5B.2 4-Dimensional Variational Data Assimilation for the Weather Research and Forecasting Model Xiang-Yu Huang*1, Qingnong Xiao1, Xin Zhang2, John Michalakes1, Wei Huang1, Dale M. Barker1, John Bray1, Zaizhong Ma1, Tom Henderson1, Jimy Dudhia1, Xiaoyan Zhang1, Duk-Jin Won3, Yongsheng Chen1, Yongrun Guo1, Hui-Chuan Lin1, Ying-Hwa Kuo1 1National Center for Atmospheric Research, Boulder, Colorado, USA 2University of Hawaii, Hawaii, USA 3Korean Meteorological Administration, Seoul, South Korea 1. Introduction The 4D-Var prototype was built in 2005 and has under continuous refinement since then. Many single observation experiments have been carried out to The 4-dimensional variational data assimilation validate the correctness of the 4D-Var formulation. A (4D-Var) (Le Dimet and Talagrand, 1986; Lewis and series of real data experiments have been conducted to Derber, 1985) has been pursued actively by research assess the performance of the 4D-Var (Huang et al. community and operational centers over the past two th 2006). Another year of fast development of 4D-Var has decades. The 5 generation Pennsylvania State led to the completion of a basic system, which will be University – National Center for Atmospheric Research described in section 3. mesoscale model (MM5) based 4D-Var (Zou et al. 1995; Ruggiero et al. 2006), for example, has been widely used for more than 10 years. There are also 2. The WRF 4D-Var Algorithm successful operational implementations of 4D-Var (e.g. Rabier et al. 2000). The WRF 4D-Var follows closely the incremental The 4D-Var technique has a number of advantages 4D-Var formulation of Courtier et al. (1994), Veersé over 3-dimensional schemes including the abilities to: and Thépaut (1998), and Lorenc (2003). The data flow 1) Use observations at the almost exact times (to and program structure of WRF 4D-Var are given in Fig. the width of the observation windows, see the 1. discussion in the next section) that they are The input to WRF 4D-Var is as the following. The observed, which suits most asynoptic data, observations are grouped into K windows, yk (k=1,K). 2) Implicitly use flow-dependent background A short-range forecast is used as the background, xb. errors, which ensures the analysis quality for The background error covariance matrix, B, and the fast developing weather systems, and observation error covariance matrix, R, are known by 3) Use a forecast model as a constraint, which assumption. The lateral boundaries, WRFBDY, are enhances the dynamic balance of the final required to integrate the WRF model over a time period. analysis. The 3D-Var solution can be obtained by setting K=1 The last mentioned advantage also implies that the and removing WRF model related components. current Weather Research and Forecasting model (WRF) Both 3D-Var and 4D-Var techniques within WRF- based 3-dimensional variational data assimilation Var include outer-loops and inner-loops. The outer- system (Barker et al. 2004b), which is developed from loops solve the nonlinear aspects of the assimilation MM5 3D-Var (Barker et al. 2004a), should be problem, which for 4D-Var includes the integration of enhanced with a 4-dimensional capability, using the the full nonlinear model, while the inner-loops run a WRF forecast model as a constraint, in order to provide minimization algorithm for a quadratic problem. Using the best initial conditions for the WRF model. superscript n for the outer-loop index, the analysis The 4D-Var component of the expanded 3/4D-Var vector, xn, is the final output of WRF 4D-Var. system (known as WRF-Var, Barker et al. 2005), For the inner-loops, the minimization starts from a hereafter referred to as WRF 4D-Var, has undergone guess vector, xn-1 (the analysis vector from the previous extensive development since 2004. It uses the WRF outer-loop). For the first outer-loop, n=1, xb is normally model and 3D-Var as its basic components (Huang et al. taken as the guess vector, x0. It should be stressed that 2005). in the incremental formulation the background vector and the guess vector should not be mixed. They are the * Corresponding author: Dr. Xiang-Yu Huang, NCAR/MMM, same only during the first outer-loop. P.O. Box 3000, Boulder, CO 80307, USA. Email: Mathematically WRF 4D-Var minimizes a cost [email protected] function J, 1 where B = UUT (Barker et al. 2005); superscripts -1 J = Jb + Jo + Jc (1) and T denote inverse and adjoint of a matrix or a linear where Jb is the background constraint which penalizes operator; dk are the innovation vectors for observation the analysis towards the background, Jo is the observation constraint penalizing the analysis towards window k: n-1 the observations, and Jc is the balancing constraint d = y ! H S "M x $ (4) k k k { W !V # k ( )%} penalizing the analysis towards a balanced state. The Jc T formulation implemented in WRF 4D-Var follows Hk, Hk and Hk are the nonlinear, tangent linear and closely the form in Gustafsson (1992), Gauthier and adjoint observation operators over observation window Thépaut (2001), and Wee and Kuo (2004). k, which transform atmospheric variables between the gridded analysis space and observation space; Mk, Mk For the preconditioning, a variable transform, T n !1 n n !1 and Mk are the nonlinear, tangent linear and adjoint v = U (x ! x ) (2) models, which propagate in time the guess vector xn-1, is chosen and the cost function gradient J’ with respect analysis increments Uvn and analysis residual, (.) in n T T to the control variable v is Equation (3), respectively; SW-V, SV-W, SW-V and SV-W n-1 n i n are the WRF 4D-Var specific operators which J '(v ) = ! v + v transform variables (e.g. between T and q) and grids i=1 + K (3) (between A-grid and C-grid) between VAR and WRF ; T T T T T -1 n f is the modified coefficients for the digital filter +U SV -W !Mk SW -V Hk R {HkSW -V MkSV -W Uv - dk } i k =1 (Lynch and Huang, 1992; Gauthier and Thépaut, 2001), N N T T T -1 # n & γdf is the weight assigned to Jc term. +U SV -W !Mi fi " df C % ! fi Mi SV -W Uv ( i=0 $ i=0 ' WRF, WRF+, VAR and COM are the 4 major components of WRF 4D-Var in terms of software structure (Fig. 1): Fig. 1. The data flow and program structure of WRF 4D-Var. 2 When the disk I/O is used, the following files are I. WRF used for the communication: WRFINPUT: the full model state at the beginning of each outerloop, written out by The Advanced Research WRF model (ARW, VAR and read in by WRF as initial model Skamarock et al. 2005) is referred to here as state; WRF_NL. The ARW solves the compressible, NL(1),…,NL(K): K model states, one for each nonhydrostatic Euler equations which are cast in flux observation window, produced by WRF and form and conserve both mass and energy. The model read in by VAR before computing the has terrain-following vertical coordinate and innovation vector d ; Arakawa C-grid staggering horizontal grid. In k BS(0),…,BS(N): N+1 model states, one for each addition to the wide range of physics options, the time step, produced by WRF and read in by high-order numerical methods including a 3rd order WRF+ as basic states; Runge-Kutta time-split integration scheme and the TL00: the initial model state for the tangent 2nd to 6th order advection options make the ARW linear model, written out by VAR after the suitable for multi-scale numerical simulations and U and S transforms and read in by WRF+; forecasts. V-W TL(1),…,TL(K): K (tangent linear) model states, one for each observation window, produced II. WRF+ by WRF+ during the tangent linear integration and read in by VAR before + WRF comprises two models in one framework, computing the adjoint forcing (AF), defined namely WRF tangent linear model (WRF_TL) and below; WRF adjoint model (WRF_AD), which are compiled TLDF: the digital filter forcing [the last together into a single executable. The Transformation summation in Equation (3)], of Algorithms in Fortran (Giering and Kaminski, N n (5) 2003) is used to construct the tangent linear model ! fi Mi SV -W Uv and its adjoint from a simplified subset of nonlinear i=0 WRF model (WRF_NL). The tangent linear and written out by WRF+ at the end of the adjoint codes passed the standard gradient tests and tangent linear integration and read in by TL/AD tests following Zou et al. (1997). Sensitivities WRF+ at the beginning of the adjoint studies using WRF_AD have been carried out and integration; reported by Xiao et al. (2007). Results in Section 4 AF(K),…,AF(1): K files with AF for each may also be used as a check for the accuracy of observation window k, T T -1 n (6) WRF_TL. SW -V Hk R {HkSW -V MkSV -W Uv - dk } written by VAR and read in by WRF+ III. VAR during the adjoint integration; AD00: the output of WRF+ after the adjoint model integration, read in by VAR before VAR contains all the components of WRF 3D- the S T and UT transforms. Var (Barker et al. 2005) plus the 4-dimensional V-W related enhancements. Among the enhancements are the grouping of observations (i.e., splitting y into yk) and their related calculations (replacing H, H and HT T by Hk, Hk and Hk ) according to the observation windows (k); the calls to WRF_NL, WRF_TL and WRF_AD; and the grid/variable transform operators.