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<h1 class="post-title-main">CilmMinusRhoHDH (Python)</h1>
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<p>Calculate the gravitational potential interior to relief with lateral variations in density referenced to a spherical interface using the finite-amplitude algorithm of Wieczorek and Phillips (1998).</p>
<h2 id="usage">Usage</h2>
<p><code class="highlighter-rouge">cilm</code>, <code class="highlighter-rouge">d</code> = CilmMinusRhoHDH (<code class="highlighter-rouge">gridin</code>, <code class="highlighter-rouge">nmax</code>, <code class="highlighter-rouge">mass</code>, <code class="highlighter-rouge">rho</code>, [<code class="highlighter-rouge">lmax</code>, <code class="highlighter-rouge">n</code>, <code class="highlighter-rouge">sampling</code>])</p>
<h2 id="returns">Returns</h2>
<dl>
<dt><code class="highlighter-rouge">cilm</code> : float, dimension (2, <code class="highlighter-rouge">lmax</code>+1, <code class="highlighter-rouge">lmax</code>+1)</dt>
<dd>The real spherical harmonic coefficients (geodesy normalized) of the gravitational potential corresponding to constant density relief referenced to a spherical interface of radius <code class="highlighter-rouge">d</code>.</dd>
<dt><code class="highlighter-rouge">d</code> : float</dt>
<dd>The mean radius of the relief in meters.</dd>
</dl>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><code class="highlighter-rouge">gridin</code> : float, dimension (<code class="highlighter-rouge">nin</code>, <code class="highlighter-rouge">sampling</code>*<code class="highlighter-rouge">nin</code>)</dt>
<dd>The radii of the interface evaluated on a grid, determined by a call to <code class="highlighter-rouge">MakeGridDH</code>.</dd>
<dt><code class="highlighter-rouge">nmax</code> : integer</dt>
<dd>The maximum order used in the Taylor-series expansion used in calculating the potential coefficients.</dd>
<dt><code class="highlighter-rouge">mass</code> : float</dt>
<dd>The mass of the planet in kg.</dd>
<dt><code class="highlighter-rouge">rho</code> : float, dimension (<code class="highlighter-rouge">nin</code>, <code class="highlighter-rouge">sampling</code>*<code class="highlighter-rouge">nin</code>)</dt>
<dd>The density contrast of the relief in kg/m^3.</dd>
<dt><code class="highlighter-rouge">lmax</code> : optional, integer, default = <code class="highlighter-rouge">n/2-1</code></dt>
<dd>The maximum spherical harmonic degree of the output spherical harmonic coefficients. <code class="highlighter-rouge">lmax</code> must be less than or equal to <code class="highlighter-rouge">n/2-1</code>.</dd>
<dt><code class="highlighter-rouge">n</code> : optional, integer, default = <code class="highlighter-rouge">nin</code></dt>
<dd>The number of samples in latitude when using Driscoll-Healy grids. For a function bandlimited to <code class="highlighter-rouge">lmax</code>, <code class="highlighter-rouge">n=2(lmax+1)</code>.</dd>
<dt><code class="highlighter-rouge">sampling</code> : optional, integer, default determined by dimensions of <code class="highlighter-rouge">gridin</code></dt>
<dd>If 1 the output grids are equally sampled (<code class="highlighter-rouge">n</code> by <code class="highlighter-rouge">n</code>). If 2, the grids are equally spaced (<code class="highlighter-rouge">n</code> by 2<code class="highlighter-rouge">n</code>).</dd>
</dl>
<h2 id="description">Description</h2>
<p><code class="highlighter-rouge">CilmMinusRhoHDH</code> will calculate the spherical harmonic coefficients of the gravitational potential exterior to relief with lateral variations in density referenced to a spherical interface. This is a generalization of equation 30 of Wieczorek (2007) and equation 12 of Wieczorek and Phillips (1998). 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 <code class="highlighter-rouge">gridin</code> must correspond to absolute radii. The parameter <code class="highlighter-rouge">nmax</code> is the order of the Taylor series used in the algorithm to approximate the potential coefficients. The spherical harmonic coefficients will be referenced to the mean radius of <code class="highlighter-rouge">gridin</code>.</p>
<p>As an intermediate step, this routine calculates the spherical harmonic coefficients of the relief referenced to the mean radius of <code class="highlighter-rouge">gridin</code> raised to the nth power, i.e., <code class="highlighter-rouge">(gridin-d)\*\*n</code>. As such, if the input function is bandlimited to degree <code class="highlighter-rouge">L</code>, the resulting function will be bandlimited to degree <code class="highlighter-rouge">L*nmax</code>. This subroutine implicitly assumes that the <code class="highlighter-rouge">gridin</code> has an effective spherical harmonic bandwidth greater or equal to this value. (The effective bandwidth is equal to <code class="highlighter-rouge">n/2-1</code>.) 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 <code class="highlighter-rouge">L</code>, though this should be verified for each application.</p>
<p>This routine uses geodesy 4-pi normalized spherical harmonics that exclude the Condon-Shortley phase.</p>
<h2 id="references">References</h2>
<p>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.</p>
<p>Wieczorek, M. A., Gravity and topography of the terrestrial planets, Treatise on Geophysics, 10, 165-206, doi:10.1016/B978-044452748-6/00156-5, 2007.</p>
<h2 id="see-also">See also</h2>
<p><a href="pycilmminusdh.html">cilmminusdh</a> <a href="pycilmplusdh.html">cilmplusdh</a>, <a href="pycilmplusrhohdh.html">cilmplusrhohdh</a>, <a href="pymakegriddh.html">makegriddh</a></p>
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