Ascending
Geosat
Ascending
Geosat
Descending
Geosat
Descending
Geosat
Ascending
ERS-1
Ascending
ERS-1
Descending
ERS-1
Descending
ERS-1
Fig. 3. Flow diagram for constructing north and east grids of vertical deflection from
ascending and descending along-track slope profiles. Iteration provides a
communication among the diverse data sets [Menke, 1991].
7) Apply isotropic low-pass, convolution filter [ kei(r/a) ] to the north and east grids. The
filter width a is proportional to the square. root of the relative error shown in Figure 4
in order to equalize the noise level. For example at the equator, the ERS-l and Geosat
tracks provide relatively poor control on the east component of vertical deflection so a
is increased (wider filter) to suppress noise.
8) Decompose the north and east grids into the original pass-oriented grids using
equations (3) and (4) in Sandwell [1984].
9) Reset bins constrained by along-track slope observations to their original values and go
to step (6).
Steps (6) through (9) are repeated until the values of unconstrained grid cells converge.
Exit from the iteration after step (7). The goal is to produce north and east vertical
deflection grids that are consistent with the original observations to within their assigned
noise level and where unconstrained cells reflect nearby values.
After generating grids of north and east vertical deflection one can generate various
derivatives of the gravitational potential. In all cases one should restore the appropriate
derivatives of the spherical harmonic reference model that was removed in step (2). One
can compute the vertical gravity gradient as the sum of the x-derivative of the east VD
component and the y-derivative of the north VD component. See equation (23) of
Sandwell [1992] which is based on Laplace's equation. Finally if one approximates the
spherical earth as a flat earth over a distance corresponding to the cutoff wavelength of the
spherical harmonic model removed (800 km for complete removal at degree 50), the two
vertical deflection grids can be converted to a gravity anomaly grid using the formula given
in Haxby et al. [1983] and Sandwell [1992]. This requires fourier transformation of each
vertical deflection grid, combination in the wavenumber domain, and inverse fourier
15
Geosat
Ascending
Geosat
Descending
Geosat
Descending
Geosat
Ascending
ERS-1
Ascending
ERS-1
Descending
ERS-1
Descending
ERS-1
Fig. 3. Flow diagram for constructing north and east grids of vertical deflection from
ascending and descending along-track slope profiles. Iteration provides a
communication among the diverse data sets [Menke, 1991].
7) Apply isotropic low-pass, convolution filter [ kei(r/a) ] to the north and east grids. The
filter width a is proportional to the square. root of the relative error shown in Figure 4
in order to equalize the noise level. For example at the equator, the ERS-l and Geosat
tracks provide relatively poor control on the east component of vertical deflection so a
is increased (wider filter) to suppress noise.
8) Decompose the north and east grids into the original pass-oriented grids using
equations (3) and (4) in Sandwell [1984].
9) Reset bins constrained by along-track slope observations to their original values and go
to step (6).
Steps (6) through (9) are repeated until the values of unconstrained grid cells converge.
Exit from the iteration after step (7). The goal is to produce north and east vertical
deflection grids that are consistent with the original observations to within their assigned
noise level and where unconstrained cells reflect nearby values.
After generating grids of north and east vertical deflection one can generate various
derivatives of the gravitational potential. In all cases one should restore the appropriate
derivatives of the spherical harmonic reference model that was removed in step (2). One
can compute the vertical gravity gradient as the sum of the x-derivative of the east VD
component and the y-derivative of the north VD component. See equation (23) of
Sandwell [1992] which is based on Laplace's equation. Finally if one approximates the
spherical earth as a flat earth over a distance corresponding to the cutoff wavelength of the
spherical harmonic model removed (800 km for complete removal at degree 50), the two
vertical deflection grids can be converted to a gravity anomaly grid using the formula given
in Haxby et al. [1983] and Sandwell [1992]. This requires fourier transformation of each
vertical deflection grid, combination in the wavenumber domain, and inverse fourier
15