title | keywords | sidebar | permalink | summary | toc | folder |
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Grid formats |
spherical harmonics software package, spherical harmonic transform, legendre functions, multitaper spectral analysis, fortran, Python, gravity, magnetic field |
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grid-formats.html |
pyshtools supports equally sampled, equally spaced, and Gauss-Legendre quadrature grids. |
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pyshtools makes use of grid formats that accommodate exact quadrature. These include regularly spaced grids that satisfy the Driscoll and Healy (1994) sampling theorem, and grids for exact quadrature using Gauss-Legendre quadrature [e.g., Press et al. 1992]. The grids are input as arrays where the rows and columns correspond to equal values of latitude and longitude, respectively. The first row corresponds to the latitude band closest to the north pole, and the first column corresponds to 0 degrees E.
For the case of Gauss-Legendre quadrature (GLQ
), the quadrature is exact when the function extend
.
The second type of grid is for data that are sampled on regular grids. As shown by Driscoll and Healy [1994], an exact quadrature exists when the function DH
), the grids make use of the longitude band at 90$$^{\circ}$$ N, but not 90$$^{\circ}$$ S, and the number of samples is
For geographic data, it is common to work with grids that are equally spaced in degrees latitude and longitude. pyshtools provides the option of using grids of size DH2
), the coefficients extend
.
The properties of the Driscoll and Healy [1994] and Gauss-Legendre Quadrature grids are summarized in the following table:
DH1 | DH2 | GLQ | |
---|---|---|---|
Name | Driscoll and Healy | Driscoll and Healy | Gauss-Legendre Quadrature |
Shape ( |
|||
Variable | |||
The figure below demonstrates how these grids sample an arbitrary function that has a maximum spherical harmonic degree of 10. The DH1
and DH2
grids are seen to have the same sampling in latitude, but the DH2
grid has twice as many samples in longitude than does the DH1
grid. The GLQ
grid is regularly sampled in longitude, but is irregularly sampled in latitude. Given the freedom associated with choosing the latitude coordinates for the GLQ
grids, these grids have about half as many latitudinal points as do the more regular DH
grids. The red points at the south pole and 360$$^{\circ}$$ E are not required when performing the spherical harmonic transforms, but can be computed by specifying the optional argument extend
.
{% include image.html file="grids.png" alt="Spherical harmonic grid formats" caption="Schematic diagram illustrating the properties of the grids used with the Gauss-Legendre quadrature and Driscoll and Healy routines. The red points are not required by the spherical harmonic transform routines, but can be computed by specifying the optional argument extend
." %}
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Driscoll, J. R. and D. M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, doi:10.1006/aama.1994.1008, 1994.
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Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, "Numerical Recipes in FORTRAN: The Art of Scientific Computing," 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1992.