makegravgradgriddh.1

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.TH "makegravgradgriddh" "1" "2021-02-15" "Fortran 95" "SHTOOLS 4.10"
.hy
.SH MakeGravGradGridDH
.PP
Create 2D cylindrical maps on a flattened ellipsoid of the components of
the gravity \[lq]gradient\[rq] tensor in a local north-oriented
reference frame.
.SH Usage
.PP
call MakeGravGradGridDH (\f[V]cilm\f[R], \f[V]lmax\f[R], \f[V]gm\f[R],
\f[V]r0\f[R], \f[V]a\f[R], \f[V]f\f[R], \f[V]vxx\f[R], \f[V]vyy\f[R],
\f[V]vzz\f[R], \f[V]vxy\f[R], \f[V]vxz\f[R], \f[V]vyz\f[R], \f[V]n\f[R],
\f[V]sampling\f[R], \f[V]lmax_calc\f[R], \f[V]extend\f[R],
\f[V]exitstatus\f[R])
.SH Parameters
.TP
\f[V]cilm\f[R] : input, real(dp), dimension (2, \f[V]lmax\f[R]+1, \f[V]lmax\f[R]+1)
The real 4-pi normalized gravitational potential spherical harmonic
coefficients.
The coefficients \f[V]c1lm\f[R] and \f[V]c2lm\f[R] refer to the cosine
and sine coefficients, respectively, with \f[V]c1lm=cilm(1,l+1,m+1)\f[R]
and \f[V]c2lm=cilm(2,l+1,m+1)\f[R].
.TP
\f[V]lmax\f[R] : input, integer(int32)
The maximum spherical harmonic degree of the coefficients
\f[V]cilm\f[R].
This determines the number of samples of the output grids,
\f[V]n=2lmax+2\f[R], and the latitudinal sampling interval,
\f[V]90/(lmax+1)\f[R].
.TP
\f[V]gm\f[R] : input, real(dp)
The gravitational constant multiplied by the mass of the planet.
.TP
\f[V]r0\f[R]: input, real(dp)
The reference radius of the spherical harmonic coefficients.
.TP
\f[V]a\f[R] : input, real(dp)
The semi-major axis of the flattened ellipsoid on which the field is
computed.
.TP
\f[V]f\f[R] : input, real(dp)
The flattening of the reference ellipsoid:
\f[V]f=(R_equator-R_pole)/R_equator\f[R].
.TP
\f[V]vxx\f[R] : output, real(dp), dimension (nlat, nlong)
A 2D map of the \f[V]xx\f[R] component of the gravity tensor that
conforms to the sampling theorem of Driscoll and Healy (1994).
If \f[V]sampling\f[R] is 1, the grid is equally sampled and is
dimensioned as (\f[V]n\f[R] by \f[V]n\f[R]), where \f[V]n\f[R] is
\f[V]2lmax+2\f[R].
If sampling is 2, the grid is equally spaced and is dimensioned as
(\f[V]n\f[R] by 2\f[V]n\f[R]).
The first latitudinal band of the grid corresponds to 90 N, the
latitudinal sampling interval is 180/\f[V]n\f[R] degrees, and the
default behavior is to exclude the latitudinal band for 90 S.
The first longitudinal band of the grid is 0 E, by default the
longitudinal band for 360 E is not included, and the longitudinal
sampling interval is 360/\f[V]n\f[R] for an equally sampled and
180/\f[V]n\f[R] for an equally spaced grid, respectively.
If \f[V]extend\f[R] is 1, the longitudinal band for 360 E and the
latitudinal band for 90 S will be included, which increases each of the
dimensions of the grid by 1.
.TP
\f[V]vyy\f[R] : output, real(dp), dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the \f[V]yy\f[R]
component of the gravity tensor.
.TP
\f[V]vzz\f[R] : output, real(dp), dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the \f[V]zz\f[R]
component of the gravity tensor.
.TP
\f[V]vxy\f[R] : output, real(dp), dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the \f[V]xy\f[R]
component of the gravity tensor.
.TP
\f[V]vxz\f[R] : output, real(dp), dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the \f[V]xz\f[R]
component of the gravity tensor.
.TP
\f[V]vyz\f[R] : output, real(dp), dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the YZ component of the
gravity tensor.
.TP
\f[V]n\f[R] : output, integer(int32)
The number of samples in latitude of the output grids.
This is equal to \f[V]2lmax+2\f[R].
.TP
\f[V]sampling\f[R] : optional, input, integer(int32), default = 1
If 1 (default) the output grids are equally sampled (\f[V]n\f[R] by
\f[V]n\f[R]).
If 2, the grids are equally spaced (\f[V]n\f[R] by 2\f[V]n\f[R]).
.TP
\f[V]lmax_calc\f[R] : optional, input, integer(int32)
The maximum spherical harmonic degree used in evaluating the functions.
This must be less than or equal to \f[V]lmax\f[R].
.TP
\f[V]extend\f[R] : input, optional, integer(int32), default = 0
If 1, compute the longitudinal band for 360 E and the latitudinal band
for 90 S.
This increases each of the dimensions of \f[V]griddh\f[R] by 1.
.TP
\f[V]exitstatus\f[R] : output, optional, integer(int32)
If present, instead of executing a STOP when an error is encountered,
the variable exitstatus will be returned describing the error.
0 = No errors; 1 = Improper dimensions of input array; 2 = Improper
bounds for input variable; 3 = Error allocating memory; 4 = File IO
error.
.SH Description
.PP
\f[V]MakeGravGradGridDH\f[R] will create 2-dimensional cylindrical maps
from the spherical harmonic coefficients \f[V]cilm\f[R], equally sampled
(\f[V]n\f[R] by \f[V]n\f[R]) or equally spaced (\f[V]n\f[R] by
2\f[V]n\f[R]) in latitude and longitude, for six components of the
gravity \[lq]gradient\[rq] tensor (all using geocentric coordinates):
.PP
\f[V](Vxx,  Vxy,  Vxz)\f[R]
.PD 0
.P
.PD
\f[V](Vyx,  Vyy,  Vyz)\f[R]
.PD 0
.P
.PD
\f[V](Vzx,  Vzy,  Vzz)\f[R]
.PP
The reference frame is north-oriented, where \f[V]x\f[R] points north,
\f[V]y\f[R] points west, and \f[V]z\f[R] points upward (all tangent or
perpendicular to a sphere of radius r).
The gravitational potential is defined as
.PP
\f[V]V = GM/r Sum_{l=0}\[ha]lmax (r0/r)\[ha]l Sum_{m=-l}\[ha]l C_{lm} Y_{lm}\f[R],
.PP
where \f[V]r0\f[R] is the reference radius of the spherical harmonic
coefficients \f[V]Clm\f[R], and the gravitational acceleration is
.PP
\f[V]B = Grad V\f[R].
.PP
The gravity tensor is symmetric, and satisfies \f[V]Vxx+Vyy+Vzz=0\f[R],
though all three diagonal elements are calculated independently in this
routine.
.PP
The components of the gravity tensor are calculated according to eq.
1 in Petrovskaya and Vershkov (2006), which is based on eq.
3.28 in Reed (1973) (noting that Reed\[cq]s equations are in terms of
latitude and that the \f[V]y\f[R] axis points east):
.PP
\f[V]Vzz = Vrr\f[R]
.PD 0
.P
.PD
\f[V]Vxx = 1/r Vr + 1/r\[ha]2 Vtt\f[R]
.PD 0
.P
.PD
\f[V]Vyy = 1/r Vr + 1/r\[ha]2 /tan(t) Vt + 1/r\[ha]2 /sin(t)\[ha]2 Vpp\f[R]
.PD 0
.P
.PD
\f[V]Vxy = 1/r\[ha]2 /sin(t) Vtp - cos(t)/sin(t)\[ha]2 /r\[ha]2 Vp\f[R]
.PD 0
.P
.PD
\f[V]Vxz = 1/r\[ha]2 Vt - 1/r Vrt\f[R]
.PD 0
.P
.PD
\f[V]Vyz = 1/r\[ha]2 /sin(t) Vp - 1/r /sin(t) Vrp\f[R]
.PP
where \f[V]r\f[R], \f[V]t\f[R], \f[V]p\f[R] stand for radius, theta, and
phi, respectively, and subscripts on \f[V]V\f[R] denote partial
derivatives.
.PP
The output grid are in units of s\[ha]-2 and are cacluated on a
flattened ellipsoid with semi-major axis \f[V]a\f[R] and flattening
\f[V]f\f[R].
To obtain units of Eotvos (10\[ha]-9 s\[ha]-2), multiply the output by
10\[ha]9.
The calculated values should be considered exact only when the radii on
the ellipsoid are greater than the maximum radius of the planet (the
potential coefficients are simply downward/upward continued in the
spectral domain).
.PP
The default is to use an input grid that is equally sampled (\f[V]n\f[R]
by \f[V]n\f[R]), but this can be changed to use an equally spaced grid
(\f[V]n\f[R] by 2\f[V]n\f[R]) by the optional argument
\f[V]sampling\f[R].
The redundant longitudinal band for 360 E and the latitudinal band for
90 S are excluded by default, but these can be computed by specifying
the optional argument \f[V]extend\f[R].
.SH References
.PP
Reed, G.B., Application of kinematical geodesy for determining the short
wave length components of the gravity field by satellite gradiometry,
Ohio State University, Dept.
of Geod.
Sciences, Rep.\ No.\ 201, Columbus, Ohio, 1973.
.PP
Driscoll, J.R.
and D.M.
Healy, Computing Fourier transforms and convolutions on the 2-sphere,
Adv.
Appl.
Math., 15, 202-250, 1994.
.PP
Petrovskaya, M.S.
and A.N.
Vershkov, Non-singular expressions for the gravity gradients in the
local north-oriented and orbital reference frames, J.
Geod., 80, 117-127, 2006.
.SH See also
.PP
makegravgriddh, makegeoidgrid, makegriddh