<p><b>duda</b> 2008-05-12 17:50:13 -0600 (Mon, 12 May 2008)</p><p>1) Added DFI section to Initialization chapter.<br>
2) Gave a label to the advection section, to which the DFI section refers.<br>
3) Added several DFI references to the bibliography.<br>
<br>
M description.bbl<br>
M initialization.tex<br>
M discretization.tex<br>
A figures/dfi_schemes.pdf<br>
</p><hr noshade><pre><font color="gray">Modified: trunk/wrf/technote/description.bbl
===================================================================
--- trunk/wrf/technote/description.bbl        2008-05-06 20:44:59 UTC (rev 65)
+++ trunk/wrf/technote/description.bbl        2008-05-12 23:50:13 UTC (rev 66)
@@ -143,6 +143,11 @@
of Clouds and Precipitation,
{\em Mon. Wea. Rev.}, {\bf 132}, 103--120.
+\bibitem[Huang and Lynch(1993)]{huanglynch93}%
+Huang, X.-Y. and P. Lynch, 1993:
+ Diabatic Digital-Filtering Initialization: Application to the HIRLAM Model.,
+ {\em Mon. Wea. Rev.}, {\bf 121}, 589--603.
+
\bibitem[Ide et al.(1997)]{ide97}%
Ide, K., P. Courtier, M. Ghil, and A. C. Lorenc, 1997: Unified notation for data
assimilation: Operational, sequential and variational.
@@ -247,7 +252,20 @@
\bibitem[Lorenc(1986)]{lorenc86}%
Lorenc, A. C., 1986: Analysis methods for numerical weather prediction.
{\em Quart. J. Roy. Meteor. Soc.}, {\bf 112}, 1177--1194.
-
+
+\bibitem[Lynch and Huang(1992)]{lynchhuang92}%,
+Lynch, P., and X.-Y. Huang, 1992: Initialization of the HIRLAM
+ Model Using a Digital Filter. {\em Mon. Wea. Rev.}, {\bf 120},
+ 1019--1034.
+
+\bibitem[Lynch and Huang(1994)]{lynchhuang94}%,
+Lynch, P., and X.-Y. Huang, 1994: Diabatic initialization using recursive filters.
+ {\em Tellus}, {\bf 46A}, 583--597.
+
+\bibitem[Lynch(1997)]{lynch97}%,
+Lynch, P., 1997: The Dolph-Chebyshev Window: A Simple Optimal Filter.
+ {\em Mon. Wea. Rev.}, {\bf 125}, 655--660.
+
\bibitem[Mellor and Yamada(1982)]{melloryamada82}%
Mellor, G. L., and T. Yamada, 1982:
Development of a turbulence closure model for geophysical fluid problems.
Modified: trunk/wrf/technote/discretization.tex
===================================================================
--- trunk/wrf/technote/discretization.tex        2008-05-06 20:44:59 UTC (rev 65)
+++ trunk/wrf/technote/discretization.tex        2008-05-12 23:50:13 UTC (rev 66)
@@ -779,6 +779,7 @@
\subsection{Advection}
+\label{advection}
The advection terms in the ARW solver are in the form of a flux divergence and
are a subset of the RHS terms in equations \eqref{u-rhs} --
Added: trunk/wrf/technote/figures/dfi_schemes.pdf
===================================================================
--- trunk/wrf/technote/figures/dfi_schemes.pdf         (rev 0)
+++ trunk/wrf/technote/figures/dfi_schemes.pdf        2008-05-12 23:50:13 UTC (rev 66)
@@ -0,0 +1,1358 @@
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+startxref
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+%%EOF
Modified: trunk/wrf/technote/initialization.tex
===================================================================
--- trunk/wrf/technote/initialization.tex        2008-05-06 20:44:59 UTC (rev 65)
+++ trunk/wrf/technote/initialization.tex        2008-05-12 23:50:13 UTC (rev 66)
@@ -422,3 +422,179 @@
the incoming data, that is not a requirement. Pre-processor {\it real} will vertically interpolate
(linear in depth below the ground) from the incoming levels to the requested soil layers to be
used within the model.
+
+\section{Digital Filtering Initialization}
+
+Version 3 of the ARW provides a digital filtering initialization (DFI) to
+remove noise, which results from imbalances between mass and wind fields,
+from the model initial state. Under the assumption that any noise is of
+higher frequency than meteorologically significant modes, DFI attempts to
+remove this noise by filtering all oscillations above a specified cutoff
+frequency. Accordingly, the filters in the ARW DFI are low-pass digital
+filters, which are applied to time series of model fields; the {\it initialized}
+model state is the output of the filter at some prescribed time,
+e.g., the analysis time. Time series of model states are generated through
+combinations of adiabatic, backward integration and diabatic, forward
+integration in the model, with the choice of DFI scheme determining the
+specific combination of integrations. Three DFI schemes --- digital filter
+launch (DFL; \cite{lynchhuang94}), diabatic DFI (DDFI; \cite{huanglynch93}),
+and twice DFI (TDFI; \cite{lynchhuang94}) --- are available.
+
+\subsection{Filter Design}
+
+In the ARW DFI, either nonrecursive (i.e., finite impulse response) digital
+filters or a recursive (i.e., infinite impulse response) digital filter may
+be used. The coefficients for the nonrecursive digital filters may be
+computed according to one of two methods, while the coefficients for the
+recursive filter are computed according to a single method.
+
+A general nonrecursive digital filter is of the form
+
+\begin{equation}
+y_n = \sum_{k=-N}^{N} h_k x_{n-k},
+\label{fir_filter}
+\end{equation}
+
+</font>
<font color="blue">oindent
+where $y_n$ is the output of the filter at time $n$, the $h_k$ are the
+coefficients of the filter, and $\{ \ldots , x_{n-1}, x_n, x_{n+1}, \ldots \}$
+is the sequence of input values to be filtered; such a filter is said to have
+span $2N+1$.
+
+One method for deriving the coefficients of a nonrecursive digital filter is
+the sinc-windowed method, described in the context of DFI by \cite{lynchhuang92}.
+In the ARW DFI, either the Lanczos, Hamming, Blackman, Kaiser, Potter, or
+Dolph-Chebyshev windows may be used; the Dolph-Chebyshev window is described
+by \cite{lynch97}. However, when a filter with a shorter span is desired,
+another nonrecursive digital filter, the Dolph filter, may be used. \cite{lynch97}
+describes the construction of the Dolph filter, and demonstrates that this
+filter has properties nearly indistinguishable from those of an optimal filter,
+which minimizes the maximum difference between a filter's transfer function
+and an ideal transfer function in the pass and stop bands.
+
+The only recursive filter in the ARW DFI is the second-order Quick-Start
+filter of \cite{lynchhuang94}. In general, a recursive digital filter that
+depends only on past and present values of the input, and on past values of
+the output, is of the form
+
+\begin{equation}
+y_n = \sum_{k=0}^{N} h_k x_{n-k} + \sum_{k=1}^{N} b_k y_{n-k}.
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent
+However, this form is inconvenient when the inputs to the filter consist of
+model states, and we wish to avoid storing many such states. \cite{lynchhuang94}
+show how this type of recursive filter can be reformulated to have the same
+form as a nonrecursive filter, and thus, we can think of the second-order
+Quick-Start filter as having the same form as (\ref{fir_filter}).
+
+\subsection{DFI Schemes}
+
+The ARW supports three different DFI schemes, illustrated graphically in
+Fig. \ref{figure:dfi_types}. The DFL scheme produces an initialized model
+state valid some time after the model analysis time, while the DDFI and TDFI
+schemes produce initialized states valid at the analysis time.
+
+%
+% Figure showing available DFI schemes
+%
+\begin{figure}
+ \centering
+ \includegraphics[width=6.5in]{figures/dfi_schemes.pdf}
+ \caption{\label{figure:dfi_types}An illustration showing the three available DFI schemes: digital filter
+ launch, diabatic digital filter initialization, and twice digital filter initialization.}
+\end{figure}
+
+\subsubsection{DFL}
+
+In the DFL scheme, forward integration with full model physics and diffusion
+begins at the initial time and continues for $2N$ time steps, during which
+time a filtered model state valid $N$ time steps beyond the analysis time is
+computed as in (\ref{fir_filter}). Then, the initialized simulation is
+launched from the midpoint of the filtering period. For any model state ${\bf X}$,
+let $\left[ {\bf X} \right]_n^D$ give the model state after diabatically
+integrating $n$ time steps forward in time; we emphasize that the superscript
+$D$ indicates diabatic integration, in contrast to adiabatic integration.
+Then, the DFL scheme is expressed as
+
+\begin{equation}
+{\bf X}_{ini} = \sum_{n=0}^{2N} h_n \left[ {\bf X}_{ana} \right]_n^D,
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent
+where ${\bf X}_{ini}$ is the initialized model state, ${\bf X}_{ana}$ is the
+model analysis or model initial state generated by {\it real} preprocessor,
+and the $h_n$ are the filter coefficients.
+
+\subsubsection{DDFI}
+
+To produce an initialized state valid at the model analysis time, the DDFI
+scheme begins with an adiabatic, backward integration for $N$ time steps,
+followed by a diabatic, forward integration for $2N$ time steps, during which
+filtering takes place. This filtered state is valid at the model analysis time.
+Letting $\left[ {\bf X}_{ana} \right]_{-n}^A$ denote the model state after
+adiabatic, backward integration for $n$ time steps from the model analysis or
+model initial state, ${\bf X}_{ana}$, the DDFI scheme is expressed as
+
+\begin{equation}
+{\bf X}_{ini} = \sum_{n=0}^{2N} h_n \left[ \left[ {\bf X}_{ana} \right]_{-N}^A \right]_n^D,
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent
+where ${\bf X}_{ini}$ is the initialized model state valid at the model analysis time.
+
+\subsubsection{TDFI}
+
+The TDFI scheme involves two applications of the digital filter. Adiabatic,
+backward integration proceeds from the model initial time for $2N$ time steps,
+during which a filter is applied. The filtered state is valid at time $-N \Delta t$;
+from this filtered state, a forward, diabatic integration is launched. The
+second integration proceeds for $2N$ time steps, during which a second filter
+is applied, giving a filtered model state valid at this model analysis time.
+The TDFI scheme is expressed as
+
+\begin{equation}
+{\bf X}_{ini} = \sum_{n=0}^{2N} h_n \left[ \sum_{n'=0}^{2N} h_{n'} \left[ {\bf X}_{ana} \right]_{-n}^A \right]_n^D.
+</font>
<font color="blue">otag
+\end{equation}
+
+\subsection{Backward Integration}
+
+To diabatically integrate backward in time, it suffices to disable all
+diabatic processes and to negate the model time step, $\Delta t$, as well as
+the sign of the horizontal velocity, $U$, in the odd-order advection
+operators of Section \ref{advection}, which become
+
+\begin{align}
+\hbox{3}^{rd} \hbox{ order:} ~~~~ &
+(\overline{q}^{x_{adv}})_{i-1/2} =
+(\overline{q}^{x_{adv}})_{i-1/2}^{4^{th}}
+</font>
<font color="blue">otag
+\\
+&~~~~~~~~~~~~~~~~~~~~
+- \hbox{sign}(U){1 \over 12} \bigl[
+(q_{i+1}-q_{i-2}) - 3(q_i - q_{i-1}) \bigr]
+</font>
<font color="blue">otag
+\\
+\hbox{5}^{th} \hbox{ order:} ~~~~ &
+(\overline{q}^{x_{adv}})_{i-1/2} =
+(\overline{q}^{x_{adv}})_{i-1/2}^{6^{th}}
+</font>
<font color="blue">otag
+\\
+&~~~~~~~~~~~~~~~~~~~~
++ \hbox{sign}(U){1 \over 60} \bigl[
+(q_{i+2}-q_{i-3}) - 5(q_{i+1} - q_{i-2})
++ 10(q_i - q_{i-1})
+\bigr].
+</font>
<font color="blue">otag
+\end{align}
+
+When specified boundary conditions are used, as in Section \ref{lbc_spec},
+the model boundaries before the model initial time are taken to be the same
+as those valid at the model initial time. We note that, with a negated time
+step, the linear ramping functions $F_1$ and $F_2$ of (\ref{lbc_relax})
+change sign, and, consequently, so does the sign of the tendency for a
+prognostic variable $\psi$.
</font>
</pre>