11 The Marine Geoid and Satellite Altimetry
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standard gravity in the denominator on the right hand side of (11.2). Since gravity
anomalies have peak values less than 10 –3 of standard gravity, while the geoid height
has peak values around 100 m, use of (11.2) results in a geoid height estimate that is
accurate to better than a decimetre in the worst case. For use with altimetry, higher
accuracy is required, and modern geoid calculations employ higher order iterative
refinement. Geoid height errors in the EGM2008 gravity field model, for example,
are around 5 cm (Pavlis et al., 2008), and these should be due to errors in the model,
not in the calculations.
When classical geodesy developed in the nineteenth century it was assumed that
the geoid would correspond to mean sea level, and topographic elevations on land
are given as heights above sea level by classical levelling surveys. However, today
one recognizes that a long-term average of sea level would include the dynamic
topography of steady flow and the permanent deformation of the solid and fluid
Earth in response to the tides. Because classical geodesy expects the disturbing
potential, T, to arise from mass anomalies within the Earth, care must be taken to
define the reference ellipsoidal model for the field appropriately.
Modern geoid models now come in various flavours, depending on how the zerofrequency tidal effects are reckoned. Since tide models are also used in altimetry,
both for correcting sea surface heights and for estimating satellite orbit ephemeredes, one must choose a geoid definition consistent with the sea surface height
corrections being used (Ekman, 1989; Rapp, 1994; Pavlis et al., 2008). Classical
geodesy also assumes that the reference ellipsoid can be defined so that the geometric center of the ellipsoid coincides with the center of mass of the Earth. Modern
observations reveal that the center of mass is in motion by small amounts, primarily
on an annual cycle. Since the satellite altimeters are also orbiting the center of mass,
these geocenter variations are handled as coordinate reference frame issues, and will
not be treated here.
11.3 Upward Continuation
The potential of the anomaly field, T, obeys Laplace’s equation outside any volume
enclosing its sources. If its sources are confined within a sphere of radius a, then T
may be expanded in spherical harmonics:
T (r, θ , λ) =
GM
r
n
a
r
n
n
m=−n
α nm Y nm (θ , λ).
(11.3)
In (11.3), r, θ, λ are the spherical coordinates radius, geocentric colatitude, and longitude, G is the Newtonian gravitational constant, M the mass of the Earth, Y is
a surface spherical harmonic of degree n and order m, and α is a dimensionless
coefficient on Y.
In dealing with anomalies whose spatial extent is small compared with the mean
radius of the Earth, a flat Earth approximation may be useful. Letting x, y, z be
183
standard gravity in the denominator on the right hand side of (11.2). Since gravity
anomalies have peak values less than 10 –3 of standard gravity, while the geoid height
has peak values around 100 m, use of (11.2) results in a geoid height estimate that is
accurate to better than a decimetre in the worst case. For use with altimetry, higher
accuracy is required, and modern geoid calculations employ higher order iterative
refinement. Geoid height errors in the EGM2008 gravity field model, for example,
are around 5 cm (Pavlis et al., 2008), and these should be due to errors in the model,
not in the calculations.
When classical geodesy developed in the nineteenth century it was assumed that
the geoid would correspond to mean sea level, and topographic elevations on land
are given as heights above sea level by classical levelling surveys. However, today
one recognizes that a long-term average of sea level would include the dynamic
topography of steady flow and the permanent deformation of the solid and fluid
Earth in response to the tides. Because classical geodesy expects the disturbing
potential, T, to arise from mass anomalies within the Earth, care must be taken to
define the reference ellipsoidal model for the field appropriately.
Modern geoid models now come in various flavours, depending on how the zerofrequency tidal effects are reckoned. Since tide models are also used in altimetry,
both for correcting sea surface heights and for estimating satellite orbit ephemeredes, one must choose a geoid definition consistent with the sea surface height
corrections being used (Ekman, 1989; Rapp, 1994; Pavlis et al., 2008). Classical
geodesy also assumes that the reference ellipsoid can be defined so that the geometric center of the ellipsoid coincides with the center of mass of the Earth. Modern
observations reveal that the center of mass is in motion by small amounts, primarily
on an annual cycle. Since the satellite altimeters are also orbiting the center of mass,
these geocenter variations are handled as coordinate reference frame issues, and will
not be treated here.
11.3 Upward Continuation
The potential of the anomaly field, T, obeys Laplace’s equation outside any volume
enclosing its sources. If its sources are confined within a sphere of radius a, then T
may be expanded in spherical harmonics:
T (r, θ , λ) =
GM
r
n
a
r
n
n
m=−n
α nm Y nm (θ , λ).
(11.3)
In (11.3), r, θ, λ are the spherical coordinates radius, geocentric colatitude, and longitude, G is the Newtonian gravitational constant, M the mass of the Earth, Y is
a surface spherical harmonic of degree n and order m, and α is a dimensionless
coefficient on Y.
In dealing with anomalies whose spatial extent is small compared with the mean
radius of the Earth, a flat Earth approximation may be useful. Letting x, y, z be