Supported grid formats
SHTOOLS 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 GaussLegendre quadrature [e.g., Press et al. 1992].
GaussLegendre Quadrature
For the case of GaussLegendre quadrature (GLQ
), the quadrature is exact when the function \(f\) is sampled in latitude at the \((L+1)\) zeros of the Legendre Polynomial of degree \((L+1)\). Since the function also needs to be sampled on \((2L+1)\) equally space grid nodes for the Fourier transforms in longitude, the function \(f\) is sampled on a grid of size \((L+1)\times(2L+1)\). The redundant data points at 360\(^{\circ}\) E longitude are not required by the spherical harmonic transformation routines, but can be computed by specifying the optional argument extend
.
Driscoll and Healy [1994]
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 \(f\) is sampled at \(N\) equally spaced nodes in latitude and \(N\) equally spaced nodes in longitude. For this sampling (DH
), the grids make use of the longitude band at 90\(^{\circ}\) N, but not 90\(^{\circ}\) S, and the number of samples is \(2(L+1)\), which is always even. Given that the sampling in latitude was imposed a priori, these grids contain almost twice as many samples in latitude as the grids used with GaussLegendre quadrature. It should be noted that for this quadrature, the longitude band at 90\(^{\circ}\) N is ultimately downweighted to zero, and hence has no influence on the returned spherical harmonic coefficients.
For geographic data, it is common to work with grids that are equally spaced in degrees latitude and longitude. SHTOOLS provides the option of using grids of size \(N\times2N\), and when performing the Fourier transforms for this case (DH2
), the coefficients \(c_{lm}\) and \(s_{lm}\) with \(m>L\) are discarded. The redundant data points at 360\(^{\circ}\) E longitude and the latitudinal band at 90\(^{\circ}\) S are not required by the spherical harmonic transformation routines, but can be computed by specifying the optional argument extend
.
Comparison of DH and GLQ grids
The properties of the Driscoll and Healy [1994] and GaussLegendre Quadrature grids are summarized in the following table:
DH1  DH2  GLQ  

Name  Driscoll and Healy  Driscoll and Healy  GaussLegendre Quadrature 
Shape (\(N_{lat} \times N_{lon}\))  \(N \times N\)  \(N \times 2N\)  \(N \times 2N\) 
\(L\)  \(N/21\)  \(N/21\)  \(N1\) 
\(N\)  \(2L+2\)  \(2L+2\)  \(L+1\) 
\(\Delta \theta\)  \(180^{\circ}/N\)  \(180^{\circ}/N\)  Variable 
\(\Delta \phi\)  \(360^{\circ}/N\)  \(180^{\circ}/N\)  \(360^{\circ}/(2N1)\) 
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
.
References

Driscoll, J. R. and D. M. Healy, Computing Fourier transforms and convolutions on the 2sphere, Adv. Appl. Math., 15, 202250, doi:10.1006/aama.1994.1008, 1994.

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.