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makegravgradgriddh.doc
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makegravgradgriddh.doc
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Create 2D cylindrical maps on a flattened ellipsoid of the components of the
gravity "gradient" tensor in a local north-oriented reference frame.
Usage
-----
vxx, vyy, vzz, vxy, vxz, vyz = MakeGravGradGridDH (cilm, gm, r0, [lmax, a, f,
sampling, lmax_calc, extend])
Returns
-------
vxx : float, dimension (nlat, nlong)
A 2D map of the xx component of the gravity tensor that conforms to the
sampling theorem of Driscoll and Healy (1994). If sampling is 1, the grid is
equally sampled and is dimensioned as (n by n), where n is 2lmax+2. If
sampling is 2, the grid is equally spaced and is dimensioned as (n by 2n).
The first latitudinal band of the grid corresponds to 90 N, the latitudinal
sampling interval is 180/n 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/n for an equally sampled and 180/n for
an equally spaced grid, respectively. If extend 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.
vyy : float, dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the yy component of the
gravity tensor.
vzz : float, dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the zz component of the
gravity tensor.
vxy : float, dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the xy component of the
gravity tensor.
vxz : float, dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the xz component of the
gravity tensor.
vyz : float, dimension (nlat, nlong)
A 2D equally sampled or equally spaced grid of the YZ component of the
gravity tensor.
Parameters
----------
cilm : float, dimension (2, lmaxin+1, lmaxin+1)
The real 4-pi normalized gravitational potential spherical harmonic
coefficients. The coefficients c1lm and c2lm refer to the cosine and sine
coefficients, respectively, with c1lm=cilm[0,l,m] and c2lm=cilm[1,l,m].
gm : float
The gravitational constant multiplied by the mass of the planet.
r0: float
The reference radius of the spherical harmonic coefficients.
a : float
The semi-major axis of the flattened ellipsoid on which the field is
computed.
f : float
The flattening of the reference ellipsoid: f=(R_equator-R_pole)/R_equator.
lmax : optional, integer, default = lmaxin
The maximum spherical harmonic degree of the coefficients cilm. This
determines the number of samples of the output grids, n=2lmax+2, and the
latitudinal sampling interval, 90/(lmax+1).
sampling : optional, integer, default = 2
If 1 the output grids are equally sampled (n by n). If 2, the grids are
equally spaced (n by 2n).
lmax_calc : optional, integer, default = lmax
The maximum spherical harmonic degree used in evaluating the functions. This
must be less than or equal to lmax.
extend : input, optional, bool, default = False
If True, compute the longitudinal band for 360 E and the latitudinal band
for 90 S. This increases each of the dimensions of griddh by 1.
Description
-----------
MakeGravGradGridDH will create 2-dimensional cylindrical maps from the spherical
harmonic coefficients cilm, equally sampled (n by n) or equally spaced (n by 2n)
in latitude and longitude, for six components of the gravity "gradient" tensor
(all using geocentric coordinates):
(Vxx, Vxy, Vxz)
(Vyx, Vyy, Vyz)
(Vzx, Vzy, Vzz)
The reference frame is north-oriented, where x points north, y points west, and
z points upward (all tangent or perpendicular to a sphere of radius r). The
gravitational potential is defined as
V = GM/r Sum_{l=0}^lmax (r0/r)^l Sum_{m=-l}^l C_{lm} Y_{lm},
where r0 is the reference radius of the spherical harmonic coefficients Clm, and
the gravitational acceleration is
B = Grad V.
The gravity tensor is symmetric, and satisfies Vxx+Vyy+Vzz=0, though all three
diagonal elements are calculated independently in this routine.
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's equations are in terms of latitude and that the y axis
points east):
Vzz = Vrr
Vxx = 1/r Vr + 1/r^2 Vtt
Vyy = 1/r Vr + 1/r^2 /tan(t) Vt + 1/r^2 /sin(t)^2 Vpp
Vxy = 1/r^2 /sin(t) Vtp - cos(t)/sin(t)^2 /r^2 Vp
Vxz = 1/r^2 Vt - 1/r Vrt
Vyz = 1/r^2 /sin(t) Vp - 1/r /sin(t) Vrp
where r, t, p stand for radius, theta, and phi, respectively, and subscripts on
V denote partial derivatives.
The output grid are in units of s^-2 and are cacluated on a flattened ellipsoid
with semi-major axis a and flattening f. To obtain units of Eotvos (10^-9 s^-2),
multiply the output by 10^9. 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).
The default is to use an input grid that is equally sampled (n by n), but this
can be changed to use an equally spaced grid (n by 2n) by the optional argument
sampling. 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 extend.
References
----------
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.
Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on
the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.
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.