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.
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