On Reducing the Large-scale Time-dependent Errors in Satellite
Altimetry While Preserving the Ocean Signal: Orbit and Tide Error
Reduction for TOPEX/POSEIDON
Chang-Kou Tai and John M. Kuhn
NOAA Tech. Memo. NOS OES 9, 1994
National Ocean Service, NOAA, Silver Spring MD 20910
ABSTRACT
A crossover adjustment scheme for reducing the orbit, tide and
any other large-scale time-dependent errors in satellite altimetry
while preserving the ocean signal is devised and tested with 130
days of TOPEX/POSEIDON data. It employs the Fourier series
truncated at 3.33 cycle per revolution (i.e., 3.33 cpr, about
12,000 km in along-track wavelength) to accurately describe these
large-scale errors. The adjustment is limited to each 10-day repeat
cycle of TOPEX/POSEIDON to ensure the preservation of the large-
scale ocean signal, which can be shown analytically (along with all
large-scale errors) to undergo transformations to well-sampled 10-
day averages after the adjustment by the ways the singularities are
handled in this scheme. The scheme (which minimizes altimetric sea
level differences at satellite ground track crossover points in the
least-squares sense) has 2 sources of singularity. First, any
geographically dependent function (that can be represented
completely by Fourier components within the 3.33 cpr spectral band)
produces no crossover difference, and therefore can be added to the
solution of the adjustment to make it non-unique, hence singular.
Fortunately the associated null space can be dissected
analytically, and the singularity can be removed by requiring the
solution to be orthogonal to this null space (i.e., the best-
fitting geographically dependent function to the data over the 10-
day span is excluded from the solution and therefore preserved as
the 10-day average). The second source comes from the presence of
land and data outages on the one hand, and from the Fourier series
with wide enough spectral band on the other hand. Note that if 1.5
cpr cutoff is used, most of the time we can avoid this singularity,
whose associated null space is spanned by eigenvectors representing
functions that almost vanish within data coverage, but may be order
one over data gaps, thus creating another source of arbitrariness.
The wide band is essential for making these rapid transitions from
within to without and vice versa possible. The damped least squares
is utilized to deal with this singularity.
The solution of the adjustment is comprised of large-scale
time-dependent orbit error, tide model error (mainly M2 tide),
environmental correction error, and ocean signal relative to their
respective 10-day means. The solution reduces the crossover
difference from 9.54 cm (rms) to 6.79 cm, while accounting for 6.40
cm. The rms value of the solution is 4.85 cm (4.34 cm if counting
solution values at crossovers only). Eliminating influences from M2
tide error (which is aliased along track into 6.26 cycles) by
forming 6-cycle differences of the solution, the non-systematic
orbit error is estimated to have a rms value of 4.08 cm; while
averaging of the solution over 12 cycles produces an estimate of
the systematic orbit error (which produces no collinear difference)
at 1.78 cm (rms), projecting an upper bound for the gravity-error-
induced orbit error at 2.5 cm, and pointing to a simple way of
using space-time averaging to reduce the orbit and tide errors for
TOPEX/POSEIDON if the signal can withstand the averaging, such as
in the tropics. The adjustment corrects for over half of the tide
model error, even though the scheme is not designed for this
purpose and there are more effective ways to remove the tide error.