Sherpa is the X-ray satellite Chandra modeling and fitting application. It enables the user to construct complex models from simple definitions and fit those models to data, using a variety of statistics and optimization methods. The issues of constraining the source position and morphology are common in X- and Gamma-ray astronomy. This notebook will show you how to apply Sherpa to CTA data.
Here we will set up Sherpa to fit the counts map and loading the ancillary images for subsequent use. A relevant test statistic for data with Poisson fluctuations is the one proposed by Cash (1979). The simplex (or Nelder-Mead) fitting algorithm is a good compromise between efficiency and robustness. The source fit is best performed in pixel coordinates.
This tutorial has 2 important parts
Since sherpa deals only with 2-dim images, the first part of this tutorial shows how to prepare gammapy maps to make classical images.
%matplotlib inline
import matplotlib.pyplot as plt
from pathlib import Path
import numpy as np
from astropy.io import fits
import astropy.units as u
from astropy.wcs import WCS
from astropy.coordinates import SkyCoord
from gammapy.data import DataStore
from gammapy.irf import EnergyDispersion, make_mean_psf
from gammapy.maps import WcsGeom, MapAxis, Map
from gammapy.cube import MapMaker, PSFKernel
We first generate the required maps using 3 simulated runs on the Galactic center, exactly as in the analysis_3d tutorial.
It is always advisable to make the maps on fine energy bins, and then sum them over to get an image.
# Define which data to use
data_store = DataStore.from_dir("$GAMMAPY_DATA/cta-1dc/index/gps/")
obs_ids = [110380, 111140, 111159]
observations = data_store.get_observations(obs_ids)
energy_axis = MapAxis.from_edges(
np.logspace(-1, 1.0, 10), unit="TeV", name="energy", interp="log"
)
geom = WcsGeom.create(
skydir=(0, 0),
binsz=0.02,
width=(10, 8),
coordsys="GAL",
proj="CAR",
axes=[energy_axis],
)
%%time
maker = MapMaker(geom, offset_max=4.0 * u.deg)
maps = maker.run(observations)
Make a PSF map and weigh it with the exposure at the source position to get a 2D PSF
# mean PSF
src_pos = SkyCoord(0, 0, unit="deg", frame="galactic")
table_psf = make_mean_psf(observations, src_pos)
# PSF kernel used for the model convolution
psf_kernel = PSFKernel.from_table_psf(table_psf, geom, max_radius="0.3 deg")
# get the exposure at the source position
exposure_at_pos = maps["exposure"].get_by_coord(
{
"lon": src_pos.l.value,
"lat": src_pos.b.value,
"energy": energy_axis.center,
}
)
# now compute the 2D PSF
psf2D = psf_kernel.make_image(exposures=exposure_at_pos)
Since sherpa image fitting works only with 2-dim images,
we convert the generated maps to 2D images using run_images()
and save them as fits files. The exposure is weighed with the spectrum before averaging (assumed to be a power law by default).
maps = maker.run_images()
Path("analysis_3d").mkdir(exist_ok=True)
maps["counts"].write("analysis_3d/counts_2D.fits", overwrite=True)
maps["background"].write("analysis_3d/background_2D.fits", overwrite=True)
maps["exposure"].write("analysis_3d/exposure_2D.fits", overwrite=True)
fits.writeto("analysis_3d/psf_2D.fits", psf2D.data, overwrite=True)
We now have the prepared files which sherpa can read. This part of the notebook shows how to do image analysis using sherpa
import sherpa.astro.ui as sh
sh.set_stat("cash")
sh.set_method("simplex")
sh.load_image("analysis_3d/counts_2D.fits")
sh.set_coord("logical")
sh.load_table_model("expo", "analysis_3d/exposure_2D.fits")
sh.load_table_model("bkg", "analysis_3d/background_2D.fits")
sh.load_psf("psf", "analysis_3d/psf_2D.fits")
In principle one might first want to fit the background amplitude. However the background estimation method already yields the correct normalization, so we freeze the background amplitude to unity instead of adjusting it. The (smoothed) residuals from this background model are then computed and shown.
sh.set_full_model(bkg)
bkg.ampl = 1
sh.freeze(bkg)
resid = Map.read("analysis_3d/counts_2D.fits")
resid.data = sh.get_data_image().y - sh.get_model_image().y
resid_smooth = resid.smooth(width=4)
resid_smooth.plot(add_cbar=True);
We then find the position of the maximum in the (smoothed) residuals map, and fit a (symmetrical) Gaussian source with that initial position:
yp, xp = np.unravel_index(
np.nanargmax(resid_smooth.data), resid_smooth.data.shape
)
ampl = resid_smooth.get_by_pix((xp, yp))[0]
# creates g0 as a gauss2d instance
sh.set_full_model(bkg + psf(sh.gauss2d.g0) * expo)
g0.xpos, g0.ypos = xp, yp
sh.freeze(g0.xpos, g0.ypos) # fix the position in the initial fitting step
# fix exposure amplitude so that typical exposure is of order unity
expo.ampl = 1e-9
sh.freeze(expo)
sh.thaw(g0.fwhm, g0.ampl) # in case frozen in a previous iteration
g0.fwhm = 10 # give some reasonable initial values
g0.ampl = ampl
%%time
sh.fit()
Fit all parameters of this Gaussian component, fix them and re-compute the residuals map.
sh.thaw(g0.xpos, g0.ypos)
sh.fit()
sh.freeze(g0)
resid.data = sh.get_data_image().y - sh.get_model_image().y
resid_smooth = resid.smooth(width=3)
resid_smooth.plot();
Instantiate additional Gaussian components, and use them to iteratively fit sources, repeating the steps performed above for component g0. (The residuals map is shown after each additional source included in the model.) This takes some time...
# initialize components with fixed, zero amplitude
for i in range(1, 10):
model = sh.create_model_component("gauss2d", "g" + str(i))
model.ampl = 0
sh.freeze(model)
gs = [g0, g1, g2]
sh.set_full_model(bkg + psf(g0 + g1 + g2) * expo)
%%time
for i in range(1, len(gs)):
yp, xp = np.unravel_index(
np.nanargmax(resid_smooth.data), resid_smooth.data.shape
)
ampl = resid_smooth.get_by_pix((xp, yp))[0]
gs[i].xpos, gs[i].ypos = xp, yp
gs[i].fwhm = 10
gs[i].ampl = ampl
sh.thaw(gs[i].fwhm)
sh.thaw(gs[i].ampl)
sh.fit()
sh.thaw(gs[i].xpos)
sh.thaw(gs[i].ypos)
sh.fit()
sh.freeze(gs[i])
resid.data = sh.get_data_image().y - sh.get_model_image().y
resid_smooth = resid.smooth(width=6)
resid_smooth.plot(add_cbar=True);
When adding a new source, one needs to check the significance of this new source. A frequently used method is the Test Statistics (TS). This is done by comparing the change of statistics when the source is included compared to the null hypothesis (no source ; in practice here we fix the amplitude to zero).
$TS = Cstat(source) - Cstat(no source)$
The criterion for a significant source detection is typically that it should improve the test statistic by at least 25 or 30. We have added only 3 sources to save time, but you should keep doing this till del(stat) is less than the required number.
from astropy.stats import gaussian_fwhm_to_sigma
from astropy.table import Table
rows = []
for g in gs:
ampl = g.ampl.val
g.ampl = 0
stati = sh.get_stat_info()[0].statval
g.ampl = ampl
statf = sh.get_stat_info()[0].statval
delstat = stati - statf
geom = resid.geom
# sherpa uses 1 based indexing
coord = geom.pix_to_coord((g.xpos.val - 1, g.ypos.val - 1))
pix_scale = geom.pixel_scales.mean().deg
sigma = g.fwhm.val * pix_scale * gaussian_fwhm_to_sigma
rows.append(
dict(delstat=delstat, glon=coord[0], glat=coord[1], sigma=sigma)
)
table = Table(rows=rows, names=rows[0])
for name in table.colnames:
table[name].format = ".5g"
table
These are good resources to learn more about Sherpa:
You could read over the examples there, and try to apply a similar analysis to this dataset here to practice.
If you want a deeper understanding of how Sherpa works, then these proceedings are good introductions: