Newtonian Nudging Research Articles

EVALUATING PREDICTIONS OF THE SOIL MOISTURE MODEL WITH DATA ASSIMILATION BY THE TRIPLE COLLOCATION METHOD

For a long time, ground measurements are considered the most accurate method of regular monitoring of soil moisture. However, ground stations are expensive, require local calibration and thus often not practical to use. Other, more affordable means of soil moisture monitoring can be developed with the recent advancement of Earth remote sensing technologies. In this paper, we describe a nonlinear problem of soil moisture transfer problem with addition of satellite soil moisture measurements. The mathematical model is based on the Richards equation for soil moisture transport, and solved with the finite difference method on implicit iterative scheme. Satellite moisture retrievals are acquired by combining active and passive sensors data with the decomposition algorithm. The satellite data are incorporated into the model with the data assimilation algorithm called Newtonian nudging. This method adds a special ‘nudging’ term into the model governing equation. This assures that the model is corrected by satellite measurements without affecting the process physics. Moreover, we look into the nudging factor problem, and propose a simple empirical relation based on the soil properties for more universally stable work of the method. For validation purpose, we conduct a massive numerical experiments over all registered ground stations in the USA. Evaluation is done by the use of triple collocation method, which allows assessing the errors of three independent data sources. The data sources used for evaluation are the model results, ground station measurements and ERA5 satellite observations. The results demonstrate that the presented model is capable of producing results with close accuracy to the ground station measurements.

Evaluation of the soil moisture model with data assimilation by the triple collocation method

This paper deals with a nonlinear soil moisture transport problem, solved with addition of satellite observed soil moisture. The satellite data are assimilated into the model using Newtonian nudging method. Evaluation is done by the triple collocation method, which involves three independent data sources: model results, ground stations and ERA5 climatic data. The results testify that model results are nearly as accurate as the ground station measurements.

Improving terrestrial evaporation estimates over continental Australia through assimilation of SMOS soil moisture

Terrestrial evaporation is an essential variable in the climate system that links the water, energy and carbon cycles over land. Despite this crucial importance, it remains one of the most uncertain components of the hydrological cycle, mainly due to known difficulties to model the constraints imposed by land water availability on terrestrial evaporation. The main objective of this study is to assimilate satellite soil moisture observations from the Soil Moisture and Ocean Salinity (SMOS) mission into an existing evaporation model. Our over-arching goal is to find an optimal use of satellite soil moisture that can help to improve our understanding of evaporation at continental scales. To this end, the Global Land Evaporation Amsterdam Model (GLEAM) is used to simulate evaporation fields over continental Australia for the period September 2010–December 2013. SMOS soil moisture observations are assimilated using a Newtonian Nudging algorithm in a series of experiments. Model estimates of surface soil moisture and evaporation are validated against soil moisture probe and eddy-covariance measurements, respectively. Finally, an analogous experiment in which Advanced Microwave Scanning Radiometer (AMSR-E) soil moisture is assimilated (instead of SMOS) allows to perform a relative assessment of the quality of both satellite soil moisture products. Results indicate that the modelled soil moisture from GLEAM can be improved through the assimilation of SMOS soil moisture: the average correlation coefficient between in situ measurements and the modelled soil moisture over the complete sample of stations increased from 0.68 to 0.71 and a statistical significant increase in the correlations is achieved for 17 out of the 25 individual stations. Our results also suggest a higher accuracy of the ascending SMOS data compared to the descending data, and overall higher quality of SMOS compared to AMSR-E retrievals over Australia. On the other hand, the effect of soil moisture data assimilation on the evaporation fields is very mild, and difficult to assess due to the limited availability of eddy-covariance data. Nonetheless, our continental-scale simulations indicate that the assimilation of soil moisture can have a substantial impact on the estimated dynamics of evaporation in water-limited regimes. Progressing towards our goal of using satellite soil moisture to increase understanding of global land evaporation, future research will focus on the global application of this methodology and the consideration of multiple evaporation models.

Application of newtonian nudging data assimilation method in hydrodynamic model of flood flow in the lower Biebrza basin

Abstract The paper presents a 1D hydrodynamic flood flow model employing a data assimilation procedure based on Newtonian nudging. Data assimilation was used to determine correctly the upstream boundary condition defined as a discharge hydrograph. In the model developed, “nudging to individual observations” method was used. The data chosen for assimilation were water table levels recorded by a D-Diver automatic sensor installed in the river channel c. 1.5 km below a computational cross-section opening the analysed stretch of the river and the adjacent valley. This hydrological model of flood flow containing the data assimilation procedure is based on a one-dimensional Saint-Venant system of equations (dynamic wave model). The calculations were performed for the 2010 spring flood event at a 20-km stretch of the river and the floodplain in the upper part of the Lower Biebrza Basin. Modifying the boundary condition by using data assimilation has dramatically improved the accuracy of water table predictions during floods in the area of the Lower Biebrza Basin.

Sensitivity and out‐of‐sample error in continuous time data assimilation

AbstractData assimilation refers to the problem of finding trajectories of a prescribed dynamical model in such a way that the output of the model (usually some function of the model states) follows a given time series of observations. Typically though, these two requirements cannot both be met at the same time–tracking the observations is not possible without the trajectory deviating from the proposed model equations, while adherence to the model requires deviations from the observations. Thus, data assimilation faces a trade‐off. In this contribution, the sensitivity of the data assimilation with respect to perturbations in the observations is identified as the parameter which controls the trade‐off. A relation between the sensitivity and the out‐of‐sample error is established, which allows the latter to be calculated under operational conditions. A minimum out‐of‐sample error is proposed as a criterion to set an appropriate sensitivity and to settle the discussed trade‐off. Two approaches to data assimilation are considered, namely variational data assimilation and Newtonian nudging, also known as synchronization. Numerical examples demonstrate the feasibility of the approach. Copyright © 2011 Royal Meteorological Society

Real-time forecasting of water table depth and soil moisture profiles

We present a method for real-time forecasting of water table depth and soil moisture profiles. The method combines a simple form of data-assimilation with a moving window calibration of a deterministic model describing flow in the unsaturated zone and local as well as regional drainage. The local drainage level is calibrated on-line using a moving window calibration. Assigning more weight to the last available measurements then yields a form of model adaptation that is in between on-line calibration and data-assimilation (i.e. a simplified form of Newtonian nudging). Five-day hydrological forecasts are performed based on 5-day weather forecasts, while on-line observations of phreatic level and soil moisture content are assimilated on a daily basis. Advantages of the proposed method are that it improves the real-time forecasts compared to off-line calibration and ordinary moving window calibration and that it yields physically consistent soil moisture profiles.

Newtonian nudging for a Richards equation-based distributed hydrological model

The objective of data assimilation is to provide physically consistent estimates of spatially distributed environmental variables. In this study a relatively simple data assimilation method has been implemented in a relatively complex hydrological model. The data assimilation technique is Newtonian relaxation or nudging, in which model variables are driven towards observations by a forcing term added to the model equations. The forcing term is proportional to the difference between simulation and observation (relaxation component) and contains four-dimensional weighting functions that can incorporate prior knowledge about the spatial and temporal variability and characteristic scales of the state variable(s) being assimilated. The numerical model couples a three-dimensional finite element Richards equation solver for variably saturated porous media and a finite difference diffusion wave approximation based on digital elevation data for surface water dynamics. We describe the implementation of the data assimilation algorithm for the coupled model and report on the numerical and hydrological performance of the resulting assimilation scheme. Nudging is shown to be successful in improving the hydrological simulation results, and it introduces little computational cost, in terms of CPU and other numerical aspects of the model’s behavior, in some cases even improving numerical performance compared to model runs without nudging. We also examine the sensitivity of the model to nudging term parameters including the spatio-temporal influence coefficients in the weighting functions. Overall the nudging algorithm is quite flexible, for instance in dealing with concurrent observation datasets, gridded or scattered data, and different state variables, and the implementation presented here can be readily extended to any of these features not already incorporated. Moreover the nudging code and tests can serve as a basis for implementation of more sophisticated data assimilation techniques in a Richards equation-based hydrological model.

Analytical investigations of horizontal meso-scale momentum equations with Newtonian nudging

This study presents an analytical investigation of the local behaviour of the solution to a mesoscale model with Newtonian nudging when observed winds are time varying. The analysis examines each Fourier component of the time series of observed winds. Unlike the case with a constant observed wind, the nudged wind vector does not asymptotically approach the observed wind. In response to sinusoidal oscillation of the observed wind, the nudged wind vector is always on a half circle connecting the vector ends of the observed and un-nudged modelled winds. When nudging parameter α→0, the nudged wind vector approaches the un-nudged wind; when α→∞, the nudged wind vector approaches the observed wind. For commonly used values of nudging parameter α, the modelled wind field always carries errors.

Impact of a Newtonian Assimilation and Physical Initialization an the Initialization and Prediction by a Tropical Mesoscale Model

This study illustrates the capability of Newtonian nudging and physical initialization in improving the initialized state and forecasts in the Florida State University high-resolution regional tropical mesoscale model. In particular it is shown that this form of initialization leads to major improvement in the precipitation forecasts. The precipitation spinup, precipitation phase shift, and overactive convection over the tropical region that are inherent characteristics of prediction models initialized through analyses produced by an intermittent data assimilation scheme are minimized by the initialization. It is shown that the coupling of Newtonian nudging of the nondivergent component of the wind and surface pressure with physical initialization of cumulus parameterization, surface latent heat flux over the rain areas, and outgoing longwave radiation enables the model to build the required temperature, moisture, and velocity divergence distribution to produce the desired precipitation. Of these, the initialization of cumulus parameterization and surface latent heat flux show the most impact. The cumulus initialization as illustrated was done through a reanalysis of the humidity fields over the cloud depth via a “reverse Kuo” algorithm. The algorithm constrained the Kuo cumulus parameterization scheme of the model to precipitate according to prescribed rates. The initialization of surface latent heat flux was done through a reanalysis of the humidity field at the lowest level of the model through a “reverse similarity” algorithm that constrained the surface latent heat flux of the model to be similar to prescribed values.

Optimal determination of nudging coefficients using the adjoint equations

The adjoint equations of a numerical model can be used for model-parameter estimation. In this study, a general computational procedure is developed to determine the size and distribution of any internal model parameter. The procedure is then applied to a one-dimensional shallow-water model in the context of analysis-nudging four-dimensional data assimilation (FDDA): the weighting coefficients used by the Newtonian nudging technique are determined such that the model error during the assimilation period is optimally reduced subject to some constraints. The sensitivity of these nudging coefficients to the optimal objectives and constraints is investigated using this simple grid-point model in an Observing Systems Simulation Experiments (OSSE) mode. The results show that in principle, it is feasible to determine a set of nudging weights which minimize the model error over the period covered by the observations. It is demonstrated, however, that the magnitude and distribution of these “optimal” nudging weights are sensitive to the prescribed estimate of the nudging weights and the corresponding coefficient matrix which define a penalty term in the cost function. The penalty term is a weak constraint on the size and distribution of the optimal nudging weights while the model is the strong constraint. The fit of the model to the data is greater when this constraint on the nudging weights is weaker, but then the nudging weights may be too large or even negative. Thus the “optimal” solution for this model parameter is not unique because specification of this penalty term in the cost function introduces a new uncertainty into the nudging FDDA framework. Nevertheless, this optimal-nudging approach does show promise, but the sensitivity of the technique to the penalty term requires further investigation under more realistic conditions. DOI: 10.1034/j.1600-0870.1993.t01-4-00003.x

An Examination of Four-Dimensional Data-Assimilation Techniques for Numerical Weather Prediction

Abstract Four-dimensional data-assimilation methods, along with the most commonly used objective analysis and initialization techniques, are examined from a historical perspective. Operational techniques, including intermittent data assimilation and Newtonian nudging, and next-generation methods (Kalman–Bucy filtering and the adjoint method) are briefly described. Several methods are compared, with primary emphasis being placed on recent papers dealing with the operational assimilation techniques. Ongoing and future research is outlined, and some important implications of this research are discussed.

Dynamic Initialization Using Observations from a Hypothetical Network of Profilers

This paper presents results from a series of observing system simulation experiments (OSSEs), designed to test a dynamic initialization procedure for continuous assimilation of observations from a hypothetical network of profilers. A 40-km adiabatic mesoscale model was used to generate a set of simulated observations from a network of 77 wind profilers over the continental United States with a station separation of 360 km. These observations were then assimilated into an 80-km model during a 12-h preforecast integration through a Newtonian nudging technique. We found that dynamic initialization by nudging can successfully assimilate the time-continuous wind profiler observations into the model. The profiler data assimilation is effective in recovering mesoscale circulations which are not properly resolved by the observing network (due to inadequate horizontal resolution), while at the same time controlling the error growth for large-scale circulations. The impact is particularly significant in the divergence field, which is crucial for an accurate precipitation forecast. The improved initial state leads to further improvement in the subsequent forecast, demonstrating the value of the time-continuous profiler observations on short-range numerical weather prediction. Assimilation of the wind field is found to be considerably more effective than assimilation of the temperature field. Specifically, wind assimilation leads to improvement in both the temperature and the wind fields, while temperature assimilation produces little improvement in the wind field. The best results are obtained when both temperature and wind fields are assimilated. The proposed demonstration network of 31 profiles is likely to have a positive impact on short-range numerical weather prediction, though mainly confined over the localized region covered by the profiler network. Further expansion of the profiler network to increase its spatial resolution and its areal coverage is needed if improved prediction is expected over a larger area.

Four-Dimensional Data Assimilation in the Monsoon Region. Part II: Role of Temperature and Moisture Data

Abstract The problem of four-dimensional data assimilation in the tropics has been studied using a limited-area primitive equation model. Of prime concern is the relative importance of different update variables and their impact on data assimilation. Five new experiments complement a set of ten previously reported experiments that investigate the feasibility of four-dimensional data assimilation in the monsoon region using only the wind observations. In addition to assessing the relative importance of update variables, the present study investigates the role of model physics in data assimilation. The assimilation experiments are carried out for the onset vortex case of the 1979 Indian summer monsoon for which many special FOGE/MONEX datasets are available. The assimilation-forecast system for all of the experiments comprises a 12-h assimilation phase followed by a 24-h forecast period. In all experiments, updating is done via the Newtonian nudging approach which, in our previous study, was found to be mor...

Four-Dimensional Data Assimilation in the Monsoon Region. Part I: Experiments with Wind Data

A limited primitive equation model has been used to study the feasibility of four-dimensional data assimilation in the monsoon region and, further, to study the applicability of several assimilation techniques currently being employed in global models. The two fundamental objectives of this research are (i) to understand how the model atmosphere responds to the insertion of asynchronous data and its impact on the assimilation-prediction cycle, and (ii) to determine what assimilation strategies work best for limited-area models in the tropics. A sequence of ten assimilation experiments are performed using different update procedures; all insertions are carried out with only the wind observations. The model is initialized with the ECMWF FGGE level III-b data for the onset vortex case of 17 June 1979, and assimilations are carried out using the summer MONEX level II-b data during the first 12 hours. From these assimilated states, 24-h forecasts are then made. The results lend support to the premise that the required initial conditions can be obtained by the process of four-dimensional updating of the prognostic variables. The results also clearly demonstrate the superiority of the continuous assimilation approach via Newtonian nudging over that by indirect insertion. Furthermore, the insertion shocks are significantly minimized by assimilating only the rotational component of the winds. On the other hand, the application of noise control measures only marginally alleviate the insertion shocks accompanying continuous indirect insertion.