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\StartSlide{Weather forecasting is difficult.  Why?}
%\NewSlide{Scientific Motivation}
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\vspace*{28pt}

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\item
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indistinguishable initial conditions of the atmosphere diverge
\ECitem

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\item
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numerical models (\textcolor{darkred}{L}) approximate true evolution
(\textcolor{blue}{H})
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\BCitem{\smbl}
\item
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ensemble solutions of \textcolor{darkred}{L} 
needed to forecast the target

\ECitem

\begin{figure}[!h]
\centering
\includegraphics[scale=.6]{../../danforth_graffle/spaghetti_1.jpg}
\end{figure}

\NewSlide{}
\begin{figure}[!h]
\centering
\includegraphics[scale=.4]{../../research_pictures/hurricane.jpg}
\end{figure}

\NewSlide{Where can the process be improved?}
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\vspace*{28pt}

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\item
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Data Assimilation
\BItem
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\item
Identify the initial condition of the model that is \\
{\itshape most like} the observed state of the atmosphere.
\item
Identify the ensemble perturbations which best capture \\
instabilities in the atmosphere.
\EItem
\item
Model Error
\BItem
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\item
Identify and correct for model deficiencies {\itshape without} \\
a degree in meteorology. 
\EItem
\item
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%\StartSlide{$\delta_x$-Shadowing Definition}
\NewSlide{$\delta$-Shadowing Definition}
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Given a system ${y_{n+1} = f(y_n)}$ and a tolerance
$\delta$, a $\sigma$-noisy 
sequence \textcolor{blue}{$\{p_n\}^b_{n=a}$}
is said to be {\itshape $\delta$-shadowed}
so long as there exists a true orbit 
\textcolor{darkred}{$\{x_n\}^b_{n=a}$}
%${\textcolor{black}{x_{n+1}} = f(
%\textcolor{black}{x_n})}$, \\
st.
${|\textcolor{darkred}{x_n} - 
\textcolor{blue}{p_n}| < \delta}$

\begin{figure}[!h]
\centering
\includegraphics[scale=.6]{../../danforth_graffle/shadow_2.jpg}
\end{figure}

\NewSlide{Noise may push the target solution outside of the ensemble
ellipse.  This is fine, so long as {\itshape some} ensemble member is 
still consistent with the target 
(\textcolor{darkred}{red} and \textcolor{blue}{blue} overlap).} 
\begin{figure}[!h]
\centering
\includegraphics[scale=.6]{../../danforth_graffle/flat_1.jpg}
\end{figure}

\NewSlide{However, if a contracting direction begins to expand, or there
is a near tangency of the stable and unstable manifolds, we
may have a shadowing failure or \textcolor{blue}{{\itshape glitch}}.}
\begin{figure}[!h]
\centering
\includegraphics[scale=.6]{../../danforth_graffle/tangency.jpg}
\end{figure}

\NewSlide{However, if a contracting direction begins to expand, or there
is a near tangency of the stable and unstable manifolds, we
may have a shadowing failure or \textcolor{blue}{{\itshape glitch}}.}
\begin{figure}[!h]
\centering
\includegraphics[scale=.4]{../../danforth_graffle/flat_4.jpg}
\end{figure}

\NewSlide{$\delta$-Stalking Definition}
\pagecolor{white}

\fontfamily{ptm}\fontsize{28.0}{30.0pt}\selectfont
Given a system ${y_{n+1} = f(y_n)}$ and a tolerance
$\delta$, a $\sigma$-noisy sequence
\textcolor{blue}{$\{p_n\}^b_{n=a}$} is 
said to be {\itshape $\epsilon$-$\delta$-stalked}
so long as there exists an 
\textcolor{darkbrown}{$\epsilon$-pseudo} orbit 
\textcolor{darkred}{$\{x_n\}^b_{n=a}$},
$|\textcolor{black}{x_{n+1}} - f(
\textcolor{black}{x_n})| <$ \textcolor{darkbrown}{$\epsilon$},
st. ${|\textcolor{darkred}{x_n} - 
\textcolor{blue}{p_n}| < \delta}$

\begin{figure}[!h]
\centering
\includegraphics[scale=.6]{../../danforth_graffle/shadow_3.jpg}
\end{figure}

\NewSlide{$\delta$-Stalking Failure}

\fontfamily{ptm}\fontsize{28.0}{30.0pt}\selectfont
Given a system ${y_{n+1} = f(y_n)}$ and a tolerance
$\delta$, a $\sigma$-noisy sequence
\textcolor{blue}{$\{p_n\}^b_{n=a}$} is 
said to be {\itshape $\epsilon$-$\delta$-stalked}
so long as there exists an 
\textcolor{darkbrown}{$\epsilon$-pseudo} orbit 
\textcolor{darkred}{$\{x_n\}^b_{n=a}$},
$|\textcolor{black}{x_{n+1}} - f(
\textcolor{black}{x_n})| <$ \textcolor{darkbrown}{$\epsilon$},
st. ${|\textcolor{darkred}{x_n} - 
\textcolor{blue}{p_n}| < \delta}$

\begin{figure}[!h]
\centering
\includegraphics[scale=.6]{../../danforth_graffle/shadow_4.jpg}
\end{figure}

\NewSlide{Goal}
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Fixing the following parameters:
  \BItem

\item
variance inflation ($\epsilon$)

\item
stalking distance ($\delta$)

\item
noise ($\sigma$), where 
$|\textcolor{blue}{p_{n+1}} - f(
\textcolor{blue}{p_n})| < \sigma$
  \EItem

\item
How long will \textcolor{red}{$x_n$} stalk \textcolor{blue}{$p_n$}?

\ECitem

\NewSlide{Stalking time in Lorenz 40-D system for parameters
$\epsilon$ (inflation),  $\sigma$ (noise),
$\delta$ (stalking distance).}

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\begin{figure}[!ht]
\centering
\includegraphics[scale=.35]{../../research_pictures/stalking_time_2.jpg}
\end{figure}  

\NewSlide{Model Error Experiment}
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\item
Initialize model with observed global wind field \\
on Jan 1, 2000
	\BItem
	\item
Integrate model for 6 hours
	\item
Compare forecast with reanalysis (observed 'truth')
	\item
Generate nudging term (synchronization)
	\item
Repeat
	\EItem

\item
Find time average model error correction

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Make zeroth order correction to forcing

\item
Does the forecast improve?

\ECitem

\NewSlide{Nudging Movie}
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\begin{figure}[!ht]
\centering
\easymovie[poster,repeat,controls]{\easybox{8.5 in}{5 in}}{nudgeplot.mov}
\end{figure}

\NewSlide{Average vorticity $(\frac{10^{-6}}{s})$ correction term, \\
model error when relaxing towards 1980-1990 reanalysis}

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\begin{figure}[!ht]
\centering
\includegraphics[scale=.45]{../sc_seminar/map.jpg}
\end{figure}  

\NewSlide{Time series average PC of 30 model integrations (1990-2000) \\
Model error correction extends forecast by 18 hours or 38\%!}
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\begin{figure}[!ht]
\centering
\includegraphics[scale=.35]{../sc_seminar/nudge_result.jpg}
\end{figure}

\NewSlide{Themes}
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Chaos $\Rightarrow$ brute force will {\itshape never} work.  We
seek clever ways to approach the limit of predictability.

\item
Computational limitations prohibit the development of techniques 
on the full blown weather model ($10^7$ variables).

\item
Use the simplest model available which still exhibits the 
complicated behavior you seek to understand.

\ECitem

\EndSlide{white}{danforth@math.umd.edu}

