Spectral analysis of extended sources#

Perform a spectral analysis of an extended source.

Prerequisites#

  • Understanding of spectral analysis techniques in classical Cherenkov astronomy.

  • Understanding the basic data reduction and modeling/fitting processes with the gammapy library API as shown in the tutorial Low level API

Context#

Many VHE sources in the Galaxy are extended. Studying them with a 1D spectral analysis is more complex than studying point sources. One often has to use complex (i.e. non-circular) regions and more importantly, one has to take into account the fact that the instrument response is non-uniform over the selected region. A typical example is given by the supernova remnant RX J1713-3935 which is nearly 1 degree in diameter. See the following article.

Objective: Measure the spectrum of RX J1713-3945 in a 1 degree region fully enclosing it.

Proposed approach#

We have seen in the general presentation of the spectrum extraction for point sources (see Spectral analysis tutorial) that Gammapy uses specific datasets makers to first produce reduced spectral data and then to extract OFF measurements with reflected background techniques: the SpectrumDatasetMaker and the ReflectedRegionsBackgroundMaker. However, if the flag use_region_center is not set to False, the former simply computes the reduced IRFs at the center of the ON region (assumed to be circular).

This is no longer valid for extended sources. To be able to compute average responses in the ON region, we can set use_region_center=False with the SpectrumDatasetMaker, in which case the values of the IRFs are averaged over the entire region.

In summary, we have to:

  • Define an ON region (a SkyRegion) fully enclosing the source we want to study.

  • Define a RegionGeom with the ON region and the required energy range (in particular, beware of the true energy range).

  • Create the necessary makers :

  • Perform the data reduction loop. And for every observation:

    • Produce a spectrum dataset

    • Extract the OFF data to produce a SpectrumDatasetOnOff and compute a safe range for it.

    • Stack or store the resulting spectrum dataset.

  • Finally proceed with model fitting on the dataset as usual.

Here, we will use the RX J1713-3945 observations from the H.E.S.S. first public test data release. The tutorial is implemented with the intermediate level API.

Setup#

As usual, we’ll start with some general imports…

import astropy.units as u
from astropy.coordinates import Angle, SkyCoord
from regions import CircleSkyRegion

# %matplotlib inline
import matplotlib.pyplot as plt
from IPython.display import display
from gammapy.data import DataStore
from gammapy.datasets import Datasets, SpectrumDataset
from gammapy.makers import (
    ReflectedRegionsBackgroundMaker,
    SafeMaskMaker,
    SpectrumDatasetMaker,
)
from gammapy.maps import MapAxis, RegionGeom
from gammapy.modeling import Fit
from gammapy.modeling.models import PowerLawSpectralModel, SkyModel

Check setup#

from gammapy.utils.check import check_tutorials_setup

check_tutorials_setup()
System:

        python_executable      : /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/bin/python
        python_version         : 3.9.18
        machine                : x86_64
        system                 : Linux


Gammapy package:

        version                : 1.3.dev60+ge5aecb334
        path                   : /home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/gammapy


Other packages:

        numpy                  : 1.26.4
        scipy                  : 1.12.0
        astropy                : 5.2.2
        regions                : 0.8
        click                  : 8.1.7
        yaml                   : 6.0.1
        IPython                : 8.18.1
        jupyterlab             : not installed
        matplotlib             : 3.8.3
        pandas                 : not installed
        healpy                 : 1.16.6
        iminuit                : 2.25.2
        sherpa                 : 4.16.0
        naima                  : 0.10.0
        emcee                  : 3.1.4
        corner                 : 2.2.2
        ray                    : 2.9.3


Gammapy environment variables:

        GAMMAPY_DATA           : /home/runner/work/gammapy-docs/gammapy-docs/gammapy-datasets/dev

Select the data#

We first set the datastore and retrieve a few observations from our source.

datastore = DataStore.from_dir("$GAMMAPY_DATA/hess-dl3-dr1/")
obs_ids = [20326, 20327, 20349, 20350, 20396, 20397]
# In case you want to use all RX J1713 data in the HESS DR1
# other_ids=[20421, 20422, 20517, 20518, 20519, 20521, 20898, 20899, 20900]

observations = datastore.get_observations(obs_ids)

Prepare the datasets creation#

Select the ON region#

Here we take a simple 1 degree circular region because it fits well with the morphology of RX J1713-3945. More complex regions could be used e.g. EllipseSkyRegion or RectangleSkyRegion.

target_position = SkyCoord(347.3, -0.5, unit="deg", frame="galactic")
radius = Angle("0.5 deg")
on_region = CircleSkyRegion(target_position, radius)

Define the geometries#

This part is especially important. - We have to define first energy axes. They define the axes of the resulting SpectrumDatasetOnOff. In particular, we have to be careful to the true energy axis: it has to cover a larger range than the reconstructed energy one. - Then we define the region geometry itself from the on region.

# The binning of the final spectrum is defined here.
energy_axis = MapAxis.from_energy_bounds(0.1, 40.0, 10, unit="TeV")

# Reduced IRFs are defined in true energy (i.e. not measured energy).
energy_axis_true = MapAxis.from_energy_bounds(
    0.05, 100, 30, unit="TeV", name="energy_true"
)

geom = RegionGeom(on_region, axes=[energy_axis])

Create the makers#

First we instantiate the target SpectrumDataset.

Now we create its associated maker. Here we need to produce, counts, exposure and edisp (energy dispersion) entries. PSF and IRF background are not needed, therefore we don’t compute them.

IMPORTANT: Note that use_region_center is set to False. This is necessary so that the SpectrumDatasetMaker considers the whole region in the IRF computation and not only the center.

maker = SpectrumDatasetMaker(
    selection=["counts", "exposure", "edisp"], use_region_center=False
)

Now we create the OFF background maker for the spectra. If we have an exclusion region, we have to pass it here. We also define the safe range maker.

bkg_maker = ReflectedRegionsBackgroundMaker()
safe_mask_maker = SafeMaskMaker(methods=["aeff-max"], aeff_percent=10)

Perform the data reduction loop.#

We can now run over selected observations. For each of them, we: - create the SpectrumDataset - Compute the OFF via the reflected background method and create a SpectrumDatasetOnOff object - Run the safe mask maker on it - Add the SpectrumDatasetOnOff to the list.

datasets = Datasets()

for obs in observations:
    # A SpectrumDataset is filled in this geometry
    dataset = maker.run(dataset_empty.copy(name=f"obs-{obs.obs_id}"), obs)

    # Define safe mask
    dataset = safe_mask_maker.run(dataset, obs)

    # Compute OFF
    dataset = bkg_maker.run(dataset, obs)

    # Append dataset to the list
    datasets.append(dataset)

display(datasets.meta_table)
   NAME           TYPE         ...        GEOLAT            ALTITUDE
                               ...
--------- -------------------- ... ------------------- ------------------
obs-20326 SpectrumDatasetOnOff ... -23.271777777777796 1834.9999999997835
obs-20327 SpectrumDatasetOnOff ... -23.271777777777796 1834.9999999997835
obs-20349 SpectrumDatasetOnOff ... -23.271777777777796 1834.9999999997835
obs-20350 SpectrumDatasetOnOff ... -23.271777777777796 1834.9999999997835
obs-20396 SpectrumDatasetOnOff ... -23.271777777777796 1834.9999999997835
obs-20397 SpectrumDatasetOnOff ... -23.271777777777796 1834.9999999997835

Explore the results#

We can peek at the content of the spectrum datasets

datasets[0].peek()
plt.show()
Counts, Exposure, Energy Dispersion

Cumulative excess and significance#

Finally, we can look at cumulative significance and number of excesses. This is done with the info_table method of Datasets.

info_table = datasets.info_table(cumulative=True)

display(info_table)
  name  counts      excess     ...   acceptance_off         alpha
                               ...
------- ------ --------------- ... ----------------- -------------------
stacked   1216           170.5 ...              18.0                 0.5
stacked   2339           270.5 ...              18.0                 0.5
stacked   3521           480.5 ...              18.0                 0.5
stacked   4684           653.0 ...              18.0                 0.5
stacked   5895 874.66650390625 ... 19.77358627319336 0.45515263080596924
stacked   6985 993.16650390625 ... 19.48602294921875  0.4618695378303528

And make the corresponding plots

fig, (ax_excess, ax_sqrt_ts) = plt.subplots(figsize=(10, 4), ncols=2, nrows=1)
ax_excess.plot(
    info_table["livetime"].to("h"),
    info_table["excess"],
    marker="o",
    ls="none",
)
ax_excess.set_title("Excess")
ax_excess.set_xlabel("Livetime [h]")
ax_excess.set_ylabel("Excess events")

ax_sqrt_ts.plot(
    info_table["livetime"].to("h"),
    info_table["sqrt_ts"],
    marker="o",
    ls="none",
)

ax_sqrt_ts.set_title("Sqrt(TS)")
ax_sqrt_ts.set_xlabel("Livetime [h]")
ax_sqrt_ts.set_ylabel("Sqrt(TS)")
plt.show()
Excess, Sqrt(TS)

Perform spectral model fitting#

Here we perform a joint fit.

We first create the model, here a simple powerlaw, and assign it to every dataset in the Datasets.

spectral_model = PowerLawSpectralModel(
    index=2, amplitude=2e-11 * u.Unit("cm-2 s-1 TeV-1"), reference=1 * u.TeV
)
model = SkyModel(spectral_model=spectral_model, name="RXJ 1713")

datasets.models = [model]

Now we can run the fit

fit_joint = Fit()
result_joint = fit_joint.run(datasets=datasets)
print(result_joint)
OptimizeResult

        backend    : minuit
        method     : migrad
        success    : True
        message    : Optimization terminated successfully.
        nfev       : 38
        total stat : 52.79

CovarianceResult

        backend    : minuit
        method     : hesse
        success    : True
        message    : Hesse terminated successfully.

Explore the fit results#

First the fitted parameters values and their errors.

 model   type    name     value         unit      ... frozen is_norm link prior
-------- ---- --------- ---------- -------------- ... ------ ------- ---- -----
RXJ 1713          index 2.1102e+00                ...  False   False
RXJ 1713      amplitude 1.3576e-11 cm-2 s-1 TeV-1 ...  False    True
RXJ 1713      reference 1.0000e+00            TeV ...   True   False

Then plot the fit result to compare measured and expected counts. Rather than plotting them for each individual dataset, we stack all datasets and plot the fit result on the result.

# First stack them all
reduced = datasets.stack_reduce()

# Assign the fitted model
reduced.models = model

# Plot the result
ax_spectrum, ax_residuals = reduced.plot_fit()
reduced.plot_masks(ax=ax_spectrum)
plt.show()
extended source spectral analysis

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