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We report here on validation and verification of tokamak equilibrium tools used for the COMPASS tokamak \cite{compass2006}. We particularly focus on fundamental global plasma parameters and the shapes of magnetic flux surfaces, which are crucial in diagnostics interpretation and other analyses.   EFIT++ \cite{efitpp2006} is used for routine equilibrium reconstruction on COMPASS. FREEBIE \cite{freebie2012} is a recent free-boundary equilibrium code; FREEBIE enables predictive equilibrium calculation consistent with the poloidal field (PF) components of the tokamak. In this study, FREEBIE is used in the so-called inverse mode, which predicts PF coils currents from a give plasma boundary and profiles. The third code employed in this study is VacTH \cite{vacthref}, which provides a fast reconstruction of the plasma boundary from magnetic measurements using a toroidal harmonics basis.  In order to verify and validate the aforementioned tolls, tools,  we analyse EFIT++ and VacTH reconstructions of equilibria constructed with FREEBIE. Synthetic diagnostics (e.g., magnetic probes or flux loops) with optional artificial errors provide inputs for the reconstructions. % section introduction (end)  diff --git a/JUrban_SOFT2014.pdf b/JUrban_SOFT2014.pdf  new file mode 100644  index 0000000..673cb1a  Binary files /dev/null and b/JUrban_SOFT2014.pdf differ diff --git a/JUrban_SOFT2014.tex b/JUrban_SOFT2014.tex  index fd10c7d..51a8cf4 100644  --- a/JUrban_SOFT2014.tex  +++ b/JUrban_SOFT2014.tex 
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\end{enumerate}  The first step employs a routine EFIT++ set-up for COMPASS with heuristically tuned parameters. In addition to the total plasma current $I_\mathrm{p}$ and the currents in individual PF circuits, 16 partial Rogowski coils and 4 flux loops are employed in this reconstruction and $p'$ and $FF'$ are assumed to be linear functions of the poloidal flux $\psi$.  In the second step, FREEBIE inputs $I_\mathrm{p}$, $p'\left( {\bar \psi } \right)$ and $FF'\left( {\bar \psi } \right)$ profiles, the plasma boundary coordinates and an initial guess for the PF coils currents. Here, $p$ is the plasma pressure, $F = RB_\phi$ and $\bar\psi$ is the normalized poloidal magnetic flux ($\bar\psi = 0$ on the magnetic axis and $\bar\psi = 1$ on the plasma boundary). $p'$ comes either from the EFIT++ reconstruction or from Thomson scattering pressure profile $p_\mathrm{TS} = 1.3 n_\mathrm{e} p_\mathrm{e}$. FREEBIE then seeks a solution to the Grad-Shafranov equation, including the PF coils currents, which minimizes the given plasma shape constraint. (This regime is called the inverse mode.) FREEBIE can naturally output arbitrary synthetic diagnostics. We use here additional 24 poloidally and 24 radially oriented partial Rogowski coils (which are actually mounted on COMPASS) and an artificial set of 16 flux loops located at the same positions as the basic magnetic probes. Hereafter, the number of magnetic probes and flux loops are denoted $n_\mathrm{mp}$ and $n_\mathrm{fl}$. $n_\mathrm{mp}=16$, $n_\mathrm{fl}=4$ refers the basic set of magnetic measurements, $n_\mathrm{mp}=64$ refers to a set of all presently mounted partial Rogowski coils on COMPASS and $n_\mathrm{fl}=16$ implies artificial flux loops positioned at the same locations as the basic magnetic probes.  In the optional third step, an artificial random noise is added to the calculated values of $I_\mathrm{p}$, magnetic probes and flux loops. In particular, for a given noise level $l$, $\tilde X = \left( {1 + U\left( { - l,l} \right)^\mathrm{T}} \right)X$, where $X$ is a row vector of the synthetic diagnostics data and $U\left( { - l,l} \right)$ is a random vector of the same shape as $X$ with a uniform distribution on $\left( { - l,l} \right)$.  The final fourth step consists of reconstructing the equilibria form synthetic FREEBIE data using EFIT++ and VacTH. The reconstructions are then compared to the original equilibrium, focusing on global parameters and geometry. VacTH does not provide a full equilibrium but the plasma shape only; The only (the  target of VacTH is to provide such reconstructions in real time for a feedback control. control).  Scans are performed over noise levels ($\epsilon$)  and selected code parameters: $p'$ and $FF'$ polynomial degrees in EFIT++ ($n_{p'}$, $n_{FF'}$)  and the number of harmonics ($n^{\mathrm e}$, $n^{\mathrm p}$) ($n_{\mathrm p}$, $n_{\mathrm q}$)  in VacTH. Following The following  quantities are used for the comparison. \begin{table}[!h]  \begin{tabular}{ll}  $R_{\mathrm ax}$, $Z_{\mathrm ax}$ & $R,Z$ coordinates of the magnetic axis \\ 
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$\kappa = \frac{\left( {{Z_{{\rm{max}}}} - {Z_{{\rm{min}}}}} \right) } { \left( {{R_{{\rm{out}}}} - {R_{{\rm{in}}}}} \right) }$ & elongation \\  ${l_{\rm{i}}} = {{\bar B_{\rm{p}}^2}} / {{B_{\rm{a}}^2}}$ & normalized internal inductance \\  ${\beta _{\rm{p}}} = {{2{\mu _0}\bar p}} / {{B_{\rm{a}}^2}}$ & poloidal beta \\  $ W = \int_0^V {3/2p{\rm{d}}V'}$ {\frac{3}{2}p{\rm{d}}V'}$  & stored plasma energy \\ $ q_0$, $q_{95}$ & safety factor at $\bar \psi = 0,\ 0.95$ \\  \end{tabular}  \end{table}  diff --git a/Results.tex b/Results.tex  index be75347..9d26a30 100644  --- a/Results.tex  +++ b/Results.tex 
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\section{Results} % (fold)  \label{sec:results}  We have selected five time slices from COMPAS shots 4275 and 6962 (i.e. 10 cases in total) for the analysis. These cases include circular, elongated and diverted plasmas with different currents. A comparison of plasma shapes for shot 4275 is shown in Fig. \ref{fig:ex4275}. Numerical values of reconstruction errors are presented in Table \ref{table:ex4275}. We can observe a very good agreement between the original equilibrium and the reconstructed shapes. In this case, FREEBIE was using linear $p'$ and $FF'$ polynomials so that the EFIT++ model agrees with the target data. VacTH uses 8 magnetic probes and 16 flux loops. As we discuss later, flux loops are essential for reliable VacTH results. Even global kinetic properties are well reconstructed in EFIT++; the largest error around 10~\% is in $l_{\mathrm i}$ (i.e. basically in the toroidal current density profile).  \begin{table*}  \centering  \begin{tabular}{lrrrrrrrrrrrrr}  \toprule  code & time & $\Delta R_{\mathrm in}$ & $\Delta R_{\mathrm out}$ & $\Delta Z_{\mathrm min}$ & $\Delta Z_{\mathrm max}$ & $\delta \kappa$ & $E_\mathrm{mp}$ & $E_\mathrm{fl}$ & $\delta W$ & $\delta l_{\mathrm i}$ & $\delta \beta_{\mathrm p}$ & $\delta q_0$ & $\delta q_{95}$ \\  \midrule  EFIT++ & 0.97 & 0 & 0.002 & 0.001 & 0.001 & 0.003 & 0.001 & 0.0009 & 0.04 & 0.09 & 0.03 & 0.03 & 0.007 \\  EFIT++ & 0.99 & 4e-05 & 9e-05 & 0.001 & 0.001 & 0.004 & 0.001 & 0.0006 & 0.04 & 0.09 & 0.04 & 0.02 & 0.005 \\  EFIT++ & 1.02 & 0.0005 & 0.0002 & 0.002 & 0.002 & 0.008 & 0.002 & 0.002 & 0.02 & 0.1 & 0.02 & 0.02 & 0.009 \\  EFIT++ & 1.05 & 0.001 & 0.0008 & 4e-05 & 0.0003 & 0.004 & 0.003 & 0.005 & 0.01 & 0.07 & 0.02 & 0.01 & 0.009 \\  EFIT++ & 1.1 & 0.001 & 0.0005 & 0.005 & 0.0002 & 0.005 & 0.004 & 0.002 & 0.04 & 0.09 & 0.03 & 0.02 & 0.05 \\  VacTH & 0.97 & 0 & 0.001 & 0.0006 & 0.0002 & 0.002 & 8e-07 & 7e-05 & nan & nan & nan & nan & nan \\  VacTH & 0.99 & 4e-05 & 0.0004 & 0.001 & 0.0008 & 0.004 & 4e-07 & 0.0001 & nan & nan & nan & nan & nan \\  VacTH & 1.02 & 0.002 & 0.0008 & 0.005 & 0.002 & 0.01 & 2e-06 & 0.002 & nan & nan & nan & nan & nan \\  VacTH & 1.05 & 0.006 & 0.002 & 5e-05 & 0.002 & 0.01 & 3e-06 & 0.02 & nan & nan & nan & nan & nan \\  VacTH & 1.1 & 0.005 & 0.002 & 0.002 & 0.003 & 0.02 & 2e-06 & 0.003 & nan & nan & nan & nan & nan \\  \bottomrule  \end{tabular}  \caption{Errors for the same cases as in Fig. \ref{fig:ex4275}.}  \label{table:ex4275}  \end{table*}  \begin{figure*}  \centering %\begin{center}  \hfill{}  \includegraphics[width=18cm]{figures/example_4275.pdf}  \hfill{}  %\end{center}  \caption{Contours of $\bar\psi=\left(0.25,0.5,0.75,1\right)$, reconstruction from FREEBIE data, shot 4275. EFIT++ parameters: $n_\mathrm{mp} = 16$, $n_\mathrm{fl} = 4$, $n_{p'} = n_{FF'} = 1$. VacTH parameters: $n_\mathrm{mp} = 8$, $n_\mathrm{fl} = 16$, $n_{\mathrm p} = n_{\mathrm q} = 5$.}  \label{fig:ex4275}  \end{figure*}  The second shot for the comparison is 6962, which has been chosen because Thomson scattering (TS) profiles are available. On top of the same exercise as for 4275, we have calculated with FREEBIE equilibria with experimental TS pressure profiles. These equilibria of course no longer feature linear $p'$ and $FF'$ profiles.  % section results (end)  diff --git a/figures/example_4275.pdf b/figures/example_4275.pdf  new file mode 100644  index 0000000..2e24571  Binary files /dev/null and b/figures/example_4275.pdf differ