pyshtools defines three major classes that simplify access to the underlying SHTOOLS functions:
The underlying data are stored in internal subclasses in different formats (e.g. kind = 'real' or 'complex' and grid = 'DH' or 'GLQ'). In this tutorial, we will demonstrate several aspects related to the SHCoeffs and SHGrid classes.
The spherical harmonic coefficient class SHCoeffs can be initialized directly from a numpy array that holds the spherical harmonic coefficient data. Spherical harmonic coefficients are stored as a numpy array of dimension (2, lmax+1, lmax+1), where the first column corresponds to the cosine [0] and sine [1] components, the second column
corresponds to the spherical harmonic degree l, and the third column corresponds to the angular order m. We will first create a numpy array of coefficients that are equal to zero, with the exception of the l=5, m=2 harmonic.
In [1]:
%matplotlib inline
from __future__ import print_function # only necessary if using Python 2.x
import matplotlib.pyplot as plt
import numpy as np
from pyshtools.shclasses import SHCoeffs, SHGrid, SHWindow
lmax = 200
coeffs = np.zeros((2, lmax+1, lmax+1))
coeffs[0, 5, 2] = 1.
To initialize a new class instance, we make use of the constructor method from_array():
In [2]:
coeffs_l5m2 = SHCoeffs.from_array(coeffs)
When initializing a new class instance, the default is to assume that the input coefficients are 4-pi normalized excluding the Condon-Shortley phase. This normalization convention can be overridden by setting the optional parameter 'normalization', which takes values of '4pi', 'ortho' or 'schmidt', along with the parameter 'csphase', which can be 1 (to exclude the Condon-Shortley phase) or -1 (to include it). The SHCoeffs class contains many methods, and here we use plot_spectrum() to plot the power spectrum:
In [3]:
fig, ax = coeffs_l5m2.plot_spectrum(xscale='log')
To plot the function that corresponds to the coefficients, we first need to expand it on a grid, which can be accomplished using the expand() method:
In [4]:
grid_l5m2 = coeffs_l5m2.expand('DH2')
This returns a new SHGrid class instance. The resolution of the grid is determined automatically to correspond to the maximum degree of the spherical harmonic coefficients in order to ensure good sampling. The optional parameter 'grid' can be 'DH2' for a Driscoll and Healy sampled grid with nlon = 2 * nlat, 'DH' for a Driscol and Healy sampled grid with nlon = nlat, or 'GLQ' for a grid used with the Gauss-Legendre quadrature expansion routines. Once the grid is created, it can be plotted using the built-in method plot().
In [5]:
fig, ax = grid_l5m2.plot()
Another constructor for the SHCoeffs class is the from_random() method. It takes a power spectrum (power per degree l of the coefficients) and generates coefficients that are independent normal distributed random variables with the provided expected power spectrum. This corresponds to a stationary and isotropic random model on the surface of the sphere whose autocorrelation function is given by the spherical harmonic addition theorem.
We initialize coefficients here with a scale-free power spectrum that has equal band power beyond the scale length that defines the size of the largest model features. The particular property of this model is that it is invariant under zoom operations.
In [6]:
a = 10 # scale length
ls = np.arange(lmax+1, dtype=np.float)
power = 1. / (1. + (ls / a) ** 2) ** 0.5
coeffs_global = SHCoeffs.from_random(power)
fig, ax = coeffs_global.plot_spectrum(unit='per_dlogl', xscale='log')
fig, ax = coeffs_global.expand('DH2').plot()
Spherical harmonics coefficients can be expressed in a different coordinate system very efficiently. Importantly, the power per degree spectrum is invariant under rotation. We demonstrate this by rotating a zonal spherical harmonic (m=0) that is centered about the north-pole to the equator.
The rotations are specified by the three Euler angles alpha, beta, and gamma. There are several different conventions for specifying these angles, and they can either provide the angles for rotating the physical body or coordinate system. Please read the documentation of this method before proceeding!
In this example, we use the constructor from_zeros(), and then use the method set_coeffs() to initialize a single spherical harmonic coefficient:
In [7]:
coeffs_l5m0 = SHCoeffs.from_zeros(lmax)
coeffs_l5m0.set_coeffs(1., 5, 0)
alpha = 0. # around z-axis
beta = 90. # around x-axis (lon=0)
gamma = 10. # around z-axis again
coeffs_l5m0_rot = coeffs_l5m0.rotate(alpha, beta, gamma, degrees=True)
fig, ax = coeffs_l5m0_rot.plot_spectrum(xscale='log', show=False)
ax.set(ylim=[0.01,10])
grid_l5m0_rot = coeffs_l5m0_rot.expand('DH2')
fig, ax = grid_l5m0_rot.plot()
In [8]:
grid_new = (2 * grid_l5m0_rot + grid_l5m2**2 - grid_l5m2 * grid_l5m2) / 2.0
grid_new.plot()
coeffs = grid_new.expand()
fig, ax = coeffs.plot_spectrum()