Multitaper Windows - Class Interface

The SHWindow class provides an interface to compute window functions that concentrate energy spatially into a given region and within a given spherical harmonic bandwidth. This allows to generate a coupling matrix that couples degrees over a range that is as little as possible.

As an example, we would like to determine the spectrum of the bathymetry of the Earth and compare it with the land spectrum. To this end, we generate window functions that are separately concentrated over the land and the oceans.

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
In [2]:
# Read Earth topography example data.
infile = '../ExampleDataFiles/topo.dat.gz'
topo = np.loadtxt(infile)
nlat, nlon = topo.shape
lmax = nlat / 2 - 1
land_mask = (topo > 0)
ocean_mask = ~land_mask

# Plot the land and ocean masks.
fig, (col1, col2) = plt.subplots(1, 2, figsize=(13, 5))
col1.imshow(land_mask, extent=(-180, 180, -90, 90), cmap='viridis')
col1.set(xlabel='longitude', ylabel='latitude', title='land mask')
col2.imshow(ocean_mask, extent=(-180, 180, -90, 90), cmap='viridis')
col2.set(xlabel='longitude', ylabel='latitude', title='ocean mask');

Optimally concentrated window functions

SHTOOLS provides a method to solve the so called concentration problem, i.e. to find a window that is optimally concentrated in a given region with band-limited coefficient energy within a range of degrees up to and including lmax. The smaller the region, the larger lmax will need to be to allow for a good concentration. Because the oceans span a larger region, lmax_ocean can therefore be smaller than lmax_land while still getting well concentrated windows. The concentration value that is indicated above the plots gives the proportion of the energy of the window that lies within the original region. The localized power spectrum is a (possibly weighted) average of the individual localized spectral estimates using these windows.

In [3]:
lmax_ocean = 7
lmax_land = 11
nwins = 10
ocean_windows = SHWindow.from_mask(ocean_mask, lmax_ocean, nwins)
land_windows = SHWindow.from_mask(land_mask, lmax_land, nwins)

print('ocean windows')
ocean_windows.info()
ocean_windows.plot_windows(nwins, show=True)
print('land windows')
land_windows.info()
land_windows.plot_windows(nwins, show=True)
ocean windows
kind = 'mask'
lwin = 7
nwin = 10
Taper weights are not set.
land windows
kind = 'mask'
lwin = 11
nwin = 10
Taper weights are not set.
Out[3]:
(<matplotlib.figure.Figure at 0x10addda90>,
 array([[<matplotlib.axes._subplots.AxesSubplot object at 0x10bbc9080>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10bc35be0>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10bd3b400>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10bd75668>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10bdc0d68>],
        [<matplotlib.axes._subplots.AxesSubplot object at 0x10be01eb8>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10be518d0>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10be8dcf8>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10bed7e10>,
         <matplotlib.axes._subplots.AxesSubplot object at 0x10befa860>]], dtype=object))

The following plot shows how power of the land windows extends to higher degrees than the power of the ocean windows. The ocean will allow for higher quality measurements with lower biasses given the lower lmax associated with these windows.

In [4]:
winpower_land = land_windows.get_powerperdegree()
winpower_ocean = ocean_windows.get_powerperdegree()

fig, ax = plt.subplots(1, 1)
for itaper in range(nwins):
    ax.plot(winpower_ocean[:, itaper], c='blue')
    
for itaper in range(nwins):
    ax.plot(winpower_land[:, itaper], c='green')

ax.set(xlabel='degree l', ylabel='power per degree l',
       title='ocean (blue) and land (green) window power');

The coupling matrix

The coupling matrix reflects the different quality of the land and ocean spectra. The higher degree part (towards the bottom right) corresponds to a simple convolution, as it becomes asymptotically symmetric around the diagonal. The land windows couple degrees over a wider range and therefore decrease the spectral resolution that can be achieved.

The second effect is the asymmetric part of the coupling matrix towards low degrees: Low input degrees map to higher output degrees. For example, the input degree 0 maps strongest to about degree 3 in the ocean case and to about degree 6 in the land case. This is a consequence of the fact that we can not estimate the global mean or very large wavelength structures from subregions that are much smaller than the feature of interest. Low degree energy maps therefore to the lowest window degrees, where they increase the overall power.

Smoothing and bias need to be considered when comparing spectra that have been obtained with two different windows.

In [5]:
lmax_data = 40

print('ocean coupling matrix')
lmax_valid1 = lmax_data - lmax_ocean
lmax_full1 = lmax_data + lmax_ocean
fig1, ax1 = ocean_windows.plot_couplingmatrix(lmax_data, nwins,
                                              mode='full', show=False)
ax1.plot([-0.5, lmax_data + 0.5], [lmax_valid1, lmax_valid1], c='red')
ax1.text(1, lmax_valid1 + 7, 'from the line on\naffected by boundary', fontsize=12, color='red')
ax1.set(xlim=(-0.5, lmax_data + 0.5), ylim=(lmax_full1, 0.))

print('land coupling matrix')
fig2, ax2 = land_windows.plot_couplingmatrix(lmax_data, nwins,
                                             mode='full', show=False)
lmax_valid2 = lmax_data - lmax_land
lmax_full2 = lmax_data + lmax_land
ax2.plot([-0.5, lmax_data + 0.5], [lmax_valid2, lmax_valid2], c='red')
ax2.text(1, lmax_valid2 + 7, 'from the line on\naffected by boundary', fontsize=12, color='red')
ax2.set(xlim=(-0.5, lmax_data+0.5), ylim=(lmax_full2, 0.))
ocean coupling matrix
land coupling matrix
Out[5]:
[(51, 0.0), (-0.5, 40.5)]

The coupling matrix describes the expected coupling between coefficients that are independent for stationary and isotropic models. Obviously, the toography of the Earth is not stationary and isotropic and a simple prediction about a regional measurement can therefore not be made. For example, if a non-stationary model is zero in the windowed region the local spectrum will also be zero, but not the global spectrum that is distorted by the coupling matrix. Nevertheless, the coupling matrix is useful, because it allows us to differentiate between the effect of the window and the effect due to a particular region.

Regional spectral estimates

We are now in a position to compute regional power spectral estimates of the land and of the ocean topography as well as the corresponding power spectra that we would expect to see on land and in the ocean if Earth's topography was stationary and isotropic.

In [6]:
# Determine the spherical harmonic coefficients and power
# spectrum of the Earth's topography.
grid_topo = SHGrid.from_array(topo)
coeffs_topo = grid_topo.expand()
power_global = coeffs_topo.get_powerperdegree()
lmax_global = coeffs_topo.lmax
ls_global = np.arange(len(power_global))

# Calculate the multitaper power spectrum estimate of the oceans.
power_ocean, dpower_ocean = ocean_windows.get_multitaperpowerspectrum(coeffs_topo, nwins)

# Determine the predicted multitaper power spectrum of the oceans
# using the global topographic power spectrum and coupling matrix.
cmatrix_ocean = ocean_windows.get_couplingmatrix(lmax_global, nwins, mode='valid')
power_ocean_predicted = np.dot(cmatrix_ocean, power_global)
ls_ocean = np.arange(len(power_ocean))

# Do the same thing for the power spectrum localized over the land.
power_land, dpower_land = land_windows.get_multitaperpowerspectrum(coeffs_topo, nwins)
cmatrix_land = land_windows.get_couplingmatrix(lmax_global, nwins, mode='valid')
power_land_predicted = np.dot(cmatrix_land, power_global)
ls_land = np.arange(len(power_land))

# Plot the ocean and land topography.
fig, (col1, col2) = plt.subplots(1, 2, figsize=(10, 5))
col1.imshow(-topo * ocean_mask, cmap='viridis')
col2.imshow(topo * land_mask, cmap='viridis')

# Plot the power per log(2) band spectra.
fig, ax = plt.subplots(1, 1, figsize=(10, 5))
ax.plot(ls_global, power_global * ls_global, c='grey', alpha=0.5, lw=2, label='global')

ax.plot(ls_ocean, power_ocean * ls_ocean, '-', c='blue', label='ocean (measured)')
ax.plot(ls_ocean, power_ocean_predicted * ls_ocean, '--', c='blue', label='ocean (predicted)')

ax.plot(ls_land, power_land * ls_land, '-', c='green', label='land (measured)')
ax.plot(ls_land, power_land_predicted * ls_land, '--', c='green', label='land (predicted)')

ax.legend(loc=3)
ax.grid(True)
ax.set_xscale('log', basex=2)
ax.set_yscale('log', basey=2)

As we would have expected, the power spectra are vastly different. The ocean spectrum is dominated by the strong low degrees. The land spectrum is entirely missing the long degree component because it doesn't deviate much from 0 altitude, except in mountain ranges. On the other hand, the ocean shows weaker small scale variations than the land spectrum.

Note that the definition of the windows largely excludes continental shelfs, which are at the boundaries of the mask and that have strong small scale power.

The difference between the predicted land and ocean spectra demonstrate, that even if Earth's topography were stationary, the window functions cause different power spectra. We illustrate this in the next plot where we replace Earth's topography with a random model that has the same expectation for the power spectrum, but that is stationary.

In [7]:
# Construct a stationary random model:
np.random.seed(0)  # fix seed for reproducable figure
coeffs_stationary = SHCoeffs.from_random(power_global)
grid_stationary = coeffs_stationary.expand('DH2')

# Determine the localized multitaper power spectrum estimate for the oceans,
# and the prediction based on the coupling matrix.
power_ocean, dpower_ocean = ocean_windows.get_multitaperpowerspectrum(coeffs_stationary, nwins)
cmatrix_ocean = ocean_windows.get_couplingmatrix(lmax_global, nwins, mode='valid')
power_ocean_predicted = np.dot(cmatrix_ocean, power_global)
ls_ocean = np.arange(len(power_ocean))

# Do the same for the land.
power_land, dpower_land = land_windows.get_multitaperpowerspectrum(coeffs_stationary, nwins)
cmatrix_land = land_windows.get_couplingmatrix(lmax_global, nwins, mode='valid')
power_land_predicted = np.dot(cmatrix_land, power_global)
ls_land = np.arange(len(power_land))

# Plot the ocean and land topography.
fig, (col1, col2) = plt.subplots(1, 2, figsize=(10, 5))
col1.imshow(grid_stationary.data * ocean_mask, cmap='viridis')
col2.imshow(grid_stationary.data * land_mask, cmap='viridis')

# Plot the power per log(2) band spectra
fig, ax = plt.subplots(1, 1, figsize=(10, 5))
ax.plot(ls_global, power_global * ls_global, c='grey', alpha=0.5, lw=2,
        label='global')

ax.plot(ls_ocean, power_ocean * ls_ocean, '-', c='blue', label='ocean (measured)')
ax.plot(ls_ocean, power_ocean_predicted * ls_ocean, '--', c='blue',
        label='ocean (predicted)')

ax.plot(ls_land, power_land * ls_land, '-', c='green', label='land (measured)')
ax.plot(ls_land, power_land_predicted * ls_land, '--', c='green',
        label='land (predicted)')

ax.set(title='regional spectra of a stationary random model')
ax.legend(loc=3)
ax.grid(True)
ax.set_xscale('log', basex=2)
ax.set_yscale('log', basey=2)

The regional estimates of the stationary model resemble closely the predicted ones. The low degrees are more sensitive to fluctuations of the random model given that these spectral estimates where constructed based on a smaller number of localized coefficients.

To compare global, ocean and land spectra, the coupling matrix can be inverted to remove the bias. In the case of the stationary model, we expect to retrieve the same original input power spectrum, because apart from random fluctuations, the differences of the power spectra are entirely due to the window effects.

In the case of the real non-stationary topography of the Earth on the other hand, we would get three power spectra that correspond to equivalent stationary global models that have the observed regional spectra. This inversion, however, has to be done with care, because the coupling matrix is singular.