Ocean calibration approach to correcting for spurious accelerations for data from the GRACE and GRACE Follow-On missions
aa r X i v : . [ phy s i c s . s p ace - ph ] J un Journal of Geodesy manuscript No. (will be inserted by the editor)
Ocean calibration approach to correcting for spuriousaccelerations for data from the GRACE and GRACEFollow-On missions
Peter L. Bender · Casey R. Betts
Received: date / Accepted: date
Abstract
The GRACE mission has been providing valu-able new information on time variations in the Earth’sgravity field since 2002. In addition, the GRACE Follow-On mission is scheduled to be flown soon after the end oflife of the GRACE mission in order to minimize the lossof valuable data on the Earth’s gravity field changes. Inview of the major benefits to hydrology and oceanogra-phy, as well as to other fields, it is desirable to investi-gate the fundamental limits to monitoring the time vari-ations in the Earth’s gravity field during GRACE-typemissions. A simplified model is presented in this paperfor making estimates of the effect of differential spu-rious accelerations of the satellites during times whenfour successive revolutions cross the Pacific Ocean. Theanalysis approach discussed is to make use of changes inthe satellite separation observed during passages acrosslow latitude regions of the Pacific and of other oceans tocorrect for spurious accelerations of the satellites. Thelow latitude regions of the Pacific and of other oceansare the extended regions where the a priori uncertain-ties in the time variations of the geopotential heightsdue to mass distribution changes are known best. Inaddition, advantage can be taken of the repeated cross-ings of the South Pole and the North Pole, since theuncertainties in changes in the geopotential heights atthe poles during the time required for four orbit revo-lutions are likely to be small.
P. L. BenderJILA, University of Colorado Boulder and NIST, UCB 440,Boulder, Colorado, USATel.: 303-492-6793Fax: 303-492-5235E-mail: [email protected]. R. BettsJILA, University of Colorado Boulder and NIST, UCB 440,Boulder, Colorado, USA
Keywords
ECCO-JPL ocean model · mass dis-tribution variations · GRACE Follow-On mission · ocean model accuracy · DART ocean bottom pressuremeasurements · geopotential variations at satellitealtitude In 2002 the Gravity Recovery and Climate Experiment(GRACE) mission was launched by NASA and the Ger-man space agency DLR (Tapley et al. 2004, 2013). Itis based on microwave measurements of changes in theroughly 200 km separation between two satellites onthe same nearly polar orbit and flying at about 500 kmaltitude. This mission has greatly improved our capa-bility for determining the Earth’s gravity field and itschanges with time. Also, in 2009 the Gravity field andsteady-state Ocean Circulation Explorer (GOCE) mis-sion was launched by ESA (Floberghagen et al. 2011),and a large amount of data from it on the short wave-length structure of the Earth’s geopotential at satellitealtitudes also is now available (Gerlach and Rummel2013).The first priority for measurements of time vari-ations in the Earth’s gravity field after the GRACEmission is to fly an improved GRACE- type mission(GRACE Follow-On) with as small a time gap as pos-sible after the end of the GRACE mission (Watkinset al. 2013). To achieve this, it wasn’t possible for theGRACE Follow-On mission to be drag-free, althoughthe mission will include laser interferometry betweenthe two satellites in order to improve the accuracy formonitoring changes in the separation. Other improve-ments in the satellite design will reduce sources of episodicspurious accelerations of the satellites. However, the re-
Peter L. Bender, Casey R. Betts
150 -180 -150 -120 -90
Fig. 1
Groundtracks for four successive upward passes ofthe GRACE or GRACE Follow-On satellites across the Pa-cific Ocean. The circles show the locations of the assumedcalibration sites in the Pacific. sults over one-revolution or longer arcs are still likely tobe limited substantially by incomplete corrections fornoise from the on-board accelerometers that are usedto correct for spurious non-gravitational forces on thesatellites.At present, for GRACE data, empirical parame-ter corrections are used to reduce the effects of spu-rious accelerations. A typical procedure is to solve foronce-per-revolution (once/rev) corrections to the along-track differential acceleration from each one revolutionarc of data, plus a few other parameters such as themean differential acceleration. However, a priori infor-mation on the geopotential height variations along theorbit at satellite altitude have to be used in solvingfor the parameters in the spurious acceleration correc-tions. This means that errors in the a priori geopoten-tial height variation models will result in errors in thefinal GRACE results. In addition, it appears possiblethat the empirical parameter set solved for with presentGRACE data is not sufficient to take out most of theeffect of differential spurious accelerations.An alternate approach that has been suggested (Ben-der et al. 2011, Bender and Betts 2013) is to make useof a priori information on geopotential height variationsonly over regions where such information is known sub-stantially better than at average worldwide locations.The main candidate for an extended region with gooda priori information is the low latitude region of the central Pacific Ocean. In particular, there is one periodper day when four successive revolutions of the satel-lite pair will pass upward across the main part of thePacific, as shown in Figure 1, and then another periodwhen four successive revolutions will pass downwardacross the region. Thus the possibility of using mainlydata over that region in correcting for spurious accelera-tions appears to be quite attractive. However, includingthe data from two or more passes across low latituderegions of the Atlantic and Indian Oceans during thesame time intervals would be desirable. The sites fromwhich the necessary data are obtained will be referredto as calibration sites.To keep the situation as simple as possible, the dis-cussion here will be limited to such four revolution arcsof data, which are assumed to start and end at theSouth Pole. Use also can be made of the fact that mea-surements can be made five times at the South Poleand four times at the North Pole during the chosen fourrevolution arcs for a polar orbiting satellite pair. At analtitude of about 500 km, this takes only 6.3 hours,so the geopotential heights at satellite altitude at thepoles are likely to have changed little during this time.Thus multiple measurements at the poles can help ininterpolating between low-latitude measurements overthe oceans, even if the mean geopotential heights atthe poles during the four revolutions are not accuratelyknown a priori.In discussing the time variations in the geopotential V ( t ), it is useful to have a measure of the changes thathas the units of distance. For this reason, the phrase“variations in geopotential height” will be used to de-scribe those changes. These variations h ( t ) are definedto be h ( t ) = δV ( t ) g , (1)where δV ( t ) is the difference of V ( t ) from a model orconstant value and g is the local acceleration of gravity(Jekeli 1999). A well-known problem in the analysis of GRACE datais the tendency for striping to occur in the results, ifprecautions against this are not taken. Striping can bedescribed as a correlation of the errors along meridians,so that maxima and minima in the errors form ridges inthe north-south direction. It will be assumed here that asubstantial part of the source of this problem is the dif-ficulty of knowing the systematic errors in the satelliteseparation at frequencies near one cycle per revolutionand at other low frequencies. Such once/rev and low- cean calibration approach for the GRACE and GRACE Follow-On missions 3 frequency variations can be caused by differential errorsin the initial conditions for the two satellite orbits, andare continuously modified by spurious acceleration er-rors. The level to which these errors need to be knownto be useful is far better than the accuracy achievableby GPS tracking. Thus variations at once/rev normallyare at least partially filtered out from the data.If the mission has a particular orbit and the resultsare analyzed, the variations in the error at once/rev andat other low frequencies during that period will lead tochanging vertical displacements of the arcs of apparentgeopotential heights with respect to the Earth’s centerof mass. This will cause correlated errors along roughlythe north-south direction in the resulting gravity fieldsolution. Thus, it appears that providing a way to im-prove our knowledge of the systematic errors in sepa-ration at once/rev and at other low frequencies, andof the spurious accelerations leading to them, would behighly desirable. Testing the suggested procedure foraccomplishing this is the main objective of this paper.In investigating this problem, three major approxi-mations will be used. One is the assumption of a verynearly spherical geopotential, with perturbations onlybecause of short period fluctuations in the surface massdistribution. The second is the neglect of tidal effects.And the third is the energy conservation approxima-tion.In the energy conservation approximation (Jekeli1999), the separation S between the satellites decreaseswhenever they fly over a region where the geopotentialheight near the satellite altitude is increased. The usualstatement is that the sum of the potential energy andthe kinetic energy has to stay constant, to the extentthat dissipative forces can be accounted for, so thatfor small changes in the potential energy, the veloc-ity change will be nearly proportional to the potentialchange. Thus, if the potential were nearly spheroidalbut with a bump in it, each of the satellites would haveabout the same decrease and then increase in velocitywhen it crossed the bump, resulting in a decrease in sep-aration of the satellites. Thus, in principle, the changesin the geopotential height can be solved for from mea-surements of changes in the separation, provided thatdifferences in the orbit parameters and uncertainties inother factors can be solved for to the extent necessary.From Eq. (1), the energy conservation approxima-tion can be written as(¯ v ) [ w − w ] + g ∗ [ h − h ] = 0 , (2)where (¯ v ) is the mean velocity over the orbit, w isthe along-track component of the velocity for satellite2, g is the acceleration of gravity at the satellite al-titude, h is the geopotential height of satellite 2, etc. The main amplitudes of the geophysical time variationsin the geopotential height are at degree 10 or less, so toa good approximation[ h − h ] = (cid:20) ( ¯ S )(¯ v ) (cid:21) ∗ dhdt (3)where h is the geopotential height at the midpoint be-tween the two satellites.Inserting (3) into (2) and integrating from t = 0 to t = T gives(¯ v ) [ S ( T ) − S (0)] + (cid:20) ( g ∗ ¯ S )(¯ v ) (cid:21) [ h ( T ) − h (0)] = 0 . (4)Eqs.(2) and (4) are clearly only approximations, andthus corrections to the satellite orbit parameters andoffsets of the velocity vectors from the satellite separa-tion vector have to be allowed for in the analysis. Since g/ (¯ v ) = 1 /a , where a is the semi-major axis of theorbit,[ S ( T ) − S (0)] + (cid:20) ( ¯ S ) a (cid:21) [ h ( T ) − h (0)] = 0 . (5)If we define δS and δh to be the variations in S and h from their initial values, we have: δS = − δhR , (6)where R is given by R = a/S , a is the orbit semi-majoraxis, and R is about 69 for an assumed value of 100 kmfor the mean separation S . This assumed separation isabout a factor of 2 less than the nominal value for theGRACE Follow-On mission, but the separation may bedecreased if the laser interferometry operating mode isbeing used.The geopotential height discussed throughout thispaper will be that near the satellite altitude, for sim-plicity. Thus, after results are obtained for a given pe-riod, they will need to be downward continued to obtainthe results at ground level. The downward continuationcan be done by expressing the results for the geopoten-tial at altitude in a spherical harmonic expansion, andthen reducing the amplitudes to those at the desiredreference surface. To be specific, the polar orbit casethat is used in this paper is a nearly circular orbit with198 revolutions in 13 sidereal days. The altitude is 489km, and there are 15.23 revolutions per sidereal day.The offset in longitude between ground tracks on suc-cessive revolutions is 23.64 ◦ . Attention will be focusedon measurements of the satellite separation when cross-ing a broad region in the Pacific Ocean between 30 ◦ Sand 30 ◦ N latitude. The basic reason is the expectedsmoothness, small amplitude, and low relative uncer-tainty in the time variations in the mass distributionat moderate latitudes over the Pacific. The inverted
Peter L. Bender, Casey R. Betts barometer effect would at least partially reduce the ef-fects of uncertainties in surface atmospheric pressure.And satellite altimetry results plus wind data derivedfrom satellites and other sources can further reduce theuncertainty.The orbit for GRACE does not have a fixed groundtrack, and neither will the orbit for the GRACE Follow-On mission. However, future missions after GRACEFollow-On seem likely to have a fixed ground track,if they are drag-free. Although such missions probablywill fly at considerably lower altitudes, it was decidedto use fixed ground track orbits in the present analysis.For short arc analyses, this is not expected to have anyeffect on the results.The main part of this paper will be based on theexpected spurious acceleration levels for the GRACEFollow-On mission. The main observables consideredwill be the satellite separations over 12 points in thechosen region in the Pacific, the separations at singlepoints in the Indian and Atlantic Oceans, and those atthe poles. The basic set of measurements proposed toestimate the differential spurious accelerations will bedescribed in 3. All of the latitudes referred to in therest of this paper will be north latitudes, and all of thelongitudes will be east longitudes.The results for the evaluation of the errors in thegeopotential heights with the ocean calibration approachwill be evaluated at satellite altitude along the orbit.In this way, the expected accuracy of the results canbe compared with those for similar 4 revolution arcsfrom other approaches for correcting for the spuriousaccelerations of the satellites, such as various empiricalparameter correction approaches, without the seriouscomplication of the limitation from temporal aliasing.The approximation of only using calibration measure-ments at 12 sites over the Pacific is not expected tolimit the accuracy substantially, but keeps the calcula-tions considerably simpler.In Sections 4, 5, and 6 the limitations on the re-sults due to two main causes will be discussed. Onelimitation results from having to interpolate over pe-riods of up to 47 minutes between the times at whichmeasurements are made at the chosen calibration sites.The other is due to uncertainties in the a priori infor-mation on geopotential height variations with time atthe calibration sites.In Section 4, a specific choice of the interpolationprocedure will be discussed. It was chosen because itdoes a good job of reducing the residual effects of thespurious acceleration noise. It is based on a specificmodel of the spurious acceleration noise as a functionof frequency that is intended to be close to what will beachieved during the GRACE Follow-On mission. How- ever, the nominal level of the spurious accelerations forthe GRACE mission is only a factor of three higher, sothe main results are also expected to be applicable tothis case.In Section 5, a specific, but ad hoc, model for themagnitudes of errors in the geopotential heights at thecalibration sites will be described. It is based mainlyon a particular model for time variations in the oceanbottom pressure in the Pacific. Then, in Section 6, theeffects of the uncertainties from this model on the spu-rious acceleration correction proceedure will be given.The results for this case where the expected spuriousacceleration levels for the GRACE Follow-On missionare assumed appear to be quite encouraging. Thus thepossibility that a similar approach could be used forsome of the data from the GRACE mission will be dis-cussed in Section 7. And finally, the general conclusionsfrom this study of the ocean calibration approach willbe reviewed in Section 8.
To keep the necessary calculations as simple as possible,the conceptual approach discussed here is based mainlyon making use of measurements of the satellite separa-tions when the satellites are at − ◦ , 0 ◦ , and +30 ◦ lat-itude over the Pacific and when they are at the SouthPole or the North Pole during the same four-revolutionarc. With the assumed 13 sidereal day repeat satelliteorbit, there will be four successive upward passes of thesatellites from − ◦ to +30 ◦ latitude each day that willcross the equator in the Pacific Ocean between the lon-gitudes of about 150 ◦ and 245 ◦ . These passes will befollowed about six hours later by four downward passesfrom +30 ◦ lat to − ◦ lat, where the equator crossingsare in the same range of longitudes. Thus the majorportions of these passes will be fairly well away fromthe regions of strong western or eastern boundary cur-rents.In addition, recent studies of uncertainties in geopo-tential heights over the oceans indicate that the uncer-tainties over the equatorial region of the Indian Oceanand a substantial range of latitudes in the Atlantic aresimilar to those in the equatorial Pacific (see Figure 1in Quinn and Ponte 2011). From the geometry, it is pos-sible to include two additional measurements near theequator over these oceans on most data arcs, one in theIndian Ocean on the first of the four revolutions and asecond measurement during the last revolution over theAtlantic. Such additional data will be represented hereby assuming additional calibration points when cross-ing the equator during the first revolution in the Indian cean calibration approach for the GRACE and GRACE Follow-On missions 5 Ocean and during the fourth revolution when crossingthe Atlantic.Finally, as discussed earlier, measurements at theSouth Pole and at the North Pole during the same 4-revolution arc will be included. This gives a total of23 calibration points during each four-revolution dataarc, including 12 in the Pacific, 1 each in the Indianand Atlantic Oceans, 5 at the South Pole, and 4 at theNorth Pole.If there were no uncertainty in the geopotential heightvariations at the calibration points in the oceans orat the South or North Pole during the four revolu-tion period, the variations in the satellite separation S due to the spurious accelerations could be interpo-lated from the measured separations at the referencelocations. However, the choice of what type of inter-polation function to use requires some care because ofthe along-track spatial gaps of up to 180 ◦ between someof the measurements. Because 23 measurements at thecalibration points are included during the four revolu-tions, up to 23 basis functions could be used to fit thereference point data. However, it has been found in thisstudy that least squares fitting on a substantially lowernumber of basis functions works better.The reason for this result is that there actually willbe differential noise in the knowledge of geopotentialheight variations at calibration points in the Pacificthat are separated by fairly short distances, down tobelow 21 ◦ . To fit such variations, basis functions withquite short wavelengths are needed, and they can am-plify the effects of the geopotential height uncertaintieswhen applied to the regions where substantial interpo-lation is needed. Thus the number of basis functionsused generally has been limited to about two-thirds ofthe number of calibration points.For convenience, the locations of the reference pointsare given in terms of the angular motion along track ofthe satellites with respect to the South Pole crossing atthe center of the four revolution arc of data. Thus theyrange from − ◦ to +720 ◦ . The crossings of − ◦ , 0 ◦ ,and +30 ◦ latitudes in the Pacific for data sets with up-ward passes there will occur at − ◦ , − ◦ , − ◦ during the first revolution, etc.The first error source to be considered is the resultof incomplete correction for the differential spurious ac-celeration noise between the two satellites. It is assumedthat equal weight will be given to the difference betweenthe observed value and the value of the geopotentialheight at satellite altitude that is used at each of thecalibration sites. Then, if the a priori value at the i -thcalibration site is Y i , an approximation function for the Y i can be defined in terms of M basis functions X j ( θ i )by Y ( θ ) = Σ j ( a j )[ X j ( θ )] . (7)If a trial set of M basis functions is chosen, plus a set of N calibration points, the best fitting set of coefficients a j can be chosen by the least squares approach. Thefunction Y ( θ ) that results from least squares fitting thecoefficients a j will be called the correction function, orthe interpolation function.The starting point for the analysis is a N × M matrix A , called the “design matrix”. Here A( i, j ) is the valueof the j -th basis function at the i -th calibration point. Ifa vector Y of the apparent geopotential height errors Y i at the i -th measurement site is assumed, and H = [A T ] ∗ A , then the vector Z of least squares fitted coefficientsof the basis functions is given by Z = [ H − ] ∗ [ A T ] ∗ Y = K ∗ Y , (8)where this equation defines the matrix K . The asteriskrepresents vector or matrix multiplication.In order to compare different possible choices of thebasis functions, a specific criterion is needed. To keepthe calculations fairly simple, the criterion used here isbased on just considering the residual noise after cor-rection at the times during the four revolutions whenthe satellites are on the opposite side of the orbit fromthe Pacific, at latitudes of 60 ◦ , 30 ◦ , − ◦ , and − ◦ .This set of points will be called the evaluation points,and the results for the specific choice of the basis func-tions that has been used will be discussed in Section4. . × − m/s / √ Hz (Foulon et al. 2013) in both the along-trackand radial directions from 0.005 Hz to 0.1 Hz, and thatit increases as [0 . /f ] . at lower frequencies. Withthese assumptions, the resulting noise in the along-trackseparation of the satellites can be calculated by inte-grating twice with respect to time, and then correctingfor the resonance with the orbital frequency at 1 cycleper revolution (cy/rev).With these assumptions, the resulting noise in thealong-track separation of the satellites without includ-ing the effect of the orbital resonance is given in the Peter L. Bender, Casey R. Betts
Table 1
Spurious acceleration noise estimates for the GRACE Follow-On missionNominal Sep. Noise Resonance Total EquivalentFrequency Amp. Without Factor Sep. Noise Geopotential(cy/rev) Res. Factor Amplitude Height Error(1 × − m) (1 × − m) (mm)3/64 2810 3.0 8430 5821/8 723 3.1 2240 1561/4 75.3 3.3 249 17.129/64 27.9 4.6 129 9.005/8 4.62 5.9 27.3 1.8911/16 3.63 7.9 28.8 1.983/4 2.92 9.9 29.1 2.0113/16 2.39 11.0 26.4 1.837/8 1.99 12.1 24.0 1.6515/16 1.67 12.7 21.3 1.471 1.42 12.9 18.3 1.2617/16 1.22 12.7 15.6 1.089/8 1.06 12.1 12.9 0.9019/16 0.924 11.0 10.2 0.695/4 0.813 9.9 8.10 0.55821/16 0.720 8.1 5.82 0.40211/8 0.639 6.3 4.02 0.27623/16 0.573 5.6 3.21 0.2223/2 0.516 4.8 2.46 0.17125/16 0.465 4.4 2.04 0.14113/8 0.420 4.0 1.68 0.11727/16 0.384 3.7 1.41 0.0967/4 0.351 3.4 1.20 0.08429/16 0.321 3.2 1.02 0.06915/8 0.294 3.0 0.87 0.06031/16 0.27 2.8 0.75 0.0512 0.252 2.6 0.66 0.045 second column of Table 1. These values are for the rootmean square noise amplitudes for 27 frequency intervalscentered on the nominal frequencies given in column 1.They range from 3/64 to 2 cy/rev. The bandwidths are1/32, 1/8, 1/8, and 9/32 cy/rev for the lowest four fre-quencies, and are 1/16 cy/rev for the rest. The sumof the frequency intervals chosen covers the range from1/32 to 65/32 cy/rev uniformly.Rough estimates of the effect of the orbital reso-nance on the satellite separation were obtained as fol-lows. The Hill equations (Kaplan 1976) were used tocalculate the effect of a given low level of along-trackperturbing force on one of the two GRACE satellites onthe range between the satellites during four revolutions.This was done for different perturbing frequencies, andthe ratio of the rms range change to that for frequenciesway above the orbital frequency was taken as the res-onance factor for the along-track perturbations. Thenthe same was done for radial perturbing forces. The rss(root sum square) combination of these factors is listedas the resonance factor in column 3 of Table 1.Finally, the error in the geopotential height at satel-lite altitude that would result from this separation noiselevel is obtained by multiplying by a factor 69, as dis- cussed in Section 2, and is given in the fifth column.These rms amplitudes are called q ( k ), where k is theindex number for the k -th noise frequency. The ampli-tudes are large at frequencies below 5/8 cy/rev, butwith suitable choices of basis functions, it was foundthat the contributions to the interpolation errors forthe different frequency intervals could be made to de-crease at the lowest frequencies.The corresponding sine and cosine functions usedto represent the range noise at specific frequencies willbe called ws k ( t ) and wc k ( t ), with ws ( t ) and wc ( t )being the sine and cosine functions for the first noisefrequency, ws ( t ) and wc ( t ) those for the second noisefrequency, etc. The maximum amplitudes of the noisefunctions are 1, except for the sines of any frequencyless than 1/8 cy/rev. The values of the noise functionsat the calibration points are given by matrices BS and BC , where BS( i, k ) is the value of the k -th sine noisefunction at the i -th calibration point and BC( i, k ) isthe value of the k -th cosine noise function at the i -thcalibration point.Let CS = K ∗ BS and CC = K ∗ BC . Then CS( i, k )and CC( i, k ) are the amplitudes of the i -th basis func-tion coefficients due to unit amplitude for the k -th sine cean calibration approach for the GRACE and GRACE Follow-On missions 7 and cosine noise functions. Also, let J( m, i ) be the am-plitude of the correction function at the m -th evaluationsite due to a unit coefficient for the i -th basis function.And, let the matrices DS ( m, k ) and DC ( m, k ) be thevalues at the m -th evaluation site due to unit amplitudefor the k -th sine and cosine noise functions. Then, if thematrices ES = J ∗ CS − DS and EC = J ∗ CC − DC ,then ES ( m, k ) is the error in the correction function atthe m -th evaluation site due to a unit error in the k -thsine noise function, etc.Let k = 1 to 4 correspond to the evaluation points atalong-track angular locations of − ◦ , − ◦ , − ◦ ,and − ◦ with respect to the mid-point of the four-revolution arc, k = 5 to 8 correspond to − ◦ , − ◦ , − ◦ , and − ◦ , etc. The different realizations of thenoise function for different sets of four revolutions ondifferent days can be evaluated by randomizing the signsof the amplitude coefficients q ( k ) of the noise functions.If the values of the q ( k ) are those given in Table 1, thenthe square of the error in the correction function at the m th evaluation point due to the rms error q ( k ) in boththe sine and cosine noise functions for the k th noisefrequency is given byE( m, k ) = [ q ( k ) ∗ ES( m, k )] + [ q ( k ) ∗ EC( m, k )] . (9)For a given set of basis functions, E( m, k ) containsall of the information needed to evaluate how well theresulting least squares fit correction function will doin minimizing the mean square errors at the 16 cho-sen evaluation sites in the hemisphere opposite to thePacific.4.2 ResultsThe investigation of different choices of basis functionsfor the interpolation process consisted of calculating thematrix E( m, k ), and looking at the amplitudes in itsdifferent columns. From these amplitudes, the contri-butions to the geopotential height uncertainty due tothe interpolation process can be seen for each frequencyinterval separately. Thus changes in the choice of basisfunctions can be considered, based on what frequencyregions appear to need better coverage.With the estimates of amplitude coefficients q ( k )given in the fifth column of Table 1, several differentcombinations of up to 16 basis functions were chosen totry to fit the noise at the calibration points in such away that the residual noise at distant locations wouldbe small. As discussed earlier, the criterion used wasto minimize the residual noise at the times during thefour revolutions when the satellites were at the 16 cho-sen evaluation points. With this criterion, a reasonable choice of M basis functions was found to be as follows: X = 1 and X = θ/ π , where θ is the angular posi-tion with respect to the center of the four-revolutionarc; X to X are the sines of 5/4, 1, 3/4, 7/16, 7/32,and 13/128 cy/rev; X to X are the cosines of thesefrequencies; X is a function that is 1 at the times ofcrossing the South Pole and is 0 at other times; and X is similarly defined for the North Pole.For this set of basis functions, the matrix E( m, k )was calculated. From it, the values of F( m ) were ob-tained, where F( m ) is the sum of E( m, k ) over k . Thevalues of F( m ) are the mean square interpolation er-rors at the individual evaluation sites that would beobtained if there were no errors in the knowledge of thegeopotential heights at the calibration sites. They aregiven in Table 2.The average of F( m ) over the 16 evaluation sites is < F( m ) > = 1 .
32 mm . Thus the expected root meansquare error in the correction function at the evaluationsites due to the interpolation process is 1.15 mm in thegeopotential height at satellite altitude.As will be seen later, this measure of the rms errordue to interpolation of the correction function for thespurious acceleration noise is quite small compared withthe effect of uncertainties in the geopotential heighttime variations at the calibration sites. However, it isof some interest to see if a simpler set of basis functionscould produce nearly as small an error due to inter-polation. As a measure of this, the calculations wererepeated for 6 different cases. In each, one of the sixbasis function frequencies was left out, so that only 14basis functions remained. The results were that in eachcase the overall rms interpolation error increased by afactor of 2 or more. Thus it appeared worthwhile tokeep all 16 of the basis functions in the remaining partsof the study.Also of interest is how the expected mean squareerrors over the evaluation sites vary with the noise fre-quency. To see this, the average of E( m, k ) over m ,G( k ) = < E( m, k ) > , is given in Table 3 for the 27different noise frequencies considered.The error contributions for 3/4, 1, and 5/4 cy/revare less than 10 − mm , since these are three of thebasis function frequencies. Also, the error contributionsare very small for the frequency bands from 3/64 to1/4 cy/rev. The main peaks are at 7/8, 9/8, and 25/16cy/rev, and the errors go down smoothly for frequenciesabove 25/16 cy/rev.At an early stage in the processing of real data, the16 parameters in the correction function for a particular4 revolution arc would be determined. Then, this cor-rection to the satellite separation during the arc wouldbe made, before the main part of the processing. The Peter L. Bender, Casey R. Betts
Table 2
Mean square interpolation error at the 16 evaluation sitesEval. site ) 0.549 0.234 0.261 0.738 2.05 2.91 2.41 1.27 0.855 1.85 2.50 1.66 0.963 0.459 0.810 1.63 Table 3
Mean square error in geopotential height for different noise frequency bandsFreq. (cy/rev) 3/64 1/8 1/4 29/64 5/8 11/16 3/4Error (mm ) 5e-6 3e-6 3e-6 0.001 0.001 0.001 4e-18Freq. (cy/rev) 13/16 7/8 15/16 1 17/16 9/8 19/16Error (mm ) 0.001 0.010 0.003 6e-14 0.005 0.271 0.015Freq. (cy/rev) 5/4 21/16 11/8 23/16 3/2 25/16 13/8Error (mm ) 2e-19 0.020 0.065 0.135 0.171 0.185 0.160Freq. (cy/rev) 27/16 7/4 29/16 15/8 31/16 2Error (mm ) 0.112 0.078 0.033 0.028 0.016 0.011 processing could be done either in terms of the range orthe range rate, since sufficiently precise numerical pro-ceedures for either approach are now available (Daraset al. 2015). <
60 days is essential for minimizingaliasing in satellite gravity missions.” Among the bestof the currently available detailed models for variationsin the properties of the oceans are the models pro-duced by the Estimating the Circulation and Climateof the Ocean consortium (the ECCO models) (Wunschet al. 2009). Results from a particular one of these mod-els, called dr080, are available at the ECCO-JPL web-site (http://ecco.jpl.nasa.gov/ external). Among the re-sults are the ocean bottom pressure variations at a gridof points covering almost all of the oceans. We usethese results in this paper to estimate the variationsin the geopotential at sites in the central part of thePacific. This model includes assimilation of data fromthe oceans.Results for ocean bottom pressure variations in theECCO model at a number of sites in different oceanswhere there are ocean bottom pressure gauges, as wellas comparisons of these two measures of variations, havebeen given in Quinn and Ponte (2011). Also includedare comparisons with variations from an alternate oceancirculation model at these sites. These results were mostencouraging for the North Atlantic and the eastern partof the Pacific. Our results in a later part of this paper will be compared with the Pacific results of Quinn andPonte (2011).The model we are using for the geopotential heightvariations at 500 km altitude at the calibration sitesin the Pacific has four components: (1) random varia-tions at each of the sites at the calibration times; (2)a uniform offset at all of the sites during the roughly6 hour calibration period; (3) a north-south gradientduring this period across the area covered by the cali-bration sites; and (4) an east-west gradient across thearea during this period. While this model is quite crude,we believe that it will give a fairly good description ofhow the geopotential variation uncertainties will affectthe acceleration correction approach we are pursuing ifthe amplitudes of the four terms listed above are wellchosen.As a basis for estimating the parameters to use inthe geopotential variation uncertainty model, it wouldbe desirable if we could start from knowledge of theuncertainties at different locations over some period oftime. However, there are limits on how well those un-certainties are known. One check would be to comparethe results from the ECCO-JPL model with the geopo-tential variation results from present analyses of datafrom the GRACE mission. However, since errors in theanti-aliasing geopotential models over both land andoceans used in analysis of the GRACE data can af-fect the GRACE results substantially, it seems quitepossible that the comparison would not give a usefulevaluation of the actual uncertainty in the geopotentialvariation information available from sources other thanthe GRACE data.In view of the above, we decided to start by look-ing at the full geopotential variations predicted by theECCO-JPL model. The way in which this was done isdescribed in the next section. cean calibration approach for the GRACE and GRACE Follow-On missions 9 ◦ in both latitude and longitude. The points includedwere most of those at 3 ◦ intervals which were oceanpoints between − ◦ and +60 ◦ in latitude and between135 ◦ and 285 ◦ longitude, and not east of Mexico or Cen-tral America. The bottom pressure results at 0 and 12hours UT during December 2010 were used.To provide a check on the values at each grid point,we first subtracted the mean for the month and thencalculated the rms variations from the mean. The largemajority of the rms values were less than 3.0, in cmof water, and grid points with values greater than thishave not been included in the calculations. In addition,120 grid points with values between 2.0 and 3.0 wereexcluded, and 14 with values less than 2.0. These weremostly at latitudes between − ◦ and − ◦ and be-tween +36 ◦ and +45 ◦ , where the variability appearedto be much greater than at lower latitudes. These pointswere excluded because of their variability being sub-stantially higher than for the typical grid points within35 degrees of the equator, which contributed most ofthe variations according to the ECCO-JPL model, andthus not being representative of how the model wouldbe used in practice. This left about 1360 grid points tobe used in the calculations.The next step was to use the differences from themean for the month for all the included grid points ata given latitude to calculate directly their contributionto the geopotential height variation at satellite altitudeat each of the 12 calibration points. These were cho-sen to be at latitudes of − ◦ , 0 ◦ , and +30 ◦ , and ateast longitudes of 234 ◦ , 211 ◦ , 188 ◦ , and 165 ◦ . For ac-tual satellite passes northward across the Pacific, thelongitudes would be about 2 degrees higher at − ◦ and 2 ◦ lower at +30 ◦ N. However, these offsets wouldbe reversed for southward passes, and these differencesare not likely to be large enough to affect the geopoten-tial variation model results appreciably. Also, on differ-ent days, the longitudes of the passes across the Pacificwould be shifted by up to about 12 ◦ in either direction,but not including these shifts is not expected to make asubstantial effect on the results. Including them wouldmake the calculations considerably more complicated. For the contributions from each latitude band ofgrid points and for each of the 62 times during themonth, the results at the 12 calibration sites were in-spected to look for outliers. However, the results ap-peared to vary quite smoothly with latitude and withtime. Thus the total variations from all of the latitudebands at each time were calculated. This gave the val-ues as a function of time shown in Figure 2, in units ofmillimeters of geopotential height at satellite altitude.The values for the times during the month and the var-ious calibration sites range from − − ◦ and those at − ◦ . These values -4mm4mm Day in December 2010Site -4mm4mm-4mm4mm-4mm4mm
Fig. 2
Variations at the 12 calibration sites in the geopotential height at satellite altitude from the ECCO-JPL ocean model. range from − ◦ minusthe average of those at 165 ◦ , plus one third of the aver-age of the values at 211 ◦ minus those at 188 ◦ . These val-ues at the different times range from − ◦ widthof the area at the equator.These values for the rms variations in the geopoten-tial heights at the individual calibration sites, the aver-ages over the calibration sites, and the differences acrossthe calibration sites, are expected to be substantiallylarger than the errors in these quantities. However, wehave used them to provide an intentionally somewhatpessimistic model for the errors in these quantities. If no better information on the expected model errors be-comes available, these values can be used as the basisfor a model of the geopotential height variation uncer-tainties to be used in the ocean calibration approach tocorrecting for spurious accelerations in future GRACE-type missions, like the GRACE Follow-On mission, andpossibly for GRACE itself. However, before discussingthis application of the results further, a partial compar-ison of the ECCO model results with another source ofinformation on ocean mass distribution changes will bedescribed.5.4 Comparison with Ocean Bottom Pressure GaugeResultsAs a partial check on the ECCO results for ocean bot-tom pressure variations, we have made a comparisonwith data from 15 of the ocean bottom pressure (BPR)gauges in the NOAA Deep-ocean Assessment and Re-porting of Tsunamis (DART) network. About 35 of cean calibration approach for the GRACE and GRACE Follow-On missions 11 Fig. 3
Locations of the NOAA Deep-ocean Assessment and Reporting of Tsunamis (DART) ocean bottom pressure gaugesused in the comparison with values from the ECCO-JPL ocean model. − ◦ to +32.46 ◦ andin longitude from 145.59 ◦ to 273.61 ◦ . This range of lo-cations was chosen to stay away from high latitudesand from western or eastern boundary current regions.DART site Table 4
Locations of the DART and ECCO sitesDART sites ECCO sitesSite Lat Long Nor. East Nor. EastLat Long Lat LongGr lack of other DART sites in the northeastern corner ofthe region of interest.The results of the comparisons of the DART andECCO variations are given in Table 5. For each DARTsite, the rms values of the differences from the meanare given for that site and for the nearby ECCO site.In addition, the correlation coefficient ρ between thevariations from the two sources is given.It is interesting that the first and last of the siteslisted have substantially larger rms variations in equiv- Table 5
Comparisons of the DART and ECCO rms waterheight variationsDART DART σ ECCO σ Corr.Site (cm) (cm) Coeff.21413 2.42 2.16 0.85232411 1.08 0.75 0.49432412 1.32 0.80 0.57932413 1.36 0.62 0.59443412 1.32 0.75 0.65943413 1.52 0.62 0.50046412 1.37 0.62 0.37951406 1.38 0.74 0.76751407 1.56 0.77 0.42951425 1.41 0.57 0.44451426 1.63 0.61 0.58052402 1.86 0.86 0.55252403 1.42 0.82 0.58452406 1.17 0.62 0.42854401 2.39 1.38 0.726 alent water depth, both for the DART and ECCO re-sults. The correlation coefficients at these sites have 2of the 3 largest values for all the sites. Also, 14 of the15 sites have correlation coefficients above 0.40. Thusthere is a substantial contribution to the results that iscommon between the two sets of measurements.At essentially all of the DART sites, the rms vari-ations in the DART results are very roughly twice thesize of the rms variations in the ECCO results. This islikely to be due to a combination of the noise in theDART results being higher than for the ECCO modeland the variations in the ocean bottom pressure in theECCO model being somewhat smaller than the realvariations. Since the correlation coefficients between thetwo types of variation results are typically about 0.5,there is enough correlation to suggest that the magni-tude of the actual variations is somewhere in between.In the paper mentioned earlier (Quinn and Ponte2011), the ECCO data were compared with ocean bot-tom pressure gauge (BPR) data and with another oceanmodel at gauge locations in the Atlantic and IndianOceans, as well as in the Pacific. The other ocean modelused was the Ocean Model for Circulation and Tides(OMCT), that has been used as a de-aliasing model forGRACE data. They found that the variance was onlyabout 1 cm of equivalent water depth variation in thePacific. The correlations of BPR results with the oceanmodel results had a wide range of values, but with val-ues of 0.3 or more at many sites. The rms variationsfor the BPR data in the eastern Pacific were only 2 to2.5 cm, which is consistent with the variations found inthis paper.Our results and those of Quinn and Ponte don’t givedirect estimates of the uncertainty in the mass distribu-tion variations in the ECCO model. However, it seems unlikely that the uncertainty is appreciably larger thanthe variations in the model. The uncertainty also couldbe substantially smaller, without this being apparentfrom the comparisons that have been done.In view of the above, we have made what we regardas somewhat conservative estimates of the values to usein our model for the four types of errors for the Pacificsites included in model. The estimated errors are asfollows: (1) 1.5 mm for the random errors at each ofthe Pacific calibration sites; (2) 1.5 mm for the commonerror at the sites during the 4 revolution observationperiod; (3) 1.0 mm for the N-S gradient; and (4) 1.0 mmfor the E-W gradient. For (1) and (2), the value chosenis about 35% higher than the rms value for the ECCOdata, to allow for the possibility of that model actuallynot including some of the real ocean variability. For (3)and (4), the values chosen are near the maximum valuesfound from the ECCO data for a similar reason.As emphasized earlier, the uncertainties in geopo-tential height time variations are believed to be lessover relatively low latitude regions in the oceans thanover land or at higher latitudes. However, to evaluatethe effect of these uncertainties, some sort of model forthem is needed. What has been done in the past is totake two different models for the variations in the oceanmass distribution with time, and compare their results,or to compare one model against other types of data.The problem with comparing different models is thatthey share some of their input data, and thus their er-rors may be considerably correlated. For comparisonsagainst other types of data, the problem is the limita-tion of coverage of data from other sources, except forcomparisons of the ocean surface height variations withaltimetry results.As an alternate approach, it was decided to evaluatethe variations in geopotential height at satellite altitudeaccording to one ocean model, and then assume thatthe uncertainty is equal to some fixed fraction of thevariations obtained from the ocean model. Although the situation would be slightly different forfour revolution arcs with downward crossings across thePacific from those with upward crossings, the resultsare expected to be nearly the same. Thus only the up-ward crossing case has been studied. Also, the choice ofmeasurements at only three latitudes in the Pacific isclearly not realistic, but it is expected that it will give agood indication of what to expect. The errors in geopo- cean calibration approach for the GRACE and GRACE Follow-On missions 13
Table 6
Ad hoc error model for the geopotential height vari-ation uncertaintiesPacific sites:Uniform error at 12 sites: 1.5 mmRandom error at sites: 1.5 mmLinear N-S variations: 1.0 mmLinear E-W variations: 1.0 mmRandom errors at IndianOcean and Atlantic Ocean sites: 2.5 mmRandom errors at NorthPole and South Pole crossingsdue to time variations: 0.5 mm tential height estimates are expected to be quite wellcorrelated over distances considerably less than 30 ◦ .Our simplified error model based on uncertainties inthe geopotential heights at satellite altitude is summa-rized in Table 6. For the sites in the Pacific, a correlatederror of 1.5 mm is assumed for all 12 sites, plus random1.5 mm errors. In addition, an error of 1.0 mm ampli-tude is assumed for the N-S gradient between the sitesat 30 ◦ N and those at 30 ◦ S, and a 1.0 mm error for theE-W gradient. For the remaining sites, random errors of2.5 mm are assumed for the Indian and Atlantic Oceansites, which is similar to the rss of the different errorsat a given Pacific Ocean site. And random errors of 0.5mm for each of the four North Pole and five South Poleobservations are included, to allow for the variationsduring the times between those observations.A study of the variability of the geopotential heightover favorable low-latitude sites in the Atlantic and In-dian Oceans based on the ECCO-JPL model has notbeen carried out. However, from Figures 1b and 1c inQuinn and Ponti (2011), it does not appear that thevariability at such sites would be substantially worsethan at central Pacific sites. Also, from their Figure 2b,the difference between the OMCT ocean model and theECCO model appears to be fairly small for both the At-lantic and Indian Ocean sites. Thus it seems reasonableto adopt 2.5 mm as the estimate for the variability forthe calibration sites in these locations.To give some indication of the scale of the errorsassumed in the error model, it is useful to consider thechange in geopotential height at satellite altitude dueto a Gaussian disk of water with 1 cm height at thecenter and a half-mass radius of 39 ◦ . This much ad-ditional water would increase the geopotential heightby 1.0 mm, equal to two-thirds of the correlated andthe random error assumed at the Pacific sites. How-ever, validation of the assumptions in the error modelwill require improved evaluations of the uncertainties inpresent ocean circulation models and their correlations Table 7
Average mean square errors at the evaluation sitesin mm Independent errors at the Pacific sites: 1.62Common error at the Pacific sites: 0.10North-south variations: 2.01East-west variations: 0.29Indep. error at Indian site: 2.74Indep. error at Atlantic site: 1.10Indep. errors at Pole crossings: 0.42Total (mm ) 8.28 over the distances between the chosen calibration sites,as discussed in Section 5.1.It should be noted that the uncertainties in thegeopotential time variations assumed here are equal toall of the time variations from the ECCO-JPL model.This may be regarded as a fairly pessimistic assump-tion, since the uncertainties actually could be substan-tially less. However, it is difficult to find evidence for themodel being better than this over the time scales of in-terest. Comparisons with ocean bottom pressure gaugeresults from the DART network, as discussed earlier,indicate that the uncertainties aren’t likely to be sub-stantially worse than assumed. But the noise level inthose data is large enough to prevent a more precisecomparison from being made.The final step in the evaluation of the ocean calibra-tion approach was to use the set of 16 basis functionsdiscussed in Sec. 4.2 along with the ad hoc geopotentialheight variation error model to see how large the result-ing errors in the correction function at the evaluationsites would be. If a particular case of the error modelfrom Table 6 is chosen, such as the common error of1.5 mm at the 12 Pacific sites, this defines the vector Y of geopotential height errors described just before eq. 3in Section 3. Then, if K is the matrix defined in eq. 4and J is the matrix defined in Section 4.1, the resultingerrors at the 16 evaluation sites will be given by thevector L , where L = J ∗ K ∗ Y (10)The calculation of L is repeated for each of the indepen-dent errors in Table 6, and the squares of the 16 entriesin each resulting vector L are averaged to give the meansquare error at the evaluation sites due to the errors inthe ad hoc error model. The resulting contributions tothe average mean square error at the evaluation sitesare given in Table 7.With the contribution of 1.32 mm from the spuri-ous acceleration noise, this gives a mean square errorof 9.60 mm at the evaluation sites, or an rms error ofabout 3.1 mm. If the values in the geopotential varia-tion error model were half as large as assumed, the rms error at the evaluation sites would be reduced to about1.8 mm. These values appear to give a reasonable rangefor the estimated geopotential height uncertainties fromthe GRACE Follow-On mission, if other comparable orlarger sources of low-frequency range uncertainty do notshow up in the data.It unfortunately is true that only about half of eachday would be covered by the 4 revolution arcs that crossthe central Pacific. Thus, other approaches to correct-ing for spurious accelerations or other low frequencyerrors would have to be used the rest of the time. How-ever, these other approaches could be compared withthe results from ocean calibration during the 4 revo-lution arcs that do cross, in order to possibly help indetermining which of the other approaches to use therest of the time. For the GRACE mission, the acceleration noise levelrequirements were a factor of 3 less severe than forGRACE Follow-On. Thus, the overall requirement onthe range acceleration noise level due to the accelerom-eters in the two satellites wasPSD / < × − (cid:20) .
005 Hzf (cid:21) . m / s / √ Hz (11)Based on this nominal acceleration noise level forGRACE, the mean square contribution to the geopo-tential height uncertainty at the evaluation sites wouldbe 12 mm . With the 8.3 mm contribution from theuncertainties in the geopotential heights at the calibra-tion sites, this gives a total mean square uncertainty of20.3 mm at the evaluation sites, or an rms uncertaintyof 4.5 mm. This is just a factor 1.5 higher than wasfound earlier for GRACE Follow-On.In view of this result, it appears useful to considerthe possibility that the ocean calibration approach couldbe tested using selected subsets of the GRACE data.But, unfortunately, it is difficult to tell how much ofa change this approach might make, since a number ofother error sources are present besides the nominal levelof accelerometer noise. The sources of real or appar-ent extra low-frequency range acceleration noise in theearly GRACE data were discussed quite early by Fluryet al. (2008), and in other papers that they refer to. Oneof these sources is related to the fairly frequent thrusterfiring for attitude control. A second is short mechanicaldisturbances called twangs that are triggered by tem-perature changes somewhere in the satellite, and a thirdis short pulses apparently due to electrical disturbances.Flury et al. (2008) were able to find some periodsof 70 to 300 seconds during which none of these distur- bances appeared to be present. Based on the data fromsuch periods, they concluded that the different com-ponents of the acceleration noise for the two GRACEsatellites met or slightly exceeded the requirements atfrequencies of 30 to 300 mHz. Also, Figure 5 of Fromm-knecht et al. (2006) indicates only a moderate increasein the level of low-frequency range noise at frequenciesdown to 1 mHz. This figure was based on data free fromthruster firings, but not other disturbances, in order tomake use of somewhat longer data arcs.More recently, the total low-frequency range noisebased on GRACE KBRR range data in 2006 was con-sidered by Ditmar et al. (2012). The quantity plottedin Figure 2 in that paper is the second derivative of theKBRR range minus the calculated range from six-hoursatellite dynamical orbits fit to the data. A curve show-ing the expected noise due to one component of thenominal accelerometer noise level is included also, butwithout an allowance for resonance at 1 cycle/rev. If afactor of about 13 is allowed for resonance, the actualnoise near 1 cycle/rev is about 20 times higher thanthat due to the nominal total accelerometer noise.A substantial amount of the excess low-frequencynoise may be due to limitations in the orbit calcula-tions and the static geopotential that were used whenthe calculations were done. However, temporal aliasingalso is believed to be a basic limitation on all anal-ysis methods. Since known noise limitations preventthe ocean calibration approach from being extendedbeyond favorable 4 revolution arcs, for GRACE data,other analysis methods would need to be evaluated oversimilar length arcs in order to achieve a fair comparison.Thus a combination of the different approaches wouldbe needed in order to give useable monthly solutionswith GRACE data. From the results for the case of GRACE Follow-On inSection 6, the prospects appear to be good for beingable to correct for low frequency range noise to a levelof about 3 mm in the geopotential height if the uncer-tainties in the geopotential heights at the calibrationsites are as small as specified in the model describedin Section 5. This model was based mainly on the as-sumption that the uncertainties at ocean calibrationsites would not be larger than the rms amplitudes ofthe variations from the monthly means as derived fromthe ECCO-JPL ocean model. It seems plausible thatthe actual variations would be smaller than this, butthere does not appear to be any fairly firm evidenceavailable on this question. cean calibration approach for the GRACE and GRACE Follow-On missions 15
An equally large or larger question is the uncertaintyin the geopotential heights over the less favorable re-gions of the globe that would result if the ocean cal-ibration approach is not used. The various sources ofmass variations have been reviewed recently by Gruberet al. (2011). If the methods of correcting for noise in theaccelerometers that has been used for GRACE is usedfor GRACE Follow-On also, models for what is knownabout the mass variations from other sources have to beapplied before the corrections are made. Thus the errorsin those models will be built into the final geopotentialresults. These models are usually called anti-aliasingmodels, and their accuracy unfortunately is not wellknown.From Gruber et al. (2011), Figure 13, the monthlymean variations in the global mass distribution due tohydrology are the largest ones, followed over a substan-tial range of harmonic degrees by the ice signal, theatmospheric signal, and the ocean signal. The hydro-logical mass variations are quite well known over someareas, but not over others. And for the oceans, the un-certainty in the mass variations at high latitudes ap-pears to be a lot higher than at low latitudes. Thus itappears plausible that the geopotential variation resultsobtained with the ocean calibration approach may bemore accurate than those obtained otherwise, but wedon’t have any estimate of how much of an improve-ment might be expected.A number of simulations of possible future Earthgravity missions have been carried out recently in Eu-rope in support of the project “Satellite Gravimetryof the Next Generation (NGGM-D)”. The results ofthe project, also referred to as the “ e .motion” project,have been published recently by Gruber et al. (2014).Some studies for a single pair of satellites in the samenearly polar orbit were included, and the results for thecumulative geoid error for this case are shown in the leftpart of Fig. 7-6 in the report.However, it is unfortunately very difficult to com-pare the expected results for these roughly 32 day sim-ulations with what has been found for 4 revolutionarcs across the Pacific with the ocean calibration ap-proach. The assumptions made are quite different, andthe much longer simulations are affected quite stronglyby temporal aliasing, because of time variations in thegeoid during the time taken for the the satellite groundtracks to cover the Earth. On the other hand, fitting offairly large numbers of empirical parameters over themuch longer simulation period in the e .motion studiesto correct for various error sources may have led to theloss of some real geophysical variation information.It is hoped that matched simulations of the differ-ent approaches can be done soon in order to see if the ocean calibration approach really would produce im-proved results for those 4 revolution arcs of GRACEFollow-On data that cross the equatorial Pacific. Also,for future missions after GRACE Follow-On that aredrag-free and have somewhat reduced acceleration noiseat low frequencies, it appears possible to extend theocean calibration approach to 12 revolution arcs wherethe middle 4 revolutions don’t cross the Pacific. Suchsolutions could be combined to give continuous resultson the variations in the Earth’s mass distribution. Acknowledgements
It is a pleasure to thank the manypeople who have contributed to discussions of the accuracylimitations for GRACE-type missions, including particularlyDavid Wiese, John Wahr, Steve Nerem, Oscar Colombo, JakobFlury, Pieter Visser, and Helen Quinn. We would also like tothank George Mungov of the NOAA National GeograpicalData Center for providing the tidally corrected DART oceanbottom pressure results.