## 1. Introduction

Since the 1980s, the four-dimensional variational data assimilation (4D-Var) technique (Le Dimet and Talagrand 1986; Lewis and Derber 1985) has become one of the most widely used advanced analysis methods in atmospheric and oceanic research and operational centers. The European Centre for Medium-Range Weather Forecasts (ECMWF) was the first operational center to implement the 4D-Var system (Courtier et al. 1994; Rabier et al. 2000). Following ECMWF, other national centers implemented 4D-Var in their operational applications, including Météo-France (Gauthier and Thépaut 2001), the Met Office (Lorenc and Rawlins 2005; Rawlins et al. 2007), the Japan Meteorological Agency (JMA; Honda et al. 2005; Kawabata et al. 2007), Environment Canada (Gauthier et al. 2007), the Swedish Meteorological and Hydrological Institute (Huang et al. 2002; Gustafsson et al. 2012), and the Naval Research Laboratory (Xu et al. 2005; Rosmond and Xu 2006).

The success of the 4D-Var system in improving global forecasts provided enough encouragement to develop a regional 4D-Var for mesoscale and thunderstorm scales. Several regional 4D-Var research and/or operational systems have been developed so far, including 1) one based on the Fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5; Zou et al. 1995, 1997; Ruggiero et al. 2006); 2) one based on the National Centers for Environmental Prediction (NCEP) Eta Model (Zupanski 1993); 3) the 4D-Var-based Regional Atmospheric Modeling Data Assimilation System (RAMDAS; Zupanski et al. 2005); 4) the Variational Doppler Radar Assimilation System (VDRAS) for convective-scale assimilation of radar data (Sun and Crook 1997); 5) the JMA Nonhydrostatic Model (NHM) 4D-Var (Kawabata et al. 2007); 6) a 4D-Var system for the High Resolution Limited Area Model (HIRLAM) forecasting system (Gustafsson et al. 2012); 7) the four-dimensional variational data assimilation for the Canadian Regional Deterministic Prediction System (REG-4D; Tanguay et al. 2012); and finally 8) a 4D-Var system for the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008; Huang et al. 2009, hereafter H09) that has been under development at NCAR since 2005. We will discuss the framework of the current WRF 4D-Var system in this paper.

We recently redeveloped the WRF tangent linear and adjoint model (WRFPLUS) based on the latest WRF Model (Zhang et al. 2013). Compared to the previous version of the WRF tangent linear and adjoint models [denoted WRF adjoint modeling system (WAMS) in Xiao et al. (2008)], the performance of WRFPLUS has been dramatically improved. The WRF 4D-Var system described in H09 is a Multiple Program Multiple Data (MPMD) system with a loose coupling of the variational data assimilation algorithm (WRF-VAR), WRF, and WAMS. In the MPMD system, different processors execute different programs on different data. With the loose coupling approach adopted in the WRF 4D-Var of H09, the information to be communicated among the components is written to and read from disk files. The disk input–output (IO) is easy to implement when coupling several stand-alone components for a new system. However, this disk IO communication is highly inefficient on modern distributed-memory high-performance computers. Moreover, the MPMD framework uses only a subset of the total number of processors at any moment, owing to the sequential nature of the 4D-Var minimization algorithm. Leveraging the development of WRFPLUS, we revisited the software design of WRF 4D-Var and coupled WRF-VAR with WRFPLUS seamlessly into a Single Program Multiple Data (SPMD) system. The current work presents a significant improvement in computational efficiency for the WRF 4D-Var system.

The 4D-Var approach requires many more computations than the three-dimensional variational data assimilation (3D-Var) approach does. To reduce the computational cost of 4D-Var and to complete the computations within the operational time constraint, most operational centers have adopted an approximation of 4D-Var, namely, the multi-incremental 4D-Var. The multi-incremental approach achieves reductions in computational cost in a manner consistent with the nonlinear estimation theory. The key strategy of this approach is to linearize the problem along high-resolution reference trajectories and to use the tangent linear model (TLM) and the adjoint model (ADM) at lower resolutions. This method involves calculating innovations on a high-resolution grid with the nonlinear forward model for outer loops, while minimizing the cost function on a lower-resolution grid for inner loops. Different inner loops may have different resolutions. To address the need of an operational WRF 4D-Var implementation, we developed the capability of multi-incremental configuration and carried out preliminary experiments to validate the multi-incremental WRF 4D-Var against its full-resolution counterpart. Compared to other 4D-Var systems, the WRF 4D-Var system has several unique features: 1) it is the only publicly released and supported 4D-Var system for community users; 2) it is built by the coupling of two stand-alone systems (WRF-VAR and WRFPLUS), which is described in this paper; 3) to the best of our knowledge, this method addresses for the first time some of the practical issues with multi-incremental 4D-Var in a regional gridpoint model, such as the control variable transfer between different horizontal resolutions for different outer loops, updates to lateral boundary conditions (LBCs), and basic-state interpolation in multi-incremental configuration.

This article is organized as follows. In section 2, we present the details for constructing a single executable WRF 4D-Var system and its computational benefits over the MPMD WRF 4D-Var system. This section also introduces the gradient check capability that ensures the correctness and accuracy of the calculated gradient in the new system. Section 3 provides the technical details of the multi-incremental WRF 4D-Var configuration. The computational benefits of applying multi-incremental WRF 4D-Var are demonstrated in this section, along with results from two week-long experiments that demonstrate the scientific impact of adopting this method for regional-scale analysis and forecasting. Results are also presented from a second set of cycling experiments using the multi-incremental system to produce convective-permitting analyses of Hurricane Sandy (2012). The summary and discussion are given in section 4.

## 2. Single executable 4D-Var system

### a. Coupling WRF-VAR and WRFPLUS

NCAR began developing the first versions of the TLM and ADM for the Advanced Research WRF (ARW) dynamic core in 2005 (Xiao et al. 2008). The WRF TLM and ADM and the WRF nonlinear forward model [NLM (also known as FWM)] were originally coupled with WRF-VAR using the MPMD method to form the WRF 4D-Var system. Three multiprocessed programs—WRF-VAR, NLM, and WAMS (includes the TLM and ADM)—were required for launching the older MPMD 4D-Var system. The WRF-VAR program calls WRF NLM, TLM, and ADM via system calls, and they communicate information with each other via disk file IO; refer to Fig. A1 in H09.

For high-resolution applications that require modern supercomputers with a large number of distributed-memory nodes, the MPMD 4D-Var system has severe limitations on the number of processors that can be used effectively, which leads to poor scalability. During the MPMD 4D-Var minimization, the NLM integration saves the basic-state trajectories to disk files at certain time intervals [refer to BS(0), …, BS(*N*) in Fig. A1 of H09]; these basic states are read in to derive the nonlinear coefficients for the TLM and the ADM integrations, respectively. The TLM integration then saves the instantaneous perturbation snapshots to disk files at the observation times [refer to TL(1), …, TL(*K*) in Fig. A1 of H09], which are read in WRF-VAR to calculate the residuals. Finally, the residuals are converted to the adjoint forcing files [refer to AD(1), …, AD(*K*) in Fig. A1 of H09] in WRF-VAR, which are input into the ADM for calculating the gradient. Even though the basic-state trajectories [BS(0), …, BS(*N*)] are read in once and stored in memory for the rest of the iterations, the cost of the remaining disk IO across a large number of distributed processors is still high.

Figure 1 (top) illustrates the logic for running the MPMD WRF 4D-Var; it requires dividing the set of processors into three subsets for WRF-VAR, NLM, and TLM–ADM, respectively (the vertical axis represents the processing core space). Because of the sequential nature of variational data assimilation, only one program of the MPMD 4D-Var system runs at any moment, which means that only the subset of cores associated with the running program are used while the rest of the processors wait. One aspect of the MPMD system that is especially troublesome is the fact that the NLM runs only once for each outer loop; therefore, the cores allocated for the NLM are idle during most of the minimization process.

WRF-VAR, NLM, and WAMS share the same variable definitions, coordinate system, and software infrastructure, so it is natural to propose a single executable WRF 4D-Var system by coupling each component seamlessly with the SPMD method. The strategy for the new SPMD version of the WRF 4D-Var system is as follows. WRF NLM, TLM, and ADM are denoted by WRF-VAR with simple interfaces consisting of the initialization, integration, and finalizing stages of running the model. Routines are developed to exchange data via memory among WRF-VAR, NLM, TLM, and ADM. WRF-VAR is also modified to add interfaces to call NLM, TLM, and ADM. Finally, NLM, TLM, and ADM are compiled as a library that is linked with WRF-VAR to build an SPMD WRF 4D-Var system. Figure 1 (bottom) illustrates the logic of the SPMD WRF 4D-Var. The disc IO communication is eliminated using this approach, but the most important improvement over the MPMD method is that all required computing cores participate in the calculations from end to end, so the computing resources are fully utilized.

The WRF Model and WRF-VAR have evolved considerably since they were first introduced, but no updates have been made to the WAMS since it was completed in 2008. To accommodate the requirements of building a single executable WRF 4D-Var system, the WAMS required some upgrades. We redeveloped the TLM and ADM of the ARW core based on the latest WRF Model, starting from version 3.2 in 2010. We call this upgraded version WRFPLUS, since it not only includes the full-physics ARW but also includes the TLM and ADM (Zhang et al. 2013). Compared to WAMS, WRFPLUS has the following major improvements: 1) it is consistent with the latest WRF Model developments; the upgraded TLM and ADM are always synchronized with changes to the WRF code; 2) it has significantly improved parallel efficiency; the Registry is augmented to generate tangent and adjoint codes for halo exchanges automatically; and 3) it has improved physics packages. In addition to vertical diffusion and surface drag, we also include a simplified microphysics parameterization scheme and a simplified cumulus parameterization scheme. The WRF NLM already has well-defined interfaces that initialize, advance, and finalize the model. New interface routines are coded to initialize, advance, and finalize the TLM and ADM.

By using a consistent model infrastructure between WRF-VAR and WRFPLUS, we no longer need regridding algorithms, which are usually required in coupling systems because of different coordinate systems and parallel strategies between components. Similar to the function of a coupler, a set of global data structures was introduced in WRFPLUS, which sits between WRF-VAR and WRFPLUS to exchange data between the coupled components. The data to be exchanged include the basic-state trajectories, the perturbation snapshots, and the adjoint forcing and gradient data that should be accessible from both sides of the system via simple memory copying. Because 4D-Var users may change the assimilation time window length, time step, or 4D-Var subwindow length to fit the available observation types, domain size, and horizontal grid resolution, the global data structure should be flexible enough to accommodate the lengths of different trajectories on the fly. The linked-list data structure was adopted for this purpose. The first linked list being allocated stores the basic-state trajectories from the NLM. The required basic-state variables are packed into a vector at every time step of the NLM integration, which is stored on one node of the linked list. Because the basic-state trajectory is used to derive the nonlinear coefficients for the TLM and ADM integration repeatedly at every iteration, this linked list is designed to be persistent for an outer loop. It is destroyed at the end of each outer loop and reallocated and initialized at the beginning of the next outer loop. The other two allocated linked lists store the model perturbation snapshots from the TLM integration and the adjoint forcing inputs for ADM integration, respectively. These two linked lists are persistent for only one inner-loop iteration. They are initialized at the beginning of each inner-loop iteration and are destroyed at the end of the iteration. The linked list to store the basic-state trajectory is the main memory consumer for the 4D-Var minimization, as it stores the snapshots of the atmospheric basic states. The number of stored snapshots depends on how many time step integrations occur within the assimilation window. Therefore, the memory requirement to run this single executable WRF 4D-Var is increased dramatically with increased horizontal resolution.

The interface and the coupling technique implemented in the upgraded WRFPLUS have been designed for coupling the TLM and ADM with other systems easily. We have also coupled WRFPLUS with the Community Gridpoint Statistical Interpolation (GSI; http://www.dtcenter.org/com-GSI/users/) to build the GSI-based WRF 4D-Var (Zhang and Huang 2013). Because of the different grids and parallel strategies in WRFPLUS and GSI, a parallel regridding algorithm is required for the GSI-based WRF 4D-Var.

### b. Performance and scalability of the single executable WRF 4D-Var system

To compare the computational performance of the single executable WRF 4D-Var system with the MPMD WRF 4D-Var system, a series of benchmarking experiments are carried out over a continental U.S. (CONUS) domain. This configuration uses a 222 × 129 gridpoint domain with 45 vertical *σ* levels. The domain grid spacing is 45 km and is centered at 40°N, 98°W. The 4D-Var analysis background is valid at 0900 UTC 8 June 2012, and the assimilation time window is 6 h from 0900 to 1500 UTC 8 June 2012. The assimilated observations include conventional Global Telecommunication System (GTS) data and GPS radio occultation data from NCEP. The basic-state trajectory produced by the NLM is a full-physics model integration with WRF single-moment 3-class (WSM3) microphysics scheme, the Kain–Fritsch cumulus parameterization scheme (Kain 2004), the Yonsei University (YSU) planetary boundary layer (PBL) scheme (Hong et al. 2006), and the Rapid Radiative Transfer Model (RRTM) and Dudia schemes for long- and shortwave radiation. The integrations of the TLM and ADM use simplified microphysics, cumulus, and PBL schemes that are included in the current version of WRFPLUS. The integration time step for the NLM, TLM, and ADM is 240 s.

The performance tests are carried out on the NCAR “Lynx” machine and all executables are compiled by Intel compilers with default optimization levels. Lynx is a single-cabinet Cray XT5m supercomputer, comprising 76 computing nodes, each with two hex-core AMD 2.2-GHz Opteron chips for 12 total computing cores. Each Lynx computing node has 16 GB of memory (https://www2.cisl.ucar.edu/supercomputer/lynx). To run the MPMD WRF 4D-Var, the required computing cores are divided into three subsets among WRF-VAR, NLM, and TLM–ADM. TLM and ADM are well known for their high computing costs; therefore, most of the computing resources are allocated to WRF TLM–ADM. In general, finding the best allocation strategy for parallel performance requires experimenting with various configurations. In this paper, we allocate 5% of the processing cores to WRF NLM and another 5% to WRF-VAR, while the WRF TLM–ADM takes the remaining 90% of the processing cores. Running the SPMD WRF 4D-Var is straightforward; all requested computing cores will be dedicated to computation at all times.

To accommodate the memory requirement of this 6-h 4D-Var case, we run the experiment starting with 12 nodes and use only one core per node to take advantage of all 192 GB of memory across the nodes. The maximum number of CPU cores that we tried is 192 cores for this medium size case. Figure 2 shows the wall clock time and speedup relative to 12 cores for five iterations of the minimization as a function of the number of CPU cores. Compared to the MPMD version, the SPMD WRF 4D-Var is approximately 40%–200% faster using 12–192 cores, with the greatest efficiency gains corresponding to higher numbers of cores. The results confirm that the MPMD 4D-Var, which suffers from the overhead of disk IO communications and the inefficient usage of requested processing cores, performs worse with increasing parallelization. The cost of gathering and broadcasting operations associated with the disk IO tends to be worse with larger numbers of cores/nodes, and the parallel scalability heavily depends on the bandwidth and latency of the network connecting the nodes.

### c. Gradient check in the WRF 4D-Var system

**f**(

**x**),

*g*_

**f**(

**x**,

*g*_

**x**), and

*a*_

**f**(

**x**,

*a*_

**x**) denote an NLM, a TLM, and an ADM, respectively, where

**x**,

*g*_

**x**, and

*a*_

**x**are the column vectors of model state variables, perturbations of state variables, and adjoint forcings, respectively. The TLM and ADM must satisfy

To check the accuracy of the model and observation operator adjoints in the WRF 4D-Var system, we run the adjoint check on the same 45-km grid-spaced 6-h assimilation window case as in section 2b. In addition to the conventional observations and GPS radio occultation observations, satellite radiance observations from Advanced Microwave Sounding Unit-A (AMSU-A), Advanced Microwave Sounding Unit-B (AMSU-B), High Resolution Infrared Radiation Sounder-3 (HIRS-3), High Resolution Infrared Radiation Sounder-4 (HIRS-4), and Microwave Humidity Sounder (MHS) are also assimilated. In total, there are 350 329 observations (233 476 from radiances) assimilated between 0009 and 0015 UTC 8 June 2012. The left- and right-hand sides of the adjoint relationship [Eq. (1)] are 1.909 375 417 539 27 × 10^{7} and 1.909 375 417 539 29 × 10^{7}, respectively, representing a 13-digit accuracy on the 64-bit machines. Results from the adjoint check confirm the mathematical accuracy of the WRFPLUS, variable transformation, and observation operator codes.

*J*is the 4D-Var cost function,

**x**

_{0}is the initial model state vector, and

*δ*

**x**is the model perturbation vector given bywhere

*α*is the scaling factor to control the magnitude of the perturbation. Therefore, the quantity ofapproaches unity with the

*α*repeatedly decreased by one order of magnitude until the machine accuracy is reached, where angle brackets with a comma denote the inner product. Table 1 shows the gradient check results for the 45-km case using the WRF 4D-Var system. There are six satisfactory digits in this gradient test, which implies a good approximation of the gradient. Any coding errors in the TLM, ADM, or data assimilation system may cause the gradient check to fail. However, as pointed out by Järvinen (1998), the gradient may pass the test if an error in the adjoint code creates only a relatively small error in the gradient. Therefore, it is important to check both the gradient and the adjoint before a production run.

Ratio of norms between the differences of the two cost functions and the gradient of the cost function. The norm is defined as the summation of the squares of all elements of the vector. Here, *α* is the perturbation scaling factors of the initial perturbation.

## 3. Multi-incremental WRF 4D-Var

### a. Implementation

The WRF 4D-Var system employs the incremental 4D-Var formulation that is commonly used in operational analysis systems (section 2 in H09). An iterative algorithm is used to solve the quadratic minimization problem. When the solution is found by linearizing the model and observation operators about a basic state, the minimization in 4D-Var is called an inner loop. To approximate the nonlinear problem that exists in data assimilation, multiple inner-loop minimizations might be carried out. After every inner-loop minimization, both the model and observation operators are relinearized based on the solution of the previous inner-loop minimization, the basic-state trajectory, and the innovations (observations minus first guess) are recalculated as well. We call the updating of the nonlinear solution the outer loop, which can be repeated several times. In summary, the incremental formulation of 4D-Var includes 1) the basic-state trajectory produced by the NLM integration; 2) the innovations calculated by comparing the basic-state trajectory to the observations; and 3) the inner loops to minimize the cost function.

To further reduce the computing cost of the 4D-Var algorithm, operational centers typically perform the inner-loop minimizations at a lower horizontal resolution than the outer loop and forecast. For example, at the time of this writing, the JMA mesoscale 4D-Var system runs outer loops with 5-km horizontal resolution and inner loops with 15 km. The JMA’s global 4D-Var analysis runs outer loops on T959 and inner loops on T319 (Takeuchi et al. 2013). If there is more than one outer loop, then different inner loops may have different horizontal resolutions (Veersé and Thépaut 1998; Rabier et al. 2000; Trémolet 2007; Kawabata et al. 2007; Gustafsson et al. 2012). We call this a multi-incremental 4D-Var algorithm. For example, the ECMWF operational 4D-Var now runs outer loops with T1279L91 (≈16 km) and three inner loops with T159, T255, and T255 (≈125, 80, 80 km, respectively). Another advantage of introducing the different horizontal resolutions stepwise into the minimization is to take nonlinear processes into account with reinitializations in the linearized minimization algorithm (Gustafsson et al. 2012).

Similar to the ECMWF 4D-Var configuration, illustrated schematically by Fig. 1 in Trémolet (2007), we implement the multi-incremental algorithm in the WRF 4D-Var system as well. The key feature of multi-incremental 4D-Var is that the outer loops use a high-resolution first guess to calculate the nonlinear model basic-state trajectory, which is compared with observations to generate the innovations. This ensures the highest possible accuracy for calculating the innovations, which are the primary input for the data assimilation. The innovations calculated by the high-resolution model trajectory are used in the low-resolution inner-loop minimization. For implementation simplicity, the low-resolution domain for the inner loop is required to have exactly the same vertical levels as the outer loop, as well as similar physical properties, in order to directly use the innovations calculated from the high-resolution grid. Before entering into the inner-loop minimization, the high-resolution first-guess fields are interpolated horizontally to the low-resolution domain to generate the first guess for the inner loops. Advanced interpolation methods, such as bilinear or spline interpolation, can be used here to include the high-resolution features that can be represented (without aliasing) on the coarse-resolution grids. All 2D and 3D fields are interpolated horizontally, which includes the static fields representing the land surface characteristics. Because the control of the LBCs is an optional capability in WRF 4D-Var (Zhang et al. 2011), the low-resolution LBCs are also needed for the integrations of the low-resolution TLM and ADM during the 4D-Var inner-loop minimization. For a regional model, the input data from a previously run external analysis or forecast model are available at every LBC update interval up to the forecast length, so the low-resolution LBCs can be calculated from two low-resolution states that are valid at the beginning and end of the assimilation time window from interpolated high-resolution states. In the ECMWF’s multi-incremental 4D-Var implementation, the low-resolution basic-state trajectory used to derive the nonlinear coefficients for the TLM and ADM in inner loops is interpolated spectrally from the high-resolution basic-state trajectory produced by the high-resolution NLM integration. In the WRF 4D-Var implementation, we have two options designed to generate the low-resolution basic-state trajectory for running the TLM and ADM. The first option follows the method used by the ECMWF in which the high-resolution basic-state trajectory is saved to files and interpolated to a low-resolution domain before the inner loop starts. The second option uses a low-resolution trajectory from an additional low-resolution WRF NLM run. The JMA’s operational mesoscale 4D-Var system also uses a lower-resolution simplified NLM (inner model) to provide trajectories (Takeuchi et al. 2013). We compared the impact of the two options on the 4D-Var analyses with extensive experiments and have not found any significant differences. The second option does not require the storage of high-resolution trajectory files on disk, nor does this option necessitate interpolating files to low-resolution domain or reading files in TLM and ADM. For these reasons, we chose the second option as the default for the production run. After the inner-loop minimization, the low-resolution analysis increments are interpolated back to high-resolution grids and added to the high-resolution first guess to produce the analysis solution for this outer loop. This analysis solution is the first guess for the next outer loop, if needed.

The multi-incremental minimization requires the control variables to be transferred between the different outer-loop iterations. The low-resolution control variables are saved at the end of the inner-loop minimization and are used to initialize the starting point of control variables for the next inner-loop minimization. If two consecutive inner loops have different horizontal resolutions, such as the first (T159) and second (T255) inner loops of the ECMWF operational 4D-Var run, then the saved control variables from the first inner loop have to be transformed to the higher horizontal resolution of the second inner loop. For control variables on spectral space, it is very easy to use different resolutions in different outer loops. If we go from a low-resolution increment (control variable) in spectral space to a full-resolution increment, we just fill in zeroes in the higher-resolution wave coefficients, carry out the inverse FFT, and add the increment to the full-resolution gridpoint model background. (N. Gustafsson 2014, personal communication). Among the current operational regional 4D-Var systems, the reference HIRLAM is a gridpoint model, but the HIRLAM 4D-Var is based on a spectral version of HIRLAM, so the TLM, ADM, and control variables are in spectral space (Gustafsson et al. 2012). The JMA mesoscale 4D-Var system uses gridpoint models, but it does not apply a multi-incremental minimization (Takeuchi et al. 2013). Therefore, the appropriate method of transfer for control variables on gridpoint space is critical for the implementation of multi-incremental WRF 4D-Var.

Starting with WRF-VAR, version 3.1, users have three choices to define the background error covariance (BE), called CV3, CV5, and CV6 (where CV stands for control variable). CV3 is the NCEP global BE, which is estimated in grid space using the National Meteorological Center (NMC) method (Parrish and Derber 1992). While CV3 can be used for any domain, CV5 and CV6 are domain dependent, and must be generated using forecast data from the same domain that is used for performing the data assimilation. With CV3 and CV5, the background errors are applied to the same set of the control variables that are used by WRF-VAR: streamfunction, unbalanced potential velocity, unbalanced temperature, unbalanced surface pressure, and pseudorelative humidity (Barker et al. 2004). However, for CV6 the moisture control variable is the unbalanced part of pseudorelative humidity. With CV3 the control variables are in physical space, while with CV5 and CV6, the control variables are in eigenvector space. The major difference between these two kinds of BE is the vertical covariance, which is modeled with a recursive filter in CV3, and with empirical orthogonal functions (EOFs) in CV5 and CV6.

*n*is the outer-loop index,

**B**is the BE,

**U**is defined as

**B**=

**UU**

^{T}, and

**v**

^{n}is the control variable vector after the

*n*th outer loop. The analysis increment from outer loop (

*n*− 1) to

*n*isEquation (7) of H09 shows that, during the calculation of the cost function gradient for the background error part, the contributions from the earlier outer loops to the control variable must be added. If the

*n*th outer loop has a different horizontal resolution than the (

*n*+ 1) outer loop, then the

**v**

^{n}has to be transferred to the resolution of the (

*n*+ 1) outer loop.

Because the control variables with CV3 are in physical space, it is straightforward to linearly or bilinearly interpolate the saved control variables from one horizontal resolution to another. To validate the horizontal interpolation of **v**^{n}, a multi-incremental experiment is designed. The case in section 2 is used with the same domain size, location, and map projection but with the horizontal grid spacing and time step reduced to 15 km and 60 s, respectively. The grid size used for the experiment is 676 × 379 with 45 vertical *σ* levels. The multi-incremental configuration has a 15-km outer loop and three inner loops of 135, 45, and 45 km. The maximum number of inner-loop iterations is limited to 20, and the analysis time is at 0000 UTC 3 June 2012. We compare the analysis increments [**x**^{n} − **x**^{n−1}; left side of Eq. (6)] from the first (135 km) outer loop with the **Uv**^{n} for the second (45 km) outer loop. If the interpolation of control variables is reasonable, they should be similar or comparable. Figures 3 and 4 show the analysis increments from the first outer loop and **Uv**^{n} for the second outer loop, respectively. The zonal (*U*) component of wind at level 17 (≈500 hPa), temperature at level 9 (≈850 hPa), sea level pressure, and specific humidity at level 9 are plotted for visual comparison. All four variables are similar in both magnitude and pattern, which indicates that the horizontal interpolation of CV3 control variables transfers effectively between different outer loops with different horizontal resolutions.

**U**as follows:where

**U**

_{h}is the horizontal transform,

**U**

_{υ}is the vertical transform, and

**U**

_{p}is the physical transform. For CV5,

**U**

_{υ}is via EOFs and the control variables are in eigenvector space. These control variables are the coefficients of the EOF expansion representing the vertical covariances of the streamfunction, unbalanced potential velocity, unbalanced temperature, unbalanced surface pressure, and pseudorelative humidity. It is not straightforward to horizontally interpolate CV5 control variables between different horizontal resolutions. One may argue that the horizontal interpolation should be conducted in physical space. However, in the WRF-VAR framework employing the incremental formula, we have the transformations from control variables to physical variables, but we do not have a transformation in the opposite direction. Specifically,

**U**

_{h}is represented via the recursive filters, and its inverse

### b. Computational performance and scientific impact

A series of experiments are carried out to evaluate the computational performance and scientific impact of applying multi-incremental 4D-Var for model initialization. Two configurations with three inner loops are tested. One uses the full-resolution configuration in which the outer and inner loops use the same resolution (15, 15, and 15 km), and the other uses the multi-incremental configuration with a 15-km outer loop and three inner loops of 135-, 45-, and 45-km grid spacing. Each inner loop has up to 20 iterations. The computing cost is estimated and compared between the two configurations on NCAR’s Yellowstone supercomputer (http://www2.cisl.ucar.edu/resources/yellowstone). The Yellowstone system is based on IBM’s iDataPlex architecture with 74 592 processor cores on 4662 IBM dx360 M4 nodes at the time of this study. Each node has dual 2.6-GHz Intel Sandy Bridge efficient performance (EP) processors and 32 GB of memory. The 14 data rate (FDR) Mellanox InfiniBand is used for interconnection.

Table 2 shows the wall clock time for completing 20 iterations for each inner loop with different numbers of processing cores for 15-, 15-, and 15-km and 135-, 45-, and 45-km configurations. For the full-resolution configuration, 32 nodes with one core per node (32 cores in total) is the minimum number of cores we used due to the constrained memory on each node and the large memory requirement for this case. The 60 total iterations need approximately 70 h to finish with 32 cores. Nevertheless, this case scales very well with an increasing number of cores, and it is able to complete the 60 iterations in 4.3 h with 1024 cores. The case that uses a multi-incremental configuration starts by using eight nodes with one core per node (eight cores in total). With a reasonable maximum of 256 cores (16 cores per node), this case executes within 37 min. The timing results show that the computational cost has been substantially reduced, because we are able to run 4D-Var using a 6-h window on a relatively large 15-km grid spacing domain with minimal computing resources.

Computational performance comparison (min) of the full-resolution WRF 4D-Var and multi-incremental WRF 4D-Var on NCAR Yellowstone. Each test has three outer loops with up to 20 iterations for each outer loop.

We use additional data assimilation experiments to evaluate the performance of the multi-incremental WRF 4D-Var configuration. Two one-week experiments are carried out to compare verification scores of the analyses and forecasts from the full-resolution and multi-incremental configurations. We use the same configurations as the CV3 test case described above, but the analyses are performed every 12 h between 0000 UTC 1 June and 1200 UTC 8 June 2012 with 24-h forecasts run from them. The analyses and 24-h forecasts are verified against the NCEP final analyses (FNL) that are valid at the same times. Figure 5 shows the average root-mean-square error (RMSE) profiles for the two configurations, which measure the differences of wind WV, temperature *T*, geopotential height *Z*, and water vapor *Q* between the analyses or forecasts and FNL. In terms of the quality of the analyses, Fig. 5a indicates that the multi-incremental configuration only slightly degrades the analyses of all variables compared to full-resolution runs. Even though the analysis is degraded slightly, Fig. 5b shows that the 24-h forecasts from the multi-incremental configuration are comparable to those of the full-resolution configuration. In this case, the multi-incremental configuration has slightly better scores for winds below 200 hPa and geopotential height below 300 hPa. Figure 6 shows the time series of 24-h RMSE at 850 and 500 hPa. This experiment confirms that the multi-incremental 4D-Var configuration consistently produces 24-h forecasts that are as accurate as the full-resolution configuration over the 15 cycles. Please note that there are many factors that could impact the analysis quality and subsequent forecast performance, such as background error statistics, observations, quality control, and the consistency between the analysis system and the model. We cannot claim that either configuration outperforms the others. The conclusion we can draw based on the one-week experiments is that the multi-incremental WRF 4D-Var configuration does not exhibit negative impacts on forecast performance with the benefit of reduced computational cost.

### c. Application to convection-permitting analyses and forecasts for Hurricane Sandy (2012)

The newly developed single executable, multi-incremental 4D-Var system is further examined through application to the convection-permitting analysis of Hurricane Sandy (2012), which was nearly unaffordable computationally with the predecessor WRF 4D-Var system. Sandy was the most destructive hurricane of the 2012 Atlantic hurricane season. The experiment design is based on a cloud-permitting hurricane analysis and prediction system including an ensemble Kalman filter for the WRF Model (Weng and Zhang 2014). The WRF Model has three two-way-nested domains with 379 × 244 (D01), 304 × 304 (D02), and 304 × 304 (D03) horizontal grid points with horizontal grid spacings of 27, 9, and 3 km, respectively. All model domains use 44 vertical levels and a pressure top at 50 hPa. D01 is fixed to cover the central to eastern three-quarters of the CONUS, the tropical and subtropical North Atlantic, and the two inner domains follow the observed hurricane position between data assimilation cycles using the preset moves option in WRF. In deterministic forecasts from the 4D-Var analyses, the two inner domains follow the storm center using the WRF vortex-following technique. These experiments use the same physical parameterization schemes described in section 2 but use the Grell–Devenyi scheme for cumulus parameterization (D1 only; D2 and D3 fully explicit), the WRF single-moment 6-class microphysics scheme (WSM6), and the five-layer thermal diffusion land surface model [for the details of these schemes, refer to Skamarock et al. (2008) and references therein]. An empirical scheme implemented in Green and Zhang (2013) is used to estimate the bulk drag and enthalpy coefficients (CD/CK). This ad hoc scheme has been found to improve the tropical cyclone (TC) wind pressure relationship forecasts.

The cycling 4D-Var system is initialized at 1200 UTC 20 October and is ended at 1200 UTC 28 October 2012; the first background field is taken from a 12-h forecast at 0000 UTC 21 October 2012. The assimilation window is 3 h and all conventional observations, satellite-derived winds, airborne reconnaissance dropsondes, and flight-level in situ measurements are assimilated with 4D-Var using two outer loops for the minimization. We picked a 1:9 grid size ratio between the first outer loop and its inner loops, as well as a 1:3 grid size ratio between the second outer loop and its inner loops for each domain. Every 6 h, a 144-h deterministic forecast is run from the 4D-Var analysis, using the Global Forecast System (GFS) for lateral boundary conditions. The experiments are conducted on the National Oceanic and Atmospheric Administration (NOAA) Global Systems Division’s Jet, which has 340 nodes with 5440 2.6-GHz Intel Sandy Bridge processors.

Figure 7 demonstrates the simulated hurricane tracks and intensities from 0000 UTC 21 October to 0000 UTC 28 October 2012. To make the figure more simple and clear, only the simulations at 0000 UTC are presented in Fig. 7. While future studies are needed to systematically evaluate the performance of the newly developed WRF 4D-Var for cloud-permitting hurricane analysis and forecasting, the performance of this initial application to Hurricane Sandy is certainly very promising. Although no satellite radiance observations were assimilated and only the operational GFS forecasts were used as the lateral boundary conditions, the deterministic WRF track hindcasts initialized as early as 0000 UTC 25 October 2012 were able to predict the storm's landfall at the New Jersey coast with as much as a 5-day lead time (Fig. 7a). Also encouraging were the intensity forecasts initialized with the 4D-Var analyses, especially after an initial spinup period (Fig. 7b). These WRF 4D-Var hindcasts also compared favorably against the other real-time and operational forecast products as shown in Munsell and Zhang (2014), though again it is beyond the scope of the current study to systematically evaluate the performance of the new 4D-Var for hurricane track and intensity prediction. The performance of the newly developed multi-incremental WRF 4D-Var was also recently shown to have promising performance for hurricane analysis and prediction in a coarser-resolution study of Hurricane Karl (2010), which can be further improved through coupling with an ensemble Kalman filter system under the WRF-VAR framework (Poterjoy and Zhang 2014).

Using the 4D-Var analysis of 0000–0300 UTC 25 October as an example, with approximately 40 iterations in total for two outer loops, the D01 analysis takes approximately 8 min 41 s with 72 computing cores, whereas D02 and D03 take 26 min 59 s, and 43 min 39 s, respectively. Please note that the 4D-Var analyses on the three domains can be ran independently of each other. Therefore, a cloud-permitting 4D-Var analysis is attainable within reasonable computational cost with the multi-incremental configuration.

## 4. Summary and discussion

This paper discussed the technical implementation and the computational performance of single executable and multi-incremental WRF 4D-Var. In general, the computing cost of a Single Program Multiple Data WRF 4D-Var system is much lower than the predecessor Multiple Program Multiple Data version.

In the single executable WRF 4D-Var system, the NLM, TLM, and ADM in WRFPLUS are compiled as a library and linked with WRF-VAR to build an SPMD system. The interfaces to call the TLM and ADM from the minimization algorithm were added into WRF-VAR, and the routines to initialize, advance, and finalize the TLM and ADM were also constructed in the WRFPLUS. A set of global data structures between WRF-VAR and WRFPLUS exchanges the data via memory copying, which is crucial for improving the computational efficiency of WRF 4D-Var. The execution of the single executable system is simpler because processors no longer need to be allocated for separate executables.

To validate the single executable WRF 4D-Var system, a gradient check was used to ensure the mathematical correctness of the tangent linear and adjoint code in the variational data assimilation system. The gradient check not only tests the correctness of the tangent linear and adjoint coding but also checks for errors in the variational assimilation system. Testing the tangent linear and adjoint codes, plus the gradient check, ensures the accuracy of the variational data assimilation system.

The multi-incremental 4D-Var configuration has been implemented to further reduce the computational cost of WRF 4D-Var. The initial implementation works well and the computational cost is reduced dramatically. Week-long experiments using a CONUS model domain indicate that the performance of multi-incremental 4D-Var is comparable to the full-resolution configuration. A second set of experiments for Hurricane Sandy shows that the multi-incremental WRF 4D-Var can be performed for a cloud-permitting grid spacing with an affordable computational cost, and both the track and intensity forecasts were promising. In addition to applying a stepwise increase in horizontal resolution for the inner loops, different quality control strategies may also be introduced, as well as more advanced physics packages at various stages of the minimization. Leveraging the developments in this study, the multi-incremental configuration for the GSI-based WRF 4D-Var (Zhang and Huang 2013) should be implemented easily because the GSI uses similar CV3 control variables.

Regarding the difficulty of the interpolation of the CV5 control variables discussed in section 3a, one proposed solution is to exchange the order of **U**_{υ} and **U**_{h} in Eq. (7), which means the vertical transformation should be applied to control variables before the horizontal transformation in Eq. (6). Therefore, it is reasonable to conduct the horizontal interpolation on the intermediate product **U**_{υ}**v**^{n}, followed by a

## Acknowledgments

This work is supported by the Air Force Weather Agency. Partial support also comes from NSF Grant 1305798 and ONR Grant N000140910526. We thank Dong-Kyou Lee and Gyu-Ho Lim of Seoul National University for their comments on the manuscript and generous support through the Korea–U.S. Weather and Climate Center. Junmei Ban helped to plot most of the figures. Michael Kavulich and Steven Olson helped to edit the manuscript.

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