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<h1 class="post-title-main">MakeMagGradGridDH (Fortran)</h1>
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<div class="post-content">
<p>Create 2D cylindrical maps on a flattened ellipsoid of the components of the magnetic field tensor in a local north-oriented reference frame.</p>
<h2 id="usage">Usage</h2>
<p>call MakeMagGradGridDH (<code class="highlighter-rouge">cilm</code>, <code class="highlighter-rouge">lmax</code>, <code class="highlighter-rouge">r0</code>, <code class="highlighter-rouge">a</code>, <code class="highlighter-rouge">f</code>, <code class="highlighter-rouge">vxx</code>, <code class="highlighter-rouge">vyy</code>, <code class="highlighter-rouge">vzz</code>, <code class="highlighter-rouge">vxy</code>, <code class="highlighter-rouge">vxz</code>, <code class="highlighter-rouge">vyz</code>, <code class="highlighter-rouge">n</code>, <code class="highlighter-rouge">sampling</code>, <code class="highlighter-rouge">lmax_calc</code>, <code class="highlighter-rouge">exitstatus</code>)</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><code class="highlighter-rouge">cilm</code> : input, real(dp), dimension (2, <code class="highlighter-rouge">lmax</code>+1, <code class="highlighter-rouge">lmax</code>+1)</dt>
<dd>The real Schmidt semi-normalized spherical harmonic coefficients of the magnetic potential. The coefficients <code class="highlighter-rouge">c1lm</code> and <code class="highlighter-rouge">c2lm</code> refer to the cosine and sine coefficients, respectively, with <code class="highlighter-rouge">c1lm=cilm(1,l+1,m+1)</code> and <code class="highlighter-rouge">c2lm=cilm(2,l+1,m+1)</code>. The coefficients are assumed to have units of nT.</dd>
<dt><code class="highlighter-rouge">lmax</code> : input, integer</dt>
<dd>The maximum spherical harmonic degree of the coefficients <code class="highlighter-rouge">cilm</code>. This determines the number of samples of the output grids, <code class="highlighter-rouge">n=2lmax+2</code>, and the latitudinal sampling interval, <code class="highlighter-rouge">90/(lmax+1)</code>.</dd>
<dt><code class="highlighter-rouge">r0</code>: input, real(dp)</dt>
<dd>The reference radius of the spherical harmonic coefficients.</dd>
<dt><code class="highlighter-rouge">a</code> : input, real(dp)</dt>
<dd>The semi-major axis of the flattened ellipsoid on which the field is computed.</dd>
<dt><code class="highlighter-rouge">f</code> : input, real(dp)</dt>
<dd>The flattening of the reference ellipsoid: <code class="highlighter-rouge">f=(R_equator-R_pole)/R_equator</code>.</dd>
<dt><code class="highlighter-rouge">vxx</code> : output, real(dp), dimension (2*<code class="highlighter-rouge">lmax</code>+2, <code class="highlighter-rouge">sampling</code>*(2*<code class="highlighter-rouge">lmax</code>+2))</dt>
<dd>A 2D equally sampled (<code class="highlighter-rouge">n</code> by <code class="highlighter-rouge">n</code>) or equally spaced (<code class="highlighter-rouge">n</code> by 2<code class="highlighter-rouge">n</code>) grid of the <code class="highlighter-rouge">xx</code> component of the magnetic field tensor. The first latitudinal band corresponds to 90 N, the latitudinal band for 90 S is not included, and the latitudinal sampling interval is 180/<code class="highlighter-rouge">n</code> degrees. The first longitudinal band is 0 E, the longitudinal band for 360 E is not included, and the longitudinal sampling interval is 360/<code class="highlighter-rouge">n</code> for an equally sampled and 180/<code class="highlighter-rouge">n</code> for an equally spaced grid, respectively.</dd>
<dt><code class="highlighter-rouge">vyy</code> : output, real(dp), dimension (2*<code class="highlighter-rouge">lmax</code>+2, <code class="highlighter-rouge">sampling</code>*(2*<code class="highlighter-rouge">lmax</code>+2))</dt>
<dd>A 2D equally sampled or equally spaced grid of the <code class="highlighter-rouge">yy</code> component of the magnetic field tensor.</dd>
<dt><code class="highlighter-rouge">vzz</code> : output, real(dp), dimension (2*<code class="highlighter-rouge">lmax</code>+2, <code class="highlighter-rouge">sampling</code>*(2*<code class="highlighter-rouge">lmax</code>+2))</dt>
<dd>A 2D equally sampled or equally spaced grid of the <code class="highlighter-rouge">zz</code> component of the magnetic field tensor.</dd>
<dt><code class="highlighter-rouge">vxy</code> : output, real(dp), dimension (2*<code class="highlighter-rouge">lmax</code>+2, <code class="highlighter-rouge">sampling</code>*(2*<code class="highlighter-rouge">lmax</code>+2))</dt>
<dd>A 2D equally sampled or equally spaced grid of the <code class="highlighter-rouge">xy</code> component of the magnetic field tensor.</dd>
<dt><code class="highlighter-rouge">vxz</code> : output, real(dp), dimension (2*<code class="highlighter-rouge">lmax</code>+2, <code class="highlighter-rouge">sampling</code>*(2*<code class="highlighter-rouge">lmax</code>+2))</dt>
<dd>A 2D equally sampled or equally spaced grid of the <code class="highlighter-rouge">xz</code> component of the magnetic field tensor.</dd>
<dt><code class="highlighter-rouge">vyz</code> : output, real(dp), dimension (2*<code class="highlighter-rouge">lmax</code>+2, <code class="highlighter-rouge">sampling</code>*(2*<code class="highlighter-rouge">lmax</code>+2))</dt>
<dd>A 2D equally sampled or equally spaced grid of the YZ component of the magnetic field tensor.</dd>
<dt><code class="highlighter-rouge">n</code> : output, integer</dt>
<dd>The number of samples in latitude of the output grids. This is equal to <code class="highlighter-rouge">2lmax+2</code>.</dd>
<dt><code class="highlighter-rouge">sampling</code> : optional, input, integer, default = 1</dt>
<dd>If 1 (default) 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>
<dt><code class="highlighter-rouge">lmax_calc</code> : optional, input, integer</dt>
<dd>The maximum spherical harmonic degree used in evaluating the functions. This must be less than or equal to <code class="highlighter-rouge">lmax</code>.</dd>
<dt><code class="highlighter-rouge">exitstatus</code> : output, optional, integer</dt>
<dd>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.</dd>
</dl>
<h2 id="description">Description</h2>
<p><code class="highlighter-rouge">MakeMagGradGridDH</code> will create 2-dimensional cylindrical maps from the spherical harmonic coefficients <code class="highlighter-rouge">cilm</code>, equally sampled (<code class="highlighter-rouge">n</code> by <code class="highlighter-rouge">n</code>) or equally spaced (<code class="highlighter-rouge">n</code> by 2<code class="highlighter-rouge">n</code>) in latitude and longitude, for six components of the magnetic field tensor (all using geocentric coordinates):</p>
<p><code class="highlighter-rouge">(Vxx, Vxy, Vxz)</code><br />
<code class="highlighter-rouge">(Vyx, Vyy, Vyz)</code><br />
<code class="highlighter-rouge">(Vzx, Vzy, Vzz)</code></p>
<p>The reference frame is north-oriented, where <code class="highlighter-rouge">x</code> points north, <code class="highlighter-rouge">y</code> points west, and <code class="highlighter-rouge">z</code> points upward (all tangent or perpendicular to a sphere of radius r). The magnetic potential is defined as</p>
<p><code class="highlighter-rouge">V = r0 Sum_{l=0}^lmax (r0/r)^(l+1) Sum_{m=-l}^l C_{lm} Y_{lm}</code>,</p>
<p>where <code class="highlighter-rouge">r0</code> is the reference radius of the spherical harmonic coefficients <code class="highlighter-rouge">Clm</code>, and the vector magnetic field is</p>
<p><code class="highlighter-rouge">B = - Grad V</code>.</p>
<p>The magnetic field tensor is symmetric, and satisfies <code class="highlighter-rouge">Vxx+Vyy+Vzz=0</code>, though all three diagonal elements are calculated independently in this routine.</p>
<p>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 <code class="highlighter-rouge">y</code> axis points east):</p>
<p><code class="highlighter-rouge">Vzz = Vrr</code><br />
<code class="highlighter-rouge">Vxx = 1/r Vr + 1/r^2 Vtt</code><br />
<code class="highlighter-rouge">Vyy = 1/r Vr + 1/r^2 /tan(t) Vt + 1/r^2 /sin(t)^2 Vpp</code><br />
<code class="highlighter-rouge">Vxy = 1/r^2 /sin(t) Vtp - cos(t)/sin(t)^2 /r^2 Vp</code><br />
<code class="highlighter-rouge">Vxz = 1/r^2 Vt - 1/r Vrt</code><br />
<code class="highlighter-rouge">Vyz = 1/r^2 /sin(t) Vp - 1/r /sin(t) Vrp</code></p>
<p>where <code class="highlighter-rouge">r</code>, <code class="highlighter-rouge">t</code>, <code class="highlighter-rouge">p</code> stand for radius, theta, and phi, respectively, and subscripts on <code class="highlighter-rouge">V</code> denote partial derivatives.</p>
<p>The output grid are in units of nT / m and are cacluated on a flattened ellipsoid with semi-major axis <code class="highlighter-rouge">a</code> and flattening <code class="highlighter-rouge">f</code>. 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).</p>
<p>The default is to calculate grids for use in the Driscoll and Healy (1994) routines that are equally sampled (<code class="highlighter-rouge">n</code> by <code class="highlighter-rouge">n</code>), but this can be changed to calculate equally spaced grids (<code class="highlighter-rouge">n</code> by 2<code class="highlighter-rouge">n</code>) by setting the optional argument <code class="highlighter-rouge">sampling</code> to 2. The input value of <code class="highlighter-rouge">lmax</code> determines the number of samples, <code class="highlighter-rouge">n=2lmax+2</code>, and the latitudinal sampling interval, 90/(<code class="highlighter-rouge">lmax</code>+1). The first latitudinal band of the grid corresponds to 90 N, the latitudinal band for 90 S is not calculated, and the latitudinal sampling interval is 180/<code class="highlighter-rouge">n</code> degrees. The first longitudinal band is 0 E, the longitudinal band for 360 E is not calculated, and the longitudinal sampling interval is 360/<code class="highlighter-rouge">n</code> for equally sampled and 180/<code class="highlighter-rouge">n</code> for equally spaced grids, respectively.</p>
<h2 id="references">References</h2>
<p>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.</p>
<p>Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.</p>
<p>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.</p>
<h2 id="see-also">See also</h2>
<p><a href="makemaggriddh.html">makemaggriddh</a>, <a href="makegriddh.html">makegriddh</a></p>
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