<p><b>dmbarker</b> 2007-01-17 09:52:14 -0700 (Wed, 17 Jan 2007)</p><p>Update SE radar. Tidy. DB<br>
</p><hr noshade><pre><font color="gray">Deleted: trunk/wrfvar/technote/description.lof
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Modified: trunk/wrfvar/technote/description.tex
===================================================================
--- trunk/wrfvar/technote/description.tex 2007-01-16 23:36:03 UTC (rev 30)
+++ trunk/wrfvar/technote/description.tex 2007-01-17 16:52:14 UTC (rev 31)
@@ -77,6 +77,7 @@
\include{obs}
\include{radiances}
\include{radar}
+\include{gps}
\include{diagnostics}
\include{suites}
\include{se}
Deleted: trunk/wrfvar/technote/description.toc
===================================================================
--- trunk/wrfvar/technote/description.toc 2007-01-16 23:36:03 UTC (rev 30)
+++ trunk/wrfvar/technote/description.toc 2007-01-17 16:52:14 UTC (rev 31)
@@ -1,37 +0,0 @@
-\contentsline {chapter}{\hbox to\@tempdima {\hfil }Acknowledgments}{vii}{chapter*.5}
-\contentsline {chapter}{</font>
<font color="red">umberline {1}Introduction}{1}{chapter.1}
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<font color="red">umberline {1.1}Overview of Variational Data Assimilation}{1}{section.1.1}
-\contentsline {section}{</font>
<font color="red">umberline {1.2}Overview of WRF-Var}{1}{section.1.2}
-\contentsline {chapter}{</font>
<font color="red">umberline {2}Minimization Algorithm}{3}{chapter.2}
-\contentsline {section}{</font>
<font color="red">umberline {2.1}Inner/Outer Loop Structure}{3}{section.2.1}
-\contentsline {section}{</font>
<font color="red">umberline {2.2}Conjugate Gradient Minimization}{3}{section.2.2}
-\contentsline {chapter}{</font>
<font color="red">umberline {3}Background Error Covariance Estimation}{5}{chapter.3}
-\contentsline {section}{</font>
<font color="red">umberline {3.1}Overview}{5}{section.3.1}
-\contentsline {section}{</font>
<font color="red">umberline {3.2}Scientific Background}{6}{section.3.2}
-\contentsline {section}{</font>
<font color="red">umberline {3.3}WRF's Forecast Error Covariance Estimator - gen\_be}{7}{section.3.3}
-\contentsline {subsection}{</font>
<font color="red">umberline {3.3.1}gen\_be\_stage0\_wrf - Calculate perturbations of standard fields}{7}{subsection.3.3.1}
-\contentsline {subsection}{</font>
<font color="red">umberline {3.3.2}gen\_be\_stage1 - Remove Mean}{8}{subsection.3.3.2}
-\contentsline {subsection}{</font>
<font color="red">umberline {3.3.3}gen\_be\_stage2 - Calculate balance regressions coefficients}{8}{subsection.3.3.3}
-\contentsline {subsection}{</font>
<font color="red">umberline {3.3.4}gen\_be\_stage2a - Calculate unbalanced control variable fields}{9}{subsection.3.3.4}
-\contentsline {subsection}{</font>
<font color="red">umberline {3.3.5}gen\_be\_stage3 - Eigenvectors/values of Vertical Error Covariances}{9}{subsection.3.3.5}
-\contentsline {subsection}{</font>
<font color="red">umberline {3.3.6}gen\_be\_stage4 - Horizontal Error Correlations}{9}{subsection.3.3.6}
-\contentsline {section}{</font>
<font color="red">umberline {3.4}Example Results From An Application Of gen\_be}{10}{section.3.4}
-\contentsline {chapter}{</font>
<font color="red">umberline {4}Background Error Covariance Modeling}{11}{chapter.4}
-\contentsline {chapter}{</font>
<font color="red">umberline {5}Conventional Observations}{13}{chapter.5}
-\contentsline {section}{</font>
<font color="red">umberline {5.1}Observation Preprocessing}{13}{section.5.1}
-\contentsline {section}{</font>
<font color="red">umberline {5.2}Observation Quality Control}{13}{section.5.2}
-\contentsline {section}{</font>
<font color="red">umberline {5.3}Observation Data Formats}{13}{section.5.3}
-\contentsline {subsection}{</font>
<font color="red">umberline {5.3.1}BUFR}{13}{subsection.5.3.1}
-\contentsline {subsection}{</font>
<font color="red">umberline {5.3.2}ASCII format}{13}{subsection.5.3.2}
-\contentsline {section}{</font>
<font color="red">umberline {5.4}First Guess at Appropriate Time}{13}{section.5.4}
-\contentsline {chapter}{</font>
<font color="red">umberline {6}Radiance Data Assimilation}{15}{chapter.6}
-\contentsline {chapter}{</font>
<font color="red">umberline {7}Radar Data Assimilation}{17}{chapter.7}
-\contentsline {chapter}{</font>
<font color="red">umberline {8}Data Assimilation Diagnostics}{19}{chapter.8}
-\contentsline {chapter}{</font>
<font color="red">umberline {9}WRF Suite Scripts}{21}{chapter.9}
-\contentsline {chapter}{</font>
<font color="red">umberline {10}Software Engineering}{23}{chapter.10}
-\contentsline {chapter}{</font>
<font color="red">umberline {A}Physical Constants}{25}{appendix.A}
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<font color="red">umberline {B}List of Symbols}{27}{appendix.B}
-\contentsline {chapter}{</font>
<font color="gray">umberline {C}Acronyms}{31}{appendix.C}
-\contentsline {chapter}{\hbox to\@tempdima {\hfil }References}{34}{appendix.C}
Added: trunk/wrfvar/technote/figures/gen_be_results.pdf
===================================================================
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Added: trunk/wrfvar/technote/gps.tex
===================================================================
--- trunk/wrfvar/technote/gps.tex 2007-01-16 23:36:03 UTC (rev 30)
+++ trunk/wrfvar/technote/gps.tex 2007-01-17 16:52:14 UTC (rev 31)
@@ -0,0 +1,5 @@
+\chapter{GPS Data Assimilation}
+\label{radar}
+
+Guo, Kuo, ?
+
Modified: trunk/wrfvar/technote/radar.tex
===================================================================
--- trunk/wrfvar/technote/radar.tex 2007-01-16 23:36:03 UTC (rev 30)
+++ trunk/wrfvar/technote/radar.tex 2007-01-17 16:52:14 UTC (rev 31)
@@ -1,4 +1,421 @@
\chapter{Radar Data Assimilation}
\label{radar}
-Xiao, Sun, Lim, etc?
\ No newline at end of file
+A Doppler radar data assimilation scheme was developed within
+WRF-Var (3D-Var system). The dynamic balance between atmospheric wind
+and thermodynamic fields, based on Richardsons equation, is introduced
+to the 3D-Var system. Vertical velocity ($w$) increments are produced to
+enable the assimilation of vertical velocity component of Doppler radial
+velocity observation. In order to assimilate radar reflectivity data,
+the model total water mixing ratio is used as a control variable, and
+a warm rain process, its linear, and its adjoint are
+incorporated into the system to partition the moisture and hydrometeor
+increments. The observation operators for Doppler radial velocity and
+radar reflectivity are developed and implemented within the 3D-Var system.
+Numerical experiments indicate that the WRF-Var (3D-Var) system can
+successfully extract useful informations from multiple radar data at
+different times (via cycling) and different locations for initiation of
+mesoscale weather systems, such as squll lines, fronts, and hurricanes.
+
+\section{Introduction}
+\label{radar-intro}
+
+Doppler radar observation is an important data source for mesoscale and
+microscale weather analysis and forecasting. Early work on Doppler radar
+data analysis mainly focused on two aspects: one is rainfall analysis
+using radar reflectivity via Z-R relation (Jorgenson and Willis 1982;
+Fujiyoshi et al. 1990), and the other is synthesis of two independent
+Doppler velocities (Ray et al. 1980; 1981). Techniques to estimate
+the velocity field by objectively determining the motion of radar echo
+patterns have also received much attention (Tuttle and Foote 1990).
+In recent years, assimilation of Doppler radar data for short-term
+numerical weather forecasting or nowcasting has become a focal point of
+research (Sun and Crook 2001; Weygandt et al. 2002a, b).
+
+There are several technical and scientific challenges in Doppler radar
+data assimilation. First of all, the radar data are normally at the
+resolution of one km or higher. Yet, at the current stage, most
+operational or even research models have much lower resolutions than
+radar data. A technique of data thinning (with the radar observations
+at reduced resolution, compatible with the analysis system) has to be
+developed. Secondly, data quality control is a very important step
+before radar data are assimilated into WRF-Var analysis. Some undesired
+radar artifacts, such as ground clutter and anomalously
+propagated (AP) clutter, sea clutter, range folding, velocity folding,
+and other noise have to be removed or corrected. These errors are
+embedded in the normal weather returns, so properly recovering or
+removing them is essential to the success of the data assimilation.
+Thirdly, the error variance of the Doppler radar data needs to be
+determined and specified in order to assimilate the radar data
+observations effectively, together with other sources of information.
+
+Although there are challenges, radar data assimilation could be
+very promising for short-range numerical weather prediction.
+To assimilate Doppler radial velocities, the WRF 3D-Var has been included
+vertical velocity ($w$) increments (Xiao et al. 2006). This is important
+when radar data are included in the analysis and for small-scale
+convective weather systems. Mixing ratios of cloud water ($q_c$) and
+rainwater ($q_r$) are included in the background fields. When 3D-Var
+system is run in cycling mode, passing $q_c$ and $q_r$ to the next cycle can
+alleviate the spin-up problems for the subsequent forecast.
+For radar reflectivity assimilation, the inclusion of the analyses
+(increments) of rainwater and cloud water mixing ratios in the
+3D-Var system is vital (Xiao et al. 2007). In the continuous
+cycling mode, 3D-Var assimilation of radar reflectivity data can
+produce relatively rational analyses of the hydrometeor fields.
+A significant advantage of 3D-Var is its greater computational
+efficiency than other assimilation techniques (e.g. 4D-Var or
+ensemble Kalman filter).
+
+In this chapter, we will briefly introduce the radar data preprocessing
+procedure, including data quality control, data thinning, and a
+simple observation error statistics in Section 2. This description of the
+preprocessing is in conceptual level, and the users should have their
+own preprocessing of the radar data before assimilation. However, we
+will present details of the techniques for Doppler radar data
+assimilation. The assimilation of Doppler radial vedlocity will be
+described in Section 3, and that of reflectivity will be in Section 4.
+
+\section{Preprocessing of Doppler Radar Data}
+\label{radar-prep}
+
+The raw Doppler radar data require to be processed before they
+can be assimilated in WRF-Var analysis. The main steps include data
+quality control, data thinning, and estimating radar
+observation errors.
+
+\subsection{Quality control of Doppler radar data}
+
+There are several commonly seen non-meteorological returns in
+the raw radar data, including ground clutter, anomalously-propagated
+ground clutter, second trip echo, sea clutter and other noise.
+We should conduct quality control using manual editting or automatic
+software. For example, the radar data can be interactively edited
+using NCAR SOLO software (Oye et al. 1995) to conduct the data
+quality control. Each scan was carefully examined in order to
+identify unwanted radar echoes. These unwanted echoes were either
+recovered (e.g., velocity folding) or removed
+(all other unwanted returns).
+
+Data quality control is an important step in data assimilation.
+Too many bad-quality data could ruin the 3D-Var analyses.
+Without removal of the unwanted radar data, the minimization of
+the 3D-Var system does not converge well, or the analysis is wired
+even it converges. Information from the bad-quality data entered
+the 3D-Var minimization, which could make it fail to converge
+or produce large analysis departure from other good observations.
+In the operational environment, an objective approach for
+Doppler radar data quality control is necessary.
+
+\subsection{Data thinning or super-observations}
+
+Radar data are sampled on radar spherical coordinates (range,
+azimuth, and elevation) with a resolution much higher than
+the numerical model. A strategy for data thinning or super-observations
+should be developed to process the data to a
+regular grid at a resolution compatible with the analysis system.
+For example, the radar data can be mapped to the Cartesian grids
+with the same map projection as the model before being injected into
+the 3D-Var system. This reduces redundant data (especially near
+the radar site) and high frequency features that cannot be resolved by
+the numerical model.
+
+The thinning of radar data can be performed using NCARs software
+SPRINT (Sorted Position Radar INTerpolation) and CEDRIC (Custom
+Editing and Display of Reduced Information in Cartesian space)
+developed by Mohr and Vaughan (1979) and Mohr (1981). Data for each
+Cartesian grid are interpolated using the surrounding eight points
+from the four beams (two on either side, above and below). Each
+volume data are put at the same time. To get rid of grid data that
+might be affected by noise during the interpolation, a local standard
+deviation of the eight points used in the interpolation is
+calculated and the radial velocity data with standard deviation
+greater than 14 m/s are removed (The threshold of 14 m/s is chosen
+because the radial velocity histogram dips at 14 m/s). Spatial
+data filtering (with the influence radius of 2km) is performed
+for the grid radar data to remove small-sale features that cannot
+be represented in the model analyses.
+
+\subsection{Estimate of the radar observation errors}
+
+If the NCAR software SPRINT is used, the standard deviation calculated
+in SPRINT during the interpolation can be used as the observation errors.
+Since the four adjacent data points each representing a pulse volume
+are used to perform the interpolation, the standard deviation
+computed in SPRINT represents the radial velocity errors in m/s.
+The horizontal distribution of the standard deviation represents
+the error structure. However, larger values are sometimes found due
+to some voids of the raw radar data. Empirical rescaling of the
+calculated errors is usually necessary before the error statistics
+are used in 3D-Var. The Gaussian distribution of the errors is
+assumed at each level during rescaling. We believe data
+thinning is necessary, which can results in more benefit even
+the spatial error correlations are neglected. After the thinned
+observations and related errors are calculated, they are converted
+to 3D-Var input format.
+
+\section{Doppler Radial Velocity Assimilation}
+\label{radar_rv}
+
+The configuration of the WRF 3D-Var system is based on a multivariate
+incremental formulation (Courtier et al. 1994). The preconditioned
+control variables are described in Chapter 2. For radail velocity
+assimilation, we added a new balance equation, the Richardson's
+equation, for generating the vertical velocity increments (Xiao et al.
+2005).
+
+\subsection{Vertical velocity increments}
+
+In order to include a capability to assimilate the vertical component of
+Doppler radial velocity data, the WRF 3D-Var includes vertical velocity
+increments. Based on Richardson (1922) and White (2000), a balance
+equation that combines the continuity equation, adiabatic
+thermodynamic equation, and hydrostatic relation is derived. Details of
+the derivation can be seen in Xiao et al. 2005.
+
+\begin{equation}
+{\gamma p} {\partial w \over \partial z} =
+-\gamma p {\bf </font>
<font color="black">abla \cdot V_h} - {V_h \cdot </font>
<font color="blue">abla} p +
+g \int _z ^\infty {\bf </font>
<font color="blue">abla \cdot (\rho {\bf v_h})dz}
+\label{richdson_b}
+\end{equation}
+
+</font>
<font color="blue">oindent where $w$ is vertical velocity, $\bf V_h$ is the vector of horizontal
+velocity (components $u$ and $v$), $\gamma$ the ratio of specific
+heat capacities of air at constant pressure/volume, $p$ pressure,
+$\rho$ density, $T$ temperature, $c_p$ specific heat capacity of air
+at constant pressure, $z$ height, and $g$ the acceleration due to
+gravity. For simplicity, hereafter Eq. (1) will be referred to as
+Richardsons equation. Linearizing Eq. (1) by writing each variable
+in terms of a basic state (overbar) plus a small increment (prime)
+gives
+
+\begin{eqnarray}
+ &{\gamma \overline p} {\partial w' \over \partial z} =
+ -{\gamma p'} {\partial \overline w \over \partial z} -
+ {\gamma \overline p} {\bf </font>
<font color="blue">abla \cdot {V_h}'} - \gamma p'
+ {\bf </font>
<font color="black">abla \cdot \overline V_h -\overline V_h \cdot </font>
<font color="blue">abla} p'
+ -{\bf {V_h}'} \cdot </font>
<font color="black">abla \overline p </font>
<font color="blue">onumber \\
+ & + g \int _z ^\infty {\bf </font>
<font color="blue">abla}
+ \cdot (\overline \rho {\bf V_h}') dz + g \int _z ^\infty {\bf </font>
<font color="blue">abla}
+ \cdot ({\rho}' {\bf \overline V_h}) dz
+\end{eqnarray}
+
+The basic state (overbar) variables satisfy Eq. (1). They also
+satisfy the continuity equation, adiabatic equation and hydrostatic
+equation. The linear equation (2) is discretized, and its adjoint code
+is developed according to the code of the linearized equation. The
+correctness of the adjoint check following the method proposed by
+Navon et al. (1992) is verified.
+
+The Richardsons equation is chosen due to the following reasons:
+a). It is a higher order approximation of the continuity equation
+than incompressible continuity equation or anelastic continuity
+equation, and the computation is affordable; b). It can build
+efficient linkage between dynamic and thermodynamic fields because
+the thermodynamic equation is directly involved in the derivation.
+The analysis fields should be more balanced than using simple
+incompressible continuity equation or anelastic continuity equation.
+c). In the Richardsons equation, the local derivative of air density
+is dropped from continuity equation by assuming hydrostatic and
+adiabatic atmosphere. It therefore avoids the difficulties in
+implementation of the continuity equation in the 3D-Var system, which
+is not designed to take local time derivates into account. Although
+4D-Var can handle the time derivates with integrations, it is much
+more expensive and complicate than 3D-Var. It must be noted that
+Richardsons equation (with adiabatic assumption) can still induce
+errors for heavy rain event where latent heat release and evaporation
+are important. However, in the 3D-Var cycling procedure, the influence
+of latent heat release, evaporation and other microphysics process
+is reflected in the model forecast. The 3D-Var cycling uses the model
+forecast as the first guess for the next cycle. This can alleviate
+the problem. In the future, it is necessary to further assess the
+impact of the adiabatic assumption on the 3D-Var analyses by
+comparing the adiabatic and diabatic Richardsons equations
+in the 3D-Var system.
+
+\subsection{Inclusion of background $q_c$ and $q_r$ in the 3D-Var system}
+
+In WRF-Var, the background fields can be WRF input (analysis),
+or WRF forecasts. Usually the input (analysis) does not contain
+cloud water and rainwater fields. However, if
+WRF 3D-Var is set up for a continuous update cycle, inclusion of
+cloud water and rainwater mixing ratios $q_c$ and $q_r$ in analysis
+is possible. With cloud water and rainwater produced by the previous
+cycles, inclusion of background $q_c$ and $q_r$ in the 3D-Var
+system enables the information of these variables to be passed on
+to the next cycle. This can reduce the time required for cloud water
+and rainwater spin-up when the model is integrated from the analysis
+as part of the cycling procedure. Moreover, observation operators
+for Doppler velocity and reflectivity require inclusion of cloud
+water and rainwater information. In the Doppler radial velocity
+operator, rainwater terminal velocity is included, and terminal
+velocity is calculated based on the rainwater mixing
+ratio (Sun and Crook 1998).
+
+\subsection{Observation operator for Doppler radial velocity}
+
+The observation operator for Doppler radial velocity is
+
+\begin{equation}
+V_r = u {x-x_i \over r_i} + v {y-y_i \over r_i} +
+(w-v_T){z-z_i \over r_i}
+\label{rv_operator}
+\end{equation}
+
+</font>
<font color="blue">oindent where ($u,v,w$) are the wind components, ($x,y,z$) are the radar
+location, ($x_i,y_i,z_i$) are the location of the radar observation,
+$r_i$ is the distance between the radar and the observation,
+and $v_T$ is terminal velocity. For radar scans at nonzero
+elevation angles, the fall speed of precipitation particles has
+to be taken into account. There are different ways to calculate
+terminal velocity. Here, we use the algorithm of Sun and Crook (1998)
+to calculate terminal velocity $v_T$ (m/s),
+
+\begin{equation}
+v_T = 5.40 a \cdot q_r^{0.125} ,
+\label{v_T}
+\end{equation}
+
+</font>
<font color="blue">oindent where $q_r$ is rainwater mixing ratio ($g/kg$). The quantity $a$ is
+a correction factor defined by
+
+\begin{equation}
+a = (p_0/\overline p ) ^{0.4} ,
+\label{v_T_a}
+\end{equation}
+
+</font>
<font color="blue">oindent where $\overline p$ is the base-state pressure and $p_0$
+is the pressure at the ground.
+
+\section{Assimilation of Radar Reflectivity}
+\label{radar_rf}
+
+Radar reflectivity measures the radar signal reflected by
+precipitation hydrometeors. To assimilate radar reflectivity directly,
+the WRF 3D-Var system should be able to produce the increments of
+the hydrometeors (at least the rainwater mixing ratio). However,
+it is not appropriate to perform the background error
+statistics for the rainwater mixing ratio, because it will result in
+zero errors in most of the domain grid points. Instead, we chose
+total water mixing ratio $q_t$ as a control variable and
+conducted background error statistics for the WRF 3D-Var when
+radar reflectivity is to be assimilated (Xiao et al. 2007).
+
+\subsection{Partitioning of moisture and hydrometeor increments}
+
+Because total water mixing ratio $q_t$ is used as a control variable,
+partitioning of the moisture and hydrometeor increments is necessary
+in the 3D-Var system. We introduced the warm rain process of
+Dudhia (1989), which includes condensation of water vapor into
+cloud ($P_{CON}$), accretion of cloud by rain ($P_{RA}$), automatic
+conversion of cloud to rain ($P_{RC}$), and evaporation of rain to
+water vapor ($P_{RE}$). These are the major processes of hydrometeor
+cycle in the summer season. For the winter season, the partitioning
+scheme should include ice-phase microphuysics. This will be developed
+in the future release.
+
+The autoconversion term, $P_{RC}$, is represented by
+
+\begin{equation}
+P_{RC} = \cases{ \ k_1 (q_c - q_{crit}),
+ &if $q_c \ge q_{crit}$; \cr
+ \ 0. &if $q_c < q_{crit}$, \cr}
+\label{P_rc}
+\end{equation}
+
+</font>
<font color="blue">oindent where $q_c$ is the cloud water mixing ratio. According to
+Kessler (1969), $k_1=10^{-3}\ s^{-1}$, $q_{crit}=0.5 g\cdot kg^{-1}$.
+The accretion of cloud water by rain is parameterized by
+
+\begin{equation}
+P_{RA} = \frac{1}{4}\pi \rho a q_c EN_0(\frac{p_0}{p})^{0.4}
+ \frac{\Gamma(3+b)}{\lambda^{3+b}},
+\label{P_ra}
+\end{equation}
+
+</font>
<font color="blue">oindent where $\Gamma$ is the gamma-function; $E$ is the collection efficiency;
+$p$ is pressure; $p_0$=1000 $hPa$, $N_0$=8X106 $m^{-4}$, $a$=841.99667 and
+$b$=0.8. The evaporation of rain can be determined from the equation:
+
+\begin{equation}
+P_{RE} = \frac{2 \pi N_0 (S-1)}{A+B} \Biggl \lbrack \frac{f_1}{\lambda^2}+
+ f_2(\frac{a \rho}{\mu})^{1 \over 2} S_c^{1 \over 3}(\frac{p_0}{p})^{0.2}
+ \frac{\Gamma ({{5+b} \over 2})}{\Lambda^{{5+b} \over 2}} \Biggl \rbrack,
+\label{P_re}
+\end{equation}
+
+</font>
<font color="blue">oindent where f1=0.78, f2=0.32. The condensation $P_{CON}$ is determined
+according to Asai (1965) by
+
+\begin{equation}
+P_{CON} = \frac{q_v - q_{vs}} {1 + \frac{L_v^2 q_{vs}} {R_v C_{pm} T^2}}
+\label{P_con}
+\end{equation}
+
+</font>
<font color="blue">oindent where $q_{vs}$ is saturated water vapor mixing ratio,
+$L_v$, $R_v$, and $C_{pm}$ are latent heat of condensation, gas constant
+for water vapor, and specific heat at constant pressure for moist air,
+respectively.
+
+The tangent linear and its adjoint of the scheme were developed and
+incorporated into the WRF 3D-Var system. Although the control variable
+is $q_t$, the $q_v$, $q_c$ and $q_r$ increments are produced through the
+partitioning procedure during the 3D-Var minimization. The warm rain
+parameterization builds a constraint: the relation among rainwater,
+cloud water, moisture, and temperature. When rainwater information
+(from reflectivity) enters the minimization iteration procedure, the
+forward warm rain process and its backward adjoint distribute this
+information to the increments of other variables (under the constraint
+of the warm rain scheme).
+
+\subsection{Observation operator for radar reflectivity}
+
+Once the 3D-Var system can produce $q_c$ and $q_r$ increments, the setup
+of the observation operator for assimilation of reflectivity is
+straightforward. Simply, we adopted the observation operator from Sun
+and Crook 1997:
+
+\begin{equation}
+ Z = 43.1+17.5log(\rho q_r),
+\label{Z_operator}
+\end{equation}
+
+</font>
<font color="gray">oindent where $Z$ is reflectivity in the unit of $dBZ$ and $q_r$ is
+the rainwater mixing ratio. The relation is derived analytically
+by assuming the Marshal-Palmer distribution of raindrop size.
+
+\subsection{Summary of the implementation procedure}
+
+Great efforts were made to incorporate the reflectivity assimilation into
+the WRF 3D-Var system. The flow chart of computations in the WRF 3D-Var
+minimization procedure can be found in Xiao et al. 2007. For clarity of
+the procedure, we summarized some implementation details in the following.
+
+First of all, the control variable transform was built to bridge
+the control variables $V$ and the analyzed variable increments $X$.
+The partitioning of the moisture and hydrometeor increments is added
+to the transform. Since we use $q_t$ as a control variable, development
+of the linear warm rain process is important to partition the $q_v$,
+$q_c$ and $q_r$ increments. At the first iteration of the WRF 3D-Var
+minimization, the $q_c$ and $q_r$ increments are zero, and the $q_v$
+increment is equal to the $q_t$ increment. We treated the warm rain
+process as a column model. If any column is detected updraft and
+super-saturation, an average 30 minutes of updraft time is taken
+to produce the cloud water and rainwater using the one-dimensional
+column model. During the process, moisture and temperature should
+also be changed due to the condensation and evaporation involved
+in the process. When reflectivity observation is assimilated,
+the adjoint of the reflectivity operator will produce the cost function
+gradient with respect to the $q_r$ by the adjoint of the reflectivity
+operator. The adjoint of moisture and hydrometeor partitioning scheme will
+distribute the information to the gradients with respect to $q_c$, $q_v$
+and temperature $T$, and produce the gradient with respect to $q_t$.
+Afterwards, the adjoint of the control variable transform will
+propagate the information to other variables and produce the cost
+function gradient of other control variables. The process is iterated
+back and forth in the WRF 3D-Var minimization procedure. When it
+converges, the increments of $q_c$, $q_r$, $q_v$ and $T$ are
+produced by the reflectivity assimilation. The radar reflectivity
+observations is therefore assimilated into the 3D-Var analusis.
+
Modified: trunk/wrfvar/technote/se.tex
===================================================================
--- trunk/wrfvar/technote/se.tex 2007-01-16 23:36:03 UTC (rev 30)
+++ trunk/wrfvar/technote/se.tex 2007-01-17 16:52:14 UTC (rev 31)
@@ -22,14 +22,14 @@
Powerpc Mac with g95 0.91
- Linux i686 or x86_64 with PGI 6.1-4, g95 0.91, pathscale 2.4
+ Linux i686 or x86\_64 with PGI 6.1-4, g95 0.91, pathscale 2.4
The following systems are supported but not fully tested due to lack of
access. We believe the the code works on:
Cray with ftn 5.6.0.0.28
- Linux EMT_64 PGI 6.1-4, g95 0.91
+ Linux EMT\_64 PGI 6.1-4, g95 0.91
ifort
@@ -96,10 +96,10 @@
interprocessor communication. Currently only MPI and the RSL communication
library are supported.
-Model data can be read in both GRIB (1&2) and Netcdf formats.
+Model data can be read in both GRIB (1 and 2) and Netcdf formats.
Conventional observations (Airep, Buoy,,GeoAMV,Metar,Pilot,PolarAMV,Profiler,Qscat,
-SATEM, Ship,Synop) can be read in GTS ASCII 'little_r' and BUFR formats.
+SATEM, Ship,Synop) can be read in GTS ASCII 'little\_r' and BUFR formats.
Radiance data (AMSUA, AMSUB) can be read in BUFR format
@@ -131,8 +131,8 @@
large binary files
index.html Summary of run with links to other files
-cost_fn
-grad_fn
+cost\_fn
+grad\_fn
statistics
rsl/ run output from each processor
rsl.error.0000.html
@@ -163,16 +163,16 @@
Makefile
README
README.NMM
-README_test_cases
+README\_test\_cases
Registry
arch - some new files for WRF-Var compilation
chem
clean
compile
configure
-dyn_em
-dyn_exp
-dyn_nmm
+dyn\_em
+dyn\_exp
+dyn\_nmm
external
frame
inc
@@ -187,15 +187,15 @@
build/ - new compilation system work directory
da/ - source code
- da_synop/
+ da\_synop/
doc/
changes/
external/ - advice on compiling external libaries
-clean_new - cleans new compilation system
+clean\_new - cleans new compilation system
convertor/ - convert model dumps to other formats
-compile_new - new compilation script
-configure_new - new configuration script
-gen_be/ - background error generation
+compile\_new - new compilation script
+configure\_new - new configuration script
+gen\_be/ - background error generation
graphics/ - graphical display tools
scripts/ - Control scripts
wrappers/ - Sample wrapper scripts for different kinds of suites
</font>
</pre>