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Ring Background Estimation

Context:

One of the challenges of IACT analysis is accounting for the large residual hadronic emission. An excess map, assumed to be a map of only gamma-ray events, requires a good estimate of the background. However, in the absence of a solid template bkg model it is not possible to obtain reliable background model a priori. It was often found necessary in classical cherenkov astronomy to perform a local renormalization of the existing templates, usually with a ring kernel. This assumes that most of the events are background and requires to have an exclusion mask to remove regions with bright signal from the estimation. To read more about this method, see here.

Objective:

Create an excess (gamma-ray events) map of MSH 15-52 as well as a significance map to determine how solid the signal is.

Proposed approach:

The analysis workflow is roughly - Compute the sky maps keeping each observation separately using the Analysis class - Estimate the background using the RingBackgroundMaker - Compute the correlated excess and significance maps using the CorrelatedExcessMapEstimator

The normalised background thus obtained can be used for general modelling and fitting.

Setup

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

[1]:
%matplotlib inline
import astropy.units as u
from astropy.coordinates import SkyCoord
import matplotlib.pyplot as plt
import numpy as np
from regions import CircleSkyRegion
from scipy.stats import norm

import logging

log = logging.getLogger(__name__)

Now let’s import gammapy specific classes and functions

[2]:
from gammapy.analysis import Analysis, AnalysisConfig
from gammapy.makers import RingBackgroundMaker
from gammapy.estimators import ExcessMapEstimator
from gammapy.maps import Map
from gammapy.datasets import MapDatasetOnOff

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.

In this example, we will use a few HESS runs on the pulsar wind nebula, MSH 1552

[3]:
# source_pos = SkyCoord.from_name("MSH 15-52")
source_pos = SkyCoord(228.32, -59.08, unit="deg")
[4]:
config = AnalysisConfig()
# Select observations - 2.5 degrees from the source position
config.observations.datastore = "$GAMMAPY_DATA/hess-dl3-dr1/"
config.observations.obs_cone = {
    "frame": "icrs",
    "lon": source_pos.ra,
    "lat": source_pos.dec,
    "radius": 2.5 * u.deg,
}
[5]:
config.datasets.type = "3d"
config.datasets.geom.wcs.skydir = {
    "lon": source_pos.ra,
    "lat": source_pos.dec,
    "frame": "icrs",
}  # The WCS geometry - centered on MSH 15-52
config.datasets.geom.wcs.fov = {"width": "3 deg", "height": "3 deg"}
config.datasets.geom.wcs.binsize = "0.02 deg"

# The FoV radius to use for cutouts
config.datasets.geom.selection.offset_max = 3.5 * u.deg

# We now fix the energy axis for the counts map - (the reconstructed energy binning)
config.datasets.geom.axes.energy.min = "0.5 TeV"
config.datasets.geom.axes.energy.max = "5 TeV"
config.datasets.geom.axes.energy.nbins = 10

# We need to extract the ring for each observation separately, hence, no stacking at this stage
config.datasets.stack = False
[6]:
print(config)
AnalysisConfig

    general:
        log: {level: info, filename: null, filemode: null, format: null, datefmt: null}
        outdir: .
    observations:
        datastore: $GAMMAPY_DATA/hess-dl3-dr1
        obs_ids: []
        obs_file: null
        obs_cone: {frame: icrs, lon: 228.32 deg, lat: -59.08 deg, radius: 2.5 deg}
        obs_time: {start: null, stop: null}
    datasets:
        type: 3d
        stack: false
        geom:
            wcs:
                skydir: {frame: icrs, lon: 228.32 deg, lat: -59.08 deg}
                binsize: 0.02 deg
                fov: {width: 3.0 deg, height: 3.0 deg}
                binsize_irf: 0.2 deg
            selection: {offset_max: 3.5 deg}
            axes:
                energy: {min: 0.5 TeV, max: 5.0 TeV, nbins: 10}
                energy_true: {min: 0.1 TeV, max: 10.0 TeV, nbins: 30}
        map_selection: [counts, exposure, background, psf, edisp]
        background:
            method: null
            exclusion: null
            parameters: {}
        safe_mask:
            methods: [aeff-default]
            parameters: {}
        on_region: {frame: null, lon: null, lat: null, radius: null}
        containment_correction: true
    fit:
        fit_range: {min: 0.1 TeV, max: 10.0 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 to do the initial data reduction which will then be used for a ring extraction

[7]:
%%time
# create the config
analysis = Analysis(config)

# for this specific case,w e do not need fine bins in true energy
analysis.config.datasets.geom.axes.energy_true = (
    analysis.config.datasets.geom.axes.energy
)

# `First get the required observations
analysis.get_observations()
Setting logging config: {'level': 'INFO', 'filename': None, 'filemode': None, 'format': None, 'datefmt': None}
Fetching observations.
Number of selected observations: 20
CPU times: user 307 ms, sys: 1.01 ms, total: 308 ms
Wall time: 297 ms
[8]:
print(analysis.config)
AnalysisConfig

    general:
        log: {level: INFO, filename: null, filemode: null, format: null, datefmt: null}
        outdir: .
    observations:
        datastore: $GAMMAPY_DATA/hess-dl3-dr1
        obs_ids: []
        obs_file: null
        obs_cone: {frame: icrs, lon: 228.32 deg, lat: -59.08 deg, radius: 2.5 deg}
        obs_time: {start: null, stop: null}
    datasets:
        type: 3d
        stack: false
        geom:
            wcs:
                skydir: {frame: icrs, lon: 228.32 deg, lat: -59.08 deg}
                binsize: 0.02 deg
                fov: {width: 3.0 deg, height: 3.0 deg}
                binsize_irf: 0.2 deg
            selection: {offset_max: 3.5 deg}
            axes:
                energy: {min: 0.5 TeV, max: 5.0 TeV, nbins: 10}
                energy_true: {min: 0.5 TeV, max: 5.0 TeV, nbins: 10}
        map_selection: [counts, exposure, background, psf, edisp]
        background:
            method: null
            exclusion: null
            parameters: {}
        safe_mask:
            methods: [aeff-default]
            parameters: {}
        on_region: {frame: null, lon: null, lat: null, radius: null}
        containment_correction: true
    fit:
        fit_range: {min: 0.1 TeV, max: 10.0 TeV}
    flux_points:
        energy: {min: 0.1 TeV, max: 10.0 TeV, nbins: 30}
        source: source
        parameters: {}

[9]:
%%time
# Data extraction
analysis.get_datasets()
Creating geometry.
Creating datasets.
No background maker set for 3d analysis. Check configuration.
Processing observation 20365
Processing observation 20366
Processing observation 20367
Processing observation 20368
Processing observation 20136
Processing observation 20137
Processing observation 20151
Processing observation 20282
Processing observation 20283
Processing observation 20301
Processing observation 20302
Processing observation 20303
Processing observation 20322
Processing observation 20323
Processing observation 20324
Processing observation 20325
Processing observation 20343
Processing observation 20344
Processing observation 20345
Processing observation 20346
CPU times: user 18.6 s, sys: 1.07 s, total: 19.7 s
Wall time: 19.7 s

Extracting the ring background

Since the ring background is extracted from real off events, we need to use the wstat statistics in this case. For this, we will use the MapDatasetOnOFF and the RingBackgroundMaker classes.

Create exclusion mask

First, we need to create an exclusion mask on the known sources. In this case, we need to mask only MSH 15-52 but this depends on the sources present in our field of view.

[10]:
# get the geom that we use
geom = analysis.datasets[0].counts.geom
energy_axis = analysis.datasets[0].counts.geom.get_axis_by_name("energy")
geom_image = geom.to_image().to_cube([energy_axis.squash()])

# Make the exclusion mask
regions = CircleSkyRegion(center=source_pos, radius=0.3 * u.deg)
exclusion_mask = Map.from_geom(geom_image)
exclusion_mask.data = geom_image.region_mask([regions], inside=False)
exclusion_mask.sum_over_axes().plot();
[10]:
(<Figure size 432x288 with 1 Axes>,
 <WCSAxesSubplot:xlabel='Right Ascension', ylabel='Declination'>,
 None)
../_images/tutorials_ring_background_17_1.png

For the present analysis, we use a ring with an inner radius of 0.5 deg and width of 0.3 deg.

[11]:
ring_maker = RingBackgroundMaker(
    r_in="0.5 deg", width="0.3 deg", exclusion_mask=exclusion_mask
)

Create a stacked dataset

Now, we extract the background for each dataset and then stack the maps together to create a single stacked map for further analysis

[12]:
#%%time
energy_axis_true = analysis.datasets[0].exposure.geom.get_axis_by_name(
    "energy_true"
)
stacked_on_off = MapDatasetOnOff.create(
    geom=geom_image, energy_axis_true=energy_axis_true, name="stacked"
)

for dataset in analysis.datasets:
    # Ring extracting makes sense only for 2D analysis
    dataset_on_off = ring_maker.run(dataset.to_image())
    stacked_on_off.stack(dataset_on_off)
No background model defined for dataset stacked
No background model defined for dataset Q-yWRn_F
/usr/share/miniconda/envs/gammapy-dev/lib/python3.7/site-packages/astropy/units/quantity.py:477: RuntimeWarning: invalid value encountered in true_divide
  result = super().__array_ufunc__(function, method, *arrays, **kwargs)
No background model defined for dataset stacked
No background model defined for dataset rd8ihQmI
No background model defined for dataset stacked
No background model defined for dataset 2U01PzQV
No background model defined for dataset stacked
No background model defined for dataset DPTLwNJp
No background model defined for dataset stacked
No background model defined for dataset 7Pp4bHgq
No background model defined for dataset stacked
No background model defined for dataset CTM0gLlG
No background model defined for dataset stacked
No background model defined for dataset Y_yNGeYb
No background model defined for dataset stacked
No background model defined for dataset ngfPMyTS
No background model defined for dataset stacked
No background model defined for dataset 88ei0-JZ
No background model defined for dataset stacked
No background model defined for dataset leYifmSa
No background model defined for dataset stacked
No background model defined for dataset 4DF4xJXq
No background model defined for dataset stacked
No background model defined for dataset 8tLi1mVG
No background model defined for dataset stacked
No background model defined for dataset Ux-qfHfu
No background model defined for dataset stacked
No background model defined for dataset SZ1H3Tuj
No background model defined for dataset stacked
No background model defined for dataset VuIdh3N1
No background model defined for dataset stacked
No background model defined for dataset u6vQ2E3L
No background model defined for dataset stacked
No background model defined for dataset Iihj72LH
No background model defined for dataset stacked
No background model defined for dataset V0trdLKX
No background model defined for dataset stacked
No background model defined for dataset 7L_BSB-7
No background model defined for dataset stacked
No background model defined for dataset qNUj99o-
No background model defined for dataset stacked

This stacked_on_off has on and off counts and acceptance maps which we will use in all further analysis. The acceptance and acceptance_off maps are the system acceptance of gamma-ray like events in the on and off regions respectively.

[13]:
print(stacked_on_off)
MapDatasetOnOff
---------------

  Name                            : stacked

  Total counts                    : 40051
  Total predicted counts          : 0.00
  Total background counts         : nan

  Exposure min                    : 1.11e+09 m2 s
  Exposure max                    : 1.30e+10 m2 s

  Number of total bins            : 22500
  Number of fit bins              : 22500

  Fit statistic type              : wstat
  Fit statistic value (-2 log(L)) : 26392.57

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

    Total counts_off                : 88113384
    Acceptance                      : 22500
    Acceptance off                  : 49447599

Compute correlated significance and correlated excess maps

We need to convolve our maps with an apprpriate smoothing kernel. The significance is computed according to the Li & Ma expression for ON and OFF Poisson measurements, see here. Since astropy convolution kernels only accept integers, we first convert our required size in degrees to int depending on our pixel size.

[14]:
# Using a convolution radius of 0.04 degrees
estimator = ExcessMapEstimator(0.04 * u.deg, selection_optional=[])
lima_maps = estimator.run(stacked_on_off)
No background model defined for dataset stacked-0.500 TeV-5.000 TeV
No background model defined for dataset stacked-0.500 TeV-5.000 TeV
No background model defined for dataset stacked-0.500 TeV-5.000 TeV
[15]:
significance_map = lima_maps["significance"]
excess_map = lima_maps["excess"]
[16]:
# We can plot the excess and significance maps
plt.figure(figsize=(10, 10))
ax1 = plt.subplot(221, projection=significance_map.geom.wcs)
ax2 = plt.subplot(222, projection=excess_map.geom.wcs)

ax1.set_title("Significance map")
significance_map.plot(ax=ax1, add_cbar=True)

ax2.set_title("Excess map")
excess_map.plot(ax=ax2, add_cbar=True)
[16]:
(<Figure size 720x720 with 4 Axes>,
 <WCSAxesSubplot:title={'center':'Excess map'}, xlabel='Right Ascension', ylabel='Declination'>,
 <matplotlib.colorbar.Colorbar at 0x7f0415e2a278>)
../_images/tutorials_ring_background_27_1.png

It is often important to look at the signficance distribution outside the exclusion region to check that the background estimation is not contaminated by gamma-ray events. This can be the case when exclusion regions are not large enough. Typically, we expect the off distribution to be a standard normal distribution.

[17]:
# create a 2D mask for the images
significance_map_off = significance_map * exclusion_mask
significance_all = significance_map.data[np.isfinite(significance_map.data)]
significance_off = significance_map_off.data[
    np.isfinite(significance_map_off.data)
]

plt.hist(
    significance_all,
    density=True,
    alpha=0.5,
    color="red",
    label="all bins",
    bins=21,
)

plt.hist(
    significance_off,
    density=True,
    alpha=0.5,
    color="blue",
    label="off bins",
    bins=21,
)

# Now, fit the off distribution with a Gaussian
mu, std = norm.fit(significance_off)
x = np.linspace(-8, 8, 50)
p = norm.pdf(x, mu, std)
plt.plot(x, p, lw=2, color="black")
plt.legend()
plt.xlabel("Significance")
plt.yscale("log")
plt.ylim(1e-5, 1)
xmin, xmax = np.min(significance_all), np.max(significance_all)
plt.xlim(xmin, xmax)

print(f"Fit results: mu = {mu:.2f}, std = {std:.2f}")
Fit results: mu = -0.02, std = 1.00
../_images/tutorials_ring_background_29_1.png
[18]: