188
W.H.F. Smith
The degree variance spectrum for Earth’s topography, h, decays as n –2 at all
harmonic degrees, indicating that |∇ 1 h|
R would have a white spectrum. This is
consistent with the notion that topography is a random walk or a fractal. The gravity spectrum shows the same trend at wavelengths shorter than a few hundred km,
and gravity and topography are mostly coherent at these scales, indicating that the
source of these gravity anomalies is primarily uncompensated topography. The rootmean-square amplitude of gravity anomalies averaged over the surface of the sphere
is about 35 mGal. Since the gravity spectrum decays slowly with n, one may conjecture that the RMS geoid slope can be inferred from the RMS gravity anomaly,
and that it is approximately 35 μrad.
11.6 Using Satellite Altimetry to Determine the Geoid
Figure 11.2 shows that there is significant power in the geoid at wavelengths too
short to be determined by GRACE or GOCE, due to upward continuation. In
order to determine the geoid at these scales one needs gravity anomalies measured
on the geoid, not 400 km above it. If ships carrying gravity meters covered the
oceans densely and uniformly enough, and if shipborne gravimetry were accurate
enough, one could construct a geoid gravimetrically, independently of altimetry.
Unfortunately, neither the accuracy nor the coverage of shipborne gravimetry is
adequate to this task (Wessel and Watts, 1988).
The best hope of determining the short-wavelength marine geoid is therefore to
use satellite altimetry itself. The question arises then, how can one be sure that
the result will be the geoid and not some hybrid mean sea surface height? If a
geoid determined by altimetry is used to obtain the dynamic topography, will one
be accused of circular reasoning? How can this possibly work?
Geoid slopes come to the rescue and furnish a “trick”. Although the satellite
altimeters measure the instantaneous sea surface height, which is not the geoid, the
slope along an altimeter height profile is very nearly (within 1 μrad, in most cases)
equal to the slope of the geoid along the altimeter’s ground track. Most errors in
the altimeter measurement have long-enough correlation distances that they produce sub-microradian errors in sea surface slope. The slope of the tide and dynamic
topography is likewise sub-microradian, except in areas where there are energetic
mesoscale geostrophic currents or wide shallow shelves with large tide gradients.
Sandwell and Smith (2009) tabulate the various causes that may make the slope
of an altimeter profile depart from the geoid slope, and show that nearly all are
sub-microradian. They also describe an iterative procedure by which the significant
departures can be filtered out. One first makes an initial model estimating north and
east geoid slopes to best-fit many ascending and descending satellite tracks from
many inclinations (Sandwell, 1984; Sandwell and Smith, 1997). These initial north
and east slopes are used to build an initial gravity field model. Then each profile
is compared against the initial model, and the residual is filtered to remove slopes
due to ocean dynamics. The filtered residual slopes are then used to refine the initial
model. After a few iterations, the results agree with adjusted altimeter slope profiles
W.H.F. Smith
The degree variance spectrum for Earth’s topography, h, decays as n –2 at all
harmonic degrees, indicating that |∇ 1 h|
R would have a white spectrum. This is
consistent with the notion that topography is a random walk or a fractal. The gravity spectrum shows the same trend at wavelengths shorter than a few hundred km,
and gravity and topography are mostly coherent at these scales, indicating that the
source of these gravity anomalies is primarily uncompensated topography. The rootmean-square amplitude of gravity anomalies averaged over the surface of the sphere
is about 35 mGal. Since the gravity spectrum decays slowly with n, one may conjecture that the RMS geoid slope can be inferred from the RMS gravity anomaly,
and that it is approximately 35 μrad.
11.6 Using Satellite Altimetry to Determine the Geoid
Figure 11.2 shows that there is significant power in the geoid at wavelengths too
short to be determined by GRACE or GOCE, due to upward continuation. In
order to determine the geoid at these scales one needs gravity anomalies measured
on the geoid, not 400 km above it. If ships carrying gravity meters covered the
oceans densely and uniformly enough, and if shipborne gravimetry were accurate
enough, one could construct a geoid gravimetrically, independently of altimetry.
Unfortunately, neither the accuracy nor the coverage of shipborne gravimetry is
adequate to this task (Wessel and Watts, 1988).
The best hope of determining the short-wavelength marine geoid is therefore to
use satellite altimetry itself. The question arises then, how can one be sure that
the result will be the geoid and not some hybrid mean sea surface height? If a
geoid determined by altimetry is used to obtain the dynamic topography, will one
be accused of circular reasoning? How can this possibly work?
Geoid slopes come to the rescue and furnish a “trick”. Although the satellite
altimeters measure the instantaneous sea surface height, which is not the geoid, the
slope along an altimeter height profile is very nearly (within 1 μrad, in most cases)
equal to the slope of the geoid along the altimeter’s ground track. Most errors in
the altimeter measurement have long-enough correlation distances that they produce sub-microradian errors in sea surface slope. The slope of the tide and dynamic
topography is likewise sub-microradian, except in areas where there are energetic
mesoscale geostrophic currents or wide shallow shelves with large tide gradients.
Sandwell and Smith (2009) tabulate the various causes that may make the slope
of an altimeter profile depart from the geoid slope, and show that nearly all are
sub-microradian. They also describe an iterative procedure by which the significant
departures can be filtered out. One first makes an initial model estimating north and
east geoid slopes to best-fit many ascending and descending satellite tracks from
many inclinations (Sandwell, 1984; Sandwell and Smith, 1997). These initial north
and east slopes are used to build an initial gravity field model. Then each profile
is compared against the initial model, and the residual is filtered to remove slopes
due to ocean dynamics. The filtered residual slopes are then used to refine the initial
model. After a few iterations, the results agree with adjusted altimeter slope profiles