1. Introduction
Numerically solving the atmospheric governing equations has been heavily relied upon to make accurate weather forecasts. Lorenz (1975) pointed out that weather forecast uncertainties generally come from errors in the initial conditions and numerical forecast models. With a given model, predicting the weather is largely an initial-value problem where future atmospheric states are predicted from the current atmospheric state, i.e., the initial conditions (Zou et al. 2001). The accurate estimation of initial conditions is thus critical for making reliable weather forecasts. The technique of four-dimensional variational (4D-Var) data assimilation (DA) has been one of the most sought-after methods in the field of atmospheric science because it offers a rigorous mathematical foundation for obtaining the optimal estimate of model initial conditions based on prior knowledge of the atmospheric state and observed features (Lewis and Derber 1985; Le Dimet and Talagrand 1986; Navon et al. 1992; Zou et al. 1995). The use of adjoint models in the 4D-Var DA system brings about the advantages of 4D-Var, including 1) efficient and accurate calculations of the gradients of the cost function that have the time of the observations accounted for, 2) implicit evolutions of the forecast error covariances following the nonlinear flow trajectories, and 3) dynamical consistency in the analyzed solutions (Navon et al. 1992; Huang et al. 2009). Given these strengths of the 4D-Var technique, numerous previous studies and operational centers have sought to adopt 4D-Var into their most sophisticated weather forecast applications. Zou et al. (1995) and Zou et al. (1997) developed the adjoint model of the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (NCAR) Mesoscale Model and formulated a 4D-Var regional DA system. Zou et al. (2001) proposed a global 4D-Var DA system based on the National Centers for Environmental Prediction global spectral model. The 4D-Var DA system in the Integrated Forecasting System at the European Centre for Medium-Range Weather Forecasts (ECMWF) marked the first successful operational implementation (Courtier et al. 1994). Other operational centers that adopted 4D-Var for operational applications include, but are not limited to, Météo-France (Gauthier and Thepaut 2001), the Met Office (Lorenc and Rawlins 2005), the Swedish Meteorological and Hydrological Institute (Gustafsson 2006; Gustafsson et al. 2012), and the China Meteorological Administration (Zhang et al. 2019).
The Model for Prediction Across Scales-Atmosphere (MPAS-A) is a global modeling framework released by NCAR (Skamarock et al. 2012; Park et al. 2013). Over the global domain, the MPAS-A employs finite-volume irregular centroidal Voronoi meshes on a staggered C-grid (Fig. 1) that allows for numerically solving atmospheric governing equations at both uniform (Fig. 1a) and smoothly variable (Fig. 1b) resolutions. Hagos et al. (2013) demonstrated the advantages of the smoothly variable resolution over the previous abruptly changing resolution in the nesting approach of the Weather Research and Forecasting (WRF) Model, which effectively avoided the erroneous wave signals generated in WRF simulations with a two-way nesting configuration. The option of doing simulations with variable resolutions implies that high-resolution regional climatological simulations may be achieved at an economical computational cost without depending on any lateral boundary conditions (Michaelis et al. 2019).
Spatial distributions of the spherical centroidal Voronoi tessellation meshes with (a) 2562 grid cells at a quasi-uniform resolution of 480 km and (b) 10 242 grid cells at resolutions ranging from 100 to 400 km over the globe in the MPAS-Atmosphere. The center of the area with a refined resolution in (b) is located at 35°N, 95°W.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
Tian and Zou (2020) documented the development and verification of the tangent linear (TL) and adjoint (AD) models of the MPAS-A and showed their usefulness in a sensitivity analysis using the MPAS AD model. Note that the TL/AD models of MPAS-A, like the original nonlinear forecast model, were developed in FORTRAN for its fast computational speed, while the top-layer structure was written in Python for its ease and flexibility in input/output, controlling simulation flows, and manipulating matrices (Lin 2012). In this study, the dynamic core of the MPAS-A nonlinear forecast model in Skamarock et al. (2012), i.e., the nonlinear forward model, and the AD model in Tian and Zou (2020) are used to formulate a global continuous 4D-Var DA system. No physics is included for the time being because the system is still under development. Similar to the structure in Tian and Zou (2020), the heavy-duty nonlinear forward and adjoint calculation processes are maintained in FORTRAN because of its efficiency, and the 4D-Var framework is coded in Python for its convenience. With such a framework, obtaining the gradients of the cost function and minimizations are still efficient, while the DA system’s overall Python coordinating structure is easily readable and adjustable. The FORTRAN codes are compiled with a Python utility f2py to build the interfaces between the two programming languages. With such interfaces, the computation results from the components of the nonlinear forward, adjoint, and minimization are communicated directly in memory. Section 2 of this study briefly introduces the TL/AD models of the MPAS-A and the formulation of a 4D-Var DA system. Section 3 describes the configurations of three idealized experiments that are done to validate and demonstrate the MPAS-A 4D-Var DA system’s performance. Numerical results of the three experiments for both the uniform- and variable-resolution meshes are shown and described in section 4. Section 5 gives a summary and conclusions.
2. MPAS-A 4D-Var DA system
a. Development of the MPAS-A TL/AD models
b. Variational analysis
providing a first-guess field xb;
computing the cost function J and its gradient ∇x0J;
passing J and ∇x0J to a minimization algorithm of choice that will compute an updated x0 following the gradient of the cost function;
repeating steps ii and iii until a satisfactory convergence is achieved.
Variations in the gradient-check results for the uniform resolution (solid curve) and variable-resolution (dashed curve) meshes following Eq. (6), i.e., log(|Φ(α) − 1|), as a function of the log of the scaling factor α.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
3. 4D-Var experimental design
The first guess and observations are generated from the ECMWF’s ERA5 reanalysis (Hoffmann et al. 2019). The ERA5 datasets at 0900 and 1200 UTC 1 February 2020 are first interpolated onto a global mesh with a uniform 480-km resolution (Fig. 1a). The mesh consists of 2562 grid cells horizontally over the global domain, upon which interpolated atmospheric profiles have 55 vertical levels. The forecasts valid at 0000 UTC 2 February 2020 made by the nonlinear MPAS-A with ERA5 at 0900 UTC 1 February as initial conditions will serve as the first guess in the 4D-Var experiments. Forecasts with ERA5 at 1200 UTC 1 February as initial conditions and valid at 0000 and 0600 UTC 2 February will serve as the “truth” to be referenced by the analysis and “observations.” No additional noise is added to the referenced “truth” or observations. Since the pseudo-observations are the model prognostic variables themselves, the observation operator Hr is simply an identity matrix
single-point observation assimilation, where θ at one location is observed and assimilated;
full-state-vector-observation assimilation to reversely infer the initial state, where “observations” of all five model prognostic variables over the entire global domain are assimilated;
mass-observation assimilation for wind-field reconstructions, where only values of θ over the global domain or a region are observed and assimilated.
A schematic illustration of the generation of the first guess, “observations,” and referenced “truth” in the three idealized experiments.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
4. Analysis results
a. Experiments with the UR mesh
1) Single-point observation
As described in section 3, a single-point observation valid at 0600 UTC 2 February 2020 is assimilated to solve for the analysis at 0000 UTC 2 February. Figure 4 shows the spatial distributions of θ and wind vectors at 500 hPa in the first guess at 0000 UTC and the observations at 0600 UTC. During the 6-h assimilation window, the trough at the latitude band of about 45°N slowly propagated eastward. The cyan cross in Fig. 4b marks the one observation point’s location, where only the value of θ will be assimilated in this experiment. Figure 5a shows the evolution of the cost function and the norm of the gradient. The value of the cost function decreased by more than one order of magnitude, and the gradient’s norm decreased by nearly three orders of magnitude, suggesting an effective convergence. Figure 5b illustrates the increments in the resulting analysis. Even though only the θ at a single location is assimilated, in addition to increments in θ, anticyclonic-patterned adjustments to the wind field can also be found upstream from the location of the observation. Flow-dependent features are seen in both fields because the gradient of the cost function is calculated by the MPAS-A AD model following the trajectory generated by the nonlinear MPAS-A model (Gustafsson 2007). Figure 6 shows the differences between the nonlinear forecasts taking the first guess and analysis as initial conditions, demonstrating how the analysis increment evolved within the assimilation window, specifically at 2, 4, and 6 h. Both θ and wind vector fields gradually propagated toward the observation location, settling down around the observation at the end of the assimilation window. As Bannister (2008) described, in 4D-Var, the BE covariance is implicitly propagated following the atmospheric flow to the time of observations within the assimilation window. However, the
Spatial distributions of potential temperature (shaded; K) and wind vectors at 500 hPa in the forecast valid for (a) 0000 UTC (the first guess) and (b) 0600 UTC (“observations”) 2 Feb 2020. The first guess is the forecast by MPAS taking ERA5 at 0900 UTC 1 Feb 2020 as the initial conditions and the “observations” taking ERA5 at 1200 UTC 1 Feb 2020 as the initial conditions. The location of the single observation assimilated is indicated by the cyan cross in (b).
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
(a) Evolutions of J (solid curve) and the norm of ∇J (dashed curve) as a function of iteration in the experiment of assimilating a single observation. Both J and |∇J| have been normalized by their initial values. (b) Analysis increments of potential temperatures (shaded; K) and of wind vectors after assimilating the observation located at the cyan cross.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
Evolution of the analysis increments shown in Fig. 5b after (a) 2, (b) 4, and (c) 6 h of nonlinear forecasts. The location of the observation is marked by the cyan cross.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
2) Inference of the initial state
The purpose of such a setup is to try to reconstruct the atmospheric state at the analysis time based on future observations. In this study, the complete atmospheric state at the time of observation is generated with the same MPAS-A nonlinear forecast model as in the 4D-Var DA system. The solved analysis and the known referenced “truth” should thus be similar. Figure 7a shows the spatial distributions of θ and wind vectors over the global domain in the first guess. Differences between the first guess and the known reference are given in Fig. 7b. Although random in general, the largest magnitudes of the differences in both mass and wind fields tend to appear in mid and high-latitude regions, where most planetary waves are located.
(a) Spatial distributions of potential temperatures (shaded; K) and wind vectors at 0000 UTC 2 Feb 2020 from the forecast with 1200 UTC 1 Feb 2020 as initial conditions, i.e., the reference. (b) Differences between the first guess and the reference shown in (a).
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
Figure 8 shows variations in the cost function and the norm of the gradient in the minimization process. The convergence is not as rapid as in the case of the single-point observation. The “observations” include all five model variables over the entire globe, so the gradients of the cost function are bound to be much more complicated than in the case of one observation. Overall, after 300 iterations, the cost function decreased by more than three orders of magnitude, and the norm of the gradient decreased by about three orders of magnitude, comparable to results reported by Thepaut and Courtier (1991) using a global primitive and adiabatic equation model. Figure 9a shows differences like those in Fig. 7b, except that the observations 6 h after the analysis time are assimilated. The magnitudes of the differences in both mass and wind fields are substantially reduced. Statistically, the histogram of the differences shows that the majority of the differences in θ are within the range of [−0.4, 0.4 K]. The wind field also shrinks noticeably toward zero, indicating that both fields are well reconstructed (Fig. 9b).
Variations in J (solid curve) and the norm of ∇J (dashed curve) as a function of iteration in the experiment assimilating full-vector observations. Both J and |∇J| have been normalized by their initial values.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
(a) Differences between the analysis after assimilating the full-vector observations and the reference shown in Fig. 6a. (b) Histograms of the differences in potential temperature (black curves) and wind (red curves) before (dashed) and after (solid) assimilating the full-vector observations.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
This experiment’s configuration is highly ideal because observations rarely cover all model variables and the entire global domain. At the same time, the assimilation of full-vector observations 6 h after the analysis time under such settings is exceedingly difficult, as evidenced by the convergence rate in the minimization process. However, the final convergence and a close reconstruction of the analysis with respect to the referenced “truth” validates the numerical feasibility of the global 4D-Var DA system proposed in this study.
3) Wind reconstruction with the mass field
Unlike the second experiment, only the θ field over the global domain 6 h after the analysis time is assimilated. Temperature is among the most commonly observed variables in realistic situations (Tian and Zou 2016, 2018; Zou and Tian 2018). Figure 10 shows the convergence rate of the minimization. After 300 iterations, the norm of the gradient of the cost function decreased by more than two orders of magnitude. Figure 11a shows the spatial distributions of the differences between the resulting analysis and the reference. Compared with Fig. 7b, differences in θ are slightly smaller in magnitude, and differences in wind vectors appear negligible. Figure 11b shows the histogram of the differences before and after assimilation. The differences in both variables shrink toward zero to some extent, more in the case of θ than in wind. Figure 12 shows the analysis increments of θ and wind compared to the first guess (Fig. 12a) and the mean and standard deviations of the analysis increments of wind with respect to latitude in 5° bins (Fig. 12b). Both the means and standard deviations of the increments in the tropics are smaller than those at higher latitudes. These statistics agree with the geostrophic adjustment theory that adjustments of wind by assimilating only mass observations should be slower close to the equator (Holton 1973; Žagar et al. 2004). Because only θ are “observed,” the wind field is still reconstructed to a statistically meaningful extent, one of the highly sought-after advantages of 4D-Var, namely, that dynamical consistency following the atmospheric governing equations can be maintained in the solved analysis.
Variations in J (solid curve) and the norm of ∇J (dashed curve) as a function of iteration in the experiment assimilating mass field observations. Both J and |∇J| have been normalized by their initial values.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
(a) Differences between the analysis after assimilating the potential temperatures and the reference shown in Fig. 6a. (b) Histograms of differences in potential temperature (black curves) and wind vectors (red curves) before (dashed) and after (solid) assimilating potential temperatures.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
(a) Analysis increments of potential temperature (shaded) and wind vectors (arrows) after assimilating only potential temperature. (b) Means (black curve) and standard deviations (shaded area) within each 5° latitude bin of the analysis increments of winds shown in (a).
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
b. Experiments with the VR mesh
Experiments A, B, and C with some modifications are conducted using a smoothly VR mesh. For experiment A, θ values over two individual points at 500 hPa are assimilated, one inside a finely resolved area and the other inside a coarsely resolved area (Fig. 13b with the Voronoi grid distributions overlaid). Figure 13a shows the evolution of the cost function and the norm of its gradient with respect to iteration. The gradient of the cost function decreased by more than two orders of magnitude after 19 iterations. The locations of the observations are marked by cyan crosses in Fig. 13a. Although only θ values at these two locations are assimilated, adjustments to the initial conditions are manifested in both the θ and wind fields upstream of the observation locations. With the VR mesh, flow dependency in the adjustments following atmospheric dynamics can be obtained regardless of the observation locations and the underlying resolutions. Figure 14 shows the evolution of the analysis increments in the nonlinear forecasts after 2, 4, and 6 h. The θ and wind vectors in both patterns gradually moved eastward following the nonlinear trajectory, stopping at the observation locations.
(a) Evolutions of J (solid curve) and the norm of ∇J (dashed curve) as a function of iteration in the experiment of assimilating two observations using the VR mesh. Both J and |∇J| have been normalized by their initial values. (b) Analysis increments of potential temperature (shaded) and wind vectors after assimilating the observations marked by the two cyan crosses. The actual Voronoi grid cells are overlaid in gray.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
Evolution of the analysis increments of potential temperatures (shaded) and wind vectors shown in Fig. 13b after (a) 2, (b) 4, and (c) 6 h of nonlinear forecasts. The locations of the observations are marked by the two cyan crosses.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
Figure 15 shows the results of assimilating full-vector observations 6 h after the analysis time using the VR mesh. As demonstrated in Fig. 15a, after 300 iterations, the cost function is reduced by about three orders of magnitude and its gradient by about two orders of magnitude. Similar to Fig. 9b, the histogram of the differences with respect to the referenced truth before and after DA is shown in Fig. 15b. The distribution of the differences in both θ and wind under initial conditions shrink considerably toward zero when compared with the reference after DA, indicating an effective inference of the complete initial state of the atmosphere from observations available 6 h later.
(a) Variations in J (solid curve) and the norm of ∇J (dashed curve) as a function of iteration in the experiment of assimilating full-vector observations using the VR mesh. Both J and |∇J| have been normalized by their initial values. (b) Histograms of the differences in potential temperature (black curves) and wind (red curves) before (dashed) and after (solid) assimilating the full-vector observations.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
Different from experiment C using the UR mesh, here with the VR mesh, the θ values of only one region are assimilated (Fig. 16b). Figure 16 shows the resulting cost functions, norms of their gradients, and analysis increments. Similar to that shown in Fig. 10, the norms of the gradients in Fig. 16a decreased by more than two orders of magnitude but only after 35 iterations. The region selected here (outlined by the oval in Fig. 16b) covers an area with grid cells that are both at fine resolution and at resolutions that gradually coarsen. This was done to demonstrate coherent analysis increment features in MPAS-A 4D-Var. Areas under initial conditions that are sensitive to the selected region 6 h after are found both inside and outside the oval. The analysis increment patterns also span various resolutions, indicating that the simulations within the oval are subject to influence from signals resolved at different scales.
(a) Variations in J (solid curve) and the norm of ∇J (dashed curve) as a function of iteration in the experiment of assimilating potential temperatures in the region outlined by the purple oval in (b). Both J and |∇J| have been normalized by their initial values. (b) Analysis increments of potential temperatures (shaded) and wind vectors.
Citation: Monthly Weather Review 149, 8; 10.1175/MWR-D-20-0408.1
5. Summary and conclusions
This study briefly documents the development of a global 4D-Var DA system with the MPAS-A model. The MPAS-A nonlinear forecast model was combined with the MPAS-A AD model described by Tian and Zou (2020) to formulate a continuous 4D-Var DA framework. A scalar cost function was defined, measuring the distances of the analysis vector to the background state vector and to the observations potentially distributed at different times. The gradient of the cost function with respect to the analysis was then calculated using the MPAS-A AD model. A necessary correctness procedure verifying the gradient calculations was implemented to ensure the accuracy of the gradients so that minimization algorithms could search in the right direction. The limited-memory BFGS was adopted for finding the minima of the cost functions in the experiments carried out in this study.
Three idealized experiments were conducted to validate and demonstrate the numerical feasibility of the MPAS-A 4D-Var DA system. The first guess, the “observations” 6 h after the analysis time, and the referenced “truth” valid for the analysis time were generated using the MPAS-A nonlinear forecast model taking ERA5 from the ECMWF as initial conditions. In the first experiment, the observation consisted of only θ at a single point. The analysis increment demonstrates a flow-dependent feature in both the wind vectors and θ values even though only one θ was “observed” and assimilated. In the second experiment, observations spanned the full analysis vector, i.e., all model prognostic variables over the entire global domain. In this experiment, perfect observations were assumed, implying that no background error covariances were involved. By assimilating full-vector observations 6 h after the analysis time, the initial state used to generate the observations was expected to be reconstructed by the 4D-Var DA system. After minimizations, the cost function (norm of the gradient) decreased by more than (about) three orders of magnitude. The differences between the analysis and the reference in wind vectors and θ were significantly smaller than those between the first guess and the reference. This experiment, although idealized, suggests that the 4D-Var DA system is numerically capable of solving for the analysis in challenging situations. The third experiment shows that the assimilation of mass field observations can also contribute to the reconstruction of the wind field, retaining dynamical consistency in the analysis. Results show that both the wind and θ in the analysis statistically agree more with the reference than the first guess. A similar set of experiments was conducted using a smoothly VR mesh. Flow-dependent features corresponding to individual observations in both fine- and coarse-resolution grids were found in the initial conditions. In the case of assimilating full-vector observations, initial conditions agreeing considerably better with the reference could be inferred from observations available 6 h after. In the third experiment, only θ values within a selected region were assimilated to demonstrate the coherence as well as the flow dependency in the analysis increment in regional DA applications.
The 4D-Var DA system proposed in this study is structured with a Python driver layer that can conveniently manage components such as input/output, control of simulation time flows, and matrix manipulations. A FORTRAN modeling layer is responsible for the heavy-duty numerical calculations. Such a structure retains a fast computational efficiency while bringing in the advantageous features of Python. Any future additions to this DA system, such as more efficient optimization modules, observation operators, radiative transfer models, and physical parameterizations, can be readily incorporated into the current structure.
Acknowledgments
The authors thank NCAR for releasing the source code of MPAS-Atmosphere at https://mpas-dev.github.io/. Thanks also go to Michael Duda and Darren Engwirda for their help in preparing the variable-resolution mesh. The authors appreciate the review comments from the editor and reviewers that helped to improve this study. The second author is supported by the National Key R&D Program of China (Grant 2018YFC1507004). Data for producing the plots in this manuscript are available at https://www.xiaoxutian.com/products/.
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