IACT background models¶

The following sections present the model components that are available in ctools for the modelling of instrumental background in data from Imaging Air Cherenkov Telescopes (IACTs) such as CTA, H.E.S.S., VERITAS, and MAGIC.

IACT background models are factorised into an optional spatial tag (tags <spatialModel> or <radialModel>) and a spectral tag (tag <spectrum>) using

\[M(p',E') = M_{\rm spatial}(p'|E') \times M_{\rm spectral}(E')\]

where \(M(p',E')\) is given in units of \({\rm events} \,\, {\rm s}^{-1} {\rm MeV}^{-1} {\rm sr}^{-1}\).

For the spectral components, all spectra described in Spectral model components may be used.

Note

Spatial directions \(p'\) and energies \(E'\) are now the reconstructed quantities, hence no convolution of the model with the Instrument Response Functions is performed.

General IACT background¶

The general IACT background model is factorised in a spatial and spectral component and has the type CTABackground. It has the following XML structure:
<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="...">
  ...
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>
Warning

You need to specify the instrument label in the source tag that corresponds to the instrument label of the observation to which the model should apply. Supported instrument labels are CTA, HESS, VERITAS and MAGIC.

For example, if you analyse a H.E.S.S. observation you need to specify instrument="HESS" in the source tag, while the model type is still CTABackground. So don’t get confused!
<source name="Background" type="CTABackground" instrument="HESS">
The same logic applies to the radial acceptance, IRF, effective area and cube background models.
Warning

In case that a background model should be used for the analysis of On/Off data, the corresponding OnOff instrument label needs to be selected. Supported OnOff instrument labels are CTAOnOff, HESSOnOff, VERITASOnOff and MAGICOnOff.

For example, if you analyse a H.E.S.S. On/Off observation you need to specify
<source name="Background" type="CTABackground" instrument="HESSOnOff">
The same logic applies to the radial acceptance, IRF, effective area and cube background models.
The following sections describe the spatial model components that are available.

Gaussian¶

The Gaussian model describes a 2D Gaussian shape in offset angle squared
<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="Gaussian">
    <parameter name="Sigma" scale="1.0" value="3.0" min="0.01" max="10.0" free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>
and implements

\[M_{\rm spatial}(\theta) = \exp \left(-\frac{1}{2} \left( \frac{\theta^2}{\sigma} \right)^2 \right)\]

where

\(\sigma\) = Sigma (degrees)

and

\[\theta = \sqrt{\mathrm{DETX} \times \mathrm{DETX} + \mathrm{DETY} \times\mathrm{DETY}}\]

with \(\mathrm{DETX}\) and \(\mathrm{DETY}\) being the detector coordinates in the nominal system.

Profile¶

The Profile model describes a radial profile

<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="Profile">
    <parameter name="Width" scale="1.0" value="1.5" min="0.1" max="1000.0" free="1"/>
    <parameter name="Core"  scale="1.0" value="3.0" min="0.1" max="1000.0" free="1"/>
    <parameter name="Tail"  scale="1.0" value="5.0" min="0.1" max="1000.0" free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\theta) = (1 + (\theta/c_0)^{c_1})^{-c_2/c_1}\]

where

\(c_0\) = Width (degrees)
\(c_1\) = Core
\(c_2\) = Tail

Polynom¶

The Polynom model describes a polynomial with an arbitrary number of coefficients

<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="Polynom">
    <parameter name="Coeff0" scale="1.0" value="+1.00000"   min="-10.0" max="10.0" free="0"/>
    <parameter name="Coeff1" scale="1.0" value="-0.1239176" min="-10.0" max="10.0" free="1"/>
    <parameter name="Coeff2" scale="1.0" value="+0.9751791" min="-10.0" max="10.0" free="1"/>
    <parameter name="Coeff3" scale="1.0" value="-3.0584577" min="-10.0" max="10.0" free="1"/>
    ...
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\theta) = \sum_{i=0}^m c_i \theta^i\]

where

\(c_0\) = Coeff0
\(c_1\) = Coeff1
\(c_2\) = Coeff2
\(c_3\) = Coeff3
…

Gradient¶

The Gradient model describes a bilinear gradient over the field of view
<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="Gradient">
    <parameter name="Grad_DETX" scale="1.0" value="0.0" min="-10.0" max="10.0" free="1"/>
    <parameter name="Grad_DETY" scale="1.0" value="0.0" min="-10.0" max="10.0" free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>
and implements

\[M_{\rm spatial}(\mathrm{DETX},\mathrm{DETY}) = 1 + \nabla_\mathrm{x} \mathrm{DETX} + \nabla_\mathrm{y} \mathrm{DETY}\]

where

\(\nabla_\mathrm{x}\) = Grad_DETX (per degree)

\(\nabla_\mathrm{y}\) = Grad_DETY (per degree)

Multiplicative¶

The Multiplicative model describes a multiplication of spatial models

<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="Multiplicative">
    <spatialModel type="...">
      ...
    </spatialModel>
    <spatialModel type="...">
      ...
    </spatialModel>
    ...
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\mathrm{DETX},\mathrm{DETY}) = \prod_{i=0}^{N-1} M^{(i)}_{\rm spatial}(\mathrm{DETX},\mathrm{DETY})\]

where \(M^{(i)}_{\rm spatial}(\mathrm{DETX},\mathrm{DETY})\) is any spatial model component, including another multiplicative model, and \(N\) is the number of model components that are multiplied. For example, the default model for a H.E.S.S. data analysis is specified by

<source name="Background" type="CTABackground" instrument="CTA">
  <spatialModel type="Multiplicative">
    <spatialModel type="Gaussian">
      <parameter name="Sigma" scale="1.0" value="3.0" min="0.01" max="10.0" free="1"/>
    </spatialModel>
    <spatialModel type="Gradient">
      <parameter name="Grad_DETX" scale="1.0" value="0.0" min="-10.0" max="10.0" free="1"/>
      <parameter name="Grad_DETY" scale="1.0" value="0.0" min="-10.0" max="10.0" free="1"/>
    </spatialModel>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Radial acceptance background¶

For legacy reasons, there exists a class of radially symmetric background models of the type RadialAcceptance with the following XML structure:
<source name="Background" type="RadialAcceptance" instrument="CTA">
  <radialModel type="Gaussian">
    ...
  </radialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>
These models require a <radialModel> tag as the spatial component and accept all spatial model types that take the offset angle \(\theta\) as variable, such as Gaussian, Profile and Polynom.

Warning

The use of the radial acceptance model is deprecated, and the CTABackground model should be used instead.

IRF background¶

The Instrument Response Functions (IRFs) contain a template that predicts the background rate as function of position in the field of view and measured energy in units of \({\rm events} \, {\rm s}^{-1} {\rm MeV}^{-1} {\rm sr}^{-1}\). This template can be used by specifying a model of type CTAIrfBackground. No spatial component will be specified explicitly since the spatial (and spectral) information is already contained in the template.

The model will be multiplied by a spectral component to allow for the adjustment of the energy distribution of the background rate.
<source name="Background" type="CTAIrfBackground" instrument="CTA">
  <spectrum type="...">
    ...
  </spectrum>
</source>
If the observation is an On/Off observation, do not forget to switch the instrument to CTAOnOff:
<source name="Background" type="CTAIrfBackground" instrument="CTAOnOff">
  <spectrum type="...">
    ...
  </spectrum>
</source>

Effective area background¶

Instead of using the background template the effective area for gamma rays can also be used to model the instrumental background. Note that in this case the effective area has to be scaled to a reasonable background rate by selecting appropriate values for the spectral model component.
<source name="Background" type="CTAAeffBackground" instrument="CTA">
  <spectrum type="...">
    ...
  </spectrum>
</source>

Cube background¶

For a stacked analysis, the background rates are predicted by a so called background cube. The FITS file name of the background cube is specified either as ctools task parameter, or using the BkgCube parameter in the the observation definition XML file.

The background cube model is used by specifying a model of type CTACubeBackground. Similar to the CTAIrfBackground model, the background cube is multplied with a spectral model to allow for the adjustment of the energy distribution of the background rate.
<source name="Background" type="CTACubeBackground" instrument="CTA">
  <spectrum type="...">
    ...
  </spectrum>
</source>

Table of Contents

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IACT background models¶

General IACT background¶

Gaussian¶

Profile¶

Polynom¶

Gradient¶

Multiplicative¶

Radial acceptance background¶

IRF background¶

Effective area background¶

Cube background¶