Now that we know how to work with the spherical harmonic coefficient and grid classes, let's move on to the other major feature of pyshtools: localization windows and localized spectral analysis. pyshtools allows for localized spectral analyses using either spherical-cap localization windows or localization windows constructed from an arbitrary mask. The windows are constructed to solve the same optimization problem as the Slepian windows that are used in Cartesian multitaper spectral analysis problems. A single class, SHWindow, bundles the relevant shtools functions. In this notebook, we will demonstrate how to construct optimal localization windows, and how to obtain localized spectral estimates for arbitrary regions on the sphere.
In [1]:
import matplotlib.pyplot as plt
import numpy as np
import pyshtools as pysh
pysh.utils.figstyle(rel_width=0.75)
%config InlineBackend.figure_format = 'retina' # if you are not using a retina display, comment this line
To demonstrate the localized spectral analysis routines, we will make use of a pathological data set. In particular, we will construct a function on the sphere whose power spectrum is different in the northern and southern hemispheres. We first create two grids using random coefficients that have power spectra following power laws with exponents of -2 and -3, up to and including degree 300:
In [2]:
ldata = 300
degrees = np.arange(ldata+1, dtype=float)
degrees[0] = np.inf
power_north = degrees**(-2)
power_south = degrees**(-3)
clm_north = pysh.SHCoeffs.from_random(power_north, seed=12345)
grid_north = clm_north.expand()
clm_south = pysh.SHCoeffs.from_random(power_south, seed=54321)
grid_south = clm_south.expand()
Next, create a new grid that contains data from the northern hemisphere of one grid and the southern hemisphere of the other. The SHGrid method from_array() returns a copy of the data, so you don't need to worry about modifying the original data set:
In [3]:
grid = pysh.SHGrid.from_array(grid_north.data)
nlat = grid_north.nlat
grid.data[nlat//2:nlat+1,:] = grid_south.data[nlat//2:nlat+1,:].copy()
grid.plot(show=False)
clm = grid.expand()
Here, we made use of the grid attribute nlat, which is the number of rows of the gridded data, and we modified the raw gridded data for grid that is stored in the data class attribute.
Next, let's construct some spherical-cap localization windows. This is done using the constructor from_cap() that is associated with the SHWindow class. The other option, as we will demonstrate later, is from_mask() and takes as input a user defined mask. For spherical-cap windows, we need to specify both the angular radius of the cap and its spherical harmonic bandwidth, and for this example we will use an angular radius of 30 degrees and a bandwidth of 29.
In [4]:
lwin = 29
capwin = pysh.SHWindow.from_cap(theta=30.,
lwin=lwin)
This command calculates the spherical harmonic coefficients of the spherical cap windows along with their associated angular orders and concentration factors. If you ever need to access these directly, they are stored in the class attributes tapers, orders and eigenvalues. By default, theta should be provided in degrees, but this can be input in radians by specifying the optional parameter theta_degrees=False. If you are curious to see what these windows look like, you can plot them using the built in method plot_windows(). Let's plot the first 12 of them:
In [5]:
fig, axes = capwin.plot_windows(12, loss=True, show=False)
As is seen, the windows are centered oven the north pole, 1 minus the concentration factor is noted above the window as the loss factor, and all of these windows are very well localized. To instead provide the concentration factor in the title, just set loss=False. If you want to know how many windows have concentration factors greater than a specified value, you can use the method number_concentrated():
In [6]:
capwin.number_concentrated(0.99)
Out[6]:
You can return an SHGrid class instance of an individual window using the method to_shgrid(), you can return the spherical harmonic coefficients of individual windows using to_array() and to_shcoeffs(), and you can also return as a numpy array the power spectra of one or more windows using the method spectra(). If you want to plot the power spectra of the windows, this can be done using the built in method plot_spectra():
In [7]:
fig, axes = capwin.plot_spectra(12, loss=True, show=False)
In general, we will not perform localized spectral analyses at the north pole, so let's rotate the windows so that they are centered at 45 degrees north and 90 degrees east. This is easily accomplished using the rotate() method. To speed things up, we will rotate only the first 50 best-localized windows, and not all 900 of them:
In [8]:
capwin.rotate(clat=45.,
clon=90.,
nwinrot=50)
fig, axes = capwin.plot_windows(12, loss=True, show=False)
In rotating the windows, the spherical harmonic coefficients of the initial tapers centered at the north pole are not modified, but a second set of rotated coefficients are created and stored in the attribute coeffs. If you ever forget how you constructed your windows, you can always use the info() method to print out all the metadata:
In [9]:
capwin.info()
Next, let's calculate the localized power spectrum at two different locations, one in the northern hemisphere and the other in the southern hemisphere. We will use all 34 windows that have concentration factors greater than 0.99:
In [10]:
k = capwin.number_concentrated(0.99)
mtse_north, sd_north = capwin.multitaper_spectrum(clm, k)
mtse_south, sd_south = capwin.multitaper_spectrum(clm, k, clat=-55., clon=90.)
If you don't provide the central latitude and longitude of the analysis with the parameters clat and clon, the coefficients that were created previously using the rotate() method will be employed. If these coefficients are not available, the analysis will be performed at the north pole.
multitaper_spectrum returns a power spectrum whose maximum degree is ldata - lwin. This is because localized degrees beyond this limit depend upon coefficients of the global data beyond ldata, and these are not known. If the global function was truly zero beyond ldata, you could simply construct the spherical harmonic coefficients of clm to have zeros beyond that degree.
When a function is multiplied by a window, the power spectrum of the localized function is modified. For the case of stationary and random data, the expectation of this biasing effect can be estimated using two methods. First, we could calculate the "coupling matrix" using the method coupling_matrix(). The expectation of the output windowed power spectrum is then simply the multiplication of this matrix by the input global spectrum. If you are interested, you can plot the coupling matrix using the built in method plot_coupling_matrix():
In [11]:
fig, ax = capwin.plot_coupling_matrix(ldata, k=k, show=False)
We will instead estimate the expectation of the windowed spectrum using biased_spectrum() and the global power spectra used to create the data in the northern and southern hemispheres:
In [12]:
power_north_biased = capwin.biased_spectrum(power_north, k)
power_south_biased = capwin.biased_spectrum(power_south, k)
Note that this function returns a spectrum with a maximum degree ldata+lwin. In calculating the expectation of the biased power spectrum from the global spectrum, the routine assumes implicitly that the global power spectrum is exactly zero for all degrees greater than ldata.
Finally, lets plot the global power spectra, the localized spectra, and the localized expectations. Assuming that the data are unknown above degree ldata, we plot only the valid localized spectral estimates up to degree ldata - lwin.
In [13]:
degrees2 = np.arange(ldata+capwin.lwin+1)
degrees3 = np.arange(ldata-capwin.lwin+1)
fig, ax = plt.subplots(1,1)
ax.plot(degrees, power_north, '-k', label='$l^{-2}$')
ax.plot(degrees, power_south, '--k', label='$l^{-3}$')
ax.plot(degrees2, power_north_biased, '-r', label='Localized expectation')
ax.plot(degrees2, power_south_biased, '--r', label='Localized expectation')
ax.plot(degrees3, mtse_north, '-b', label='Observed')
ax.plot(degrees3, mtse_south, '--b', label='Observed')
ax.set(yscale='log', xlabel='Spherical harmonic degree', ylabel='Power', xlim=(0,200-capwin.lwin), ylim=(1.e-7, 1.))
ax.legend()
ax.axvline(lwin, color='k', linestyle=':')
Out[13]:
As is seen, the expectation of the localized spectra (red) are significantly biased at low degrees in comparison to the unwindowed global fields (black). This is not necessarily a problem, because this effect is quantifiable. The actual localized power spectra (blue) are found to closely resemble the expected value, especially for degrees greater than the window bandwidth of 29 (vertical dashed line).
In this example, we calculated only the localized power spectrum of a single function. If you were interested in the localized cross-power spectrum of two functions, this could be caluculated using the method multitaper_cross_spectrum().
We next demonstrate how to perform a localized spectral analysis over an irregular region. For this example, we will first create a function whose power spectrum is different over the oceans and landmass of Earth, we will then calculate windows that localize these two regions, and finally we will perform a multitaper spectral analysis similar to what we did with the spherical cap localization windows above.
First, let's read in the spherical harmonic coefficients of Earth's topography (referenced to mean sea level):
In [14]:
topo_coeffs = pysh.SHCoeffs.from_file('../ExampleDataFiles/srtmp300.msl', lmax=300) / 1000.
topo = topo_coeffs.expand()
fig, ax = topo.plot(colorbar='right', cb_label='Elevation, km', show=False)
Note that instead of reading the coefficients from a local file on your computer, it is also possible to specify a URL to read the file remotely. Next, create a boolean mask that is True for elevations that are greater than zero:
In [15]:
land_mask = pysh.SHGrid.from_array(topo.data > 0)
ocean_mask = pysh.SHGrid.from_array(topo.data <= 0)
fig, ax = land_mask.plot(show=False)
fig2, ax2 = ocean_mask.plot(show=False)
Constructing localization windows from a boolean mask is simple using the from_mask() constructor of the SHWindow class. Here, we will use a spherical harmonic bandwidth of 30, and then plot maps and power spectra of the first 12 windows that are concentrated on land:
In [16]:
lwin = 30
land_windows = pysh.SHWindow.from_mask(land_mask.data, lwin)
ocean_windows = pysh.SHWindow.from_mask(ocean_mask.data, lwin)
fig, axes = land_windows.plot_windows(12, show=False)
fig, axes = land_windows.plot_spectra(12, show=False)
To demonstrate that these windows actually concentrate their power over the land and oceans, we next plot the cumulative energy of all windows that have concentration factors greater than 0.99. In this case, because the surface area of the oceans is larger than that of the land, there are many more ocean windows than land windows.
In [17]:
k_land = land_windows.number_concentrated(0.99)
k_ocean = ocean_windows.number_concentrated(0.99)
print('k_land = {:d}'.format(k_land))
print('k_ocean = {:d}'.format(k_ocean))
land_energy = (land_windows.to_shgrid(0).to_array())**2
for i in range(1,k_land):
land_energy += (land_windows.to_shgrid(i).to_array())**2
ocean_energy = (ocean_windows.to_shgrid(0).to_array())**2
for i in range(1,k_ocean):
ocean_energy += (ocean_windows.to_shgrid(i).to_array())**2
land_energy = pysh.SHGrid.from_array(land_energy)
ocean_energy = pysh.SHGrid.from_array(ocean_energy)
fig, ax = land_energy.plot(show=False)
fig, ax = ocean_energy.plot(show=False)
Next, let's construct a pathological function that has different power spectra over the oceans and land. In this example, we will make use of the same grids that we constructed for our demonstation using spherical cap localization windows. In particular, the land mass will use the grid constructed from random coefficients with a power spectrum that followed a power law with exponenent -2, and the oceans will use the grid constructed with the power law exponent of -3.
In [18]:
grid_earth = grid_north.copy()
grid_earth.data[topo.data<=0.] = grid_south.data[topo.data<=0.].copy()
clm_earth = grid_earth.expand()
fig, ax = grid_earth.plot(show=False)
Finally, we calculate localized power spectra over the land and oceans, as well as the expection of the windowed spectra. For demonstration purposes, we will use the same number of localization windows for both regions.
In [19]:
mtse_land, sd_land = land_windows.multitaper_spectrum(clm_earth, k_land)
mtse_ocean, sd_ocean = ocean_windows.multitaper_spectrum(clm_earth, k_land)
power_land_biased = land_windows.biased_spectrum(power_north, k_land)
power_ocean_biased = ocean_windows.biased_spectrum(power_south, k_land)
degrees2 = np.arange(ldata+land_windows.lwin+1)
degrees3 = np.arange(ldata-land_windows.lwin+1)
fig, ax = plt.subplots(1, 1)
ax.plot(degrees, power_north, '-k', label='$l^{-2}$, land')
ax.plot(degrees, power_south, '--k', label='$l^{-3}$, oceans')
ax.plot(degrees2, power_land_biased, '-r', label='Localized expectation')
ax.plot(degrees2, power_ocean_biased, '--r', label='Localized expectation')
ax.plot(degrees3, mtse_land, '-b', label='Observed')
ax.plot(degrees3, mtse_ocean, '--b', label='Observed')
ax.set(yscale='log', xlabel='Spherical harmonic degree', ylabel='Power', xlim=(0,200-land_windows.lwin), ylim=(1.e-7, 1.))
ax.legend()
ax.axvline(lwin, color='k', linestyle=':')
Out[19]:
As with our example using spherical cap localization windows, the localized power spectra (blue) over the oceans and land match closely their theoretical expectations (red). Both of these are significantly biased from the unwindowed global spectra (black) at low degrees.