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# 2D map fitting¶

## Prerequisites:¶

• To understand how a generel modelling and fiiting works in gammapy, please refer to the analysis_3d tutorial

## Context:¶

We often want the determine the position and morphology of an object. To do so, we don’t necessarily have to resort to a full 3D fitting but can perform a simple image fitting, in particular, in an energy range where the PSF does not vary strongly, or if we want to explore a possible energy dependance of the morphology.

## Objective:¶

To localize a source and/or constrain its morphology.

## Proposed approach:¶

The first step here, as in most analysis with DL3 data, is to create reduced datasets. For this, we will use the Analysis class to create a single set of stacked maps with a single bin in energy (thus, an image which behaves as a cube). This, we will then model with a spatial model of our choice, while keeping the spectral model fixed to an integrated power law.

## Setup¶

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

[1]:

%matplotlib inline
import astropy.units as u
import numpy as np
from astropy.coordinates import SkyCoord
from astropy.time import Time
from regions import CircleSkyRegion
from astropy.coordinates import Angle

import logging

log = logging.getLogger(__name__)


Now let’s import gammapy specific classes and functions

[2]:

from gammapy.analysis import Analysis, AnalysisConfig


## Creating the config file¶

Now, we create a config file for out analysis. You may load this from disc if you have a pre-defined config file.

Here, we use 3 simulated CTA runs of the galactic center.

[3]:

config = AnalysisConfig()
# Selecting the observations
config.observations.datastore = "$GAMMAPY_DATA/cta-1dc/index/gps/" config.observations.obs_ids = [110380, 111140, 111159]  Technically, gammapy implements 2D analysis as a special case of 3D analysis (one one bin in energy). So, we must specify the type of analysis as 3D, and define the geometry of the analysis. [4]:  config.datasets.type = "3d" config.datasets.geom.wcs.skydir = { "lon": "0 deg", "lat": "0 deg", "frame": "galactic", } # The WCS geometry - centered on the galactic center config.datasets.geom.wcs.fov = {"width": "8 deg", "height": "6 deg"} config.datasets.geom.wcs.binsize = "0.02 deg" # The FoV radius to use for cutouts config.datasets.geom.selection.offset_max = 2.5 * u.deg config.datasets.safe_mask.methods = ["offset-max"] config.datasets.safe_mask.parameters = {"offset_max": 2.5 * u.deg} config.datasets.background.method = "fov_background" config.fit.fit_range = {"min": "0.1 TeV", "max": "30.0 TeV"} # We now fix the energy axis for the counts map - (the reconstructed energy binning) config.datasets.geom.axes.energy.min = "0.1 TeV" config.datasets.geom.axes.energy.max = "10 TeV" config.datasets.geom.axes.energy.nbins = 1 config.datasets.geom.wcs.binsize_irf = 0.2 * u.deg  [5]:  print(config)  AnalysisConfig general: log: {level: info, filename: null, filemode: null, format: null, datefmt: null} outdir: . observations: datastore:$GAMMAPY_DATA/cta-1dc/index/gps
obs_ids: [110380, 111140, 111159]
obs_file: null
obs_cone: {frame: null, lon: null, lat: null, radius: null}
obs_time: {start: null, stop: null}
datasets:
type: 3d
stack: true
geom:
wcs:
skydir: {frame: galactic, lon: 0.0 deg, lat: 0.0 deg}
binsize: 0.02 deg
fov: {width: 8.0 deg, height: 6.0 deg}
binsize_irf: 0.2 deg
selection: {offset_max: 2.5 deg}
axes:
energy: {min: 0.1 TeV, max: 10.0 TeV, nbins: 1}
energy_true: {min: 0.1 TeV, max: 10.0 TeV, nbins: 30}
map_selection: [counts, exposure, background, psf, edisp]
background:
method: fov_background
exclusion: null
parameters: {}
methods: [offset-max]
parameters: {offset_max: 2.5 deg}
on_region: {frame: null, lon: null, lat: null, radius: null}
containment_correction: true
fit:
fit_range: {min: 0.1 TeV, max: 30.0 TeV}
flux_points:
energy: {min: 0.1 TeV, max: 10.0 TeV, nbins: 30}
source: source
parameters: {}



## Getting the reduced dataset¶

We now use the config file and create a single MapDataset containing counts, background, exposure, psf and edisp maps.

[6]:

%%time
analysis = Analysis(config)
analysis.get_observations()
analysis.get_datasets()

Setting logging config: {'level': 'INFO', 'filename': None, 'filemode': None, 'format': None, 'datefmt': None}
Fetching observations.
Number of selected observations: 3
Creating geometry.
Creating datasets.
Processing observation 110380
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Processing observation 111140
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Processing observation 111159
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)
Invalid unit found in background table! Assuming (s-1 MeV-1 sr-1)

CPU times: user 9.12 s, sys: 3 s, total: 12.1 s
Wall time: 12.1 s

[7]:

print(analysis.datasets["stacked"])

MapDataset
----------

Name                            : stacked

Total counts                    : 85625
Total background counts         : 85624.99
Total excess counts             : 0.01

Predicted counts                : 85625.00
Predicted background counts     : 85624.99
Predicted excess counts         : nan

Exposure min                    : 1.57e+08 m2 s
Exposure max                    : 1.87e+10 m2 s

Number of total bins            : 120000
Number of fit bins              : 96602

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

Number of models                : 0
Number of parameters            : 0
Number of free parameters       : 0



The counts and background maps have only one bin in reconstructed energy. The exposure and IRF maps are in true energy, and hence, have a different binning based upon the binning of the IRFs. We need not bother about them presently.

[8]:

analysis.datasets["stacked"].counts

[8]:

WcsNDMap

geom  : WcsGeom
axes  : ['lon', 'lat', 'energy']
shape : (400, 300, 1)
ndim  : 3
unit  :
dtype : float32

[9]:

analysis.datasets["stacked"].background

[9]:

WcsNDMap

geom  : WcsGeom
axes  : ['lon', 'lat', 'energy']
shape : (400, 300, 1)
ndim  : 3
unit  :
dtype : float32

[10]:

analysis.datasets["stacked"].exposure

[10]:

WcsNDMap

geom  : WcsGeom
axes  : ['lon', 'lat', 'energy_true']
shape : (400, 300, 30)
ndim  : 3
unit  : m2 s
dtype : float32


We can have a quick look of these maps in the following way:

[11]:

analysis.datasets["stacked"].counts.reduce_over_axes().plot(vmax=5)

/usr/share/miniconda/envs/gammapy-dev/lib/python3.7/site-packages/astropy/visualization/wcsaxes/core.py:211: MatplotlibDeprecationWarning: Passing parameters norm and vmin/vmax simultaneously is deprecated since 3.3 and will become an error two minor releases later. Please pass vmin/vmax directly to the norm when creating it.
return super().imshow(X, *args, origin=origin, **kwargs)

[11]:

(<Figure size 432x288 with 1 Axes>,
<WCSAxesSubplot:xlabel='Galactic Longitude', ylabel='Galactic Latitude'>,
None)


## Modelling¶

Now, we define a model to be fitted to the dataset. The important thing to note here is the dummy spectral model - an integrated powerlaw with only free normalisation. Here, we use its YAML definition to load it:

[12]:

model_config = """
components:
- name: GC-1
type: SkyModel
spatial:
type: PointSpatialModel
frame: galactic
parameters:
- name: lon_0
value: 0.02
unit: deg
- name: lat_0
value: 0.01
unit: deg
spectral:
type: PowerLaw2SpectralModel
parameters:
- name: amplitude
value: 1.0e-12
unit: cm-2 s-1
- name: index
value: 2.0
unit: ''
frozen: true
- name: emin
value: 0.1
unit: TeV
frozen: true
- name: emax
value: 10.0
unit: TeV
frozen: true
"""

[13]:

analysis.set_models(model_config)

Reading model.
Models

Component 0: SkyModel

Name                      : GC-1
Datasets names            : None
Spectral model type       : PowerLaw2SpectralModel
Spatial  model type       : PointSpatialModel
Temporal model type       :
Parameters:
amplitude               :   1.00e-12  1 / (cm2 s)
index        (frozen)   :   2.000
emin         (frozen)   :   0.100  TeV
emax         (frozen)   :  10.000  TeV
lon_0                   :   0.020  deg
lat_0                   :   0.010  deg

Component 1: FoVBackgroundModel

Name                      : stacked-bkg
Datasets names            : ['stacked']
Spectral model type       : PowerLawNormSpectralModel
Parameters:
norm                    :   1.000
tilt         (frozen)   :   0.000
reference    (frozen)   :   1.000  TeV



We will freeze the parameters of the background

[14]:

analysis.datasets["stacked"].background_model.parameters["tilt"].frozen = True

[15]:

# To run the fit
analysis.run_fit()

Fitting datasets.
OptimizeResult

backend    : minuit
method     : minuit
success    : True
message    : Optimization terminated successfully.
nfev       : 168
total stat : 169967.40


[16]:

# To see the best fit values along with the errors
analysis.fit_result.parameters.to_table()

[16]:

Table length=9
typenamevalueunitminmaxfrozenerror
str8str9float64str8float64float64boolfloat64
spectralamplitude2.9853e-11cm-2 s-1nannanFalse1.469e-12
spectralindex2.0000e+00nannanTrue0.000e+00
spectralemin1.0000e-01TeVnannanTrue0.000e+00
spectralemax1.0000e+01TeVnannanTrue0.000e+00
spatiallon_0-4.8873e-02degnannanFalse2.375e-03
spatiallat_0-5.2356e-02deg-9.000e+019.000e+01False2.367e-03
spectralnorm9.9273e-01nannanFalse3.411e-03
spectraltilt0.0000e+00nannanTrue0.000e+00
spectralreference1.0000e+00TeVnannanTrue0.000e+00
[ ]: