedto establishthe height of the satelliteabovethe referenceellipsoid(dashedcurve). We assumethe sea surfaceheightabovethe ellipsoidis equal to the geoid height so permanentsea surfaceslopesassociatedwith currentswill appearasfalseanomaliesin our gravitysolution.
anomaly at the General BathymetricChart of the Oceansmap scale. The companionelectronicpaper also containslinks to scriptsillustratingthe useof the gravityanomalygridswith GMT software[Wesseland Smith, 1995].
Satellite Altimetry
roughnessof the oceansurfacelimits the accuracyof the shortwavelengthslopeestimate. We showthat oceantides,which can also have a short-wavelengthcomponentespecially on the continentalshelves,are a third limiting factor in the recoveryof the gravity field. It is surprisingthat otherfactorssuchas basinscaledynamicoceantopographyare not important;in this case the slopeof this erroris usuallylow (e.g., 1 m over say 3000 km which translatesinto 0.33 mGal). Sharperoceanographic steps associatedwith westernboundarycurrentscan be a significant error source[Gille, 1994].
The repeatperiodof the satelliteorbit governsthe spacingof the altimetertrackson the oceansurface(Figure 2). Very long repeatcyclessuchas 168-dayERS 1 geodeticphaseor the nonrepeat (drifting) orbit of the Geosat/Geodetic Mission (Geosat/GM) provide the high-density coverage needed for complete resolutionof the gravity field. The shorter repeat periodsof 10 daysfor TOPEX, 17 daysfor Geosat,and 35 days for ERS 1 do not provide densetrack coverage. However, the repeatedprofilescan be averagedto improvethe signal-to-noise ratio as well as to assessthe noise propertiesof the altimeter measurements.
Here we do not use the TOPEX
or POSEIDON
A satellitealtimeterusesa pulse-limitedradar to measurethe altimetermeasurements becausethe wide track spacingprovides little new information. altitude of the satelliteabove the closestsea surfacepoint to a very high precision(Figure 1). Global precisetrackingcoupled with orbit dynamic calculations provide an independent Along-Track Preprocessing measurementof the height of the satellite above the ellipsoid. The differencebetweenthesetwo measurementsis equal to the The startingpoint for the Geosatdata are geophysicaldata instantaneous sea surface height minus any delays in the records (GDR) from National OceanographicData Center propagation of the radar echo due to the ionosphere and [Cheney et al., 1991] where the orbital information for the troposphere.There are many errorsin thesemeasurements that GeodeticMissiondatawasupgradedto the JointGravity Model 3 are discussedand evaluatedin a number of excellent papers (JGM-3) orbits[Neremet al., 1994]. The ERS 1 oceanproduct [Neremet al., 1994; Tapleyet al., 1994;Ma et al., 1994; Visseret (OPR) data [Dumontet al., 1995] were usedwith their original al., 1993]. Most of these errors occur over length scales of orbits as were the GeosatExact RepeatMission (Geosat/ERM) greaterthan a few hundredkilometers. They are importantfor data. The along-trackprocessingof the raw geophysicaldata preciseoceanographic studiesor studieswherethe geoidheightis recordsconsistsof a numberof steps: edit, apply corrections, needed. However,becausewe are only interestedin the gradient divide into passes, low-pass filter, resample at 5 Hz, and of the seasurface,the short-wavelengthaltimeternoisedominates differentiate.The editingcriteriawere established by examining the error budget. raw altimeterdatafrom a varietyof satellitesandundera variety There are at least two factors that impose limits on the of conditions. Care is taken to remove outliers in areas of ice or accuracyand resolutionof gravity field recoveryfrom satellite land prior to any filtering. In the openoceanwe try to retain all altimetry. First the oceandepth (-4 km) attenuatesthe short- pointsexceptwhen the shapeof the returnpulsesuggests that the
SANDWELL
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20'N
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Figure 2. (a) Tracks of stackedGeosat/ERM (17-day repeat cycle) (22.5o-25ø N), Geosat/GM (20o-22.5' N), ERS 1 GeodeticPhase(168-dayrepeatcycle)(17.5ø-20 øN), andstackedERS 1 (35-dayrepeatcycle)(15ø-17.5N). Thesetracks showdata remainingafter editing and filtering. (b) Vertical gravity gradient(i.e., curvatureof oceansurface)around Hawaii derived from all four data sets. Contoursat 50 and 100 Eotvosunits are shownto highlight seamount/island signatures.
significantwave height (SWH) is high; thesedata are usually noisy [Yale et al., 1995]. Unfortunately, much of our documentation
on the choice of edit thresholds is located in a
varietyof old notebooksandtapesandthuscannotbe completely justified.
comparedwith the bestfit line. If its deviationfrom the line was more than 5 times the rms about the line, then the point was eliminated,a new bestfit line was computed,andthe pointswere testedagainuntil they all passedor until only six pointsremained.
Our primaryfocusis on the recoveryof the short-wavelength Our editingapproach examinestheindividual10-Hz samples gravity field information,and thusnot all correctionsare relevant
in relation to the best fit straightline throughthe 10 pointsas describedby Cheneyet al. [1991]. If the rms aboutthe line is greaterthan0.15 m for Geosat (0.25 m for ERS 1) or the SWH exceeds8 m for Geosat (6 m for ERS 1), then all 10 points are edited. In addition,a new 2-min griddedland/watermask was developedusing the Generic Mapping Tools (GMT) software package[Wesseland Smith,1995] to eliminateframesnearland where stray echos can contaminate the ocean data, especially when low-passfilters are applied. Finally, each 10-Hz point was
or evenuseful. For example,corrections basedon globalmodels (i.e., wet troposphere,dry troposphere,ionosphere,and inverted barometer)typicallydo not have wavelengthcomponentsshorter than 1000 km, and if their amplitudevariationsare lessthan 1 m they do not contributemore than 1 grad to the noise. Some correctionssuch as the electromagneticbias can actually add noise[Gille, 1994]. We have found that the most important correction to the sea surface slope is the ocean tide and have
determined thattheCenterfor SpaceResearch (CSR) composite
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Figure 3. Absolutevalue of tide model slope[Eanesand Bettadpur,1995]. Contoursare at 0.5, 1, 2, 3, 4, and 5 grad. One grad of tidal slopeerror will map into -1 mGal of gravity anomalyerror. Tide modelsare mostimportanton some shallow continentalmargins.
oceantide modelV3.0 [Bettadpurand Eanes,1994; Eanesand Bettadpur,1995] is superiorto the tidessuppliedwith eitherthe GeosatGDRs or the ERS 10PR. Figure 3 showsthe rms of the slopeof theCSR V3.0 tide correctionfor a 35-dayrepeatcycleof ERS 1. Tidal slopecorrectionsare usuallysmall over the deep ocean but can be up to 6 grad over some of the shallow continentalmargins.Previousversionsof the globalgravitygrid did not incorporatethis tide modelcorrectionand sufferedfrom track line noiseon the shallowareassurroundingGreat Britain as
After editing and application of corrections, the data are brokeninto passeswhenevera time gap exceeds2 s (-14 km) or there is a changein pass orientation (e.g., from ascendingto descending). The passes are low-pass filtered to suppress altimeternoisehaving wavelengthsshorterthan 5 km for Geosat and 10 km for ERS 1 as shownin Figure 4 (solid curves)and the filtered heights are subsampledat 5 Hz. Note that the 1-Hz boxcarfilter (dashed-dotcurves)followedby a 1-Hz decimation (i.e., the normal 1-Hz data suppliedon both the GDR and OPR)
well as on the Falkland Plateau;the CSR V3.0 tide model vastly
will cause the sinc function sidelobes to fold from wavenumbers
improvesthegravityfield recoveryin theseandotherareas.
greater that 0.075 (13.4 km wavelength) to much longer wavelengthsso that there is more than 20% aliasing at 20 km wavelength; thus a boxcar filter should not be used and the samplingrate shouldbe greaterthan2 Hz andpreferably5 Hz to retainall signalswith wavelengthsgreaterthan20 km. The final andperhapsmostimportantstepin the preprocessing is to differentiatecontinuousprofilesalongtrack with respectto time. As shownin previousstudies[e.g., Sandwelland Zhang, 1989], this suppresseslong-wavelengtherrors and reference frame shiftsso data adjustmentsareunnecessary.For simplicity, and to retain the ability to integratethe profiles,a first difference is used for the slope estimates. For example,the difference in heightbetweenpoints1 and 2 is dividedby their time difference and the slopeis storedin location 1. This introducesa 1/2 phase shift (670 m) that is removed later with a secondalong-track
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A study of the along-track resolutioncapabilitiesof stacked Geosat, ERS 1, andTOPEX altimeterswasrecentlypublishedby Yale et al. [1995] using the coherencemethod developed by Figure 4. Prestackfilters usedon ERS 1 and Geosataltimeter Brammer[ 1979] andMarks and Sailor [1986]. Exampleprofiles data (solid). The poststackfilter (dashed)is appliedto all stacked are shownin Figure 5 and an overview of the resultsis given in and nonstackedprofilesprior to griddingandhasa 0.5 gain at a Table 1. All available data were loaded into three-dimensional wavelengthof 18 km. The 1-Hz boxcarfilter (dashed-dot)is not suitable for resolving wavelengths shorter than about 30 km files where repeat profiles were aligned along track. Outliers were detected by a comparison with the median of available because of the large-amplitude sidelobes and slow roll-off. Filters were designed using the Remez routine in MatlabTM cycles, and then the stack was computedas the average of the software. The routine is based on the Parks-McClellan finite survivors. To determinethe resolutionimprovementgained by stacking,the first half of available repeatprofiles was averaged impulseresponsefilter design. 0
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SANDWELL AND SMITH' MARINE GRAVITY
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Figure5. Individual andstacked vertical deflection profiles fora trackcrossing theMid-Atlantic Ridge: (left)Geosat, (middle) ERS1,(right) TOPEX.Onlyeight oftheavailable cycles areshown foreach satellite [fromYaleetal.,1995]. independently from the secondhalf prior to the coherence Gridding: Accumulation,Interpolation, analysis. Two areaswere initially selectedfor coherence and Gravity Conversion analysis: theequatorial Atlantic (Area1inTable1),aregion with of griddedgravityanomalies from hightectonic signal andlowoceanographic noise; andtheSouth Ourrecipefor construction is lessthan Pacific(Area2 in Table1), a regionwithlow tectonicsignaland altimeterprofilesworksbestwhenthetrackspacing resolution of thealtimeter data.Themethodwas high oceanographic variability. In all cases,along-track thealong-track large radial orbit error, longresolution is betterin the equatorial Atlanticthanin the South designedto accommodate
tide modelerror,andshiftsin referenceassociated Pacific;Geosatand TOPEX dataresolveshorterwavelengths wavelength than ERS 1. Global maps of along-trackresolutionshow with different tracking networks. In addition, it can data with differing accuracy,resolution,and pass considerable geographic variation.On averageglobally,the accomrnodate orientation (Appendix B andFigure6). Finally,thealgorithm is along-track resolution (0.5 coherence) of GeosatandTOPEX sotheentireword canbegridded ona workstation in stacks areapproximately thesame(24km),whiletheresolution fastenough and 68 of ERS 1 stacksare slightlyworse(30 km). However,when a reasonabletime (-2 days)(206 million observations million grid cells). We start with along-track slope estimates that equalnumbers of repeatcyclesarestacked, all threealtimeters andstacked asdescribed above.Thegoalis to have aboutthe sameresolutionlimit of 28-30 km. The resolution werepreprocessed northandeastverticaldeflection gridsthatareconsistent estimatesshow that the shortestwavelengthrecoverablein the produce to withintheirassigned noiselevel gravityfieldfromsatellite altimetry is about20 km. Later,we withtheoriginalobservations
applyanalong-track filtertoalloftheprofile datawitha 0.5gain and whereunconstrainedcells reflect nearbyvalues. The basic at a wavelength of 18km (Figure4, dashed curves).Thenafter stepsare(Figure7) thefollowing. the northand eastverticaldeflectiongridsare created,they are
1. Edit nonstacked databy comparingthe heightvalue(i.e.,
slope)with a Gaussian-filtered prediction fromthe low-pass filtered in two-dimensions withafilterhaving a 0.5gain integrated
surrounding points(0.5 gainat 36 km). If the suspect height
at 19 km.
Table 1. Summaryof Along-TrackResolutionEstimates Area
Global Average
Area 2
1
Cycle
Stack8
Stack31
Cycle
Stack 8
Stack 31
Geosat
33
26
20
52
38
27
ERS 1
38
26
-
50
33
TOPEX
34
24
19
43
31
Units in kilometers
_
23
Cycle
Stack 8
Stack 31 24
38
29
43
30
37
28
_
22
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Figure6. Propagation of along-track vertical deflection errors fromdense Geosat andERS1 profiles intonorthandeast components of verticaldeflection.At theequator, satellite tracksrunmainlyN-S sotheE-W component of vertical deflection ismore poorly determined thantheN-Scomponent. Thiscovariance information isused inboththeblending during theiteration (Figure 7) andthefiltering following theiteration (Figure8).
estimatedifferedby morethan0.3 m fromtheprediction thenthat pointwaseditedandtheslopeprofilewasrecomputed. 2. Filter all of the along-trackdatawith the samefilter (0.5 gain at 18 km wavelength,Figure 4, dashedcurves)to ensurea commonbandwidth. The cutoff wavelengthis basedon the coherenceanalysisof Yale et al. [1995] which showsthat the shortestpossiblewavelengthresolvableis about20 km. These two editing/filteringstepswere not executedin the version6.2
gravityfield whichresultedin a verynoisygravitymodel.
3. Removefromtheprofilesa reference geoidmodelbasedon JGM-3 [Neremet al., 1994]to degree70 wherecoefficients are cosinetaperedbetweendegrees50 and 70. This samefield is restored later on; the reference field is removed so the vertical
deflectionto gravityanomalyconversion canbe accurately performedwith a flat-Earthapproximation.
4. Bin dataon an equidimensional Mercatorgrid withcell dimensions of 2 min in longitudeandcos(O)x 2 min in latitude. Theworldis dividedinto54 overlapping areasthatare1440cells
SANDWELL AND SMITH'
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wherez0,0is the interpolated valuebasedon the surrounding
ERS1
I
values zi,j.Steps (6) through (8) arerepeated (Figure 7) untilthe
(hit)
I
valuesof unconstrainedgrid cells convergeand the interpolated grids vary smoothlyto zero in continentalareas. Exit from the iterationafter step(6). 9. The final stepis to apply an isotropic,low-pass,convolution
Reset•
filter [kei(x/2r/a)]to the northandeastgridsof vertical IAscending I IDescending I I Ascending I IDescending I deflection. The filter width a is proportionalto the fourthroot of
ERS 1 I
the relative error shownin Figure 6 in an attemptto equalizethe noise level between the north and east slope grids. The exponentialupwardcontinuationmodelpredictsthat halving the
errorincreases theresolution byx/2.Anadditional x/2resolution
/
Figure 7. Flow diagramfor constructing northandeastgridsof vertical deflectionfrom ascendingand descendingalong-track slope profiles. Iteration providesa communicationamongthe diverse data sets[Menke, 1991].
wide (48ø)and 1296 cellstall, sothereis about400 km of overlap on the perimeters.The areasarelargeenoughsothatthe longest wavelengthsignalremainingin the residualprofiles(-800 km) is
is gainedby averagingtwice the amountof data. This is an armwaving argumentfor usingthe fourthroot of the relativeerror as the filter radius. In practice,the fourth-rootmodel was found to work best. The wavelengthat whichthe isotropicfilter attenuates the signal by 0.5 is shown in Figure 8. For example, at the equator the ERS 1 and Geosat tracks provide relatively poor controlon the east componentso the 0.5-gain wavelengthis 26 km while the 0.5 gain for the northcomponentis 20 km. After generatinggrids of north and east vertical deflection, various other derivatives of the gravitational potential can be computed. In all cases one should restore the appropriate derivativesof the sphericalharmonicreferencemodel that was removedin step3. For example,the vertical gravity gradientis the
sum
of
the x derivative
of the east vertical
deflection
component and the y derivative of the north component(equation (A6)) (Figure 2). The gravity anomaly is computed using (A10). The accuracyof this flat-Earth approximationis relatedto the cutoff wavelengthof the sphericalharmonicmodel removed (800 km for complete removal at degree 50). This requiresFourier transformationof each vertical deflectiongrid, combination in the wavenumber domain, and inverse Fourier
transformation of the sum. If the grid pixelsare equidimensional, thenthe transformation from verticaldeflectionto gravitylargely passorientationare binnedinto thesegrid cells;unconstrained avoids latitude-dependentlength-scaleproblems [Haxby and cells are left with an empty flag. Binning is done using the Hayes, 1991]. For example,the operatoron the Fouriertransform medianvalue within the bin, and stackedprofiles are counted5 of the eastcomponent of verticaldeflectionis kx/Ikl,sothe length timessothey will dominatethe median. scale largely cancels. Maximum error introduced by this 5. Fill emptycellswith reasonable valuesfor thefirstiteration approximationwill correspondto the changein lengthscalethat using a weighted averageof surroundingdata based on an
less than the area dimension (> 2000 km). Data with common
(a + r)-3weight function wherea setsthewidthof thekerneland r is distance.
6. Blendpass-oriented gridsusingequation(B8) to form grids of north and eastvertical deflection. Note that eachdata type is
28
assigneda different weight accordingto expectednoiselevel
26
(ERS 1 noise= 1.41 x Geosatnoise). Also note that the blending operationaccounts for the passorientationasillustratedin Figure 6.
east slope
•24
v
._c
• 22
7. Decomposethe northand eastgridsinto the originalpassorientedgridsusingequations (B 1) and(B2). 8. Resetbins constrainedby along-trackslopeobservationsto theiroriginalvaluesunlesstheydeviatefrom predictionby more thana threshold(15 grad,Geosat;21 grad, ERS 1) in whichcase, edit the bin. Use biharmonicoperator[Briggs, 1974;Smithand Wessel,1990]to interpolateemptycellsandgo to step6:
;2o •18 •16
•14 12
2 (z],0+ z0,]+z-i,0+ z0,-]) ZO,O = •1 (Zl,1-t-Zl,-1 -t-Z-l,1 -t-Z-l,-1 )
10
1 (z0,2 + z2,0 +z-2,0 + z-0,-2 )
20
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(1)
Latitude(deg)
Figure 8. Cutoff wavelength(0.5 gain) of isotropickei filter used to equalizethe noiselevel in the north andeastverticaldeflection grids.
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occursover a latitude range correspondingto about 400 km of distance. At the equatorthis changeis less than 0.2%, while at 70øN this changeis 6%. We expectthat the actualerrorsare smallerthan this becausemostof the contributionto the gravity anomaly will be nearby. The error due to this "flat Earth" approximationis reducedas the sphericalharmonicdegreeof the referencemodel is increasedas long as there are no errorsin the referencemodel [Haxby and Hayes, 1991]. Also note that to safelyavoid edgeeffectsthe dimensionsof the verticaldeflection grids must be severaltimes larger than the longestwavelength remaining in the residual data. The final step in the gravity computationis to restorethe gravity anomalycorresponding to the sphericalharmonicmodelremovedin step3.
Accuracy and Resolution The accuracyandresolutionof gravitygridsconstructed using a similar approach were recently established through a comparison with accurate shipboard gravity measurements [Neumannet al., 1993]. For a small region along the southern Mid-Atlantic Ridge where there is a denseshipboardsurveyand large gravity anomalies (140 mGal total variation), the rms difference is 7-8 mGal when only Geosat/GM data are used. Marks [1996] has recently compared seven well-navigated shipboardprofileswith thislatestversionof our grid aswell as a Geosat-onlygrid [Marks et al., 1993] to assessthe resolutionof the satellite-derivedgravity and improvementgainedby adding ERS 1 altimeterdata. She found agreement(0.5 coherence)to wavelengthsof 23-30 km for this Geosat/ERS1 grid and 26-30 km for the Geosat-onlygrid; rms differencesrangedfrom 3 to 9 mGal. This resolution analysis compareswell with estimates derivedfrom repeattrack analysis[Yale et al., 1995] and further justifiesthe use of along-trackand two-dimensionalfilters to cut (0.5 gain) wavelengthsshorterthan20 km. We alsocompareshipboardgravity profileswith the satellitederivedgravity grid and find that individualship profilesshow
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rmsdifferences of 3-6 mGaldepending on proximityto a stacked altimeterprofile. The first two examplesarefrom shipprofiles that follow the track line of two Geosat/ERMprofilesin the SouthAtlantic [Jungand Vogt, 1992]. Along the Conrad2802 profile,the shiptrackdeviatesfrom the Geosattrackline at-20.5 ø latitude to avoid a small island. The mean difference between the
Geosat-derived gravityandthe shipboard gravityis -0.57 mGal, and the rms difference is 2.76 mGal when the island/seamount
data are omitted(Figure 9a). The differenceat the crestof the seamountis about 50 mGal, which is probablydue to lack of short-wavelength componentsin the satellite-derivedgravity field. The secondexample(easternSouthAtlantic) crossesthe
Walvisridgeandhasa similarrmsmisfit(3.03mGal)buta large mean difference(16.82 mGal) (Figure 9b). This large mean difference probably reflects an error in the ship gravity measurement at the tie point, or perhapsno tie pointcorrection wasmade.It shouldbe notedthata 3-mGalerrorcorresponds to a relativeheightaccuracyof only 15 mm over a distanceof 5 km
(i.e., 1/4 of theresolution wavelength).Considering thattypical surfacewave heightsare a meteror moretall, thisis a remarkable achievement.When repeatingprofiles are stacked,the vertical deflectionerror decreasesas the squareroot of the numberof profilesusedin the stack[ Yale et al., 1995].
To establish the accuracyof the satellite-derived gridin areas away from repeat altimeterprofiles, we selecteddata from two well-navigated Conrad cruises on the western flank of the northernMid-Atlantic Ridge (Figure 10a). Note thisis the same ship, gravimeter,and navigationsystemthat was usedfor the SouthAtlanticcomparison (Figure9); thusdifferenceswill reflect proximity to a stackedGeosat/ERM profile. All three cruises stopin the Azoreswherewe assumethe tie pointcorrectionwas established.Three segmentswere tested,c2912 outboundfrom Spainto the Mid-Atlantic Ridge (Figure 10b, mean 13.01 mGal, rms 5.80 mGal), c2912 from Mid-Atlantic Ridge to the Azores (Figure 9c, mean 13.92 mGal, rms 5.84 mGal), and c3001 from the Azoresto the Mid-Atlantic Ridge (Figure9c, mean 12.43
i
i
i
-20
-15
-10
150
50
0 -5O
I
-25
•
lO
'
mean=-0.5'7mGal ! rms=2.•6mGal ,t ' ,
E
c•
. ,
0
I
ß• -lO
'
-25
-
-1
'
'
-10
Latitude(deg) Figure 9a. Comparison betweena shipboard gravityprofileConrad2802in thewesternSouthAtlantic(solid[seeJung and Vogt,1992])withgravityprofilealonga trackcorresponding to a 62-foldstackof Geosataltimeterprofiles(dashed). Thedifference wasnotcomputed at -21.5øSlatitudewheretheshiptrackdeviates fromthesatellitetrackto avoidanisland.
SANDWELL AND SMITH: MARINE GRAVITY 100
'
10,047
i
'
•5
\X,..I I \ "'1 \// -16 -14
\1
i
!
mean ='16.82 mGal
-22
i
rms =3.03mGal
lO
ß
o
ß ,.• t•.
•.-. •..'•, •.,• .•q,•s..,: '" ,,. '
, ..
,.'.
,',
.
•
-lO
-14
-32
-22
Latitude(deg) Figure 9b. Similar comparisonfor Conrad2711 alongGeosat/ERMsuborbitaltrackin the easternSouthAtlantic.
mGal, rms 6.70 mGal). Note that for a short distancethe ship of 13 mGal betweenthe cruisesc2912 and c3001 suggests that tracksare nearly coincidentand thereis a goodmatchamongall thereis a 13-mGalerrorin a commontie point. The implication six profiles(Figure 10c). From thesecomparisons we conclude is that one could correctthe DC level of all shipboardgravity the following: (1) Away from a stackedGeosat/ERMprofilethe profilesthroughcomparisonwith the gravity grid [Wesseland accuracyof the satellite-derivedgravity field degradesto 4-7 Watts, 1988; Smith and Sandwell, 1995]. Moreover, instrument mGal. Given a typicalcombinedGeosat/ERS1 track spacingof drift and other more subtle errors such as navigation-induced about2.5 km and a 1/2 wavelengthresolutionof 10 km thereis a error could be corrected. redundancyof aboutfour tracksovermostof the grid. This can Based on these ground truth analyses,one could expect be comparedto a typical redundancyof 40 along the stacked perhapsa factorof 2 improvementin the accuracyof the satelliteGeosat/ERM profile [Yale et al., 1995]. Thus a 3.2 times derivedgravityfield by collectingat least4 timesmoredata. The improvementin accuracy near the stackedprofile is to be Geosat/GMandERS 1/GM missionsreflect2.5 yearsof data,so expectedbut only a 2 timesimprovementis observed;of course a 10-year dedicatedmappingmissionwould be needed. In the oceandepthand typical seastateare othermajor considerations. deepoceanstheupwardcontinuation causesthe amplitudesof the (2) The goodmatchbetweenthe satellite-derived gravityandthe anomaliesto decreaseexponentiallywith decreasing wavelength, shipboardgravity over the sharp 350-mGal anomaly at a so a factorof 2 improvementin accuracywould only yield a 1.4 longitudeof 344ø (Figure 10b) demonstrates thereare no major increase in resolution to about 18 km. Such a resolution increase problemswith either the dynamicrangeof the altimeteror the would be geophysicallysignificant,especiallyfor hydrocarbon dataprocessing.(3) The systematic disagreement in meanlevel explorationwherethe currentsatellite-derivedgravity fields are
35øN
30øN
40øW
35øW
30øW
25øW
20øW
15øW
10øW
5øW
Figure 10a. Comparisonbetweensatellite-derived gravityandgravityalongrandomshiptracks. (a) Locationof Conrad 2912 Spainto Azoresto Mid-AtlanticRidgeandrepeatingtrackbackto Azores. Locationof Conrad3001 from Azoresto Mid-AtlanticRidgealongsametrackline.
10,048
SANDWELL
AND SMITH:
MARINE
GRAVITY
4OO
i
300
•E200 100 0
320
325
330
335
340
345
i
i
i
i
i
i
35O
I
355
i
ß
•
mean =13.01 mGal rms =5.84 mGal =1
20 !
•
II
.
ß •'
ß
-i
-20
:!
I
I
I
i I
I
I
I
320
325
330
335
340
345
350
355
Longitude(deg) Figure10b.Comparison alongoutbound trackof C2912,outbound.
marginally useful.Perhapsa newtechnology suchasa scanning determine yourposition asa function of time. (Of course, your laseraltimeteror a multibeamradaraltimetercouldprovidethis startingpositionand velocitymustalso be known.) If the enhanced gravityfield in a shortertimeperiod. windowsof your vehicleare closed,a true accelerationcannotbe
distinguished froma variationin thepull of gravity.Thusthe gravity data are needed for correction of inertial navi-
Applications
gation/guidance systems.The militaryapplications areobvious and providedthe rationalefor the $80 million costof the Geosat
Navigation
mission aswell astheclassification of thesedata,especially
TheGeosatdatawerecollected by theU.S.Navyto fulfill their during the cold war when nuclearsubmarineswere more active navigationalandmappingrequirements.Considermeasuring thantheyaretoday.On thecommercial side,Honeywell Inc.is accelerationsin a moving submarineor aircraft in order to usingthesedatatoupdate theirinertialnavigation systems aboard
i
i
i
i
i
i
i
•200 150 ß
50 0
318
20
I
I
I
I
I
I
I
I
320
322
324
326
328
330
332
334
i
!
i
i
i
mean =13.92 mGal I
rms =4.20 mGal
0
-20
318
i
i
320
322
i
i
i
324 326 328 Longitude(deg)
i
i
i
330
332
334
Figure10c.Comparison wherethreeshiptracksarenearlycoincident (top,c3001'middle,c2912outbound; bottom, c2912 inbound).
SANDWELL
AND SMITH:
aircraft. In particular,whenthiscorrection is not applied,they havefoundlargenavigationalerrorsalongPacificOceanflight pathswhichfollow the majoroceantrenches(K. Vanderwerf, Honeywell Inc., Coon Rapids, Minnesota, personal communication, 1995).
MARINE
GRAVITY
10,049
chainswhich are believedto be formed as the plate movesover a stationarymantleplume. The hot plumeheadmeltsthe mantle rocks which erupt on the surfaceas a hot spot. Becauseall of thesemajorfeaturesare evidentin the gravitymaps,the geologic history of the oceanbasinscan now be establishedin greater detail.
Prediction of Seafloor Depth Undersea
Volcanoes
We are usingthesedensesatellitealtimetermeasurements in combinationwith sparsemeasurementsof seafloor depth to constructa uniform resolutionmap of the seafloortopography [Smithand Sandwell,1994]. Thesemapsdo not have sufficient accuracy andresolution to beusedto assess navigational hazards, but they are usefulfor suchdiverseapplications as locatingthe obstructions/constrictions to the majoroceancurrentsandlocating
The global gravity grids reveal all volcanoeson the seafloor greaterthan about1000 m tall. Approximately30-50% of these volcanoes were not charted previously. One of the more importantaspectsof thesenew data will be to locateall of these volcanoesand identify spatialpatternsthat may help determine how they formed. Many volcanoesappear in chains, perhaps shallowseamounts wherefish and lobstermay be abundant.An associatedwith mantleplumes,but there are many more that do intermediate stepin the depthpredictionprocessdetermines the not fit this simple model. Moreover, numerous undersea correlationbetweengravity and depthand a numberinversely volcanoesare long linear ridgeswith aspectratiosof 20 or more. proportional to the seafloordensitycontrast;theseparameters These featuressuggestthat the plates are not exactly rigid as predictedby the simpleplate tectonictheory. Using thesedata, maybe usedasproxiesfor sediment thickness. On a broadscalethe topography of the oceanfloor reflectsthe we are exploring the internal deformations of the plates, coolingandsubsidence of theplatesastheymoveawayfromthe especiallyoutboardof trencheswherethe forcesgeneratedby the platesare greatest. spreading center.While thisprocess is fairly well understood, slab-pullforceof the subducted
thereareinterruptions in thisnormalsubsidence caused by mantle Petroleum Exploration plumesandothertypesof solid-state convection in themantleof the Earth.
Planning Shipboard Surveys
The satellite-derivedgravity grids reveal all of the major structures of the oceanfloor havingwidthsgreaterthan 10-15 km. This resolutionmatchesthe total swathwidth of the muchhigher resolution(100 m) multibeammappingsystemon a ship so the
All of the majorpetroleumexplorationcompaniesusesatellite altimetergravity data from Geosatand ERS 1 to locateoffshore sedimentary basins in remote areas. This information is combined with other reconnaissancesurvey information to determine where to collect or purchasemultichannel seismic surveydata. Currently,the regionsof most intenseexploration interest are the continental
shelves of Australia
and the former
SovietUnion; recently,companieshave expressedinterestin the gravitymapsaretheperfectreconnaissance tool for planningthe CaspianSea. Developmentsin offshoredrilling technologynow more detailed shipboardsurveys. Scientistsaboardresearch make it economicalto recoveroil from continentalslopeareasin vesselsusethe gravitygridsalongwith othermeasurements to wateroncethoughtprohibitivelydeep. optimizetheirsurveystrategy; in manycasesthisis donein real While we are not directly involved in this activity, we fill data time. The costto operatea researchvesselis typically $20,000 requests from many large exploration companies including
perday,sothesegravitydatahavebecomeanessential item.
Plate Tectonics
These satellite altimeter data provide an important and definitiveconfirmationof the theoryof plate tectonics. Indeed, almost everythingapparentin the marine gravity field was
Unocal, Amoco, Exxon, Arco, Mobil, Texaco, Shell, Conoco, and
British Petroleum, as well as many smaller exploration companies. Lithospheric Structure There are numerousotherscientificapplicationsthatcannotbe
createdby the formationand motionof the plates. Spreading describedin a shortreport. One of the traditionalusesof marine ridgesare characterized by an orthogonal patternof ridgesand gravitymeasurements is to estimatethe thicknessof the elastic transformfaults. The scarproducedin the activetransformvalley is carriedby seafloorspreading out ontoolderseafloor,leaving
evidenceof the pastplate motions. The Indian Oceantriple junction(27øSlatitude,70øElongitude) is a textbookexampleof seafloorspreading.The satellite-derived gravityfield showsthe intersectionof the threespreadingridgesas describedby plate tectonic theory. The theory predictsthat the ridges would intersectat 120ø anglesif the three ridgeswere spreadingat exactly the same rate. In this case, one can measurethe intersectionanglesand infer the relativespreadingratesof each ridge. Plates are created at spreading ridges and destroyed (subducted)at the deepoceantrenches. All of the major ocean trenchesare evidentin the gravity map as linear troughs. The
deepoceanbasinsaway from the trenchesare characterized by fracturezone gravity signaturesinheritedat the spreadingridge axis. This patternis sometimesoverprintedby linear volcanic
portion of the tectonicplates [Watts, 1979]. When a volcano formson the oceanfloor it impartsa large downwardload on the plate causingit to deform. This deformationappearsin the gravity field as a donut-shapedgravity low surroundingthe gravityhigh associated with the volcanoitself. By measuringthe amplitudeand width of the gravity low and relating this to the size of the volcanoas measuredby a shipwith an echosounder, one can establishthe thicknessand strengthof the elasticplate. The new satellite-derived gravity data enable researchersto perform this type of analysiseverywherein the oceans. Thus scientistscan now probe the outermostpart of the Earth using these and other methods.
A griddedfile of gravity anomalies(version7.2) is available by anonymousftp (baltica.ucsd.edu).A large-format,laminated posteris availablefrom the ScrippsInstitutionof Oceanography Geological Data Center. Also visit our web site http://tøpex'ucsd'edu/mar-grav'html'
10,050
SANDWELL AND SMITH: MARINE GRAVITY
computationwhich does not involve sphericalharmonicsor Fourier transforms. Indeed, given two orthogonal satellite altimeterprofiles,theverticalgravitygradientat theirintersection The geoidheightN(x) andothermeasurable quantitiessuchas point is the sum of the curvaturesof each profile times the gravity anomalyzig(x) are relatedto the gravitationalpotential averageaccelerationof gravity. The simplicityof this calculation V(x,z) [Heiskanenand Moritz, 1967]. We assumethatall of these is particularlydesirablefor computingthe gravity gradientnear quantitiesare deviationsfrom a sphericalharmonicreference coastlineswhere the altimeterprofilesterminate; the calculation Earth model so a flat-Earth approximationcan be used for the of the vertical gravity gradientfrom (A6) has no edge effect, gravitycomputation (A10). In the followingequations theboldx while the Fourier computationof the gravity field can have a denotes thecoordinate (x,y);similarly k denotes (kx,ky)wherekx significantedgeeffect. = 1/&,where •,xis wavelength. To a first approximation, the In contrastto the simple formulationof the gravity gradient, geoid heightis relatedto the gravitationalpotentialby Brun's computationof the gravity anomalyis much more difficult and formula, error prone. Following Haxby et al. [1983], the differential equation(A5) is reducedto an algebraicequationby Fourier
AppendixA' GeoidHeight, Vertical Deflection, Gravity Gradient,and Gravity Anomaly
transformation.
N(x)_= 3-V(x,0)
(A•)
go
The forward and inverse Fourier transforms are
defined as 7oo
F(k)=l_ool_•f(x)e-i2•k'x)d2x (A7)
where goiStheaverage acceleration ofgravity (9.81ms-2).The gravityanomalyis theverticalderivativeof thepotential
aV(x,0)
zig(x) = -
(A2)
The Fourier transformof (A6) is
•zig(k,z)
theeastcomponent of verticaldeflectionis the slopeof thegeoid
•z
= -i2•rgo [kxr/(k)+ ky•(k) 1.
(A8)
in the x direction
r/(x) -- •N_ -1•V 3x
go 3x
From the solution to Laplace's equation in the wavenumber domain the upward continuation formula relates the gravity anomaly at the surface of the Earth to the gravity anomaly at
(A3)
some elevation z.
andthe northcomponentof verticaldeflectionis the slopeof the geoidin the y direction.
•(x)-- 3N_ -13V _
3y
.
go •y
Ag(k,z)= Ag(k,0)exp(-2•{klz)
(A9)
where Ikl= 1/•x 2+•y2. Taking thederivative of(A9)with respect (A4)to z andevaluatingthe resultat z = 0, one arrivesat an algebraic formula relatingthe Fourier transformof the gravity anomalyto the sum of the Fourier transformsof the two componentsof
These quantitiesare relatedto one anotherthroughLaplace's
vertical deflection:
equation
Ag(k,0) = igo[kxr/(k) +ky•(k) 1. +
•x 2
= 0.
+
•y 2
(AlO)
(AS)
3z2
To compute gravity anomalies from a dense network of satellitealtimeterprofilesof geoidheight,oneconstructs a grid of
Substitutionof (A2), (A3), and(A4) into Laplace'sequation(A5)
eastr/and north• verticaldeflection(AppendixB). The grids
yieldsa relationship betweentheverticalgravitygradientandthe sum of the x and y derivativesof the east and north vertical
are then Fourier transformedusing a discreteapproximationto (A7). Finally, one performsthe multiplicationsgiven in (A10) and inverse Fourier transforms the result to obtain gravity anomaly. At thispointonecouldalsoaddthe sphericalharmonic gravity model back to the gridded gravity values in order to recoverthe long wavelengthgravity field.
deflection
-
go
+
ß
(A6)
This expressionis used to computevertical gravity gradient (Figure 2) from grids of east and north vertical deflection [Rummel and Haagmans, 1990]. Note that this is a local
Appendix B: Vertical DeflectionsFrom Along-Track Slopes To avoidany adjustmentof the data,ascendinganddescending satellitealtimeterprofilesare first differentiatedin the along-track
SANDWELL AND SMITH: MARINE GRAVITY direction resulting in geoid slopes or along-track vertical deflections. These along-track vertical deflections are then
combined to produceeastr/and north• components of vertical
10,051
nearly perpendicularso that the east and north componentsof geoidslopehavethe samesignal-to-noise ratio. When two or more satellites with different
orbital inclinations
deflection [Sandwell, 1984]. Finally, the east and north vertical deflectionsareusedto computebothgravityanomalyandvertical
are available,the situationis slightlymorecomplexbut alsomore stable. Consider the intersection of four passesas shown in
gravitygradient(AppendixA). The algorithmusedfor gridding the altimeterprofilesis an iterationschemethat relieson rapid transformation from ascending/descendinggeoid slopes to
below.
north/east vertical deflection
and vice versa.
2
Consider for the
4
momentthe intersectionpoint of an ascendingand a descending satellitealtimeterprofile. The derivativeof the geoid heightN with respectto time t alongthe ascending profileis
3•a=aNa_ aNOa ß+ •aN;a at
30
(B1)
3q)
andalongthe descending profile is 1
•
aN;d
3
(B2)
The along-trackderivativeof eachpasscanbe computedfrom the geoid gradientat the crossoverpoint
where is0 geodetic latitude and• islongitude. Thefunctions • and½arethelatitudinal andlongitudinal components of the
P•I
satellitegroundtrackvelocity(AppendixC). It is assumedthat the satellitealtimeterhas a nearly circularorbit so that its velocity
=
dependsmainlyon latitude;at the crossover pointthe following relationships areaccurateto betterthan0.1%.
Oa =-Od
q)a= q)d.
(B3)
(Analytic expressionsfor satellite velocities are given in AppendixC usingorbit parametersin Table C1.) The geoid gradientis obtainedby solving(B1) and(B2) using(B3).
3N
I (/•a+ ß
_ 1 (/•a-/•d).
(B4)
(BS)
ao It is evident from this formulation
.
3N
'
(B6)
P•4 or in matrix notation
lq= © VN.
(B7)
Sincethisis an overdetermined system,thefour along-trackslope measurements cannot be matched exactly unless the measurements are error free. In addition,an a priori estimateof the errorin the along-trackslopetyi measurements canbe usedto weight each equation in (B6) (i.eo, divide each of the four equationsby tYi). The leastsquaressolutionto (B7) is
vm=((•t(•)-lot1•
(U8)
that there are latitudes where
where t and -1 are the transpose and inverse operations, respectively. In this casea 2 by 4 systemmustbe solvedat each crossoverpoint, althoughthe methodis easily extendedto three or more satellites. Later we will assumethat every grid cell • goes tozeroandthus(B5)becomes singular. In theabsence of correspondsto a crossoverpoint of all the satellitesconsidered,so noisethisis not a problembecausethe ascendinganddescending this small systemmustbe solvedmany times. profilesare nearlyparallelsothat theirdifferencegoesto zero at In additionto the estimatesof geoid gradient,the covariances the samerate that the latitudinalvelocity goesto zero. Of course, of these estimates are also obtained: in practice,altimeter profiles containnoise, so that the north componentof geoid slopewill have a signal-to-noise ratio that (B9) decreasesnear+72ølatitude. Similarly, for an altimeterin a near either the eastor north componentof geoidslopemay be poorly determined.For example,at +72ølatitudethe SeasatandGeosat altimetersreachtheir turningpointswherethe latitudinalvelocity
polar orbit the ascendingand descendingprofiles are nearly antiparallelat the low latitudes; the east componentof geoid slope is poorly determinedand the north componentis well determined. The optimal situationoccurswhen the tracks are
We use the covariance estimates to low-pass filter the north componentof geoid slope differently from the east component
10,052
SANDWELL AND SMITH:
MARINE GRAVITY
depending on latitude.An exampleof covariance estimates for a combination of GeosatandERS 1 is shownin Figure6 wherethe
andlatitudinal velocity.Assuming theEarthis anoblateellipse withflattening f, theconversion fromgeocentric 0c to geodetic
standard deviation of ERS 1 was set to 1.4 times the standard
latitude 0 is
deviation of Geosat.SinceGeosatandERS1 arehigh-inclination satellites, theestimated uncertainty of theeastcomponent is about 3 times greaterthan the estimateduncertaintyof the north
tan0 = (l-f)-2 tanOc.
(C1)
componentat the equator. At higherlatitudesof 600-700where
andat thepoles,thetwolatitudes areequal,butat thetracksarenearlyperpendicular, thenorthandeastcomponentsAt theequator latitudes(e.g.,450) theydifferby up to 0.20. The are equally well determined. At 72øN where the Geosat tracks intermediate of (C1) withrespect to timeprovides thecorrection to runin a westerlydirection, theuncertainty of theeastcomponent derivative from the geocentric is low and the higher-inclination ERS 1 trackspreventthe the latitudinalvelocitywhenconverting systemto the geodeticsystem. estimateof thenorthcomponent frombecoming singularat 720. Finally, the east r/ and north • componentsof vertical deflectionare relatedto the two geoidslopes[Heiskanenand 0 = (1_f)-2cos20 (C2)
Moritz, 1967]by
Oc
r/-
1 aN
cøs20c
(B10)
Equationsfor the relative positionof the satelliteversustime
a cos0 a•p
werederivedfollowingKaula[1966]. Thebasicproblemis to map the positionof a satellitein a circularorbit aboutthe Earth
• = 1aN
(Bll)
a 30
intoanEarth-fixedcoordinate system.Let t = 0 be thetime when
the satelliteorbit crosses the Earth'sequatorialplaneon an ascending passat a longitudeof •Po.To developformulas,one first representsthe position of the satellite in a Cartesian
coordinate system q wheretheqxaxisis thelineconnecting the
where a is the mean radius of the Earth.
center of theEarthandtheascending equator crossing. Theqz Appendix C: Approximate SatellitePosition and Velocity
axisis perpendicular to theorbitplaneandtheqy axisis orthogonal totheqxandqzaxes.Inthisframe theqx,qyandqz
positions arecos(cost ), sin(cost), and0, respectively.Next the satellite frame is rotated about the qxaxisby theinclination of the The exact satellitegroundtrack velocitycouldbe calculated planeI. A third directlyfrom thegroundtrackprofilessuppliedwith the satellite orbitplanerelativeto theEarth'sequatorial rotation abouttheEarth's spinaxismapsthesatellite planeintoan altimeter data records. However, later on we will need to
evaluate (B1), (B2), and (B6) at grid cells that were not Earth-fixedsystem. This final rotationinvolvesthe rotationrate relative totheprecessing orbitplaneroe'=roe- con. necessarilyintersectedby a satelliteprofile. Thusit is desirable of theEarth
thethreerotations andtransforming theresults tohaveaccurate formulae forcomputing 0 and• versus latitude. Afterperforming
coordinates intospherical coordinates, oneobtains In addition,the accumulation of the along-track repeatprofiles fromCartesian expressions for the latitude and longitude versus time. The into the stackfiles requirean approximatetrajectoryfor the geocentriclatitude is satellite.Thuswe deriveexpressions for boththe approximate position and velocity of a satellite in a circular orbit about an
Oc(t)= sift1[sincost sinI ]. (C3) ellipsoidal Earth. The important parametersare the orbit frequencyco s, theEarthrotationrateroe, theprecession rateof the orbit planeaboutthe Earth'sspinaxis con,the inclinationof the latitude is converted to geodetic latitude using satelliteorbit/, theflattening of theEarthf, thestarting longitude Thisgeocentric equation (C1). In addition, (C3) can be inverted to yield the time of the satelliteC)o,the geocentriclatitude0c,and the geodetic sincetheequatorcrossing latitude0. Numericalvaluesof theconstant parameters aregiven in Table C1.
To attainthedesiredlevelof approximation, it is necessary to accountfor the flatteningof the Earthwhencomputing latitude
t(Oc)= co•l sin-1 [sin0c ]. L sin I
Table C1. Orbit Geometric Parameters r
Description
GEOS 3
Seasat
Geosat
TOPEX
ERS 1
ms orbitfrequency, s-1 On precess frequency, s-1
1.0420 x 10-3 -4.143 x 10-7
1.0407x 10-3 -6.743x 10-5
1.0407x 10-3
9.3143x 10-4
1.0379x 10-3
(roe - co s) 17/244
(roe - co s) 10/127
(roe- Cøs) 35/501
I
114.980 ø
108.0584 ø
108.0584 ø
66.010 ø
98.5557 ø
inclination
roe Earth rotation frequency, s-1 7.29212 x 10-5 f
flattening
1/298.25
a
earth radius, rn
6371000
go
acceleration ofgravity, rns-2 9.81
(C4)
SANDWELL AND SMITH: MARINE GRAVITY The cosineand sineof the longitude(relativeto •Po)at somelater time are given by
cosq(t) =[cos co•t cos COst +sin COe't sin COst cos I (C5)
cos Oc(t)
and
sin•t) =
-sinCO•t cosCOst + cosCOe't sinCOst cosI
ß(C6)
cos Oc(t)
10,053
References Andersen,O. B., P. Knudsen,and C. C. Tscherning,Investigationof methodsfor global gravity field recovery from the denseERS-1 GeodeticMission altimetry,paperpresentedat IUGG XXI General Assembly,Int. Union of Geol. and (3eophys.,Boulder,Colo., July, 1995.
Bettadpur,S. V., and R. J. Eanes,Geographicalrepresentation of radial orbit perturbationsdue to oceantides: Implicationsfor satelliye altimetry,J. Geophys. Res.,99, 24883-24898,1994. Brainmet,R. F., Estimationof theoceangeoidneartheBlakeEscarpment using(3eos-3satellitealtimetrydata,J. Geophys. Res.,84, 3843-3860, 1979.
By combiningthesetwo expressions, the longitudeat a later time is
Briggs,I. C., Machinecontouring usingminimumcurvature,Geophysics, 39, 39-48, 1974.
Cazenave,A., B. Parsons,and P. Calcagno,Geoid lineationsof 1000 km wavelengthoverthe CentralPacific,Geophys.Res.Lett., 22, 97-100, 1995.
q(t) =tan-l[-sin co•t cos cost +cos coe't sin cost cosl I+•po(C7 ) cosCO•t cosCOst +sinCOe't sinCOst cosI
Given theseequationsfor positionversustime, one can derive expressions for the latitudinalandlongitudinalcomponents of the satellite velocity versus latitude. The latitudinal velocity is obtainedby differentiating(C3) with respectto time and using (C4) to relatevelocityto latitudeinsteadof time. The resultis
Oc(Oc)=COs 1 cos2i 1/2. cos20c
Cheney,R. E., N. S. Doyle, B.C. Douglas,R. W. Agreen,L. L. Miller, E. L. Timmerman,and D.C. McAdoo, The CompleteGeosatAltimeter GDR Handbook,Natl. (3eod.Surv./Natl. Oceanicand Atmos.Admin., Silver Spring,Md., 1991. Dumont, J.P., F. Ogor, and J. Stum, Quality Assessment of Cersat
AltimeterOPRProducts: 168-DayRepeatPeriod,FrenchProcessing andArchiveFacility, Toulouse,France,1995. Eanes,R. andS. Bettadpur,The CSR 3.0 globaloceantide model,Center for SpaceResearch,TechnicalMemorandum,CSR-TM-95-06, 1995. Freedman,A. P., and B. Parsons,Seasat-derivedgravity over the Musicianseamounts, J. Geophys.Res.,91, 8325-8340,1986.
(C8) Gille,S. T., Meanseasurfaceheightof theAntarcticcircumpolarcurrent from (3eosatdata: Methodand application,J. Geophys.Res., 99, 18255-18273, 1994.
Of course,the sign of the velocity will dependon whether the satellite profile is ascending(+) or descending(-). To convert from geocentricvelocityto geodeticvelocity,(C2) is used. The longitudinalvelocityof the satelliteis mosteasilydeterminedby usingthe fact thatthe total angularvelocityof the satellite(in the satelliteframe)is constant(COs). Thenthelongitudinalvelocityof the satellite relative to the Earth is
COs cosI
o•.
(C9)
cos20c
Haxby,W. F., andD. E. Hayes,Free-airgravityof the SouthernOcean derived from Seasat and Geosat altimeter data: Circum-Antarctic to
30øS.Antarct.Res.Ser.,volo54,MarineGeological andGeophysical Atlasof theCircum-Antarctic to30øS,editedby D. E. Haynes,pp. 1119 A(3U, Washington,D.C., 1991.
Haxby,W. F., (3. D. Karner,J. L. LaBrecque, andJ. K. Weissel,Digital imagesof combinedoceanicand continentaldata setsand their usein tectonicstudies,Eos Trans.AGU, 64 (52), 995-1004, 1983. Haxby, W. F., and J. K. Weissel, Evidence for small-scaleconvection
fromSeasataltimeterdata,J. Geophys. Res.,91, 3507-3520,1986. Heiskanen,W. A., andH. Moritz, PhysicalGeodesy,W. H. Freeman, New York, 1967.
To establish the accuracy of these approximate satellite velocities,(C8) and (C9) were comparedwith the trajectoryof a GeosatExact RepeatMissionprofile [Cheneyet al., 1991]. The model velocities lie within 1 grad/s of the actual velocites [Sandwell,1992]. The greatesterror in total velocity occursat 72 ø latitude
where the difference
is 7 rn/s or 0.1%.
Other
Hwang,C., andB. Parsons, An optimalprocedure for derivingmarine gravityfrom multi-satellite altimetry,J. Geophys. Int., 125, 705-718, 1996.
Hwang, C. W., and B. Parsons,Gravity anomaliesderivedfrom Seasat, Geosat,ERS-1andTopex/Poseidon altimetryandshipgravity:A case studyovertheReykjanesridge,Geophys. J. Int., 122,551-568, 1995. Jung,W. Y., andP. R. Vogt, Predictingbathymetryfrom (3eosat-ERM
numerical tests show that the position estimatesfrom (C3) and andshipborne profilesin the SouthAtlanticOcean,Tectonophysics, (C7) are accurateto better than 1 km as long as the predicted 210, 235-253, 1992. positionis lessthat 1/4 an orbit from theknownequatorcrossing Kaula, W. K., Theoryof SatelliteGeodesy,Blaidsel,Waltham, Mass., 1966. position. Laxon,S., andD. McAdoo,ArcticOceangravityfieldderivedfromERSAcknowledgments. We thank Bruce Douglas,Bob Cheney,Laury Miller, and Russ Agreen for their safe storageand distributionof the GeosatGM GDRs; no byteswerelost. ERS 1 datawereprovidedby ESA throughtheFrenchProcessing andArchiveFacility. Mara Yale supplied stackedprofilesof GeosatERM and ERS 1 data. RichardEanesandC. K. Shumprovidedthe softwareneededto computethe tide corrections. SteveNeremprovidedtheJGM-3 gravityfield coefficients thatwereused as a referencemodel. Dick Rapp suppliedsoftwareto constructthe referencegravity model. Countlessengineersand scientistsworked to make both the (3eosatand ERS 1 missionsgreat successes. This paper benefited from the reviews of Steve Nerem and Anny Cazenave. This researchwas supportedby the NASA (31obal(3eophysicsProgram NA(3W-3035
and UNOCAL
Co.
1 satellitealtimetry,Science,265, 621-624, 1994.
Ma, X. C., C. K. Shum,R. J. Eanes,andB. D. Tapley,Determination of ocean tides from the first year of TOPEX/POSEIDON altimeter measurements, J. Geophys.Res.,99, 24809-24820, 1994.
Marks,K. M., Resolution of theScripps/NOAA marinegravityfield from satellitealtimetry,Geophys.Res.Lett., 23, 2069-2072, 1996. Marks, K. M., and R. V. Sailor, Comparisonof (3eos-3 and Seasat altimeterresolutioncapabilities,Geophys.Res.Lett., 13, 697-700, 1986.
Marks, K. M., D.C. McAdoo,andW. H. F. Smith,Marine Gravity Derivedfrom Geosat,Natl. (3eophys.Data Cent., Boulder,Colo., 1993.
Menke,W., Applications of thePOCSinversion methodto interpolating
10,054
SANDWELL AND SMITH: MARINE GRAVITY
topography andothergeophysical fields,Geophys. Res.Lett., 18, 435438, 1991.
Nerem, R. S., et al., Gravity model development for the TOPEX/POSEIDON: Jointgravity models1 and 2, J. Geophys.Res., 99, 24421-24447, 1994.
Neumann,G. A., D. W. Forsyth,andD. Sandwell,Comparison of marine gravityfrom shipboardand high-densitysatellitealtimetryalongthe Mid-AtlanticRidge,30.5ø-35.5øS, Geophys. Res.Lett., 20, 1639-1642, 1993.
Noreus, J.P., Improvedresolutionof Geosataltimetry using dense samplingandpolynomialadjusted averaging, Int. J. RemoteSens.,16, 2843-2862, 1995.
Roest,W. R., Seafloorspreadingpatternof the Noah Atlantic between 10ø and 40øN, Geol. Ultraiectina, 48, 1-121, 1987.
Rummel, R., and R. H. N. Haagmans,Gravity gradientsfrom satellite altimetry,Mar. Geod.14, 1-12, 1990. Sandwell, D. T., A detailed view of the South Pacific from satellite
altimetry,J. Geophys. Res.,89, 1089-1104,1984. Sandwell,D. T., Ant_arctic marinegravityfield fromhigh-densitysatellite altimetry,Geophys. J. Int., 109,437-448, 1992. Sandwell, D. T., and B. Zhang, Global mesoscalevariability from the Geosatexactrepeatmission:Correlationwithoceandepth,J. Geophys. Res., 94, 17971-17984, 1989.
Stewart,R. H., Methodsof SatelliteOceanography,Univ. of Calif. Press, Berkeley,1985. Tapley, B. D., D. P. Chambers,C. K. Shum,R. J. Eanes,J. C. Ries, and R. H. Stewart, Accuracy assessmentof large-scaledynamicocean topography fromTOPEX/POSEIDON altimetry,J. Geophys. Res.,99, 24605-24617, 1994.
¾isser,P. N. A.M., K. F. Wakker, andB. A. C. Ambrosius,Dynamicsea surfacetopographyfrom Geosataltimetry,Mar. Geod. 16, 215-239, 1993.
Wakker, K. F., M. C. Naeije, E. Wisse, R. Scharroo,P. Visser, and B. Ambrosius,GeosatandERS-1 radar altimetryover the North Atlantic, Adv.SpaceRes., 13, 305-314, 1993. Watts, A. B., On geoidheightsderivedfrom Geos3 altimeterdataalong the Hawaiian-Emperorseamountchain,J. Geophys.Res., 84, 38173826, 1979.
Wessel,P., andW. H. F. Smith,New versionof genericmappingtools, Eos Trans. AGU Electron. Suppl., Aug. 15, 1995. (Available as http://www.agu.org/eos_elec/95154e.html) Wessel, P., and A. B. Watts, On the accuracy of marine gravity measurements, J. Geophys.Res.,93, 393-413, 1988. Yale, M. M., D. T. Sandwell,andW. H. F. Smith,Comparisonof alongtrack resolution of stacked Geosat, ERS-1, and TOPEX satellite
altimeters,J. Geophys.Res.,100, 15117-15127,1995.
Smith,W. H. F., On the accuracyof digitalbathymetric data,J. Geophys. Res., 98, 9591-9603, 1993.
Smith,W. H. F., andD. T. Sandwell,Bathymetricpredictionfron dense satellitealtimetryand sparseshipboardbathymetry,J. Geophys.Res., 99, 21803-21824, 1994.
Smith, W. H. F., and D. T. Sandwell, Global comparisonof gravity anomalies measured by ships and derived from dense satellite altimetry (abstract),Eos Trans.A GU, 76(16), SpringMeet. Suppl.,
D. T. Sandwell,ScrippsInstitutionof Oceanography, IGPP-0225, University of California, 9500 Gilman Drive, La Jolla, CA 92093-0225. (e-mail:
[email protected]) W. H. F. Smith, GeosciencesLaboratory,NOAA, NOES-12, Silver Spring,MD 20910-3281. (e-mail:
[email protected]. gov)
S89, 1995.
Smith, W. H. F., and P. Wessel, Gridding with continuouscurvature splinesin tension,Geophysics, 55, 293-305, 1990.
(Received June3, 1996; revised October 11, 1996;
acceptedOctober15, 1996.)
