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Poseidon and ERS-1 crossover altimetry
Adv. Space Res. Vol. 16, No. 12, pp. (12)131-(12)141. 1995 Copyright Q 1995 COSPAR Printedin Great Britain. All ri hts reserved. 0273-I 177/9P $9.50 + 0.00 0273-1177(95)00435-l
EVALUATION OF JGM 2 GEOPOTENTIAL ERRORS FROM GEOSAT, TOPEX/POSEIDON AND ERS- 1 CROSSOVER ALTIMETRY C. A. Wagner,*
J. Klokocnik**
and C. K. Tai*
* NOAA, Code N/OES 11, 1305 East- West Highway, Silver Spring, IUD 20910. U.S.A. **Astronomical Institute, Czech Academy of Sciences, CZ-251 65 Ondrejov Observatory, Czech Republic
ABSTRACT World-ocean distribution of the crossover altimetry data from Geosat, TOPEX/Poseidon (T/P) and the ERS 1 missions have provided strong independent evidence that NASA’s/CSR’s JGM 2 geopotential model (70 x 70 in spherical harmonics) yields accurate radial ephemerides for these satellites. In testing the sea height crossover differences found from altimetry and JGM 2 orbits for these satellites, we have used the sea height differences themselves (of ascending minus descending passes averaged at each location over many exact repeat cycles) and the Lumped Latitude Coefficients (LLC) derived from them. For Geosat we find the geopotential-induced LLC errors (exclusive of non-gravitational and initial state discrepancies) mostly below 6 cm, for TOPEX the corresponding errors are uslually below 2 cm, and for ERS 1 (35-day cycle) they are generally below 5 cm. In addition, we have found that these observations agree well overall with predictions of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected LLC errors for Geosat, T/P, and ERS 1 are usually between 1 and 4 cm, 1 - 2 cm, and 1 - 4 cm, respectively (they depend on the filtering of long-periodic perturbations and on the order of the LLC). This agreement is especially impressive for ERS 1 since no data of any kind from this mission was used in forming JGM 2. The observed crossover differences for Geosat, T/P and ERS 1 are 8, 3, and 11 cm (rms), respectively. These observations also agree well with prediction of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected crossover errors for Geosat and T/P are 8 cm and 2.3 cm, respectively. The precision of our mean difference observations is about 3 cm for Geosat (approx. 24,000 observations), 1.5 cm for T/P (approx. 6,000 observations) and 5 cm for ERS 1 (approx. 44,000 observations). Thus, these “global” independent data should provide a valuable new source for improving geopotential models. Our results show the need for further correction of the low order JGM 2 geopotential as well as certain resonant orders for all 3 satellites. METHOD The method used within this paper has been suggested by Klokotnik and Wagner (1994) and tested first with the Geosat crossovers and the GEM T2 model (ibid). The method is based on the Latitude Lumped Coefficients (LLC) defined separately for latitudinal belts (with the crossover data) and orders of the harmonic geopotential coefficients, see equation (2) below. The power of the method is its indication of specific orders of a geopotential model in need of correction. Along a latitude belt the crossover values (properly corrected), AX, from a satellite altimetry mission are used in a least-squares adjustment and the “observed” errors in the LLC (presumably) for the geopotential used in the trajectory are derived, equation (1). They are then confronted with the errors of the LLC projected from the relevant variance-covariance matrix (already calibrated or
(12)132
C.A.Wagmetal.
said to be calibrated) for the same range of latitudes and orders, equations (3). If the LLC errors “observed” and the LLC errors “computed” (from the covariance matrix) are independent and if the LLC “observed” are accurate and extensive enough, this comparison results in a fair calibration of the non zonal part of the variance-covariance matrix of the given spherical harmonic gravity field model. (Its sensitivity is greatest at the lower degrees generally because of the diminution of the higher degree field at satellite altitude). The method shows details (the comparison is possible for each latitudinal belt and for each order m separately) but is simultaneously “global” (valid over the whole area with “measured” LLC). We repeat only a minimum of information concerning the definitions and the method. can be found in KlokoEnik and Wagner (1994) and Wagner and KlokoEnik (1994). geopotential effects with respect to a single satellite with a nearly circular orbit (same axis and inclination), it follows from Rosborough’s (1986) theory and from our previous (papers quoted above) that the error of the single-satellite crossovers is given by
All details For purely semimajor derivations
77d77LaX
AX(4,
X) =
c
(dC,
sin mX + 0,
cos mx) ,
(1)
m=l
where the quantities
C,(a,
I, 4),
,!?,(a, I, 9)):
l=lmar C,
=
i=lmaX
c 2Qff,$,m I=772
Sm = -
c 2Q;S,m 1=97l
(2)
are the latitude lumped coefficients (LLC) of the geopotential model, and dC,(a, I, $), dS,(a, I, 4) their errors. In equation (2), the orbit dynamic influence functions Qfm(u, I, 4) (Rosborough, 1986) 1, m, a, I, 4, X are with the same order m were “lumped” together, into one linear combination; degree and order of the harmonics, (mean) semimajor axis of the satellie’s orbit and its inclination, and the geocentric latitude and longitude of the crossovers, respectively. The standard are
deviations
where COVAR(.) covariance matrix
of (2), called errors of “computed”
belong to the particular order to a maximum degree Imar).
LLC, as derived from their definitions,
m only (a submatrix
of the whole
variance-
The BD-diagrams (in the next section) that display our results are based on equation (1) for the “observed” and on equations (3) for the “computed” LLC; they show their powers, i.e. dP&‘” = ,/m
and dPgmp=
&Fz
respectively.
The data for the computations
are of two
types, the crossovers (that need various corrections to be applied before their use), and the calibrated variance-covariance matrices. The Geosat, T/P and ERS 1 crossover data were processed by NOAA. The orbits for the first two satellites were computed by NASA-GSFC and the ERSl orbits were computed at Delft. The covariances for GEM T2 and JGM 2 were kindly provided by NASA-GSFC. Software is partly Czech, US and German. The adjustments of dC,, any additional constraints
dS,
from AX(.) in equation (1) can be performed most simply without put on the crossovers data by using the discrete Fourier transform (on
Evaluatiooof JGM 2 GeopotentialErrors
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filling
the missing data over the continents with zero AX). But in the presence of extensive land areas within a latitude belt, a more accurate approximating signal then zero over the continents may be used to stabilize the least-squares procedure. Finally and more generally, with significant data gaps, apriori values of zero and initial errors for alI Fourier coefficients may be prescribed. Obviously we prefer results in all-marine areas (i.e. the far southern oceans) to avoid any “contamination” by apriori constraints. However this would limit a reasonable survey for significant order-errors too severely. Under these conditions, the adjustments over a wider area with some apriori condition is preferable to achieve a fairer calibration. The apriori information we used generally assumed initial where Rms(z) is the runs of the corrected “white noise” coefficient errors of Rms(z)/~~, crossovers over all latitudinal belts (after the 1 cpr correction), and M is the maximum order of the solution. The value of Rms(z) was (empirically) found to be 30-35 cm for the 2 years of Geosat ERM crossover data set (KlokoEnik and Wagner, 1994)) and 3 cm (rms over the southern oceans between -29 and -66 degrees) here for T/P. For T/P, however, we relaxed the “white noise” value to ratio for the 9 cm, because 3 cm was found to be too restrictive considering the low signal/noise JGM 2 “observations”. With this apriori the total power of the solution over all lumped coefficients was approx. 3 cm (rms), agreeing with the power of the actual data. Similarly we had to relax the apriori errors in the Geosat-JGM 2 solutions to 21 cm to achieve realistic solution values of about 8 cm (rms).
ANALYSES Geosat
(GEM T2 based orbit)
KlokoEnik and Wagner, (1994) examined discrepancies in altimetrically determined sea surface heights at over 30 000 crossover positions (cycles l-63) of Geosat (GEM T2 based orbit) during its Exact Repeat Mission (ERM, 244/17-orbit) in 1986-1989, after removal of many variable media and surface effects as well as the initial condition orbit error. This last error was removed empirically by fitting a secular 1-cpr sinusoid to the altimeter derived crossover height differences. In addition to the %tandard” media corrections (Cheney et al, 1991)) corrections to 4 constituents of the ocean tide model were applied and the effects of a time-tag bias of 3.9 msec with 1-cpr and 2-cpr signature were also removed. Finally the mean of the residual crossover measurements from cycles 1-44 of the Geosat ERM were used (as AX) for the GEM T2 calibration. In this work we tested both types of least-squares adjustments (with and without apriori information) to find the LLC errors for GEM T2, by order, for the southern oceans. Because the signal/noise was so large for this data set the results were similar for most of the latitude bands. Just for comparison purposes with our present results, we reproduce from (Klokocnik and Wagner, 1994) on Figures la to lc the “observed” LLC errors from the unconstrained and constrained/apriori adjustment, and the “computed” LLC errors as projected from the GEM T2 calibrated full variancecovariance matrix (Marsh et al., 1989). GEM T2’s performace was found to be as expected with the notable exception for the low order harmonic (m=3), probably due to a systematic error in the Doppler tracking data of Geosat used in GEM T2. (It does not occur in subsequent models which corrected for this Doppler bias). The “wail” of the “computed” LLC at m=43 (Figure lc) is due to the amplification of resonant perturbations as predicted by the analytical theory (Rosborough, 1986). By contrast, the observed crossover data show little power at 43rd order because of the filtering effect of 1 cpr “orbit error” adjustments. In recent analyses, we prefer to exclude (filter out) these resonant effects (in the computed predictions) before performing the legitimate comparisons. The test of GEM T2 by the Geosat ERM crossover data, using the LLC, was strong crossovers were independent of GEM T2, and precise enough.
because
the
(12)134
C. A. Wagner er al.
Fig. la. The discrepancies of the observed latitude lumped coefficients as adjusted without any apriori constraints from the 2-year mean values of the corrected crossover data of Geosat ER?vI (GEM T2 based orbit). Plotted on the z-axis a.re the powers dl$ in [cm].
Fig, lb. The discrepancies of the observed latitude lumped coeffiricnts apriori constrabrts from t.he P-year mean values of the corrected crossover (GE&f T2 based orbit). Plotted on the z-axis are the powers dP,f” in jcm].
as adjusted with the data of Geosat ER.LI
Fig. lc. The powers of errors of the LLC dPzmp computed from the GEM T2 calibrated variance-covariance matrix for orbit of Geosat (using a 4.day cut filter). To be compared with Figures la and lb.
Evaluation of JGM 2 Gecpoteotial Errors
(12)135
Fig. 2a. The discrepancies of the observed latitude Iumped coefficients as adjusted (without any apriori condition) from selected mean values of the corrected crossover data of Geosat ERM (JGh12 based orbit). The powers dPAbSare in [cm]; notice a finer scale on the z-axis than on Figures 1 a,b,c.
Fig. 2b. The discrepancies of the observed latitude lumped coefficients as adjusted (constrained) (JGhl 2 based orbit). from selected mean values of the corrected crossover data of Ceosat ERbf
T 2.
Fig. 2c. The powers of errors of the LLC dP,?‘p compmed from the JGM 2 calibrated covariance matrix for the Geosat orbit (using a 4-day cut filter).
variance-
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C. A. Wagner et al.
Geosat (JGM 2 based orbit) A choice of the best mean crossover data set (cycles l-22 of ERM) derived from a similar reduction scheme as mentioned above was used with an orbit of Geosat recomputed with JGM 2 (Nerem et al, 1994). But the Geosat/JGM2 crossovers are based on the much denser global coverage of the TRANET system observations for orbit determination and newer ionosphere and tide models. Wagner’s (1994) tide corrections (M2, Kl, 5’2, 01) to the Cartwright and Ray (1991) tides model, as derived from T/P data, were used (instead of the earlier Geosat-based tide models). The initial condition induced error was again reduced using secular 1-cpr sinusoids in each cycle (but in addition a small fraction of 4 0 residual data was also removed). The errors of the “observed” LLC, adjusted without apriori constraints, are plotted on Figure 2a, those with the apriori constraints on Figure 2b. (Notice the finer scale on the z-axis, as compared to Figures 1 a-c). The surfaces on Figures 2 a,b are without sharp peaks, only the errors of the 1st and 3rd orders are consistently higher over all bands. The “observed” LLC errors in Figure 2a show a north-south tilt (higher errors to the north) and another tilt to smaller power at high order. The north-south tilt with this no-apriori adjustment is probably due to the increasing large “continental” data gaps to the north. The results of the mild application of apriori information to the adjustment (including weighting by the variable error of the mean crossover data) is shown in Figure 2b. The power of this adjustment has been made roughly equal to the power of the mean crossover data itself (3 cm, rms). Notice it agrees fairly well at all orders with the “computed” errors projected from the JGM2 covariance matrix (Figure 2~). While this was expected overall from previous analyses, the agreement by order is also satisfying. However, the test of JGM 2 by the Geosat ERM crossover data, using the LLC, can not be as strong as for the GEM T2, because the crossovers are not fully independent of JGM 2 which used direct Geosat altimetry in its formation. In addition, as mentioned previously, JGM 2 is much more accurate than GEM T2 was, so that the signal/noise ratio in this test limited to only the first year of Geosat ERM data is significantly smaller than previously. As a test of consistency Figures 2 a-c show that the LLC errors derived from the Geosat-JGM2 crossovers and the LLC errors computed from the JGM 2 calibrated full variance-covariance matrix are mutually consistent. TOPEX/Poseidon
(JGM
2 based orbit)
Higher expectations were put on the calibration of the JGM 2 variance-covariance matrix by means of wholly independent T/P crossovers (though T/P tracking was used in JGM 2’s development, no altimetry was employed). We have available mean crossover data from 47 ERM cycles of of T/P (127/10-orbit). The average and rms values of these at some 9000 locations are 0.7 and 3.0 cm, respectively. In contrast to Geosat, no empirical orbit or time-tag corrections were found to be necessary for T/P but the data was corrected with the Wagner (1994) T/P-derived tides. Figure 3a shows the “observed” LLC errors (due presumably to JGM 2) from the no-apriori adjustment. As in the Geosat-JGM 2 case the signal tends to increase towards the equator and decrease towards higher orders. There is also a sharp cut-off at the highest order resolved m = 40. It is obvious that the adjustment should involve a higher order, but the latitude belts of T/P crossovers contain at most 127 crossover locations and most have considerably less representation for a reasonable unconstrained adjustment to higher order. Thus to allow a solution to m=50 we would need 101 locations which is available only for a small number of bands in the far southern oceans. To permit high order solutions we need the apriori constrained adjustment more for T/P than for Geosat (whose ERM of 244 revolutions in 17 nodal days yields a maximum of 244 locations in the far southern all-marine bands). The unconstrained solutions in Figure 3a already excludes
Evaluatioo of JGM 2 Geopotential Errors
(WI37
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(‘. A. Wagner cf al.
unsatisfactory results with large values of the unknowns and very large formal standard deviations. These often appeared at the low southern and very high southern latitudes with poor longitude representation. Figure 3b shows the “observed” LLC errors from the apriori adjustment. This case is more promising, the surface of the error power is very flat, without any inclination (notice the very fine scale on the z-axis in contrast to Figures 2a,b or la,b). The least-squares solutions provided LLC values to m=60, but some values are not significant (their formal standard deviations are larger than the values themselves). This consequence of the relatively small signal/noise for this data is responsible for much of the “roughness” of this surface compared to the Geosat-GEM T2 results in Figure lb. Figure 3c contains the powers of the “computed” LLC errors, as computed by means of the previously calibrated variance-covariance matrix of JGM2, complete to degree and order 70 (plotted to m = 50, but insensitive to the zonal terms). The cut in this calculation affects perturbation periods longer than the repeat period (P= 10 days, also the period of each T/P orbit determination) which were ignored. This reduced amplified shallow resonances at m = 13, 25, and 38, but significant shorter-periodic signals at other orders remain. The highest LLC errors are for the lowest orders, as expected, here for m = 2. The relevant harmonic coefficients still need some improvement even in JGM2, but the data for this purpose is difficult to find. The comparison of Figure 3c with 3b reveals very good agreement. This is a strong indication that the variance-covariance matrix of JGMB, provided to us, has already been well calibrated (Nerem et al, 1994). A similar calibration with ERS 1 crossover data has been made that is especially valuable since neither ERS 1 altimetry nor its tracking was used in JGM 2 and the orbit of ERS 1 differs considerably from those of T/P, Geosat and most of the other satellites included in JGM 2.
ERS 1 (JGM 2 based orbit 501/35) From April 1992 to December 1993, the ERS 1 satellite was placed in a 501/35 repeat orbit (501 nodal revolutions per 35 solar days) to obtain roughly twice the spatial resolution of the Geosat ERM at the sacrifice of twice the temporal resolution (35 instead of 17 day repetition). Since much oceanographically interesting activity takes place on seasonal time scales and longer, this sacrifice of time for space resolution was a reasonable compromise for a limited mission. For our purposes, the resolution of geopotential-orbit errors, it provided an ideal testing ground because the maximum 501 longitude locations for each latitude band far exceeds the requirements for all the errors in harmonics likely to be seen at satellite altitude. Indeed while it is true that specific high orders of resonant harmonics may be theoretically visible in our data, in practice we have found that the short periods used for the orbit adjustments of the low altitude satellites (e.g., Geosat and ERS 1) often effectively renders these errors invisible to us. Using the JGM 2 orbits provided by Delft, determined in 3-4 day overlapping periods from Doppler and laser tracking, we have reduced the altimetric sea heights (on the Interim Geophysical Data Records) with similar models for media and tides as T/P and Geosat here (for their JGM 2 orbits). The major difference was that for ERS 1 we used the SSMI wet troposphere correction (from U.S. Defense Dept. satellites) since the ERS 1 on-board radiometer values were not yet available. Prior to the empirical secular 1-cpr orbit error reductions (using the sea heights at crossover locations) we found height differences which ranged from 15 to 35 cm(rms), examining the data in monthly segments. After the correction crossover differences ranged from 13 to 15 cm (T~s). After averaging
(12)139
Evaluation of JGM 2 Geopotential Errors
at nearly 44,000 locations, the mean crossover differences had a power of only 10.8 cm (rms), close to the worldwide value projected by the calibrated JGM 2 covariance matrix. In the ERS 1 empirical orbit error reduction we also solved for (and removed) a varying time tag bias from the data. This bias was small however (compared to Geosat), averaging 1 ms (amounting to a maximum of 3 cm in crossover difference). Figure 4 shows a typical analysis of a well represented southern ocean latitude band of mean ERS 1 crossover residuals (there are 535 such bands, not all visible to marine altimetry). Again, as with Geosat and T/P, there is a clear indication of improvement possible in low order and certain resonant harmonics of JGM 2. In contrast with T/P, there are a fairly large number of orders which have significant signals here at the level of a few centimeters. The power of the LLC errors, computed from the crossover data, is shown on Figure 5a (the unconstrained adjustment, unit weights) and on Figure 5b (the constrained adjustment). Both figures reveal higher power at the 14th-order and for the lowest orders. This is a measure of the remaining error of the harmonic coefficients of JGM 2. An inspection of the formal standard deviations of the adjusted LLC errors (individually for the C and S terms as well as for their power spectra) confirms that most of the powers on Figures 5a and 5b are significant (e.g., Figure 4). Comparison with the projected standard commission errors for this data from the covariance matrix of JGM 2 (cut at 4 days) again shows a good correspondence (Fig. 5c) even for this orbit which is foreign to JGM 2.
Spectrum
of Mecn Apriori
4
-1 L
’ a
=
10
Era l/JGMZ 11.0.
Xover
Diffs.(A-D):Lot:
Power (cm.rms) Actual Data - 9.2,
20
30
so
Solution
50
56.21
South
= 7.3
60
Order
Fig. 4. The spectrum of mean ERSl/JGM2 crossover sea height differences (Ascending - Descending) at 56.21 degrees South. From 1 2/3 years of IGDR data at 498 longitude locations. Apriori errors of 11 cm (white noise) were assumed for the harmonic solutions (maximum order 60, using 121 parameters). A weighted least squares adjustment was performed to find the spectrum. The apriori information has only slight influence on this well represented latitude band. Note the many orders with significant power, especially the 14th order which has a shallow resonance on ERS 1.
DISCUSSION Two notes of caution should be observed with all three analyses. All three orbits were determined with additional empirical l-cpr accelerations to absorb poorly modeled along-track effects on the satellites ostensibly of non-gravitational origin. These terms may have additional filtering effects on the radial residuals we have evaluated. Secondly, there may be other non-gravitational systematic effects in the long term mean crossover residuals that form the basis for our analysis. Previously we have identified unresolved average 1-cpr errors and time tag biases as possible sources of these. While we have removed these sources when they are clearly evident there may be other more subtle biases from media errors which do not average out over these long but still time limited records. Note however that while the additional filtering makes it easier for the “normal” covariance predictions (say with only “cut” filtering) to agree with the “measured” data, the presence of non-gravitational bias makes it harder. But while these two effects tend to cancel in an overall assessment they may introduce unacceptable errors into the data as a source for future geopotential improvement.
(12)140
C. A. Wagner et al.
Fig. 5a. The discrepancies of the observed latitude lumped coeficicnts apriori condition) from the mean values of the corrected crossover data usinrr JGhl 2.
as adjusted (ivithout anv of ERS 1 (501/35-orbitj.
Fig. 5~. The powers of errors of the LLC dPmcomp computed from the JGM 2 calibrated variancecoyariance matrix for the ERS 1 orbit (using a 4-da>! cut filt.er). ‘To be compared to Figures 5 a.b.
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Evaluation of JGM 2 Geopotential Errors
CONCLUSION A test of consistency and a re-calibration of the variance-covariance matrix of the harmonic geopotential coefficients of JGM2 gravity solution has been performed by means of the crossover data sets from altimetry ERMs of Geosat, TOPEX/P oseidon (T/P) and ERS 1, respectively. The latitude lumped coefficients (LLC) derived from the crossovers and from the covariance matrix of JGM2 on southern mostly land-free latitudinal belts were used for the analyses. We find the performace of JGM 2 is as expected; its variance-covariance matrix seems to be sufficiently conservative.
Acknowledgment Our thanks are due to Nancy Doyle for additional Geosat crossover processing. The research has partly been supported by grant # 205/93/0894 of the Grant Agency of the Czech Republic.
REFERENCES 1. D.E. Cartwright, R.D. Ray, Energetics phys. Res. 96 (C9), 16,897 (1991).
of Global
Ocean Tides from Geosat Altimetry,
GEOSAT Altimeter Crossover Difference 2. R.E. Cheney, B.C. Douglas, R.W. Agreen, NOAA Manual NOS, NGS 6, National Ocean Service, Rockville (1991). A Test of GEM T2 from GEOSAT 3. J. KlokoEnik, C.A. Wagner, Lumped Coefficients, Bull. Geod. 68, 100 (1994).
Crossovers
J. Geo-
Handbook,
using
Latitude
4. J.G. Marsh, and 16 others, The GEM-T2 Gravitational Model, NASA TM 100746, Greenbelt (1989); see also: in Sea Surface Topography and the Geoid, IAG Symp. 104, eds.: H.Sfinkel and T.Baker, Springer-Verlag, pp. l-10, (1990). 5. R. S. Nerem, and 19 others, Gravity Model Development for TOPEX/Poseidon: Joint Gravity Models 1 and 2, J. Geophys. Res. (Oceans), TOPEX/Poseidon Special Issue, (in press). 6. G.W. Rosborough, Satellite Orbit Perturbations due to the Geopotential, Texas at Austin, Center for Space Research (1986). Accuracy of the GEM T2 Geopotential 7. C.A. Wagner, J. Klokocnik, Crossover Altimetry, J. Geophys Res. 99 (B5), 9179 (1994).
CSR-86-l
from GEOSAT
Univ. of
and ERS-1
EVALUATION OF JGM 2 GEOPOTENTIAL ERRORS FROM GEOSAT, TOPEX/POSEIDON AND ERS- 1 CROSSOVER ALTIMETRY C. A. Wagner,*
J. Klokocnik**
and C. K. Tai*
* NOAA, Code N/OES 11, 1305 East- West Highway, Silver Spring, IUD 20910. U.S.A. **Astronomical Institute, Czech Academy of Sciences, CZ-251 65 Ondrejov Observatory, Czech Republic
ABSTRACT World-ocean distribution of the crossover altimetry data from Geosat, TOPEX/Poseidon (T/P) and the ERS 1 missions have provided strong independent evidence that NASA’s/CSR’s JGM 2 geopotential model (70 x 70 in spherical harmonics) yields accurate radial ephemerides for these satellites. In testing the sea height crossover differences found from altimetry and JGM 2 orbits for these satellites, we have used the sea height differences themselves (of ascending minus descending passes averaged at each location over many exact repeat cycles) and the Lumped Latitude Coefficients (LLC) derived from them. For Geosat we find the geopotential-induced LLC errors (exclusive of non-gravitational and initial state discrepancies) mostly below 6 cm, for TOPEX the corresponding errors are uslually below 2 cm, and for ERS 1 (35-day cycle) they are generally below 5 cm. In addition, we have found that these observations agree well overall with predictions of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected LLC errors for Geosat, T/P, and ERS 1 are usually between 1 and 4 cm, 1 - 2 cm, and 1 - 4 cm, respectively (they depend on the filtering of long-periodic perturbations and on the order of the LLC). This agreement is especially impressive for ERS 1 since no data of any kind from this mission was used in forming JGM 2. The observed crossover differences for Geosat, T/P and ERS 1 are 8, 3, and 11 cm (rms), respectively. These observations also agree well with prediction of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected crossover errors for Geosat and T/P are 8 cm and 2.3 cm, respectively. The precision of our mean difference observations is about 3 cm for Geosat (approx. 24,000 observations), 1.5 cm for T/P (approx. 6,000 observations) and 5 cm for ERS 1 (approx. 44,000 observations). Thus, these “global” independent data should provide a valuable new source for improving geopotential models. Our results show the need for further correction of the low order JGM 2 geopotential as well as certain resonant orders for all 3 satellites. METHOD The method used within this paper has been suggested by Klokotnik and Wagner (1994) and tested first with the Geosat crossovers and the GEM T2 model (ibid). The method is based on the Latitude Lumped Coefficients (LLC) defined separately for latitudinal belts (with the crossover data) and orders of the harmonic geopotential coefficients, see equation (2) below. The power of the method is its indication of specific orders of a geopotential model in need of correction. Along a latitude belt the crossover values (properly corrected), AX, from a satellite altimetry mission are used in a least-squares adjustment and the “observed” errors in the LLC (presumably) for the geopotential used in the trajectory are derived, equation (1). They are then confronted with the errors of the LLC projected from the relevant variance-covariance matrix (already calibrated or
(12)132
C.A.Wagmetal.
said to be calibrated) for the same range of latitudes and orders, equations (3). If the LLC errors “observed” and the LLC errors “computed” (from the covariance matrix) are independent and if the LLC “observed” are accurate and extensive enough, this comparison results in a fair calibration of the non zonal part of the variance-covariance matrix of the given spherical harmonic gravity field model. (Its sensitivity is greatest at the lower degrees generally because of the diminution of the higher degree field at satellite altitude). The method shows details (the comparison is possible for each latitudinal belt and for each order m separately) but is simultaneously “global” (valid over the whole area with “measured” LLC). We repeat only a minimum of information concerning the definitions and the method. can be found in KlokoEnik and Wagner (1994) and Wagner and KlokoEnik (1994). geopotential effects with respect to a single satellite with a nearly circular orbit (same axis and inclination), it follows from Rosborough’s (1986) theory and from our previous (papers quoted above) that the error of the single-satellite crossovers is given by
All details For purely semimajor derivations
77d77LaX
AX(4,
X) =
c
(dC,
sin mX + 0,
cos mx) ,
(1)
m=l
where the quantities
C,(a,
I, 4),
,!?,(a, I, 9)):
l=lmar C,
=
i=lmaX
c 2Qff,$,m I=772
Sm = -
c 2Q;S,m 1=97l
(2)
are the latitude lumped coefficients (LLC) of the geopotential model, and dC,(a, I, $), dS,(a, I, 4) their errors. In equation (2), the orbit dynamic influence functions Qfm(u, I, 4) (Rosborough, 1986) 1, m, a, I, 4, X are with the same order m were “lumped” together, into one linear combination; degree and order of the harmonics, (mean) semimajor axis of the satellie’s orbit and its inclination, and the geocentric latitude and longitude of the crossovers, respectively. The standard are
deviations
where COVAR(.) covariance matrix
of (2), called errors of “computed”
belong to the particular order to a maximum degree Imar).
LLC, as derived from their definitions,
m only (a submatrix
of the whole
variance-
The BD-diagrams (in the next section) that display our results are based on equation (1) for the “observed” and on equations (3) for the “computed” LLC; they show their powers, i.e. dP&‘” = ,/m
and dPgmp=
&Fz
respectively.
The data for the computations
are of two
types, the crossovers (that need various corrections to be applied before their use), and the calibrated variance-covariance matrices. The Geosat, T/P and ERS 1 crossover data were processed by NOAA. The orbits for the first two satellites were computed by NASA-GSFC and the ERSl orbits were computed at Delft. The covariances for GEM T2 and JGM 2 were kindly provided by NASA-GSFC. Software is partly Czech, US and German. The adjustments of dC,, any additional constraints
dS,
from AX(.) in equation (1) can be performed most simply without put on the crossovers data by using the discrete Fourier transform (on
Evaluatiooof JGM 2 GeopotentialErrors
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filling
the missing data over the continents with zero AX). But in the presence of extensive land areas within a latitude belt, a more accurate approximating signal then zero over the continents may be used to stabilize the least-squares procedure. Finally and more generally, with significant data gaps, apriori values of zero and initial errors for alI Fourier coefficients may be prescribed. Obviously we prefer results in all-marine areas (i.e. the far southern oceans) to avoid any “contamination” by apriori constraints. However this would limit a reasonable survey for significant order-errors too severely. Under these conditions, the adjustments over a wider area with some apriori condition is preferable to achieve a fairer calibration. The apriori information we used generally assumed initial where Rms(z) is the runs of the corrected “white noise” coefficient errors of Rms(z)/~~, crossovers over all latitudinal belts (after the 1 cpr correction), and M is the maximum order of the solution. The value of Rms(z) was (empirically) found to be 30-35 cm for the 2 years of Geosat ERM crossover data set (KlokoEnik and Wagner, 1994)) and 3 cm (rms over the southern oceans between -29 and -66 degrees) here for T/P. For T/P, however, we relaxed the “white noise” value to ratio for the 9 cm, because 3 cm was found to be too restrictive considering the low signal/noise JGM 2 “observations”. With this apriori the total power of the solution over all lumped coefficients was approx. 3 cm (rms), agreeing with the power of the actual data. Similarly we had to relax the apriori errors in the Geosat-JGM 2 solutions to 21 cm to achieve realistic solution values of about 8 cm (rms).
ANALYSES Geosat
(GEM T2 based orbit)
KlokoEnik and Wagner, (1994) examined discrepancies in altimetrically determined sea surface heights at over 30 000 crossover positions (cycles l-63) of Geosat (GEM T2 based orbit) during its Exact Repeat Mission (ERM, 244/17-orbit) in 1986-1989, after removal of many variable media and surface effects as well as the initial condition orbit error. This last error was removed empirically by fitting a secular 1-cpr sinusoid to the altimeter derived crossover height differences. In addition to the %tandard” media corrections (Cheney et al, 1991)) corrections to 4 constituents of the ocean tide model were applied and the effects of a time-tag bias of 3.9 msec with 1-cpr and 2-cpr signature were also removed. Finally the mean of the residual crossover measurements from cycles 1-44 of the Geosat ERM were used (as AX) for the GEM T2 calibration. In this work we tested both types of least-squares adjustments (with and without apriori information) to find the LLC errors for GEM T2, by order, for the southern oceans. Because the signal/noise was so large for this data set the results were similar for most of the latitude bands. Just for comparison purposes with our present results, we reproduce from (Klokocnik and Wagner, 1994) on Figures la to lc the “observed” LLC errors from the unconstrained and constrained/apriori adjustment, and the “computed” LLC errors as projected from the GEM T2 calibrated full variancecovariance matrix (Marsh et al., 1989). GEM T2’s performace was found to be as expected with the notable exception for the low order harmonic (m=3), probably due to a systematic error in the Doppler tracking data of Geosat used in GEM T2. (It does not occur in subsequent models which corrected for this Doppler bias). The “wail” of the “computed” LLC at m=43 (Figure lc) is due to the amplification of resonant perturbations as predicted by the analytical theory (Rosborough, 1986). By contrast, the observed crossover data show little power at 43rd order because of the filtering effect of 1 cpr “orbit error” adjustments. In recent analyses, we prefer to exclude (filter out) these resonant effects (in the computed predictions) before performing the legitimate comparisons. The test of GEM T2 by the Geosat ERM crossover data, using the LLC, was strong crossovers were independent of GEM T2, and precise enough.
because
the
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C. A. Wagner er al.
Fig. la. The discrepancies of the observed latitude lumped coefficients as adjusted without any apriori constraints from the 2-year mean values of the corrected crossover data of Geosat ER?vI (GEM T2 based orbit). Plotted on the z-axis a.re the powers dl$ in [cm].
Fig, lb. The discrepancies of the observed latitude lumped coeffiricnts apriori constrabrts from t.he P-year mean values of the corrected crossover (GE&f T2 based orbit). Plotted on the z-axis are the powers dP,f” in jcm].
as adjusted with the data of Geosat ER.LI
Fig. lc. The powers of errors of the LLC dPzmp computed from the GEM T2 calibrated variance-covariance matrix for orbit of Geosat (using a 4.day cut filter). To be compared with Figures la and lb.
Evaluation of JGM 2 Gecpoteotial Errors
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Fig. 2a. The discrepancies of the observed latitude Iumped coefficients as adjusted (without any apriori condition) from selected mean values of the corrected crossover data of Geosat ERM (JGh12 based orbit). The powers dPAbSare in [cm]; notice a finer scale on the z-axis than on Figures 1 a,b,c.
Fig. 2b. The discrepancies of the observed latitude lumped coefficients as adjusted (constrained) (JGhl 2 based orbit). from selected mean values of the corrected crossover data of Ceosat ERbf
T 2.
Fig. 2c. The powers of errors of the LLC dP,?‘p compmed from the JGM 2 calibrated covariance matrix for the Geosat orbit (using a 4-day cut filter).
variance-
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C. A. Wagner et al.
Geosat (JGM 2 based orbit) A choice of the best mean crossover data set (cycles l-22 of ERM) derived from a similar reduction scheme as mentioned above was used with an orbit of Geosat recomputed with JGM 2 (Nerem et al, 1994). But the Geosat/JGM2 crossovers are based on the much denser global coverage of the TRANET system observations for orbit determination and newer ionosphere and tide models. Wagner’s (1994) tide corrections (M2, Kl, 5’2, 01) to the Cartwright and Ray (1991) tides model, as derived from T/P data, were used (instead of the earlier Geosat-based tide models). The initial condition induced error was again reduced using secular 1-cpr sinusoids in each cycle (but in addition a small fraction of 4 0 residual data was also removed). The errors of the “observed” LLC, adjusted without apriori constraints, are plotted on Figure 2a, those with the apriori constraints on Figure 2b. (Notice the finer scale on the z-axis, as compared to Figures 1 a-c). The surfaces on Figures 2 a,b are without sharp peaks, only the errors of the 1st and 3rd orders are consistently higher over all bands. The “observed” LLC errors in Figure 2a show a north-south tilt (higher errors to the north) and another tilt to smaller power at high order. The north-south tilt with this no-apriori adjustment is probably due to the increasing large “continental” data gaps to the north. The results of the mild application of apriori information to the adjustment (including weighting by the variable error of the mean crossover data) is shown in Figure 2b. The power of this adjustment has been made roughly equal to the power of the mean crossover data itself (3 cm, rms). Notice it agrees fairly well at all orders with the “computed” errors projected from the JGM2 covariance matrix (Figure 2~). While this was expected overall from previous analyses, the agreement by order is also satisfying. However, the test of JGM 2 by the Geosat ERM crossover data, using the LLC, can not be as strong as for the GEM T2, because the crossovers are not fully independent of JGM 2 which used direct Geosat altimetry in its formation. In addition, as mentioned previously, JGM 2 is much more accurate than GEM T2 was, so that the signal/noise ratio in this test limited to only the first year of Geosat ERM data is significantly smaller than previously. As a test of consistency Figures 2 a-c show that the LLC errors derived from the Geosat-JGM2 crossovers and the LLC errors computed from the JGM 2 calibrated full variance-covariance matrix are mutually consistent. TOPEX/Poseidon
(JGM
2 based orbit)
Higher expectations were put on the calibration of the JGM 2 variance-covariance matrix by means of wholly independent T/P crossovers (though T/P tracking was used in JGM 2’s development, no altimetry was employed). We have available mean crossover data from 47 ERM cycles of of T/P (127/10-orbit). The average and rms values of these at some 9000 locations are 0.7 and 3.0 cm, respectively. In contrast to Geosat, no empirical orbit or time-tag corrections were found to be necessary for T/P but the data was corrected with the Wagner (1994) T/P-derived tides. Figure 3a shows the “observed” LLC errors (due presumably to JGM 2) from the no-apriori adjustment. As in the Geosat-JGM 2 case the signal tends to increase towards the equator and decrease towards higher orders. There is also a sharp cut-off at the highest order resolved m = 40. It is obvious that the adjustment should involve a higher order, but the latitude belts of T/P crossovers contain at most 127 crossover locations and most have considerably less representation for a reasonable unconstrained adjustment to higher order. Thus to allow a solution to m=50 we would need 101 locations which is available only for a small number of bands in the far southern oceans. To permit high order solutions we need the apriori constrained adjustment more for T/P than for Geosat (whose ERM of 244 revolutions in 17 nodal days yields a maximum of 244 locations in the far southern all-marine bands). The unconstrained solutions in Figure 3a already excludes
Evaluatioo of JGM 2 Geopotential Errors
(WI37
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(‘. A. Wagner cf al.
unsatisfactory results with large values of the unknowns and very large formal standard deviations. These often appeared at the low southern and very high southern latitudes with poor longitude representation. Figure 3b shows the “observed” LLC errors from the apriori adjustment. This case is more promising, the surface of the error power is very flat, without any inclination (notice the very fine scale on the z-axis in contrast to Figures 2a,b or la,b). The least-squares solutions provided LLC values to m=60, but some values are not significant (their formal standard deviations are larger than the values themselves). This consequence of the relatively small signal/noise for this data is responsible for much of the “roughness” of this surface compared to the Geosat-GEM T2 results in Figure lb. Figure 3c contains the powers of the “computed” LLC errors, as computed by means of the previously calibrated variance-covariance matrix of JGM2, complete to degree and order 70 (plotted to m = 50, but insensitive to the zonal terms). The cut in this calculation affects perturbation periods longer than the repeat period (P= 10 days, also the period of each T/P orbit determination) which were ignored. This reduced amplified shallow resonances at m = 13, 25, and 38, but significant shorter-periodic signals at other orders remain. The highest LLC errors are for the lowest orders, as expected, here for m = 2. The relevant harmonic coefficients still need some improvement even in JGM2, but the data for this purpose is difficult to find. The comparison of Figure 3c with 3b reveals very good agreement. This is a strong indication that the variance-covariance matrix of JGMB, provided to us, has already been well calibrated (Nerem et al, 1994). A similar calibration with ERS 1 crossover data has been made that is especially valuable since neither ERS 1 altimetry nor its tracking was used in JGM 2 and the orbit of ERS 1 differs considerably from those of T/P, Geosat and most of the other satellites included in JGM 2.
ERS 1 (JGM 2 based orbit 501/35) From April 1992 to December 1993, the ERS 1 satellite was placed in a 501/35 repeat orbit (501 nodal revolutions per 35 solar days) to obtain roughly twice the spatial resolution of the Geosat ERM at the sacrifice of twice the temporal resolution (35 instead of 17 day repetition). Since much oceanographically interesting activity takes place on seasonal time scales and longer, this sacrifice of time for space resolution was a reasonable compromise for a limited mission. For our purposes, the resolution of geopotential-orbit errors, it provided an ideal testing ground because the maximum 501 longitude locations for each latitude band far exceeds the requirements for all the errors in harmonics likely to be seen at satellite altitude. Indeed while it is true that specific high orders of resonant harmonics may be theoretically visible in our data, in practice we have found that the short periods used for the orbit adjustments of the low altitude satellites (e.g., Geosat and ERS 1) often effectively renders these errors invisible to us. Using the JGM 2 orbits provided by Delft, determined in 3-4 day overlapping periods from Doppler and laser tracking, we have reduced the altimetric sea heights (on the Interim Geophysical Data Records) with similar models for media and tides as T/P and Geosat here (for their JGM 2 orbits). The major difference was that for ERS 1 we used the SSMI wet troposphere correction (from U.S. Defense Dept. satellites) since the ERS 1 on-board radiometer values were not yet available. Prior to the empirical secular 1-cpr orbit error reductions (using the sea heights at crossover locations) we found height differences which ranged from 15 to 35 cm(rms), examining the data in monthly segments. After the correction crossover differences ranged from 13 to 15 cm (T~s). After averaging
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Evaluation of JGM 2 Geopotential Errors
at nearly 44,000 locations, the mean crossover differences had a power of only 10.8 cm (rms), close to the worldwide value projected by the calibrated JGM 2 covariance matrix. In the ERS 1 empirical orbit error reduction we also solved for (and removed) a varying time tag bias from the data. This bias was small however (compared to Geosat), averaging 1 ms (amounting to a maximum of 3 cm in crossover difference). Figure 4 shows a typical analysis of a well represented southern ocean latitude band of mean ERS 1 crossover residuals (there are 535 such bands, not all visible to marine altimetry). Again, as with Geosat and T/P, there is a clear indication of improvement possible in low order and certain resonant harmonics of JGM 2. In contrast with T/P, there are a fairly large number of orders which have significant signals here at the level of a few centimeters. The power of the LLC errors, computed from the crossover data, is shown on Figure 5a (the unconstrained adjustment, unit weights) and on Figure 5b (the constrained adjustment). Both figures reveal higher power at the 14th-order and for the lowest orders. This is a measure of the remaining error of the harmonic coefficients of JGM 2. An inspection of the formal standard deviations of the adjusted LLC errors (individually for the C and S terms as well as for their power spectra) confirms that most of the powers on Figures 5a and 5b are significant (e.g., Figure 4). Comparison with the projected standard commission errors for this data from the covariance matrix of JGM 2 (cut at 4 days) again shows a good correspondence (Fig. 5c) even for this orbit which is foreign to JGM 2.
Spectrum
of Mecn Apriori
4
-1 L
’ a
=
10
Era l/JGMZ 11.0.
Xover
Diffs.(A-D):Lot:
Power (cm.rms) Actual Data - 9.2,
20
30
so
Solution
50
56.21
South
= 7.3
60
Order
Fig. 4. The spectrum of mean ERSl/JGM2 crossover sea height differences (Ascending - Descending) at 56.21 degrees South. From 1 2/3 years of IGDR data at 498 longitude locations. Apriori errors of 11 cm (white noise) were assumed for the harmonic solutions (maximum order 60, using 121 parameters). A weighted least squares adjustment was performed to find the spectrum. The apriori information has only slight influence on this well represented latitude band. Note the many orders with significant power, especially the 14th order which has a shallow resonance on ERS 1.
DISCUSSION Two notes of caution should be observed with all three analyses. All three orbits were determined with additional empirical l-cpr accelerations to absorb poorly modeled along-track effects on the satellites ostensibly of non-gravitational origin. These terms may have additional filtering effects on the radial residuals we have evaluated. Secondly, there may be other non-gravitational systematic effects in the long term mean crossover residuals that form the basis for our analysis. Previously we have identified unresolved average 1-cpr errors and time tag biases as possible sources of these. While we have removed these sources when they are clearly evident there may be other more subtle biases from media errors which do not average out over these long but still time limited records. Note however that while the additional filtering makes it easier for the “normal” covariance predictions (say with only “cut” filtering) to agree with the “measured” data, the presence of non-gravitational bias makes it harder. But while these two effects tend to cancel in an overall assessment they may introduce unacceptable errors into the data as a source for future geopotential improvement.
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C. A. Wagner et al.
Fig. 5a. The discrepancies of the observed latitude lumped coeficicnts apriori condition) from the mean values of the corrected crossover data usinrr JGhl 2.
as adjusted (ivithout anv of ERS 1 (501/35-orbitj.
Fig. 5~. The powers of errors of the LLC dPmcomp computed from the JGM 2 calibrated variancecoyariance matrix for the ERS 1 orbit (using a 4-da>! cut filt.er). ‘To be compared to Figures 5 a.b.
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Evaluation of JGM 2 Geopotential Errors
CONCLUSION A test of consistency and a re-calibration of the variance-covariance matrix of the harmonic geopotential coefficients of JGM2 gravity solution has been performed by means of the crossover data sets from altimetry ERMs of Geosat, TOPEX/P oseidon (T/P) and ERS 1, respectively. The latitude lumped coefficients (LLC) derived from the crossovers and from the covariance matrix of JGM2 on southern mostly land-free latitudinal belts were used for the analyses. We find the performace of JGM 2 is as expected; its variance-covariance matrix seems to be sufficiently conservative.
Acknowledgment Our thanks are due to Nancy Doyle for additional Geosat crossover processing. The research has partly been supported by grant # 205/93/0894 of the Grant Agency of the Czech Republic.
REFERENCES 1. D.E. Cartwright, R.D. Ray, Energetics phys. Res. 96 (C9), 16,897 (1991).
of Global
Ocean Tides from Geosat Altimetry,
GEOSAT Altimeter Crossover Difference 2. R.E. Cheney, B.C. Douglas, R.W. Agreen, NOAA Manual NOS, NGS 6, National Ocean Service, Rockville (1991). A Test of GEM T2 from GEOSAT 3. J. KlokoEnik, C.A. Wagner, Lumped Coefficients, Bull. Geod. 68, 100 (1994).
Crossovers
J. Geo-
Handbook,
using
Latitude
4. J.G. Marsh, and 16 others, The GEM-T2 Gravitational Model, NASA TM 100746, Greenbelt (1989); see also: in Sea Surface Topography and the Geoid, IAG Symp. 104, eds.: H.Sfinkel and T.Baker, Springer-Verlag, pp. l-10, (1990). 5. R. S. Nerem, and 19 others, Gravity Model Development for TOPEX/Poseidon: Joint Gravity Models 1 and 2, J. Geophys. Res. (Oceans), TOPEX/Poseidon Special Issue, (in press). 6. G.W. Rosborough, Satellite Orbit Perturbations due to the Geopotential, Texas at Austin, Center for Space Research (1986). Accuracy of the GEM T2 Geopotential 7. C.A. Wagner, J. Klokocnik, Crossover Altimetry, J. Geophys Res. 99 (B5), 9179 (1994).
CSR-86-l
from GEOSAT
Univ. of
and ERS-1