Four-dimensional Variational Data Assimilation Method Research Articles

An Enhanced Long-Term Wind Speed Prediction Using Dynamic Unified Ensemble Learning and Data Assimilation Techniques: A Case Study in Tamil Nadu, India

Reliable wind speed prediction is essential for effective grid management since wind energy is a key component of renewable energy generation. However, controlling wind speed just right is no easy feat since the wind flow changes constantly. This paper presents an innovative method for predicting wind speeds in the future. It uses state-of-the-art data assimilation methods and a dynamic unified ensemble learning model. For efficient wind energy planning and monitoring, WSP accuracy is crucial. Data from a single site limited the accuracy of WSPs in previous research. An improved accuracy of long-term wind speed predictions is achieved by the proposed model via the integration of data assimilation and ensemble learning. To increase forecast accuracy, the model uses sophisticated data assimilation methods like the Kalman filter to combine data from many sources. Specifically, the model employs the Stacked CNN + BiLSTM with Data Assimilator (SCBLSTM+DA) technique, which integrates Wind Speed (WS) data from adjacent areas with the CNN + BiLSTM-based Ensemble Learning Model (ELM) and the Four-Dimensional Variational and Ensemble Kalman Filter (4DVar/EnKF) Data Assimilation method. Using real-world wind speed data from nine meteorological stations in Tamil Nadu, India, we find that current prediction models, including both classical statistical and cutting-edge machine learning models, perform better. Further, unlike standalone models, the suggested model shows less susceptibility to changes in prediction time scales. Promising a solution to improve long-term wind speed predictions accuracy, this study has significant consequences for wind energy management and production.

Hybrid IFDMB/4D-Var inverse modeling to constrain the spatiotemporal distribution of CO and NO2 emissions using the CMAQ adjoint model

We performed a hybrid approach that combines the iterative finite difference mass balance (IFDMB) and four-dimensional variational data assimilation (4D-Var) methods to effectively constrain the spatiotemporal distribution of emissions. To quantitatively compare the performance of inverse modeling in constraining CO and NO2 emissions in South Korea spatiotemporally, we conducted a model-based twin experiment for three inverse modeling methods: IFDMB, 4D-Var, and hybrid inversions. We performed numerical modeling using the Community Multi-scale Air Quality (CMAQ) and its adjoints to calculate the values required for inverse modeling. As a result, the IFDMB inversion can effectively constrain the average spatial distribution of emissions. Meanwhile, the 4D-Var inversion can help estimate temporal variations in emissions, but it is not effective in regions with large prior emission errors. The hybrid inversion showed the best performance in constraining the spatiotemporal distribution of emissions because it combined the strengths of the two aforementioned methods. Furthermore, to compare the performance of the inverse modeling of pollutants with different chemical properties, we conducted additional inverse modeling for highly reactive NO2. After the application of inverse modeling, the emission errors of NO2 (18.339%) were larger than those of CO (10.593%). This difference in inverse modeling errors was due to the greater nonlinear relationship between emissions and concentrations in the inverse modeling process for NO2, which is more reactive compared to CO. In this study, the ideal modeling tests were performed to quantitatively assess the performance of inverse modeling. In future studies, we expect to apply the hybrid inversion approach used in this study to inverse modeling using actual observations.

Combined assimilation of hourly rainfall data and every 10-min AHI radiance with WRF 4DVar for the short-range heavy rainfall forecast in Eastern China

Rainfall observations and geostationary satellite water vapor radiances account for complementary humidity information over rainy areas and large-scale environments, respectively. This study investigates the simultaneous assimilation of China Hourly Merged Precipitation Analysis (CHMPA) data and clear-sky water vapor radiances provided by the Himawari-8 Advanced Himawari Imager (AHI) with Four-Dimensional Variational (4DVar) data assimilation method using Weather Research and Forecasting (WRF) model with an hourly rapid-update cycling data assimilation configuration. The rapid-update 4DVar assimilation is configured as 10-min interval for AHI radiances and 1-h interval for CHMPA rainfall data in a same 1-h assimilation time window. Generally positive impacts from assimilating CHMPA rainfall and AHI water vapor radiance observations are achieved for the 12-h short-term precipitation forecast. Error reductions are found after assimilating CHMPA rainfall and AHI water vapor radiance observations for both analyses and forecasts when compared to ERA5 reanalysis. It is found that combined assimilation of CHMPA rainfall and AHI water vapor radiance data has clearly positive impacts on heavy rainfall forecasts. A heavy rainfall case that occurred in Jiangsu Province on 28 June 2020 is examined in details. The combined assimilation of infrared radiances from AHI water vapor channels together with CHMPA rainfall data improves the vertical and horizontal transport of humidity conditions, which better adjusts the humidity distribution. The over-estimation of rainfall forecast caused by the assimilation of CHMPA is alleviated. This result can provide reference for quantitative precipitation forecasts of regional numerical weather model.

Role of Ocean Initialization in Skillful Prediction of Sahel Rainfall on the Decadal Time Scale

Abstract Sahel summer rainfall has undergone persistent drought from the 1970s to 1980s, causing severe human life and economic losses. Many studies pointed out that the decadal variations of Sahel rainfall are mainly modulated by low-frequency sea surface temperature (SST) variations in different ocean basins. However, how this modulation contributes to the decadal prediction skill of Sahel rainfall remains unknown. This study provided an affirmative response using the decadal hindcasts initialized by a dimensional-reduced projection four-dimensional variational (DRP-4DVar) data assimilation method to incorporate only ocean analysis data into the gridpoint version 2 of the Flexible Global Ocean–Atmosphere–Land System Model (FGOALS-g2). The hindcasts reveal the benefits of the DRP-4DVar approach for improving the Sahel rainfall decadal prediction skill measured by the anomaly correlation coefficient (ACC), root-mean-square error, ACC difference, and mean square skill score. The decadal variations of SSTs in the Atlantic, Mediterranean Sea, Indian Ocean, and Pacific as well as correct representations of the associated Sahel rainfall–SST relationships are well predicted, thus leading to skillful predictions of Sahel rainfall. In particular, the initialization of SSTs in the Atlantic and Mediterranean Sea plays a more important role in skillful Sahel rainfall predictions than in the other basins. The prediction skill of Sahel rainfall by the FGOALS-g2 prediction system is significantly higher than those by most phase 5 and 6 of the Coupled Model Intercomparison Project (CMIP5&6) prediction systems initialized only with ocean analysis data. This result is likely attributed to a more accurate relationship between Sahel rainfall and SST by the FGOALS-g2 prediction system than by the CMIP5&6 prediction systems. Significance Statement Previous studies have shown limited success in predicting Sahel rainfall. By using an advanced coupled data assimilation method constrained only by ocean observational data, we achieve a high decadal prediction skill for Sahel rainfall. The successful prediction is attributed to accurately predicted decadal variations of sea surface temperatures in the Atlantic, Mediterranean Sea, Indian Ocean, and Pacific as well as their relationships with Sahel rainfall. Our results can provide references for future decadal predictions of Sahel rainfall and motivate the need to evaluate the contributions of the initialization of the ocean versus the other climate components (e.g., atmosphere or land) to Sahel rainfall predictions.

4D-Var data assimilation using satellite sea surface temperature to improve the tidally-driven interior ocean dynamics estimates in the Indo-Australian Basin

Estimating ocean processes in regions with sparse observations, and where dynamics are dominated by strong barotropic and baroclinic tides, poses a significant modelling challenge. In this study, a Four-Dimensional Variational (4D-Var) data assimilation method, incorporating remotely sensed sea surface temperature (SST) data observations only, was used to improve estimates of the internal tide dynamics in the ocean between Australia and Indonesia. In this region, local internal tide generation, propagation and interference of the internal tides is highly variable and dependent on the 3D density field and mesoscale dynamics. The results of this study show that assimilation of remotely sensed SST data not only reduced the surface temperature root mean squared error (RMSE) from 0.8 °C to 0.3 °C, and removed a 0.4 °C cold bias (compared to a non-assimilative model), but also improved estimates of through-water-column variables — with the latter evaluated against ARGO profiling floats and two independent mooring sites. Overall, the depth-averaged temperature RMSE was reduced by as much as 38%, and ocean currents by ∼20% at the two mooring sites. The improvements were most significant in the top 100 m, where the internal tide signature was significantly corrected despite no depth-varying variables being assimilated. The study also showed that assimilating 2 km resolution SST observations resulted in more realistic SST wavenumber power spectral density estimates than those without assimilating data; in particular, the scheme reduced the energy at mesoscale length scales and improved the spectral slope towards the smaller submesoscale dynamics. It is concluded that 4D-Var assimilation using only SST improves estimates of upper 100 m ocean dynamics in tropical regions where there is sensitivity to the air–sea fluxes.

The Impact of Length-Scale Variation When Diagnosing the Standard Deviations of Background Error in a 4D-Var System and Filtering Method Investigation

The four-dimensional variational data assimilation (4D-Var) method has been widely employed as an operational scheme in mainstream numerical weather prediction (NWP) centers. In addition to the ensemble data assimilation method, the randomization technique is still used to diagnose the standard deviations of background error in variational data assimilation (VAR) systems; however, such randomization techniques induce sampling noise, which may contaminate the quality of the standard deviations. First, this paper studies the properties of the sampling noise induced by the randomization technique. The results show that the sampling noise is on a small scale displaying high-frequency oscillations around the estimate compared with the estimate and this difference motivates the use of filtering techniques to eliminate the sampling noise effects. The characteristics of the standard deviation field of the control variables are also investigated, and the standard deviation fields of different model parameters have different scales and vary with the vertical model levels. To eliminate such sampling noise, the spectral filtering method used widely in the operational system and a modified spatial averaging approach are investigated. Although both methods have splendid performance in eliminating sampling noise, the spatial averaging approach is more efficient and easier to implement in operational systems. In addition, the optimal filtered results from the spatial averaging approach are dependent on model parameters and vertical levels, which is consistent with the variation in the standard deviation field. Finally, the spatial averaging approach is tested on the operational system at the global scale based on the YH4DVAR and the global NWP system, and the results indicate that the spatial averaging approach has positive effects on both analysis and forecast quality.

Implicit Equal-Weights Variational Particle Smoother

Under the motivation of the great success of four-dimensional variational (4D-Var) data assimilation methods and the advantages of ensemble methods (e.g., Ensemble Kalman Filters and Particle Filters) in numerical weather prediction systems, we introduce the implicit equal-weights particle filter scheme in the weak constraint 4D-Var framework which avoids the filter degeneracy through implicit sampling in high-dimensional situations. The new variational particle smoother (varPS) method has been tested and explored using the Lorenz96 model with dimensions N x = 40 , N x = 100 , N x = 250 , and N x = 400 . The results show that the new varPS method does not suffer from the curse of dimensionality by construction and the root mean square error (RMSE) in the new varPS is comparable with the ensemble 4D-Var method. As a combination of the implicit equal-weights particle filter and weak constraint 4D-Var, the new method improves the RMSE compared with the implicit equal-weights particle filter and LETKF (local ensemble transformed Kalman filter) methods and enlarges the ensemble spread compared with ensemble 4D-Var scheme. To overcome the difficulty of the implicit equal-weights particle filter in real geophysical application, the posterior error covariance matrix is estimated using a limited ensemble and can be calculated in parallel. In general, this new varPS performs slightly better in ensemble quality (the balance between the RMSE and ensemble spread) than the ensemble 4D-Var and has the potential to be applied into real geophysical systems.

The demand analysis of oceanic T-S-V 3D reconstruction on wide-swath SSH data features based on ROMS and 4DVAR

Summary Future wide-swath altimetry missions will provide high-resolution information about ocean surface elevation, and facilitate the characterization of meso- and sub-mesoscale ocean activities. In this study, the demand analysis of three-dimensional (3D) oceanic state reconstruction on wide-swath SSH data features was evaluated using a data assimilation strategy. Three groups of experiments were performed to determine if the wide-swath altimetry observations would improve the three-dimensional (3D) field estimates of ocean temperature-salinity-velocity (T-S-V), and to evaluate how the spatial and temporal resolution and accuracy of the wide-swath altimetry observations affected the ocean state estimation. The Regional Ocean Modeling System and the four-dimensional variational data assimilation method were used in the experiments, with numerical simulation for the Taiwan region at a resolution of 1/10° as the example. The sensitivity of the 3D ocean state construction to the wide-swath altimetry measurements was also investigated. The results showed that the wide-swath sea surface height (SSH) measurements would have an overall positive impact on the 3D T-S-V field and that the positive effect would increase as the resolution and accuracy of the observations increased, but the net benefits would gradually decrease. Among the three examined features of the wide-swath altimetry observations, the temporal resolution had the most influence on the 3D ocean state analysis.

Assimilation of Middepth Velocities from Argo Floats in the Western South China Sea

AbstractPrevious studies are mainly limited to temperature and salinity (T/S) profiling data assimilation, while data assimilation based on Argo float trajectory information has received less research focus. In this study, a new method was proposed to assimilate Argo trajectory data: the middepth (indicates the parking depth of Argo floats in this study, ~1200 m) velocities are estimated from Argo trajectories and subsequently assimilated into the Regional Ocean Model System (ROMS) using four-dimensional variational data assimilation (4DVAR) method. This method can avoid a complicated float trajectory model in direct position assimilation. The 2-month assimilation experiments in South China Sea (SCS) showed that this proposed method can effectively assimilate Argo trajectory information into the model and improve middepth velocity field by adjusting the unbalanced component in the velocity increments. The assimilation of the Argo trajectory-derived middepth velocity with other observations (satellite observations and T/S profiling data) together yielded the best performance, and the velocity fields at the float parking depth are more consistent with the Argo float trajectories. In addition, this method will not decrease the assimilation performance of other observations [i.e., sea level anomaly (SLA), sea surface temperature (SST), and T/S profiles], which is indicative of compatibility with other observations in the 4DVAR assimilation system.

An ensemble of perturbed analyses to approximate the analysis error covariance in 4dvar

The analysis error covariance is not readily available from four-dimensional variational (4dvar) data assimilation methods, not because of the complexity of mathematical derivation, but rather its computational expense. A number of techniques have been explored for more readily obtaining the analysis error covariance such as using Monte–Carlo methods, an ensemble of analyses, or the adjoint of the assimilation method; but each of these methods retain the issue of computational inefficiency. This study proposes a novel and less computationally costly, approach to estimating the 4dvar analysis error covariance. It consists of generating an ensemble of pseudo analyses by perturbing the optimal adjoint solution. An application with a nonlinear model is shown.

An adjoint-free four-dimensional variational data assimilation method via a modified Cholesky decomposition and an iterative Woodbury matrix formula

In this paper, we propose an efficient and practical implementation of a four-dimensional variational ensemble Kalman filter (4D-EnKF) via a modified Cholesky decomposition. The main scope of our method is to avoid the intrinsic needed of adjoint models in the four-dimensional context. As it is well known, in practice, adjoint models can be labor-intensive to develop and computationally expensive to run. We avoid the use of adjoint models by taking snapshots of an ensemble of model realizations at observation times. Then, we employ a modified Cholesky decomposition on those ensembles to build control spaces, which in turn are employed to estimate analysis increments and to mitigate the impact of sampling noise. We discuss a matrix-free implementation of our 4D-EnKF formulation via the Woodbury matrix identity. Experimental tests are performed by using the Lorenz 96 model and an atmospheric general circulation model. The numerical results reveal that the accuracy of our proposed filter implementation outperforms those of traditional 4D-EnKF formulations in terms of L-2 error norms and root-mean-square error values.

Evaluation and assimilation of various satellite-derived rainfall products over India

ABSTRACT Accurate prediction of rainfall from the numerical weather prediction model is one of the major objectives over tropical regions. In this study, four different satellite-derived rainfall products (viz. merged-rainfall product from TRMM (Tropical Rainfall Measuring Mission) 3B42 and IMERG (Integrated Multi-satellitE Retrievals for GPM (Global Precipitation Measurement)), and Indian meteorological satellite INSAT-3D retrieved HEM (Hydro-Estimator Method) and IMSRA (INSAT Multi-Spectral Rainfall Algorithm) rainfall) are assimilated in the Weather Research and Forecasting (WRF) model using variational method. Before assimilation of satellite retrieved rainfall product in the WRF model, selected rainfall products are compared with ground rainfall from India Meteorological Department during Indian summer monsoon (June–September) 2015. Preliminary validation results show root-mean-square-difference (mean difference) of 18.1 (2.1), 21.3 (2.1), 15.4 (−0.72), and 14.4 (0.5) mm day−1 in IMSRA, HEM, IMERG, and TRMM 3B42 rainfall, respectively. Further, the four-dimensional variational data assimilation method is used daily to assimilate selected rainfall products in the WRF model during the entire month of August 2015. Results suggest that assimilation of satellite rainfall improved the WRF model analyses and subsequent temperature and moisture forecasts. Moreover, rainfall prediction is also improved with the maximum positive impact from TRMM rainfall assimilation followed by IMERG rainfall assimilation. Similar nature of improvements is also seen in rainfall prediction when INSAT-3D retrieved rainfall products (HEM and IMSRA) are used for assimilation.

A regional data assimilation system for estimating CO surface flux from atmospheric mixing ratio observations-a case study of Xuzhou, China.

Carbon monoxide (CO) emission inventory data are crucial for air quality control. However, the emission inventories are labor-intensive and time-consuming and generally have large uncertainties. In this study, we developed a new regional data assimilation system (TracersTracker) for estimating the surface CO emission flux from continuous mixing ratio observations using the proper orthogonal decomposition (POD)-based four-dimensional variational (4D-VAR) data assimilation method (POD-4DVar) and a coupled regional model (Weather Research and Forecasting model (WRF) with the Models-3 Community Multi-scale Air Quality (CMAQ) model). This system was applied to estimate CO emissions in Xuzhou city, China. An experiment was conducted with the continuous hourly surface CO mixing ratio observations from 21 monitoring towers in January and July of 2016. The experimental results of the system were examined and compared with the continuous surface CO observations (a priori emission). We found that the retrieved CO emission fluxes were higher than the a priori emission and were mainly distributed in urban and industrial areas, which were 104% higher in January (winter) and 44% higher in July (summer).

Evaluating the Role of the EOF Analysis in 4DEnVar Methods

The four-dimensional variational data assimilation (4DVar) method is one of the most popular techniques used in numerical weather prediction. Nevertheless, the needs of the adjoint model and the linearization of the forecast model largely limit the wider applications of 4DVar. 4D ensemble-variational data assimilation methods (4DEnVars) exploit the strengths of the Ensemble Kalman Filter and 4DVar, and use the ensemble trajectories to directly estimate four-dimensional background error covariance. This study evaluates the role of the empirical orthogonal function (EOF) analysis in 4DEnVars. The widely-recognized 4DEnVar method DRP-4DVar (the Dimension-reduced projection 4DVar) is adopted as the representation of EOF analyses in this study. The performance of the Dimension-reduced projection 4DVar (DRP-4DVar), 4DEnVar (i.e., another traditional 4DEnVar scheme without EOF transformation), and the Ensemble Transform Kalman Filter (ETKF) was compared to demonstrate the effect of the EOF analysis in DRP-4DVar. Sensitivity experiments indicate that EOF analyses construct basis vectors in eigenvalue space and the dimension reduction in the DRP-4DVar approach helps improve computational efficiency and analysis accuracy. When compared with 4DEnVar and the ETKF, the DRP-4DVar demonstrates similar analysis root-mean-square error (RMSE) to 4DEnVar, whereas it surpasses the ETKF by 22.3%. In addition, sensitivity experiments of DRP-4DVar on the ensemble size, the assimilation window length, and the standard deviation of the initial perturbation imply that the DRP-4DVar with the optimized EOF truncation number is robust to a wide range of the parameters, but extremely low values should be avoided. The results presented here suggest the potential wide application of EOF analysis in the hybrid 4DEnVar methods.

Estimation of the Tropical Pacific Ocean State 2010–13

AbstractA data-assimilating ⅓° regional dynamical ocean model is evaluated on its ability to synthesize components of the Tropical Pacific Ocean Observing System. The four-dimensional variational data assimilation (4DVAR) method adjusts initial conditions and atmospheric forcing for overlapping 4-month model runs, or hindcasts, that are then combined to give an ocean state estimate for the period 2010–13. Consistency within uncertainty with satellite SSH and Argo profiles is achieved. Comparison to independent observations from Tropical Atmosphere Ocean (TAO) moorings shows that for time scales shorter than 100 days, the state estimate improves estimates of TAO temperature relative to an optimally interpolated Argo product. The improvement is greater at time scales shorter than 20 days, although unpredicted variability in the TAO temperatures implies that TAO observations provide significant information in that band. Larger discrepancies between the state estimate and independent observations from Spray gliders deployed near the Galápagos, Palau, and Solomon Islands are attributed to insufficient model resolution to capture the dynamics in strong current regions and near coasts. The sea surface height forecast skill of the model is assessed. Model forecasts using climatological forcing and boundary conditions are more skillful than climatology out to 50 days compared to persistence, which is a more skillful forecast than climatology out to approximately 20 days. Hindcasts using reanalysis products for atmospheric forcing and open boundary conditions are more skillful than climatology for approximately 120 days or longer, with the exact time scale depending on the accuracy of the state estimate used for initializing and on the reanalysis forcing. Estimating the model representational error is a goal of these experiments.

Impact of single-point GPS integrated water vapor estimates on short-range WRF model forecasts over southern India

Specifying physically consistent and accurate initial conditions is one of the major challenges of numerical weather prediction (NWP) models. In this study, ground-based global positioning system (GPS) integrated water vapor (IWV) measurements available from the International Global Navigation Satellite Systems (GNSS) Service (IGS) station in Bangalore, India, are used to assess the impact of GPS data on NWP model forecasts over southern India. Two experiments are performed with and without assimilation of GPS-retrieved IWV observations during the Indian winter monsoon period (November–December, 2012) using a four-dimensional variational (4D-Var) data assimilation method. Assimilation of GPS data improved the model IWV analysis as well as the subsequent forecasts. There is a positive impact of ∼10 % over Bangalore and nearby regions. The Weather Research and Forecasting (WRF) model-predicted 24-h surface temperature forecasts have also improved when compared with observations. Small but significant improvements were found in the rainfall forecasts compared to control experiments.

Testing a four-dimensional variational data assimilation method using an improved intermediate coupled model for ENSO analysis and prediction

Abstract A four-dimensional variational (4D-Var) data assimilation method is implemented in an improved intermediate coupled model (ICM) of the tropical Pacific. A twin experiment is designed to evaluate the impact of the 4D-Var data assimilation algorithm on ENSO analysis and prediction based on the ICM. The model error is assumed to arise only from the parameter uncertainty. The “observation” of the SST anomaly, which is sampled from a “truth” model simulation that takes default parameter values and has Gaussian noise added, is directly assimilated into the assimilation model with its parameters set erroneously. Results show that 4D-Var effectively reduces the error of ENSO analysis and therefore improves the prediction skill of ENSO events compared with the non-assimilation case. These results provide a promising way for the ICM to achieve better real-time ENSO prediction.

Estimation of Finescale Rainfall Fields Using Broadcast TV Satellite Links and a 4DVAR Assimilation Method

AbstractThis study proposes a method based on the use of a set of commercial satellite-to-Earth microwave links to rebuild finescale rainfall fields. Such microwave links exist all over the world and can be used to estimate the integrated rain attenuation over the links’ first 5–7 km with a very high temporal resolution (10 s in the present case). The retrieval algorithm makes use of a four-dimensional variational data assimilation (4DVAR) method involving a numerical advection scheme. The advection velocity is recovered from the observations or from radar rainfall fields at successive time steps.This technique has been successively applied to simulated 2D rain maps and to real data recorded in the autumn of 2013 during the Hydrological Cycle in the Mediterranean Experiment (HyMeX), with one sensor receiving microwave signals from four different satellites. The performance of this system is assessed and is compared to an operational Météo-France radar and a network of 10 rain gauges. Because of the limitations of the propagation model, this study is limited to the events with strong advective characteristics (four out of eight recorded events). For these events (only), the method produces rainfall fields that are highly correlated with the radar maps at spatial resolutions greater than . The point-scale results are also satisfactory for temporal resolutions greater than 10 min (mean correlation with rain gauge data equal to approximately 0.8, similar to the correlation between radar and rain gauge data).This method can also be adapted to the fusion of a rain gauge with microwave link measurements and, through the use of several sensors, it has the potential of being applied to larger areas.

A Review of Measurement-Integrated Simulation of Complex Real Flows

In spite of the inherent difficulty, reproducing the exact structure of real flows is a critically important issue in many fields, such as weather forecasting or feedback flow control. In order to obtain information on real flows, extensive studies have been carried out on methodology to integrate measurement and simulation, for example, the four-dimensional variational data assimilation method (4D-Var) or the state estimator such as the Kalman filter or the state observer. Measurement-integrated (MI) simulation is a state observer in which a computational fluid dynamics (CFD) scheme is used as a mathematical model of the physical system instead of a small dimensional linear dynamical system usually used in state observers. A large dimensional nonlinear CFD model makes it possible to accurately reproduce real flows for properly designed feedback signals. This review article surveys the theoretical formulations and applications of MI simulation. Formulations of MI simulation are presented, including governing equations of a flow field observer, those of a linearized error dynamics describing the convergence of the observer, and stabilization of the numerical scheme, which is important in implementation of MI simulation. Applications of MI simulation are presented ranging from fundamental turbulent flows in pipes and Karman vortices in a wind tunnel to clinical application in diagnosis of blood flows in a human body.

A joint data assimilation system (Tan-Tracker) to simultaneously estimate surface CO<sub>2</sub> fluxes and 3-D atmospheric CO<sub>2</sub> concentrations from observations

Abstract. We have developed a novel framework ("Tan-Tracker") for assimilating observations of atmospheric CO2 concentrations, based on the POD-based (proper orthogonal decomposition) ensemble four-dimensional variational data assimilation method (PODEn4DVar). The high flexibility and the high computational efficiency of the PODEn4DVar approach allow us to include both the atmospheric CO2 concentrations and the surface CO2 fluxes as part of the large state vector to be simultaneously estimated from assimilation of atmospheric CO2 observations. Compared to most modern top-down flux inversion approaches, where only surface fluxes are considered as control variables, one major advantage of our joint data assimilation system is that, in principle, no assumption on perfect transport models is needed. In addition, the possibility for Tan-Tracker to use a complete dynamic model to consistently describe the time evolution of CO2 surface fluxes (CFs) and the atmospheric CO2 concentrations represents a better use of observation information for recycling the analyses at each assimilation step in order to improve the forecasts for the following assimilations. An experimental Tan-Tracker system has been built based on a complete augmented dynamical model, where (1) the surface atmosphere CO2 exchanges are prescribed by using a persistent forecasting model for the scaling factors of the first-guess net CO2 surface fluxes and (2) the atmospheric CO2 transport is simulated by using the GEOS-Chem three-dimensional global chemistry transport model. Observing system simulation experiments (OSSEs) for assimilating synthetic in situ observations of surface CO2 concentrations are carefully designed to evaluate the effectiveness of the Tan-Tracker system. In particular, detailed comparisons are made with its simplified version (referred to as TT-S) with only CFs taken as the prognostic variables. It is found that our Tan-Tracker system is capable of outperforming TT-S with higher assimilation precision for both CO2 concentrations and CO2 fluxes, mainly due to the simultaneous estimation of CO2 concentrations and CFs in our Tan-Tracker data assimilation system. A experiment for assimilating the real dry-air column CO2 retrievals (XCO2) from the Japanese Greenhouse Gases Observation Satellite (GOSAT) further demonstrates its potential wide applications.