<p><b>duda</b> 2008-05-19 12:30:30 -0600 (Mon, 19 May 2008)</p><p>Add text to DFI section, and list DFI as an acronym in Appendix C.<br>
<br>
M initialization.tex<br>
M appenc.tex<br>
</p><hr noshade><pre><font color="gray">Modified: trunk/wrf/technote/appenc.tex
===================================================================
--- trunk/wrf/technote/appenc.tex 2008-05-15 16:04:15 UTC (rev 71)
+++ trunk/wrf/technote/appenc.tex 2008-05-19 18:30:30 UTC (rev 72)
@@ -11,6 +11,7 @@
CGM & Conjugate Gradient Method \\
COAMPS & Coupled Ocean / Atmosphere Mesoscale Prediction System \\
COMET & Cooperative Program for Operational Meteorology, Education, and Training \\
+DFI & Digital Filtering Initialization \\
DTC & Developmental Testbed Center \\
ECMWF & The European Centre for Medium-Range Weather Forecasts \\
EOF & Empirical Orthogonal Function \\
@@ -33,11 +34,11 @@
NFS & Non-hydrostatic Forecast System (Central Weather Bureau of Taiwan) \\
NMM & Nonhydrostatic Mesoscale Model \\
NOAA & National Oceanographic and Atmospheric Administration \\
-NRL & Navy Research Laboratory \\
\end{tabular}
</font>
<font color="black">ewpage
</font>
<font color="gray">ormalsize
\begin{tabular}[t]{ll}
+NRL & Navy Research Laboratory \\
NWP & Numerical Weather Prediction \\
OSU & Oregon State University \\
PBL & Planetary Boundary Layer \\
Modified: trunk/wrf/technote/initialization.tex
===================================================================
--- trunk/wrf/technote/initialization.tex 2008-05-15 16:04:15 UTC (rev 71)
+++ trunk/wrf/technote/initialization.tex 2008-05-19 18:30:30 UTC (rev 72)
@@ -1,600 +1,605 @@
-\chapter{Initial Conditions}
-\label{initialization_chap}
-
-The ARW may be run with initial conditions that are defined
-analytically for idealized simulations, or it may be run using
-interpolated data from either a large-scale analysis or forecast for
-real-data cases. Both 2D and 3D tests cases for idealized
-simulations are provided.
-Several sample cases for real-data simulations are provided, which
-rely on pre-processing from an external package that converts
-the large-scale GriB data into a format suitable for ingest by the ARW's
-real-data processor.
-
-The programs that generate the specific initial conditions for the selected
-idealized or real-data case function similarly. They provide the ARW with:
-\begin{itemize}\setlength{\parskip}{-5pt}
-\item input data that is on the correct horizontal and vertical staggering;
-\item hydrostatically balanced reference state and perturbation fields; and
-\item metadata specifying such information as the date, grid physical characteristics,
-and projection details.
-\end{itemize}
-</font>
<font color="red">oindent For neither the idealized nor the real-data cases
-are the initial conditions enhanced with observations. However, output from
-the ARW system initial condition programs is suitable as input to the WRF variational
-assimilation package (see Chapter \ref{var_chap}).
-
-\section{Initialization for Idealized Conditions}
-
-The ARW comes with a number of test cases using idealized
-environments, including large eddy simulations (em\_les),
-sea breezes ( em\_seabreeze), mountain waves (em\_hill2d\_x), squall lines
-(em\_squall2d\_x, em\_squall2d\_y), supercell thunderstorms
-(em\_quarter\_ss), gravity currents (em\_grav2d\_x), baroclinic
-waves (em\_b\_wave), and global systems (em\_heldsuarez).
-A brief description of these test cases can be
-found in the README\_test\_cases file provided in the ARW release.
-The test cases include examples of atmospheric
-flows at fine scales (e.g., the gravity current example has a grid-spacing of
-100 meters and a time step of 1 second) and examples of flow at large
-scales (e.g., the Held Suarez global test case uses a grid-spacing around 600 km and
-a time step of 1800 s), in addition to the traditional mesoscale and
-cloudscale model simulations. The test suite allows an ARW user to
-easily reproduce these known solutions. The test suite is also the
-starting point for constructing idealized flow simulations by modifying
-initializations that closely resemble a desired initialization.
-
-All of these tests use as input a 1D sounding specified as a function of
-geometric height $z$ (except for the baroclinic wave case that uses a 2D
-sounding specified in $[y,z]$), and, with the exception of the baroclinic
-wave test case, the sounding files are in text format that can be
-directly edited by the user. The 1D specification of the sounding in
-these test files requires the surface values of pressure, potential
-temperature, and water vapor mixing ratio, followed by the potential
-temperature, vapor mixing ratio, and horizontal wind components at some
-heights above the surface. The initialization programs for each case
-assume that this moist sounding represents an atmosphere in hydrostatic
-balance.
-
-Two sets of thermodynamic fields are needed for the model--- the
-reference state and the perturbation state (see Chapter
-\ref{equation_chap} for further discussion of the equations). The
-reference state used in the idealized initializations is computed using
-the input sounding from which the moisture is discarded (because the
-reference state is dry). The perturbation state is computed using the full
-moist input sounding. The procedure for computing the hydrostatically-balanced
-ARW reference and perturbation state variables from the input
-sounding is as follows. First, density and both a dry and full
-hydrostatic pressure are computed from the input sounding at the input
-sounding levels $z$. This is accomplished by integrating the
-hydrostatic equation vertically up the column using the surface pressure
-and potential temperature as the lower boundary condition. The
-hydrostatic equation is
-%
-\begin{equation} \delta_z p = - {\overline
-\rho}^z g (1 + (R_d/R_v) {\overline q_v}^z),
-\label{init_hydro}
-\end{equation}
-%
-</font>
<font color="red">oindent
-where $\overline{\rho}^z$ is a two point average between input sounding
-levels, and $\delta_z p$ is the difference of the pressure between input
-sounding levels divided by the height difference. Additionally, the
-equation of state is needed to close the system:
-%
-\begin{equation} \alpha_d = {1 \over \rho_d} = {R_d
-\theta \over p_o} \biggl(1 + {R_d \over R_v}q_v\biggr)
-\biggl({p \over p_o}\biggr)^{-{c_v \over c_p}},
-\label{init_state}
-\end{equation}
-%
-</font>
<font color="red">oindent
-where $q_v$ and $\theta$ are given in the input sounding.
-\eqref{init_hydro} and \eqref{init_state} are a coupled set of nonlinear
-equations for $p$ and $\rho$ in the vertical integration, and they are
-solved by iteration. The dry pressure on input sounding levels is
-computed by integrating the hydrostatic relation down from the top,
-excluding the vapor component.
-
-Having computed the full pressure (dry plus vapor) and dry air pressure
-on the input sounding levels, the pressure at the model top ($p_{dht}$)
-is computed by linear interpolation in height (or possibly
-extrapolation) given the height of the model top (an input variable).
-The column mass $\mu_d$ is computed by interpolating the dry air
-pressure to the surface and subtracting from it $p_{dht}$. Given the
-column mass, the dry-air pressure at each $\eta$ level is known from the
-coordinate definition \eqref{eta_def}, repeated here
-%
-\begin{equation}
-\eta = (p_{dh}-p_{dht})/\mu_d ~~~~~~~{\rm where
- }~~~ \mu_d = p_{dhs}-p_{dht},
-</font>
<font color="red">otag
-\end{equation}
-%
-</font>
<font color="red">oindent
-and the pressures $p_{dhs}$ and $p_{dht}$ refer to the dry atmosphere.
-The potential
-temperature from the input sounding is interpolated to
-each of the model pressure levels, and the equation of state
-\eqref{init_state} is used to compute the inverse density
-$\alpha_d$. Finally, the
-ARW's hydrostatic relation \eqref{hydrostatic_relation},
-in discrete form
-%
-\begin{equation}
-\delta_\eta \phi = - \alpha_d \mu_d
-</font>
<font color="red">otag
-\end{equation}
-</font>
<font color="red">oindent
-is used to compute the geopotential. This procedure is used to compute
-the reference state (assuming a dry atmosphere) and the full state
-(using the full moist sounding). The perturbation variables are
-computed as the difference between the reference and full state. It
-should also be noted that in the nonhydrostatic model integration,
-the inverse density $\alpha_d$ is diagnosed from the geopotential using
-this equation of state, and the pressure is diagnosed from the equation
-of state using the inverse density $\alpha_d$ and the prognostic potential
-temperature $\theta$. Thus, the ARW's prognostic variables $\mu_d$,
-$\theta$, and $\phi$ are in exact hydrostatic balance for the model
-equations (to machine roundoff).
-
-\section{Initialization for Real-Data Conditions}
-
-The initial conditions for the real-data cases are pre-processed through a separate
-package called the WRF Preprocessing System (WPS, see Fig. \ref{figure:WPS_real_wrf}).
-The output from WPS is passed to the
-real-data pre-processor in the ARW--- program {\it real}--- which generates initial and lateral boundary
-conditions. This section is primarily about the steps taken to build the
-initial and the lateral boundary conditions for a real-data case. Even though the
-WPS is outside of the ARW system, a brief description is appropriate to see how the
-raw meteorological and static terrestrial data are brought into the model
-for real-data cases.
-
-\subsection{Use of the WRF Preprocessing System by the ARW}
-
-The WPS is a set of programs that takes
-terrestrial and meteorological data (typically in GriB format) and transforms them for input to
-the ARW pre-processor program for real-data cases ({\it real}).
-Figure \ref {figure:WPS_real_wrf} shows the flow of data into and out of the WPS system.
-The first step for the WPS is to define a physical grid (including
-the projection type, location on the globe,
-number of grid points, nest locations, and grid distances) and
-to interpolate static fields to the prescribed domain.
-Independent of the domain configuration,
-an external analysis or forecast is processed by the WPS GriB decoder,
-which diagnoses required fields and
-reformats the GriB data into an internal binary format.
-With a specified domain,
-WPS horizontally interpolates the meteorological data onto the projected domain(s).
-The output data from WPS supplies a complete 3-dimensional snapshot of the atmosphere
-on the selected model grid's horizontal staggering at the selected time slices,
-which is sent to the ARW pre-processor program for real-data cases.
-
-%
-% Figure showing WPS and real and ARW
-%
-\begin{figure}
- \centering
- \includegraphics[width=6in]{figures/WPS_real_wrf.pdf}
- \caption{\label{figure:WPS_real_wrf}Schematic showing
-the data flow and program components in WPS, and how WPS feeds initial data to the ARW.
-Letters in the rectangular boxes indicate program names.
-GEOGRID: defines the model domain and creates static files of terrestrial data. UNGRIB:
-decodes GriB data. METGRID: interpolates meteorological data to the model domain.}
-\end{figure}
-
-The input to the ARW real-data processor from
-WPS contains 3-dimensional fields of temperature (K), relative humidity
-(%), geopotential height (m), pressure (Pa),
-and the horizontal components of momentum (m/s, already rotated to the model
-projection).
-The 2-dimensional static terrestrial fields include:
-albedo, Coriolis parameters, terrain elevation, vegetation/land-use type,
-land/water mask, map scale factors, map rotation angle, soil texture category, vegetation greenness fraction,
-annual mean temperature,
-and latitude/longitude.
-The 2-dimensional time-dependent fields from the external model, after processing by WPS, include:
-surface pressure and sea-level pressure (Pa), layers of soil temperature (K) and soil moisture (kg/kg,
-either total moisture, or
-binned into total and liquid content),
-snow depth (m), skin temperature (K), and sea ice.
-
-\subsection{Reference State}
-\label{initialization_real_base_section}
-Identical to the idealized initializations, there is a partitioning of some of the
-meteorological data into reference and perturbation fields.
-For real-data cases, the reference state is defined by terrain elevation and three constants:
-\begin{itemize}\setlength{\parskip}{-5pt}
-\item $p_{0}$ ($10^5$ Pa) reference sea level pressure;
-\item $T_{0}$ (usually 270 to 300 K) reference sea level temperature; and
-\item $A$ (50 K) temperature difference between the pressure levels of $p_{0}$ and $p_{0}/e$.
-\end{itemize}
-
-</font>
<font color="red">oindent Using these parameters, the dry reference state surface pressure is
-\begin{equation}
-p_{dhs} = p_{0}~exp\Bigg({-T_{0} \over A} +
- \sqrt{ {\bigg( {T_{0} \over A } \bigg)}^2 - ~
- { 2\phi_{sfc} \over { A~R_d}} } \Bigg).
-\label{init_psurf}
-\end{equation}
-
-</font>
<font color="red">oindent From \eqref{init_psurf}, the 3-dimensional reference pressure (dry hydrostatic pressure $p_{dh}$)
-is computed as
-a function of the vertical coordinate $\eta$ levels and the model top $p_{dht}$:
-\begin{equation}
-p_{dh} = \overline{p}_d = \eta ~( p_{dhs} - p_{dht} ) + p_{dht}.
-\label{init_pbar}
-\end{equation}
-
-</font>
<font color="red">oindent With \eqref{init_pbar}, the reference temperature is a straight line on a skew-T plot, defined as
-\begin{equation}
-T = T_0 + A~ln {\overline{p}_d \over p_0}.
-</font>
<font color="red">otag
-\end{equation}
-
-</font>
<font color="red">oindent From the reference temperature and pressure,
-the reference potential temperature is then defined as
-
-</font>
<font color="red">oindent
-\begin{equation}
-\overline{\theta}_d = {\bigg( T_{0} + A~ln{\overline{p}_d \over p_{0} } \bigg) }
-{\bigg( {p_0 \over \overline{p}_d } \bigg) }
-^{R_d \over C_p},
-\label{init_thetad}
-\end{equation}
-
-</font>
<font color="red">oindent and the reciprocal of the reference density using
-\eqref{init_pbar} and \eqref{init_thetad} is given by
-\begin{equation}
-\overline{\alpha}_d = {1 \over \overline{\rho}_d} = {{R_d ~\overline{\theta}_d}\over p_{0} }~\bigg(
-{\overline{p}_d \over p_{0} } \bigg)^{-{C_v \over C_p}}.
-\label{init_alphabar}
-\end{equation}
-
-</font>
<font color="red">oindent The base state difference of the dry surface pressure
-from \eqref{init_psurf} and the model top is
-given as
-\begin{equation}
-\overline{\mu}_d = p_{dhs} - p_{dht}.
-\label{init_mubar}
-\end{equation}
-
-</font>
<font color="red">oindent
-From \eqref{init_alphabar} and \eqref{init_mubar},
-the reference state geopotential defined from the hydrostatic relation is
-\begin{equation}
-\delta_{\eta} \overline{\phi} = -\overline{\alpha}_d~\overline{\mu}_d.
-</font>
<font color="red">otag
-\end{equation}
-
-
-\subsection{Vertical Interpolation and Extrapolation}
-The ARW real-data preprocessor vertically interpolates using functions of dry pressure.
-The input data from WPS contains both a total pressure and a moisture field (typically
-relative humidity). Starting at the top each column of input pressure data, the integrated moisture
-is subtracted from the pressure field step-wise down to the surface.
-Then, by removing the pressure at the model
-lid, the total dry surface pressure $p_{sd}$ diagnosed from WPS defines the
-model total dry column pressure
-\begin{equation}
-\mu_d = \overline{\mu}_d + \mu_d' = p_{sd} - p_{dht}.
-\label{init_mutotal}
-\end{equation}
-
-</font>
<font color="red">oindent
-With the ARW vertical coordinate $\eta$, the model lid $p_{dht}$, and the column dry
-pressure known at each $(i,j,k)$ location, the 3-dimensional arrays are interpolated.
-
-In the free atmosphere up to the model lid, the vertical calculations are always interpolations.
-However, near the model surface, it is possible to have an inconsistency between the input
-surface pressure (based largely on the input surface elevation) and the ARW surface
-pressure (possibly with a much higher resolution topography). These inconsistencies
-may lead to an extrapolation. The default behavior for extrapolating the horizontal winds and
-the relative humidity below the known surface is to keep the values constant, with zero vertical gradient.
-For the potential temperature, by default a -6.5 $K/km$ lapse rate for the temperature is applied.
-The vertical interpolation of the geopotential field is optional and is
-handled separately. Since a known lower boundary condition exists
-(the geopotential is defined as zero at the pressure at sea-level), no extrapolation is required.
-
-
-
-\subsection{Perturbation State}
-In the real-data preprocessor, first a topographically defined reference state is computed,
-then the input 3-dimensional data are vertically
-interpolated in dry pressure space. With the potential temperature $\theta$ and mixing ratio
-$q_v$ available on each $\eta$ level, the pressure, density, and height diagnostics are
-handled.
-</font>
<font color="red">oindent The perturbation dry column pressure
-field given the reference dry column pressure \eqref{init_mubar} is
-\begin{equation}
-\mu_d' = \mu_d - \overline{\mu}_d,
-\label{init_muprime}
-\end{equation}
-
-</font>
<font color="red">oindent where $\mu_d$ is the column total dry pressure.
-Starting with the reference state fields
-(\ref{init_pbar}, \ref{init_alphabar}, and \eqref{init_mubar}) and the
-dry surface pressure perturbation (\ref{init_muprime}),
-the perturbation fields for pressure and inverse density are diagnosed.
-The pressure perturbation includes moisture and is diagnosed from
-the hydrostatic equation
-%
-\begin{equation}
-\delta_{\eta} p' = \mu'_d \bigg(1 + {\overline{q_v}^\eta} \bigg) +
- \overline{q_v}^\eta~\overline \mu_d,
-%\label{init_pprime}
-</font>
<font color="red">otag
-\end{equation}
-%
-</font>
<font color="red">oindent
-which is
-integrated down from
-at the model top (where $p'= 0$) to recover $p'$.
-The total dry inverse density is given as
-\begin{equation}
-\alpha_d = {R_d \over p_{0} } ~ \theta~ \bigg( 1 + {R_v \over R_d}q_v \bigg)~
- \bigg( {{p'_d + \overline {p}_d} \over p_{0} } \bigg )^{-{C_v \over C_p}},
-</font>
<font color="red">otag
-\end{equation}
-
-</font>
<font color="red">oindent which defines the perturbation field for inverse density
-
-\begin{equation}
-\alpha'_d = \alpha_d - \overline{\alpha}_d.
-</font>
<font color="red">otag
-\end{equation}
-
-</font>
<font color="red">oindent
-The perturbation geopotential
-is diagnosed from the hydrostatic relation
-\begin{equation}
-\delta_{\eta} \phi' = - \big ( {\mu}_d \alpha'_d + \mu'_d
-\overline{\alpha}_d \big )
-</font>
<font color="red">otag
-\end{equation}
-%
-by upward integration using the terrain elevation as the lower boundary condition.
-
-\subsection{Generating Lateral Boundary Data}
-
-This section deals with generating the separate lateral boundary condition file used
-exclusively for the real-data cases. For information
-on which lateral boundary options are available for both the idealized and real-data
-cases, see Chapter \eqref{lbc_chap}.
-
-The specified
-lateral boundary condition for the coarse grid for real-data cases is supplied by an external file that is
-generated by program {\it real}.
-This file contains
-records for the fields $u$, $v$, $\theta$, $q_v$, $\phi'$, and $\mu'_d$ that are used by the ARW to
-constrain the lateral boundaries (other fields are in the boundary file, but are not used).
-The lateral boundary file holds one less time period than was processed by WPS.
-Each of these variables has both
-a valid value at the initial time of the lateral boundary time and a tendency term to get to the
-next boundary time period. For example, assuming a 3-hourly availability of data from WPS,
-the first time period of the lateral boundary file
-for $u$ would contain data for both coupled $u$ (map scale factor and $\mu_d$ interpolated to
-the variable's
-staggering) at the 0 h time
-
-\begin{equation}
-%U_{0h} = {{\mu_u~u}\over{m_u}} \bigg | _{0h}
-U_{0h} = {{\overline{\mu_d}^x u}\over{\overline{m}^x}} \bigg | _{0h},
-</font>
<font color="red">otag
-\end{equation}
-</font>
<font color="red">oindent and a tendency value defined as
-\begin{equation}
-U_t = { U_{3h} - U_{0h} \over 3h},
-</font>
<font color="red">otag
-\end{equation}
-
-</font>
<font color="red">oindent which would take a grid point from the initial value to the value at the next large-scale time
-during 3 simulation hours.
-The horizontal momentum fields are coupled both with $\mu_d$ and the inverse map factor. The
-other 3-dimensional fields ($\theta$, $q_v$, and $\phi'$) are coupled only with $\mu_d$.
-The 2-dimensional $\mu'_d$ lateral boundary field is not coupled.
-
-Each lateral boundary field
-is defined along the four sides of the
-rectangular grid (loosely referred to as the north, south, east, and west sides).
-The boundary values and tendencies for vertical velocity and the non-vapor moisture species are included
-in the external lateral boundary file, but act as
-place-holders for the nested boundary data for the fine grids.
-The width of the lateral
-boundary along each of the four sides is user selectable at run-time.
-
-\subsection{Masking of Surface Fields}
-
-Some of the meteorological and static fields are ``masked''. A masked field is one in which
-the values are typically defined only over water (e.g., sea surface temperature) or defined
-only over land (e.g., soil temperature).
-The need to match all of the masked fields consistently to each other requires additional steps
-for the real-data cases due to the masked data's presumed use in various physics packages in the soil,
-at the surface, and in the boundary layer.
-If the land/water
-mask for a location is flagged as a water point, then the vegetation and soil categories must also
-recognize the location as the special water flag for each of their respective categorical indices.
-
-The values for the soil temperature and soil moisture come from WPS on the
-native levels originally defined for those variables
-in the large-scale model. WPS does no vertical interpolation for the
-soil data. While it is typical to try to match the ARW soil scheme with
-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="red">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="red">otag
-\end{equation}
-
-</font>
<font color="red">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="red">otag
-\end{equation}
-
-</font>
<font color="red">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="red">otag
-\end{equation}
-
-</font>
<font color="red">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="red">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="red">otag
-\\
-&~~~~~~~~~~~~~~~~~~~~
-- \hbox{sign}(U){1 \over 12} \bigl[
-(q_{i+1}-q_{i-2}) - 3(q_i - q_{i-1}) \bigr]
-</font>
<font color="red">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="red">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="red">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 $\psi$.
+\chapter{Initial Conditions}
+\label{initialization_chap}
+
+The ARW may be run with initial conditions that are defined
+analytically for idealized simulations, or it may be run using
+interpolated data from either a large-scale analysis or forecast for
+real-data cases. Both 2D and 3D tests cases for idealized
+simulations are provided.
+Several sample cases for real-data simulations are provided, which
+rely on pre-processing from an external package that converts
+the large-scale GriB data into a format suitable for ingest by the ARW's
+real-data processor.
+
+The programs that generate the specific initial conditions for the selected
+idealized or real-data case function similarly. They provide the ARW with:
+\begin{itemize}\setlength{\parskip}{-5pt}
+\item input data that is on the correct horizontal and vertical staggering;
+\item hydrostatically balanced reference state and perturbation fields; and
+\item metadata specifying such information as the date, grid physical characteristics,
+and projection details.
+\end{itemize}
+</font>
<font color="blue">oindent For neither the idealized nor the real-data cases
+are the initial conditions enhanced with observations. However, output from
+the ARW system initial condition programs is suitable as input to the WRF variational
+assimilation package (see Chapter \ref{var_chap}).
+
+\section{Initialization for Idealized Conditions}
+
+The ARW comes with a number of test cases using idealized
+environments, including large eddy simulations (em\_les),
+sea breezes ( em\_seabreeze), mountain waves (em\_hill2d\_x), squall lines
+(em\_squall2d\_x, em\_squall2d\_y), supercell thunderstorms
+(em\_quarter\_ss), gravity currents (em\_grav2d\_x), baroclinic
+waves (em\_b\_wave), and global systems (em\_heldsuarez).
+A brief description of these test cases can be
+found in the README\_test\_cases file provided in the ARW release.
+The test cases include examples of atmospheric
+flows at fine scales (e.g., the gravity current example has a grid-spacing of
+100 meters and a time step of 1 second) and examples of flow at large
+scales (e.g., the Held Suarez global test case uses a grid-spacing around 600 km and
+a time step of 1800 s), in addition to the traditional mesoscale and
+cloudscale model simulations. The test suite allows an ARW user to
+easily reproduce these known solutions. The test suite is also the
+starting point for constructing idealized flow simulations by modifying
+initializations that closely resemble a desired initialization.
+
+All of these tests use as input a 1D sounding specified as a function of
+geometric height $z$ (except for the baroclinic wave case that uses a 2D
+sounding specified in $[y,z]$), and, with the exception of the baroclinic
+wave test case, the sounding files are in text format that can be
+directly edited by the user. The 1D specification of the sounding in
+these test files requires the surface values of pressure, potential
+temperature, and water vapor mixing ratio, followed by the potential
+temperature, vapor mixing ratio, and horizontal wind components at some
+heights above the surface. The initialization programs for each case
+assume that this moist sounding represents an atmosphere in hydrostatic
+balance.
+
+Two sets of thermodynamic fields are needed for the model--- the
+reference state and the perturbation state (see Chapter
+\ref{equation_chap} for further discussion of the equations). The
+reference state used in the idealized initializations is computed using
+the input sounding from which the moisture is discarded (because the
+reference state is dry). The perturbation state is computed using the full
+moist input sounding. The procedure for computing the hydrostatically-balanced
+ARW reference and perturbation state variables from the input
+sounding is as follows. First, density and both a dry and full
+hydrostatic pressure are computed from the input sounding at the input
+sounding levels $z$. This is accomplished by integrating the
+hydrostatic equation vertically up the column using the surface pressure
+and potential temperature as the lower boundary condition. The
+hydrostatic equation is
+%
+\begin{equation} \delta_z p = - {\overline
+\rho}^z g (1 + (R_d/R_v) {\overline q_v}^z),
+\label{init_hydro}
+\end{equation}
+%
+</font>
<font color="blue">oindent
+where $\overline{\rho}^z$ is a two point average between input sounding
+levels, and $\delta_z p$ is the difference of the pressure between input
+sounding levels divided by the height difference. Additionally, the
+equation of state is needed to close the system:
+%
+\begin{equation} \alpha_d = {1 \over \rho_d} = {R_d
+\theta \over p_o} \biggl(1 + {R_d \over R_v}q_v\biggr)
+\biggl({p \over p_o}\biggr)^{-{c_v \over c_p}},
+\label{init_state}
+\end{equation}
+%
+</font>
<font color="blue">oindent
+where $q_v$ and $\theta$ are given in the input sounding.
+\eqref{init_hydro} and \eqref{init_state} are a coupled set of nonlinear
+equations for $p$ and $\rho$ in the vertical integration, and they are
+solved by iteration. The dry pressure on input sounding levels is
+computed by integrating the hydrostatic relation down from the top,
+excluding the vapor component.
+
+Having computed the full pressure (dry plus vapor) and dry air pressure
+on the input sounding levels, the pressure at the model top ($p_{dht}$)
+is computed by linear interpolation in height (or possibly
+extrapolation) given the height of the model top (an input variable).
+The column mass $\mu_d$ is computed by interpolating the dry air
+pressure to the surface and subtracting from it $p_{dht}$. Given the
+column mass, the dry-air pressure at each $\eta$ level is known from the
+coordinate definition \eqref{eta_def}, repeated here
+%
+\begin{equation}
+\eta = (p_{dh}-p_{dht})/\mu_d ~~~~~~~{\rm where
+ }~~~ \mu_d = p_{dhs}-p_{dht},
+</font>
<font color="blue">otag
+\end{equation}
+%
+</font>
<font color="blue">oindent
+and the pressures $p_{dhs}$ and $p_{dht}$ refer to the dry atmosphere.
+The potential
+temperature from the input sounding is interpolated to
+each of the model pressure levels, and the equation of state
+\eqref{init_state} is used to compute the inverse density
+$\alpha_d$. Finally, the
+ARW's hydrostatic relation \eqref{hydrostatic_relation},
+in discrete form
+%
+\begin{equation}
+\delta_\eta \phi = - \alpha_d \mu_d
+</font>
<font color="blue">otag
+\end{equation}
+</font>
<font color="blue">oindent
+is used to compute the geopotential. This procedure is used to compute
+the reference state (assuming a dry atmosphere) and the full state
+(using the full moist sounding). The perturbation variables are
+computed as the difference between the reference and full state. It
+should also be noted that in the nonhydrostatic model integration,
+the inverse density $\alpha_d$ is diagnosed from the geopotential using
+this equation of state, and the pressure is diagnosed from the equation
+of state using the inverse density $\alpha_d$ and the prognostic potential
+temperature $\theta$. Thus, the ARW's prognostic variables $\mu_d$,
+$\theta$, and $\phi$ are in exact hydrostatic balance for the model
+equations (to machine roundoff).
+
+\section{Initialization for Real-Data Conditions}
+
+The initial conditions for the real-data cases are pre-processed through a separate
+package called the WRF Preprocessing System (WPS, see Fig. \ref{figure:WPS_real_wrf}).
+The output from WPS is passed to the
+real-data pre-processor in the ARW--- program {\it real}--- which generates initial and lateral boundary
+conditions. This section is primarily about the steps taken to build the
+initial and the lateral boundary conditions for a real-data case. Even though the
+WPS is outside of the ARW system, a brief description is appropriate to see how the
+raw meteorological and static terrestrial data are brought into the model
+for real-data cases.
+
+\subsection{Use of the WRF Preprocessing System by the ARW}
+
+The WPS is a set of programs that takes
+terrestrial and meteorological data (typically in GriB format) and transforms them for input to
+the ARW pre-processor program for real-data cases ({\it real}).
+Figure \ref {figure:WPS_real_wrf} shows the flow of data into and out of the WPS system.
+The first step for the WPS is to define a physical grid (including
+the projection type, location on the globe,
+number of grid points, nest locations, and grid distances) and
+to interpolate static fields to the prescribed domain.
+Independent of the domain configuration,
+an external analysis or forecast is processed by the WPS GriB decoder,
+which diagnoses required fields and
+reformats the GriB data into an internal binary format.
+With a specified domain,
+WPS horizontally interpolates the meteorological data onto the projected domain(s).
+The output data from WPS supplies a complete 3-dimensional snapshot of the atmosphere
+on the selected model grid's horizontal staggering at the selected time slices,
+which is sent to the ARW pre-processor program for real-data cases.
+
+%
+% Figure showing WPS and real and ARW
+%
+\begin{figure}
+ \centering
+ \includegraphics[width=6in]{figures/WPS_real_wrf.pdf}
+ \caption{\label{figure:WPS_real_wrf}Schematic showing
+the data flow and program components in WPS, and how WPS feeds initial data to the ARW.
+Letters in the rectangular boxes indicate program names.
+GEOGRID: defines the model domain and creates static files of terrestrial data. UNGRIB:
+decodes GriB data. METGRID: interpolates meteorological data to the model domain.}
+\end{figure}
+
+The input to the ARW real-data processor from
+WPS contains 3-dimensional fields of temperature (K), relative humidity
+(%), geopotential height (m), pressure (Pa),
+and the horizontal components of momentum (m/s, already rotated to the model
+projection).
+The 2-dimensional static terrestrial fields include:
+albedo, Coriolis parameters, terrain elevation, vegetation/land-use type,
+land/water mask, map scale factors, map rotation angle, soil texture category, vegetation greenness fraction,
+annual mean temperature,
+and latitude/longitude.
+The 2-dimensional time-dependent fields from the external model, after processing by WPS, include:
+surface pressure and sea-level pressure (Pa), layers of soil temperature (K) and soil moisture (kg/kg,
+either total moisture, or
+binned into total and liquid content),
+snow depth (m), skin temperature (K), and sea ice.
+
+\subsection{Reference State}
+\label{initialization_real_base_section}
+Identical to the idealized initializations, there is a partitioning of some of the
+meteorological data into reference and perturbation fields.
+For real-data cases, the reference state is defined by terrain elevation and three constants:
+\begin{itemize}\setlength{\parskip}{-5pt}
+\item $p_{0}$ ($10^5$ Pa) reference sea level pressure;
+\item $T_{0}$ (usually 270 to 300 K) reference sea level temperature; and
+\item $A$ (50 K) temperature difference between the pressure levels of $p_{0}$ and $p_{0}/e$.
+\end{itemize}
+
+</font>
<font color="blue">oindent Using these parameters, the dry reference state surface pressure is
+\begin{equation}
+p_{dhs} = p_{0}~exp\Bigg({-T_{0} \over A} +
+ \sqrt{ {\bigg( {T_{0} \over A } \bigg)}^2 - ~
+ { 2\phi_{sfc} \over { A~R_d}} } \Bigg).
+\label{init_psurf}
+\end{equation}
+
+</font>
<font color="blue">oindent From \eqref{init_psurf}, the 3-dimensional reference pressure (dry hydrostatic pressure $p_{dh}$)
+is computed as
+a function of the vertical coordinate $\eta$ levels and the model top $p_{dht}$:
+\begin{equation}
+p_{dh} = \overline{p}_d = \eta ~( p_{dhs} - p_{dht} ) + p_{dht}.
+\label{init_pbar}
+\end{equation}
+
+</font>
<font color="blue">oindent With \eqref{init_pbar}, the reference temperature is a straight line on a skew-T plot, defined as
+\begin{equation}
+T = T_0 + A~ln {\overline{p}_d \over p_0}.
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent From the reference temperature and pressure,
+the reference potential temperature is then defined as
+
+</font>
<font color="blue">oindent
+\begin{equation}
+\overline{\theta}_d = {\bigg( T_{0} + A~ln{\overline{p}_d \over p_{0} } \bigg) }
+{\bigg( {p_0 \over \overline{p}_d } \bigg) }
+^{R_d \over C_p},
+\label{init_thetad}
+\end{equation}
+
+</font>
<font color="blue">oindent and the reciprocal of the reference density using
+\eqref{init_pbar} and \eqref{init_thetad} is given by
+\begin{equation}
+\overline{\alpha}_d = {1 \over \overline{\rho}_d} = {{R_d ~\overline{\theta}_d}\over p_{0} }~\bigg(
+{\overline{p}_d \over p_{0} } \bigg)^{-{C_v \over C_p}}.
+\label{init_alphabar}
+\end{equation}
+
+</font>
<font color="blue">oindent The base state difference of the dry surface pressure
+from \eqref{init_psurf} and the model top is
+given as
+\begin{equation}
+\overline{\mu}_d = p_{dhs} - p_{dht}.
+\label{init_mubar}
+\end{equation}
+
+</font>
<font color="blue">oindent
+From \eqref{init_alphabar} and \eqref{init_mubar},
+the reference state geopotential defined from the hydrostatic relation is
+\begin{equation}
+\delta_{\eta} \overline{\phi} = -\overline{\alpha}_d~\overline{\mu}_d.
+</font>
<font color="blue">otag
+\end{equation}
+
+
+\subsection{Vertical Interpolation and Extrapolation}
+The ARW real-data preprocessor vertically interpolates using functions of dry pressure.
+The input data from WPS contains both a total pressure and a moisture field (typically
+relative humidity). Starting at the top each column of input pressure data, the integrated moisture
+is subtracted from the pressure field step-wise down to the surface.
+Then, by removing the pressure at the model
+lid, the total dry surface pressure $p_{sd}$ diagnosed from WPS defines the
+model total dry column pressure
+\begin{equation}
+\mu_d = \overline{\mu}_d + \mu_d' = p_{sd} - p_{dht}.
+\label{init_mutotal}
+\end{equation}
+
+</font>
<font color="blue">oindent
+With the ARW vertical coordinate $\eta$, the model lid $p_{dht}$, and the column dry
+pressure known at each $(i,j,k)$ location, the 3-dimensional arrays are interpolated.
+
+In the free atmosphere up to the model lid, the vertical calculations are always interpolations.
+However, near the model surface, it is possible to have an inconsistency between the input
+surface pressure (based largely on the input surface elevation) and the ARW surface
+pressure (possibly with a much higher resolution topography). These inconsistencies
+may lead to an extrapolation. The default behavior for extrapolating the horizontal winds and
+the relative humidity below the known surface is to keep the values constant, with zero vertical gradient.
+For the potential temperature, by default a -6.5 $K/km$ lapse rate for the temperature is applied.
+The vertical interpolation of the geopotential field is optional and is
+handled separately. Since a known lower boundary condition exists
+(the geopotential is defined as zero at the pressure at sea-level), no extrapolation is required.
+
+
+
+\subsection{Perturbation State}
+In the real-data preprocessor, first a topographically defined reference state is computed,
+then the input 3-dimensional data are vertically
+interpolated in dry pressure space. With the potential temperature $\theta$ and mixing ratio
+$q_v$ available on each $\eta$ level, the pressure, density, and height diagnostics are
+handled.
+</font>
<font color="blue">oindent The perturbation dry column pressure
+field given the reference dry column pressure \eqref{init_mubar} is
+\begin{equation}
+\mu_d' = \mu_d - \overline{\mu}_d,
+\label{init_muprime}
+\end{equation}
+
+</font>
<font color="blue">oindent where $\mu_d$ is the column total dry pressure.
+Starting with the reference state fields
+(\ref{init_pbar}, \ref{init_alphabar}, and \eqref{init_mubar}) and the
+dry surface pressure perturbation (\ref{init_muprime}),
+the perturbation fields for pressure and inverse density are diagnosed.
+The pressure perturbation includes moisture and is diagnosed from
+the hydrostatic equation
+%
+\begin{equation}
+\delta_{\eta} p' = \mu'_d \bigg(1 + {\overline{q_v}^\eta} \bigg) +
+ \overline{q_v}^\eta~\overline \mu_d,
+%\label{init_pprime}
+</font>
<font color="blue">otag
+\end{equation}
+%
+</font>
<font color="blue">oindent
+which is
+integrated down from
+at the model top (where $p'= 0$) to recover $p'$.
+The total dry inverse density is given as
+\begin{equation}
+\alpha_d = {R_d \over p_{0} } ~ \theta~ \bigg( 1 + {R_v \over R_d}q_v \bigg)~
+ \bigg( {{p'_d + \overline {p}_d} \over p_{0} } \bigg )^{-{C_v \over C_p}},
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent which defines the perturbation field for inverse density
+
+\begin{equation}
+\alpha'_d = \alpha_d - \overline{\alpha}_d.
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent
+The perturbation geopotential
+is diagnosed from the hydrostatic relation
+\begin{equation}
+\delta_{\eta} \phi' = - \big ( {\mu}_d \alpha'_d + \mu'_d
+\overline{\alpha}_d \big )
+</font>
<font color="blue">otag
+\end{equation}
+%
+by upward integration using the terrain elevation as the lower boundary condition.
+
+\subsection{Generating Lateral Boundary Data}
+
+This section deals with generating the separate lateral boundary condition file used
+exclusively for the real-data cases. For information
+on which lateral boundary options are available for both the idealized and real-data
+cases, see Chapter \eqref{lbc_chap}.
+
+The specified
+lateral boundary condition for the coarse grid for real-data cases is supplied by an external file that is
+generated by program {\it real}.
+This file contains
+records for the fields $u$, $v$, $\theta$, $q_v$, $\phi'$, and $\mu'_d$ that are used by the ARW to
+constrain the lateral boundaries (other fields are in the boundary file, but are not used).
+The lateral boundary file holds one less time period than was processed by WPS.
+Each of these variables has both
+a valid value at the initial time of the lateral boundary time and a tendency term to get to the
+next boundary time period. For example, assuming a 3-hourly availability of data from WPS,
+the first time period of the lateral boundary file
+for $u$ would contain data for both coupled $u$ (map scale factor and $\mu_d$ interpolated to
+the variable's
+staggering) at the 0 h time
+
+\begin{equation}
+%U_{0h} = {{\mu_u~u}\over{m_u}} \bigg | _{0h}
+U_{0h} = {{\overline{\mu_d}^x u}\over{\overline{m}^x}} \bigg | _{0h},
+</font>
<font color="blue">otag
+\end{equation}
+</font>
<font color="blue">oindent and a tendency value defined as
+\begin{equation}
+U_t = { U_{3h} - U_{0h} \over 3h},
+</font>
<font color="blue">otag
+\end{equation}
+
+</font>
<font color="blue">oindent which would take a grid point from the initial value to the value at the next large-scale time
+during 3 simulation hours.
+The horizontal momentum fields are coupled both with $\mu_d$ and the inverse map factor. The
+other 3-dimensional fields ($\theta$, $q_v$, and $\phi'$) are coupled only with $\mu_d$.
+The 2-dimensional $\mu'_d$ lateral boundary field is not coupled.
+
+Each lateral boundary field
+is defined along the four sides of the
+rectangular grid (loosely referred to as the north, south, east, and west sides).
+The boundary values and tendencies for vertical velocity and the non-vapor moisture species are included
+in the external lateral boundary file, but act as
+place-holders for the nested boundary data for the fine grids.
+The width of the lateral
+boundary along each of the four sides is user selectable at run-time.
+
+\subsection{Masking of Surface Fields}
+
+Some of the meteorological and static fields are ``masked''. A masked field is one in which
+the values are typically defined only over water (e.g., sea surface temperature) or defined
+only over land (e.g., soil temperature).
+The need to match all of the masked fields consistently to each other requires additional steps
+for the real-data cases due to the masked data's presumed use in various physics packages in the soil,
+at the surface, and in the boundary layer.
+If the land/water
+mask for a location is flagged as a water point, then the vegetation and soil categories must also
+recognize the location as the special water flag for each of their respective categorical indices.
+
+The values for the soil temperature and soil moisture come from WPS on the
+native levels originally defined for those variables
+in the large-scale model. WPS does no vertical interpolation for the
+soil data. While it is typical to try to match the ARW soil scheme with
+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. DFI is applied to the output of the {\it real}
+preprocessor before the model simulation begins. If data assimilation is
+performed using WRF-Var, DFI is applied to the analysis produced by the
+WRF-Var system, rather than the output of program {\it real}.
+
+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 windowed-sinc 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 the {\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>