Global marine gravity field from the ERS-1 and Geosat geodetic mission altimetry

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. C4, PAGES , APRIL 15, 1998 Global marine gravity field from the ERS-1 and Geosat geodetic mission altimetry Ole Baltazar Andersen and Per Knudsen Kort- og Matrikelstyrelsen, Copenhagen, Denmark Abstract. Satellite altimetry from the Geosat and the ERS-1 Geodetic Missions provide altimeter data with very dense spatial coverage. Therefore the gravity field may be recovered in great detail. As neighboringround tracks are very closely distributed, cross-track variations in the sea surface heights are extremely sensitive to sea surface variability. To avoid errors in the gravity field caused by such effects, sea surface variability needs to be carefully eliminated from the observations. Initially, a careful removal of gross errors and outliers was performed, and the tracks were fitted individually to a geoid model and crossover adjusted using bias and tilt. Subsequently, sea surface heights were gridded using local collocation in which residual ocean variability was considered. The conversion of the heights into gravity anomalies was carried out using the fast Fourier transform (FF ). In this process, filtering was done in the spectral domain to avoid the so-called "orange skin" characteristics. Comparison with marine gravity was finally carried out in three different regions of the Earth to evaluate the accuracy of the global marine gravity field from ERS-1 and Geosat. 1. Introduction atmospheric path delay using values provided in the ERS-1 Ocean Product Records (OPR) and in the Geophysical Data Satellite altimetry from the geodetic missions of the Geosat Records (GDR) for Geosat for wet and dry tropospheric path and the ERS-1 provides the opportunity for geodesists to make delay and ionospheric path delay. Similarly, the provided very detailed mappings of the marine gravity field. The newly correction for Earth tides was applied. Finally, the ocean and released Geosat geodetic mission altimetry contains data from the first 18 months of the Geosat satellite mission ( ) load tide contributions were updated using the global Andersencovering marine regions at latitudes between -72 ø and 72 ø. The Grenoble (AG95.1) ocean tide model [Andersen et al., 1995]. satellite was operated in a nonrepeating orbit yielding a very The AG95.1 model was recently investigated by Shum et al. dense, though not completely homogeneous coverage of [ 1997]. It was chosen as it has been carefully tapered onto the observations. ERS-1 completed its 336-day geodetic mission in Finite Element Solution (FES94.1) ocean tide model [Le March Altimetry from this mission covers all oceans Provost et al., 1994] around the 66 ø parallels to avoid between -82 ø and 82 ø latitude and provides a very dense and discrepancies at this latitude. Such discrepancies can be found homogeneous coverage. At the equator the spacing between the in many recent global ocean tide models based on ERS-1 satellite's ground tracks is constantly 8.3 km. TOPEX/POSEIDON altimetry (e.g., the Center of Space Mapping of the Earth gravity fields from different data Research (CSR3.0) model [Shum et al., 1997]). sources has previously been presented by, e.g., Haxby [ 1987], Initially, data were removed if any of the applied range Balmino et al. [1987], Sandwell [1992], Sandwell and Smith corrections were absent except the ocean tide correction. [1997], Knudsen [1991], Knudsen et al. [1992], Tscherning et Subsequently, sea surface height observations differing by al. [ 1993], and Andersen et al. [ 1995]. In this study the method more, than 10 m from the joint National Aeronautics and Space used by Andersen and Knudsen [1995, 1996] is used with Administration (NASA) Goddard Space Flight Center (GSFC) special emphasis on the latest improvement in data availability and National Imagery and Mapping Agency (NIMA) Earth and in the determination of the global marine gravity field. Gravity Model (EGM96) geoid model complete to degree and order 360 [Lemoinet al., 1997] were removed. Finally, data 2. Data Editing and Initial Data Processing were removed if the standard deviation of the height observations exceeded 0.3 m. This resulted in about 30 million The altimeter data from both geodetic missions are obtained altimeter data from the Geosat and about 20 million altimeter as the usual 1 s mean values with an along-track spacing of data from the ERS-1 geodetic missions. about 7 km. To enhance the quality of the altimetry, both data From this point on all subsequent processing and derivation sources applied the newest set of orbits similar for the two of the gravity field was carried out in small cells of the size of geodetic missions. Here Geosat applied the new recomputed 2 o latitude by 10 ø longitude. Globally, this mosaic has 80 times orbits based on the JGM-3 gravity model, [Tapley et al., 1996] 36 cells (latitude by longitude) equivalent to 2880 cells. and ERS-1 applied the JGM-3 orbits provided by the Delft However, observations were extracted in cells extended by a Institute of Technology. The observations were corrected for border zone of 0.5 ø latitude by 1 o longitude to a size of 3 o latitude by 12 ø longitude. The selection of such small subareas Copyright 1998 by the American Geophysical Union. was essential to the modeling of orbit errors and sea surface variability. A choice of larger cells caused problems in the Paper number 97JC removal of these signals, while smaller cells may corrupt parts /98/97JC of the regional geoid signal. 8129

2 8130 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD Initially, the number of tracks was limited at high latitude where the track distance becomes extremely narrow. Here, the Then the edited and adjusted altimeter data within each extended cell used for processing (3 ø latitude by 12 ø longitude) number of tracks can be limited without much loss of were interpolated onto a regular grid using the collocation information. Therefore only the altimeter data along the 800 longestracks were selected. To reduce the effects of residual orbit errors and sea surface variability, the tracks were then fitted individually to the joint NASA GSFC and DMA geoid model EGM96 by estimating bias and tilt terms to each track, thus removing all signals with a wavelength longer than about 30-4 ø. Subsequently, a technique described. To filter out remaining sea surface variability that may cause erroneous cross-track gradients between parallel tracks, an additional covariance function for this error was introduced. This error covariance function was applied to observations on the same track only, assuming the error to be temporally uncorrelated. Hence, for observations on the same track, a covariance function like crossover adjustment of the tracks was carried out, also using bias and tilt terms [e.g., Knudsen and Brovelli, 1993]. To avoid r (-r/a) r (-rl[ ) problems with rank deficiency, a minimum variance criterion c(r) = Co(1 + --)e + O0(1 + - )e (3) was applied. Hereafter, each height observation was compared with an estimated height to detect outliers and gross errors. As in work by Tscherning [1990], the estimate was obtained using least squares collocation. That is, was used. The parameters D O and [3 were empirically determined to a variance of (0.1 m) 2 and a correlation length of 100 km, respectively. For observations on differentracks, D O was set to zero yielding an expression similar to (2). In both / p = c (C+D)- h, (1) cases, C o was fixed at (0.2 m) 2, and the parameter a was fixed so that the correlation length was 15 km. To enhance the filtering of the sea surface variability the where C and D are signal and error covariance matrices, % is a estimation was carried out relative to a local weighted average vector of the covariances between the estimate and the of the nearest 100 observations. A weight function 1/(Ia+r0 2) observations, and h contains the 20 nearest observations (5 in with r0 = 0.1 o was applied so the local weighted average is a each quadrant around the estimation location). The observation smooth representation of the surface heights that is less affected itself entered the estimation with its standard deviation by the sea surface variability. In the collocation estimation the increased to 1 m. As a covariance function, a second-order 48 nearest observations are used to secure redundant geoid Markov covariance function is used like information at crossing tracks. The result of the gridding is a 1/16ø by 1/16ø grid, having a size of 4ø latitude by 16ø F (-r/a) longitude. Compared with the area used to select data, this grid c(r) = Co(1 + --)e, (2) was extended by a border zone of 1 o latitude by 3 o longitude to avoid spectraleakage in the following steps. This grid of sea where r is the lag and Co is the signal variance. The parameter surface heights will be treated as geoid undulations through the a was fixed so that the correlation length (where a 50% correlation is obtained) was 30 km to ensure a relatively smooth interpolation. Then a screening for gross errors, e.g., observations affected following steps. The gravity anomalies, Ag, was derived from the geoid undulations, N, using FFF techniques [Schwarz et al., 1990]. In the frequency domain (u,v), that is, by sea ice, was carried out. This was done by removing observations with a discrepancy between the observed and the estimated height larger than 1 m. Finally, the fitting to the geoid model EGM96 and the Ag(u,v) = (o yn(u,v) F(to), (4) crossover minimization were repeated, this time using the original data edited for outliers and gross errors that might have affected the initial adjustment. Signals of wavelengths longer where 6]=u:+v: and y is the normal gravity. Such a transformation from geoid undulations into free air gravity than 30-4 ø are removed through this process. Consequently, anomalies is a differentiation that enhances the high most of the permanent parts of the sea surface topography have been reduced. frequencies. Consequently, it is sensitive to noise. Therefore a Wiener filtering function, F(w), was introduced in (4). This 3. Mapping of the Gravity Field The estimation procedure for the marine gravity field was chosen to derive a global map of the gravity field within a reasonable computational effort. Because of the very large amount of observations, rigorous methodsuch as least squares collocation is not feasible. Hence a less rigorous but very efficient approximation method based on the fast Fourier transform (FFr) was chosen, though it requires that the data are distributed in a regular grid. The result may be sensitive to cross-track gradients caused by sea surface variability arising as the distance between parallel tracks becomes very small in the geodetic mission altimetry. Such effects may be reduced by using altimetric slopes or second-order derivatives [e.g., Hwang and Parsons, 1995; Rurnrnel and Haagrnans, 1990]. However, it was decided to stay with the heights as the observations and 't elaborate existing software in the GRAVSOFr package Figure 1o Location of test area south of Greenland in the North [Tscherning et al., 1992] for gridding and FFF manipulations. Atlantic Oceanø 65

3 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD 8131

4 8132 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD 6O corrections from ERS-1 + Geosat B) No aim track sea level variabilit O D' at 20 km O eodetic mission data onl F eodetic mission data onl, r Figure 2. Comparison of different gravity field solutions in the test area shown in Figure 1. All units are mgal. 2(a) The "final" derived gravity residuals from ERS-1 and Geosat with respecto the Earth Gravity Model (EGM96) [Lemoinet al., 1997]. The greyscale should be multiplied by a factor of 10 to range between -30 and 30 mgal for this picture. 2(b) Gravity difference between the final gravity field in Figure 2a and a gravity field estimated without along-track sea level variability modeling. 2(c) Gravity difference between the final gravity field in Figure 2a and a gravity field estimated with a filtering at 8 km. 2(d) Gravity difference between the final gravity field in Figure 2a and a gravity field estimated with a filtering at 20 km. 2(e) Gravity difference between the final gravity field in Figure 2a and a gravity field estimated using data from the Geosat Geodetic Mission (GM) data only. 2(f) Gravity difference between the final gravity field in Figure 2a and a gravity field estimated using data from the ERS-1 GM only. filter function is equivalento a collocation filter that assumes Kaulas rule to be valid and that assumes uncorrelated noise [details by Forsberg and Solhe#n, 1988]. That is, 4 Wc F(ro) :. (5) In this case the "cutoff' frequency, ro c, where the filter is 0.5, was empirically determined to a wavelength of 12 km, which roughly corresponds to about 10 cycles per degree or harmonic degree Before the Fourier transform of the geoid grid was computed, a cosine taper was applied in the outer 0.5 o parts of the grid.

5 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD 8133 Table 1. Statistics of Different Gravity Field Solutions in a Test Area South of Greenland Mean Standard Deviation Minimum Maximum Final residual gravity to EGM96 from ERS-1 and Geosat; the final residual gravity field is shown in Figure 2a -O.8O Gravity difference between solutions computed with and without along-track sea level variability modeling; the gravity differences are shown in Figure 2b Gravity at wavelength between 8 and 12 km as shown in Figure 2c Gravity at wavelength between 12 and 20 km as shown in Figure 2d Gravity difference between final gravity field and gravity field computed using Geosat geodetic mission (GM) data only; the gravity differences are shown in Figure 2e Gravity difference between final gravity field and gravity field computed using ERS-1 GM data only; the gravity differences are shown in Figure 2f The location of the test area is shown in Figure 1. All units are mgal. This was done to avoid spectral leakage caused by wavelengths Figure 2a shows the gravity field from ERS-1 and Geosat. The that are not periodic within the area. Subsequently, the gravity anomalies are shown with respecto the EGM96 as it contribution of the EGM96 gravity field was restored to obtain comes out when the fine tuned procedure is used. This field will the free air gravity anomalies, and the gravity field was isolated in the following be called the final gravity field. within the small 2 ø by 10 ø cell. Finally, all cells were collected Figure 2b illustrates the importance of considering the to give the global gravity field. residual sea surface variability in the gridding process. It shows the difference between a solution obtained without the error 4. Results covariance function in (3) and the final gravity field in Figure The procedure described above is well known, but in this 2a. The statistics of the difference are shown in Table 1. Even analysis the various parameters have been fine-tuned to obtain though, the standardeviation of the difference is only around an optimal product from a combination of Geosat and ERS-1 1 mgal. Figure 2b clearly shows anomalies with a clear track geodetic mission altimeter data in most of the world' s oceans. pattern. This so-called "trackiness" affects the gravity field The tuning of the parameters was based on empirical criteria recovery if the residual sea surface variability was ignored in with the scope of recovering as much detail and as much the gridding process. reliable gravity field information as possible. To demonstrate The role of the filter, F(to) in (5), is sketched in Figures 2c important phases in the derivation of the gravity field process, and 2d. Figure 2c shows which parts of the solution filtered out a 2ø by 10 ø cell south of Greenland was selected. The location when the cutoff is changed from a wavelength of 8 to 12 km. of this test site can be seen in Figure 1. The differences look noisy with no coherency with the Figure 2 shows the result of the different steps in the fine- structureshown in Figure 2a. If the cutoff is changed from a tuning of parameters used for the derivation of the gravity field. wavelength of 12 to 20 km, then the pattern changes. Figure 2d Table 2. Comparison With 23,663 Selected BGI Gravity Observations in the North Atlantic Ocean Solution Altimetry Mean Standard Difference Deviation Andersen and Knudsen [this issue], computed with respect to the EGM96 geoid model ERS Geosat ERS-1 and Geosat Sandwell and Smith [ 1997] ERS-1 and Geosat Andersen and Knudsen [this issue], computed with respect to the OSU91A geoid model ERS-1 and Geosat Data are from BGI [ 1996]. Locations can be seen in Figure 3. All units are mgal.

6 8134 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD t_, 35 * " Figure 3. Location of 23,663 selected gravity observations in the North Atlantic Ocean from BGI [ 1996]. shows gravity differences that are clearly coherent with the gravity structure shown in Figure 2a. The standard deviations of the two sets of gravity differences are 3.6 and 2.0 mgal, respectively. Therefore valuable gravity field information has been removed. Wavelengths shorter than 12 km appear to be dominated by noise and cause the so-called "orange skin effect" in altimeter-derived gravity fields. Figures 2e and 2f deal with the importance of using a combination of ERS-1 and Geosat geodetic mission altimetry rather than just ERS-1 or Geosat altimetry. In Figure 2e the difference between the solution in Figure 2a from ERS-1 and Geosat altimetry and a solution from Geosat altimetry alone is shown. Figure 2f shows the similar difference with a solution from ERS-I onlyo In both figures a track pattern related to Geosatracks (Figure 2e) and ERS-1 tracks (Figure 2f) can be observed. Table I shows that the standard deviation of the differences are 1.7 and 2.0 mgal for Geosat and ERS-I data, respectively. The standard deviation is highest for the ERS-1 alone differences as more Geosat data entered the gravity field determination. However, in both comparisons (Figures 2e and 2f), the combined use of both ERS-1 and Geosat improves the crossover analysis and thus reduces the trackiness. The global recovery of the marine gravity is shown in Plate 1. This marine gravity field covers most ocean regions of the world. However, a few regions at extreme latitudes exist where no data were available. These are shown with grey in Plate 1. In most regions of the world, this gravity field method works very well. However, some marine regions exist where the method does not work as well. Problems as "trackiness" may be expected in regions with a large ocean variability and in shallow water regions where the ocean tide models may not be sufficiently accurate Figure 4. Location of 3642 gravity observations close to Antarctica The data have been measured by Bundesanstaldt ffir Geowissenshaften und Rohstoffe (BGR). 5. Comparison With Marine Gravity Comparisons with marine free air observations were made in three different regions of the world. These regions were chosen as they have very different oceanographic and gravity signals. One region is in the middle of the Atlantic Ocean toward the Azores, which has very large oceanographic variability. The second is close to the coast of Antarctica in a partly ice-covered region. The third and final region is in the eastern Mediterranean Sea, which has a very large gravity signal. The first comparison was made with a selection of gravity measurements from the Bureau Gravimetrique International (BGI) database. In the region bounded by 10 ø and 50øN and 60 ø and 10øW, 23,663 marine gravity observations were selected from BGI [ 1996]. Only validated data after 1989 were selected. The selected marine gravity anomalies ranges from- 100 mgal to 123 mgal. The result of this comparison is presented in Table 2 with the locations of the observation shown

7 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD 8135 Table 3. Comparison With 3642 Gravity Observations in the Antarctic Region Solution Altimetry Mean Standard Difference Deviation Andersen and Knudsen [this issue], computed with respect to the EGM96 geoid model Sandwell and Smith [ 1997] Andersen and Knudsen [this issue], computed with respect to the OSU91A geoid model ERS Geosat ERS- 1 and Geosat ERS- 1 and Geosat ERS- 1 and Geosat The locations are shown in Figure 4. All units are mgal. in Figure 3. An additional comparison with a previous global gravity field by Andersen and Knudsen [1995] derived with respecto the OSU91A geopotential model [Rapp et al., 1991 ] has been added. The similar comparison was done with the global gravity field by Sandwell and Smith, [ 1997]. This gravity field is one of their most recent gravity fields (version 7.2), and it will hereinafter be referred to as the Sandwell and Smith gravity field. This gravity field is derived from Geosat and ERS-1 geodetic mission altimetry using slopes. The distributed gravity field has a 1/30 ø longitude spacing and a variable latitude spacing. This was subsequently interpolated onto a 1/16 ø by 1/160grid using collocation. The standard deviations for these five global gravity field solutions range between 5.6 and 7.6 mgal as shown in Table 2. The gravity field from ERS-1 and Geosat derived with respect to the OSU91A geopotential model has the lowest standard deviation of 5.6 mgal, and the Sandwell and Smith gravity field have the highest standar deviation of 7.6 mgal. All satellitederived models agree on a systematic mean difference (ship gravity anomalies minus altimetry derived anomalies) of 4.5 mgal. However, deciding the cause of these systematic errors is difficult, as no information was available on the processing of the BGI marine gravity data. The second region of comparison was selected close to Antarctica in a partly ice-covered region between 60ø-70øS and 0ø-40øE. In this comparison, 3642 observations made by the ship Polar Stern in 1990, were provided to us by the Bundesanstaldt fiir Geowissenschaften und Rohstoffe (BGR) in Germany. Comparison with the similar data set has previously been reported by Kim [ 1996], Trimmer and Manning [ 1996], and Rapp and Yi [1997]. The observed marine gravity observations range from -70 to 85 mgal in this region. The location of the observations is shown in Figure 4 and the comparison reported in Table 3. The global gravity field from ERS-1 and Geosat has the same standard deviation with the marine data set as the Sandwell and Smith global gravity field, namely 3.8 mgal. However, the Sandwell and Smith [1997] gravity field has a mean difference of 2.2 mgal. Similar comparisons with different subsets of the same data set were carried out by Rapp and Yi [ 1997]. The final comparison with marine gravity data was done with a set of 4151 gravity observations in the eastern Mediterranean Sea by Morelli et al. [1975]. The location of the stations is Figure 5. Location of 4151 marine gravity observations in the eastern Mediterranean Sea from Morelli et a/.[1975].

8 8136 ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD Table 4. Comparison With 4151 Gravity Observations in the Eastern Mediterranean Sea Solution Altimetry Mean Standard Difference Deviation Andersen and Knudsen [this issue], computed with respecto the EGM96 geoid model Sandwell and Smith [ 1997] Andersen and Knudsen [this issue], computed with respecto the OSU91A geoid model ERS Geosat ERS- 1 and Geosat ERS- 1 and Geosat ERS- 1 and Geosat Locations can be found in Figure 5. All units are mgal. shown in Figure 5 and the comparison is tabulated in Table 4. This region has a very large gravity signal and the marine observations range from-218 to 116 mgal. Therefore it is expected that the comparison will have a larger standard deviation in this region. The standar deviations with this data set are generally around 10 mgal with the Sandwell and Smith [ 1997] data set having a very large standar deviation of 18.9 mgal. Similarly, the Sandwell and Smith gravity field has a systematic mean difference of 6.6 mgal. In the previous comparison, close to Antarctica, a systematic difference with marine gravity was also found. The cause of this has not been investigated, but the estimation of the medium to long wavelength parts of the gravity field may suffer when using slopes as done by Sandwell and Smith. Comparisons of the Andersen and Knudsen, and Sandwell and Smith gravity fields with marine gravity in the whole of the Mediterranean Sea by Behrend et al. [ 1996] shows similar result. 6. Discussion The method used by Andersen and Knudsen [1995, 1996] has been fine-tuned to obtain an optimal gravity field from a combination of Geosat and ERS-1 geodetic mission altimeter data. This technique has been used in the global recovery of the marine gravity shown in Plate 1. The comparison with three different sets of marine gravity showed very promising results. However, some marine regions may exist where the method does not work as well. Problems with trackiness may be expected in regions with a large ocean variability and in shallow water regions where the ocean tide models may not be sufficiently accurate. Much effort was focused on the removal of trackiness which primarily is caused by sea level variability. Usually, this oceanographic variability shows up as smooth signatures along a ground track and causes no problems before it is combined with the variability along other tracks. A method was derived to model this error by introducing an additional covariance function for this error in the interpolation of the data. Such signals may be filtered out more easily if sea surface slopes are used as observations instead of sea surface heights. However, the estimation of the medium to long wavelength parts of the gravity field may suffer when using slopes. Acknowledgments. This research was partly supported by the Danish Space Board. The Geosat data were kindly provided by NOAA. The ERS-1 data were obtained from ESA to which DUT/DEOS kindly provided the JGM-3 orbits. The authors wishes to thank R. H. Rapp, Ohio State University, for valuable suggestions. References Andersen, O. B., and P. Knudsen, Global gravity field from the ERS-1 geodetic mission, Earth Obs. Q., 47, 1-5, Andersen, O. B., and Po Knudsen, Altimetric gravity field from the full ERS-1 geodetic mission, Phys. Chem. of the Earth, 21(4), , Andersen, O. B., P. Knudsen, and C. C. Tscheming, Investigation of methods for global gravity field recovery from dense ERS-1 geodetic mission altimetry, in Global Gravity Field and Its Temporal Variations, lag Syrup., vol. 116, edited by R. H. Rapp, A. Cazenave, and S. Nerem, pp , Springer- Verlag, New York, Andersen, O. B., P. L. Woodworth, and R. A. Flather, Intercomparison of recent global ocean tide models, J. Geophys. Res., 100(C12), 25,261-25,282, Balmino, G., B. Moynot, M. Sarrailh, and N. Vales, Free air gravity anomalies over the oceans from Seasat and Geos 3 altimeter data, Eos Trans. AGU, 68(2), 17-18, 1987 Behrend, D., H. Denker, and K. Schmidt, Digital gravity data sets for the Mediterranean Sea derived from available maps, Bull d. Inf., 78, 32-41, Bureau Gravimetrique International (BGI), Global gravity data, CD-ROM sea data, version 1.0, CNES, Toulouse-Cedex, France, 1996 Forsberg, R., and D. Solheim, Performance of FFF methods in local gravity field modelling, Chapman Conference on Progress in the Determination of the Earth's Gravity Field, pp , AGU, Ft. Lauderdale, Fla., Haxby, W. F., Gravity field of the worlds oceans (Seasat altimetry), map, Nat. Geophys. Data Cent., Boulder, Colo., Hwang, C., and B. Parsons, Gravity anomalies derived from Seasat, Geosat, ERS-1, and TOPEX/POSEIDON altimetry and ship gravity: A case study over the Reykjanes Ridge, Geophys. J. Int., 122, , Kim, J.-H., Improved recovery of gravity anomalies from dense altimeter data, Rep. 444, 130pp. Dept. of Geod. Sci. and Surv., Ohio State Univ., Columbus, Knudsen, P., Simultaneous estimation of the gravity field and sea surface topography from satellite altimeter data by least squares collocation, Geophys. J. Int., 104, , Knudsen, P., and M. Brovelli, Collinear and cross-over adjustment of Geosat ERM and Seasat altimeter data in the Mediterranean Sea, Surv. Geophys., 14(4-5), , Knudsen, P., O. B. Andersen, and C. C. Tscheming, Altimetric gravity anomalies in the Norwegian-Greenland Sea: Preliminary results from the ERS-1 35 days repeat mission, Geophys. Res. Lett., 19(17), , Lemoine, F. G., et al., The development of the NASA GSFC and DMA joint geopotential model, in Proceedings of the International Symposium on Gravity Geoid and Marine Geodesy, Springer-Verlag, New York, in press, Le Provost, C., M. L. Genco, F. Lyard, P. Vincent, and P. Canceil, Spectroscopy of the world ocean fides from a finite element

9 ,, ANDERSEN AND KNUDSEN: GLOBAL MARINE GRAVITY FIELD 8137 hydrodynamic model, J. Geophys. Res., 99(C12), 24,777-24,797, Morelli, C., M. Pisani, and C. Gantar, Geophysical studies in the Aegean Sea and in the eastern Mediterranean, Boll. Geofis. Teor. Appl., 18, , Rapp, R. H., and Y. Yi. Role of ocean variability and sea surface topography in the recovery of the mean sea surface and gravity anomalies from satellite altimeter data, J. Geod., in press, Rapp, R. H., Y. M. Wang, and N. K. Pavlis, The Ohio State 1991 geopotential and sea surface topography harmoni coefficient models, Rep. 410, Dep. of Geod. and Sci. Surv., Ohio State Univ., Columbus, Rumreel, R., and R. H. N. Haagmans, Gravity gradients from satellite altimetry, Mar. Geod., 14, 1-12, Sandwell, D.T., Antarctic marine gravity field from high-density satellite altimetry, Geophys. J. Int., 109, , 1992o Sandwell, D. T., and W. H. F. Smith, Marine gravity anomaly from Geosat and ERS-1 satellite altimetry, J. Geophys Res. 102(B5), 10,039-10, Schwarz, K.-P., M. G. Sideris, and R. Forsberg, Use of FFT methods in physical geodesy, Geophys. J. lnt., I00, , Shum, Co Kø,et al., Accuracy assessment of recent ocean tide models, J. Geophys. Res., in press, Tapley, Bø D., et al., The Joint Gravity Model 3, J. Geophys. Res., 101(B 12), 28,029-28,049, Trimmer, R., and D. Manning, The altimetry derived gravity anomalies to be used in computing the joint DMA/NASA earth gravity model, in Global Gravity Field and Its Temporal Variations, IAG Symp. vol. 116, edited by R. H. Rapp, A. Cazenave, and S. Nerem, pp , Springer-Verlag, New York, Tscheming, C. C., A strategy for gross-error detection in satellite altimeter data applied in the Baltic Area for enhanced geoid and gravity determination, Determination of the Geoid- Present and Future, IAG Symp., vol. 106, pp. 3-12, edited by R.H. Rapp and Fø Sans6, Springer-Verlag, New York, Tscherning, C. C., P. Knudsen, S. Ekholm, and O. B. Andersen, An analysis of the gravity field in the Norwegian sea using ERS-1 altimeter measurements, in Proceedings of the First ERS-1 symposium, European Space Agency Spec. Publ. ESA SP-359, , Tscheming, C. C., R. Forsberg, and P. Knudsen, The GRAVSOFT package for geoid determination, Proceedings of the First Workshop on the European Geoid, Prague, pp , Springer-Verlag, New York, Ole Baltazar Andersen and Per Knudsen, Kort- og Matrikelstyrelsen, Dk-2400 Copenhagen NV, Denmark. ( oa@kms.- min.dk; pk@kms.min.dk) (Received December 31, 1996; revised June 30, 1997; accepted July 30, 1997.)