1D spectrum simulation#

Simulate a number of spectral on-off observations of a source with a power-law spectral model using the CTA 1DC response and fit them with the assumed spectral model.

Prerequisites#

Context#

To simulate a specific observation, it is not always necessary to simulate the full photon list. For many uses cases, simulating directly a reduced binned dataset is enough: the IRFs reduced in the correct geometry are combined with a source model to predict an actual number of counts per bin. The latter is then used to simulate a reduced dataset using Poisson probability distribution.

This can be done to check the feasibility of a measurement, to test whether fitted parameters really provide a good fit to the data etc.

Here we will see how to perform a 1D spectral simulation of a CTA observation, in particular, we will generate OFF observations following the template background stored in the CTA IRFs.

Objective: simulate a number of spectral ON-OFF observations of a source with a power-law spectral model with CTA using the CTA 1DC response, fit them with the assumed spectral model and check that the distribution of fitted parameters is consistent with the input values.

Proposed approach#

We will use the following classes and functions:

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

# %matplotlib inline
import matplotlib.pyplot as plt

Setup#

from IPython.display import display
from gammapy.data import FixedPointingInfo, Observation, observatory_locations
from gammapy.datasets import Datasets, SpectrumDataset, SpectrumDatasetOnOff
from gammapy.irf import load_irf_dict_from_file
from gammapy.makers import 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

Simulation of a single spectrum#

To do a simulation, we need to define the observational parameters like the livetime, the offset, the assumed integration radius, the energy range to perform the simulation for and the choice of spectral model. We then use an in-memory observation which is convolved with the IRFs to get the predicted number of counts. This is Poisson fluctuated using the fake() to get the simulated counts for each observation.

# Define simulation parameters parameters
livetime = 1 * u.h

pointing_position = SkyCoord(0, 0, unit="deg", frame="galactic")
# We want to simulate an observation pointing at a fixed position in the sky.
# For this, we use the `FixedPointingInfo` class
pointing = FixedPointingInfo(
    fixed_icrs=pointing_position.icrs,
)
offset = 0.5 * u.deg

# Reconstructed and true energy axis
energy_axis = MapAxis.from_edges(
    np.logspace(-0.5, 1.0, 10), unit="TeV", name="energy", interp="log"
)
energy_axis_true = MapAxis.from_edges(
    np.logspace(-1.2, 2.0, 31), unit="TeV", name="energy_true", interp="log"
)

on_region_radius = Angle("0.11 deg")

center = pointing_position.directional_offset_by(
    position_angle=0 * u.deg, separation=offset
)
on_region = CircleSkyRegion(center=center, radius=on_region_radius)

# Define spectral model - a simple Power Law in this case
model_simu = PowerLawSpectralModel(
    index=3.0,
    amplitude=2.5e-12 * u.Unit("cm-2 s-1 TeV-1"),
    reference=1 * u.TeV,
)
print(model_simu)
# we set the sky model used in the dataset
model = SkyModel(spectral_model=model_simu, name="source")
PowerLawSpectralModel

type    name     value         unit        error   ... frozen is_norm link prior
---- --------- ---------- -------------- --------- ... ------ ------- ---- -----
         index 3.0000e+00                0.000e+00 ...  False   False
     amplitude 2.5000e-12 cm-2 s-1 TeV-1 0.000e+00 ...  False    True
     reference 1.0000e+00            TeV 0.000e+00 ...   True   False

Load the IRFs In this simulation, we use the CTA-1DC IRFs shipped with Gammapy.

irfs = load_irf_dict_from_file(
    "$GAMMAPY_DATA/cta-1dc/caldb/data/cta/1dc/bcf/South_z20_50h/irf_file.fits"
)

location = observatory_locations["cta_south"]
obs = Observation.create(
    pointing=pointing,
    livetime=livetime,
    irfs=irfs,
    location=location,
)
print(obs)
/home/runner/work/gammapy-docs/gammapy-docs/gammapy/.tox/build_docs/lib/python3.9/site-packages/astropy/units/core.py:2097: UnitsWarning: '1/s/MeV/sr' did not parse as fits unit: Numeric factor not supported by FITS If this is meant to be a custom unit, define it with 'u.def_unit'. To have it recognized inside a file reader or other code, enable it with 'u.add_enabled_units'. For details, see https://docs.astropy.org/en/latest/units/combining_and_defining.html
  warnings.warn(msg, UnitsWarning)
Observation

        obs id            : 0
        tstart            : 51544.00
        tstop             : 51544.04
        duration          : 3600.00 s
        pointing (icrs)   : 266.4 deg, -28.9 deg

        deadtime fraction : 0.0%

Simulate a spectra

# Make the SpectrumDataset
geom = RegionGeom.create(region=on_region, axes=[energy_axis])

dataset_empty = SpectrumDataset.create(
    geom=geom, energy_axis_true=energy_axis_true, name="obs-0"
)
maker = SpectrumDatasetMaker(selection=["exposure", "edisp", "background"])

dataset = maker.run(dataset_empty, obs)

# Set the model on the dataset, and fake
dataset.models = model
dataset.fake(random_state=42)
print(dataset)
SpectrumDataset
---------------

  Name                            : obs-0

  Total counts                    : 298
  Total background counts         : 22.29
  Total excess counts             : 275.71

  Predicted counts                : 303.66
  Predicted background counts     : 22.29
  Predicted excess counts         : 281.37

  Exposure min                    : 2.53e+08 m2 s
  Exposure max                    : 1.77e+10 m2 s

  Number of total bins            : 9
  Number of fit bins              : 9

  Fit statistic type              : cash
  Fit statistic value (-2 log(L)) : -1811.58

  Number of models                : 1
  Number of parameters            : 3
  Number of free parameters       : 2

  Component 0: SkyModel

    Name                      : source
    Datasets names            : None
    Spectral model type       : PowerLawSpectralModel
    Spatial  model type       :
    Temporal model type       :
    Parameters:
      index                         :      3.000   +/-    0.00
      amplitude                     :   2.50e-12   +/- 0.0e+00 1 / (cm2 s TeV)
      reference             (frozen):      1.000       TeV

You can see that background counts are now simulated

On-Off analysis#

To do an on off spectral analysis, which is the usual science case, the standard would be to use SpectrumDatasetOnOff, which uses the acceptance to fake off-counts. Please also refer to the Dataset simulations section in the Spectral analysis with energy-dependent directional cuts tutorial, dealing with simulations based on observations of real off counts.

SpectrumDatasetOnOff
--------------------

  Name                            : WXPFGAfD

  Total counts                    : 286
  Total background counts         : 21.80
  Total excess counts             : 264.20

  Predicted counts                : 302.85
  Predicted background counts     : 21.48
  Predicted excess counts         : 281.37

  Exposure min                    : 2.53e+08 m2 s
  Exposure max                    : 1.77e+10 m2 s

  Number of total bins            : 9
  Number of fit bins              : 9

  Fit statistic type              : wstat
  Fit statistic value (-2 log(L)) : 9.46

  Number of models                : 1
  Number of parameters            : 3
  Number of free parameters       : 2

  Component 0: SkyModel

    Name                      : source
    Datasets names            : None
    Spectral model type       : PowerLawSpectralModel
    Spatial  model type       :
    Temporal model type       :
    Parameters:
      index                         :      3.000   +/-    0.00
      amplitude                     :   2.50e-12   +/- 0.0e+00 1 / (cm2 s TeV)
      reference             (frozen):      1.000       TeV

    Total counts_off                : 109
  Acceptance                      : 9
  Acceptance off                  : 45

You can see that off counts are now simulated as well. We now simulate several spectra using the same set of observation conditions.

 name  counts excess ... acceptance   acceptance_off         alpha
                     ...
------ ------ ------ ... ---------- ----------------- -------------------
 obs-0    317  298.6 ...        9.0              45.0                 0.2
 obs-1    275  253.0 ...        9.0              45.0                 0.2
 obs-2    293  272.4 ...        9.0              45.0                 0.2
 obs-3    280  257.6 ...        9.0 45.00000000000001 0.19999999999999998
 obs-4    337  316.4 ...        9.0              45.0                 0.2
 obs-5    283  258.6 ...        9.0              45.0                 0.2
 obs-6    330  307.6 ...        9.0 44.99999999999999 0.20000000000000004
 obs-7    283  257.2 ...        9.0              45.0                 0.2
 obs-8    308  284.6 ...        9.0              45.0                 0.2
 obs-9    299  278.6 ...        9.0 45.00000000000001 0.19999999999999998
   ...    ...    ... ...        ...               ...                 ...
obs-90    286  267.0 ...        9.0 44.99999999999999 0.20000000000000004
obs-91    285  259.8 ...        9.0              45.0                 0.2
obs-92    313  289.4 ...        9.0              45.0                 0.2
obs-93    302  283.2 ...        9.0              45.0                 0.2
obs-94    322  300.2 ...        9.0              45.0                 0.2
obs-95    305  280.6 ...        9.0 44.99999999999999 0.20000000000000004
obs-96    301  277.4 ...        9.0              45.0                 0.2
obs-97    290  271.2 ...        9.0              45.0                 0.2
obs-98    301  280.6 ...        9.0              45.0                 0.2
obs-99    323  302.2 ...        9.0              45.0                 0.2
Length = 100 rows

Before moving on to the fit let’s have a look at the simulated observations.

fix, axes = plt.subplots(1, 3, figsize=(12, 4))
axes[0].hist(table["counts"])
axes[0].set_xlabel("Counts")
axes[1].hist(table["counts_off"])
axes[1].set_xlabel("Counts Off")
axes[2].hist(table["excess"])
axes[2].set_xlabel("excess")
plt.show()
spectrum simulation

Now, we fit each simulated spectrum individually

results = []

fit = Fit()

for dataset in datasets:
    dataset.models = model.copy()
    result = fit.optimize(dataset)
    results.append(
        {
            "index": result.parameters["index"].value,
            "amplitude": result.parameters["amplitude"].value,
        }
    )

We take a look at the distribution of the fitted indices. This matches very well with the spectrum that we initially injected.

fig, ax = plt.subplots()
index = np.array([_["index"] for _ in results])
ax.hist(index, bins=10, alpha=0.5)
ax.axvline(x=model_simu.parameters["index"].value, color="red")
ax.set_xlabel("Reconstructed spectral index")
print(f"index: {index.mean()} += {index.std()}")
plt.show()
spectrum simulation
index: 3.0036925550381217 += 0.08081469527619482

Exercises#

  • Change the observation time to something longer or shorter. Does the observation and spectrum results change as you expected?

  • Change the spectral model, e.g. add a cutoff at 5 TeV, or put a steep-spectrum source with spectral index of 4.0

  • Simulate spectra with the spectral model we just defined. How much observation duration do you need to get back the injected parameters?

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