11 The Marine Geoid and Satellite Altimetry
185
Upward continuation prevents satellite gravity observing systems such as
GRACE or GOCE from obtaining useful information on the marine geoid at scales
shorter than a few hundred km. Although satellite altimeters orbit at 800 to more
than 1,300 km above the Earth, their radars measure the sea surface height, which
reflects gravity at sea level, not at orbital altitude. It turns out that altimetry is the
best way to get information about marine gravity anomalies at scales shorter than a
few hundred km.
11.4 Geoid Slopes and Gravity Anomalies
Anomalies in atmospheric mass contribute negligibly to marine geoid height anomalies, and so we can assume that T satisfies Laplace’s equation on and above the sea
surface. In Cartesian coordinates
∂ 2
∂x 2 +
∂ 2
∂y 2 +
∂ 2
∂z 2
T = 0.
(11.5)
Substituting the geoid slope components η = −∂N
∂x, ξ = −∂N
∂y, and the
gravity anomaly g = −∂T
∂z (the negative signs come from sign conventions in
classical geodesy) one has
γ 0
∂η
∂x
+
∂ξ
∂y
=
∂g
∂z
(11.6)
which couples the geoid slopes to the gravity anomaly.
The Fourier transform of this expression
i2πγ 0
u ˜
η + v ˜
ξ
= −2π q˜ g
(11.7)
may be used to compute the gravity anomaly from geoid slopes, and vice versa
(Haxby et al., 1983; Sandwell and Smith, 1997). The tilde over a quantity in (11.7)
indicates the Fourier transform in the z = 0 plane, as in (11.4).
Geoid heights are measured from the ellipsoid, while the gravity anomaly is the
difference between actual gravity on the geoid and reference gravity on the ellipsoid.
If z = 0 is the local flat Earth approximation to the ellipsoid then the expressions
(11.6) and (11.7) are correct apart from a small error in the gravity anomaly known
as the “indirect effect” (Chapman and Bodine, 1979). This effect is negligible if
the flat-Earth approximation is used only for local and short-wavelength anomalies, with geoid slopes and gravity anomalies of larger scale treated by spherical
harmonics, as in the “remove-restore” procedure (Sandwell and Smith, 2009).
The geoid slopes η and ξ are known as “deflections of the vertical” in geodesy.
Historically, on land, they were measured astronomically, and were given in
arc-seconds (a.s.); in the context of altimetry of the marine geoid it is convenient
to measure them in micro-radians (μrad, 1 a.s. is about 4.8 μrad). The μrad is
dimensionless. Gravity anomalies are traditionally expressed in milliGals (mGal;
1 mGal = 10 –5 m/s 2 ). Since γ 0 is about 9.8 m/s 2 , or 0.98 times 10 6 mGal, the mGal
185
Upward continuation prevents satellite gravity observing systems such as
GRACE or GOCE from obtaining useful information on the marine geoid at scales
shorter than a few hundred km. Although satellite altimeters orbit at 800 to more
than 1,300 km above the Earth, their radars measure the sea surface height, which
reflects gravity at sea level, not at orbital altitude. It turns out that altimetry is the
best way to get information about marine gravity anomalies at scales shorter than a
few hundred km.
11.4 Geoid Slopes and Gravity Anomalies
Anomalies in atmospheric mass contribute negligibly to marine geoid height anomalies, and so we can assume that T satisfies Laplace’s equation on and above the sea
surface. In Cartesian coordinates
∂ 2
∂x 2 +
∂ 2
∂y 2 +
∂ 2
∂z 2
T = 0.
(11.5)
Substituting the geoid slope components η = −∂N
∂x, ξ = −∂N
∂y, and the
gravity anomaly g = −∂T
∂z (the negative signs come from sign conventions in
classical geodesy) one has
γ 0
∂η
∂x
+
∂ξ
∂y
=
∂g
∂z
(11.6)
which couples the geoid slopes to the gravity anomaly.
The Fourier transform of this expression
i2πγ 0
u ˜
η + v ˜
ξ
= −2π q˜ g
(11.7)
may be used to compute the gravity anomaly from geoid slopes, and vice versa
(Haxby et al., 1983; Sandwell and Smith, 1997). The tilde over a quantity in (11.7)
indicates the Fourier transform in the z = 0 plane, as in (11.4).
Geoid heights are measured from the ellipsoid, while the gravity anomaly is the
difference between actual gravity on the geoid and reference gravity on the ellipsoid.
If z = 0 is the local flat Earth approximation to the ellipsoid then the expressions
(11.6) and (11.7) are correct apart from a small error in the gravity anomaly known
as the “indirect effect” (Chapman and Bodine, 1979). This effect is negligible if
the flat-Earth approximation is used only for local and short-wavelength anomalies, with geoid slopes and gravity anomalies of larger scale treated by spherical
harmonics, as in the “remove-restore” procedure (Sandwell and Smith, 2009).
The geoid slopes η and ξ are known as “deflections of the vertical” in geodesy.
Historically, on land, they were measured astronomically, and were given in
arc-seconds (a.s.); in the context of altimetry of the marine geoid it is convenient
to measure them in micro-radians (μrad, 1 a.s. is about 4.8 μrad). The μrad is
dimensionless. Gravity anomalies are traditionally expressed in milliGals (mGal;
1 mGal = 10 –5 m/s 2 ). Since γ 0 is about 9.8 m/s 2 , or 0.98 times 10 6 mGal, the mGal