\documentclass[main.tex]{subfiles}
\begin{document}

\section*{Fri Dec 20 2019}

\section{Gravitational waves}

They are solutions of the linearized EFE on a given fixed background. Today we only treat GW on a Minkowski background, but we could do it for any number of other (cosmological) background.

So, we say:
%
\begin{align}
g_{\mu \nu } = \eta_{\mu \nu } + h_{\mu \nu }
\,,
\end{align}
%
with \(h_{\mu \nu } \ll 1\), so we neglect anything which is \(\mathcal{O}(h^2)\).

GW are very topical today: they could allow us to see beyond the last scattering surface, when the universe stopped being opaque to EM radiation: then, it had been transparent to gravitational radiation from a long time before.

We can do statistics to the stochastic GW background, for now however we have only detected GW from localized event.

We must start to write down the EFE from the Christoffel symbols: in general they are
%
\begin{align}
\Gamma^{\alpha }_{\mu \nu } = \frac{1}{2} g^{\alpha \lambda } \qty(g_{\lambda \nu , \mu } + g_{\mu \lambda , \nu } - g_{\mu \nu , \lambda })
\,,
\end{align}
%
but in the derivatives of the metric we only have \(\mathcal{O}(h)\) terms, so in the inverse metric we neglect all the \(\mathcal{O}(h)\) terms since their global contribution would be \(\mathcal{O}(h^2)\): so we get
%
\begin{align}
    \Gamma^{\alpha }_{\mu \nu } = \frac{1}{2} \eta ^{\alpha \lambda } \qty(h_{\lambda \nu , \mu } + h_{\mu \lambda , \nu } - h_{\mu \nu , \lambda })
\,,
\end{align}
%
and when we compute the Ricci tensor we will only keep the \(\partial \Gamma \) terms, since the \(\Gamma \Gamma \) terms are \(\mathcal{O}(h^2)\). So our expression becomes
%
\begin{subequations}
\begin{align}
R_{\mu \nu } &= \partial_{\alpha } \Gamma^{\alpha }_{\mu \nu } - \partial_{\nu } \Gamma^{\alpha }_{\mu \alpha }  \\
&= \frac{1}{2} \eta^{\alpha \lambda } \partial_{\alpha } \qty(h_{\lambda \nu , \mu } + \cancelto{}{h_{\mu \lambda , \nu }} - h_{\mu \nu , \lambda }) - \frac{1}{2} \eta^{\alpha \lambda } \partial_{\nu  }\qty(h_{\lambda \alpha , \mu } + \cancelto{}{h_{\mu \lambda , \alpha }} - h_{\mu \alpha , \lambda })  \\
&= \frac{1}{2} \eta^{\alpha \lambda } \qty(h_{\lambda \nu , \mu \alpha } - h_{\mu \nu , \alpha \lambda } - \frac{1}{2} h_{\lambda \alpha , \mu \nu } - \frac{1}{2} h_{\lambda \alpha , \mu \nu } + h_{\mu \alpha , \lambda \nu } )
\,,
\end{align}
\end{subequations}
%
where we split a term in two in order to collect, so we get
%
\begin{align}
R_{\mu \nu } = \frac{1}{2} \eta^{\alpha \lambda } \partial_{\mu } \qty(h_{\lambda \nu , \alpha } - \frac{1}{2} h_{\lambda \alpha , \nu })
+ \frac{1}{2} \eta^{\alpha \lambda } \partial_{\nu }
\qty(h_{\mu \alpha , \lambda } - \frac{1}{2} h_{\lambda \alpha , \mu })- \frac{1}{2} \eta^{\alpha \lambda } \partial_{\alpha } \partial_{\lambda } h_{\mu \nu }
\,,
\end{align}
%
where we recognize the Dalambertian \(\square = \eta^{\alpha \lambda  }  \partial_{\alpha } \partial_{\lambda }\) and the trace of \(h_{\mu \nu }\): \(h \equiv h^{\mu }_{\mu }\). So we get
%
\begin{align}
R_{\mu \nu } = - \frac{1}{2} \square h_{\mu \nu }
+ \frac{1}{2} \partial_{\mu } \qty(\partial_{\lambda } h^{\lambda }_{\nu } - \frac{1}{2} \partial_{\nu }h)
+ \frac{1}{2} \partial_{\nu }\qty(\partial_{\lambda}  h^{\lambda }_{\mu } - \frac{1}{2} \partial_{\mu} h)
\,,
\end{align}
%
so we need to solve \(R_{\mu \nu }  =0 \) with some initial conditions, since we need to solve the EFE \(G_{\mu \nu } = M_P^{-2} T_{\mu \nu } = 0\), which implies \(R_{\mu \nu }= 0 \).
The problem is that the solution is not unique: we have the freedom of choosing a gauge, since different metrics connected to each other through changes of coordinates represent the same physical scenario: \(h_{\mu \nu } \) \emph{has no direct physical meaning}.

In general we have a change of coordinates \(x \to \widetilde{x}\), and if we do this
%
\begin{align}
g_{\mu \nu } \to \widetilde{g}_{\mu \nu } = \pdv{x^{\alpha }}{\widetilde{x}^{\mu  }} \pdv{x^{\beta }}{\widetilde{x}^{\nu }} g_{\alpha \beta }
\,,
\end{align}
%
but we can only do changes of coordinates which leave the Minkowski part of the metric invariant, that is \(\widetilde{g}_{\mu \nu } = \eta_{\mu \nu } + \widetilde{h}_{\mu \nu }\). We can actually restrict ourselves to infinitesimal changes of coordinates: \(x \to \widetilde{x} = x + \epsilon \). So we do
%
\begin{align}
\pdv{\widetilde{x}^{\mu }}{x^{\alpha }} = \partial_{\alpha } \qty( x^{ \mu } + \epsilon^{\mu })
= \delta_{\alpha }^{\mu } + \pdv{\epsilon^{\mu }}{x^{\alpha }} = \delta_{\mu }^{\alpha } + \partial_{\alpha } \epsilon^{\mu }
\,.
\end{align}

Inserting this inside the inverse change of coordinates formula for the metric (the one for \(g_{\mu \nu }\) in terms of \(\widetilde{g}_{\mu \nu }\), the inverse of the one we wrote down before) we get
%
\begin{subequations}
\begin{align}
\qty(\delta^{\mu}_{\alpha } + \partial_{\alpha } \epsilon^{\mu })
\qty(\delta^{\nu }_{\beta } + \partial_{\beta } \epsilon^{\nu })
\qty(\eta_{\mu \nu } + \widetilde{h}_{\mu \nu })
&= \eta_{\alpha \beta }
+h_{\alpha \beta }  \\
\qty(\delta^{\mu }_{\alpha } \delta^{\nu }_{\beta } + \delta^{\mu }_{\alpha } \partial_{\beta } \epsilon^{\nu }
+ \delta^{\nu  }_{\beta  } \partial_{\alpha  } \epsilon^{\mu  } + \mathcal{O} (\epsilon^2))  \qty(\eta_{\mu \nu } + \widetilde{h}_{\mu \nu }) &= \eta_{\alpha \beta }  + h_{\alpha \beta }  \\
\widetilde{h}_{\alpha \beta }+
\partial_{\alpha } \epsilon_{\beta } + \partial_{\beta } \epsilon_{\alpha } &= h_{\alpha \beta }
\,.
\end{align}
\end{subequations}

This freedom of changing \(h_{\mu \nu }\) is called gauge freedom: if \(h_{\alpha \beta }\) solves the linearized EFE, then \(\widetilde{h}_{\alpha \beta } = h_{\alpha \beta } + \partial_{(\alpha } \epsilon_{\beta )}\) also does (there is a factor 2 missing, but it does not matter since \(\epsilon \) is generic and still infinitesimal).

We can solve \(R_{\mu \nu } \) together with some extra conditions: this means \emph{choosing a gauge}.
We can do this as long as, starting from a generic \(h_{\mu \nu }\), we can find a transformation \(\epsilon_{\mu }\) which leaves \(\eta_{\mu \nu } \) invariant and for which \(\widetilde{h}_{\mu \nu } = h_{\mu \nu } - \partial_{(\mu } \epsilon_{\nu )}\).

For example, a condition which cannot be imposed is \(h_{\mu \nu } =0\): if we could impose it, then it would be equivalent to have a wave or not to have it.
This can be shown from degrees of freedom: we only have 4 degrees of gauge freedom.

So when we set a condition we must prove that we \emph{can actually} do so.

Then, any gauge we are actually allowed to impose will give the same results for any physical experiment.

An electromagnetic analogy: since \(\vec{\nabla} \cdot \vec{B} = 0\), we know that we can locally choose a potential \(\vec{A} \) such that \(\vec{B} = \vec{\nabla} \times \vec{A}\). This has no intrinsic physical meaning, as any \(\vec{A}' = \vec{A} + \vec{\nabla} \lambda \) for a scalar function \(\lambda \) also gives the exact same magnetic field.
Some gauge fixing choices are \(\vec{\nabla} \cdot \vec{A} = 0\) (Coulomb), \(A_{3} = 0\) (axial).

We do, however, use potentials since they are convenient.
Analogously, \(h_{\mu \nu }\) is convenient to use but it is not measurable nor unique.

\todo[inline]{Is there a gauge-independent tensorial quantity we can define starting from \(h_{\mu \nu }\), a gravitational ``field strength''?}

The last parentheses in the expression for \(R_{\mu \nu } \) is annoying: so we choose the gauge
%
\begin{align}
\partial_{\lambda } h^{\lambda  }_{\nu } - \frac{1}{2} \partial_{\nu } h = 0
\,,
\end{align}
%
and then in this gauge we have \(R_{\mu \nu } = - \frac{1}{2}  \square h_{\mu \nu }\).

\begin{bluebox}
This gauge choice can also be written in terms of the Christoffel symbols:
%
\begin{align}
g^{\mu \nu } \Gamma^{\rho }_{\mu \nu } = 0
\,.
\end{align}
%
\end{bluebox}

There are two questions we must ask: can we actually impose this condition, and why is it called harmonic?
% The answers are yes, and because it gives a harmonic equation \(\square h_{\mu \nu } = 0\).

Let us prove the first one:
%
\begin{subequations}
\begin{align}
0 \overset{?}{=} \partial_{\lambda } \widetilde{h}^{\lambda }_{\nu } - \frac{1}{2} \partial_{\nu }\widetilde{h}
&= \partial_{\lambda } \qty( h^{\lambda }_{\nu } - \partial^{\lambda } \epsilon_{\nu } - \cancelto{}{\partial^{\nu } \epsilon_{\lambda }} ) - \frac{1}{2} \partial_{\nu } \qty(h - 2 \cancelto{}{\partial_{\lambda } \epsilon^{\lambda }})  \\
&= \partial_{\lambda } h^{\lambda }_{\nu } - \frac{1}{2} \partial_{\nu } h  - \square \epsilon_{\nu }
\,,
\end{align}
\end{subequations}
%
which can be solved by setting an \(\epsilon_{\nu }\) such that:
%
\begin{align}
  \square  \epsilon_{\nu } = \partial_{\lambda } h^{\lambda }_{\nu } - \frac{1}{2} \partial_{\nu } h
\,.
\end{align}

Notice that in this gauge \(\square \widetilde{h}_{\mu \nu } = 0 \): the wave equation, for a wave which propagates with velocity 1.

We have residual gauge freedom! As long as \(\square \widetilde{\epsilon}_{\mu} = 0\), we can still change \(h_{\mu \nu }\).
So the solution is not unique. The gauge is called harmonic because we can do harmonic residual gauge fixing.
We have 4 more degrees of freedom: we choose \(h_{0 \mu } \equiv 0\). Now we prove that this can be done.

We do not prove this in general (although it can be done), because we want to move towards a more specific example.

We then consider plane wave solutions: waves that only depend on one spatial coordinate, and are constant with respect to both other spatial coordinates.

If the direction of motion is the \(z\) axis, then the solution looks like
%
\begin{align}
h_{\mu \nu } (t, x, y, z) = C_{\mu \nu } \exp(-i (k t - k z ))
\,.
\end{align}

For a wave propagating in a generic direction we will have
%
\begin{align}
h_{\mu \nu } (t, \vec{x}) = C_{\mu \nu } \exp(i k_{\alpha } x^{\alpha })
\,,
\end{align}
%
where \(k^{\alpha } k_{\alpha } = 0\) (which comes from the Fourier expression of the Dalambertian: \(\square = -k^{\alpha } k_{\alpha }\)) is a constant vector. The scalar product in the exponent must be computed with the Minkowski metric since we are working at  first order in \(h\) and \(C_{\mu \nu }\) is already first order.
%
\begin{subequations}
\begin{align}
\square h_{\mu \nu } &= \eta^{\alpha \beta } \partial_{\alpha } \partial_{\beta } \qty(C_{\mu \nu } \exp(i k_{\alpha }x^{\alpha }))  \\
&= \eta^{\alpha  \beta } C_{\mu \nu } \partial_{\alpha } \qty(\exp(i k_{\alpha } x^{\alpha }) \partial_{\beta } \qty(i k_{\alpha } x^{\alpha }))  \\
&= \eta^{\alpha  \beta } C_{\mu \nu } i k_{ \beta } \partial_{\alpha } \qty(\exp(i k \cdot x )) \\
&= \eta^{\alpha \beta } (i k_{ \beta }) (ik_{\alpha }) C_{\mu \nu } \exp(i k \cdot x) = - k^2 C_{\mu \nu } \exp(i k \cdot x)
\,,
\end{align}
\end{subequations}
%
then we see that this is always zero as long as \(k^2=0\), which is what we impsed for \(k^{\alpha }\).

What is the particle analog of a gravitational wave? it is a so-called \emph{graviton}.
For it we will have a momentum operator \(\vec{p} = - i \hbar \vec{\nabla}\) and an energy operator \(E = i \hbar \partial_{t}\). We can apply both of these to our wave to find that it is an eigenvector of both:
%
\begin{align}
E h_{\mu \nu } = i \hbar \partial_{t} \qty(C_{\mu \nu } \exp(i k \cdot x))= \hbar k^{t} h_{\mu \nu }
\,,
\end{align}
%
and we know that \(k^{t} = \abs{\vec{k}}\).

If we compute the momentum, analogously we find \(\vec{p} h_{\mu \nu } = \hbar \vec{k}\).

This means that \(p \cdot p  = 0\), which means that the mass of the graviton is equal to zero.

\end{document}


\section{Related Work}

Quantum algorithms have drawn much attention and there are many problems for which specialized algorithms have been designed, such as searching \cite{Grover96} \cite{BoyerBHT98}, Element Distinctness \cite{Santha08}, Minimum-Finding \cite{DurrH96} and many others (e.g., \cite{DurrHHM04} \cite{BerzinaDFLS04}, \cite{Zhangthesis06}).

Evolutionary Algorithms (EA) aim not to look specifically into a certain problem, but rather provide some wide-applicable algorithms which may be used if it is not possible to analyze the problem thoroughly - either because the problem complexity is too high, or because the algorithm designer can not spend much time and resources into investigating the problem.

Quantum Evolutionary Algorithms (QEA) have the same objective, but are algorithms that run on a quantum computer. They are not to be confused with Quantum-Inspired Evolutionary Algorithms (QIEA), which use ideas from quantum computing, but are algorithms that run on a classical machine.

It is difficult to test QEAs in practice, because quantum computers do not exist apart from proof-of-concept machines. It is possible to run QEAs by simulating a quantum computer on a classical machine (e.g., \cite{SpectorBBS99}), but this approach is cumbersome since such simulations have exponential running time.

Santha and Szegedy have shown \cite{SanthaS09} that quantum and classical query complexities of local search are polynomially related. Moreover, Magniez et al. have shown \cite{MagniezNRS09} that the speed-up in terms of hitting times is exactly quadratic for a large class of algorithms (``quantum random walks'' in the sense of \cite{Szegedy04} - note that there are different notions of quantum random walks, cf. \cite{MagniezNRS07} for an overview).

For some classes of local search problems, more specific and superior algorithms are available: In particular for search spaces with transition probabilities that are symmetric \cite{Szegedy04}, or ergodic \cite{MagniezNRS07}.

In our paper, we do not prove lower bounds. Conjectures of Aaronson \cite{Aaronson06} imply that the EA-algorithms considered in our paper are asymptotically optimal.  He has shown his conjecture for hypercubes, and Zhang \cite{Zhang06} has generalized the result to product graphs.

%\cite{MagniezNRS09}, \cite{Aaronson06}, \cite{DurrHHM04},
%\cite{DurrH96}, \cite{DurrHHM06}, \cite{HoyerNS02}, \cite{SanthaS04},
%\cite{MagniezNRS07}, \cite{Santha08}, \cite{SanthaS09},
%\cite{SpectorBBS99}, \cite{Zhang06}, \cite{LiR07}, \cite{ZhouZHW05},
%\cite{NarayananM96}, \cite{HanK02}, \cite{LiLR07}, \cite{XiaoYLYZ08},
%\cite{ZhangZPW08}, \cite{AraujoNM08}, \cite{MahdabiJA08},
%\cite{ZhangGW08}, \cite{LiSGS09}, \cite{WangWF09}, \cite{XingJBLQW09},
%\cite{ZhangS06}, \cite{Ambainis06}, \cite{Ambainis08},
%\cite{AmbainisSW09}, \cite{AmbainisCGT09}, \cite{HoyerNS01}


\input{header.tex}
\begin{document}

\frame{\maketitle}

%\begin{frame}
%\tableofcontents
%\end{frame}

\begin{frame}{ $IV$ curve: $0$ to $\SI{130}{\volt}$}

  \begin{columns}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./iv_curve_reverse_bias_pixel_detectors.png}
       \caption{ Theoretical $IV$ curve \cite{pixel_detectors}. }
       \label{ fig: iv_curve_theoretical}
     \end{figure}
   \end{column}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./iv_curve_0_130_V.pdf}
       \caption{Measured $IV$ curve. }
       \label{ fig: iv_curve_measured}
     \end{figure}
   \end{column}

  \end{columns}

\end{frame}

\section{Laserwidth}

\begin{frame}{ Measurment - $\SI{5}{\milli\meter}$ }

  \begin{figure}
    \centering
    \includegraphics[width=\textwidth]{./5_mm_measurment_plot.pdf}
    \caption{Measured Laserbeam }
    \label{ fig: 5_mm }
  \end{figure}

\end{frame}

\begin{frame}{ Fit - $\SI{5}{\milli\meter}$ }
  \begin{columns}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./5_mm_erorbar_plot_col.pdf}
       \caption{ Fit Columns }
       \label{ fig: iv_curve_theoretical}
     \end{figure}
   \end{column}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./5_mm_erorbar_plot_row.pdf}
       \caption{ Fit Rows }
       \label{ fig: iv_curve_measured}
     \end{figure}
   \end{column}

  \end{columns}

\end{frame}

\begin{frame}{ Measurment - $\SI{9}{\milli\meter}$ }

  \begin{figure}
    \centering
    \includegraphics[width=\textwidth]{./9_mm_measurment_plot.pdf}
    \caption{Measured Laserbeam }
    \label{ fig: 5_mm }
  \end{figure}

\end{frame}

\begin{frame}{ Fit - $\SI{9}{\milli\meter}$ }
  \begin{columns}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./9_mm_erorbar_plot_col.pdf}
       \caption{ Fit Columns }
       \label{ fig: iv_curve_theoretical}
     \end{figure}
   \end{column}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./9_mm_erorbar_plot_row.pdf}
       \caption{ Fit Rows }
       \label{ fig: iv_curve_measured}
     \end{figure}
   \end{column}

  \end{columns}

\end{frame}

\begin{frame}{ Measurment - $\SI{15}{\milli\meter}$ }

  \begin{figure}
    \centering
    \includegraphics[width=\textwidth]{./15_mm_measurment_plot.pdf}
    \caption{Measured Laserbeam }
    \label{ fig: 5_mm }
  \end{figure}

\end{frame}

\begin{frame}{ Fit - $\SI{15}{\milli\meter}$ }

  \begin{columns}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./15_mm_erorbar_plot_col.pdf}
       \caption{ Fit Columns }
       \label{ fig: iv_curve_theoretical}
     \end{figure}
   \end{column}

   \begin{column}{0.48\textwidth}
     \begin{figure}
       \centering
       \includegraphics[width=1.05\textwidth]{./15_mm_erorbar_plot_row.pdf}
       \caption{ Fit Rows }
       \label{ fig: iv_curve_measured}
     \end{figure}
   \end{column}

  \end{columns}

\end{frame}

\begin{frame}{ First Result }

     \begin{figure}
       \centering
       \includegraphics[width=0.8\textwidth]{./perfect_height.pdf}
       \caption{ Sigma against height }
       \label{ fig: iv_curve_theoretical}
     \end{figure}

\end{frame}

\end{document}


\chapter{Introduction}
\section{Introduction}
The \textit{algorithm selection} (AS) problem presents the question, "Given a set of algorithms, how to choose one that best solves a given problem?" AS is best applied in solving intractable problems, whose solutions often require brute-force or heuristic approaches. An effective method of performing AS has broad implications on solving real-world problems which often are intractable in nature.

\section{Problem Background}
A survey of AS techniques \citep{kotthoff2016algorithm} reported a majority of approaches that used supervised learning in training an AS model. Can reinforcement learning (RL) techniques be used? Recently, RL emerged as a superior approach to learning as demonstrated in applications such as automatic design of neural network architecture \citep{zoph2016neural} and board game AI exhibiting superhuman performance \citep{silver2016mastering}. This study explores the potential of RL in training AS models with superior performance. The AS models in this study focus on solving a set of hard computational tasks called subgraph isomorphism problems. This is an NP-complete problem under graph theory where the objective is to determine if a graph contains a pattern of a much smaller graph. A recent study tackled this problem by building and training an AS model using pairwise random forest regression \citep{kotthoff2016portfolios}. This study tries to improve upon the results of the previous study by proposing RL as an alternative to training AS models.

\section{Problem Statement}
This study addresses how reinforcement learning can be applied to algorithm selection for subgraph isomorphism problems. Specifically, it attempts to answer the following questions:
\begin{itemize}
	\item How can the algorithm selection problem be viewed from the perspective of reinforcement learning?
	\item Can reinforcement learning be used to train an algorithm selection model in solving subgraph isomorphism problems?
	\item Does an algorithm selection model perform better when trained using reinforcement learning as compared to supervised learning?
\end{itemize}

\section{Research Objectives}
This study aims to use reinforcement learning in training an algorithm selection model for subgraph isomorphism problems. Specifically, it attempts to meet the following objectives:
\begin{itemize}
	\item To prepare the subgraph isomorphism dataset for training the algorithm selection model.
	\item To implement a reinforcement learning algorithm for training an algorithm selection model.
	\item To evaluate whether reinforcement learning can improve the performance of the algorithm selection model as compared to supervised learning.
\end{itemize}

\section{Scope of the Study}
The constraints observed in this study are outlined below:
\begin{itemize}
	\item The AS models are trained and tested using the subgraph isomorphism dataset generated from a previous related study by Kotthoff \citep{kotthoff2016portfolios}
	\item The characteristics of the dataset impose restrictions on how training and testing can proceed.
	\begin{itemize}
		\item The problems focus only on subgraph isomorphism problems.
		\item The set of algorithms are predefined and fixed.
		\item The model can only be trained to match one algorithm to one problem at a time.
	\end{itemize}
	\item Only one RL algorithm is considered in this study.
\end{itemize}

\section{Significance of the Study}
This study contributes to the following:

\begin{itemize}
	\item Investigates the feasibility of training algorithm selection models using reinforcement learning.
	\item Provides insights on how reinforcement learning algorithms can be applied to algorithm selection.
	\item Introduces the potential of reinforcement learning in training effective algorithm selection models.
\end{itemize}


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\begin{document}

\title{Homework 21, Section 4.3: 3, 4, 7, 10, 15, 25}
\author{Alex Gordon}
\date{\today}
\maketitle
\section*{Homework}
\subsection*{3.}
This set does not form a basis for $R^3$. The set is linearly dependent and does not span $R^3$
\subsection*{4.}
The determinant equals 1, so these vectors are linearly independent. These vectors form a basis for $R^3$.
\subsection*{7. }
This set does not form a basis for $R^3$ because it is linearly independent and  $m \geq n$.
\subsection*{10.}
The RREF is $ \begin{bmatrix}  1 & 0 & 0 & 2 & 3 & 0 \\ 0 & 1 & 0 & -1 & -2 & 0 \\ 0 & 0 & 1 & 0 & -2 & 0  \end{bmatrix} $. The span is then $ \begin{bmatrix}  -2 \\ 1 \\ 0 \\ 1 \\ 0  \end{bmatrix} $, $ \begin{bmatrix}  -3 \\ 2 \\ 2 \\ 0 \\ 1  \end{bmatrix} $
\subsection*{15.}
The RREF is $ \begin{bmatrix}  1 & 0 & 2 & 2 & 3 \\ 0 & 1 & -2 & -1 & -1 \\ -2 & 2 & -8 & 10 & -6 \\ 3 & 3 & 0 & 3 & 9   \end{bmatrix} $. This reduces to \\
$ \begin{bmatrix}  1 & 0 & 2 & 2 & 3 \\ 0 & 1 & -2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}    $
\subsection*{25.}
Yes it is a basis a typical element in H is $sv1 + (t-s)v2 + sv3$.

we can re-write this as $s(v1+v3) + (t-s)(v2), so \{v1+v3,v2\}$ is a basis for H.

the problem is that the span of \{v1,v2,v3\} is much bigger than H, it is all of $R^3$.
This means that just because it is linearly independent, it is not the basis for H. Since the basis of the subspace is not isomorphic to R or $R^2$, it's too big for the span.

\end{document}


\typeout{new file: Band_Models_Chapter.tex}

\chapter{Narrow Band Models and Approximations}
\label{chap:SNB}

This chapter briefly presents the models and their formulations that are used in RadCal. The reader seeking further details on the rather complex theory behind these models is invited to read the exhaustive monographs from Penner~\cite{Penner1959}, Tien~\cite{Tien1968}, and Modest~\cite{Modest2013}, the reviews from De Ris~\cite{DeRis1979} and from Viskanta and Menguc~\cite{vm_87}, and the report from Ludwig \textit{et al.}~\cite{Ludwig1973}. Most of the equations used in RadCal can be found in the latter.

RadCal was designed for uses at moderate gas pressure (near atmospheric pressure) and for ambient to high gas temperatures (typically from around 300~K to 2500~K). As such, collision broadening is the main broadening mechanism considered. However, the effects of Doppler line broadening, which are common at low pressure (typically less than 0.01 atm) and/or at high temperature, are still accounted for in the overall expression of narrow-band transmissivity. The Doppler broadening effect is accounted using the method presented in Ludwig \textit{et al.}~\cite{Ludwig1973}. The Voigt profile, commonly used in atmospheric sciences~\cite{Chu1994}, is not considered here. Unlike the Lorentz profile the Voigt profile does not have a closed analytical expression. Effects of scattering or reflection are not included in RadCal; as such, RadCal considers only absorption and emission.

RadCal can be used to model homogeneous and non-homogeneous conditions. Following Ludwig~\cite{Ludwig1973}, the Curtis-Godson approximation is then used with the assumption of Single Line Group. In the case of a mixture of different radiative participating gases, the contribution for each species is calculated independently and their contribution on the total spectral optical depth is assumed to be additive. Finally, it is important to recall the important assumption of local thermodynamic equilibrium used in establishing the Radiative Transfer Equation as solved by RadCal. This justifies consideration of the Planck blackbody distribution law for the emission term.

\section{Introduction}

\subsection{Radiative Transfer Equation (RTE)}
The expression of the equation of transfer in non-scattering, participating media in local thermodynamic equilibrium is recalled. This equation is the mathematical foundation of RadCal. The Eulerian point of view is adopted here. The assumption of local thermodynamic equilibrium has for consequences that the spectral blackbody emission can be modeled with the Planck blackbody distribution - denoted from here and in the rest of this manuscript $I_b$, and that the Kirchoff law applies at the spectral level, \textit{i.e.} for a monochromatic light the absorption and emission coefficients are equal. The amount of radiative energy emitted by a participating medium of local thermodynamic temperature $T$, over an infinitesimal spectrum $\d\nu$, in an arbitrary direction $\hat{\textbf{s}}$, and along an infinitesimal length $\d s$, is expressed by:
\begin{equation}\label{eq:emission}
 j_{e,\nu} \d s \d\nu = \kappa_{\nu} I_{\rm b,\nu}(T)\d\nu \d s
\end{equation}
where $j_{e,\nu}$ is the emission at the frequency $\nu$, $\kappa_{\nu}$ is the absorption coefficient at the considered frequency, $I_{b,\nu}$ denotes the spectral intensity emitted by a blackbody of temperature $T$. Naturally, the absorption coefficient $\kappa_{\nu}$ is proportional to the amount of local participating molecule.

Note: it is common in spectroscopy to use $B_b$ and $\tilde{\nu}$ as the Planck function and wavenumber, respectively. However in the continuation of the previous work on RadCal, we keep the previously used nomenclature. In the rest of the manuscript, $\omega$ denotes the wavenumber. A common units employs to characterize radiative spectrum is the wavelength, denoted $\la$. Wavelength $\la$ represents the distance traveled during one cycle when propagating at the speed of light in vacuum. When dealing with infrared radiation, its units are commonly expressed in ${\rm \mu m}$. Wavenumber $\om$ is the reciprocal of the wavelength. It represents the number of cycles per unit of length. It is expressed in units of ${\rm cm^{-1}}$. It is convenient to use wavelength $\la$ or wavenumber $\om$ instead of frequency $\nu$ when dealing with thermal radiation. Wavenumber and wavelength are related to frequency through:
\be
 \la = c/\nu \: \: \rm{and} \: \: \om = \nu /c,
\ee
 and one can easily switch from wavenumber in units of ${\rm cm^{-1}}$ to wavelength in units of ${\rm \mu m}$ with:
\be
  \la \; {\rm \mu m}  \; = 10000/\om \; \; {\rm cm^{-1}}.
\ee

Similarly to Eq.~\ref{eq:emission}, an expression characterizing the amount of energy of an incident beam, of propagating direction $\hat{\textbf{s}}$ absorbed by local participating molecules over an infinitesimal spectrum $\d\nu$ and along an infinitesimal length $\d s$ can be derived:
\begin{equation}\label{eq:absorption}
  j_{a,\nu} \d\nu \d \rm s = - \kappa_{\nu} I_{\nu} \d\nu \d \rm s
\end{equation}
where $I_{\nu}$ is the spectral intensity at frequency $\nu$ of the incident beam of direction $\vec{\textbf{s}}$.

Using Eqs.~\ref{eq:emission} and \ref{eq:absorption}, the change in the spectral intensity of an incident beam of direction $\hat{\textbf{s}}$ penetrating a participating medium over a distance $\d s$ is then expressed by:
\be
\d I_{\nu}\d \nu = (j_{e,\nu} + j_{a,\nu}) \d s \d\nu.
\ee
The local Radiative Transfer Equation (neglecting scattering) is then expressed as:
\begin{equation}\label{eq:localRTE}
 \dfrac{\d I_{\nu}}{\d s}\d \nu = \kappa_{\nu} (I_{\rm b,\nu}(T) - I_{\nu}) \d \nu.
\end{equation}
Equation~\ref{eq:localRTE} is a first-order ordinary differential equation; it is assumed here that the speed of light is very large compared to local time and length scales involved. The functional form of the spectral intensity $I_{\nu}(s)$ at an arbitrary depth $s$ within the participating medium, assuming that the participating medium starts at $s=0$, is expressed as:
\begin{equation}\label{eq:RTE_Kappa}
I_{\nu}(s) =  I_{\nu}(0) \exp\left(-\displaystyle\int_{0}^{s}{\kappa_{\nu}\d s}'\right) + \displaystyle\int_0^{s}{I_{\rm b,\nu}\left(T(s')\right)\exp\left(-\displaystyle\int_{s'}^{s}{\kappa_{\nu}\d s''}  \right)\kappa_{\nu} \d s'}.
\end{equation}
Note that $\kappa_{\nu}$ is a function of the local thermodynamic conditions and hence is an implicit function of $s$.

The first right-hand side term of Eq.~\ref{eq:RTE_Kappa} represents the fraction of the incident spectral intensity that is transmitted unto the depth $s$. The second right-hand side term corresponds to the contribution of the local emission and accounts for self-absorption.
The spectral transmissivity, denoted $\tau(\nu; 0 \rightarrow s)$, is defined as the fraction of the spectral energy of frequency $\nu$ incident upon a participating medium which is transmitted by it. It can be seen from Eq.~\ref{eq:RTE_Kappa}, that an explicit formulation of the spectral transmissivity is:
\begin{equation}\label{eq:tau_definition}
 \tau(\nu; 0 \rightarrow s) = \exp\left(-\displaystyle\int_{0}^{s}{\kappa_{\nu}\d s'}\right).
\end{equation}
It can be shown that:
\be
\exp\left(-\displaystyle\int_{s'}^{s}{\kappa_{\nu}\d s''}\right) \kappa_{\nu} = \dfrac{\partial \tau}{\partial s'}(\nu; s' \rightarrow s).
\ee
Therefore, the RTE can be rewritten as:
\be\label{eq:RTE_Final}
I_{\nu}(s) =  I_{\nu}(0) \tau(\nu; 0 \rightarrow s) + \displaystyle\int_0^{s}{I_{\rm b,\nu}\left(T(s')\right)\dfrac{\partial \tau  }{\partial s'}(\nu; s' \rightarrow s) \d s'}.
\ee

Equation~\ref{eq:RTE_Final} represents the fundamental governing equation for the radiative transfer into a non-scattering, non-reflective participating medium at local thermodynamic equilibrium. This equation is valid regardless whether the model is homogeneous or not. It is of course assumed here that the spectral distribution of the incident intensity, $I_{\nu}(s=0)$, is known. In RadCal, $I_{\nu}(0)$ is the spectral distribution of a blackbody at a temperature defined by the user. This temperature is referred to as the \textit{wall} temperature, denoted $T_w$.

Equation~\ref{eq:RTE_Final} can be used to calculate the incident energy at the location $s$ comprised in a spectral range $\Delta \nu$:

\be\label{eq:RTE_Final_int}
\displaystyle\int_{\Delta \nu }{I_{\nu}(s) \d \nu}  = \displaystyle\int_{\Delta \nu}{I_{\nu}(0) \tau(\nu; 0 \rightarrow s)}\d \nu + \displaystyle\int_{\Delta \nu}{\displaystyle\int_0^{s}{I_{\rm b,\nu}\left(T(s')\right)\dfrac{\partial \tau  }{\partial s'}(\nu; s' \rightarrow s) \d s'} \d \nu}.
\ee
Some simplifications can be made if one assumes that $\Delta \nu$ is small enough so that $I_{\nu}$ and $I_{\rm b,\nu}$ can satisfactorily be assumed constant over $\Delta \nu$. In this case, Eq.~\ref{eq:RTE_Final_int} can be written as:
\be
I_{\nu_0}(s)  = {I_{\nu_0}(0) \bar{\tau}(\nu_0; 0 \rightarrow s)} + {\displaystyle\int_0^{s}{I_{\rm b,\nu_0}\left(T(s')\right)\dfrac{\partial \bar{\tau}  }{\partial s'}(\nu_0; s' \rightarrow s) \d s'}},
\ee
where $\nu_0$ is the center of the narrow-band $\Delta \nu$ and with
\be
\bar{\tau} = \dfrac{1}{\Delta \nu}\displaystyle\int_{\Delta \nu}{\tau \d \nu} = \dfrac{1}{\Delta \nu} \displaystyle\int_{\Delta \nu}{\exp \left( -\displaystyle\int{\kappa_{\nu} \d s'} \right) \d \nu}.
\ee
It is very important, in the narrow-band framework, to note that in general:
\be\label{eq:NB_frame_work}
\bar{\tau} \neq \exp\left(-\displaystyle\int{\bar{\kappa}_{\nu} \d s'} \right),
\ee
where the mean absorption coefficient $\kappa_{\nu}$ is defined as the average of the spectral absorption coefficient over the spectral range $\Delta \nu$:
\be
\bar{\kappa}_{\nu} = \dfrac{1}{\Delta \nu} \displaystyle\int_{\Delta \nu}{{\kappa}_{\nu} \d \nu}.
\ee
Equation~\ref{eq:NB_frame_work} arises from:
\be
\displaystyle\int_{\Delta}{\exp f(x) \d x} \neq  \exp\left(\displaystyle\int_{\Delta}{f(x) \d x} \right),
\ee
where the equality is only verified for some particular functions and integration domain.

One of the main difficulties in calculating the incident radiative intensity over a given spectral range is the rapid variation of $\kappa_{\nu}$ with $\nu$ for all gas phase species. Indeed, the spectroscopic study of participating species in gas-phase shows that their infrared spectrum consists of millions of narrow peaks. The discrete nature of the infrared spectrum is a direct consequence of the quantification of the energy associated with a species molecular vibration and rotation motions. Figure~\ref{fig:H2O_Detailed_spectrum} below plots the spectral absorption coefficient for $\rm H_2O$ in the spectral range 1050 -- 1100~cm$^{\rm -1}$. This spectrum was generated using $\rm H_2O$ line specifications from the HITRAN 2012 database \cite{Rothman2013}.

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/H20_Line_Strength.png}
\end{center}
 \caption{Spectral $\rm H_2O$ absorption coefficient $\kappa_{\om}$ for the spectral range 1000 -- 1050~cm$^{\rm -1}$. This synthetic spectrum calculated using HITRAN 2012 line specifications assumes a temperature of 296~K and 10\% of water in a total pressure of 1~atm.\label{fig:H2O_Detailed_spectrum}}
\end{figure}

\subsection{Planck blackbody distribution law}

The Planck blackbody distribution law, sometime called the Planck function, but the term is misleading as it is a distribution, describes the spectral distribution of the equilibrium rate of radiant energy emitted from a blackbody at temperature $T$. Its expression is given, as a function of frequency $\nu$ and for an unit of solid angle, by:
\begin{equation}\label{eq:Plank_freqency}
I_{\rm b,\nu}(T) = \displaystyle\frac{2 h\nu^3}{c^2}\displaystyle\frac{1}{\exp\left(\displaystyle\frac{h\nu}{k_{\rm B}T}\right)-1}.
\end{equation}
Here, $h$ is the Planck constant ($6.626 \times 10^{-34}$~J$\cdot$s), and $k_{\rm B}$ is the Boltzmann constant ($1.381 \times 10^{-23}$~J/K). $I_{\rm b,\nu}(T)$ is in units of $\rm{W/m^{2}/str/s^{-1}}$.

In terms of wavenumber, the Planck blackbody distribution law, $I_{\rm b,\om}$, is written:
\be\label{eq:Planck_WN}
I_{\rm b,\om}(T) = \dfrac{2 \, h \, c^2 \, \om^{3}}{\exp\left(\dfrac{h \, c\, \om}{k_{\rm B} \, T }\right)-1} .
\ee
$I_{\rm b,\om}(T)$ is in units of $\rm{W/m^2/str/m^{-1}}$; the wavenumber $\om$ in Eq.~\ref{eq:Planck_WN} is in units of $\rm{m^{-1}}$.

The user who wishes to express the Planck blackbody distribution law as a function of wavelength should take caution when performing the change of variables. One should start by expressing that the radiant energy emitted at a wavelength $\la$ over an infinitesimal spectral range $\d\la$ is the same as the radiant energy emitted at the corresponding wavenumber $\om$ over an infinitesimal spectral range $\d\om$:
\be \label{eq:Planck_WN_WL}
I_{\rm b,\la}(T) \d \la = -I_{\rm b,\om}(T)\d \om,
\ee
the negative sign is introduced because $\om$ is the reciprocal of $\la$. Since $\la$ = 1/$\om$, it comes:
\be
\dfrac{\d \la}{\d \om} = -\dfrac{1}{\om^2}.
\ee
Equation \ref{eq:Planck_WN_WL} can be rewritten, after the appropriate change of variables:
\be \label{eq:Planck_WL}
I_{\rm b,\la}(T) = \dfrac{2 \, h \, c^2}{\la^5}\dfrac{1}{\exp\left(\dfrac{h \, c}{\la \, k_{\rm B} \, T }\right)-1},
\ee
where $I_{\rm b,\la}$ is in units of $\rm{W/m^2/str/m}$; the wavelength $\la$ in Eq.~\ref{eq:Planck_WL} is in unit of $\rm m$.

\begin{figure}
\begin{center}
 \includegraphics[width=6.5in]{Figures/Black_body.png}
\end{center}
 \caption{Normalized spectral Blackbody distributions at T = 500~K (blue line) and T = 1800~K (red line) with the wavelength in units of $\mu m$ (left) and with the wavenumber in units of $\rm cm^{-1}$ (right). \label{fig:bb_spectrum}}
\end{figure}

Figure~\ref{fig:bb_spectrum} plots profiles of normalized spectral Blackbody distribution for temperatures of 500~K and 1800~K using wavelength and wavenumber. It can be seen that an increase of temperature will shift the mode of the distribution toward lower wavelength and higher wavenumber. Figure~\ref{fig:bb_spectrum} indicates that the shape of the distribution is not invariant with the units chosen to characterize the spectrum. Indeed, while an elevation of blackbody temperature corresponds to a narrowing of the normalized distribution when using wavelength, the same elevation of blackbody temperature leads to a spreading of the profile when using wavelength. Finally, an important remark is that the location of the distribution mode (corresponding to the location of maximum emission) does not relate between wavelength and wavenumber. This location can be calculated using the Wien's displacement law. Using wavelength, the Wien's displacement law is expressed by:
\begin{equation}\label{eq:Wien_WL}
\la_{\rm max} T = 2898 \; \mu \rm m . \rm K,
\end{equation}
while using wavenumber, the Wien's displacement law is expressed by:
\begin{equation}\label{eq:Wien_WN}
 \dfrac{\om_{\rm max}}{T} = 1.961 \; \rm cm^{-1}.K^{-1}.
\end{equation}
Using Eq.~\ref{eq:Wien_WL}, the maximum of emission at 500~K is located at $\la_{\rm max} = \rm 5.8 \; \mu m$. The same calculation using Eq.~\ref{eq:Wien_WN} gives $\om_{\rm max} =  980 \; \rm cm^{-1}$. It is crucial to note that $10000/\la_{\rm max} = 1725 \; \rm cm^{-1} \; \neq \om_{\rm max}$. Hence caution must me taken when converting spectral variable from wavenumber to wavelength and vice versa. In RadCal, the wavenumber is the variable of choice for spectral quantities.

\section{Single line emission}
This section briefly recalls the characteristics of a single line emission. The expression for an isolated line with pressure broadening mechanism (also called Lorentz lines) is recalled along with the expression of lines broadened by Doppler effects. Note that RadCal mixes both lines expression in the calculation of the RTE. Hence, it is important to recall some of their properties. The underlying mechanisms responsible for the emission and/or absorption of electromagnetic radiation by a molecule or atom are not recalled here as they are out of the scope of this work. It is just recalled here that the discrete nature of a species spectrum in the infrared region is a direct consequence of the discretization of the vibrational and rotational energy level admissible for a given species. The location and intensity of these lines can be obtained from the solution of the time-dependent Schr\"{o}dinger wave equation of a molecule with an incident radiation field. The lines location and intensity depend on the molecule geometry, mass, and its electric dipole moments. For further details, see Chapter 7 of Penner, Ref.~\cite{Penner1959}; Herzberg Ref.~\cite{Herzberg1949}; Chapter 11 of Modest, Ref.~\cite{Modest2013}; and Tien monograph Ref.~\cite{Tien1968}.

\subsection{Lorentz lines}
A spectral line is never truly monochromatic as different line broadening mechanisms are present. The most fundamental one is the natural line broadening which is a consequence of the Heisenberg's uncertainty principle. While this effects is always present, it is usually omitted in engineering applications as the collision broadening and the Doppler broadening mechanisms are much more important.

The collision broadening mechanism originates from disruptions during the emission or absorption of energy due to the collision between molecules. The broadening of the line increases as the collision frequency increases, hence as the local pressure is increased. The shape of such broadened lines can be calculated using the electron theory of Lorentz or from quantum mechanism. The shape of the line is given by:
\begin{equation}\label{eq:lorentz_line}
 \kappa_{\om} = \dfrac{S}{\pi}\dfrac{\gamma_c}{\left( \om - \om_o\right)^2 + \gamma_c^2},
\end{equation}
where $\gamma_c$ is the collision broadening half-width at half the maximum (HWHM), $\omega_0$ is the wavenumber at the line center, and $S$ is called the line intensity or line strength and is defined as:
\begin{equation}
 S = \displaystyle\int{\kappa_{\om} \d \om}.
\end{equation}
Note that $S$ is a function of the temperature alone while $\gamma_c$ is a function of the temperature, pressure, and mixture composition. Because the effects of collision depend on the molecular diameters of the colliding species, the quantity $\gamma_c$ varies with the collisional partner. Among the collision broadening, distinction is made between the foreign gas broadening, \textit{i.e.} due to collisions among dissimilar species, and self-broadening, \textit{i.e} due to collisions among like species. The self-broadening collisions can be further differentiated between resonant and non-resonant collisions. Resonant collisions are more effective than non-resonant collisions but they have a different temperature dependence than non-resonant collisions. Non-resonant collisions are similar to collisions with a foreign gas. In RadCal, when not tabulated, the Lorentz HWHM $\gamma_{c,i}$ of a given species $i$ belonging to a mixture, is calculated according to:
\begin{equation}\label{eq:gamma_L}
 \gamma_{c,i} = \displaystyle\sum_{j} \gamma_{c0,(i,j)} \left(\dfrac{P_j}{P_0}\right)\displaystyle\sqrt{\dfrac{T_0}{T}} + \gamma^*_{c0,i}\dfrac{P_i}{P_0}\left(\dfrac{T_0}{T}\right),
\end{equation}
where $\gamma_{c0,(i,j)}$ denotes the value of non-resonant collision broadening HWHM with species $j$ at conditions of standard pressure (denoted $P_0$) and temperature (denoted $T_0$), and $\gamma^*_{c0,i}$ demotes the value of resonant self-collision broadening HWHM at conditions of standard pressure and temperature. Note that Eq.~\ref{eq:gamma_L} first right-hand side term includes the effects of foreign gas collision and non-resonant self-collision.

Profile of a normalized Lorentz line ($\kappa_{\omega} (\pi \gamma_c)/S$ as a function of $(\omega-\omega_0)/\gamma_c $) is plotted in Fig.~\ref{fig:Lorentz_line} together with the profile of a normalized Doppler line.

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/Lorentz_Doppler.png}
\end{center}
 \caption{Profiles of a normalized Lorentz line (in red) and a normalized Doppler line (in blue). \label{fig:Lorentz_line}}
\end{figure}

\subsection{Doppler lines}
The motion of radiating particles in the line of sight may change the apparent frequency due to the Doppler effect. The apparent frequency increases when the particle moves toward the observer and decreases when it moves away. The Doppler line profile is calculated considering the Maxwell velocity distribution (which is a consequence of the local thermodynamic equilibrium assumption). The absorption coefficient is then expressed as:
\begin{equation}
\kappa_{\omega} = S \dfrac{\displaystyle\sqrt{\rm ln(2) }}{ \gamma_D \displaystyle\sqrt{\pi} } \exp{\left[- \rm ln(2) \left( \dfrac{\omega-\omega_0}{\gamma_D} \right)^2 \right]},
\end{equation}
where $S$ and $\omega_0$ are the line strength and the wavenumber at the line center, respectively. The quantity $\gamma_D$ is the Doppler HWHM. It is calculated from:
\begin{equation}\label{eq::DopplerHWHM}
  \gamma_D = \displaystyle\frac{\omega_0}{c} \displaystyle\sqrt{\rm{ ln(2)} \displaystyle\frac{2 k_B T}{m} },
\end{equation}
where $m$ is the mass of the radiating species. Note that the Doppler HWHM depends linearly on the wavenumber; it increases with elevated wavenumber.
The Doppler broadening mechanism is important at high temperature ($T > 2000\;K$) \cite{Modest2013} and/or low pressure, typically lower than 0.01 atm.
In RadCal both broadening mechanisms are included in the calculation of the RTE; see Section~\ref{sec:RTE_NB}.

\subsection{Emission from an isolated line}
This subsection recalls the results of the emission and absorption of radiation for an isolated Lorentz line. This ideal case helps to bring an understanding of a single line absorption variation with the pressure path length as it is a very fundamental concept.

The spectral variation of the absorption coefficient $\kappa_{\om}$ of a single line centered in $\om_0$, of line strength $S$, and of HWHM $\gamma_c$ with the wavenumber $\om$ is recalled:
\begin{equation}
 \kappa_{\om} = \dfrac{S}{\pi}\dfrac{\gamma_c}{\left( \om - \om_o\right)^2 + \gamma_c^2}. \tag{\ref{eq:lorentz_line}}
\end{equation}
The shape of the line is plotted in Fig.~\ref{fig:Lorentz_line}.

Assuming a path going through an isothermal gas, of temperature $T_g$, and homogeneous layer of participating species of partial pressure $P_i$, and of physical thickness $L$, the RTE, Eq.~\ref{eq:RTE_Final}, can be simplified as:
\begin{equation}
 I_{\om}(L) =  I_{\om}(0) \exp\left( -\kappa_{\om} P_i L\right) + I_{\rm b,\om}\left(T_g\right) \left(1-\exp\left(-\kappa_{\om} P_i L\right)\right).
\end{equation}
Note that in this equation, the local amount of participating species is expressed by the partial pressure $P_i$; hence in the equation above, $\kappa_{\om}$ has the dimension of an inverse pressure times an inverse length. In RadCal, $\kappa_{\om}$ is expressed in $\rm (cm^{-1} \cdot atm^{-1})$, while $P_i$ is in $\rm atm$ and the path physical length $L$ is in $\rm cm$. The product $P_i\;L$, the pressure path length, is referred to in RadCal as a species optical thickness and is also denoted $U$, and the product $\kappa_{\om}\;P_i\;L$ is referred to as the optical depth.

Considering only the emission term (\textit{i.e.} $I_{\om}(0)$ = 0), the exiting intensity integrated over the whole spectrum, emitted by a single line, denoted $I(L)$, is expressed as:
\begin{equation}
I(L) =  I_{\rm b,\om}\left(T_g\right) \displaystyle\int_{\Delta \om}\left( 1-\exp\left(-\kappa_{\om'} P_i L\right)\d \om'\right),
\end{equation}
this assumes that $I_{\rm b,\om}$ does not vary much over the $\Delta \om$ range. The integrand is often called the equivalent line width \cite{Modest2013} and is usually denoted $W$,
\be
W = \displaystyle\int_{\Delta \om}\left( 1-\exp\left(-\kappa_{\om'} P_i L\right)\d \om'\right).
\ee

The analytical expression of $W$ for a Lorentz line can be derived, see for example Penner, \cite{Penner1959}:
\be
W(P_i L) = 2 \pi \gamma_c L(x) = 2  \pi \gamma_c x \;\exp\left( -x\right) \left(I_0(x) + I_1(x)\right), \; x = \dfrac{SP_iL}{2 \pi \;\gamma_c}.
\ee
The function $L(x)$ is named the Ladenburg-Reiche function, and $I_0$ and $I_1$ are the modified Bessel functions of the first kind. The graph of the Ladenburg-Reiche function is
given in Fig.~\ref{fig:Ladenburg}. Note that for small and large values of $x$, the equivalent line has the following asymptotic forms:
\begin{align}
W(x) \sim SP_iL ,\: {\rm for}\: x \ll 1 \\
W(x) \sim 2\sqrt{S\gamma_c P_iL} ,\: {\rm for}\: x \gg 1.
\end{align}

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/Ladenburg.png}
\end{center}
 \caption{Profile the Ladenburg-Reiche function. \label{fig:Ladenburg}}
\end{figure}

It is worth noting that the parameter $x$ gives an indication of the optical thickness of the gas layer for a single line. When $x$ is small, this indicates that the medium is weakly participating (either $S$ or the product $P_iL$ is small) and the absorption and emission of a single line is linear with the product $P_iL$. On the contrary, when $x$ is much greater than unity, the absorption or emission of a single line is not linear anymore but has a square-root dependency.

\section{Narrow band models}\label{Sec::SNBM}
This section briefly describes the different models used to obtain most of the tabulated species IR spectral mean absorption coefficients at different temperatures, $\bar{\kappa}_i(\om,T)$. Narrow band models are used in lieu of line-by-line models to represent the IR spectra of radiating species in engineering applications. In the narrow-band approach, the whole spectrum is divided into small spectral bands (typically several $\rm cm^{-1}$), and different statistical approaches are used to compute the average radiative properties over these narrow-bands. What is of interest is to give functional expression of the integrand:
\be
\bar{\tau}_{\om} = \displaystyle\int_{\Delta \om}\left( \exp\left(-\kappa_{\om'} P_i L\right)\d \om'\right),
\ee
when $\Delta \om$ is large enough so it includes hundreds, even thousands, of single lines, but small enough to assume $I_{\rm b,\om}\left(T_g\right)$ and the incident spectral intensity $I_{\om}(0)$ constant over it. It is important to state that generally:
\be
\bar{\tau}_{\om} \neq \displaystyle\int_{\Delta \om}\left( \exp\left(-\bar{\kappa}_{\om'} P_i L\right)\d \om'\right).
\ee
Equality is verified only is some specific situations (\textit{e.g.} $P_iL \ll 1$). An important assumption to recall is that it is assumed that the Kirchoff law, \textit{i.e.} $\bar{\alpha} = \bar{\epsilon}$, holds over a narrow-band.

Three main narrow-band models are presented below: the Elsasser model, the Goody model, and the Malkmus model. All assume Lorentz lines.

\subsection{Elsasser model}

The Elsasser model assumes all the lines to have the same HWHM $\gamma_c$, the same line strength $S$, and to be equally spaced every $d$ wavenumber from each other.
As a consequence, the absorption coefficient can be expressed by the infinite sequence:
\be\label{eq:kappa_El}
\kappa_{\om} = \displaystyle\sum_{n = -\infty}^{\infty}{\dfrac{S}{\pi}\dfrac{\gamma_c}{\left(\om - \left(\om_0+n\,d\right)\right)^2 + \gamma_c^2}}.
\ee
This function is periodic, with a period $d$. Following Elsasser derivation (see Ref.~\cite{Penner1959}), Eq.~\ref{eq:kappa_El} can be expressed as:
\be\label{eq:Elsasser_periodic}
\kappa_{\om}  = \dfrac{S}{d} \dfrac{\rm sinh (8\beta)}{{\rm cosh (8\beta)} - \cos(\frac{2\pi}{d}(\om-\om_0))},
\ee
where the overlap parameter $\beta$, which is also called the (Lorentz) fine structure parameter and denoted $a_c$ in Ref.~\cite{Ludwig1973} and in the code, is defined here as:
\be\label{eq::beta}
\beta = \dfrac{\pi}{4}\dfrac{\gamma_c}{d}.
\ee
The mean absorption coefficient $\bar{\kappa}$ over the narrow-band centered in $\om$ is:
\be\label{eq:kappa_def}
\bar{\kappa}_{\om} = \dfrac{S}{d}.
\ee
Figure~\ref{fig:Elsasser_profile} plots the variations of the Elsasser absorption coefficient $\kappa_{\om}$ normalized by $\bar{\kappa}$ as a function of the ratio $\om/d$ for two different values of the overlap parameter $\beta$. Strong overlap effects are seen for the largest value of $\beta$.

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/Elsasser_profile.png}
\end{center}
 \caption{Variations of the Elsasser absorption coefficient normalized by $\bar{\kappa}$ plotted as a function of the ratio $\om/d$ for two different values of the overlap parameter $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red). \label{fig:Elsasser_profile}}
\end{figure}
The mean transmissivity $\bar{\tau}_{\om}$ of the narrow-band centered in $\om$ can be calculated using Eq.~\ref{eq:Elsasser_periodic}, averaging over one period $d$:
\be
\bar{\tau}_{\om} = \frac{1}{d}\displaystyle\int_{-\frac{d}{2}}^{\frac{d}{2}}{\exp\left(-\bar{\kappa}_{\om} P_i L\dfrac{\rm sinh (8\beta)}{{\rm cosh (8\beta)} - \cos\left(\frac{2\pi}{d}(\om'-\om)\right)} \right)\d\om'}.
\ee
This expression can be assessed using the Godson and Tien approximations \cite{Brosmer1985b,Kunitomo1975}:
\be\label{eq::Elsasser}
    \bar{\tau}_{\om} = 1- \erf \left( \dfrac{\sqrt{\pi}}{2} \, \frac{\bar{\kappa}_{\om} \, U }{\displaystyle \sqrt{1 + \frac{\pi \, \bar{\kappa}_{\om} \, U}{16 \, \beta}}}  \right),
\ee
where $U = P_iL$ is the optical thickness. Equation~\ref{eq::Elsasser} is used in RadCal to compute $\bar{\tau}_{\om}$ when using the Elsasser narrow-band model, which is only used for Methane. Note that the Godson and Tien approximation is reasonably accurate for values of the overlap parameter lower or of the order of unity, \textit{i.e.} $\beta < 1$.
Figure~\ref{fig::Elsasser_curve_growth} plots the quantity $-\ln(\bar{\tau}_{\om})$ (this quantity is also referred to as the curved of growth) versus the product $\bar{\kappa}_{\om} U$ for two different values of the overlap parameter: $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red). The back dashed curve corresponds to the ``linear'' behavior, \textit{i.e.} $\bar{\tau}_{\om} = \exp\left(\bar{\kappa} U\right)$. It is noteworthy that this linear behavior corresponds to either ``weak lines regimes'' (situations with small product $\bar{\kappa} U$) or to situations with a strong overlap, $\beta > 1$). Outside of these regimes, assuming linear behavior might lead to overestimation of the curve of growth. Figure~\ref{fig::Elsasser_curve_growth} also illustrates a case where the Godson and Tien approximation looses its accuracy and leads to unphysical results, as seen for the case $\beta=0.5$, which overestimates the absorption.

It can be seen from Eq.~\ref{eq::Elsasser} that $\bar{\tau}_{\om}$ is fully characterized once the narrow-band parameters $\bar{\kappa}_{\om}$ and $\beta$ are known. Note that these two parameters are by construction independent of the physical length of radiation propagation but depend on the local temperature, pressure, and local mixture composition.

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/Elsasser_curve_of_growth.png}
\end{center}
 \caption{Elsasser model curve of growth ($\ln \bar{\tau}$) as a function of the product $\bar{\kappa}_{\om} U$. Two different values of the overlap parameter are plotted: $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red). The curve of growth for the ``weak line regime'' is plotted in the bash dashed curve.\label{fig::Elsasser_curve_growth}}
\end{figure}

\subsection{Generalities on statistical narrow-band models}
The Goody and Malkmus models have been developed to provide a better modeling of the narrow-band spectral properties than that predicted by the Elsasser model, which is based on strong assumptions: uniformity of the line strength and of line spacing. While the Elsasser model works fine for some light, diatomic  species (\textit{i.e} HCl), most of the more complex polyatomic species cannot be modeled appropriately with the Elsasser model, since the very detailed experimental characterization of their spectral lines does not exhibit regularity \cite{Modest2013}.

The Goody and Malkmus models are statistical narrow-band models, the former preceding the latter by a couple of decades and was first developed by Mayer and Goody \cite{Penner1959}. Statistical narrow-band models are based on two main assumptions: first, there is no correlation between lines position and line intensities, and the location of the line can be modeled by any random arrangement~\cite{Penner1959,Young1977d}; second, while all the lines have the same shape, their line strength within the narrow-band of spectral range $\Delta \nu$ is a random variable assigned to a continuous probability density function $P(\bar{S},S)$, with $\bar{S}$ the mean value of the line strength, \textit{i.e.} $\bar{S} = \displaystyle\int_{0}^{+\infty}{S P(\bar{S},S)\d S}$.

A consequence of these assumptions for a very large number of lines contained within the narrow-band is that the transmissivity of a narrow-band $\bar{\tau}_{\om}$, and of optical thickness $U$, is expressed by:
\be\label{eq:SNB_tau_equiline}
\bar{\tau}_{\om}(U) = \exp\left(-\displaystyle\frac{\bar{W}(U)}{d}\right),
\ee
where $d$ is the average line spacing and $\bar{W}(U)$ is the average equivalent line width of the narrow-band considered and is calculated as:
\begin{align}\label{eq::SNB_Equiline}
\bar{W}(U) &= \displaystyle\int_{0}^{+\infty}{P(\bar{S},S)W(S,U) \d S} \nonumber \\
           &= \displaystyle\int_{0}^{+\infty}{P(\bar{S},S)\displaystyle\int_{-\infty}^{+\infty}{1 - \exp\left(-\kappa_{\om} P_i L\right) \d\om} \d S}
\end{align}
where $\kappa_{\om}$ is the absorption coefficient of a Lorentz line, given by Eq.~\ref{eq:lorentz_line}. An example of the derivation of Eq.~\ref{eq::SNB_Equiline} is presented in Penner's book~\cite{Penner1959}. By their different choice of the probability density function $P(\bar{S},S)$, the Goody and Malkmus models provide an explicit and simple evaluation of the average equivalent line width of a narrow-band, $\dfrac{\bar{W}}{d}$.

\subsection{Goody model}\label{sec:goody_model}

The Goody model assumes an exponential distribution of the line strength:
\begin{align}
 P(\bar{S},S) &= \dfrac{1}{\bar{S}}\exp\left(-\dfrac{S}{\bar{S}} \right) \\
              &= \dfrac{4}{\pi S_E}\exp\left(-\dfrac{4S}{\pi S_E} \right),
\end{align}
where $S_E$ is an effective equivalent line strength which relates to the average line strength $\bar{S}$ with \cite{Malkmus1967,Ludwig1973}:
\be\label{eq::S_E_barS}
\bar{S} = \dfrac{\pi}{4} S_E.
\ee
Solving Eq.~\ref{eq:SNB_tau_equiline} and introducing the effective average lines spacing $d_E = \dfrac{4}{\pi}d$, Ref.~\cite{Malkmus1967}, it comes:
\begin{align}\label{eq::Goody}
    \bar{\tau}_{\om} = \exp\left(-\dfrac{S_E}{d_E} \dfrac{U}{\displaystyle \sqrt{1+\frac{S_E U}{4 \, \gamma_c }}}\right) \nonumber \\
    \bar{\tau}_{\om} = \exp\left(-\frac{\bar{\kappa}_{\om} \, U} {\displaystyle \sqrt{1+\frac{\bar{\kappa}_{\om} \, U}{4 \, \beta}}}\right)
\end{align}
where the band overlap parameter $\beta$ is here defined as:
\be
\beta = \dfrac{\gamma}{d_E} = \dfrac{\pi}{4}\dfrac{\gamma}{d}.
\ee
It is important to remark that the band overlap parameter $\beta$, while constant over a narrow-band, varies with the wavenumber. In addition, it has a strong dependence on the total pressure. This dependence varies from species to species. The mean absorption coefficient $\bar{\kappa}$ in Eq.~\ref{eq::Goody} keeps the same definition as given in Eq.~\ref{eq:kappa_def}. It is noteworthy to realize that:
\be
\bar{\kappa} = \dfrac{S}{d} = \dfrac{S_E}{d_E},
\ee
from the definition of $d_E$ and $S_E$. The effective parameter has been introduced to be consistent with the definition from Ludwig \textit{et al.}, Ref.~\cite{Ludwig1973}, from which most of RadCal is formulated after.

Figure~\ref{fig::Goody_curve_growth} plots the curve of growth, $-\ln(\bar{\tau}_{\om})$, versus the product $\bar{\kappa}_{\om} U$ for two different values of the overlap parameter: $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red) using the Goody model. The back dashed curve corresponds to the ``linear'' behavior, \textit{i.e.} $\bar{\tau}_{\om} = \exp\left(\bar{\kappa} U\right)$.

Again, it can be seen from Eq.~\ref{eq::Goody} that $\bar{\tau}_{\om}$ is fully characterized once the narrow-band parameters $\bar{\kappa}_{\om}$ and $\beta$ are known.

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/Goody_curve_of_growth.png}
\end{center}
 \caption{Goody model curve of growth ($\ln \bar{\tau}$) as a function of the product $\bar{\kappa}_{\om}\;U$. Two different values of the overlap parameter are plotted: $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red). The curve of growth for the ``weak line regime'' is also plotted (dashed curve).\label{fig::Goody_curve_growth}}
\end{figure}

\subsection{Malkmus model}\label{sec:malkmus_model}

Like the Goody model, the Malkmus model assumes the line strength of all the lines in a given narrow-band model follows a particular distribution. This model was developed because the exponential distribution assumed in the Goody model substantially underestimates the number of low-intensity lines. The Malkmus model is based on a more physical approach that assumes that the line strength distribution $P(S)$ varies proportionally to $S^{-1}$. This assumption is based on physical arguments and is empirically verified for some molecules, see Malkmus~\cite{Malkmus1967} for more details.

The problem of non-normalization of the $S^{-1}$ distribution is circumvented by cutting-off the $S^{-1}$ distribution below some very small and above some very large values of $S$. A continuous distribution can then be constructed based on some assumptions relating the energy of a transition with the line strength and assuming that the ratio between the maximum and the minimum line strengths considered is very large. The line strength probability distribution used in the Malkmus model is expressed by an exponential-tailed $S^{-1}$ distribution \cite{Malkmus1967,Young1977d}:
\begin{equation}
 P(S_E,S) = \dfrac{1}{S}\exp\left(-\dfrac{4}{\pi}\dfrac{S}{S_E} \right),
\end{equation}
where the effective line strength $S_E$ relates to the average line strength $\bar{S}$ through Eq.~\ref{eq::S_E_barS}. Solving Eq.~\ref{eq:SNB_tau_equiline}, and using the same definition for the effective average lines spacing $d_E$ as given in the above section, it comes:
\begin{align}\label{eq::Malkmus}
    \bar{\tau}_{\om} = \exp\left( -2 \dfrac{\gamma_c}{d_E} {\displaystyle \sqrt{1+\frac{S_E U}{\gamma_c }} -1 }\right) \nonumber \\
    \bar{\tau}_{\om} =  \exp\left(-2  \beta \left[\sqrt{1+\dfrac{\bar{\kappa}_{\omega}U}{\beta} }- 1\right] \right)
\end{align}
where the band overlap parameter $\beta$ is here again defined as:
\be
\beta = \dfrac{\gamma}{d_E} = \dfrac{\pi}{4}\dfrac{\gamma}{d}.
\ee

The Malkmus band model is recognized as the most suitable statistical narrow-band model for polyatomic gases \cite{Modest2013}. In this new version of RadCal, this model has been introduced to model the newly implemented fuel species. Note that the Malkmus and Goody models do not differ when considering the extreme cases of optically thin and optically thick medium. In such cases, both models asymptote to:
\begin{align}
\lim_{U \to 0} \dfrac{\bar{W}}{d}/U &= \bar{\kappa} \\
\lim_{U \to +\infty} \dfrac{\bar{W}}{d}/\sqrt{U} &= 2\displaystyle\sqrt{\bar{\kappa}\beta}
\end{align}

Figure~\ref{fig::Malkmus_curve_growth} plots the curve of growth, $-\ln(\bar{\tau}_{\om})$, versus the product $\bar{\kappa}_{\om} U$ for two different values of the overlap parameter: $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red) using the Malkmus model. The back dashed curve corresponds to the ``linear'' behavior, \textit{i.e.} $\bar{\tau}_{\om} = \exp\left(-\bar{\kappa} U\right)$.

Again, it can be seen from Eq.~\ref{eq::Malkmus} that $\bar{\tau}_{\om}$ is fully characterized once the narrow-band parameters $\bar{\kappa}_{\om}$ and $\beta$ are known. These two narrow-band spectral quantities can be obtained either from line-by-line calculations, by fitting experimental data, or by physical considerations.

\begin{figure}
\begin{center}
 \includegraphics[width=4.0in]{Figures/Malkmus_curve_of_growth.png}
\end{center}
 \caption{Malkmus model curve of growth ($-\ln \bar{\tau}$) as a function of the product $\bar{\kappa}_{\om} U$. Two different values of the overlap parameter are plotted: $\beta = 0.05$ (in green) and $\beta = 0.5$ (in red). The curve of growth for the ``weak line regime'' is also plotted (dashed curve).\label{fig::Malkmus_curve_growth}}
\end{figure}

\textbf{Note}: For all the tabulated data, a linear interpolation of $\bar{\kappa}_{\om}$ and $\beta_{\om}$ in temperature and/or in wavenumber is performed by RadCal when necessary. If the temperature sought is out of the tabulated data range, then the data at the nearest temperature are used.

\section{Treatment of the RTE in the scope of narrow-band model}\label{sec:RTE_NB}
This section presents the expression of the different calculations performed by RadCal to numerically solve the Radiative Transfer Equation for a narrow-band. The Radiative Transfer Equation for a narrow-band is recalled below using wavenumbers:
\be\label{eq::RTE_Wavenumber}
\underbrace{I_{\om_0}(s)}_{\rm Received\:intensity\:in\:s}  = \underbrace{{I_{\rm b,\om_o}(T_w) \bar{\tau}(\om_0; 0 \rightarrow s)}}_{\rm transmitted\:incident\:intensity} + \underbrace{{\displaystyle\int_0^{s}{I_{\rm b,\om_0}\left(T(s')\right)\dfrac{\partial \bar{\tau}  }{\partial s'}(\om_0; s' \rightarrow s) \d s'}}}_{\rm intensity\:emitted\:by\:the\:medium\:between\:0\:and\:s}.
\ee
where $\Delta \om$ is the narrow-band centered in $\om_0$, $I_{\om_0}(s)$ is the received flux at the location $s$, $I_{b,\om_o}(T_w)$ is the incident flux penetrating the participating medium and which is modeled in RadCal as a flux from a blackbody of temperature $T_w$ ($T_w$ is an user input). The second term of the equation right-hand side accounts for the medium radiative emission and self-absorption.

\subsection{Treatment of Homogeneous pressure-path}
\label{sec::homogeneous_path}
If the participating medium is homogeneous -- \textit{i.e.} of constant composition, temperature $T$, and total pressure $P_T$ -- Eq.~\ref{eq::RTE_Wavenumber} can be simplified as:
\be\label{eq::RTE_wavenumber_H}
I_{\om_0}(s) = I_{\rm b,\om_o}(T_w) \bar{\tau}(\om_0; 0 \rightarrow s) + I_{\rm b,\om_o}(T)(1-\bar{\tau}(\om_0; 0 \rightarrow s)).
\ee
RadCal computes the value of $\bar{\tau}(\om_0; 0 \rightarrow s)$ combining Lorentz and Doppler lines. While usually for cases at atmospheric pressure and at moderate temperature the Doppler lines are not important, they are however included for the sake of completeness. The different steps of the calculation of $\bar{\tau}(\om_0; 0 \rightarrow s)$ are presented below.

For each participating species present in the medium, RadCal first retrieve the band mean absorption coefficient $\bar{\kappa}_{\om}$, the band overlap parameter $\beta$ (for Lorentz lines), and the Doppler fine structure parameter, denoted $a_D$, which is defined as the ratio of the Doppler HWHM $\gamma_D$, given by Eq.~\ref{eq::DopplerHWHM}, over the mean line spacing $d$.

Then RadCal computes the optical depth from Lorentz lines, denoted $X_C$, using Eqs.~\ref{eq::Elsasser}, \ref{eq::Goody}, and \ref{eq::Malkmus}, depending on the species appropriate model, using  $\bar{\kappa}_{\om}$ and $\beta$. Note that:
\be\label{eq::Collision_optical_depth}
X_C = \dfrac{\bar{W}}{d}
\ee

RadCal computes the optical depth from Doppler lines, denoted $X_D$. The expression of $X_D$ depends of the model chosen to compute $X_C$. If the Elsasser or Goody model is used to compute $X_C$, then the optical depth from Doppler lines is calculated by \cite{Ludwig1973}:
\be\label{eq::Doppler_optical_path_Goody}
X_D = \sqrt{\dfrac{2}{\ln 2}} a_D \sqrt{\ln \left(1+\dfrac{\ln 2}{2} \left(\dfrac{\bar{\kappa}_{\om_0}U}{a_D}\right)^2\right)},
\ee
where $U$ is the optical thickness defined by $U = P_i\,s$, with $P_i$ the partial pressure (in atm) of the participating species considered and $s$ is the distance given in cm.
If the Malkmus model is used to compute $X_C$, then the optical depth from Doppler lines is calculated by \cite{Ludwig1973}:
\be\label{eq::Doppler_optical_path_Malkmus}
X_D = \sqrt{\dfrac{3}{2 \ln 2}} a_D \left[ \ln\left(1 + \left(\sqrt{\dfrac{2 \ln 2}{3}} \dfrac{\bar{\kappa}_{\om_0}U}{a_D} \right)^{2/3} \right)\right]^{3/2}.
\ee
The combined optical depth, $Y$, is then calculated:
\be\label{eq::combined_optical_path}
Y  = \left(1 - \left(\dfrac{X_C}{\bar{\kappa}_{\om_0}U}\right)^2 \right)^{-2} + \left(1 - \left(\dfrac{X_D}{\bar{\kappa}_{\om_0}U}\right)^2 \right)^{-2} - 1.
\ee
Finally, the narrow-band transmissivity, $\bar{\tau}(\om_0; 0 \rightarrow s)$, is obtained by:
\be\label{eq::trans_HPP_final}
\bar{\tau}(\om_0; 0 \rightarrow s) = \exp\left(-\bar{\kappa}_{\om_0}U \sqrt{1 - \dfrac{1}{\sqrt{Y}}}\right).
\ee
If there are more than one participating species, then $\bar{\tau}(\om_0; 0 \rightarrow s)$ is calculated from:
\be\label{eq::trans_HPP_final_multispecies}
\bar{\tau}(\om_0; 0 \rightarrow s) = \exp\left(-\displaystyle\sum_i{\bar{\kappa}_{i,\om_0}U \sqrt{1 - \dfrac{1}{\sqrt{Y_{i}}}}}\right),
\ee
the optical depth for each species $i$ is first calculated separately, and then summed altogether.

\subsection{Treatment of non-Homogeneous pressure-path - Curtis-Godson approximation}
\label{sec::Curtis_Godson}
The treatment of the RTE in the case of non-homogeneous medium (where gradients of species, temperature, or pressure are present) is explicitly detailed below. The non-homogeneous case is first treated by discretizing the depth of penetration of the incident beam of direction $\hat{\textbf{s}}$ into a set of $n$ smaller segments $\{[s_{i-1};s_i]\}_n$ over which the local composition, temperature, and pressure can reasonably be assumed constant. Hence, the expression of the RTE over a narrow-band $\Delta \om$, centered in $\om_0$ and given by Eq.~\ref{eq::RTE_Wavenumber} can be simplified as:
\begin{equation}\label{eq::RTE_wavenumber_NH}
I_{\om_0}(s) = I_{\rm b,\om_o}(T_w) \bar{\tau}(\om_0; 0 \rightarrow s) + \displaystyle\sum_{i=1}^n{I_{\rm b,\om_o}(T_i)\left(\bar{\tau}(\om_0; s_i \rightarrow s)-\bar{\tau}(\om_0; s_{i-1} \rightarrow s)\right)}.
\end{equation}
In the expression, it assumed that $s_0 = 0$, which is coincided with the location of the blackbody wall of temperature $T_w$, and $s_n = s$. The temperature $T_i$ is the temperature of the small segment $\{[s_{i-1};s_i]\}_n$. The difficulty in solving Eq.~\ref{eq::RTE_Wavenumber} lies in evaluating  $\bar{\tau}(\om_0; s_i \rightarrow s)$ as it is correlated with the cells crossed by the beam $\widehat{\textbf{sis}}$.

To circumvent this difficulty, the Curtis-Godson approach is used. In this approach, it is assumed that
each $\bar{\tau}(\om_0; s_i \rightarrow s)$ for a given species can still be calculated using Eqs.~\ref{eq::Doppler_optical_path_Goody}, \ref{eq::Doppler_optical_path_Malkmus}, \ref{eq::combined_optical_path}, \ref{eq::trans_HPP_final}, and the models presented in Section~\ref{Sec::SNBM} but their parameters $\bar{\kappa}$, $\beta$, and $a_D$ are replaced by some effective parameters, $\bar{\kappa}^*$, $\beta^*$, $a_{D}^*$, respectively. See Young \cite{Young1977d} and Ludwig \textit{et al.} \cite{Ludwig1973} for additional details on the Curtis-Godson approach. It was found that using path-averaged parameters as effective parameters work best \cite{Young1977d} to calculate the transmissivity of non-homogeneous medium. These parameters are defined as:
\begin{align}
\bar{\kappa}^* &= \dfrac{\displaystyle\int_{s_i}^{s} \bar{\kappa}(s')P_i(s')\d s'}{\displaystyle\int_{s_i}^{s} P_i(s')\d s'}, \label{eq::effective_kappa} \\
\beta^* &= \dfrac{\displaystyle\int_{s_i}^{s} \beta(s')\bar{\kappa}(s')P_i(s')\d s'}{\displaystyle\int_{s_i}^{s} \bar{\kappa}(s')P_i(s')\d s'}, \label{eq::effective_beta}\\
a_D^* &= \dfrac{\displaystyle\int_{s_i}^{s} a_D(s')\bar{\kappa}(s')P_i(s')\d s'}{\displaystyle\int_{s_i}^{s}, \bar{\kappa}(s')P_i(s')\d s'},\label{eq::effective_ad}
\end{align}
with $P_i$ is the partial pressure of the participating gas considered. The quantity:
\be
U = \displaystyle\int_{s_i}^{s}{P_i(s')\d s'},
\ee
is the optical thickness, in {\rm atm.cm}, between the points $s_i$ and $s$.

\section{Output quantities: effective coefficients and integrated quantities}\label{sec::Output}
RadCal solves the RTE based on the input parameters found in the input file \verb=RADCAL.in= and returns in a Tecplot file (\verb=<CASE ID>.tec=, where \verb=<CASE ID>= is an user input defined in the input file) the spectral profile of the spectral transmissivity of the medium $\bar{\tau}(\om_0; 0 \rightarrow s)$ and the incident spectral intensity $I_{\om_0}(s)$ calculated using either Eq.~\ref{eq::RTE_wavenumber_H} or \ref{eq::RTE_wavenumber_NH}.

In addition to the Tecplot file, RadCal also generates an output file \verb=RADCAL.out=, that contains several effective coefficients and integrated quantities. Some of these integrated quantities were already included in the previous version of RadCal. This is the case of the effective absorption coefficient, the Planck mean absorption coefficient, and the received total directional radiative energy flux (or received total intensity). Two new integrated quantities have been added: the total emissivity, and the total transmissivity. These effective coefficients and integrated quantities are described formally in this section. Note that the integration is performed using a Simpson rule over non regular abscissa using a 3-point Lagrangian interpolation (quadratic interpolation).

\subsection{Effective absorption coefficient}
The path-averaged or effective absorption coefficient, denoted here $\kappa_{\rm e}$ and denoted \verb=Amean= in the output file, is defined such that the received total intensity in $s = L$ can be expressed as a sum of a transmitted total intensity coming from the blackbody wall, set at temperature $T_w$, and a total intensity emitted by the gas at the local temperature $T_g$:
\be\label{eq:effective_epsilon1}
   \int_{\om_{\rm min}}^{\om_{\rm max}}{I_{\om}(L) \; \d \om} = \exp\left(-\kappa_{\rm e}\, L\right) \int_{\om_{\rm min}}^{\om_{\rm max}}{ I_{\rm b,\om}(T_w) \; \d \om} + \left(1-\exp\left(-\kappa_{\rm e} \, L\right)\right) \int_{\om_{\rm min}}^{\om_{\rm max}}{I_{\rm b,\om}(T_g) \; \d \om}
\ee
where $L$ is the total path length given in cm. The bounds of integration in Eq.~\ref{eq:effective_epsilon1} are fixed, with $\om_{min} = 5$ cm$^{\rm -1}$, and $\om_{max} = 25000$ cm$^{\rm -1}$. However, the bounds of the integration for the gas-phase species are specified by the user. The remain outside spectral domain is integrated then considering only the contribution of soot (if present). See Eq.~\ref{eq::received_flux}. The large bounds of integration (5 -- 25000 cm$^{\rm -1}$) is large enought such that:
\be
\int_{\om_{\rm min}}^{\om_{\rm max}}{I_{b,\om}(T) \; \d \om} \approx \dfrac{\sigma}{\pi}T^4.
\ee

Formally, $\kappa_{\rm e}$ is calculated from:
\be\label{eq:effective_epsilon}
   \kappa_{\rm e} = -\dfrac{1}{L} \ln\left(\dfrac{\displaystyle\int_{\om_{\rm min}}^{\om_{\rm max}}{I_{\om}(L)  - I_{\rm b,\om}(T_g)\; \d \om}   }{\displaystyle\int_{\om_{\rm min}}^{\om_{\rm max}}{ I_{\rm b,\om}(T_w)-I_{\rm b,\om}(T_w) \; \d \om}  } \right).
\ee
Note that $\kappa_{\rm e}$ has the units of cm$^{-1}$. The effective absorption coefficient is calculated for homogeneous and non-homogeneous cases. For the latter, the gas temperature $T_g$ used in Eq.~\ref{eq:effective_epsilon} is the path-average temperature:
\be\label{eq::Path_avg_temp}
T_g = \dfrac{\displaystyle\int_{0}^{L}{T(s) \d s }}{L}.
\ee
While the effective absorption coefficient works well for a given length $L$, one has to be careful when using it to estimate the received total intensity from a similar homogeneous isothermal medium but with a different thickness as the calculated results using Eq.~\ref{eq:effective_epsilon} might deviate significantly from the actual results unless the medium considered is gray (no spectral variation of the absorption coefficient).

\subsection{Planck mean absorption coefficient}
The Planck mean coefficient, denoted here $\kappa_{\rm Planck}$, is calculated by:
\begin{equation}\label{eq::planck_mean}
\kappa_{\rm Planck} = \dfrac{\pi}{\sigma {T_g}^4}
\displaystyle\int_{\om_{\min}}^{\om_{\max}}{I_{\rm b,\om}(T_g)
\displaystyle\sum_i \bar{\kappa}_{i,\om} \, P_i \; \d \om}
\end{equation}
where $P_i$ is the partial pressure of participating species $i$, in units of ${\rm atm}$; and $\bar{\kappa}_{\om}$ is the narrow-band mean absorption coefficient of participating species $i$, in units of ${\rm atm^{-1}.cm^{-1}}$. Note that the temperature used in this expression is the local gas temperature; thus, $\kappa_{\rm Planck}$ is a function of the gas phase temperature and of the species partial pressure. It is independent of the path physical length. Its units are in ${\rm cm^{-1}}$. The integration bounds, $\om_{min}$, and $\om_{max}$ are defined by the user in the input file.

The Planck mean absorption coefficient is calculated for homogeneous and non-homogeneous cases. For the latter, the gas temperature used to calculate $I_{\rm b,\om}(T_g)$ is the path-average temperature as defined by Eq.~\ref{eq::Path_avg_temp}. The term $T_g^4$ is also calculated using a path-average value:
\be
T_g^4 = \dfrac{\displaystyle\int_{0}^L{ T(s)^4 \d s}}{L},
\ee
where $L$ is the total path length given in cm.

For non-homogeneous cases, the mean absorption coefficient used in Eq.~\ref{eq::planck_mean}, $\bar{\kappa}_{i,\om}$, corresponds to the effective mean absorption coefficient $\bar{\kappa}^*$ defined by Eq.~\ref{eq::effective_kappa}. The partial pressure used is a path-average partial pressure:
\be
P_i = \dfrac{\displaystyle\int_{0}^L{P_i(s)\d s}}{L}.
\ee

\subsection{Total transmissivity}

Radcal returns the total transmissivity, denoted $\tau_{T}$, of the spectral range bounded between the user defined $\omega_{min}$ and $\omega_{max}$. The total transmissivity is calculated as:
\begin{equation}\label{eq::total_transmissivity}
 \tau_{T} = \dfrac{\displaystyle\int_{\omega_{min}}^{\omega_{max}}\tau(0 \rightarrow s;\omega)I_{\rm b,\om}(T_w) \d \omega }{\displaystyle\int_{\omega_{min}}^{\omega_{max}}I_{\rm b,\om}(T_w)\d \omega }.
\end{equation}
This quantity is dimensionless. It represents the fraction of transmitted incident intensity from the blackbody wall set at temperature $T_w$. It is worth noting that since it depends on $T_w$, changing $T_w$ while keeping constant all the other simulation parameters will yield different values of $\tau_{T}$.

\subsection{Total emissivity}

Radcal returns the total emissivity, denoted $\varepsilon_{T}$, for the spectral range bounded by the user defined $\omega_{min}$ and $\omega_{max}$. It is derived from the effective absorption coefficient $\kappa_{\rm e}$ through:
\begin{equation}\label{eq:total_emissivity}
\varepsilon_{T} = 1 - \exp\left(- \kappa_{\rm e} L \right),
\end{equation}
where $L$ is the total physical length of the participating layer, given in $\rm cm$. The total emissivity is dimensionless.

\subsection{Received flux}
RadCal returns the integrated value of the received total directional radiative energy flux (or received total intensity), denoted \verb=Received Flux= in \verb=RADCAL.out= file. The integration is performed between the user defined bounds $\omega_{min}$ and $\omega_{max}$. Note that contribution of soot (if present) is added in the ranges 5~$\rm cm^{-1}$ to $\om_{min}$ and $\om_{\max}$ to 25,000~$\rm cm^{-1}$. Its units are in $\rm W.m^{-2}.str^{-1}$. The received total intensity, denoted $I(L)$, is calculated by:
\be\label{eq::received_flux}
I(L) = \underbrace{\displaystyle\int_{\om_{min}}^{\om_{max}}{I_{\om}(L)\d \om}}_{\rm All\:species} + \underbrace{\displaystyle\int_{5}^{\om_{min}}{I_{\om}(L)\d \om}}_{\rm Soot\:only} +  \underbrace{\displaystyle\int_{\om_{max}}^{25,000}{I_{\om}(L)\d \om}}_{\rm Soot\:only},
\ee
where $I_{\om}(L)$ is obtained from Eq.~\ref{eq::RTE_wavenumber_H} for homogeneous cases, and Eq.~\ref{eq::RTE_wavenumber_NH} for non-homogeneous cases.


% !TEX root = ../bachlor-arbeit.tex
\begin{tabular}{ll}
    \toprule
    Input: &
    \begin{tabular}[t]{@{}l@{}}
        transmission spectrum $I=(I_\s{x}, \,I_\s{y})$ a $\lambda \times 2$ array\\
        $I_\s{x/y}$ ... X- and Y-transmission spectra,
        $\lambda$ ... number of wavelengths
    \end{tabular}\\
    Output: &
    two sets layer parameters $\mc L_1$ and $\mc L_2$,
    stack parameters $\mc S$ \\
    \bottomrule
\end{tabular}

\paragraph{Network Architecture}
This module is a 1D Convolutional Neural Network instead of the basic Multi Layer Perceptron. It was chosen to utilize the translational invariance of ConvNets. For example the concept "peak" should be learned independent of its position in the spectrum. As described in section \ref{sec:NN_bg}, a ConvNet provides this functionality. Another constraint on the network architecture arises from the different kind of outputs.
Most of the outputs are continuous but the choices about material $m$ and geometry $g$ are discrete/categorical.
These need different activation functions $\sigma$ to reach the different value ranges. The continuous outputs are mostly bounded by physical constraints and $m, \, g \in [0, \, 1]$ as they are \textit{one hot encoded}, meaning $1 \rightarrow$ "The layer has this property" and
$0 \rightarrow$ "The layer does not have this property".

\indent The different outputs also need different cost functions $C(y, y')$ during training where $y'$ is the networks output and $y$ is the known solution. For the continuous output one can simply use the mean squared error
\begin{equation}
    C_\s{mse}(\vb y, \, \vb y') = \sum_i \qty(y_i - y_i')^2
\end{equation}

\noindent
as all outputs are equally important and the cost function should be indifferent on whether the networks prediction is over or under target. For the categorical output the network learns quicker with the \textit{Categorical Cross-Entropy} error

\begin{equation}
    C_\s{ce}(\vb y, \, \vb y') = - \sum_i y_i \log y'_i.
\end{equation}

\noindent
This error treats false positives ($y_i = 0, \, y_i' = 1$) and false negatives ($y_i = 1, \, y_i' = 0$) differently. A false positive does not increase the overall cost as $y_i = 0 \Rightarrow C_\s{ce} = 0$ but for a false negative $C_\s{ce} \rightarrow \infty$. This is favorable behavior because it does not matter if the network outputs some probability for a wrong class as long as it outputs a higher probability for the correct class. The final architecture, seen in figure \ref{fig:al:NN_architecture}, is similar to the example given in figure \ref{fig:bg:NN_example} while meeting the above-mentioned constraints.

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{al_NN_architecture}
    \caption{The network starts with 4 pairs of convolutional and pooling layers $C^i P^i   $. The convolutions are characterized by (\textit{number of kernels}, \textit{kernel size}). The kernel size is always 5 and the number of kernels is gradually increased. Then the Network splits into a discrete and a continuous branch via two Dense layers with (\textit{number of neurons}). In the discrete branch a dropout is applied to the dense layer where (0.5) is (\textit{fraction of neurons to drop}).
    All the internal activations $\sigma_\s{r}$ are ReLu's and the final activations $\sigma_\s{s}$ and $\sigma_{l}$ are a sigmoid and a linear function.}
    \label{fig:al:NN_architecture}
\end{figure}

\newpage
\paragraph{Network Training}~\\
To train a Neural Network, one needs a training set $(X, \, Y)$ of known input output pairs. In this case they are generated using the pre-simulated single layers in the database which are randomly combined into stacks. Then the stacks X- and Y-transmission spectra $(I_x, \, I_y)$ are calculated via SASA.
This means $I = (I_x, \, I_y) \in X$ are the networks input and the random design parameters ${\mc D = (\mc S, \mc L_1, \mc L_2) \in Y}$ are the output. For the first test we used squares and square holes of Aluminum and Gold. During the training one epoch is defined as one loop over all training samples. After every epoch the network is validated on a data set $(X_\s v, \, Y_\s v)$ of samples it has not seen before. This is done to check wether the network actually learns something rather than just memorizing the input data. It is common to rely on the cost function for training but use a separate function called \textit{metric} to evaluate the Network. This is done because there are occasions when the cost of a network drops but the performance does not increase. The metric should be a function where the output can be easily understood by the human evaluating the network. For discrete outputs we are going to use the metric \textit{accuracy}, that is what fraction of choices were correct and for the continuous outputs we will use the \textit{mean absolute error}

\begin{equation} \label{eq:al:mae}
    M_\s{mae}(\vb y, \vb y') = \sum_i \qty|y_i - y'_i|.
\end{equation}

\indent
The results of this first training on the square geometry can be seen in figure \ref{fig:al:square_results}.
Training and validation results are very similar which indicates that there is no overfitting or memorization. The discrete accuracy quickly reaches a maximum of $\sim 76\%$ which is less than expected and also the speed at which this value is reached is suspicious. The network does not improve much after the fourth epoch and this does not change by tuning the network architecture. This is because the issue lies not within the architecture but in the training data. In section \ref{sec:SASA} we have shown that for the used two layer stacks the transmission spectrum is reciprocal, that is the same for both directions. Therefor the data generation can result in two different stacks which produce the same spectrum. Consider a stack where one layer is Aluminum and the other is Gold as seen in figure \ref{fig:al:same_spec}. As both of them produce the same spectrum, one time the network is taught that the first layer is Gold and the second is Aluminum but another time it is taught the complete opposite. Actually, if the network is trained this way it only ever predicts stacks with layers of equal materials because this is the only setup it can get right.
\begin{figure}[H]
    \centering
    \captionsetup[subfigure]{position=b}
    \begin{subfigure}{.5\textwidth}
        \centering
        \includegraphics[width=\linewidth]{al_square_dis}
        \caption{}
    \end{subfigure}%
    \begin{subfigure}{.5\textwidth}
        \centering
        \includegraphics[width=\linewidth]{al_square_mae}
        \caption{}
    \end{subfigure}

    \caption{Training only with squares. (a) shows the accuracy of the discrete branch. That is, what percentage of choices for material and geometry were correct. In (b) we can see the performance of the continuous branch evaluated using the mean absolute error \ref{eq:al:mae}.}
    \label{fig:al:square_results}
    \end{figure}

\newpage
This is a well known problem when trying to solve an inverse problem.
In this case the function $\mc D \rightarrow I$ is well defined as one stack can only produce one spectrum but for the inverse problem we are trying to solve $I \rightarrow \mc D$ and there might be multiple designs which produce the same spectrum. We can solve the issue simply by allowing only one of the orientations into the training data. By doing this the training  results change as seen in figure \ref{fig:al:squares_fix}.
The discrete accuracy is much better at $\sim 98 \%$ and the training curve looks more natural in the sense that improvements diminish over time but do not hit a sudden barrier as they did in figure \ref{fig:al:square_results}.
Having resolved this issue, we can train the model on the more complex rectangle geometry reaching 94\% accuracy and 11 mae.

\begin{figure}[H]
    \floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,top}}}]{figure}[\FBwidth]
    {\caption{A stack of one Gold and one Aluminium layer separated by a glass spacer. Both stacks produce the same spectrum which leads to issues when using completely random stacks to train the network.}
    \label{fig:al:same_spec}}
    {\includegraphics[width=.36\textwidth]{al_flipped_stack}}
\end{figure}

\begin{figure}[H]
\centering
\begin{subfigure}{.5\textwidth}
    \centering
    \includegraphics[width=\linewidth]{al_square_fix_dis}
    \caption{}
    \label{}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
    \centering
    \includegraphics[width=\linewidth]{al_square_fix_mae}
    \caption{}
    \label{}
\end{subfigure}
\caption{Training on the square geometry after only allowing one of the equivalent stacks shown in figure \ref{fig:al:same_spec} into the training data.
Again, (a) shows the accuracy of the discrete branch and (b) the average cost of the continuous branch. The discrete accuracy has improved significantly compared to figure \ref{fig:al:square_results} but the continuous loss remains similar.}
\label{fig:al:squares_fix}
\end{figure}


%!TEX root = ./jctt.tex

\section{Conclusions and Future Work}
We have presented a VEF method for the one-group slab-geometry neutron transport equation coupling \SN equations having a Lumped Linear Discontinuous Galerkin spatial discretization with a drift-diffusion equation having a constant-linear Mixed Finite-Element discretization. We have numerically demonstrated that our LLDG/MFEM VEF method is as effective as consistently-differenced S$_2$SA;
that both the \SN and drift-diffusion equations exhibit second-order accuracy and thus approach each other with second-order accuracy as the spatial mesh is refined; and that the thick diffusion limit is preserved.  We also investigated two methods for updating the \SN scattering source given the drift-diffusion solution for the scalar flux. The first was to simply use the flat scalar flux dependence of the drift-diffusion solution in the \SN scattering source.  The other was to reconstruct a linear discontinuous scattering source
using the flat or cell-average MFEM fluxes together with standard slope reconstruction methods. For a homogeneous problems, the linear update was more accurate, but there was little difference between the flat and linear updates for a highly inhomogeneous problem.  All of the desired VEF properties such as order accuracy and preservation of the diffusion limit were obtained independent of the scattering source update procedure. Most importantly, it is clear that \SN and and drift-diffusion solutions that differ in proportion to truncation
error is a small price to pay for the versatility afforded by the VEF method for multiphysics calculations. The conservative drift-diffusion equation can be coupled to the other physics equations and discretized in a manner compatible with those equations.  Furthermore, the difference between the \SN and drift-diffusion equations is a measure of the truncation error of the drift-diffusion solution, which is a very useful error estimator naturally provided by the VEF method.

% add rational polynomial conclusions: expected to be important in rad transfer
% mfem is conservative and can be paired to multiphysics
% take out consistency talk

In the future we intend to extend the VEF method presented in this paper to the radiative transfer equations, with higher order discretizations in 2-D and 3-D geometries.  A major question that we intend to address in the near term is the impact of the linear
scattering source update technique in radiative transfer calculations, particularly in Marshak wave problems, where slope reconstruction
for the emission source is known to be important to mitigate a numerical artifact referred to as the ``teleportation'' effect  \cite{tele}.