fix docs, optimise forward model

README.md

@@ -10,15 +10,15 @@ _Probabilistic 3D Reconstruction of Spectral Line Observations._
 
 ## About
 
-p3droslo is a software library for probabilistic 3D reconstruction of astronomical spectral line observations.
-Observations of [spectral lines](https://en.wikipedia.org/wiki/Spectral_line) are indespensible in astronomy, since they encode a lot of information about the physical and chemical conditions of the medium in which they were formed.
-For instance, their narrow extend in frequency space make them very sensitive to Doppler shifts, such that their shape encodes the motion of the medium along the line of sight.
-Given a good model for the line formation process and a good inversion method, these physical and chemical properties can be retrieved.
+**p3droslo** is a software library for **probabilistic 3D reconstruction** of astronomical **spectral line observations**.
+Observations of [spectral lines](https://en.wikipedia.org/wiki/Spectral_line) are indespensible in astronomy, since they encode a wealth of information about the physical and chemical conditions of the medium in which they were formed.
+For instance, their narrow extent in frequency space make them very sensitive to Doppler shifts, such that their shape encodes the motion of the medium along the line of sight.
+As a result, given a good model for the line formation process and a good inversion method, these physical and chemical properties can be retrieved from observations.
 Currently, we mainly focus on retrieving the distributions of the abundance of the chemical species producing the line, the velocity field, and its kinetic temperature.
 
 More information about the model for [spectral line formation](https://p3droslo.readthedocs.io/en/latest/background/spectral_line_formation.html) and the [probabilistic reconstruction methods](https://p3droslo.readthedocs.io/en/latest/background/probabilistic_reconstruction.html) can be found in the [background](https://p3droslo.readthedocs.io/en/latest/background/index.html) pages.
 
-p3droslo is built on top of [PyTorch](https://pytorch.org) and benefits a lot from functionality provided by [Astropy](https://www.astropy.org).
+**p3droslo** is built on top of [PyTorch](https://pytorch.org) and benefits a lot from functionality provided by [Astropy](https://www.astropy.org).
 It is currently developed and maintained by [Frederik De Ceuster](https://freddeceuster.github.io) at [KU Leuven](https://www.kuleuven.be/english/kuleuven/index.html).
 
 

docs/src/background/probabilistic_reconstruction.ipynb

@@ -4,7 +4,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Probabilistic Reconstruction"
+    "# Probabilistic reconstruction"
    ]
   },
   {

docs/src/background/spectral_line_formation.ipynb

@@ -4,7 +4,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Spectral Line Formation"
+    "# Spectral line formation"
    ]
   },
   {
@@ -43,6 +43,7 @@
     "\\begin{equation*}\n",
     "\\Delta \\tau \\ = \\ \\Delta x \\int_{0}^{1} \\text{d}\\lambda \\ \\chi (\\lambda) .\n",
     "\\end{equation*}\n",
+    "(Note: in the implementation we included the factor $\\Delta x$ in the definition of $a$, for efficiency.)\n",
     "Using the linear interpolation scheme,\n",
     "\\begin{align*}\n",
     "a(\\lambda) \\ &= \\ (1-\\lambda) a_{0} \\ + \\ \\lambda a_{1}, \\\\\n",
@@ -104,25 +105,25 @@
     "\\end{align*}\n",
     "for the source function, $S$, and the optical depth, $tau$, this yields,\n",
     "\\begin{equation*}\n",
-    "\\Delta I \\ = \\ \\frac{1}{2\\Delta\\tau^{2}} \\Big( S_{0} \\, e^{-\\tau_{0}} \\left( e^{-\\Delta\\tau} - (1 - \\Delta\\tau) \\right) \\ + \\\n",
+    "\\Delta I \\ = \\ \\frac{1}{2\\Delta\\tau} \\Big( S_{0} \\, e^{-\\tau_{0}} \\left( e^{-\\Delta\\tau} - (1 - \\Delta\\tau) \\right) \\ + \\\n",
     "S_{1} \\, e^{-\\tau_{1}} \\left( e^{+\\Delta\\tau} - (1 + \\Delta\\tau) \\right) \\Big),\n",
     "\\end{equation*}\n",
     "where $\\Delta\\tau \\equiv \\tau_{1} - \\tau_{0}$. This expression is numerically stable as long as $\\Delta \\tau$ is not too small, but will suffer from cancellation errors otherwise.\n",
     "\n",
-    "Therefore, for $\\Delta \\tau < 10^{-2} we use the first four terms in the Taylor expansion,\n",
+    "Therefore, for $\\Delta \\tau < 10^{-2}$ we use the first four terms in the Taylor expansion,\n",
     "\\begin{equation*}\n",
-    "\\begin{split}\n",
-    "& \\frac{1}{2\\Delta\\tau^{2}} \\left(e^{-\\Delta\\tau} - (1 - \\Delta\\tau) \\right) \\ \\approx \\ \\frac{1}{2} \\ - \\ \\frac{1}{6}\\Delta\\tau \\ + \\ \\frac{1}{120}\\Delta\\tau^{2} \\ - \\  \\frac{1}{24}\\Delta\\tau^{3}, \\\\\n",
-    "& \\frac{1}{2\\Delta\\tau^{2}} \\left(e^{+\\Delta\\tau} - (1 + \\Delta\\tau) \\right) \\ \\approx \\ \\frac{1}{2} \\ + \\ \\frac{1}{6}\\Delta\\tau \\ + \\ \\frac{1}{120}\\Delta\\tau^{2} \\ + \\  \\frac{1}{24}\\Delta\\tau^{3},\n",
-    "\\end{split}\n",
+    "\\frac{1}{2\\Delta\\tau} \\left(e^{-\\Delta\\tau} - (1 - \\Delta\\tau) \\right) \\ \\approx \\ \\frac{1}{2}\\Delta\\tau \\ - \\ \\frac{1}{6}\\Delta\\tau^{2} \\ + \\ \\frac{1}{120}\\Delta\\tau^{3} \\ - \\  \\frac{1}{24}\\Delta\\tau^{4},\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\frac{1}{2\\Delta\\tau} \\left(e^{+\\Delta\\tau} - (1 + \\Delta\\tau) \\right) \\ \\approx \\ \\frac{1}{2} \\Delta\\tau \\ + \\ \\frac{1}{6}\\Delta\\tau^{2} \\ + \\ \\frac{1}{120}\\Delta\\tau^{3} \\ + \\  \\frac{1}{24}\\Delta\\tau^{4},\n",
     "\\end{equation*}"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "\n",
     "in which you can recognize the expansion of the exponential minus the first two terms.\n",
     "The implementation of this intensity increment can be found in [src/p3droslo/forward.py](https://github.com/Magritte-code/p3droslo/blob/main/src/p3droslo/forward.py)."
    ]