Combining multiple altimeter missions

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 102, NO. C10, PAGES 23,187-23,206, OCTOBER 15, 1997 Combining multiple altimeter missions G. A. Jacobs Naval Research Laboratory, Stennis Space Center, Mississippi J. L. Mitchell Colorado Center for Astrodynamics Research, University of Colorado, Boulder Abstract. Viewing altimeter data only at the points where separate altimeter missions' ground tracks cross provides a method to observe long time period sea surface height (SSH) variations and avoids many of the problems inherent in combining separate altimeter data sets through an independently determined geoid. TOPEX/POSEDON (T/P) data over the time period from January 1, 1993, to December 31, 1995, form a mean SSH that is used as a reference by other altimeter data sets. A least squares analysis of the mean T/P SSH determines the portion of the Geographically Correlated Orbit Error (GCOE) that may be observed through crossover differences and removes this portion of the GCOE. The analysis removes errors of 0.86 cm RMS at 1 cycle per orbit revolution (cpr) and indicates negligible e ors at higher frequencies. After the GCOE removal, the accuracy of the T/P reference mean is better than 1 cm RMS as measured by crossover differences. The GCOE contained in the Geosat-Exact Repeat Mission (ERM) and ERS 1 data with orbit solutions using the Joint Gravity Model (JGM) 3 is evaluated through an adjustmento the T/P reference mean surface. The Geosat-ERM data indicate a bias of about 28 cm averaged over the globe, and the ERS 1 bias is 44 cm. The T/P data used here is not corrected for the oscillator drift correction error so that the actual bias is less by about 13 cm. Both the Geosat-ERM and ERS 1 GCOE are mainly 1 cpr. GCOE estimates at frequencies above 1 cpr indicate little actual orbit error but are more correlated to instrument correction errors (particularly water vapor). Simultaneous T/P and ERS 1 SSH anomalies to the T/P mean indicate good correlation. 1. ntroduction this is the most accurate among the data sets [Fu and Cheney, 1995]. Future reprocessing of these data sets will aid in There is a great interest in monitoring eddy- to global-scale mitigating some of the error sources observed here. changes in the ocean environment, and satellite altimeters The main obstacle to merging data sets from separate have demonstrated the capability for monitoring ocean current altimeter satellites is the different marine geoid sampled along variations with global coverage and timescale spanning the the ground tracks of the various altimeter missions. Recent order of 3 years via TOPEX/POSEDON (T/P) [Chelton and altimeters (Geosat-ERM, ERS 1, and T/P) have been in Schlax, 1996; Wunsch and Stammer, 1995; Nerern, 1995] and different repeat orbits, sampling different geographical points the Geosat-Exact Repeat Mission (ERM) [Wagner and and hence different locations on the marine geoid. The Cheney, 1992; Chelton et al., 1990; Zlotnicki et al., 1989]. distance from Earth's referenc ellipsoid to the surface of the Ongoing and planned altimeter missions such as T/P, ERS ocean is defined as sea level (SL). The SL can be viewed as 2, Geosat Follow On (GFO), and Jason ensure the necessary being composed of two parts: the geoid, which is constant in data for continued global studies and monitoring. While each time (or at least on the timescales of interest here), and of these global data sets is very useful for studying dynamics deviations from the geoid due to oceanographic effects such of the order of years, they are not individually extensive as tides, currents, and thermal expansion. The ocean surface enough to study global variations over time periods of height above the geoid is the sea surface height (SSH). Over decades. Thus the current challenge is to link together data distances greater than a few kilometers, the marine geoid is from different altimeter missions to produce a unified data set capable of causing SL variations which are orders of capable of global coverage with timescales potentially magnitude greater than the contribution to SL caused by spanning decades. There have been efforts to combine past oceanographic effects (Figure 1). altimeter data sets such as Seasat and the Geosat-ERM data For an altimeter in a repeat orbit, observing SL variations [Haines, 1992]. n this paper we examine the main error due to changes in the ocean about the mean state is sources involved in combining the present T/P, Geosat-ERM, straightforward. Since the geoid is constant in time, and ERS 1 data sets by using the T/P data as a reference, since subtracting the mean over time at each point along the satellite's ground track removes the geoid influence. This process (known as collinear analysis) also removes the mean This paper is not subjecto U.S. copyright. Published in 1997 by the American Geophysical Union. SSH (the oceanographicomponent of SL) but retains oceanic SSH variations about the mean. Efforts to recover the mean Paper number 97JC SSH caused by mean ocean currents have been undertaken 23,187

2 23,188 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS ascending descending 40N (g 20N o 0 2os E 40S 60S!. _ '' -'--., ß o,,,. '¾') 40N ;'.? ø''';""'' OP 20N 2: 40S os.' - 60S =----, - 40N (L) 20N -0 O 0 2os E 40S 60S L 40N (D 20N -0 O 0 2os E 40S J 60S OE 60E 120E W 60E 120E W 60W Plate 1. The first five modes' (left) ascending and (fight) descending components. An eigenvector decomposition of the least squares matrix determines these modes. Mode numbers 1 and 2 have equal ascending and descending components, thus no crossover differences can be generated by including these modes in a GCOE solution. These modes are in the null space of the least squares matrix that was designed to minimize crossover differences. Modes 3 and 4 are badly measured as determined by the relative amplitude of their eigenvalues (Figure 4). Thus, the first four modes are not included in a solution to determine the GCOE.

3 . ß JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23,189 (a) A s c ending track c orre c tions 40N 20N 20S 40S 60S OE 60E 120E W 60W 0 (b) Descending track corrections 40N, 20N[ o 2os! 40S 60S OE 60E 120E W 60W Plate 2 The T/P GCOE solution for (a) ascending ground tracks and (b) descending ground tracks. The GCOE reduces RMS crossover differences from 1.87 cm to 1.42 cm (Figure 3). Thus the GCOE implies an RMS orbit error of 0.86 cm RMS as measured by the crossover differences.

4 23,190 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS over limited regions with some success. These methods usually assume an a priori shape function of the mean flow and infer the function parameters from the altimeter's variability about the mean [Qiu et al., 1991 ] or use limited in situ measurements at points coincident in space and time with the altimeter data [Mitchell et al., 1990]. Observing SSH variability about its mean (collinear analysis) has been the primary mode of applying altimetry to oceanographic studies. One method for combining multiple altimeter data sets is to use a geoid produced from independent data sources. Many geoids exist on the basis of gravimetric data [Fukuda and Yoichi, 1990; Rapp, 1991]. However, on the scales of the ground track spacing of different altimeter missions (up to 157 km for the T/P and the Geosat-ERM or the T/P and the ERS 1 ground tracks) (Figure 2), the errors in the gravimetric geoids can be larger than the signal due to SSH features. The geoid errors are due to bathymetric features such as seamounts and trenches which have large spatial gradients and are poorly resolved by existing data sets. To remove geoid errors to an acceptable level, the SSH must be smoothed to scales larger than 2000 km [Wunsch, 1991]. While significant ocean circulation changes do exist on scales greater than 2000 km, information on the smaller scales would be lost, and large-scale change does not imply isotropically large scale change. This is certainly the case for the oceans which have much shorter meridional length scales than zonal length scales because of balances between relative and absolute vorticity. The loss in small-scale change is undesirable and can be avoided. The geoid at points where ground tracks of different missions cross is the same for both missions. Thus SSH changes at these points may be accurately observed. The most serious error source contaminating the observed SSH change between separate altimeter missions is the Geographically Correlated Orbit Error (GCOE) of each satellite. This error is constant in time, and the GCOE at any point along an altimeter ground track is an integration of orbit solution errors (mainly due to gravity model errors) along the ground track up to that point. At a point where two ground Ascending c orre c tion Descending correction 40N, - ' -' o -' l a 20N ' 4os J 6os ' ii!1 g 20N 20S 40S 60S 40N Plate 3. The GCOE corrections applied to the Geosat-ERM data set. (a) The corrections required if the GCOE is modeled as a linear trend and a 1 cpr sinusoid. The average bias is about -28 cm. (b) The corrections using frequencies up to 2 cpr to model the GCOE minus the corrections based on using only a 1 cpr GCOE as in (Plate 3a). (c) The corrections using frequencies up to 4 cpr to model the GCOE minus the corrections based on using a 2 cpr GCOE as in (Plate 3b). The GCOE is principally 1 cpr, and the higherfrequency corrections contain the zonal structure of the differences in wet troposphere corrections (Plate 10).

5 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23,191 loo 40N 20N 0 20S 40S S OE 60E W 60W 0! ' Position along track Figure 1. The mean sea level (SL) along one TOPEX/K)SEDON (T/P) ground track. The heights above the referenc ellipsoid produce spatial variations which are orders of magnitude larger than the spatial variations produced by mean currents in the oceans. Present independent geoids have errors which are larger than sea surface height (SSH) variations due to ocean circulation on spatial scales less than a few hundred kilometers. Using data from separate altimeter satellites only at the multimission crossover points circumvents this difficulty. tracks of a single satellite cross, the kn_tegra! of errors along track to the crossover point is different for the two tracks, and so the GCOE is different for the two tracks. Thus crossover differences provide one method of measuring GCOE. However, certain spatial functions can not be observed by crossover differences. For example, a constant bias error along all ground tracks produces no crossover differences. We first estimate the T/P GCOE structure through a crossover difference analysis and at the same time determine the spatial GCOE structure that is not observable (and thus not removed) through the crossover analysis. The T/P mean SL corrected for the observable GCOE used as reference truth surface for estimating the Geosat-ERM and ERS 1 mission GCOE structures by examining the differences between the altimeter mean SL and the T/P reference mean SL at the multimission crossover points. The T/P mean SL surface contains GCOE not observable by the crossover analysis. However, the residual GCOE errors have negligible effects for two reasons. First, the remaining GCOE (expected to be less than 1 cm RMS [Tapley et al., 1996]) is much smaller than many long time period oceanographic signals of interest. Second, since the Geosat-ERM and ERS 1 is

6 23,192 JACOBS AND MTCHELL: COMBN]'qG MULTPLE ALTMETER MSSONS (a) TOPEX/POSEDON ground tracks 50N 40N 30N 20N 1 20W 1 40W 1 60W 1 80 (b) T/P crossovers of the Oeosaf-ERM ground tracks 50N 40N 30N 20N 120W 1 40W 1 60W 180 (c) T/P crossovers of the ERS-1 50N ground tracks 40N 30N 20N 120W 140W 160W 180 Figure 2. (a) Ground track pattern of T/P over the Kuroshio Extension region. Ground tracks are separated by a distance of about 315 km at the equator. Measurements exist about every 6.5 km along the ground tracks. (b) Points at which the Geosat-Exact Repeat Mission (ERM) ground track intersects the T/P ground track. The spacing of the Geosat-ERM ground tracks is about 164 km at the equator. SSH changes from the Geosat-ERM epoch to the T/P epoch are observable at these multimission crossover locations, since the geoid is constant in time. (c) Points at which the ERS 135 day repeat ground track intersects the T/P ground track. The spacing of the ERS 1 ground tracks is about 80 km at the equator.

7 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23,193 mean SL data are fit to the T/P mean SL, the spatial structure errors. The difference in water vapor corrections between of the unobservable T/P GCOE errors (such as a spatially constant field) are identically embedded in the Geosat-ERM T/P, ERS 1, and the Geosat-ERM data does produce some false A SSH. Some of the water vapor correction difference and ERS 1 data and thus will not affect the difference is removed through the GCOE estimation process. between the T/P mean SL and the Geosat-ERM and ERS 1 Explaining the A SSH in terms of changes in ocean mean SL. circulation is beyond the scope of this paper, though We must also be aware that the process used here removes any oceanographic changes between satellite missions that are correlated to the functions that represent the GCOE. One important example is the globally averaged SSH which is correlated to the constant function along every track. Thus the assumption that the T/P mean SL forms a truth surface is reasonable for the purposes of observing long time period SSH changes between altimeter missions that are not correlated to the GCOE function representation. The spatial structure of the oceanographic changes removed by the GCOE estimation process are examined through a collinear analysis of T/P data. The main feature removed through this process is the zonal gradient in the equatorial Pacific. While the mean SL of each data set is corrected, the variability of each data set about its mean does not change through this process. For nonrepeating missions such as the Geosat or ERS 1 geodetic mission phases, the data from each repeat pass would be fit to the T/P reference mean. After significant structures have been studied in other papers [Jacobs et al., 1994; Jacobs and Mitchell, 1996]. n this paper we present the method, examine the mean SL error structure of T/P, the Geosat-ERM, and ERS 1, and verify that other altimeter missions give consistent results relative to the T/P mean SL. We also consider possible contamination of the results from changes in instruments and processing. The observed global changes do not necessarily indicate a long-term trend, nor do we know how representative of longterm change they might be, as accurate satellite altimetry is not available for earlier periods. The decades of archived hydrography are arguably not of sufficient data density or consistency to observe such interannual changes. However, results do give an indication of the amplitudes and types of changes which may be expected on large scales over decadal periods. The construction of the T/P reference mean is covered in section 2, and the fitting of the Geosat-ERM and ERS 1 data correcting the Geosat-ERM and ERS 1 means for GCOE, the to this mean is discussed in section 3. Section 4 discusses T/P mean is subtracted, and the Geosat-ERM and ERS 1 data become anomalies relative to the mean over the 3 years of some of the effects of changes in the corrections applied to the altimeter data sets as well as the ocean signal removed in T/P. The SL anomalies to the T/P reference mean at the the orbit error estimation process. multimission crossover points are mainly oceanographic SSH variations. Some differences in instrument corrections produce errors in the observed SSH change. 2. Construction of a Global Mean The orbit solutions in all the altimeter data sets are based on the Joint Gravity Model (JGM) 3 [Tapley et al., 1996]. T/P data over the time period form the mean SL that serves as a reference for examining global A SSH. After Using different gravity models for the different satellites producing a mean SL along each T/P ground track, GCOE produce slightly different results. For example, if we were to observable by crossover differences is removed by a least use the orbit solutions for the Geosat-ERM based on the squares minimization of mean SL crossover differences Goddard Earth Model (GEM) T2 gravity model and the orbit solutions for T/P based on the JGM-3 gravity model, significant differences between the two satellites' mean SL appear at 2 cycles per orbital revolution (cpr) and not just 1 cpr. The multimission crossover point spacing is roughly of the order of the larger of the ground track spacing of the two missions being combined. When using T/P as the reference, similar to the processing of Fu and Vazquez [1988]. The least squares process used here is not initially solvable because of a set of unobservable modes with particular spatial structure. The amplitude of these modes cannot be determined by the least squares process, so a solution is made by requiring the unobservable mode amplitudes to be zero. Thus only a portion of the GCOE is removed by the crossover analysis, and the spatial structure of the portion not removed is scales of the order of 200 km at midlatitudes are observable indicated by the least squares analysis. (Figure 2). Since the T/P data forms the reference mean SSH, all T/P data along the ground tracks may be used relative to the reference SSH. However, for the Geosat-ERM and ERS 1, examining only multimission crossover differences to T/P eliminates about 95% of the data, since data are used at 200 km intervals along ground tracks instead of 7 km intervals. The results provide a verification of several items. First, the T/? GCOE structure is within expected values based on gravity model and orbit computation estimates [Marshall et al., 1995]. The SSH changes (A SSH) from the Geosat-ERM to the T/P time period are zonally coherent across scales of 4000 km showing the GCOE is sufficiently correctable by referencing to the T/P mean. Simultaneous SSH anomalies from T/P and ERS 1 (both relative to the T/P mean) indicate good correlation. The mean changes in corrections ( A CORR) applied to the T/P and ERS 1 data are compared to determine if observed A SSH is significantly influenced by instrument correction The T/P data are first broken into arcs composed of one satellite revolution beginning and ending at the satellite's southernmost extent (66øS). Arcs that fall on the same ground track are grouped into a set of repeat (or collinear) passes. Computed orbits for T/P are based on the JGM-3 gravity field. Atmosphericorrections (dry troposphere, wet troposphere, and ionosphere) are applied, as well as solid earth tides, an ocean tide estimate [LeProvost et al., 1994], tidal loading, an electromagnetic (EM) bias, and an inverse barometer correction. SSH from all repeat passes is interpolated along ground tracks to the same points spaced by 1 s (i.e., approximately 6.5 km along track). Mean SL along one particular ground track is plotted in Figure 1. Most of this signal is due to the marine geoid. Although the mean ocean currents cause deviations from the geoid, these oceanographic deviations are not separable from the geoid without further independent information. Thus mean ocean currents are not available from the altimeter data

8 23,194 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS... œ epr corrected TOPfiX X-over. RMiS' 1.89 ern... elar correcteel TOPfiX X-overa. RM ' 1.42 ern or ir [ TOPfiX X-overs. RMS' 1.87 ern ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' i Crossover differences Figure 3. The distribution of T/P mean SL crossover differences before and after COlXecting for Geographically Correlated Orbit Error (GCOE) through crossover minimization. The initial RMS crossover difference of 1.87 cm is reduced to 1.42 cm after a 1 cpr GCOE removal. The distribution is not significantly improved if higher frequencies (1 and 2 cpr) are included in the fit. Thus the principle GCOE is at 1 cpr with negligible contributions at higher frequencies. The RMS GCOE removed is about.86 cm RMS. Some GCOE is not measurable by crossover differences (Plate 1), and this remains in the T/P mean SL. alone. The SL at a crossover point is found by a second-order polynomial interpolation of the mean SL along the track to the _ i i, i crossover point. The histogram of crossover differences before removing the GCOE (Figure 3) indicates an RMS value of about 1.87 cm. The GCOE estimate for each T/P ground track is modeled as a combination of four functions' a constant plus a linear trend in time plus a 1 cpr sinusoid with unknown cosine and sine coefficients. The amplitudes of these functions are estimated by a weighted least squares process that minimizes the total mean SL crossover differences along all the T/P -2 i i i! i,1 i! i i i i ground tracks simultaneously. There are certain combinations of the four functions used to model the GCOE for each Mode number i 15 20,, i,, i i ground track that are not measurable by crossover differences. Figure 4. The eigenvalues for the least squares matrix to For example, if the same constant function is added to the solve the GCOE. The eigenvalues for modes 3 and 4 are 4 mean SL along each ground track, the crossover differences orders of magnitude less than the largest eigenvalues and are are not changed. We use the terminology "unobservable 2 orders of magnitude less than the next larger eigenvalues. mode" [Resborough and Marshall, 1990] while others refer to These modes are badly measured. Small errors in the data it as "singularities"[fu and Vazquez, 1988]. t is a vector in produce unrealistically large solutions for these modes, and the null space of the least squares process. We show shortly thus these modes should not be included in a solution.

9 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23,195 OE 6 E 12,0E 18,0 12 _. 60W O, 6 i '" ' '- ' '' - '' ' ',. -r. ii, ; 40N '-. _ ON " - 20N. ':' ' N. ' '-. ' o ' - 40S *,, 6os OE 60E 120E W 60W 0., _1 --.,i, J --, t' :., '. _;. _ ß 40N i ß. --._ OS 40S..., m ß ' - -' -' S OE 60E 120E W 60W 0-10 crr 0 Plate 4. (a) The Geosat-ERM mean SL relative to the T/P reference mean SL after a 1 cpr GCOE correction. (b) The same as (Plate 4a), but including frequencies up to 2 cpr in the GCOE correction. (c) The same as (Plate 4a), but including frequencies up to 4 cpr in the GCOE correction. Most anomalies here are zonally oriented as opposed to GCOE errors which are oriented along ground tracks.

10 23,196 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS A sc ending c orre c tion D e s c ending c orre c { 4ON 20N!. e 0 4os 60S 40N g 20N 0 J.. 20S e J 40S 60S 1, ' 120E W 60W 00E Plate 5. The GCOE corrections applied to the ERS 1 data set. (a) The corrections required by modeling the GCOE as a linear trend plus a 1 cpr sinusoid. (b) The corrections using frequencies up to 2 cpr to model the GCOE minus the corrections based on using only a 1 cpr GCOE as in (Plate 5a). (c) The corrections using frequencies up to 4 cpr to model the GCOE minus the corrections based on using a 2 cpr GCOE as in (Plate 5b). The GCOE errors are mainly 1 cpr. The higher-frequency orbit errors (up to 4 cpr) do not appear to be the results of oceanography. The oceanographic effects on orbit error calculation (Plate 8) do not indicate the same spatial structures. The errors in instrument corrections (Plate 10) do indicate some of the same spatial structure in the changes in wet troposphere correction. Thus a portion of what is labeled GCOE here is actually removing differences in wet troposphere corrections. that there are additional free modes determined from the measurement scheme, and other modes are badly measured by the crossover sampling. The least squares process that estimates the amplitudes ( x ) sinusoids is uncorrelated from one crossover pointo another (the distance between crossover points is about 315 km at the equator and 200 km at midlatitudes). The uncorrelated error assumption is not completely realistic, but it is necessary to of the functions required to minimize the crossover make the problem solvable on available computers. differences ( y ) is defined by Let A = H rwh; in an eigenvalue decomposition of the H r whx = H r wy (1) A matrix of the least squares process, the unobservable modes are eigenvectors with corresponding eigenvalues of where H is the design matrix of the problem (i.e., the /j th zero. The unobservable modes' eigenvectors provide the component of H is the j th function measured at the i th combinations of the four functions for each track which result point). Note that only 2 of the 127 ground tracks (each in no crossover differences. For example, the smallest having four functions representing the GCOE) are measured eigenvalue of the matrix solved here has a corresponding at each crossover point, so each row of H has eight nonzero eigenvector indicating that the constant function for each elements. n (1), W is the weighting matrix which is track has an equal magnitude and the other functions have assumed to be diagonal with the diagonal elements given by zero amplitude for each track. Spatial maps of the ascending the reciprocal of the sum of the variance of the mean SSH of and descending components of the first five modes (Plate 1) the two tracks at the measurement point. This scheme implies indicate that the first two modes have no crossover that error in the mean not modeled by the linear trend and differences, and thus the first two modes are unobservable

11 .. JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23,197 -.½,, 40N, ' ß >. ø '- 2.0S,,...4os N C 20N o i- OE 60E 120E W 60W os 4os i " 4D 6os 40N OE " 20N J o oj 60E mm mm; mmm m mmmmmmmmmmmm m m 120E OE 60E 120E W 60W 0-10 cm 10 Plate 6. (a) The ERS 1 mean SL relative to the T/P reference mean SL after a 1 cpr GCOE correction. (b) The same as (Plate 6a), but including frequencies up to 2 cpr in the GCOE correction. (c) The same as (Plate 6a), but including frequencies up to 4 cpr in the GCOE correction.

12 23,198 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS (-40N i - ',, o x ½)20N,,._. C) ' -, ry Ld40S,. 60S j' '",, -i OE 60E 120E W 60W 40N i :":-, t' td20n E o 2os 40S ' i, ' 60S OE 60E 120E W 60W Plate 7. (a) The ERS 1 SL anomalies to the T/P reference mean averaged over the 1 year time period from December 1, 1992, through November 30, 1993, with a GCOE based on 1 through 4 cpr removed. (b) The T/P anomalies to the T/P reference mean averaged over the same 1 year period as (Plate 7a) with frequencies from 1 to 4 cpr removed. ERS 1 data are available only at multimission crossover points, and T/P data are available at all points along track. The two results indicate good correlation between anomalies relative to a common mean. modes. The two modes do not have eigenvalues exactly equal to zero because of numerical round-off errors, but the eigenvalues are 10 orders of magnitude smaller than the largest eigenvalue. Mode numbers 3 and 4 have eigenvalues that are 2 orders of magnitude less than the next modes (Figure 4), and the eigenvalues are 4 orders of magnitude less than the largest eigenvalue. From an examination of eigenvalues, we determine that even though modes 3 and 4 are not technically unobservable modes they are "badly measured" modes. Noise in the measurements (the mean SL crossover) can produce unreasonably large amplitudes in the badly measured modes. This can be seen through the solution of (1). f the matrix A is written as its eigenvalue decomposition, A = Ar/ A, where A contains the eigenvectors of A as columns and / is a diagonal matrix of eigenvalues, then the solution of (1) is written as x = A rz- z (2) where Z = AHrwy. n (2), each eigenvector, column of A, is divided by its corresponding eigenvalue. f an eigenvalue is very small, noise in y will amplify the eigenmode. To avoid this, these modes are excluded from the solution by setting the corresponding elements of Z-! to 0. This is similar to the Moore-Penrose inverse. Normally, the matrix A is not invertable, and the least squares process does not have a unique solution because A is positive in&finite. Exclusion of the free and badly measured modes results in additional constraints which make the problem solvable. As mentioned above, the first 4 modes of Plate 1 are determined to be free and badly measured modes, while the fifth and subsequent modes are not. The T/P GCOE solution is shown in Plate 2. The histogram of crossover differences after GCOE correction is

13 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23, ' cm RMS' o o.o eosat -- (JCM-3) 6.83 cm RMS' ;g cm RMS' 120, too ! cpr leakage cm RMS' Figure 5. The along track amplitude spectrum of (a) T/P, (b) the Geosat-ERM based on the Joint Gravity Model (JGM) 3, (c) ERS 1 based on the JGM-3 gravity model, and (d) a 1 cpr sinusoid Sampled at the T/P sample points. The spectrare calculated through a least squares estimation because of data outage due to land masking. The 1 cpr test function (Figure 5d) indicates minor leakage of the main spectral peak because of the nonuniform sampling. The spectrare produced by first calculating a spectrum of collinearepeat pass SSH anomaly for each cycle of the data sets and then averaging the spectra over all cycles. The amplitude peaks of the Geosat-ERM and ERS 1 data sets indicate a 1 cpr time-varying orbit error estimation is required.

14 23,200 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS plotted in Figure 3. The RMS crossover difference decrease from 1.87 cm prior to GCOE correction to 1.42 cm RMS after correction implies a GCOE crossover error of fibout 1.22 cm RMS and a corrected mean SL accuracy of 1 cm RMS. Thus the GCOE itself implies an orbit error of about 0.86 cm RMS which is within expected values [Tapley et al., 1994]. To demonstrate that the GCOE is primarily 1 cpr, we calculate the GCOE based on a linear trend and 1 cpr and 2 cpr sinusoid functions. The RMS crossover differences after removing the 1 and 2 cpr frequency GCOE (Figure 3) indicate only a very slight improvement over the 1 cpr frequency GCOE. 3. Fitting the Geosat-ERM and ERS 1 Data to the T/P Mean To observe A SSH, we use data from the Geosat-ERM and ERS 1 altimeter satellites. Both data sets are obtained from the National Oceanic and Atmospheric Administration (NOAA) and are processed similarly. Atmospheric i osat (JGML3) ' ' ' zo, clays o 500 ooo Figure 6. The RMS crossover differences for (a) T/P, (b) the Geosat-ERM based on the JGM-3 gravity model, and (c) ERS 1 based on the JGM-3 gravity model. The RMS crossover differences are calculated for each repeat cycle. Satellites with a long repeat cycle time (such as ERS 1) indicate a slightly higher RMS due to oceanographic SSH changes.

15 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS 23,201 corrections (dry troposphere, wet troposphere, and ionosphere) are applied, as well as solid and ocean tide estimates [LeProvost et al., 1994], an EM bias correction of 2.5% of the significant wave height [Witter and Chelton, 1991 ], and a static inverse barometer correction. The first high-accuracy orbit solutions for the Geosat-ERM were based on the GEM-T2 gravity model [Haines et al., 1990]. The Geosat-ERM orbit solutions have recently been recomputed on the basis of the JGM-3 gravity model and the ERS 1 orbit solutions based on the JGM-3 gravity model have been produced by the Delft nstitute for Earth-Oriented Space Research (DEOS). Significant time-varying orbit errors exist in the Geosat- ERM and ERS 1 data. The relative amplitude of the timevarying orbit error in each data set is examined through two methods. The first method computes an average SSH anomaly spectrum (Figure 5). For this, the mean SSH over time is removed from the collinear repeat samples at each point along all the ground tracks to create SSH anomalies. The SSH anomaly of all ground tracks for a given cycle is used in a least squares estimation of the spectrum. A least squares approach is necessary because of land masking that causes the samples not to be evenly spaced in time. The spectra from all cycles are then averaged. The land masking causes some leakage which can be estimated by sampling a 1 cpr sinusoid with a 1 m amplitude using the T/P sampling scheme and calculating the spectrum (Figure 5). The results indicate only small leakage peaks (about 10 cm). The total RMS variability is inflated by about 5%. The along-track spectrum method does not include variability due to the GCOE but includes variability due to large-scale ocean processes. The second method used to evaluate the time- Because the Geosat-ERM and ERS 1 spectra indicate significantly higher RMS orbit error than the T/P data, a timevarying long wavelength orbit error estimate (modeled as a constant plus a linear trend in time and a sinusoid with 1 cpr frequency) is removed from each repeat pass of Geosat-ERM and ERS 1 data by a weighted least squares method. The one revolution cutoff for the arc length is chosen so that the arc is long enough to well determine the orbit error and short enough so that the orbit error parameters may be assumed constant over the arc. The estimated time-varying orbit error is then removed from each repeat pass of SSH. More details on this processing may be found in work by Jacobs et al. [1992]. The process to remove the time-varying orbit error does nothing to change the mean SSH or the GCOE of the Geosat-ERM or ERS 1. To estimate the GCOE of the Geosat-ERM and ERS 1, the total of the mean SL differences to the T/P reference mean at the multimission crossover points is minimized separately for each track composed of one satellite revolution. This process only corrects the mean of the Geosat-ERM and ERS 1. The SSH anomalies about the mean are not changed. The GCOE energy contained within particular frequency bands is examined for the Geosat-ERM (Plates 3 and 4) and for ERS 1 (Plates 5 and 6). We first model the Geosat-ERM and ERS 1 GCOE as a linear trend in time plus a 1 cpr sinusoid (Plates 3a and 5a). To obtain the energy at 2 cpr, we model the GCOE as a linear trend plus 1 and 2 cpr sinusoids and subtracthe GCOE solution using a linear trend and only a 1 cpr fit (Plates 3b and 5b). The GCOE at 3 and 4 cpr is calculated by modeling the GCOE as a linear trend plus sinusoids from 1 to 4 cpr then subtracting the GCOE using a linear trend and 1 and 2 cpr sinusoids (Plates 3c and 5c). We have referred to the solution as GCOE; however, the GCOE is principally 1 cpr, and higher-frequency solutions are correlated to the changes in atmospheric corrections (particularly water vapor). Thus we expect that the 1 cpr GCOE solution is mainly due to actual GCOE in the Geosat- ERM and ERS 1 orbit solutions, while higher-frequency estimates are actually atmospheric errors. The bias between the Geosat-ERM and T/P (28 cm) and between ERS 1 and T/P (44 cm) is based on T/P data which is not corrected by the oscillator drift algorithm error. Correction for this error would reduce the bias between T/P and the other satellites by 13 cm [Nerern, 1997]. After removing the GCOE, the T/P SL reference mean is subtracted from the Geosat-ERM and ERS 1 mean SL at the multimission crossover points. As SL due to the geoid is the same at multimission crossover points, the remaining Geosat- ERM and ERS 1 anomalies are SSH anomalies to the mean varying orbit error is the RMS crossover differences for each SSH over the T/P reference time period. To demonstrate that cycle of data (Figure 6). The RMS crossover differences SSH variations are observable through this method, we use include variability due to the GCOE and ocean processes at ERS 1 and T/P data over the same 1 year time period and all spatial scales with time periods shorter than one cycle. observe the SSH anomalies relative to the T/P reference For T/P, the peak amplitude at long wavelengths (frequencies less than 1.5 cpr) is about 2 cm. The total RMS energy at long wavelengths contained in the spectrum is 3.56 mean. The ERS 1 data are useful only at the multimission crossover points, since it is only at these points that the T/P reference mean may be used with the ERS 1 data. However, cm. The RMS crossover difference for T/P is about 10 cm the T/P reference mean may be used with the T/P data at all throughouthe mission. The Geosat-ERM data based on the JGM-3 gravity model indicates about 6.82 cm RMS. The last ground track points. Thus the ERS 1 data used are through the multimission crossover analysis, and the T/P data are used year of the Geosat-ERM indicates peak RMS crossover through the single-satellite collinear analysis. The ERS 1 SL differences of 40 cm. The ERS 1 data set used here indicates anomalies to the T/P reference mean are averaged over the a slightly higher long wavelength RMS and crossover RMS time period from December 1, 1992, through November 30, than the Geosat-ERM data (Plate 7a), and the T/P anomalies to the T/P reference mean are averaged over the same time period (Plate 7b). Frequencies from 1 to 4 cpr have been removed from the ERS 1 mean SL relative to the T/P reference to remove GCOE and differences in water vapor corrections. For consistency, sinusoids with frequencies from 1 to 4 cpr are removed from the 1-year-averaged T/P SSH anomaly. The oceanographic signal removed through this process is examined in section 4. Zonally coherent anomalies are well correlated betwee n the two results. Thus the ERS 1 mean SL is referenced to the common mean SL provided by the 3 years of T/P. 4. Discussion of the Results Damage to the Ocean The process of removing orbit errors through empirical methods inevitably removes a portion of the A SSH. Understanding the damage caused to the ocean signal by the

16 23,202 JACOBS AND MTCHELL: COMBN G MULTPLE ALTMETER MSSONS GCOE estimation is important to accurately interpret the Geosat-ERM and the 3 years of the T/P data and difference A SSH of Plate 7. The 3 years of T/P data present an the time-averaged corrections (Plate 10). The globally opportunity to assess the ocean signal damage caused by the GCOE removal. For this study we use collinear T/P data. The time-varying orbit error is small for T/P [Fu et al., 1994]. Thus any changes in mean SSH caused by empirically removing orbit error estimates are mainly due to damage to oceanography. The 3-year mean SL is removed at each point along the T/P ground track. The resulting SSH anomaly data is averaged averaged difference is also removed. The time period over which the instrument corrections are averaged is the same for both T/P and ERS 1; however, the satellite sampling causes some differences in the corrections. Diurnal and semidiurnal variations are aliased differently for T/P and ERS 1, and this effect causes the differences in the ionospheri corrections. ERS 1 is Sun synchronous, so the large diurnal and semidiurnal ionosphere variations are over the 1 year time period used to compare ERS 1 with T/P aliased to the mean. Thus the difference in the mean (December 1, 1992, to November 30, 1993). The 1-yearaveraged SSH anomaly (Plate 8a) represents a SSH uncorrupted by orbit error removal. The effects of removing a 1 cpr function from the 1 year mean SSH anomaly (Plate 8b) are calculated by removing a linear trend plus a 1 cpr sinusoid from the 1-year-averaged SSH anomaly and subtracting the uncorrupted 1-year-averaged SSH anomaly. The corruptions caused by the 1 cpr GCOE removal are mainly global in nature. SSH is raised over the entire western Pacific with peak values of 1 cm and depressed over the eastern Pacific with peak values of 1 cm. The global RMS change is 0.6 cm. Thus these types of changes may not be observed from one altimeter mission to another when 1 cpr GCOE must be removed. The effects of removing higherfrequency functions (Plates 8c and 8d) indicate that ocean signal is removed on gradually smaller spatial scales. By including frequencies from 1 to 4 cpr, basin-scale SSH changes appear in the corrections, and the global RMS error is about 0.9 cm (Plate 8d). Though the basin-scale variations are still observable in each individual data set, the changes in corrections may not be due to actual errors in the corrections but due to aliasing characteristics of the sampling. The algorithm to compute the F_3/ bias is different for T/P and ERS 1. For ERS 1, the F_3/ bias is based on 2.5% of the significant wave height. For T/P, the EM bias from the GDR is based on a more complex algorithm [Hayne et al., 1994]. The wet troposphere correction contains the largest differences in corrections. The T/P wet troposphere correction is based on the onboard water vapor radiometer, while the ERS 1 correction is based on data from the Special Sensor Microwave mager (SSM/). The high-frequency GCOE corrections for ERS 1 (Plates 5b and 5c) indicate some of the same spatial pattern as the wet troposphere correction difference. The GCOE at the higher frequencies (2-4 cpr) is therefore biased by the difference in wet troposphere correction. The effect of the wet troposphere correction difference is reduced in the A SSH (Plate 7) as the GCOE estimation is absorbing a portion of this error. The Geosat-ERM A CORR (Plate 10) indicates that the largest difference in corrections is also due to the water vapor basin-scale circulation from one data set to another will be correction which is based on National Meteorological Center affected. Smaller-scale variations (less than 9000 km) will (NMC) analysis fields. The ionospheri correction for the not be affected. Geosat-ERM data is based on the nternatinal Reference Differences in Corrections onosphere (R) 95 model and indicates errors less than about 1 cm. Apart from the water vapor correction changes, both ERS and the Geosat-ERM corrections indicate less than 1 Observed A SSH (Plates 4 and 6) could be partially contaminated by differences in the environmental and cm peak amplitudes. The wet troposphere corrections instrumental corrections applied to the different altimeter data indicate peak errors of about 3 cm, which is less than the peak sets. Different instruments, processing methods, and SSH changes which are 10 cm. The high-frequency GCOE processing groups can all contribute to what could be estimation further reduces the errors in the wet troposphere misinterpreted as changes in SSH. By examining the correction. The damage caused to the ocean signal by corrections applied to the altimeter data sets and by removing the high-frequency GCOE estimate also indicates comparing these corrections to the observed A SSH, the peaks of about 3 cm (Plate 8d). Thus, at this frequency, the majority of the signal present in the A SSH is not correlated damage to the ocean signal is no larger than errors caused by correction differences. to changes in corrections ( A CORR). The wet troposphere, dry troposphere, ionosphere, and EM bias corrections are extracted from the Geosat-ERM, ERS 1, 5. Conclusions and T/P Geophysical Data Records (GDRs) and are decimated to one-tenth the original data set to reduce the data volume. The four principle corrections for ERS 1 and T/P are averaged over the 1 year time period used in Plate 7 (from December 1, 1992, through November 30, 1993) to facilitate direct comparison. The four principal corrections to ERS 1 and T/P are differenced to produce the A CORR, and the global A reference mean SL is constructed along T/P ground tracks within expected error limits of 1 cm RMS on the basis of crossover differences. The principal T/P GCOE is at 1 cpr with negligible errors at higher frequencies. The GCOE of the Geosat-ERM and ERS 1 indicates the largest errors at 1 cpr. t is possible that the differences in wet troposphere average of each correction difference is removed (Plate 9). corrections between the Geosat-ERM and T/P and between The A CORR may be compared to the ERS 1 A SSH in Plate ERS 1 and T/P may account for some of the apparent GCOE 7. The lack of correlation of structures indicates that the at the higher frequencies. The GCOE removal at frequencies A SSH is not severely contaminated by changes in up to 4 cpr causes errors in the oceanographic SSH changes of environmental or instrumental corrections. Since the order of 0.9 cm RMS. While small, the errors are simultaneous T/P data is not available for the Geosat-ERM correlated to oceanographic features with scales larger than data, we average the corrections over the 3 years of the 9000 km.

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20 23,206 JACOBS AND MTCHELL: COMBNNG MULTPLE ALTMETER MSSONS This analysis does not require that the ERS 1 or Geosat data be from a satellite in a repeating orbit. Thus data from nonrepeat orbit missions could be analyzed relative to the T/P reference mean using the multimission method. This is an important consideration for mission phases such as the 168 day repeat phase of ERS 1, the nonrepeatin geodetic phase of the Geosat mission, and possible future nonrepeating altimeter missions. This is not to advocate nonrepeat missions however, as only a small fraction (i.e., data at multimission crossover points) of the altimeter data returned is actually used in this method. Additionally, as data along ground tracks other than T/P improve because of new orbit solutions and improved corrections, new mean SSH surfaces will increase the multimission crossover point density for application of the adjustmentechnique to future altimeters. Acknowledgments. This work was sponsored by the Office of Naval Research (program element N) as part of the projects "Yellow and East China Seas Response to Winds and Currents" and "Dynamics of Low Latitude Western Boundary Currents". The ability to perform this research is based on the dedication of several groups to provide the science community with high-quality data. n particular, Bob Cheney and John Lillibridge of NOAA have continually updated and improved the Geosat-ERM data that is used here. The Physical Oceanography Data Acquisition and Archival Center (PO-DAAC) at the Jet Propulsion Laboratory provided the T/P data set. We also thank Bruce Haines, who having acted as reviewer of this manuscript, provided valuable assistance in this manuscript' s preparation and certainly improved the content. References Chelton, D. B., and M. G. Schlax, Global observations of oceanic Rossby waves, Science, 272, , Chelton, D. B., M. G. Schlax, L. L. Witter, and J. G. 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Res., 96(C20), , Wunsch, C., Global-scale sea surface variability from combined altimetric and tide gauge measurements, J. Geophys. Res., 96(C8), 15,053-15,082, Wunsch, C., and D. Stammer, The global frequency-wavenumber spectrum of oceanic variability estimated from TOPEX/POSEDON altimetric measurements, J. Geophys. Res., 100(C12), 24,895-24,910, Zlotnicki, V., L.-L. Fu, and W. Patzert, Seasonal variability in global sea level observed with Geosat altimetry, J. Geophys. Res., 94(C12), 17,959-17,969, G. A. Jacobs, Naval Research Laboratory, Stennis Space Center, MS ( jacobs@nrlssc.navy.mil) J. L. Mitchell, Colorado Center for Astrodynamics Research, Boulder CO (Received October 2, 1995; revised March 11, 1997; accepted March 31, 1997.)