Nonlinear Observation Operator Research Articles

Theoretical and numerical study of case reporting rate with application to epidemiology

A clear understanding of actual infection rate is imperative for control and prevention of diseases. In particular, it helps in formulating effective vaccination strategies and in assessing the level of herd immunity required to contain the virus. In this paper, we conduct theoretical and numerical study of a novel optimization procedure aimed at stable estimation of incidence reporting rate and time-dependent effective reproduction number from real data on new incidence cases, daily new deaths, and vaccination percentages. The iteratively regularized optimization algorithm can be applied to a broad class of data fitting problems constrained by various biological models, where one has to account for under-reporting of cases. To that end, general nonlinear observation operators in real Hilbert spaces are considered in the proposed convergence analysis. To illustrate theoretical findings, numerical simulations with SVIsIvRD compartmental model and real data for Delta variant of COVID-19 pandemic in different states of the US are conducted.

Reconstruction of incidence reporting rate for SARS-CoV-2 Delta variant of COVID-19 pandemic in the US

In recent years, advanced regularization techniques have emerged as a powerful tool aimed at stable estimation of infectious disease parameters that are crucial for future projections, prevention, and control. Unlike other system parameters, i.e., incubation and recovery rates, the case reporting rate, Ψ, and the time-dependent effective reproduction number, Re(t), are directly influenced by a large number of factors making it impossible to pre-estimate these parameters in any meaningful way. In this study, we propose a novel iteratively-regularized trust-region optimization algorithm, combined with SuSvIuIvRD compartmental model, for stable reconstruction of Ψ and Re(t) from reported epidemic data on vaccination percentages, incidence cases, and daily deaths. The innovative regularization procedure exploits (and takes full advantage of) a unique structure of the Jacobian and Hessian approximation for the nonlinear observation operator. The proposed inversion method is thoroughly tested with synthetic and real SARS-CoV-2 Delta variant data for different regions in the United States of America from July 9, 2021, to November 25, 2021. Our study shows that case reporting rate during the Delta wave of COVID-19 pandemic in the US is between 12% and 37%, with most states being in the range from 15% to 25%. This confirms earlier accounts on considerable under-reporting of COVID-19 cases due to the impact of ”silent spreaders” and the limitations of testing.

Assimilation of GOES-16 ABI All-Sky Radiance Observations in RRFS Using EnVar: Methodology, System Development, and Impacts for a Severe Convective Event

Abstract The Advanced Baseline Imager (ABI) aboard the GOES-16 and GOES-17 satellites provides high-resolution observations of cloud structures that could be highly beneficial for convective-scale DA. However, only clear-air radiance observations are typically assimilated at operational centers due to a variety of problems associated with cloudy radiance data. As such, many questions remain about how to best assimilate all-sky radiance data, especially when using hybrid DA systems such as EnVar wherein a nonlinear observation operator can lead to cost function gradient imbalance and slow minimization. Here, we develop new methods for assimilating all-sky radiance observations in EnVar using the novel Rapid Refresh Forecasting System (RRFS) that utilizes the Finite-Volume Cubed-Sphere (FV3) model. We first modify the EnVar solver by directly including brightness temperature (Tb) as a state variable. This modification improves the balance of the cost function gradient and speeds up minimization. Including Tb as a state variable also improves the model fit to observations and increases forecast skill compared to utilizing a standard state vector configuration. We also evaluate the impact of assimilating ABI all-sky radiances in RRFS for a severe convective event in the central Great Plains. Assimilating the radiance observations results in better spinup of a tornadic supercell. These data also aid in suppressing spurious convection by reducing the snow hydrometeor content near the tropopause and weakening spurious anvil clouds. The all-sky radiance observations pair well with reflectivity observations that remove primarily liquid hydrometeors (i.e., rain) closer to the surface. Additionally, the benefits of assimilating the ABI observations continue into the forecast period, especially for localized convective events.

Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models

Reduced-order modelling and low-dimensional surrogate models generated using machine learning algorithms have been widely applied in high-dimensional dynamical systems to improve the algorithmic efficiency. In this paper, we develop a system which combines reduced-order surrogate models with a novel data assimilation (DA) technique used to incorporate real-time observations from different physical spaces. We make use of local smooth surrogate functions which link the space of encoded system variables and the one of current observations to perform variational DA with a low computational cost. The new system, named generalised latent assimilation can benefit both the efficiency provided by the reduced-order modelling and the accuracy of data assimilation. A theoretical analysis of the difference between surrogate and original assimilation cost function is also provided in this paper where an upper bound, depending on the size of the local training set, is given. The new approach is tested on a high-dimensional (CFD) application of a two-phase liquid flow with non-linear observation operators that current Latent Assimilation methods can not handle. Numerical results demonstrate that the proposed assimilation approach can significantly improve the reconstruction and prediction accuracy of the deep learning surrogate model which is nearly 1000 times faster than the CFD simulation.

Manifold Markov Chain Monte Carlo Methods for Bayesian Inference in Diffusion Models

AbstractBayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology—borrowing ideas from statistical physics and computational chemistry—for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process. Joint configurations of the underlying process noise and of parameters, mapping onto diffusion paths consistent with observations, form an implicitly defined manifold. Then, by making use of a constrained Hamiltonian Monte Carlo algorithm on the embedded manifold, we are able to perform computationally efficient inference for a class of discretely observed diffusion models. Critically, in contrast with other approaches proposed in the literature, our methodology is highly automated, requiring minimal user intervention and applying alike in a range of settings, including: elliptic or hypo-elliptic systems; observations with or without noise; linear or non-linear observation operators. Exploiting Markovianity, we propose a variant of the method with complexity that scales linearly in the resolution of path discretisation and the number of observation times. Python code reproducing the results is available at http://doi.org/10.5281/zenodo.5796148.

Estimating initial conditions for dynamical systems with incomplete information

In this paper, we study the problem of inferring the latent initial conditions of a dynamical system under incomplete information, i.e., we assume we observe aggregate statistics of the system rather than its state variables directly. Studying several model systems, we infer the microstates that best reproduce an observed time series when the observations are sparse, noisy, and aggregated under a (possibly) nonlinear observation operator. This is done by minimizing the least-squares distance between the observed time series and a model-simulated time series using gradient-based methods. We validate this method for the Lorenz and Mackey–Glass systems by making out-of-sample predictions. Finally, we analyze the predicting power of our method as a function of the number of observations available. We find a critical transition for the Mackey–Glass system, beyond which it can be initialized with arbitrary precision.

Estimating NOx LOTOS-EUROS CTM Emission Parameters over the Northwest of South America through 4DEnVar TROPOMI NO2 Assimilation

In this work, we present the development of a 4D-Ensemble-Variational (4DEnVar) data assimilation technique to estimate NOx top-down emissions using the regional chemical transport model LOTOS-EUROS with the NO2 observations from the TROPOspheric Monitoring Instrument (TROPOMI). The assimilation was performed for a domain in the northwest of South America centered over Colombia, and includes regions in Panama, Venezuela and Ecuador. In the 4DEnVar approach, the implementation of the linearized and adjoint model are avoided by generating an ensemble of model simulations and by using this ensemble to approximate the nonlinear model and observation operator. Emission correction parameters’ locations were defined for positions where the model simulations showed significant discrepancies with the satellite observations. Using the 4DEnVar data assimilation method, optimal emission parameters for the LOTOS-EUROS model were estimated, allowing for corrections in areas where ground observations are unavailable and the region’s emission inventories do not correctly reflect the current emissions activities. The analyzed 4DEnVar concentrations were compared with the ground measurements of one local air quality monitoring network and the data retrieved by the satellite instrument Ozone Monitoring Instrument (OMI). The assimilation had a low impact on NO2 surface concentrations reducing the Mean Fractional Bias from 0.45 to 0.32, primordially enhancing the spatial and temporal variations in the simulated NO2 fields.

The Maximum Likelihood Ensemble Filter with State Space Localization

AbstractA new method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference is presented. The method is referred to as the MLEF with state space localization (MLEF-SSL) due to its similarity with the maximum likelihood ensemble filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update, the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to ensemble Kalman filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for an ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.

On the Inverse Source Identification Problem in L^{infty } for Fully Nonlinear Elliptic PDE

In this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the L^{infty } minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

Data assimilation algorithm based on an approximate solution of the optimal nonlinear filtering problem

In the ensemble Kalman filter, a currently popular data assimilation algorithm, it is assumed that the density of the estimation error distribution is a Gauss function, which is not fulfilled in the general case of a nonlinear model and a nonlinear observation operator. Statement of the optimal filtering problem in the general case is based on the Bayesian approach. One way to approximate the nonlinear optimal filtration problem is to represent the distribution density as a sum of Gaussian distributions with given mean values and covariance matrices. Then the optimal estimate is the weighted sum of estimates obtained in the Kalman filter for the corresponding parameters of the Gaussian distribution in this sum. The paper considers the algorithm for implementing this approach, which uses the previously developed effective local ensemble data assimilation algorithm (ensemble pi-algorithm). The results of numerical experiments to evaluate the properties of the proposed algorithm with the 1-dimensional nonlinear model are presented.

An Efficient Bi-Gaussian Ensemble Kalman Filter for Satellite Infrared Radiance Data Assimilation

AbstractThe introduction of infrared water vapor channel radiance ensemble data assimilation (DA) has improved numerical weather forecasting at operational centers. Further improvements might be possible through extending ensemble data assimilation methods to better assimilate infrared satellite radiances. Here, we will illustrate that ensemble statistics under clear-sky conditions are different from cloudy conditions. This difference suggests that extending the ensemble Kalman filter (EnKF) to handle bi-Gaussian prior distributions may yield better results than the standard EnKF. In this study, we propose a computationally efficient bi-Gaussian ensemble Kalman filter (BGEnKF) to handle bi-Gaussian prior distributions. As a proof-of-concept, we used the 40-variable Lorenz 1996 model as a proxy to examine the impacts of assimilating infrared radiances with the BGEnKF and EnKF. A nonlinear observation operator that constructs radiance-like bimodal ensemble statistics was used to generate and assimilate pseudoradiances. Inflation was required for both methods to effectively assimilate pseudoradiances. In both 800- and 20-member experiments, the BGEnKF generally outperformed the EnKF. The relative performance of the BGEnKF with respect to the EnKF improved when the observation spacing and time between DA cycles (cycling interval) are increased from small values. The relative performance then degraded when observation spacing and cycling interval become sufficiently large. The BGEnKF generated less noise than the EnKF, suggesting that the BGEnKF produces more balanced analysis states than the EnKF. This proof-of-concept study motivates future investigation into using the BGEnKF to assimilate infrared observations into high-order numerical weather models.

Toward a variational assimilation of polarimetric radar observations in a convective-scale numerical weather prediction (NWP) model

Abstract. This paper presents the potential of nonlinear and linear versions of an observation operator for simulating polarimetric variables observed by weather radars. These variables, deduced from the horizontally and vertically polarized backscattered radiations, give information about the shape, the phase and the distributions of hydrometeors. Different studies in observation space are presented as a first step toward their inclusion in a variational data assimilation context, which is not treated here. Input variables are prognostic variables forecasted by the AROME-France numerical weather prediction (NWP) model at convective scale, including liquid and solid hydrometeor contents. A nonlinear observation operator, based on the T-matrix method, allows us to simulate the horizontal and the vertical reflectivities (ZHH and ZVV), the differential reflectivity ZDR, the specific differential phase KDP and the co-polar correlation coefficient ρHV. To assess the uncertainty of such simulations, perturbations have been applied to input parameters of the operator, such as dielectric constant, shape and orientation of the scatterers. Statistics of innovations, defined by the difference between simulated and observed values, are then performed. After some specific filtering procedures, shapes close to a Gaussian distribution have been found for both reflectivities and for ZDR, contrary to KDP and ρHV. A linearized version of this observation operator has been obtained by its Jacobian matrix estimated with the finite difference method. This step allows us to study the sensitivity of polarimetric variables to hydrometeor content perturbations, in the model geometry as well as in the radar one. The polarimetric variables ZHH and ZDR appear to be good candidates for hydrometeor initialization, while KDP seems to be useful only for rain contents. Due to the weak sensitivity of ρHV, its use in data assimilation is expected to be very challenging.

Aerosol optical depth assimilation for a modal aerosol model: Implementation and application in AOD forecasts over East Asia

A new aerosol optical depth (AOD) data assimilation (DA) module was developed in Gridpoint Statistical Interpolation (GSI) 3-dimensional variational (3DVAR) system, named FastJ/CRTM-AOD DA module. And applied to the Modal Aerosol Dynamics Model for Europe with the Secondary Organic Aerosol Model (MADE/SORGAM) in the Weather Research and Forecasting/Chemistry model (WRF/Chem). The Fast-J optical module in WRF/Chem was used as the observation operator of AOD. The corresponding Jacobian code was modified from the one of CRTM-AOD in GSI. This way obviated the need for the Jacobian code's generation, which was complex and difficult for the highly nonlinear observation operator. During the studying period (January and April of 2014), compared to the ground AOD observations, AOD DA reduced about 20% fractional error (FE) with MADE/SORGAM. The original DA framework, which applied to the Goddard Chemistry Aerosol Radiation and Transport (GOCART) mechanism, performed slightly better than the new assimilation scheme for the low-value AOD situations (value < 0.4). However, compared to the original DA framework, the new DA scheme show a notable improvement for the high-value (0.4 < value ≤ 1.2) and extreme-high-value (value > 1.2) AOD situations. FE can be reduced by 48% and 64%, respectively. It indicates that the AOD DA impacts on AOD forecasts vary significant between different aerosol mechanisms. Moreover, FastJ/CRTM-AOD DA module can be easily and efficiently applied to the other aerosol schemes and the other optical modules, which is important to the development on AOD assimilation.

A Maximum Likelihood Ensemble Filter via a Modified Cholesky Decomposition for Non-Gaussian Data Assimilation

This paper proposes an efficient and practical implementation of the Maximum Likelihood Ensemble Filter via a Modified Cholesky decomposition (MLEF-MC). The method works as follows: via an ensemble of model realizations, a well-conditioned and full-rank square-root approximation of the background error covariance matrix is obtained. This square-root approximation serves as a control space onto which analysis increments can be computed. These are calculated via Line-Search (LS) optimization. We theoretically prove the convergence of the MLEF-MC. Experimental simulations were performed using an Atmospheric General Circulation Model (AT-GCM) and a highly nonlinear observation operator. The results reveal that the proposed method can obtain posterior error estimates within reasonable accuracies in terms of error norms. Furthermore, our analysis estimates are similar to those of the MLEF with large ensemble sizes and full observational networks.

High Spatial Resolution Soil Moisture Mapping Using a Lobe Differencing Correlation Radiometer on a Small Unmanned Aerial System

A persistent challenge in the measurement of soil moisture from satellites stems from the inherently low spatial resolution using active or passive system. The lobe differencing correlation radiometer (LDCR) on a small unmanned aerial system (sUAS) is shown to provide a capability to measure soil moisture at high spatial resolution for a range of scientific and operational purposes. Flight tests of LDCR on a fixed wing sUAS were performed at the Canton, Oklahoma Soilscape site in September 2015, and Irrigation Research Foundation (IRF) in Yuma, Colorado, in June 2016. The LDCR design and performance are discussed, and the calibration using both preflight lab test data and in-flight data over a calm pond was performed to calibrate the radiometer. Radio frequency interference (RFI) from the sUAS platform was observed and mitigated. The LDCR sampling processes are detailed and an implementation of the $\tau -\omega $ vegetation correction model along with a semiempirical surface roughness correction model incorporating a full-domain soil moisture mapping algorithm is presented. The algorithm uses a weakly nonlinear observation operator suitable for irregular sUAS flight trajectories that maps volumetric soil moisture (VSM) on a user-defined product grid from the sUAS sampling grid. Using LDCR radiometric and thermal measurements, along with Landsat-based vegetation water content (VWC) and soil texture information, soil moisture was mapped at decameter spatial resolution. The retrieved VSM data are favorably compared with in situ VSM measurements and irrigation records. A method for determining LDCR VSM estimation errors is developed to quantify the mapping algorithm accuracy and assess the impact of error sources.

An adaptive background error inflation method for assimilating all‐sky radiances

An adaptive background error inflation (ABEI) method is proposed for assimilating all‐sky satellite brightness temperatures with an ensemble Kalman filter. This empirical cloud‐scene‐dependent covariance inflation method is designed to mitigate the model's difficulties in initiating convection in the observed cloudy regions where the background prior estimated from the ensemble mean incorrectly simulates clear‐sky conditions. This new approach calculates a spatially varying, flow‐dependent, multiplicative ensemble covariance inflation factor based on error statistics produced by a well‐constructed, off‐line observing system simulation experiment (OSSE) that assimilates similar all‐sky radiance observations but were generated by the model, in which case the truth is known for all the state variables and the assimilated radiances. The adaptive inflation factor is a linear function of a cloud parameter which is only applied to the observed cloudy regions where there are less or no cloud in the prior ensemble mean estimates. The performance of ABEI is evaluated through assimilating synthetic and real‐data all‐sky radiance experiments from the Advanced Baseline Imager on board GOES‐16 for Hurricanes Karl of 2010 and Harvey of 2017. Assimilation experiments with ABEI allow adaptive inflation of the ensemble covariance in the model‐simulated clear‐sky regions when there are observed clouds while avoiding unnecessarily large ensemble spread in other cloud scenarios. This new approach alleviates the difficulty in estimating the appropriate inflation factors in the model state space using the innovation statistics in the observation space (radiance) with a highly nonlinear observation operator. It serves as an alternative to existing methods using spatially varying adaptive inflations; their relative performance and potential combinations are to be further assessed in the future.

Analysis of influence of observation operator on sequential data assimilation through soil temperature simulation with common land model

An observation operator is a bridge linking the system state vector and observations in a data assimilation system. Despite its importance, the degree to which an observation operator influences the performance of data assimilation methods is still poorly understood. This study aimed to analyze the influences of linear and nonlinear observation operators on the sequential data assimilation through soil temperature simulation using the unscented particle filter (UPF) and the common land model. The linear observation operator between unprocessed simulations and observations was first established. To improve the correlation between simulations and observations, both were processed based on a series of equations. This processing essentially resulted in a nonlinear observation operator. The linear and nonlinear observation operators were then used along with the UPF in three assimilation experiments: an hourly in situ soil surface temperature assimilation, a daily in situ soil surface temperature assimilation, and a moderate resolution imaging spectroradiometer (MODIS) land surface temperature (LST) assimilation. The results show that the filter improved the soil temperature simulation significantly with the linear and nonlinear observation operators. The nonlinear observation operator improved the UPF's performance more significantly for the hourly and daily in situ observation assimilations than the linear observation operator did, while the situation was opposite for the MODIS LST assimilation. Because of the high assimilation frequency and data quality, the simulation accuracy was significantly improved in all soil layers for hourly in situ soil surface temperature assimilation, while the significant improvements of the simulation accuracy were limited to the lower soil layers for the assimilation experiments with low assimilation frequency or low data quality.

Direct Assimilation of Radar Reflectivity without Tangent Linear and Adjoint of the Nonlinear Observation Operator in the GSI-Based EnVar System: Methodology and Experiment with the 8 May 2003 Oklahoma City Tornadic Supercell

Abstract A GSI-based EnVar data assimilation system is extended to directly assimilate radar reflectivity to initialize convective-scale forecasts. When hydrometeor mixing ratios are used as state variables (method mixing ratio), large differences of the cost function gradients with respect to the small hydrometeor mixing ratios and wind prevent efficient convergence. Using logarithmic mixing ratios as state variables (method logarithm) fixes this problem, but generates spuriously large hydrometeor increments partly due to the transform to and from the logarithmic space. The tangent linear of the reflectivity operators further contributes to spuriously small and large hydrometeor increments in method mixing ratio and method logarithm, respectively. A new method is proposed by directly adding the reflectivity as a state variable (method dBZ). Without the tangent linear and adjoint of the nonlinear operator, the new method therefore avoids the aforementioned problems. The newly proposed method is examined on the analysis and prediction of the 8 May 2003 Oklahoma City tornadic supercell storm. Both the probabilistic forecast of strong low-level vorticity and maintenance of strong updraft and vorticity in method dBZ are more consistent with reality than in method logarithm and method mixing ratio. Detailed diagnostics suggest that a more realistic cold pool due to the better analyzed hydrometeors in method dBZ than in other methods leads to constructive interaction between the surface gust front and the updraft aloft associated with the midlevel mesocyclone. Similar low-level vorticity forecast and maintenance of the storm are produced by the WSM6 and Thompson microphysics schemes in method dBZ. The Thompson scheme matches the reflectivity distribution with the observations better for all lead times, but shows more southeastward track bias compared to the WSM6 scheme.

Adaptive Observation Error Inflation for Assimilating All-Sky Satellite Radiance

An empirical flow-dependent adaptive observation error inflation (AOEI) method is proposed for assimilating all-sky satellite brightness temperatures through observing system simulation experiments with an ensemble Kalman filter. The AOEI method adaptively inflates the observation error when the absolute difference (innovation) between the observed and simulated brightness temperatures is greater than the square root of the combined variance of the uninflated observational error variance and ensemble-estimated background error variance. This adaptive method is designed to limit erroneous analysis increments where there are large representativeness errors, as is often the case for cloudy-affected radiances, even if the forecast model and the observation operator (the radiative transfer model) are perfect. The promising performance of this newly proposed AOEI method is demonstrated through observing system simulation experiments assimilating all-sky brightness temperatures from GOES-R (now GOES-16) in comparison with experiments using an alternative empirical observation error inflation method proposed by Geer and Bauer. It is found that both inflation methods perform similarly in the accuracy of the analysis and in the containment of potential representativeness errors; both outperform experiments using a constant observation error without inflation. Besides being easier to implement, the empirical AOEI method proposed here also shows some advantage over the Geer–Bauer method in better updating variables at large scales. Large representative errors are likely to be compounded by unavoidable uncertainties in the forecast system and/or nonlinear observation operator (as for the radiative transfer model), in particular in the areas of moist processes, as will be the case for real-data cloudy radiances, which will be further investigated in future studies.

Handling non-linearity in radar data assimilation using the non-linear least squares enhanced POD-4DVar

The Proper Orthogonal Decomposition (POD)-based ensemble four-dimensional variational (4DVar) assimilation method (POD4DEnVar) was proposed to combine the strengths of EnKF (i.e., the ensemble Kalman filter) and 4DVar assimilation methods. Recently, a POD4DEnVar-based radar data assimilation scheme (PRAS) was built and its effectiveness was demonstrated. POD4DEnVar is based on the assumption of a linear relationship between the model perturbations (MPs) and the observation perturbations (OPs); however, this assumption is likely to be destroyed by the highly non-linear forecast model or observation operator. To address this issue, using the Gauss-Newton iterative method, the nonlinear least squares enhanced POD4DEnVar algorithm (referred to as NLS-4DVar) was proposed. Naturally, the PRAS was upgraded to form the NLS-4DVar-based radar data assimilation scheme (NRAS). To evaluate the performance of NRAS against PRAS, observing system simulation experiments (OSSEs) were conducted to assimilate reflectivity and radial velocity individually, with one, two, and three iterations. The results demonstrated that the NRAS outperformed PRAS in improving the initial condition and the forecasting of model variables and rainfall. The NRAS, with a smaller number of iterations, can yield a convergent result. In contrast to the situation when assimilating radial velocity, the advantages of NRAS over PRAS were more obvious when assimilating reflectivity.