Carl A. Wagner

Global sea level change from satellite altimetry

We examine the feasibility of using satellite altimeter data to measure the long-term change of global sea level (estimated from tide gauge data to be a rise of approximately 0.2 cm yr−1). Two and one-half years of collinear Geosat altimeter data (1986–1989) are used together with a 17-day set of Seasat altimetry (July–August 1978) having nearly the same ground track. A consistent set of precise orbits was used throughout, and residual orbit error was removed as a sinusoidal fit to approximately 3-day arcs of sea level collinear differences. The globally averaged Geosat data show sea level falling at 1.2 ± 0.3 cm yr−1 over the first 2 years, even after removal of tide errors and instrument biases not accounted for in the Geosat geophysical data records. This unrealistic result is found to be due largely to long-term error in the ionospheric model for the single-frequency Geosat altimeter. The Geosat-Seasat comparison, based on data 10 years apart, shows an apparent sea level rise of 1.0 cm yr−1. Assuming this result is also unrealistic, a possible explanation is a biased scale to the Doppler-determined Geosat orbit which, unlike Seasat, did not have the benefit of laser tracking. It is also possible that the Geosat altimeter (without external in-orbit calibration) had a bias of the order of 10 cm. We conclude that for satellite altimetry to make a fundamental contribution to monitoring global mean sea level change, both the altimeter (including its media corrections) and the orbit model which provides a geocentric reference for the ocean surface will need continuing and careful calibration with absolute standards.

The accuracy of geopotential models

Abstract Extensive tests of two recent geopotential models (GEM 7 and 8) have been made with observations not used in the solutions. Several other recent models are also evaluated. These tests show the accuracy of the satellite derived model (GEM 7, with 400 coefficients) to be about 4.3 m (r.m.s.) with respect to the global geoid surface. The corresponding accuracy of the combined satellite and surface gravimetry model (GEM 8, with 706 coefficients) is found to be 3.9m (r.m.s.). These results include a calibration for the commission errors of the coefficients in the models and an estimate of the errors from omitted coefficients. For GEM 7, the formal precision (commission errors) of the solution gives 0.7 m for the geoid error which after calibration increases to 2.4 m. Independent observations used in this assessment include: 159 lumped coefficients from 35 resonant orbits of 1 and 9 through 15 revolutions per day, two sets of (8, 8) fields derived from optical-only and laser-only data, sets of zonal and resonant coefficients derived from largely independent sources and geoid undulations measured by satellite altimetry. In addition, the accuracy of GEM 7 has been judged by the gravimetry in GEM 8. The ratio of estimated commission to formal error in GEM 7 and 8 ranges from 2 to 5 in these tests.

An improved error assessment for the GEM‐T1 Gravitational Model

Several tests have been designed to estimate the correct error variances for the GEM-T1 gravitational solution that was derived exclusively from satellite tracking data. The basic method uses both independent and dependent subset data solutions and produces a coefficient by coefficient estimate of the model uncertainties. The GEM-T1 errors have been further analyzed using a method based on eigenvalue-eigenvector analysis, which calibrates the entire covariance matrix. Dependent satellite data sets and independent altimetric, resonant satellite, and surface gravity data sets all confirm essentially the same error assessment The calibration test results yield very stable calibration factors, which vary only by approximately 10% over the range of tests performed. Based on these calibrated error estimates, GEM-T1 is a significantly improved solution, which to degree and order 8 is twice as accurate as earlier satellite derived models like GEM-L2. Also, by being complete to degree and order 36, GEM-T1 is more complete and has significantly reduced aliasing effects that were present in previous models.

Gravitational harmonics from shallow resonant orbits

AbstractUntil very recently, there has been no identification of the significant gravitational constraints on the many common artificial earth satellite orbits in shallow resonance. Without them it is difficult to compare results derived for different sets of harmonics from different orbits. With them it is possible to extend these results to any degree without reintegration of the orbits. All such constraints are shown to be harmonic in the argument of perigee with constants determinable from tracking data:

15th order resonance terms using the decaying orbit of TETR-3

(C*,S*) = (C_0 ,S_0 ) + \sum\limits_{i = 1}^\infty {(C_{Ci} ,S_{Ci} )\cos i\omega + (C_{Si} ,S_{Si} )\sin i\omega .}

Improvements in Arctic Gravity and Geoid from CHAMP and GRACE: An Evaluation

The constants are simple linear combinations of geopotential harmonics. Five such constants (lumped harmonics) have been derived for the GEOS-2 orbit (order 13, to 30th degree) whose principal resonant period is 6 days. These five lumped harmonics are shown to account for almost all (>98%) of the resonant information in the tracking. They agree well with recent gravitational models which include substantial amounts of GEOS-2 data.

Error analyses of resonant orbits for geodesy

Abstract Spherical harmonics are the natural parameters for the Earths gravity field as sensed by orbiting satellites, but problems of resolution arise because the spectrum of effects is narrow and unique to each orbit. Comprehensive gravity models now contain many hundreds of thousands of observations from more than thirty different near-Earth artificial satellites. With refinements in tracking systems, newer data is capable of sensing the spherical harmonics of the field experienced by these satellites to very high degree and order. For example, altimeter, laser and satellite-tracking-satellite systems contain gravitational information well above present levels of satellite gravity field recovery (l = 20), but significant aliasing results because the orbital parameters are too restricted compared to the large number of spherical harmonics. It is shown however that the unique spectrum of information for each satellite contained within a comprehensive spherical harmonic model can be represented by simple gravitational constraint equations (lumped harmonics). All such constraints are harmonic in the argument of perigee (ω) with constants determinable directly from tracking data or reconstituted from the comprehensive solution: (C ∗ , S ∗ ) = (C o , S o ) + Σ i = 1 (C Ci , S Ci ) cos i ω + (C Si , S Si ) sin i ω . The constants are simple linear combinations of the geopotential harmonics. Through these lumped harmonics any satellite gravity field can be decomposed and then uniformly extended to any degree or tailored to a given orbit without reintegration of the trajectory and variational equations. They also make possible the inclusion of information into the field from special deep resonance passages, long arc zonal analyses, and satellites unique to other models. Numerous examples of the derivation, combination, extension and tailoring of the harmonics are presented. The importance of using data spanning an apsidal period is emphasized.

CHAMP and Resonances

Abstract The orbit of TETR-3 (1971-83B), inclination: 33°, passed through resonance with 15th order geopotential terms in February 1972. The resonance caused the orbit inclination to increase by 0.015°. Analysis of 48 sets of mean Kepler elements for this satellite in 1971–1972 (across the resonance) has established the following strong constraint for high degree, 15th order gravitational terms (normalized): 10 9 ( C, S ) 15 = (28.3 ± 3.0, 7.4 ± 3.0) = 0.001( C, S ) 15,15 −0.015( C, S ) 17,15 +0.073( C, S ) 19,15 −0.219( C, S ) 21,15 +0.477( C, S ) 23,15 −0.781( C, S ) 25,15 +1.000( C, S ) 27,15 −0.0963( C, S ) 29,15 +0.622( C, S ) 31,15 −0.119( C, S ) 33,15 −0.290( C, S ) 35,15 +0.403( C, S ) 37,15 −0.223( C, S ) 39,15 −0.058( C, S ) 41,15 +… This result combined with previous results on high inclination 15th order and other resonant orbits suggests that the coefficients of the gravity field beyond the 15th degree are smaller than Kaulas rule ( 10 −5 l 2 ).