Spatial background model components¶

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

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>
Note

Please speficy the instrument label in the XML file that corresponds to the instrument label of the data. Otherwise the model will not be used for your data. Valid instrument labels are HESS, VERITAS, MAGIC and CTA.

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

CTA 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 law.
<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>

CTA 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.
<source name="Background" type="CTAAeffBackground" instrument="CTA">
  <spectrum type="...">
    ...
  </spectrum>
</source>

CTA cube background¶

For a stacked analysis, the background rate is predicted by a so called background cube. The background cube is used by specifying a model of type CTACubeBackground. Similar to the CTAIrfBackground model, the background cube is multplied by a spectral model.
<source name="Background" type="CTACubeBackground" instrument="CTA">
  <spectrum type="...">
    ...
  </spectrum>
</source>

Table Of Contents

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Spatial background model components¶

General IACT background¶

Gaussian¶

Profile¶

Polynom¶

Gradient¶

Multiplicative¶

Radial acceptance background¶

CTA IRF background¶

CTA effective area background¶

CTA cube background¶