def getpolsky(self, debug=False, celestial=True):
"""Create a realisation of the *polarised* sky.
Parameters
----------
debug : boolean, optional
Return intermediate products for debugging. Default is False.
celestial : boolean, optional
Maps are returned in celestial co-ordinates.
Returns
-------
skymap : np.ndarray[freq, pol, pixel]
Notes
-----
This routine tries to make a decent simulation of the galactic
polarised emission.
"""
# Load and smooth the Faraday rotation map to get an estimate for the
# width of the distribution
sigma_phi = healpy.ud_grade(
healpy.smoothing(np.abs(self._faraday), fwhm=np.radians(10.0), verbose=False), self.nside
)
# Set the correlation length in phi
xiphi = 1.0
lmax = 3 * self.nside - 1
la = np.arange(lmax + 1)
# The angular powerspectrum of polarisation fluctuations, we will use
# this to generate the base fluctuations
def angular(l):
l[np.where(l == 0)] = 1.0e16
return (l / 100.0) ** -2.8
# Define a grid in phi that we will use to model the Faraday emission.
dphi = self._dphi
maxphi = self._maxphi
nphi = 2 * int(maxphi / dphi)
phifreq = np.fft.fftfreq(nphi, d=(1.0 / (dphi * nphi)))
# Create the weights required to turn random variables into alms
ps_weight = (angular(la[:, np.newaxis]) / 2.0) ** 0.5
# Generate random maps in the Fourier conjugate of phi. This is
# equivalent to generating random uncorrelated phi maps and FFting
# into the conjugate.
map2 = np.zeros((12 * self.nside ** 2, nphi), dtype=np.complex128)
print "SHTing to give random maps"
for i in range(nphi):
w = np.random.standard_normal((lmax + 1, 2 * lmax + 1, 2)).view(np.complex128)[..., 0]
w *= ps_weight
map2[:, i] = hputil.sphtrans_inv_complex(w, self.nside)
# Weight the conj-phi direction to give the phi correlation structure.
pcfreq = np.fft.fftfreq(nphi, d=dphi)
map2 *= np.exp(-2 * (np.pi * xiphi * pcfreq[np.newaxis, :]) ** 2)
# We need to FFT back into phi, but as scipy does not have an inplace
# transform, we can do this in blocks, replacing as we go.
chunksize = self.nside ** 2
nchunk = 12
for ci in range(nchunk):
si = ci * chunksize
ei = (ci + 1) * chunksize
map2[si:ei] = np.fft.ifft(map2[si:ei], axis=1)
# numpy's var routine is extremely memory inefficient. Use a crappy
# chunking one.
map2 /= 2.0 * chunk_var(map2) ** 0.5
w = np.exp(-0.25 * (phifreq[np.newaxis, :] / sigma_phi[:, np.newaxis]) ** 2)
# Calculate the normalisation explicitly (required when Faraday depth
# is small, as grid is too large).
w /= w.sum(axis=1)[:, np.newaxis]
print "Applying phi weighting"
map2 *= w
if not debug:
del w
def ptrans(phi, freq, dfreq):
dx = dfreq / freq
alpha = 2.0 * phi * 3e2 ** 2 / freq ** 2
return np.exp(1.0j * alpha) * np.sinc(alpha * dx / np.pi)
# return np.exp(1.0J * alpha)
fa = self.nu_pixels
df = np.median(np.diff(fa))
pta = ptrans(phifreq[:, np.newaxis], fa[np.newaxis, :], df) / dphi
print "Transforming to freq"
map4 = np.dot(map2, pta)
if not debug:
del map2
print "Rescaling freq"
map4a = np.abs(map4)
map4 = map4 * np.tanh(map4a) / map4a
del map4a
map5 = np.zeros((self.nu_num, 4, 12 * self.nside ** 2), dtype=np.float64)
print "Scaling by T"
# Unflatten the intensity by multiplying by the unpolarised realisation
map5[:, 0] = self.getsky(celestial=False)
map5[:, 1] = map4.real.T
map5[:, 2] = map4.imag.T
map5[:, 1:3] *= map5[:, 0, np.newaxis, :]
if not debug:
del map4
print "Rotating"
# Rotate to celestial co-ordinates if required.
if celestial:
map5 = hputil.coord_g2c(map5)
if debug:
return map2, map4, w, sigma_phi, pta, map5
else:
return map5
def getsky(self, debug=False, celestial=True):
"""Create a realisation of the *unpolarised* sky.
Parameters
----------
debug : boolean, optional
Return intermediate products for debugging. Default is False.
celestial : boolean, optional
Maps are returned in celestial co-ordinates.
Returns
-------
skymap : np.ndarray[freq, pixel]
"""
# Read in data files.
haslam = healpy.ud_grade(self._haslam, self.nside)
syn = FullSkySynchrotron()
lmax = 3 * self.nside - 1
efreq = np.concatenate((np.array([408.0, 1420.0]), self.nu_pixels))
## Construct map of random fluctuations
# cla = skysim.clarray(syn.angular_powerspectrum, lmax, efreq, zwidth=(self.nu_pixels[1] - self.nu_pixels[0]))
cla = skysim.clarray(syn.angular_powerspectrum, lmax, efreq, zromb=0)
fg = skysim.mkfullsky(cla, self.nside)
## Find the smoothed fluctuations on each scale
sub408 = healpy.smoothing(fg[0], fwhm=np.radians(1.0), verbose=False)
sub1420 = healpy.smoothing(fg[1], fwhm=np.radians(5.8), verbose=False)
## Make a multifrequency map constrained to look like the smoothed maps
## depending on the spectral_map apply constraints at upper and lower frequency (GSM),
## or just at Haslam map frequency
if self.spectral_map == "gsm":
fgs = skysim.mkconstrained(cla, [(0, sub408), (1, sub1420)], self.nside)
else:
fgs = skysim.mkconstrained(cla, [(0, sub408)], self.nside)
# Construct maps of appropriate resolution
sc = healpy.ud_grade(self._sp_ind[self.spectral_map], self.nside)
am = healpy.ud_grade(self._amp_map, self.nside)
## Bump up the variance of the fluctuations according to the variance
# map
vm = healpy.smoothing(fg[0], sigma=np.radians(0.5), verbose=False)
vm = healpy.smoothing(map_variance(vm, 16) ** 0.5, sigma=np.radians(2.0), verbose=False)
mv = vm.mean()
## Construct the fluctuations map
fgt = (am / mv) * (fg - fgs)
## Get the smooth, large scale emission from Haslam+spectralmap
fgsmooth = haslam[np.newaxis, :] * ((efreq / 408.0)[:, np.newaxis] ** sc)
# Rescale to ensure output is always positive
tanh_lin = lambda x: np.where(x < 0, np.tanh(x), x)
fg2 = (fgsmooth * (1.0 + tanh_lin(fgt / fgsmooth)))[2:]
## Co-ordinate transform if required
if celestial:
fg2 = hputil.coord_g2c(fg2)
if debug:
return fg2, fg, fgs, fgt, fgsmooth, am, mv
return fg2
def getpolsky(self, debug=False, celestial=True):
"""Create a realisation of the *polarised* sky.
Parameters
----------
debug : boolean, optional
Return intermediate products for debugging. Default is False.
celestial : boolean, optional
Maps are returned in celestial co-ordinates.
Returns
-------
skymap : np.ndarray[freq, pol, pixel]
Notes
-----
This routine tries to make a decent simulation of the galactic
polarised emission.
"""
# Load and smooth the Faraday rotation map to get an estimate for the
# width of the distribution
sigma_phi = healpy.ud_grade(
healpy.smoothing(np.abs(self._faraday),
fwhm=np.radians(10.0),
verbose=False), self.nside)
# Set the correlation length in phi
xiphi = 1.0
lmax = 3 * self.nside - 1
la = np.arange(lmax + 1)
# The angular powerspectrum of polarisation fluctuations, we will use
# this to generate the base fluctuations
def angular(l):
l[np.where(l == 0)] = 1.0e16
return (l / 100.0)**-2.8
# Define a grid in phi that we will use to model the Faraday emission.
dphi = self._dphi
maxphi = self._maxphi
nphi = 2 * int(maxphi / dphi)
phifreq = np.fft.fftfreq(nphi, d=(1.0 / (dphi * nphi)))
# Create the weights required to turn random variables into alms
ps_weight = (angular(la[:, np.newaxis]) / 2.0)**0.5
# Generate random maps in the Fourier conjugate of phi. This is
# equivalent to generating random uncorrelated phi maps and FFting
# into the conjugate.
map2 = np.zeros((12 * self.nside**2, nphi), dtype=np.complex128)
print "SHTing to give random maps"
for i in range(nphi):
w = np.random.standard_normal(
(lmax + 1, 2 * lmax + 1, 2)).view(np.complex128)[..., 0]
w *= ps_weight
map2[:, i] = hputil.sphtrans_inv_complex(w, self.nside)
# Weight the conj-phi direction to give the phi correlation structure.
pcfreq = np.fft.fftfreq(nphi, d=dphi)
map2 *= np.exp(-2 * (np.pi * xiphi * pcfreq[np.newaxis, :])**2)
# We need to FFT back into phi, but as scipy does not have an inplace
# transform, we can do this in blocks, replacing as we go.
chunksize = self.nside**2
nchunk = 12
for ci in range(nchunk):
si = ci * chunksize
ei = (ci + 1) * chunksize
map2[si:ei] = np.fft.ifft(map2[si:ei], axis=1)
# numpy's var routine is extremely memory inefficient. Use a crappy
# chunking one.
map2 /= (2.0 * chunk_var(map2)**0.5)
w = np.exp(-0.25 *
(phifreq[np.newaxis, :] / sigma_phi[:, np.newaxis])**2)
# Calculate the normalisation explicitly (required when Faraday depth
# is small, as grid is too large).
w /= w.sum(axis=1)[:, np.newaxis]
print "Applying phi weighting"
map2 *= w
if not debug:
del w
def ptrans(phi, freq, dfreq):
dx = dfreq / freq
alpha = 2.0 * phi * 3e2**2 / freq**2
return (np.exp(1.0J * alpha) * np.sinc(alpha * dx / np.pi))
fa = self.nu_pixels
df = np.median(np.diff(fa))
pta = ptrans(phifreq[:, np.newaxis], fa[np.newaxis, :], df) / dphi
print "Transforming to freq"
map4 = np.dot(map2, pta)
if not debug:
del map2
print "Rescaling freq"
map4a = np.abs(map4)
map4 = map4 * np.tanh(map4a) / map4a
del map4a
map5 = np.zeros((self.nu_num, 4, 12 * self.nside**2), dtype=np.float64)
print "Scaling by T"
# Unflatten the intensity by multiplying by the unpolarised realisation
map5[:, 0] = self.getsky(celestial=False)
map5[:, 1] = map4.real.T
map5[:, 2] = map4.imag.T
map5[:, 1:3] *= map5[:, 0, np.newaxis, :]
if not debug:
del map4
print "Rotating"
# Rotate to celestial co-ordinates if required.
if celestial:
map5 = hputil.coord_g2c(map5)
if debug:
return map2, map4, w, sigma_phi, pta, map5
else:
return map5
def getpolsky(self, debug=False, celestial=True):
"""Create a realisation of the *polarised* sky.
Parameters
----------
debug : boolean, optional
Return intermediate products for debugging. Default is False.
celestial : boolean, optional
Maps are returned in celestial co-ordinates.
Returns
-------
skymap : np.ndarray[freq, pol, pixel]
"""
# Load and smooth the Faraday emission
sigma_phi = healpy.ud_grade(healpy.smoothing(np.abs(self._faraday), fwhm=np.radians(10.0)), self.nside)
# Get the Haslam map as a base for the unpolarised emission
haslam = healpy.smoothing(healpy.ud_grade(self._haslam, self.nside), fwhm=np.radians(3.0))
# Create a map of the correlation length in phi
xiphi = 3.0
xiphimap = np.minimum(sigma_phi / 20.0, xiphi)
lmax = 3*self.nside - 1
la = np.arange(lmax+1)
# The angular powerspectrum of polarisation fluctuations
def angular(l):
l[np.where(l == 0)] = 1.0e16
return (l / 100.0)**-2.8
c6 = 3e2 # Speed of light in million m/s
xf = 1.5 # Factor to exand the region in lambda^2 by.
## Calculate the range in lambda^2 to generate
l2l = (c6 / self.nu_upper)**2
l2h = (c6 / self.nu_lower)**2
l2a = 0.5 * (l2h + l2l)
l2d = 0.5 * (l2h - l2l)
# Bounds and number of points in lambda^2
l2l = l2a - xf * l2d
l2h = l2a + xf * l2d
nl2 = int(xf * self.nu_num)
l2 = np.linspace(l2l, l2h, nl2) # Grid in lambda^2 to use
## Make a realisation of c(n, l2)
# Generate random numbers and weight by powerspectrum
w = (np.random.standard_normal((nl2, lmax+1, 2*lmax+1)) + 1.0J * np.random.standard_normal((nl2, lmax+1, 2*lmax+1))) / 2**0.5
w *= angular(la[np.newaxis, :, np.newaxis])**0.5
# Transform from spherical harmonics to maps
map1 = np.zeros((nl2, 12*self.nside**2), dtype=np.complex128)
print "Making maps"
for i in range(nl2):
map1[i] = hputil.sphtrans_inv_complex(w[i], self.nside)
# Weight frequencies for the internal Faraday depolarisation
map1 *= np.exp(-2 * (xiphimap[np.newaxis, :] * l2[:, np.newaxis])**2) / 100.0
if not debug:
del w
# Transform into phispace
map2 = np.fft.fft(map1, axis=0)
# Calculate the phi samples
phifreq = np.pi * np.fft.fftfreq(nl2, d=(l2[1] - l2[0]))
if not debug:
del map1
# Create weights for smoothing the emission (corresponding to the
# Gaussian region of emission).
w = np.exp(-0.25 * (phifreq[:, np.newaxis] / sigma_phi[np.newaxis, :])**2)
# Calculate the normalisation explicitly (required when Faraday depth
# is small, as grid is too large).
w /= w.sum(axis=0)[np.newaxis, :]
# When the spacing between phi samples is too large we don't get the
# decorrelation from the independent regions correct.
xiphimap = np.maximum(xiphimap, np.pi / (l2h - l2l))
xiphimap = np.minimum(xiphimap, sigma_phi)
w *= 0.2 * (sigma_phi / xiphimap)**0.5 * (10.0 / haslam)**0.5
# Additional weighting to account for finite frequency bin width
# (assume channels are gaussian). Approximate by performing in
# lambda^2 not nu.
dnu_nu = np.diff(self.nu_pixels).mean() / self.nu_pixels.mean()
# Calculate the spacing lambda^2
dl2 = 2*l2.mean() * dnu_nu / (8*np.log(2.0))**0.5
w *= np.exp(-2.0 * dl2**2 * phifreq**2)[:, np.newaxis]
# Weight map, and transform back into lambda^2
map3 = np.fft.ifft(map2 * w, axis=0)
if not debug:
del map2, w
# Array to hold frequency maps in.
map4 = np.zeros((self.nu_num, 4, 12*self.nside**2), dtype=np.float64)
## Interpolate lambda^2 sampling into regular frequency grid.
## Use a piecewise linear method
for i in range(self.nu_num):
# Find the lambda^2 for the current frequency
l2i = (c6 / self.nu_pixels[i])**2
# Find the bounding array indices in the lambda^2 array
ih = np.searchsorted(l2, l2i)
il = ih - 1
# Calculate the interpolating coefficient
alpha = (l2i - l2[il]) / (l2[ih] - l2[il])
# Interpolate each map at the same time.
mapint = map3[ih] * alpha + (1.0 - alpha) * map3[il]
# Separate real and imaginary parts into polarised matrix
map4[i, 1] = mapint.real
map4[i, 2] = mapint.imag
if not debug:
del map3
# Unflatten the intensity by multiplying by the unpolarised realisation
map4[:, 0] = self.getsky(celestial=False)
map4[:, 1:3] *= map4[:, 0, np.newaxis, :]
# Rotate to celestial co-ordinates if required.
if celestial:
map4 = hputil.coord_g2c(map4)
if debug:
return map4, map1, map2, map3, w, sigma_phi
else:
return map4
def getsky(self, debug=False, celestial=True):
"""Create a realisation of the *unpolarised* sky.
Parameters
----------
debug : boolean, optional
Return intermediate products for debugging. Default is False.
celestial : boolean, optional
Maps are returned in celestial co-ordinates.
Returns
-------
skymap : np.ndarray[freq, pixel]
"""
# Read in data files.
haslam = healpy.ud_grade(self._haslam, self.nside)
syn = FullSkySynchrotron()
lmax = 3 * self.nside - 1
efreq = np.concatenate((np.array([408.0, 1420.0]), self.nu_pixels))
# Construct map of random fluctuations
cla = skysim.clarray(syn.angular_powerspectrum, lmax, efreq, zromb=0)
fg = skysim.mkfullsky(cla, self.nside)
# Find the smoothed fluctuations on each scale
sub408 = healpy.smoothing(fg[0], fwhm=np.radians(1.0), verbose=False)
sub1420 = healpy.smoothing(fg[1], fwhm=np.radians(5.8), verbose=False)
# Make a multifrequency map constrained to look like the smoothed maps
# depending on the spectral_map apply constraints at upper and lower
# frequency (GSM), or just at Haslam map frequency
if self.spectral_map == 'gsm':
fgs = skysim.mkconstrained(cla, [(0, sub408), (1, sub1420)],
self.nside)
else:
fgs = skysim.mkconstrained(cla, [(0, sub408)], self.nside)
# Construct maps of appropriate resolution
sc = healpy.ud_grade(self._sp_ind[self.spectral_map], self.nside)
am = healpy.ud_grade(self._amp_map, self.nside)
# Bump up the variance of the fluctuations according to the variance map
vm = healpy.smoothing(fg[0], sigma=np.radians(0.5), verbose=False)
vm = healpy.smoothing(map_variance(vm, 16)**0.5,
sigma=np.radians(2.0),
verbose=False)
mv = vm.mean()
# Construct the fluctuations map
fgt = (am / mv) * (fg - fgs)
if not debug:
del fg, fgs
# Get the smooth, large scale emission from Haslam+spectralmap
fgsmooth = haslam[np.newaxis, :] * ((efreq / 408.0)[:, np.newaxis]**sc)
# Rescale to ensure output is always positive, do this inplace where
# possible to save memory
def tanh_lin(x):
return np.where(x < 0, np.tanh(x), x)
fgt /= fgsmooth
fgt = tanh_lin(fgt)
fgt += 1
fgt *= fgsmooth
fgt = fgt[2:]
# Co-ordinate transform if required
if celestial:
fgt = hputil.coord_g2c(fgt)
if debug:
return fgt, fg, fgs, fgsmooth, am, mv
return fgt
