Working with Sky Maps¶
Let’s take a look at what is inside one of the LIGO/Virgo probability sky maps. They are FITS image files and can be manipulated and viewed with many commonplace FITS tools. However, they are a little unusual in two regards. First, since they are all-sky images, they are stored in the HEALPix 1 projection, a format that is used for Planck all-sky CMB maps and by Aladin for hierarchical, progressively refined, all-sky survey images (HiPS). Second, the value stored at each pixel is the probability that the gravitational-wave source is within that pixel.
Let’s download an example FITS file with curl:
$ curl -O https://emfollow.docs.ligo.org/userguide/_static/bayestar.fits.gz,0
$ fitsheader bayestar.fits.gz,0 # HDU 0 in bayestar.fits.gz,0: SIMPLE = T / conforms to FITS standard BITPIX = 8 / array data type NAXIS = 0 / number of array dimensions EXTEND = T # HDU 1 in bayestar.fits.gz,0: XTENSION= 'BINTABLE' / binary table extension BITPIX = 8 / array data type NAXIS = 2 / number of array dimensions NAXIS1 = 32 / length of dimension 1 NAXIS2 = 50331648 / length of dimension 2 PCOUNT = 0 / number of group parameters GCOUNT = 1 / number of groups TFIELDS = 4 / number of table fields TTYPE1 = 'PROB ' TFORM1 = 'D ' TUNIT1 = 'pix-1 ' TTYPE2 = 'DISTMU ' TFORM2 = 'D ' TUNIT2 = 'Mpc ' TTYPE3 = 'DISTSIGMA' TFORM3 = 'D ' TUNIT3 = 'Mpc ' TTYPE4 = 'DISTNORM' TFORM4 = 'D ' TUNIT4 = 'Mpc-2 ' PIXTYPE = 'HEALPIX ' / HEALPIX pixelisation ORDERING= 'NESTED ' / Pixel ordering scheme: RING, NESTED, or NUNIQ COORDSYS= 'C ' / Ecliptic, Galactic or Celestial (equatorial) NSIDE = 2048 / Resolution parameter of HEALPIX INDXSCHM= 'IMPLICIT' / Indexing: IMPLICIT or EXPLICIT OBJECT = 'MS181101ab' / Unique identifier for this event REFERENC= 'https://example.org/superevents/MS181101ab/view/' / URL of this event INSTRUME= 'H1,L1,V1' / Instruments that triggered this event DATE-OBS= '2018-11-01T22:22:46.654437' / UTC date of the observation MJD-OBS = 58423.93248442613 / modified Julian date of the observation DATE = '2018-11-01T22:34:49.000000' / UTC date of file creation CREATOR = 'BAYESTAR' / Program that created this file ORIGIN = 'LIGO/Virgo' / Organization responsible for this FITS file RUNTIME = 3.24746292643249 / Runtime in seconds of the CREATOR program DISTMEAN= 39.76999609489013 / Posterior mean distance (Mpc) DISTSTD = 8.308435058808886 / Posterior standard deviation of distance (Mpc) LOGBCI = 13.64819688928804 / Log Bayes factor: coherent vs. incoherent LOGBSN = 261.0250944470225 / Log Bayes factor: signal vs. noise VCSVERS = 'ligo.skymap 0.1.8' / Software version VCSREV = 'becb07110491d799b753858845b5c24c82705404' / Software revision (Git) DATE-BLD= '2019-07-25T22:36:58' / Software build date HISTORY HISTORY Generated by calling the following Python function: HISTORY ligo.skymap.bayestar.localize(event=..., waveform='o2-uberbank', f_low=3 HISTORY 0, min_inclination=0.0, max_inclination=1.5707963267948966, min_distance HISTORY =None, max_distance=None, prior_distance_power=2, cosmology=False, mcmc= HISTORY False, chain_dump=None, enable_snr_series=True, f_high_truncate=0.95) HISTORY HISTORY This was the command line that started the program: HISTORY bayestar-localize-lvalert -N G298107 -o bayestar.multiorder.fits
There are several useful pieces of information here:
COORDSYS=C, telling you that the HEALPix projection is in the Celestial (equatorial, ICRS) frame, as all LIGO/Virgo probability sky maps will be.
OBJECT, the unique LIGO/Virgo identifier for the event.
REFERENC, a link to the candidate page in GraceDB.
INSTRUME, a list of gravitational-wave sites that triggered on the event: H1 for LIGO Hanford, L1 for LIGO Livingston, and V1 for Virgo.
DATE-OBS, the UTC time of the event. In the case of a compact binary coalescence candidate, this is the time that the signal from the merger passed through the geocenter.
MJD-OBS, same as DATE-OBS, but given as a modified Julian day.
You can view the sky map in many common FITS image viewers such as Aladin:
Now, let’s go through some examples of manipulating HEALPix sky maps programmatically. The HEALPix project provides official libraries for many languages, including C, C++, Fortran, IDL, and Java. However, since this is a Python tutorial, we are going to demonstrate how to manipulate HEALPix maps with the official Python library, Healpy.
Reading Sky Maps¶
First, if you have not already downloaded an example sky map, you can do so now by having Python call curl on the command line:
$ curl -O https://emfollow.docs.ligo.org/userguide/_static/bayestar.fits.gz,0
Next, we need to read in the file in Python with Healpy:
>>> import healpy as hp >>> import numpy as np >>> hpx = hp.read_map('bayestar.fits.gz,0') NSIDE = 2048 ORDERING = NESTED in fits file INDXSCHM = IMPLICIT Ordering converted to RING
You can suppress printing informational messages while loading the file by
passing the keyword argument
verbose=False. You can read both the HEALPix
image data and the FITS header by passing the
h=True keyword argument:
>>> hpx, header = hp.read_map('bayestar.fits.gz,0', h=True, verbose=False)
Manipulating HEALPix Coordinates¶
The image data is a 1D array of values:
>>> hpx array([2.70726059e-66, 1.27374324e-66, 2.62611513e-67, ..., 2.04700874e-40, 1.05781210e-35, 4.44174764e-31])
Each entry in the array represents the probability contained within a quadrilateral pixel whose position on the sky is uniquely specified by the index in the array and the array’s length. Because HEALPix pixels are equal area, we can find the number of pixels per square degree just from the length of the HEALPix array:
>>> npix = len(hpx) >>> sky_area = 4 * 180**2 / np.pi >>> sky_area / npix 0.0008196227004015301
>>> nside = hp.npix2nside(npix) >>> nside 2048
Let’s look up the right ascension and declination of pixel number 123. We’ll
hp.pix2ang to get the spherical polar
coordinates \((\theta, \phi)\) in radians, and then use
np.rad2deg to convert these to right ascension and declination in degrees.
>>> ipix = 123 >>> theta, phi = hp.pix2ang(nside, ipix) >>> ra = np.rad2deg(phi) >>> dec = np.rad2deg(0.5 * np.pi - theta) >>> ra, dec (129.375, 89.81725848475484)
Let’s find which pixel contains the point RA=194.95, Dec=27.98.
>>> ra = 194.95 >>> dec = 27.98 >>> theta = 0.5 * np.pi - np.deg2rad(dec) >>> phi = np.deg2rad(ra) >>> ipix = hp.ang2pix(nside, theta, phi) >>> ipix 13361492
Test if a Sky Location is in the 90% Credible Region¶
You can easily test if a given sky position is in the 90% credible region. Let’s continue using the sky position from the previous example, for which we have already determined the pixel index.
Use the following simple algorithm to construct a map that gives the credible level of each pixel:
Sort the pixels by descending probability density.
Cumulatively sum the pixels’ probability.
Return the pixels to their original order.
In Python, you can use this simple recipe:
>>> i = np.flipud(np.argsort(hpx)) >>> sorted_credible_levels = np.cumsum(hpx[i]) >>> credible_levels = np.empty_like(sorted_credible_levels) >>> credible_levels[i] = sorted_credible_levels >>> credible_levels array([1., 1., 1., ..., 1., 1., 1.])
Observe that the values in the resulting credible level map vary inversely with probability density: the most probable pixel is assigned to the credible level 0.0, and the least likely pixel is assigned the credible level 1.0.
>>> from ligo.skymap.postprocess import find_greedy_credible_levels >>> credible_levels = find_greedy_credible_levels(hpx) >>> credible_levels array([1., 1., 1., ..., 1., 1., 1.])
To check if the pixel that we identified in the previous section is within the 90% credible level, simply test if the value of the credible level map is less than or equal to 0.9 at that pixel:
>>> credible_levels[ipix] 0.9999999999947833 >>> credible_levels[ipix] <= 0.9 False
The credible level map has a value greater than 0.9 at that sky location, therefore the sky location is outside the 90% credible region.
Find the Area of the 90% Credible Region¶
Since we just found the credible level map, it’s easy to compute the 90% credible area by counting the number of pixels inside the 90% credible region and multiplying by the area per pixel.
In the Python expression below, note that
(credible_levels <= 0.9)
evaluates to a binary array; when it is summed over, true values are treated as
1 and false values are treated as 0.
>>> np.sum(credible_levels <= 0.9) * hp.nside2pixarea(nside, degrees=True) 30.979279207076633
Most Probable Sky Location¶
Let’s find the highest probability pixel.
>>> ipix_max = np.argmax(hpx)
What is the probability density per square degree at that position?
>>> hpx[ipix_max] / hp.nside2pixarea(nside, degrees=True) 0.0782516470191411
Where is the highest probability pixel on the sky? Use
>>> theta, phi = hp.pix2ang(nside, ipix_max) >>> ra = np.rad2deg(phi) >>> dec = np.rad2deg(0.5 * np.pi - theta) >>> ra, dec (194.30419921875, -17.856895095545475)
Integrated Probability in a Circle¶
How do we find the probability that the source is contained within a circle on
the sky? First we find the pixels that are contained within the circle using
hp.query_disc. Note that this function takes as its
arguments the Cartesian coordinates of the center of the circle, and its radius
in radians. Then, we sum the values of the HEALPix image array contained at
First, we define the RA, Dec, and radius of circle in degrees:
>>> ra = 213.22 >>> dec = -37.45 >>> radius = 3.1
Then we convert to spherical polar coordinates and radius of circle in radians:
>>> theta = 0.5 * np.pi - np.deg2rad(dec) >>> phi = np.deg2rad(ra) >>> radius = np.deg2rad(radius)
Then we calculate the Cartesian coordinates of the center of circle:
>>> xyz = hp.ang2vec(theta, phi)
hp.query_disc, which returns an array of
the indices of the pixels that are inside the circle:
>>> ipix_disc = hp.query_disc(nside, xyz, radius)
Finally, we sum the probability in all of the matching pixels:
>>> hpx[ipix_disc].sum() 3.655661941088471e-10
Integrated Probability in a Polygon¶
Similarly, we can use the
function to look up the indices of the pixels within a polygon (defined by the
Cartesian coordinates of its vertices), and then compute the probability that
the source is inside that polygon by summing the values of the pixels.
>>> xyz = [[-0.69601758, -0.41315628, -0.58724902], ... [-0.68590811, -0.40679797, -0.60336181], ... [-0.69106913, -0.39820114, -0.60320752], ... [-0.7011786 , -0.40455945, -0.58709473]] >>> ipix_poly = hp.query_polygon(nside, xyz) >>> hpx[ipix_poly].sum() 1.128695302404769e-12
These are all of the HEALPix functions from Healpy that we will need for the remainder of the this tutorial.
Other useful Healpy functions include
hp.ud_grade for upsampling or downsampling a sky map and
hp.get_interp_val for performing
bilinear interpolation between pixels. See the Healpy tutorial for other useful operations.