Updating and including content on the PSF introduction

shearbook/_bibliography/z_psf_models.bib

@@ -158,3 +158,19 @@ @article{LSST2009
 	primaryclass = {astro-ph.IM},
 	title = {{LSST Science Book, Version 2.0}},
 	year = 2009}
+
+@article{heymans2012,
+author = {Heymans, Catherine and Rowe, Barnaby and Hoekstra, Henk and Miller, Lance and Erben, Thomas and Kitching, Thomas and Van Waerbeke, Ludovic},
+title = {The impact of high spatial frequency atmospheric distortions on weak-lensing measurements},
+journal = {Monthly Notices of the Royal Astronomical Society},
+volume = {421},
+number = {1},
+pages = {381-389},
+keywords = {gravitational lensing: weak, cosmology: observations},
+doi = {https://doi.org/10.1111/j.1365-2966.2011.20312.x},
+url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1365-2966.2011.20312.x},
+eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1365-2966.2011.20312.x},
+abstract = {ABSTRACT High-precision cosmology with weak gravitational lensing requires a precise measure of the point spread function across the imaging data where the accuracy to which high spatial frequency variation can be modelled is limited by the stellar number density across the field. We analyse dense stellar fields imaged at the Canada–France–Hawaii Telescope to quantify the degree of high spatial frequency variation in ground-based imaging point spread functions and compare our results to models of atmospheric turbulence. The data show an anisotropic turbulence pattern with an orientation independent of the wind direction and wind speed. We find the amplitude of the high spatial frequencies to decrease with increasing exposure time as t−1/2, and find a negligibly small atmospheric contribution to the point spread function ellipticity variation for exposure times t > 180 s. For future surveys analysing shorter exposure data, this anisotropic turbulence will need to be taken into account as the amplitude of the correlated atmospheric distortions becomes comparable to a cosmological lensing signal on scales less than ∼10 arcmin. This effect could be mitigated, however, by correlating galaxy shear measured on exposures imaged with a time separation greater than 50 s, for which we find the spatial turbulence patterns to be uncorrelated.},
+year = {2012}
+}
+

shearbook/psf/1-intro/contributions.md

@@ -4,8 +4,20 @@
 ## Optic level
 
 - __Mirrors__
+
+    - _Size_
+
+    The size of the main mirror has a direct implicance to the size of the PSF. Following the diffraction theory, the larger the mirror for the lens, the smaller size the PSF should have. There are practical limitations on the maximal sizes of mirrors that are due to the manufacturing process. As an example, we can show the impact of the telescope's pupil diameter in the size of the PSF in the ideal Airy PSF we only take into account the diffraction phenomena. For this type of PSF we can write the Full Width Half Maximum (FWHM) as a function of the wavelength $\lambda$ and the pupil diameter $D$ as,
+
+    $$
+    \text{FWHM}_{\text{Airy}} = 1.028 \; \frac{\lambda}{D} .
+    $$
+
+
     - _Aberrations_
-        - There are imperfections in the curvature of mirrors producing different types of aberrations. These introduce optical path differences for the incoming light rays producing features in the PSF. 
+
+    There are imperfections in the curvature of mirrors producing different types of aberrations. These introduce optical path differences for the incoming light rays producing features in the PSF. They are usually modelled using [Zernike polynomials](https://en.wikipedia.org/wiki/Zernike_polynomials) which are a sequence of polynomials that are orthogonal in the unit disk.
+
 
 ```{figure} ../../_static/psf_wavefron_aberrations.png
 ---
@@ -26,10 +38,10 @@ Typical aberrations with simulated in-focus stars with typical seeing. Credit: A
 ```
 
 - __Mirrors__
-    - _Size_
-        - The size of the main mirror has a direct implicance to the size of the PSF. Following the diffraction theory, the larger the mirror for the lens, the smaller size the PSF should have. There are practical limitations on the maximal sizes of mirrors that are due to the manufacturing process. 
+
     - _Polishing effects_
-        - The polishing of the mirrors aims to have a perfectly smooth surface. However, the manufacturing process is not perfect and small errors remain in the surface of the mirrors. They are sometimes referred to as _surface errors_.
+
+    The polishing of the mirrors aims to have a perfectly smooth surface. However, the manufacturing process is not perfect and small errors remain in the surface of the mirrors. They are sometimes referred to as _surface errors_.
 
 ```{figure} ../../_static/psf-HST-surface-errors.png
 ---
@@ -42,7 +54,8 @@ Surface errors estimated for the Hubble Space Telescope. They are in the range o
 
 - __Optical system__
     - _Misalignment_
-        - Misalignement of optical elements can cause a defocus of the system.  For example, in Figure 7 and 8 from {cite}`jee2011` one can see the effects of Charge-Coupled Devices (CCD) misalignements on the focal plane on a simulation of the LSST mission {cite}`LSST2009`.
+
+    Misalignement of optical elements can cause a defocus of the system.  For example, in Figure 7 and 8 from {cite}`jee2011` one can see the effects of Charge-Coupled Devices (CCD) misalignements on the focal plane on a simulation of the LSST mission {cite}`LSST2009`.
 
 ```{figure} ../../_static/psf-misalignments.png
 ---
@@ -55,14 +68,21 @@ Simulated focal plane of LSST. We used the CCD assembly/fabrication specificatio
 
 - __Optical system__
     - _Scattered light_
-        - It consist of spuriours reflections within the optical system and the detector. An example for the HST mission can be found in Fig 3 in {cite}`krist1995c`.
+
+    It consist of spuriours reflections within the optical system and the detector. An example for the HST mission can be found in Fig 3 in {cite}`krist1995c`.
 
 
-- __Optical system__
     - _Wavelength dependency_
-        - The optical system is composed by several elements where some of them can be non-ideal refractive elements. This means that they introdcuce a wavelength dependance variation.
+
+    The optical system is composed by several elements where some of them can be non-ideal refractive elements. This means that they introdcuce a wavelength dependance variation.
+
+
     - _Obscurations_
-        - Depending on the optical system design it might require arms to sustain the camera or even superposing mirrors that block each other. These obscurations modify the shape of the PSF and vary depending on the source's angle of incidence. They are 3D structures that are projected on the 2D pupil plane.
+
+    Depending on the optical system design it might require arms to sustain the camera or even superposing mirrors that block each other. These obscurations modify the shape of the PSF and vary depending on the source's angle of incidence. They are 3D structures that are projected on the 2D pupil plane. As the obscurations are projected into the pupil plane they vary as a function of the angle of incidence. 
+
+    The obscurations of the pupil plane have a direct relation with the shape of the PSF, specially for space-based surveys where the PSFs are closer to the diffraction limit. The [Fraunhoffer approximation](https://en.wikipedia.org/wiki/Fraunhofer_diffraction) of diffraction is most of the time a valid approximation for astronomical telescopes. This approximation allows to relate the wavefront at the pupil plane and the observed PSF with the Fourier transform. The obscurations force a zero amplitude value for the wavefront at the obscured parts of the pupil plane. The Fourier transform of these zero values can be easily identified in diffraction-limited PSFs, being an important contributor to the PSF shape.
+
 
 ```{figure} ../../_static/psf-HST-obscurations.png
 ---
@@ -75,7 +95,9 @@ Projected obscurations of the Hubble Space Telescope. Credit: {cite}`krist2011`.
 
 - __Filters__
     - Passband
-        - The filters serve to select the wavelengths incoming to the detector. Ideally they would be a perfect band-pass filter, begin one at the desired wavelength range and zero on the other wavelengths. However, it is not possible to manufacture an ideal filter. A real one does not have an abrupt transition, meaning that there are out-of-band contributions. It is also non-flat, meanining that the passband is near the unity but it changes with the wavelength. See this [page](https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/megapipe/docs/filt.html) to visualise some examples of filters.
+
+    The filters serve to select the wavelengths incoming to the detector. Ideally they would be a perfect band-pass filter, begin one at the desired wavelength range and zero on the other wavelengths. However, it is not possible to manufacture an ideal filter. A real one does not have an abrupt transition, meaning that there are out-of-band contributions. It is also non-flat, meanining that the passband is near the unity but it changes with the wavelength. See this [page](https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/megapipe/docs/filt.html) to visualise some examples of filters. As mentioned, the observations are always polychromatic, meaning that they are integrated in wavelength in the passband of the filter being used. 
+
 
 ```{figure} ../../_static/psf-filters.png
 ---
@@ -89,8 +111,24 @@ MegaCam filter set transmission for broad band filters u, g, r, i and z. Credit:
 
 ## Ground-based
 
+- __Seeing__
+
+    In astronomy, seeing refers to the degradation of the image of an astronomical object due to the Earth's atmosphere. It is measured using three common descriptors: i) the Full Width Half Maximum (FWHM) of the PSF, ii) the Fried parameter $r_0$ and the atmospheric time constant $t_0$, and iii) the profile of the turblence strength as a function of the altitude $C_{N}^{2}$. The Fried parameter is a measure of the quality of optical transmission through the atmosphere due to random inhomogeneities in the atmosphere's refractive index.
+    In terms of FWHM, if the seeing is above $1.5$ arcsec, the observations are practically unusable for weak-lensing purposes. The best sites on Earth can reach up to $0.4$ arcsec.
+
+
 - __Atmosphere__
-    - The atmosphere plays a central role in ground-based observations. The refractive index of the atmosphere varies rapidly with the position generating different optical paths for the incoming light. As a consequence, the way it affects the PSF changes if we consider short-term exposures or long-term exposures. It also depends of the incoming wavelength. See {cite}`jee2011` for a first study on the Legacy Survey of Space and Time  (LSST) from the Vera C. Rubin Observatory.
+
+    The atmosphere plays a central role in ground-based observations. The refractive index of the atmosphere varies rapidly with the position generating different optical paths for the incoming light. The atmosphere presents different properties as we change altitude $h$ making it harder to model. Some of these properties are the turbulence strength $C_{N}^{2}(h)$, and the wind speed and diretion. The quality of the observations depend strongly on the exposure time used as well as the telescope location. For example, the best conditions are found in places like the [Mauna Kea observatory](https://en.wikipedia.org/wiki/Mauna_Kea_Observatories) in Hawaii. Most of the atmospheric simulations are simulated using parametric functions of the power spectrum. For example, in {cite}`jee2011` they simulate the atmosphere for the Legacy Survey of Space and Time (LSST) from the Vera C. Rubin Observatory by using several *phase screens* at different altitudes. Each phase screen describes a part of the atmosphere at a given altitude and how it modifies the incoming wavefront on its path towards the telescope. Each screen is characterised by a power spectrum like a Kolmogorov/Von Kármán model:
+
+    $$
+    \Psi(\nu) = 0.023 r_{0}^{-5/3} \left( \nu^{2} + \frac{1}{L_{0}^{2}} \right)^{-11/6},
+    $$
+
+    where $\nu$, $r_{0}$, and $L_{0}$ are the two-dimensional frequency, the Fried parameter, and the outer scale, respectively. The outer scale $L_0$ is infinite for Kolmogorov screens and finite for Von Kármán screens. Once the layer is simulated they are moved with a fixed wind speed and direction that will be finally integrated with the exposure time. This type of modelling requires measurements of the different parameters at the telescope's location. To have a broad picture of their values, for the Cerro Pachón in Chile, the parameters would be $L_{0} = 25\text{m}$, $r_{0} = 0.16\text{m}$, and a wind speed of at most $\sim 20 \text{m/s}$ at some random direction.
+
+    Another way to simulate the atmospheric contribution to the PSF is to model the ellipticity contributions to the PSF by the atmosphere. This approach was studied by {cite}`heymans2012` where the authors study the ellipticity residuals of the observed stars at the [Canada-France-Hawaii Telescope](https://www.cfht.hawaii.edu) at the Mauna Kea observatory. They use the Von Kármán power spectrum and they found the best-fitting outer scale of their observations at different exposure times.
+
 
 ```{figure} ../../_static/psf-atmospheric-screen.png
 ---
@@ -111,7 +149,8 @@ Variations of the PSF in short exposures due to the atmospheric turbulence. Cred
 ```
 
 - __Observing environment__
-    - The temperature differences in the telescope environment as well as the wind speed and direction on the surroundings of the telescope can affect its image quality. See {cite}`salmon2009` for a study on the Canada-France-Hawaii Telescope (CFHT).
+
+    The temperature differences in the telescope environment as well as the wind speed and direction on the surroundings of the telescope can affect its image quality. See {cite}`salmon2009` for a study on the Canada-France-Hawaii Telescope (CFHT).
 
 ```{figure} ../../_static/psf-dome-seeing.png
 ---
@@ -126,18 +165,24 @@ Frame of the [video](https://www.youtube.com/watch?v=HU1i6JzvIzY) that shows a G
 ## Space-based
 
 - __Guiding errors__
-    - The satellite should remain perfectly still during an exposure. This is not the case and there exist a pointing error making the exposure the integration of the varyign satellite pointing. It sometimes referred as _Jitter_ or _Attitude and Orbit Control System_.
+
+    The satellite should remain perfectly still during an exposure. This is not the case and there exist a pointing error making the exposure the integration of the varyign satellite pointing. It sometimes referred as _Jitter_ or _Attitude and Orbit Control System_.
+
 - __Thermal variations__
-    - The satellite experiences time where the sun is closer or were it is eclipsed by other object. This can occasionate high temperature changes in the space telescope. These temperature variations dilate and contract the optical system making it change of state as a function of temperature. It is sometimes to as _telescope's breathing_ for its repetitive pattern due to the orbits. See {cite}`nino2007` for an HST study of temperature variations.
+
+    The satellite experiences time where the sun is closer or were it is eclipsed by other object. This can occasionate high temperature changes in the space telescope. These temperature variations dilate and contract the optical system making it change of state as a function of temperature. It is sometimes to as _telescope's breathing_ for its repetitive pattern due to the orbits. See {cite}`nino2007` for an HST study of temperature variations.
+
 - __Polarisation__
-    - Starlight can be naturally polarised by galactic foreground dust. The different polarisations that the incoming light has can interact differently with refractive elements of the optical system. This occasionates variations that are polarisation dependent. This source is currently being studied. 
+
+    Starlight can be naturally polarised by galactic foreground dust. The different polarisations that the incoming light has can interact differently with refractive elements of the optical system. This occasionates variations that are polarisation dependent. This source is currently being studied. 
 
 
 ## Detector level
 
 
 - __Undersampling and pixelation__
-    - Many detectors, specially space-based ones, are undersampled or not Nyquist sampled. Generally, several exposures of the same object with small shifts are taken in order to super-resolve the object. The pixelation effect, the fact of using a limited number of pixels also affects the PSF and will impact the shear measurements. See {cite}`high2007` for a study of pixelation effects in weak lensing. See Section 3.5 and Figure 4 from {cite}`krist2011` for an analysis of these effects on the HST telescope.
+
+    Many detectors, specially space-based ones, are undersampled or not Nyquist sampled. Generally, several exposures of the same object with small shifts are taken in order to super-resolve the object. The pixelation effect, the fact of using a limited number of pixels also affects the PSF and will impact the shear measurements. See {cite}`high2007` for a study of pixelation effects in weak lensing. See Section 3.5 and Figure 4 from {cite}`krist2011` for an analysis of these effects on the HST telescope.
 
 ```{figure} ../../_static/psf-pixelation.png
 ---
@@ -150,7 +195,8 @@ On top, the computed 336 nm HST PSF prior to pixellation. Below it is the PSF ce
 
 
 - __Charge Diffusion Effect__
-    - There is a charge diffusion of the electrons between the pixels. It depends on the depth of the substrate and on the wavelength. It varies in the field of view as seen in {cite}`krist2003` on a study for the HST mission. It can be modelled as a varying convolutional kernel.
+
+    There is a charge diffusion of the electrons between the pixels. It depends on the depth of the substrate and on the wavelength. It varies in the field of view as seen in {cite}`krist2003` on a study for the HST mission. It can be modelled as a varying convolutional kernel.
 
 ```{figure} ../../_static/psf-charge-difussion-effects.png
 ---
@@ -162,13 +208,18 @@ Surface fits to the measured WFC charge diffusion blur kernel widths shown over
 ```     
 
 - __Charge Transfer Inefficiency__
-    - This phenomenon describes the inefficiency of transfering the charge when reading CCD detectors. It can be caused by charged particle radiation damage in space-based detectors. See {cite}`rhodes2010` and {cite}`massey2009`for studies of this phenomenon.
+
+    This phenomenon describes the inefficiency of transfering the charge when reading CCD detectors. It can be caused by charged particle radiation damage in space-based detectors. See {cite}`rhodes2010` and {cite}`massey2009`for studies of this phenomenon.
+
 - __Brighter Fatter Effect__
-    - Each pixel in an camera is not independent of each other. One charge starts accumulating it modifies boundary pixel's electric fields. This occasionates a variation of the PSF size as a function of the flux and the wavelength. See {cite}`guyonnet2015` for a study and a model of the effect.
+
+    Each pixel in an camera is not independent of each other. One charge starts accumulating it modifies boundary pixel's electric fields. This occasionates a variation of the PSF size as a function of the flux and the wavelength. See {cite}`guyonnet2015` for a study and a model of the effect.
+
 - __Quantum efficiency__
-    - This effect describes the ratio of created electrons from the number of incoming photons. It depends on the wavelength and on the detector's technology.
+    This effect describes the ratio of created electrons from the number of incoming photons. It depends on the wavelength and on the detector's technology.
 
 - __Other effects__
   - Analog-to-Digital Unit non-linearity
   - Thermal noise
   - Background estimation and subtraction
+