Healpix doc

fix code markdown

Update app/routes/docs.notices.healpix.mdx

Co-authored-by: Leo Singer <[email protected]>

resolve doi issue

app/routes/docs.client.samples.md

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   breadcrumb: Sample Code
 ---
 
+import { Highlight } from '~/components/Highlight'
+import { Tab, Tabs } from '~/components/tabs'
+
 # Sample Code
 
 Here is a collection of functions and example code that may be useful
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             continue
     print(message.value())
 ```
+
+## HEALPix Sky Maps
+
+HEALPix stands for <b>H</b>ierarchical, <b>E</b>qual <b>A</b>rea, and iso-<b>L</b>atitude <b>Pix</b>elisation is a scheme for indexing positions on the unit sphere.
+For localization of events, the multi-messenger community uses the standard [HEALPix](https://healpix.sourceforge.io) format with the file extension `.fits.gz`, as well as multi-resolution HEALPix format with the file extension `.multiorder.fits`. The preferred format is the multi-resolution HEALPix format, that is only format explicitly listed in the GCN.
+
+### Multi-Order Sky Maps
+
+GCN strongly encourages the use of multi-order sky maps. These are a more complex form of healpix sky maps that utilize a variable resolution, with higher probability regions having higher resolution and lower probability regions being encoded with a lower resolution. This is signicantly more efficient than single-resolution healpix sky maps with respect to storage footprint and read speed. However, interpreting these multi-order sky maps can be more complex.
+
+### Working with HEALPix Maps
+
+HEALPix data is distrubuted in standard format with the file extension `.fits.gz`. These files are FITS image files and can be utilized with many common FITS tools. Typically, the data is stored using the HEALPix projection.
+
+#### Reading Sky Maps
+
+Sky maps can be parsed using python; to start, import a handful of packages (note: while this documentation covers the use of `astropy-healpix`, there are several packages that can be used for this purpose; a number of alternatives are listed at the bottom of this page)
+
+```python
+import astropy_healpix as ah
+import numpy as np
+
+from astropy import units as u
+from astropy.table import QTable
+```
+
+A given sky map can then be read in as
+
+```python
+skymap = QTable.read('skymap.multiofits)
+```
+
+#### Most Probable Sky Location
+
+The index of the highest probability point can be found by doing the following
+
+```python
+hp_index = np.argmax(skymap['PROBDENSITY'])
+uniq = skymap[hq_index]['UNIQ']
+
+level, ipix = ah.uniq_to_level_ipix(uniq)
+nside = ah.level_to_nside(level)
+
+ra, dec = ah.healpix_to_lonlat(ipix, nside, order='nested')
+```
+
+#### Probability Density at a Known Position
+
+Similarly, one can calculate the probability density at a known position
+
+```python
+ra, dec = 197.450341598 * u.deg, -23.3814675445 * u.deg
+
+level, ipix = ah.uniq_to_level_ipix(skymap['UNIQ'])
+nside = ah.level_to_nside(level)
+
+match_ipix = ad.lonlat_to_healpix(ra, dec, nside, order='nested')
+
+match_index = np.flatnonzero(ipix == match_ipix)[0]
+
+prob_density = skymap[match_index]['PROBDENSITY'].to_value(u.deg**-2)
+```
+
+#### Find the 90% Probability Region
+
+The estimation of a 90% probability region, involves sorting the pixel, calculating the area of each pixel, and then proceeding by the probability of each pixel.
+
+```python
+#Sort the pixels by decending probability density
+skymap.sort('PROBDENSITY', reverse=True)
+
+#Area of each pixel
+level, ipix = ah.uniq_to_level_ipix(skymap['UNIQ'])
+pixel_area = ah.nside_to_pixel_area(ah.level_to_nside(level))
+
+#Pixel area times the probability
+prob = pixel_area * skymap['PROBDENSITY']
+
+#Cummulative sum of probability
+cumprob = np.cumsum(prob)
+
+#Pixels for which cummulative is 0.9
+i = cumprob.searchsorted(0.9)
+
+#Sum of the areas of the pixels up to that one
+area_90 = pixel_area[:i].sum()
+area_90.to_value(u.deg**2)
+
+```
+
+#### Flat Resolution HEALPix Sky maps
+
+Let's say you have a sky map fits file, read in Python with Healpy:
+
+```python
+import healpy as hp
+import numpy as np
+
+# Read both the HEALPix image data and the FITS header
+hpx, header = hp.read_map('*.fits.gz,0', h=True)
+
+# Plotting a Mollweide-projection all-sky map
+np.mollview(hpx)
+```
+
+Most Probable Sky Location
+
+```python
+# Find the highest probability pixel
+ipix_max = np.argmax(hpx)
+
+# Probability density per square degree at that position
+hpx[ipix_max] / hp.nside2pixarea(nside, degrees=True)
+
+# Highest probability pixel on the sky
+theta, phi = hp.pix2ang(nside, ipix_max)
+ra = np.rad2deg(phi)
+dec = np.rad2deg(0.5 * np.pi - theta)
+ra, dec
+```
+
+#### Integrated probability in a Circle
+
+We call hp.query_disc to obtain an array of indices for the pixels inside the circle.
+
+```python
+# First define the Cartesian coordinates of the center of the Circle
+ra = 180.0
+dec = -45.0
+radius = 2.5
+
+# We convert to spherical polar coordinates
+theta = 0.5 * np.pi - np.deg2rad(dec)
+phi = np.deg2rad(ra)
+radius = np.deg2rad(radius)
+
+# Calculate Cartesian coordinates of the center of the Circle
+xyz = hp.ang2vec(theta, phi)
+
+# Obtain an array of indices for the pixels inside the circle
+ipix_disc = hp.query_disc(nside, xyz, radius)
+
+# Sum the probability in all of the matching pixels:
+hpx[ipix_disc].sum()
+
+```
+
+#### Integrated probability in a Polygon
+
+We can use the `hp.query_polygon` function to find the pixels inside a polygon and compute the probability that the source is inside the polygon by adding up the pixel values.
+
+```python
+#  Indices of the pixels within a polygon (defined by the Cartesian coordinates of its vertices)
+
+xyz = [[-0.5, -0.4, -0.5],
+        [-0.4, -0.4, -0.6],
+        [-0.6, -0.3, -0.6],
+        [-0.7 , -0.4, -0.5]]
+ipix_poly = hp.query_polygon(nside, xyz)
+hpx[ipix_poly].sum()
+```
+
+##### Further documentation
+
+Additional information can be found on the [LIGO website](https://emfollow.docs.ligo.org/userguide/tutorial/multiorder_sky maps.html)
+
+[healpy](https://healpy.readthedocs.io/en/latest/): Official python library for handling the pixlated data on sphere
+
+[astropy-healpix](https://pypi.org/project/astropy-healpix/): Integrates HEALPix with Astropy for data manipulation and analysis
+
+[mhealpy](https://mhealpy.readthedocs.io/en/latest/): Object-oriented wrapper of healpy for handling the multi-resolution maps
+
+[MOCpy](https://cds-astro.github.io/mocpy/): Python library allowing easy creation, parsing and manipulation of Multi-Order Coverage maps.
+
+## Papers
+
+(1) Calabretta, M. R., & Roukema, B. F. 2007, Mon. Notices Royal Astron. Soc., 381, 865. [doi: 10.1111/j.1365-2966.2007.12297.x](https://doi.org/10.1111/j.1365-2966.2007.12297.x)
+
+(2) Górski, K.M., Hivon, E., Banday, A.J., et al. 2005, Astrophys. J., 622, 759. [doi: 10.1086/427976](https://doi.org/10.1086/427976)
+
+(3) Górski, K. M., Wandelt, B. D., et al. 1999. [doi: 10.48550/arXiv.astro-ph/9905275](https://doi.org/10.48550/arXiv.astro-ph/9905275)
+
+(4) Fernique, P., Allen, et al. 2015, Astron. Astrophys., 578, A114. [doi: 10.1051/0004-6361/201526075](https://doi.org/10.1051/0004-6361/201526075)
+
+(5) Fernique, P., Boch, T., et al. 2014, IVOA Recommendation. [doi: 10.48550/arXiv.1505.02937](https://doi.org/10.48550/arXiv.1505.02937)
+
+(6) Martinez-Castellanos, I., Singer, L. P., et al. 2022, Astrophys. J., 163, 259. [doi: 10.3847/1538-3881/ac6260](https://doi.org/10.3847/1538-3881/ac6260)
+
+(7) Singer, L. P., & Price, L. R. 2016, Phys. Rev. D, 93, 024013. [doi: 10.1103/PhysRevD.93.024013](https://doi.org/10.1103/PhysRevD.93.024013)