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cilmplusdh.doc
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cilmplusdh.doc
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Calculate the gravitational potential exterior to relief referenced to a
spherical interface using the finite-amplitude algorithm of Wieczorek and
Phillips (1998).
Usage
-----
cilm, d = CilmPlusDH (gridin, nmax, mass, rho, [lmax, n, sampling])
Returns
-------
cilm : float, dimension (2, lmax+1, lmax+1)
The real spherical harmonic coefficients (geodesy normalized) of the
gravitational potential corresponding to constant density relief referenced
to a spherical interface of radius d.
d : float
The mean radius of the relief in meters.
Parameters
----------
gridin : float, dimension (nin, sampling*nin)
The radii of the interface evaluated on a grid, determined by a call to
MakeGridDH.
nmax : integer
The maximum order used in the Taylor-series expansion used in calculating
the potential coefficients.
mass : float
The mass of the planet in kg.
rho : float
The density contrast of the relief in kg/m^3.
lmax : optional, integer, default = n/2-1
The maximum spherical harmonic degree of the output spherical harmonic
coefficients. lmax must be less than or equal to n/2-1.
n : optional, integer, default = nin
The number of samples in latitude when using Driscoll-Healy grids. For a
function bandlimited to lmax, n=2(lmax+1).
sampling : optional, integer, default determined by dimensions of gridin
If 1 the output grids are equally sampled (n by n). If 2, the grids are
equally spaced (n by 2n).
Description
-----------
CilmPlus will calculate the spherical harmonic coefficients of the gravitational
potential exterior to constant density relief referenced to a spherical
interface. This is equation 10 of Wieczorek and Phillips (1998), where the
potential is strictly valid only when the coefficients are evaluated at a radius
greater than the maximum radius of the relief. The input relief gridin must
correspond to absolute radii. The parameter nmax is the order of the Taylor
series used in the algorithm to approximate the potential coefficients. The
output spherical harmonic coefficients will be referenced to the mean radius of
gridin.
As an intermediate step, this routine calculates the spherical harmonic
coefficients of the relief referenced to the mean radius of gridin raised to the
nth power, i.e., (gridin-d)**n. As such, if the input function is bandlimited to
degree L, the resulting function will be bandlimited to degree L*nmax. This
subroutine implicitly assumes that the gridin has an effective spherical
harmonic bandwidth greater or equal to this value. (The effective bandwidth is
equal to n/2-1.) If this is not the case, aliasing will occur. In practice, for
accurate results, it is found that the effective bandwidth needs only to be
about three times the size of L, though this should be verified for each
application.
This routine uses geodesy 4-pi normalized spherical harmonics that exclude the
Condon-Shortley phase.
References
----------
Wieczorek, M. A. and R. J. Phillips, Potential anomalies on a sphere:
applications to the thickness of the lunar crust, J. Geophys. Res., 103,
1715-1724, 1998.