.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "user-guide/model-gallery/spatial/plot_gauss.py"
.. LINE NUMBERS ARE GIVEN BELOW.
.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_user-guide_model-gallery_spatial_plot_gauss.py>`
to download the full example code. or to run this example in your browser via Binder
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_user-guide_model-gallery_spatial_plot_gauss.py:
.. _gaussian-spatial-model:
Gaussian spatial model
======================
This is a spatial model parametrising a Gaussian function.
By default, the Gaussian is symmetric:
.. math::
\phi(\text{lon}, \text{lat}) = N \times \exp\left\{-\frac{1}{2}
\frac{1-\cos \theta}{1-\cos \sigma}\right\}\,,
where :math:`\theta` is the sky separation to the model center. In this case, the
Gaussian is normalized to 1 on the sphere:
.. math::
N = \frac{1}{4\pi a\left[1-\exp(-1/a)\right]}\,,\,\,\,\,
a = 1-\cos \sigma\,.
In the limit of small :math:`\theta` and :math:`\sigma`, this definition
reduces to the usual form:
.. math::
\phi(\text{lon}, \text{lat}) = \frac{1}{2\pi\sigma^2} \exp{\left(-\frac{1}{2}
\frac{\theta^2}{\sigma^2}\right)}\,.
In case an eccentricity (:math:`e`) and rotation angle (:math:`\phi`) are passed,
then the model is an elongated Gaussian, whose evaluation is performed as in the symmetric case
but using the effective radius of the Gaussian:
.. math::
\sigma_{eff}(\text{lon}, \text{lat}) = \sqrt{
(\sigma_M \sin(\Delta \phi))^2 +
(\sigma_m \cos(\Delta \phi))^2
}.
Here, :math:`\sigma_M` (:math:`\sigma_m`) is the major (minor) semiaxis of the Gaussian, and
:math:`\Delta \phi` is the difference between `phi`, the position angle of the Gaussian, and the
position angle of the evaluation point.
**Caveat:** For the asymmetric Gaussian, the model is normalized to 1 on the plane, i.e. in small angle
approximation: :math:`N = 1/(2 \pi \sigma_M \sigma_m)`. This means that for huge elongated Gaussians on the sky
this model is not correctly normalized. However, this approximation is perfectly acceptable for the more
common case of models with modest dimensions: indeed, the error introduced by normalizing on the plane
rather than on the sphere is below 0.1\% for Gaussians with radii smaller than ~ 5 deg.
.. GENERATED FROM PYTHON SOURCE LINES 51-54
Example plot
------------
Here is an example plot of the model:
.. GENERATED FROM PYTHON SOURCE LINES 54-92
.. code-block:: Python
import numpy as np
from astropy.coordinates import Angle
from gammapy.maps import WcsGeom
from gammapy.modeling.models import (
GaussianSpatialModel,
Models,
PowerLawSpectralModel,
SkyModel,
)
phi = Angle("30 deg")
model = GaussianSpatialModel(
lon_0="2 deg",
lat_0="2 deg",
sigma="1 deg",
e=0.7,
phi=phi,
frame="galactic",
)
geom = WcsGeom.create(
skydir=model.position, frame=model.frame, width=(4, 4), binsz=0.02
)
ax = model.plot(geom=geom, add_cbar=True)
# illustrate size parameter
region = model.to_region().to_pixel(ax.wcs)
artist = region.as_artist(facecolor="none", edgecolor="red")
ax.add_artist(artist)
transform = ax.get_transform("galactic")
ax.scatter(2, 2, transform=transform, s=20, edgecolor="red", facecolor="red")
ax.text(1.5, 1.85, r"$(l_0, b_0)$", transform=transform, ha="center")
ax.plot([2, 2 + np.sin(phi)], [2, 2 + np.cos(phi)], color="r", transform=transform)
ax.vlines(x=2, color="r", linestyle="--", transform=transform, ymin=-5, ymax=5)
ax.text(2.25, 2.45, r"$\phi$", transform=transform)
.. image-sg:: /user-guide/model-gallery/spatial/images/sphx_glr_plot_gauss_001.png
:alt: plot gauss
:srcset: /user-guide/model-gallery/spatial/images/sphx_glr_plot_gauss_001.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Text(2.25, 2.45, '$\\phi$')
.. GENERATED FROM PYTHON SOURCE LINES 93-96
YAML representation
-------------------
Here is an example YAML file using the model:
.. GENERATED FROM PYTHON SOURCE LINES 96-104
.. code-block:: Python
pwl = PowerLawSpectralModel()
gauss = GaussianSpatialModel()
model = SkyModel(spectral_model=pwl, spatial_model=gauss, name="pwl-gauss-model")
models = Models([model])
print(models.to_yaml())
.. rst-class:: sphx-glr-script-out
.. code-block:: none
components:
- name: pwl-gauss-model
type: SkyModel
spectral:
type: PowerLawSpectralModel
parameters:
- name: index
value: 2.0
- name: amplitude
value: 1.0e-12
unit: cm-2 s-1 TeV-1
- name: reference
value: 1.0
unit: TeV
spatial:
type: GaussianSpatialModel
frame: icrs
parameters:
- name: lon_0
value: 0.0
unit: deg
- name: lat_0
value: 0.0
unit: deg
- name: sigma
value: 1.0
unit: deg
- name: e
value: 0.0
- name: phi
value: 0.0
unit: deg
metadata:
creator: Gammapy 1.3
date: '2024-11-26T10:08:42.078571'
origin: null
.. _sphx_glr_download_user-guide_model-gallery_spatial_plot_gauss.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: binder-badge
.. image:: images/binder_badge_logo.svg
:target: https://mybinder.org/v2/gh/gammapy/gammapy-webpage/v1.3?urlpath=lab/tree/notebooks/1.3/user-guide/model-gallery/spatial/plot_gauss.ipynb
:alt: Launch binder
:width: 150 px
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: plot_gauss.ipynb <plot_gauss.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_gauss.py <plot_gauss.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: plot_gauss.zip <plot_gauss.zip>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_