This is a fixed-text formatted version of a Jupyter notebook.

Fitting 2D images with Sherpa

Introduction

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 1. Generating the Maps 2. The actual fitting with sherpa.

Since sherpa deals only with 2-dim images, the first part of this tutorial shows how to prepare gammapy maps to make classical images.

Necessary imports

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
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.extern.pathlib import Path
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

Generate the required Maps

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.

In [2]:
# Define which data to use
data_store = DataStore.from_dir("$GAMMAPY_DATA/cta-1dc/index/gps/")
obs_ids = [110380, 111140, 111159]
obs_list = data_store.obs_list(obs_ids)
In [3]:
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],
)
In [4]:
%%time
maker = MapMaker(geom, offset_max=4.0 * u.deg)
maps = maker.run(obs_list)
CPU times: user 48.3 s, sys: 4.56 s, total: 52.9 s
Wall time: 14.1 s

Making a PSF Map

Make a PSF map and weigh it with the exposure at the source position to get a 2D PSF

In [5]:
# mean PSF
src_pos = SkyCoord(0, 0, unit="deg", frame="galactic")
table_psf = make_mean_psf(obs_list, 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)

Make 2D images from 3D ones

Since sherpa image fitting works only with 2-dim images, we convert the generated maps to 2D images using make_images() and save them as fits files. The exposure is weighed with the spectrum before averaging (assumed to be a power law by default).

In [6]:
maps = maker.make_images()
In [7]:
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)

Read the maps and store them in a sherpa model

We now have the prepared files which sherpa can read. This part of the notebook shows how to do image analysis using sherpa

In [8]:
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")
WARNING: imaging routines will not be available,
failed to import sherpa.image.ds9_backend due to
'RuntimeErr: DS9Win unusable: Could not find ds9 on your PATH'
WARNING: failed to import sherpa.astro.xspec; XSPEC models will not be available

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.

In [9]:
sh.set_full_model(bkg)
bkg.ampl = 1
sh.freeze(bkg)
In [10]:
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);
../_images/notebooks_image_fitting_with_sherpa_17_0.png

Find and fit the brightest source

We then find the position of the maximum in the (smoothed) residuals map, and fit a (symmetrical) Gaussian source with that initial position:

In [11]:
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
In [12]:
%%time
sh.fit()
Dataset               = 1
Method                = neldermead
Statistic             = cash
Initial fit statistic = 290025
Final fit statistic   = 290021 at function evaluation 206
Data points           = 200000
Degrees of freedom    = 199998
Change in statistic   = 3.82422
   g0.fwhm        11.7532
   g0.ampl        3.34936
CPU times: user 10.3 s, sys: 132 ms, total: 10.5 s
Wall time: 10.4 s

Fit all parameters of this Gaussian component, fix them and re-compute the residuals map.

In [13]:
sh.thaw(g0.xpos, g0.ypos)
sh.fit()
sh.freeze(g0)
Dataset               = 1
Method                = neldermead
Statistic             = cash
Initial fit statistic = 290021
Final fit statistic   = 289981 at function evaluation 436
Data points           = 200000
Degrees of freedom    = 199996
Change in statistic   = 40.5419
   g0.fwhm        6.24068
   g0.xpos        253.153
   g0.ypos        197.932
   g0.ampl        8.66022
In [14]:
resid.data = sh.get_data_image().y - sh.get_model_image().y
resid_smooth = resid.smooth(width=3)
resid_smooth.plot();
../_images/notebooks_image_fitting_with_sherpa_23_0.png

Iteratively find and fit additional sources

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…

In [15]:
# 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)
In [16]:
%%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)
Dataset               = 1
Method                = neldermead
Statistic             = cash
Initial fit statistic = 289472
Final fit statistic   = 288943 at function evaluation 217
Data points           = 200000
Degrees of freedom    = 199998
Change in statistic   = 528.964
   g1.fwhm        41.7615
   g1.ampl        0.745676
Dataset               = 1
Method                = neldermead
Statistic             = cash
Initial fit statistic = 288943
Final fit statistic   = 288840 at function evaluation 345
Data points           = 200000
Degrees of freedom    = 199996
Change in statistic   = 103.759
   g1.fwhm        41.5589
   g1.xpos        200.032
   g1.ypos        197.884
   g1.ampl        0.798509
Dataset               = 1
Method                = neldermead
Statistic             = cash
Initial fit statistic = 288679
Final fit statistic   = 288470 at function evaluation 223
Data points           = 200000
Degrees of freedom    = 199998
Change in statistic   = 209.467
   g2.fwhm        44.3725
   g2.ampl        0.411113
Dataset               = 1
Method                = neldermead
Statistic             = cash
Initial fit statistic = 288470
Final fit statistic   = 288371 at function evaluation 329
Data points           = 200000
Degrees of freedom    = 199996
Change in statistic   = 98.6953
   g2.fwhm        42.9457
   g2.xpos        292.004
   g2.ypos        189.037
   g2.ampl        0.481605
CPU times: user 57.2 s, sys: 413 ms, total: 57.6 s
Wall time: 57.5 s
In [17]:
resid_smooth.plot(add_cbar=True);
../_images/notebooks_image_fitting_with_sherpa_27_0.png

Generating output table and Test Statistics estimation

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.

In [18]:
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
    coord = geom.pix_to_coord((g.xpos.val, g.ypos.val))
    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
Out[18]:
Table length=3
delstatglonglatsigma
float64float64float64float64
1034.2359.93-0.0313520.053003
1138.30.98937-0.0323180.35297
468.76359.15-0.209260.36475

Exercises

  1. Keep adding sources till there are no more significat ones in the field. How many Gaussians do you need?
  2. Use other morphologies for the sources (eg: disk, shell) rather than only Gaussian.
  3. Compare the TS between different models

More about sherpa

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: