4D-Var Method Research Articles

Variational assimilation of IASI SO<sub>2</sub> plume height and total column retrievals in the 2010 eruption of Eyjafjallajökull using the SILAM v5.3 chemistry transport model

Abstract. This study focuses on two new aspects of inverse modelling of volcanic emissions. First, we derive an observation operator for satellite retrievals of plume height, and second, we solve the inverse problem using an algorithm based on the 4D-Var data assimilation method. The approach is first tested in a twin experiment with simulated observations and further evaluated by assimilating IASI SO2 plume height and total column retrievals in a source term inversion for the 2010 eruption of Eyjafjallajökull. The inversion resulted in temporal and vertical reconstruction of the SO2 emissions during 1–20 May 2010 with formal vertical and temporal resolutions of 500 m and 12 h.The plume height observation operator is based on simultaneous assimilation of the plume height and total column retrievals. The plume height is taken to represent the vertical centre of mass, which is transformed into the first moment of mass (centre of mass times total mass). This makes the observation operator linear and simple to implement. The necessary modifications to the observation error covariance matrix are derived.Regularization by truncated iteration is investigated as a simple and efficient regularization method for the 4D-Var-based inversion. In the twin experiments, the truncated iteration was found to perform similarly to the commonly used Tikhonov regularization, which in turn is equivalent to a Gaussian a priori source term. However, the truncated iteration allows the level of regularization to be determined a posteriori without repeating the inversion.In the twin experiments, assimilating the plume height retrievals resulted in a 5–20 % improvement in root mean squared error but simultaneously introduced a 10–20 % low bias on the total emission depending on assumed emission profile. The results are consistent with those obtained with real data. For Eyjafjallajökull, comparisons with observations showed that assimilating the plume height retrievals reduced the overestimation of injection height during individual periods of 1–3 days, but for most of the simulated 20 days, the injection height was constrained by meteorological conditions, and assimilation of the plume height retrievals had only small impact. The a posteriori source term for Eyjafjallajökull consisted of 0.29 Tg (with total column and plume height retrievals) or 0.33 Tg (with total column retrievals only) erupted SO2 of which 95 % was injected below 11 or 12 km, respectively.

Application of HY-2 Satellite SST Data in 4D Variational Assimilation Ocean Forecast Model

The 4D variational (4DVAR) assimilation numerical ocean model research is proposed. This model for Taiwan Straits (TWS) is based on Regional Ocean Model System (ROMS). The background of the 4DVAR method is introduced and the development process of assimilation system is presented. In the present research, the model assimilated with Sea Surface Temperature (SST) data of HY-2 satellite (Qi, 2012; Xu, 2013) which is the first marine environmental monitoring satellite of China. In this paper, the model processes from Feb. 1 to Feb. 7, 2014 with one-day assimilation time window and root mean square error (RMSE) reduces averagely by 14.7%.

A 4D-Var inversion system based on the icosahedral grid model (NICAM-TM 4D-Var v1.0) – Part 1: Offline forward and adjoint transport models

Abstract. A four-dimensional variational (4D-Var) method is a popular algorithm for inverting atmospheric greenhouse gas (GHG) measurements. In order to meet the computationally intense 4D-Var iterative calculation, offline forward and adjoint transport models are developed based on the Nonhydrostatic ICosahedral Atmospheric Model (NICAM). By introducing flexibility into the temporal resolution of the input meteorological data, the forward model developed in this study is not only computationally efficient, it is also found to nearly match the transport performance of the online model. In a transport simulation of atmospheric carbon dioxide (CO2), the data-thinning error (error resulting from reduction in the time resolution of the meteorological data used to drive the offline transport model) is minimized by employing high temporal resolution data of the vertical diffusion coefficient; with a low 6-hourly temporal resolution, significant concentration biases near the surface are introduced. The new adjoint model can be run in discrete or continuous adjoint mode for the advection process. The discrete adjoint is characterized by perfect adjoint relationship with the forward model that switches off the flux limiter, while the continuous adjoint is characterized by an imperfect but reasonable adjoint relationship with its corresponding forward model. In the latter case, both the forward and adjoint models use the flux limiter to ensure the monotonicity of tracer concentrations and sensitivities. Trajectory analysis for high CO2 concentration events are performed to test adjoint sensitivities. We also demonstrate the potential usefulness of our adjoint model for diagnosing tracer transport. Both the offline forward and adjoint models have computational efficiency about 10 times higher than the online model. A description of our new 4D-Var system that includes an optimization method, along with its application in an atmospheric CO2 inversion and the effects of using either the discrete or continuous adjoint method, is presented in an accompanying paper Niwa et al.(2016).

A Case Study of Assimilating Lightning-Proxy Relative Humidity with WRF-3DVAR

Lightning network data, considered as a useful supplement to radar observations, are a good indicator of severe convection, and has high temporal and spatial resolution. In Numerical Weather Prediction (NWP) models, lightning data are a new source of data to improve the forecasting of convective systems. In this case study, lightning data assimilation is conducted by converting lightning data to water vapor mixing ratio via a simple smooth continuous function, with input variables of total flash rate and simulated graupel mixing ratio at 9 km gridded resolution. Relative humidity converted from the retrieved water vapor mixing ratio is assimilated into the background field utilizing the three-dimensional variational (3DVAR) method in WRFDA (the Weather Research and Forecasting model Data Assimilation system). The benefits of assimilating lightning data are demonstrated in a series of experiments using data from a strong convection event that affected Beijing, Tianjin, Hebei and Shandong Province, on 31 July 2007. A nested domain with resolutions of 9 km and 3 km is implemented. For this case, assimilating lightning data shows some improvements in predictions of both reflectivity and neighboring precipitation, and in the temperature, dew-point temperature and relative humidity profile after seven hours.

Oceanic data assimilation study in northern Chile: use of a 3DVAR method

We report the use of a 3-dimensional variational (3DVAR) data assimilation method as part of a numerical model off northern Chile. The numerical model is part of an ocean forecasting project that aims to understand the impact of environmental variability on the distribution of biological species in the area. We assimilated data from a simulated ocean observing system to recover a known state, obtaining a significantly smaller error when compared to a numerical run with no assimilation. Our results validate the computational implementation of the code, and allow us to evaluate the impact of the choice of data in the assimilation process: the assimilation of sea surface height being particularly important. We note that the assimilation of surface data propagates properly to greater depths and reduces the error with reference to the known state. This was possible by using covariance error matrices calculated previously for the California coastal area. The implementation of the data assimilation module is relatively simple and permits its use in operational forecasting systems, and for the design and evaluation of future ocean observational systems.

Satellite radiance data assimilation for rainfall prediction in Java Region

This study examined the influence of satellite radiance data assimilation for predicting two days of heavy rainfall in the Java region. The first case occurred from 22 to 23 on January 2015 while the second case occurred from 1 to 2 on February 2015. The analysis examined before and after data assimilation in the two cases study. The Global Forecast System (GFS) data were used as initial condition which was assimilated with several data such as surface observation data, radiance data from AMSUA sensor, radiance data from HIRS sensor, and radiance data from MHS sensor. Weather Research and Forecasting Data Assimilation (WRFDA) is a tool which is used in this study for assimilating process with Three Dimensional Variation (3D-Var) method. The Quantitative Precipitation Forecast (QPF) skill was used to evaluate influence data assimilation for rainfall prediction. The result of the study obtained different rainfall prediction with different data assimilation. In general, the surface observation data assimilation has lower QPF skill than the satellite radiance data assimilation. Even thought radiance data assimilation has slightly contribution on rainfall prediction, but it gave better accuracy on rainfall prediction for two heavy rainfall cases.

Data assimilation in a coupled physical-biogeochemical model of the California Current System using an incremental lognormal 4-dimensional variational approach: Part 2—Joint physical and biological data assimilation twin experiments

Coupled physical and biological data assimilation is performed within the California Current System using model twin experiments. The initial condition of physical and biological variables is estimated using the four-dimensional variational (4DVar) method under the Gaussian and lognormal error distributions assumption, respectively. Errors are assumed to be independent, yet variables are coupled by assimilation through model dynamics. Using a nutrient-phytoplankton-zooplankton-detritus (NPZD) model coupled to an ocean circulation model (the Regional Ocean Modeling System, ROMS), the coupled data assimilation procedure is evaluated by comparing results to experiments with no assimilation and with assimilation of physical data and biological data separately. Independent assimilation of physical (biological) data reduces the root-mean-squared error (RMSE) of physical (biological) state variables by more than 56% (43%). However, the improvement in biological (physical) state variables is less than 7% (13%). In contrast, coupled data assimilation improves both physical and biological components by 57% and 49%, respectively. Coupled data assimilation shows robust performance with varied observational errors, resulting in significantly smaller RMSEs compared to the free run. It still produces the estimation of observed variables better than that from the free run even with the physical and biological model error, but leads to higher RMSEs for unobserved variables. A series of twin experiments illustrates that coupled physical and biological 4DVar assimilation is computationally efficient and practical, capable of providing the reliable estimation of the coupled system with the same and ready to be examined in a realistic configuration.

Reduced-order modelling of the flow around a high-lift configuration with unsteady Coanda blowing

We propose a hierarchy of low-dimensional proper orthogonal decomposition (POD) models for the transient and post-transient flow around a high-lift airfoil with unsteady Coanda blowing over the trailing edge. The modal expansion comprises actuation modes as a lifting method for wall actuation following Graham et al. (Intl J. Numer. Meth. Engng, vol. 44 (7), 1999, pp. 945–972) and Kasnakoğlu et al. (Intl J. Control, vol. 81 (9), 2008, pp. 1475–1492). A novel element is separate actuation modes for different frequencies. The structure of the dynamic model rests on a Galerkin projection using the Navier–Stokes equations, simplifying mean-field considerations, and a stochastic term representing the background turbulence. The model parameters are identified with a data assimilation (4D-Var) method. We propose a model hierarchy from a linear oscillator explaining the suppression of vortex shedding by blowing to a fully nonlinear model resolving unactuated and actuated transients with steady and high-frequency modulation of blowing. The models’ accuracy is assessed through the mode amplitudes and an estimator for the lift coefficient. The robustness of the model is physically justified, and then observed for the training and the validation dataset.

Adjoint free four-dimensional variational data assimilation for a storm surge model of the German North Sea

In this paper, a data assimilation scheme based on the adjoint free Four-Dimensional Variational(4DVar) method is applied to an existing storm surge model of the German North Sea. To avoid the need of an adjoint model, an ensemble-like method to explicitly represent the linear tangent equation is adopted. Results of twin experiments have shown that the method is able to recover the contaminated low dimension model parameters to their true values. The data assimilation scheme was applied to a severe storm surge event which occurred in the North Sea in December 5, 2013. By adjusting wind drag coefficient, the predictive ability of the model increased significantly. Preliminary experiments have shown that an increase in the predictive ability is attained by narrowing the data assimilation time window.

Assimilation of MWHS radiance data from the FY‐3B satellite with the WRF Hybrid‐3DVAR system for the forecasting of binary typhoons

AbstractChan‐hom and Linfa were binary typhoons that occurred in the western North Pacific in 2015. In this study, the impacts of FY‐3B satellite Microwave Humidity Sounder (MWHS) radiance observations on the analyses and forecasts of Linfa and Chan‐hom are assessed. The regional Weather Research and Forecasting model and its data assimilation (DA) systems, using three‐dimensional variational (3DVAR) and Hybrid (ensemble/3DVAR) methods are used. Assimilation of the FY‐3B MWHS data using the 3DVAR method slightly improves the descriptive wind and temperature fields. Positive impacts on the specific humidity forecasts, for levels higher than 850 hPa, are also obvious. 3DVAR adjusts the typhoons' initial positions and their dynamic structures favorably, yielding better tracks, intensities, and precipitation forecasts, compared to the experiment run without MWHS data (control). With the Hybrid method, the water vapor information from the MWHS data better improve the analyses through multivariable correlations with the flow‐dependent background error. The Hybrid method further improves the track, intensities, and precipitation forecasts. For Typhoon Linfa, with the coexistence of Typhoon Chan‐hom, the Hybrid method provides a more descriptive background error covariance matrix, than using 3DVAR. Experiments on multiple binary typhoon cases are also provided to further validate the robustness of the results on the FY‐3B satellite MWHS radiance data assimilation.

ESTUDO DE CASOS DE PRECIPITAÇÃO UTILIZANDO O WRF-3DVAR NA REGIÃO SUL DO BRASIL

The aim of the study has been to evaluate the results of the data assimilation of INMET automatic weather stations and soundings of the southern of Brazil in precipitation and temperature forecasting of predicting of the WRF model by 3DVAR method (WRF-3DVAR), analyzing in each case, whether the WRF-3DVAR forecasts are more accurate than the WRF without data assimilation. The obtained results show the great importance of data assimilation in precipitation prediction of the WRF model. The prognoses of vertical profiles of the air and the dew point temperatures, showed that the data assimilation produced predictions with thermal and moist vertical structures more realistic in the light of the observational soundings.

On the Role of Temperature and Salinity Data Assimilation to Constrain a Coupled Physical–Biogeochemical Model in the Baltic Sea

AbstractA three-dimensional variational data assimilation (3DVAR) method is implemented in a coupled physical–biogeochemical (CPB) model in the Baltic Sea. This study carries out a 10-yr assimilation experiment with satellite sea surface temperature (SST) and observed in situ temperature (T) and salinity (S) profiles. The impact of the assimilation is assessed with the focus on how the biogeochemical model responds to the improved hydrodynamics. The assimilation of temperature and salinity data yields considerable improvements in the physical model. On a basin scale, the mean bias of SST, T, S, and mixed layer depth (MLD) is decreased by 0.18°C (57%), 0.31°C (49%), 0.34 psu (43%), and 1.8 m (43%), respectively. More importantly, the biogeochemical simulation is improved in response to the physical data assimilation. Compared with in situ observations, the mean biases of chlorophyll a (Chl), dissolved inorganic nitrogen (DIN) and phosphorus (DIP) are decreased by 0.09 mg m−3 (15.5%), 0.19 mmol m−3 (9%), and 0.15 mmol m−3 (23%). Physical data assimilation also improves the simulated variability of Chl, DIN, and DIP and their correlations with observation. Compared with satellite observations, the mean bias of surface chlorophyll is reduced by 0.10–0.32 mg m−3 especially in the Skagerrak–Kattegat area and Bornholm basin. The decrease of total Chl change is caused by different mechanisms for winter and summer. While the deepened mixed layer acts as a dilution factor in winter, strengthened stratification agrees well with the decrease of chlorophyll in summer. In the vertical, relatively large changes of DIN and DIP occur below 60 m, which corresponds to the mean permanent halocline depth (~60–80 m) of the Baltic Sea.

Estimation of Volcanic Ash Emissions Using Trajectory-Based 4D-Var Data Assimilation

Abstract Volcanic ash forecasting is a crucial tool in hazard assessment and operational volcano monitoring. Emission parameters such as plume height, total emission mass, and vertical distribution of the emission plume rate are essential and important in the implementation of volcanic ash models. Therefore, estimation of emission parameters using available observations through data assimilation could help to increase the accuracy of forecasts and provide reliable advisory information. This paper focuses on the use of satellite total-ash-column data in 4D-Var based assimilations. Experiments show that it is very difficult to estimate the vertical distribution of effective volcanic ash injection rates from satellite-observed ash columns using a standard 4D-Var assimilation approach. This paper addresses the ill-posed nature of the assimilation problem from the perspective of a spurious relationship. To reduce the influence of a spurious relationship created by a radiate observation operator, an adjoint-free trajectory-based 4D-Var assimilation method is proposed, which is more accurate to estimate the vertical profile of volcanic ash from volcanic eruptions. The method seeks the optimal vertical distribution of emission rates of a reformulated cost function that computes the total difference between simulated and observed ash columns. A 3D simplified aerosol transport model and synthetic satellite observations are used to compare the results of both the standard method and the new method.

Europeans shine in weather forecasting

The US global model lags the performance of two European competitors in predicting weather up to two weeks ahead.

A Hybrid Global Ocean Data Assimilation System at NCEP

Abstract Seasonal forecasting with a coupled model requires accurate initial conditions for the ocean. A hybrid data assimilation has been implemented within the National Centers for Environmental Prediction (NCEP) Global Ocean Data Assimilation System (GODAS) as a future replacement of the operational three-dimensional variational data assimilation (3DVar) method. This Hybrid-GODAS provides improved representation of model uncertainties by using a combination of dynamic and static background error covariances, and by using an ensemble forced by different realizations of atmospheric surface conditions. An observing system simulation experiment (OSSE) is presented spanning January 1991 to January 1999, with a bias imposed on the surface forcing conditions to emulate an imperfect model. The OSSE compares the 3DVar used by the NCEP Climate Forecast System (CFSv2) with the new hybrid, using simulated in situ ocean observations corresponding to those used for the NCEP Climate Forecast System Reanalysis (CFSR). The Hybrid-GODAS reduces errors for all prognostic model variables over the majority of the experiment duration, both globally and regionally. Compared to an ensemble Kalman filter (EnKF) used alone, the hybrid further reduces errors in the tropical Pacific. The hybrid eliminates growth in biases of temperature and salinity present in the EnKF and 3DVar, respectively. A preliminary reanalysis using real data shows that reductions in errors and biases are qualitatively similar to the results from the OSSE. The Hybrid-GODAS is currently being implemented as the ocean component in a prototype next-generation CFSv3, and will be used in studies by the Climate Prediction Center to evaluate impacts on 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.

Assimilação de Dados Via Método 3D-Var em Dinâmica Caótica do Modelo de Lorenz

This paper explores some aspects of data assimilation by 3D-Var method implemented in the Lorenz model. This system was chosen because, although simple, has similar structure to the atmosphere under certain parameters. Conceptual models allow difficult judgments to be made in primitive equation models with their various nonlinear interactions at various scales. It is shown the need to apply data assimilation techniques in highly sensitive models to initial conditions. It was also shown that 3D-Var method is very dependent on the number of cycles during the process of minimizing of the cost function. In the experiment performed on this work was found that at least 1,000 interactions are necessary to solve this minimization problem. However the technique fail to 40% noise added to initial conditions, comparing to control. Additional experiments show the difficulty of performing high quality data assimilation in chaotic dynamics for underdetermined systems, such as the Earth's atmosphere.

Assimilation of surface NO<sub>2</sub> and O<sub>3</sub> observations into the SILAM chemistry transport model

Abstract. This paper describes the assimilation of trace gas observations into the chemistry transport model SILAM (System for Integrated modeLling of Atmospheric coMposition) using the 3D-Var method. Assimilation results for the year 2012 are presented for the prominent photochemical pollutants ozone (O3) and nitrogen dioxide (NO2). Both species are covered by the AirBase observation database, which provides the observational data set used in this study. Attention was paid to the background and observation error covariance matrices, which were obtained primarily by the iterative application of a posteriori diagnostics. The diagnostics were computed separately for 2 months representing summer and winter conditions, and further disaggregated by time of day. This enabled the derivation of background and observation error covariance definitions, which included both seasonal and diurnal variation. The consistency of the obtained covariance matrices was verified using χ2 diagnostics. The analysis scores were computed for a control set of observation stations withheld from assimilation. Compared to a free-running model simulation, the correlation coefficient for daily maximum values was improved from 0.8 to 0.9 for O3 and from 0.53 to 0.63 for NO2.

Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

AbstractWe propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

Estimating the Aboveground Dry Biomass of Grass by Assimilation of Retrieved LAI Into a Crop Growth Model

This study presents a method to assimilate leaf area index (LAI) retrieved from MODIS data using a physically based method into a soil-water-atmosphere-plant (SWAP) model to estimate the aboveground dry biomass of grass in the Ruoergai grassland, China. The assimilation method consists of reinitializing the model with optimal input parameters that allow a better temporal agreement between the LAI simulated by the SWAP model and the LA! retrieved from MODIS data. The minimization is performed by a four-dimensional variational data assimilation (4D-VAR) algorithm but which is challenged by the development of the adjoint model. The automatic differentiation (AD) technique is thus used to provide the adjoint model at the level of computer language codes. After the re-initialization, the simulated aboveground dry biomass value is compared with ground measurements taken in early August2013. The results show that the biomass can be estimated with highly satisfactory accuracy level through the assimilation method with R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (the deterministic coefficient) = 0.73 and RMSE(root-mean-square error) = 617.94 kg ha <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> . The accuracy is further improved when the newly derived RMSELAI values are used as observation errors in the assimilation process, with R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0.76 and RMSE = 542.52 kgha <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> . Both assimilation strategies yield a significant improvement in SWAP model accuracy with respect to no significant correlation obtained when the SWAP model is run alone with constant values of the input parameters employed for the whole area. The validity of the 4D-VAR method for biomass estimation is well demonstrated.