Create 2D cylindrical maps on a flattened ellipsoid of the components of the magnetic field tensor in a local north-oriented reference frame.
Usage
call MakeMagGradGridDH (cilm, lmax, r0, a, f, vxx, vyy, vzz, vxy, vxz, vyz, n, sampling, lmax_calc, extend, exitstatus)
Parameters
cilm: input, real(dp), dimension (2,lmax+1,lmax+1)- The real Schmidt semi-normalized spherical harmonic coefficients of the magnetic potential. The coefficients
c1lmandc2lmrefer to the cosine and sine coefficients, respectively, withc1lm=cilm(1,l+1,m+1)andc2lm=cilm(2,l+1,m+1). The coefficients are assumed to have units of nT. lmax: input, integer(int32)- 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). r0: input, real(dp)- The reference radius of the spherical harmonic coefficients.
a: input, real(dp)- The semi-major axis of the flattened ellipsoid on which the field is computed.
f: input, real(dp)- The flattening of the reference ellipsoid:
f=(R_equator-R_pole)/R_equator. vxx: output, real(dp), dimension (nlat, nlong)- A 2D map of the
xxcomponent of the magnetic field tensor that conforms to the sampling theorem of Driscoll and Healy (1994). Ifsamplingis 1, the grid is equally sampled and is dimensioned as (nbyn), wherenis2lmax+2. If sampling is 2, the grid is equally spaced and is dimensioned as (nby 2n). The first latitudinal band of the grid corresponds to 90 N, the latitudinal sampling interval is 180/ndegrees, 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/nfor an equally sampled and 180/nfor an equally spaced grid, respectively. Ifextendis 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: output, real(dp), dimension (nlat, nlong)- A 2D equally sampled or equally spaced grid of the
yycomponent of the magnetic field tensor. vzz: output, real(dp), dimension (nlat, nlong))- A 2D equally sampled or equally spaced grid of the
zzcomponent of the magnetic field tensor. vxy: output, real(dp), dimension (nlat, nlong)- A 2D equally sampled or equally spaced grid of the
xycomponent of the magnetic field tensor. vxz: output, real(dp), dimension (nlat, nlong)- A 2D equally sampled or equally spaced grid of the
xzcomponent of the magnetic field tensor. vyz: output, real(dp), dimension (nlat, nlong)- A 2D equally sampled or equally spaced grid of the YZ component of the magnetic field tensor.
n: output, integer(int32)- The number of samples in latitude of the output grids. This is equal to
2lmax+2. sampling: optional, input, integer(int32), default = 1- If 1 (default) the output grids are equally sampled (
nbyn). If 2, the grids are equally spaced (nby 2n). lmax_calc: optional, input, integer(int32)- The maximum spherical harmonic degree used in evaluating the functions. This must be less than or equal to
lmax. extend: 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
griddhby 1. exitstatus: 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.
Description
MakeMagGradGridDH 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 magnetic field 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 magnetic potential is defined as
V = r0 Sum_{l=0}^lmax (r0/r)^(l+1) Sum_{m=-l}^l C_{lm} Y_{lm},
where r0 is the reference radius of the spherical harmonic coefficients Clm, and the vector magnetic field is
B = - Grad V.
The magnetic field tensor is symmetric, and satisfies Vxx+Vyy+Vzz=0, though all three diagonal elements are calculated independently in this routine.
The components of the magnetic field 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 nT / m and are cacluated on a flattened ellipsoid with semi-major axis a and flattening f. 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.