Nudging Algorithm Research Articles

Back-and-forth nudging moving horizon estimation for discrete-time linear systems

In this paper, we propose a novel moving horizon estimation algorithm for discrete-time linear systems with a limited number of measurements. Motivated by the idea of the back-and-forth nudging algorithm, we design the back-and-forth nudging in time moving-horizon estimation method by deploying two moving horizon estimators that move backward and forward iteratively. Based on a finite number of measurements, the proposed algorithm can be used for moving-horizon state estimation with guaranteed convergence. By using the proposed method, we show that the norm of the state estimation error is upper bounded by a sequence that converges to its steady-state in finite-time (i.e., using a finite number of measurements), provided that suitable parameters are selected. The unbiasedness properties and comparison results with the conventional (forward-in-time) moving horizon estimation are discussed. The effectiveness of the proposed approach is validated through a numerical example.

Determining the viscosity of the Navier–Stokes equations from observations of finitely many modes

In this work, we ask and answer the question: when is the viscosity of a fluid uniquely determined from spatially sparse measurements of its velocity field? We pose the question mathematically as an optimization problem using the determining map (the mapping of data to an approximation made via a nudging algorithm) to define a loss functional, the minimization of which solves the inverse problem of identifying the true viscosity given the measurement data. We give explicit a priori conditions for the well-posedness of this inverse problem. In addition, we show that smallness of the loss functional implies proximity to the true viscosity. We then present an algorithm for solving the inverse problem and provide a priori verifiable conditions that ensure its convergence.

Regional mapping of energetic short mesoscale ocean dynamics from altimetry: performances from real observations

Abstract. For over 25 years, satellite altimetry has provided invaluable information about the ocean dynamics at many scales. In particular, gridded sea surface height (SSH) maps allow us to estimate the mesoscale geostrophic circulation in the ocean. However, conventional interpolation techniques rely on static optimal interpolation schemes, hence limiting the estimation of non-linear dynamics at scales not well sampled by altimetry (i.e., below 150–200 km at mid-latitudes). To overcome this limitation in the resolution of small-scale SSH structures (and thus small-scale geostrophic currents), a back-and-forth nudging algorithm combined with a quasi-geostrophic model, a technique called BFN-QG, has been successfully applied on simulated SSH data in observing system simulation experiments (OSSEs). The result is a significant reduction in interpolation error and an improvement in the space–time resolutions of the experimental gridded product compared to those of operational products. In this study, we propose that the BFN-QG be applied to real altimetric SSH data in a highly turbulent region spanning a part of the Agulhas Current. The performances are evaluated within observing system experiments (OSEs) that use independent data (such as independent SSH, sea surface temperature and drifter data) as ground truth. By comparing the mapping performances to the ones obtained with operational products, we show that the BFN-QG improves the mapping of short, energetic mesoscale structures and associated geostrophic currents both in space and time. In particular, the BFN-QG improves (i) the spatial effective resolution of the SSH maps by a factor of 20 %, (ii) the zonal and (especially) the meridional geostrophic currents, and (iii) the prediction of Lagrangian transport for lead times up to 10 d. Unlike the results obtained in the OSSEs, the OSEs reveal more contrasting performances in low-variability regions, which are discussed in the paper.

Luenberger Observers for Infinite-Dimensional Systems, Back and Forth Nudging, and Application to a Crystallization Process

This paper deals with the observer design problem for time-varying linear infinite-dimensional systems. We address both the problem of online estimation of the state of the system from the output via an asymptotic observer, and the problem of offline estimation of the initial state via a back and forth nudging algorithm. In both contexts, we show under a weak detectability assumption that a Luenberger-like observer may reconstruct the so-called observable subspace of the system. However, since no exact observability hypothesis is required, only a weak convergence of the observer holds in general. Additional conditions on the system are required to show the strong convergence. We provide an application of our results to a batch crystallization process modeled by a one-dimensional transport equation with periodic boundary conditions, in which we try to estimate the crystal size distribution from the chord length distribution.

Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations

We study the computational efficiency of several nudging data assimilation algorithms for the 2D magnetohydrodynamic equations, using varying amounts and types of data. We find that the algorithms work with much less resolution in the data than required by the rigorous estimates in [7]. We also test other abridged nudging algorithms to which the analytic techniques in [7] do not seem to apply. These latter tests indicate, in particular, that velocity data alone is sufficient for synchronization with a chaotic reference solution, while magnetic data alone is not. We demonstrate that a new nonlinear nudging algorithm, which is adaptive in both time and space, synchronizes at a super exponential rate.

Data assimilation at local scale to improve CFD simulations of atmospheric dispersion: application to 1D shallow-water equations and method comparisons

Atmospheric dispersion modelling requires meteorological inputs over local domains with possibly complex topographies. These local wind fields may be difficult to simulate with CFD models, in particular because of their sensitivity to geometrical features and to model inputs, especially the boundary conditions which are generally provided by larger-scale models or measurements. Using data assimilation, a few measurements inside the domain could add information to the imprecise boundary conditions and thus greatly enhance the precision of the dispersion simulations. Three data assimilation techniques (3DVar, the back and forth nudging algorithm, and the iterative ensemble Kalman smoother) have been adapted to local scale simulations by taking boundary conditions into account instead of initial conditions for which they are usually applied. Their performances have been evaluated at small scales, with a simple representation of the atmosphere into two layers, using 1D solution of the shallow-water equations.

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.

On the potential of data assimilation to generate SMOS-Level 4 maps of sea surface salinity

The Soil Moisture/Ocean Salinity (SMOS) satellite, launched in November 2009, measures visibilities at L-band, from which brightness temperatures are computed. This information is used to retrieve values of the sea surface salinity (SSS) and soil moisture; two variables whose observation is a key to better understand the oceanic component of the water cycle. A hierarchy of SSS products has been defined in the SMOS data processing chain. This work focuses on the so-called Level 3 (binned maps of SSS) and Level 4 (products combining SMOS data with any other source of information). The objective is to illustrate the feasibility of using data assimilation to produce Level 4 maps of sea surface salinity. The numerical model will increase the geophysical coherence of SMOS data as a dynamical interpolator. Here, the employment of data assimilation differs from its usual applications (improving model outputs for example). Indeed, the numerical model will interpolate the observations according to the general laws of fluid mechanics and, if possible, reduce the error contained in the original observations. The data assimilation method analyzed is a nudging algorithm. The domain of application for this feasibility study is the Northeast subtropical Atlantic gyre, a challenging region due to the presence of a large amount of noise that deteriorates the SMOS data. The main sources of errors are the vicinity of large landmasses that introduce a spurious bias, and the presence of a significant amount of artificial radio frequency interferences (RFI). While the Quality Controls already set up in the SMOS processing chain do filter the retrievals containing too large errors, wrong data are still present in Level 3 maps. Despite this difficulty, the results provide meaningful SMOS SSS Level 4 products in terms of their geophysical coherence (estimated using singularity analysis) and better agreement with in-situ data than Level 3 product.

Assimilation of low-level wind in a high-resolution mesoscale model using the back and forth nudging algorithm

ABSTRACTThe performance of a new data assimilation algorithm called back and forth nudging (BFN) is evaluated using a high-resolution numerical mesoscale model and simulated wind observations in the boundary layer. This new algorithm, of interest for the assimilation of high-frequency observations provided by ground-based active remote-sensing instruments, is straightforward to implement in a realistic atmospheric model. The convergence towards a steady-state profile can be achieved after five iterations of the BFN algorithm, and the algorithm provides an improved solution with respect to direct nudging. It is shown that the contribution of the nudging term does not dominate over other model physical and dynamical tendencies. Moreover, by running backward integrations with an adiabatic version of the model, the nudging coefficients do not need to be increased in order to stabilise the numerical equations. The ability of BFN to produce model changes upstream from the observations, in a similar way to 4-D-Var assimilation systems, is demonstrated. The capacity of the model to adjust to rapid changes in wind direction with the BFN is a first encouraging step, for example, to improve the detection and prediction of low-level wind shear phenomena through high-resolution mesoscale modelling over airports.

Diffusive Back and Forth Nudging algorithm for data assimilation

In this Note, we propose an improvement to the Back and Forth Nudging algorithm for handling diffusion in the context of geophysical data assimilation. We introduce the Diffusive Back and Forth Nudging, in which the sign of the diffusion term is changed in the backward integrations. We study the convergence of this algorithm, in particular for linear transport equations.

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers' equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.

Reconstruction and analysis of the Chukchi Sea circulation in 1990–1991

The Chukchi Sea (CS) circulation reconstructed for September 1990 to October 1991 from sea ice and ocean data is presented and analyzed. The core of the observational data used in this study comprises the records from 12 moorings deployed in 1990 and 1991 in U.S. and Russian waters and two hydrographic surveys conducted in the region in the fall of 1990 and 1991. The observations are processed by a two‐step data assimilation procedure involving the Pan‐Arctic Ice‐Ocean Modeling and Assimilation System (employing a nudging algorithm for sea ice data assimilation) and the Semi‐implicit Ocean Model [utilizing a conventional four‐dimensional variational (4D‐var) assimilation technique]. The reconstructed CS circulation is studied to identify pathways and assess residence times of Pacific water in the region; quantify the balances of volume, freshwater, and heat content; and determine the leading dynamical factors configuring the CS circulation. It is found that in 1990–1991 (high AO index and a cyclonic circulation regime) Pacific water transiting the CS toward the Canada basin followed two major pathways, namely via Herald Canyon (Herald branch of circulation, 0.23 Sv) and between Herald Shoal and Cape Lisburne (central branch of circulation and Alaskan Coastal Current, 0.32 Sv). The annual mean flow through Long Strait was negligible (0.01 Sv). Typical residence time of Pacific water in the region varied between 150 days for waters entering the CS in September and 270 days for waters entering in February/March. Momentum balance analysis reveals that geostrophic balance between barotropic pressure gradient and Coriolis force dominated for most of the year. Baroclinic effects were important for circulation only in the regions with large horizontal salinity gradients associated with the fresh Alaskan and Siberian coastal currents and the Cape Lisburne and Great Siberian polynyas. In the polynyas, the baroclinic effects were due to strong salinification and convection processes associated with sea ice formation.

The back and forth nudging algorithm applied to a shallow water model, comparison and hybridization with the 4D‐VAR

AbstractWe study in this paper a new data assimilation algorithm, called the back and forth nudging (BFN). This scheme has been very recently introduced for simplicity reasons, as it does not require any linearization, or adjoint equation, or minimization process in comparison with variational schemes, but nevertheless it provides a new estimation of the initial condition at each iteration. We study its convergence properties as well as efficiency on a 2D shallow water model. All along the numerical experiments, comparisons with the standard variational algorithm (called 4D‐VAR) are performed. Finally, a hybrid method is introduced, by considering a few iterations of the BFN algorithm as a preprocessing tool for the 4D‐VAR algorithm. We show that the BFN algorithm is extremely powerful in the very first iterations and also that the hybrid method can both improve notably the quality of the identified initial condition by the 4D‐VAR scheme and reduce the number of iterations needed to achieve convergence. Copyright © 2008 John Wiley & Sons, Ltd.

An easy-to-implement and efficient data assimilation method for the identification of the initial condition: the Back and Forth Nudging (BFN) algorithm

This paper deals with a new data assimilation algorithm called the Back and Forth Nudging. The standard nudging technique consists in adding to the model equations a relaxation term, which is supposed to force the model to the observations. The BFN algorithm consists of repeating forward and backward resolutions of the model with relaxation (or nudging) terms, that have opposite signs in the direct and inverse resolutions, so as to make the backward evolution numerically stable. We then applied the Back and Forth Nudging algorithm to a simple non-linear model: the ID viscous Burgers' equations. The tests were carried out through several cases relative to the precision and density of the observations. These simulations were then compared with both the variational assimilation (VAR) and quasi-inverse (QIL) algorithms. The comparisons deal with the programming, the convergence, and time computing for each of these three algorithms.

Back and forth nudging algorithm for data assimilation problems

In this Note, we introduce a new algorithm for data assimilation problems, called the back and forth nudging (BFN) algorithm. The standard forward nudging algorithm is first studied for a linear ODE model. The backward nudging algorithm is then introduced in order to reconstruct the initial state of the system. These two algorithms are combined in the new BFN algorithm. The mathematical proof of its convergence is given for a linear ODE system. To cite this article: D. Auroux, J. Blum, C. R. Acad. Sci. Paris, Ser. I 340 (2005).