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Commit ccfb4d9e authored by robic97's avatar robic97
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Reformating and shortening text. Some graphical changes.

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...@@ -216,7 +216,7 @@ ...@@ -216,7 +216,7 @@
\def\posterbox[#1](#2,#3)<#4,#5>#6#7{ \def\posterbox[#1](#2,#3)<#4,#5>#6#7{
% Print content box % Print content box
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\ifx&#6&% \ifx&#6&%
\else \else
......
...@@ -34,6 +34,9 @@ ...@@ -34,6 +34,9 @@
%\usetikzlibrary{backgrounds} %\usetikzlibrary{backgrounds}
%\usepackage{pgfplots} %\usepackage{pgfplots}
%\newcommand{\key}[1]{\textcolor{red}{#1}}
\newcommand{\key}[1]{\textbf{#1}}
% Set project, title, and authors (use \\ for multiple lines) % Set project, title, and authors (use \\ for multiple lines)
\setproject{Project A08} \setproject{Project A08}
\settitle{Lagrangian Coherence in Quasi-Stationary Atmospheric States} \settitle{Lagrangian Coherence in Quasi-Stationary Atmospheric States}
...@@ -48,41 +51,55 @@ ...@@ -48,41 +51,55 @@
\setlength{\boxwidth}{0.455\paperwidth} \setlength{\boxwidth}{0.455\paperwidth}
\posterbox[](\margin, -\headerheight-\margin)<0.95\boxwidth,14.5cm>{Lagrangian Coherence?}{ \posterbox[](\margin, -\headerheight-\margin)<0.95\boxwidth,13cm>{Lagrangian Coherence}{
Suppose we have a set of $m$ \textcolor{red}{trajectories} evaluated at $T$ time instances $X := \{x_t^i:=\Phi_tx^i_{t_0}:i=1,..., m;t\in \{t_0, ..., t_{T-1}\} \}$, with $\Phi_t$ the \textcolor{red}{flow map} of a \textcolor{red}{dynamical system} from $t_0$ to $t$. We want to find \textcolor{red}{coherent sets}, which are regions in the phase space ($\mathbb{R}^3$ here) that keep their geometric integrity to a large extent during temporal evolution. This is equivalent to finding \textcolor{red}{tight bundles of trajectories} if the points forming the trajectories are conceived as samples from such a set. Suppose we have $m$ sample trajectories of a flow $\Phi^t$ in $\mathbb{R}^n$, evaluated at $T$ time instances
$$ x_t^i:=\Phi^t x^i_{t_0} \in \mathbb{M}_t, \quad i\in \{1,\hdots, m\};\:t\in \{t_0, \hdots, t_{T-1}\}.$$
We want to find \key{coherent sets}, which are regions in the phase space that keep their geometric integrity to a large extent
during temporal evolution.\\
$\rightarrow$ Find tight bundles of trajectories.
%This is equivalent to finding \textcolor{red}{tight bundles of trajectories} if the points forming the trajectories are conceived as samples from such a set.
} }
\posterbox[](\margin, -\headerheight-2\margin-14.5cm)<0.95\boxwidth,12cm>{Diffusion}{ \posterbox[](\margin, -\headerheight-2\margin-13cm)<0.95\boxwidth,12cm>{Diffusion maps}{
If $\Phi_{\epsilon, t}$ is the deterministic flow map plus some small random pertubation with variance $\epsilon$, \textcolor{red}{coherence implies robustness} in the sense of "$\Phi_{\epsilon, t}^{-1}(\Phi_{\epsilon, t} \mathbb{X}) \approx \mathbb{X}$", where $\mathbb{X} \subset \mathbb{R}^3$ is a coherent set at $t_0$. We make use of this by constructing a \textcolor{red}{diffusion operator on the data points}, whose eigenvectors give a low-dimensional representation which is used to extract coherent sets (\textcolor{red}{spectral clustering}) [1]. The evolution of coherent sets should be \key{insensitive to small perturbations}. Let $D_\epsilon$ denote the diffusion operator with variance $\epsilon$. We call $\mathbb{X}$ coherent if
$$D_\epsilon \circ \left(\Phi^t\right)^{-1} \circ D_\epsilon \circ \Phi^t \: \mathbb{X} \approx \mathbb{X}.$$
This heuristic is implemented using tranfer operators on the set of data-points.
%If $\Phi_{\epsilon, t}$ is the deterministic flow map plus some small random pertubation with variance $\epsilon$, \textcolor{red}{coherence implies robustness} in the sense of "$\Phi_{\epsilon, t}^{-1}(\Phi_{\epsilon, t} \mathbb{X}) \approx \mathbb{X}$", where $\mathbb{X} \subset \mathbb{R}^3$ is a coherent set at $t_0$. We make use of this by constructing a \textcolor{red}{diffusion operator on the data points}, whose eigenvectors give a low-dimensional representation which is used to extract coherent sets (\textcolor{red}{spectral clustering}) [1].
} }
\posterbox[anchor=north east](\paperwidth-\margin, -\headerheight-\margin)<1.05\boxwidth,28.8cm>{Quasi-Stationary Atmospheric States?}{ \posterbox[anchor=north east](\paperwidth-\margin, -\headerheight-\margin)<1.07\boxwidth,28.8cm>{Quasi-Stationary Atmospheric States?}{
QSAS -- a.k.a. high pressure blockings -- are \textcolor{red}{critical features of mid-latitude weather} and, importantly, associated with extreme events. \textcolor{red}{Forecasting skill} is, however, still \textcolor{red}{unsatisfying}. Physically, QSAS are characterized by \textcolor{red}{high stability}. QSAS -- a.k.a. high pressure blockings -- are \key{critical features of mid-latitude weather} and, importantly, associated with extreme events. \key{Forecasting skill} is, however, still \key{unsatisfying}. Physically, QSAS are characterized by \key{high stability}.
\vfill \vfill
\includegraphics[scale=1]{im/2016_00.pdf} \includegraphics[scale=0.8, viewport=0 0 375 450, clip]{im/blocktypes.png}\\ \includegraphics[scale=1]{im/2016_00.pdf} \includegraphics[scale=0.8, viewport=0 0 375 450, clip]{im/blocktypes.png}\\
{\footnotesize Left: Blocking example with PV field (shaded), wind $>40 \mathrm{\frac{m}{s}}$ (blue), surface pressure contours (yellow) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).} {\footnotesize Left: Blocking example with PV field (shaded), wind $>40 \mathrm{\frac{m}{s}}$ (blue), surface pressure contours (yellow) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).}
} }
\posterbox[](\margin, -\headerheight-3\margin-26.5cm)<0.95\boxwidth,16cm>{Algorithm}{ \posterbox[](\margin, -\headerheight-3\margin-25cm)<0.95\boxwidth,15.5cm>{Algorithm}{
\begin{enumerate} \begin{enumerate}
\item Calculate trajectories $X$ \item Calculate trajectories $x_t^i$.
\item Calculate pointwise diffusion distances $\mathbf{K}_{\epsilon, t}(i,j) = \exp \left( - \frac{D(x^i_t, x^j_t)}{\epsilon} \right)$, with distance metric $D(\cdot,\cdot)$ \item Calculate pointwise diffusion similarity
\item Calculate diffusion transition matrix $\mathbf{P}_{\epsilon, t}$ through normalization and boundary condition application $$\mathbf{K}_{\epsilon, t}(i,j) = \exp \left( -\epsilon^{-1}\|x^i_t - x^j_t\|^2 \right).$$ %with distance metric $D(\cdot,\cdot)$
\item Obtain space-time diffusion map transition matrix $\mathbf{Q}_{\epsilon} = \frac{1}{T} \sum_t \mathbf{P}_{\epsilon, t}$ \item Calculate diffusion transition matrix $\mathbf{P}_{\epsilon, t}$ through normalization and applying boundary conditions.
\item Perform spectral clustering on $\mathbf{Q}_{\epsilon}$ \item %Obtain space-time diffusion map
Averaged transition matrix $\mathbf{Q}_{\epsilon} = \frac{1}{T} \sum_t \mathbf{P}_{\epsilon, t}$.
\item Perform spectral clustering on $\mathbf{Q}_{\epsilon}$.
\end{enumerate} \end{enumerate}
} }
\posterbox[anchor=north east](\paperwidth-\margin, -\headerheight-2\margin-28.8cm)<1.05\boxwidth,52.7cm>{Trajectory Density}{ \posterbox[anchor=north east](\paperwidth-\margin, -\headerheight-2\margin-28.8cm)<1.05\boxwidth,52.7cm>{Trajectory Density}{
Assume (at any $t$) the $m$ points are sampled from some manifold $\mathbb{M}_t$. Then, the sum of \textcolor{red}{diffusion distances} can be interpreted as a \textcolor{red}{Monte Carlo approximation} [4] s.t. $S_{\epsilon, t} := \sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j) \approx \frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2} \int_{\mathbb{M}_t}\int_{\mathbb{M}_t} \exp (-D(x, y)/\epsilon) \dd x \dd y \approx \frac{m^2}{\mathrm{vol}(\mathbb{M}_t)} (2 \pi \epsilon)^{d/2}$, where $\mathbb{R}^d$ is the tangential space of $\mathbb{M}_t$. $S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon \to 0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon \to \infty}}S_{\epsilon, t} = m^2$. Also $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$. Assume at time $t$ the points $x_t^i$ points are sampled from some manifold $\mathbb{M}_t$. Then, the sum of the diffusion similarities can be interpreted as an \key{Monte Carlo integral approximation} [4]
\vspace{6.5in}\\ \begin{align*}S_{\epsilon, t} := \sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j) \approx \frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2} \int_{\mathbb{M}_t}\int_{\mathbb{M}_t} \exp (-\epsilon^{-1} \|x-y\|^2) \dd x \dd y \approx \frac{m^2}{\mathrm{vol}(\mathbb{M}_t)} (\pi \epsilon)^{d/2}
We obtain $d = \dim (\mathbb{M}_t)$ by maximizing $\frac{\dd \log (S_{\epsilon, t})}{\dd \log (\epsilon)}$ and calculate the point density measure $\rho (t) := \frac{1}{d}\log (\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$. \end{align*}
\vspace{5.4in}\\ %$S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon \to 0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon \to \infty}}S_{\epsilon, t} = m^2$. Also $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
The trajectories tend to be \textcolor{red}{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability. For suited $\epsilon$, this yields a linear relationship between $\log(\epsilon)$ and $\log(S_{\epsilon, t})$.
$$\log(S_{\epsilon, t}) = \frac{d}{2} \log(\epsilon) + \log(\rho(t)) + \log(m) + \frac{d}{2} \log(\pi); \quad\rho(t) := \frac{m}{\mathrm{vol}(\mathbb{M}_t)}.$$
\vspace{16cm}\\
The dimension $d$ and density $\rho(t)$ can be computed from this graph. We use $\ell(t):=\frac{1}{d} \log(\rho(t))$ as a measure of density. The trajectories tend to be \key{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
%We obtain $d = \dim (\mathbb{M}_t)$ by maximizing $\frac{\dd \log (S_{\epsilon, t})}{\dd \log (\epsilon)}$ and calculate the point density measure $\rho (t) := \frac{1}{d}\log (\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
} }
\node[anchor=north west] (epsloglog) at (\paperwidth-\margin-1.05\boxwidth, -\headerheight-2\margin-43cm) {\includegraphics[scale=1]{im/20160502_00.pdf}}; \node[anchor=north west] (epsloglog) at (\paperwidth-\margin-1.06\boxwidth, -\headerheight-2\margin-44cm) {\includegraphics[scale=1]{im/20160502_00.pdf}};
\node[anchor=north west] (trajs) at (\paperwidth-\margin-0.4007\boxwidth, -\headerheight-2\margin-46.015cm) {\includegraphics[scale=1]{im/traj_20160502_00.pdf}}; \node[anchor=north west] (trajs) at (\paperwidth-\margin-0.4\boxwidth, -\headerheight-2\margin-47.015cm) {\includegraphics[scale=1]{im/traj_20160502_00.pdf}};
%\begin{tikzpicture} %\begin{tikzpicture}
% First image (bottom layer) % First image (bottom layer)
% \node[anchor=north west,inner sep=0] (image1) at (0,0) {\includegraphics[scale=1]{im/20160502_00.pdf}}; % \node[anchor=north west,inner sep=0] (image1) at (0,0) {\includegraphics[scale=1]{im/20160502_00.pdf}};
...@@ -90,39 +107,42 @@ The trajectories tend to be \textcolor{red}{denser after passage} through the QS ...@@ -90,39 +107,42 @@ The trajectories tend to be \textcolor{red}{denser after passage} through the QS
% \node[anchor=north west,inner sep=0, opacity=1] (image2) at (10,-3) {\includegraphics[scale=1]{im/trajs.pdf}}; % \node[anchor=north west,inner sep=0, opacity=1] (image2) at (10,-3) {\includegraphics[scale=1]{im/trajs.pdf}};
%\end{tikzpicture} %\end{tikzpicture}
\node[anchor=north west] (dimension) at (\paperwidth-\margin-1.05\boxwidth, -\headerheight-2\margin-64cm) {\includegraphics[scale=1]{im/dimensionall.pdf}}; \node[anchor=north west] (dimension) at (\paperwidth-\margin-1.05\boxwidth, -\headerheight-2\margin-66cm) {\includegraphics[scale=1]{im/dimensionall.pdf}};
\node[anchor=north west] (density) at (\paperwidth-\margin-0.5\boxwidth, -\headerheight-2\margin-64cm) {\includegraphics[scale=1]{im/densityall.pdf}}; \node[anchor=north west] (density) at (\paperwidth-\margin-0.5\boxwidth, -\headerheight-2\margin-66cm) {\includegraphics[scale=1]{im/densityall.pdf}};
\posterbox[](\margin, -\headerheight-4\margin-42.5cm)<0.95\boxwidth,8cm>{Boundary Conditions}{ \posterbox[](\margin, -\headerheight-4\margin-40.5cm)<0.95\boxwidth,10cm>{Boundary Conditions}{
Flow across the boundary of $\mathbb{M}_t$ is possible. Therefore, \textcolor{red}{Dirichlet boundary conditions} $\mathbf{P}_{\epsilon, t}(i, j) = 0 \forall x^i_t \text{ or } x^j_t \in \partial \mathbb{M}_t$ have to be applied. Boundary points are detected with $\alpha$\texttt{-shapes} [3] using $\alpha = 0.001 \mathrm{km}^{-1} \approx \mathcal{O}(\epsilon^{-1/2})$. Flow across the boundary $\partial \mathbb{M}_t$ is possible. We apply \key{Dirichlet boundary conditions}
$$\mathbf{P}_{\epsilon, t}(i, j) = 0 \quad \forall x^i_t \text{ or } x^j_t \in \partial \mathbb{M}_t$$
The boundary is detected with $\alpha$\texttt{-shapes} [3] using $\alpha \approx \mathcal{O}(\epsilon^{-1/2})$.
} }
\posterbox[](\margin, -\headerheight-5\margin-50.5cm)<0.95\boxwidth,\paperheight-\headerheight-6\margin-50.5cm>{Spectral Clustering}{ \posterbox[](\margin, -\headerheight-5\margin-50.5cm)<0.95\boxwidth,\paperheight-\headerheight-6\margin-50.5cm>{Spectral Clustering}{
To study the \textcolor{red}{airstreams entering} the QSAS, we construct $\mathbf{Q}_{\epsilon}$ for $t_0, ..., t_{72}$. Eigenvaluedecay for various $\epsilon$ reveals a \textcolor{red}{spectral gap} after the 6\textsuperscript{th} eigenvalue, so we expect to find 6 coherent sets. To study the \key{airstreams entering} the QSAS, sample $x_0^i$ from the QSAS. Construct $\mathbf{Q}_{\epsilon}$ and compute its dominant eigenvalues.\\
\vspace{14.6cm} $\rightarrow$ Find spectral gap.\\
The clustered trajectories differ not only wrt their \textcolor{red}{geometric}, but also in their \textcolor{red}{dynamic properties}. Specifically, two coherent sets are Using $k$-means, group the points into $k$ clusters, using the $k-1$ dominant eigenvectors. \vspace{14.2cm}\\
\parbox{.42\boxwidth}{ \parbox{.42\boxwidth}{
identified, which feature strong vertical, cross-isentropical motion (latent heating) and \textcolor{red}{stabilize the QSAS} via injection of low-PV air masses. Objectively identifying this process presents an advancement of existing research [5].} The clustered trajectories differ not only wrt their \key{geometric}, but also in their \key{dynamic properties}. Specifically, two coherent sets are identified, which feature strong vertical, cross-isentropical motion (latent heating) and \key{stabilize the QSAS} via injection of low-PV air masses. Objectively identifying this process presents an advancement of existing research [5].}
} }
\node[anchor=north west] (spectrum) at (\margin+0.1cm, -\headerheight-5\margin-60cm) {\includegraphics[scale=1]{im/spectrum.pdf}};
\node[anchor=north west] (clustering) at (\margin+17.2cm, -\headerheight-5\margin-58.2cm) {\includegraphics[scale=1, trim={11cm 6.5cm 5.8cm 3.5cm}, clip]{im/3d.pdf}}; \node[anchor=north west] (spectrum) at (\margin+0.1cm, -\headerheight-5\margin-59cm) {\includegraphics[scale=1]{im/spectrum.pdf}};
\node[anchor=north west] (clustering) at (\margin+17.2cm, -\headerheight-5\margin-57.5cm) {\includegraphics[scale=1, trim={11cm 6.5cm 5.8cm 3.5cm}, clip]{im/3d.pdf}};
\node[anchor=south east] (theta) at (\margin+0.95\boxwidth, -\paperheight+\margin+0.1cm) {\includegraphics[scale=1]{im/test.pdf}}; \node[anchor=south east] (theta) at (\margin+0.95\boxwidth, -\paperheight+\margin+0.1cm) {\includegraphics[scale=1]{im/test.pdf}};
\posterbox[anchor=north east](\paperwidth-\margin, -\headerheight-3\margin-81.5cm)<1.05\boxwidth,\paperheight-\headerheight-4\margin-81.5cm>{References}{ \posterbox[anchor=north east](\paperwidth-\margin, -\headerheight-3\margin-81.5cm)<1.07\boxwidth,\paperheight-\headerheight-4\margin-81.5cm>{References}{
\scriptsize \scriptsize
\hspace*{0.5cm} [1] \textsc{Banisch, Ralf and P\' eter Koltai}, \textit{Understanding the Geometry of \hspace*{0.5cm} [1] \textsc{Banisch, Ralf and Koltai, P\'eter}, \textit{Understanding the Geometry of
Transport: Diffusion Maps for Lagrangian Trajectory Data Unravel Coherent Sets}, Chaos 27.3, 2017\\[0.5cm] Transport: Diffusion Maps for Lagrangian Trajectory Data Unravel Coherent Sets}, Chaos 27.3, 2017\\[0.5cm]
\hspace*{0.5cm} [2] \textsc{Detring, Carola et al}, \textit{Occurrence and Transition Probabilities \hspace*{0.5cm} [2] \textsc{Detring, Carola et al.}, \textit{Occurrence and Transition Probabilities
of Omega and High-over-Low Blocking in the Euro-Atlantic Regio}, Weather Clim. Dynam. 2.4, 2021\\[0.5cm] of Omega and High-over-Low Blocking in the Euro-Atlantic Regio}, Weather Clim. Dynam. 2.4, 2021\\[0.5cm]
\hspace*{0.5cm} [3] \textsc{Edelsbrunner, Herbert and Ernst P. Mücke}, \textit{Three-Dimensional \hspace*{0.5cm} [3] \textsc{Edelsbrunner, Herbert and Mücke, Ernst P.}, \textit{Three-Dimensional
Alpha Shapes}, ACM Trans. Graph. 13.1, 1994\\[0.5cm] Alpha Shapes}, ACM Trans. Graph. 13.1, 1994\\[0.5cm]
\hspace*{0.5cm} [4] \textsc{Koltai, P\' eter and Stephan Weiss}, \textit{Diffusion Maps Embedding and \hspace*{0.5cm} [4] \textsc{Koltai, P\' eter and Weiss Stephan}, \textit{Diffusion Maps Embedding and
Transition Matrix Analysis of the Large-Scale Flow Structure in Turbulent Transition Matrix Analysis of the Large-Scale Flow Structure in Turbulent
Rayleigh–B\' enard Convectio}, Nonlinearity 33.4, 2020 \\[0.5cm] Rayleigh–B\' enard Convectio}, Nonlinearity 33.4, 2020 \\[0.5cm]
\hspace*{0.5cm} [5] \textsc{Pfahl, Stephan, C. Schwierz, M. Croci-Maspoli, C. M. Grams, and H. Wernli}, \textit{Importance of Latent Heat Release in Ascending Air Streams for Atmospheric Blocking}, Nature Geoscience 8, no. 8, 2015 \hspace*{0.5cm} [5] \textsc{Pfahl, Stephan et al.}, \textit{Importance of Latent Heat Release in Ascending Air Streams for Atmospheric Blocking}, Nature Geoscience 8, no. 8, 2015
} }
\end{document} \end{document}
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