Accounting for the beam shape¶
Warning
The beam size of an observation introduces artificial correlations into the data on scales near to and below the beam size. This affects the shape of various turbulence statistics that measure spatial properties (Spatial Power-Spectrum, MVC, VCA, Delta-Variance, Wavelets, SCF).
The beam size is typically expressed as the full-width-half-max (FWHM). However, it is important to note that the data will still be correlated beyond the FWHM. For example, consider a randomly-drawn image with a specified power-law index:
>>> import matplotlib.pyplot as plt
>>> from turbustat.simulator import make_extended
>>> from turbustat.io.sim_tools import create_fits_hdu
>>> from astropy import units as u
>>> # Image drawn from red-noise
>>> rnoise_img = make_extended(256, powerlaw=3.)
>>> # Define properties to generate WCS information
>>> pixel_scale = 3 * u.arcsec
>>> beamfwhm = 3 * u.arcsec
>>> imshape = rnoise_img.shape
>>> restfreq = 1.4 * u.GHz
>>> bunit = u.K
>>> # Create a FITS HDU
>>> plaw_hdu = create_fits_hdu(rnoise_img, pixel_scale, beamfwhm, imshape, restfreq, bunit)
>>> plt.imshow(plaw_hdu.data)
The power-spectrum of the image should give a slope of 3:
>>> from turbustat.statistics import PowerSpectrum
>>> pspec = PowerSpectrum(plaw_hdu)
>>> pspec.run(verbose=True, radial_pspec_kwargs={'binsize': 1.0},
... fit_kwargs={'weighted_fit': True}, fit_2D=False,
... low_cut=1. / (60 * u.pix))
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 1.000
Model: OLS Adj. R-squared: 1.000
Method: Least Squares F-statistic: 8.070e+06
Date: Thu, 21 Jun 2018 Prob (F-statistic): 0.00
Time: 11:43:47 Log-Likelihood: 701.40
No. Observations: 177 AIC: -1399.
Df Residuals: 175 BIC: -1392.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 0.0032 0.001 3.952 0.000 0.002 0.005
x1 -2.9946 0.001 -2840.850 0.000 -2.997 -2.992
==============================================================================
Omnibus: 252.943 Durbin-Watson: 1.077
Prob(Omnibus): 0.000 Jarque-Bera (JB): 26797.433
Skew: -5.963 Prob(JB): 0.00
Kurtosis: 62.087 Cond. No. 4.55
==============================================================================
Now we will smooth this image with a Gaussian beam. The easiest way to do this is to use the built-in
tools from the spectral-cube and
radio_beam packages.
We will convert the FITS HDU to a spectral-cube Projection, and define a pencil beam for the
initial image:
>>> from spectral_cube import Projection
>>> from radio_beam import Beam
>>> pencil_beam = Beam(0 * u.deg)
>>> plaw_proj = Projection.from_hdu(plaw_hdu)
>>> plaw_proj = plaw_proj.with_beam(pencil_beam)
Next we will define the beam to smooth to. A 3-pixel wide FWHM is reasonable:
>>> new_beam = Beam(3 * plaw_hdu.header['CDELT2'] * u.deg)
>>> plaw_conv = plaw_proj.convolve_to(new_beam)
>>> plaw_conv.quicklook()
How has smoothing changed the shape of the power-spectrum?
>>> # Change the colours and comment these lines if you don't use seaborn
>>> import seaborn as sb
>>> col_pal = sb.color_palette()
>>> pspec2 = PowerSpectrum(plaw_conv)
>>> pspec2.run(verbose=True, xunit=u.pix**-1, fit_2D=False,
... low_cut=0.025 / u.pix, high_cut=0.1 / u.pix,
... radial_pspec_kwargs={'binsize': 1.0},
... apodize_kernel='tukey')
>>> plt.axvline(np.log10(1 / 3.), color=col_pal[3], linewidth=8, alpha=0.8,
... zorder=1)
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.988
Model: OLS Adj. R-squared: 0.988
Method: Least Squares F-statistic: 2059.
Date: Thu, 21 Jun 2018 Prob (F-statistic): 1.54e-25
Time: 14:23:19 Log-Likelihood: 35.997
No. Observations: 27 AIC: -67.99
Df Residuals: 25 BIC: -65.40
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const -1.0626 0.098 -10.848 0.000 -1.264 -0.861
x1 -3.5767 0.079 -45.378 0.000 -3.739 -3.414
==============================================================================
Omnibus: 3.417 Durbin-Watson: 0.840
Prob(Omnibus): 0.181 Jarque-Bera (JB): 2.072
Skew: -0.650 Prob(JB): 0.355
Kurtosis: 3.391 Cond. No. 15.7
==============================================================================
The slope of the power-spectrum is significantly steepened on small scales by the beam (see the reported result in variable x1 above).
And this steepening occurs on scales much larger than the beam FWHM, which is indicated by
the thick purple vertical line in the left-hand side of the plot. The fitting was restricted to scales much larger than three times the beam width. However, the recovered slope is still steeper than the original -3.
Also note that convolving the image with the beam causes some tapering at the edges of the image, breaking the periodicity at the edges. The image was apodized with a Tukey window, which causes some of the deviations at large scales (small frequencies). See the tutorial page on apodizing kernels for more.
The beam size must be corrected for in the image prior to fitting the power-spectrum. This can be done by (1) including a Gaussian beam component in the model used to fit the power-spectrum, or (2) divide the power-spectrum of the image by the power-spectrum of the beam response. The former requires using a non-linear model, and is not currently implemented in TurbuStat (see Martin et al. 2015 for an example). The latter method can be applied prior to fitting, allowing a linear model to still be used for fitting.
The beam correction in TurbuStat requires the optional package radio_beam to be installed. radio_beam allows the beam response for any 2D elliptical Gaussian to be returned. For statistics that create a power-spectrum (Spatial Power-Spectrum, VCA, MVC), the beam correction can be applied by specifying beam_correct=True:
>>> pspec3 = PowerSpectrum(plaw_conv)
>>> pspec3.run(verbose=True, xunit=u.pix**-1, fit_2D=False,
... low_cut=0.025 / u.pix, high_cut=0.4 / u.pix,
... apodize_kernel='tukey', beam_correct=True)
>>> plt.axvline(np.log10(1 / 3.), color=col_pal[3], linewidth=8, alpha=0.8,
... zorder=1)
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.998
Model: OLS Adj. R-squared: 0.998
Method: Least Squares F-statistic: 8.828e+04
Date: Thu, 21 Jun 2018 Prob (F-statistic): 5.55e-192
Time: 14:38:33 Log-Likelihood: 268.87
No. Observations: 137 AIC: -533.7
Df Residuals: 135 BIC: -527.9
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const -0.2247 0.008 -27.671 0.000 -0.241 -0.209
x1 -2.9961 0.010 -297.116 0.000 -3.016 -2.976
==============================================================================
Omnibus: 7.089 Durbin-Watson: 1.500
Prob(Omnibus): 0.029 Jarque-Bera (JB): 9.274
Skew: 0.285 Prob(JB): 0.00969
Kurtosis: 4.140 Cond. No. 5.50
==============================================================================
The shape of the power-spectrum has been restored and we recover the correct slope. The deviation on small scales (large frequencies) occurs on scales smaller than about the FWHM of the beam where the information has been lost by the spatial smoothing applied to the image. If the beam is over-sampled by a larger factor — say with a 6-pixel FWHM instead of 3 — the increase in power on small scales will affect a larger region of the power-spectrum. This region should be excluded from the power-spectrum fit. A reasonable lower-limit to fit the power-spectrum to is the FWHM of the beam. Additional noise in the image will tend to flatten the power-spectrum to larger scales, so setting the lower fitting limit to a couple times the beam width may be necessary. We recommend visually examining the quality of the fit.
Here are the three power-spectra shown above overplotted to highlight the shape changes from spatial smoothing:
>>> pspec.plot_fit(color=col_pal[0], label='Original')
>>> pspec2.plot_fit(color=col_pal[1], label='Smoothed')
>>> pspec3.plot_fit(color=col_pal[2], label='Beam-Corrected')
>>> plt.legend(frameon=True, loc='lower left')
>>> plt.axvline(np.log10(1 / 3.), color=col_pal[3], linewidth=8, alpha=0.8, zorder=-1)
>>> plt.ylim([-2, 7.5])
>>> plt.tight_layout()
Similar fitting restrictions apply to the MVC and VCA, as well. The beam correction can be applied in the same manner as described above. For other spatial methods which do not use the power-spectrum, the scales of the beam should at least be excluded from any fitting. For example, lag scales smaller than the beam in the Delta-Variance, Wavelets, and SCF should not be fit. The spatial filtering used to measure Statistical Moments should be set to a width of at least the beam size.