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 2`n`

).

## 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.

## See also

cilmplusrhohdh, cilmminusdh, cilmminusrhohdh, makegriddh

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