Continuous Data Assimilation Research Articles

Continuous data assimilation for global numerical weather prediction

AbstractA new configuration of the European Centre for Medium‐Range Weather Forecasts (ECMWF) incremental 4D‐Var data assimilation (DA) system is introduced which builds upon the quasi‐continuous DA concept proposed in the mid‐1990s. Rather than working with a fixed set of observations, the new 4D‐Var configuration exploits the near‐continuous stream of incoming observations by introducing recently arrived observations at each outer loop iteration of the assimilation. This allows the analysis to benefit from more recent observations. Additionally, by decoupling the start time of the DA calculations from the observational data cut‐off time, real‐time forecasting applications can benefit from more expensive analysis configurations that previously could not have been considered. In this work we present results of a systematic comparison of the performance of a Continuous DA system against that of two more traditional baseline 4D‐Var configurations. We show that the quality of the analysis produced by the new, more continuous configuration is comparable to that of a conventional baseline that has access to all of the observations in each of the outer loops, which is a configuration not feasible in real‐time operational numerical weather prediction. For real‐time forecasting applications, the Continuous DA framework allows configurations which clearly outperform the best available affordable non‐continuous configuration. Continuous DA became operational at ECMWF in June 2019 and led to significant 2 to 3% reductions in medium‐range forecast root mean square errors, which is roughly equivalent to 2–3 hr of additional predictive skill.

Error analysis of fully discrete mixed finite element data assimilation schemes for the Navier-Stokes equations

In this paper we consider fully discrete approximations with inf-sup stable mixed finite element methods in space to approximate the Navier-Stokes equations. A continuous downscaling data assimilation algorithm is analyzed in which measurements on a coarse scale are given represented by different types of interpolation operators. For the time discretization an implicit Euler scheme, an implicit and a semi-implicit second-order backward differentiation formula are considered. Uniform-in-time error estimates are obtained for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements. For the spatial discretization we consider both the Galerkin method and the Galerkin method with grad-div stabilization. For the last scheme error bounds in which the constants do not depend on inverse powers of the viscosity are obtained.

Case Study of a Retrieval Method of 3D Proxy Reflectivity from FY-4A Lightning Data and Its Impact on the Assimilation and Forecasting for Severe Rainfall Storms

As the first Geostationary Satellite with the LMI (Lightning Mapping Imager) instrument aboard running over the eastern hemisphere, FY-4A (Feng-Yun-4A) can better indicate severe convection and compensate for the limitations of radar observation in temporal and spatial resolution. In order to realize the application of FY-4A lightning data in numerical weather prediction (NWP) models, a logarithmic relationship between FY-4A lightning density and maximum radar reflectivity is presented to convert FY-4A lightning data into maximum FY-4A proxy reflectivity. Then, according to the profiles of radar reflectivity, the maximum FY-4A proxy reflectivity is extended to 3D FY-4A proxy reflectivity. Finally, the 3D FY-4A proxy reflectivity is assimilated in RMAPS-ST (Rapid-refresh Multi-scale Analysis and Prediction System—Short Term) to compare with radar assimilation. Four groups of continuous cycling data assimilation and forecasting experiments are carried out for a severe rainfall case. The results demonstrate that cycling assimilation of 3D FY-4A proxy reflectivity can adjust the moisture condition effectively, and indirectly affects the temperature and wind fields, then makes the thermal and dynamic analysis more reasonable. The Fractions Skill Scores (FSSs) show that the rainfall forecasts are improved significantly within 6 h by assimilating 3D FY-4A proxy reflectivity, which is similar to the parallel experiments in assimilating radar reflectivity. In addition, other cycling data assimilation experiments are carried out in mountainous areas without radar data. The improvement of precipitation forecasts in mountainous areas further proves that the application of assimilating 3D FY-4A proxy reflectivity can be considered a useful substitute where observed radar data are missing. Through the two severe rainfall cases, this method could be framed as an example of how to use lightning for data assimilation.

Uniform in Time Error Estimates for a Finite Element Method Applied to a Downscaling Data Assimilation Algorithm for the Navier--Stokes Equations

In this paper we analyze a finite element method applied to a continuous downscaling data assimilation algorithm for the numerical approximation of the two- and three-dimensional Navier--Stokes equ...

Parameter Recovery for the 2 Dimensional Navier--Stokes Equations via Continuous Data Assimilation

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown viscosity. We determine the large-time error between the true solution of t...

Data Assimilation in Large Prandtl Rayleigh--Bénard Convection from Thermal Measurements

This work applies a continuous data assimilation scheme---a framework for reconciling sparse and potentially noisy observations to a mathematical model---to Rayleigh--Bénard convection at infinite or large Prandtl numbers using only the temperature field as observables. These Prandtl numbers are applicable to the earth's mantle and to gases under high pressure. We rigorously identify conditions that guarantee synchronization between the observed system and the model, then confirm the applicability of these results via numerical simulations. Our numerical experiments show that the analytically derived conditions for synchronization are far from sharp; that is, synchronization often occurs even when sufficient conditions of our theorems are not met. We also develop estimates on the convergence of an infinite Prandtl model to a large (but finite) Prandtl number generated set of observations. Numerical simulations in this hybrid setting indicate that the mathematically rigorous results are accurate, but of practical interest only for extremely large Prandtl numbers.

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the \begin{document}$ L^2 $\end{document} and \begin{document}$ H^1 $\end{document} norms of error are bounded by a constant times a power of the Voigt-regularization parameter \begin{document}$ \alpha>0 $\end{document} , plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as \begin{document}$ \alpha $\end{document} goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the \begin{document}$ H^2 $\end{document} norm.

Continuous Data Assimilation with Blurred-in-Time Measurements of the Surface Quasi-Geostrophic Equation

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

Continuous data assimilation reduced order models of fluid flow

We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement data into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs.

Efficient dynamical downscaling of general circulation models using continuous data assimilation

AbstractContinuous data assimilation (CDA) is successfully implemented for the first time for efficient dynamical downscaling of a global atmospheric reanalysis. A comparison of the performance of CDA with the standard grid and spectral nudging techniques for representing long‐ and short‐scale features in the downscaled fields using the Weather Research and Forecast (WRF) model is further presented and analysed. The WRF model is configured at 0.25° × 0.25° horizontal resolution and is driven by 2.5° × 2.5° initial and boundary conditions from NCEP/NCAR reanalysis fields. Downscaling experiments are performed over a one‐month period in January 2016. The similarity metric is used to evaluate the performance of the downscaling methods for large (2,000 km) and small (300 km) scales. Similarity results are compared for the outputs of the WRF model with different downscaling techniques, NCEP/NCAR reanalysis, and NCEP Final Analysis (FNL, available at 0.25° × 0.25° horizontal resolution). Both spectral nudging and CDA describe better the small‐scale features compared to grid nudging. The choice of the wave number is critical in spectral nudging; increasing the number of retained frequencies generally produced better small‐scale features, but only up to a certain threshold after which its solution gradually became closer to grid nudging. CDA maintains the balance of the large‐ and small‐scale features similar to that of the best simulation achieved by the best spectral nudging configuration, without the need of a spectral decomposition. The different downscaled atmospheric variables, including rainfall distribution, with CDA is most consistent with the observations. The Brier skill score values further indicate that the added value of CDA is distributed over the entire model domain. The overall results clearly suggest that CDA provides an efficient new approach for dynamical downscaling by maintaining better balance between the global model and the downscaled fields.

Continuous data assimilation for the 3D primitive equations of the ocean

In this article, we show that the continuous data assimilation algorithm is valid for the 3D primitive equations of the ocean. Namely, the assimilated solution converges to the reference solution in $L^2$ norm at an exponential rate in time. We also prove the global existence of strong solution to the assimilated system.

Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier–Stokes equations

We study numerical schemes for incompressible Navier–Stokes equations using IMEX temporal discretizations, finite element spatial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani et al. (2014)). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical.

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

We introduce a continuous (downscaling) data assimilation algorithm for the 2D Benard convection problem using vorticity or local circulation measurements only. In this algorithm, a nudging term is added to the vorticity equation to constrain the model. Our numerical results indicate that the approximate solution of the algorithm is converging to the unknown reference solution (vorticity and temperature) corresponding to the measurements of the 2D Benard convection problem when only spatial coarse-grain measurements of vorticity are assimilated. Moreover, this convergence is realized using data which is much more coarse than the resolution needed to satisfy rigorous analytical estimates.

Continuous data assimilation algorithm for simplified Bardina model

We present a continuous data assimilation algorithm for three-dimensional viscous simplified Bardina turbulence model, based on the fact that dissipative dynamical systems possess finite degrees of freedom. We construct an approximating solution of simplified Barbina model through an interpolant operator which is obtained using observational data of the system. This interpolant is inserted to theoric model coupled to a relaxation parameter, and our main result provides conditions on the finite-dimensional spatial resolution of collected measurements sufficient to ensure that the approximating solution converges to the theoric solution of the model. Global well-posedness of approximating solutions and related results with degrees of freedom are also presented.

Uniform-in-Time Error Estimates for the Postprocessing Galerkin Method Applied to a Data Assimilation Algorithm

We apply the postprocessing Galerkin method to a recently introduced continuous data assimilation (downscaling) algorithm for obtaining a numerical approximation of the solution of the two-dimensional Navier--Stokes equations corresponding to given measurements from a coarse spatial mesh. Under suitable conditions on the relaxation (nudging) parameter, the resolution of the coarse spatial mesh, and the resolution of the numerical scheme, we obtain uniform-in-time estimates for the error between the numerical approximation given by the postprocessing Galerkin method and the reference solution corresponding to the measurements. Our results are valid for a large class of interpolant operators, including low Fourier modes and local averages over finite volume elements. Notably, we use here the two-dimensional Navier--Stokes equations as a paradigm, but our results apply equally to other evolution equations, such as the Boussinesq system of Benard convection and other oceanic and atmospheric circulation models.

Downscaling the 2D Bénard convection equations using continuous data assimilation

We consider a recently introduced continuous data assimilation (CDA) approach for downscaling a coarse resolution configuration of the 2D Benard convection equations into a finer grid. In this CDA, a nudging term, estimated as the misfit between some interpolants of the assimilated coarse grid measurements and the fine grid model solution, is added to the model equations to constrain the model. The main contribution of this study is a performance analysis of CDA for downscaling measurements of temperature and velocity. These measurements are assimilated either separately or simultaneously and the results are compared against those resulting from the standard point-to-point nudging approach (NA). Our numerical results suggest that the CDA solution outperforms that of NA, always converging to the true solution when the velocity is assimilated as has been theoretically proven. Assimilation of temperature measurements only may not always recover the true state as demonstrated in the case study. Various runs are conducted to evaluate the sensitivity of CDA to noise in the measurements, the size and the time frequency of the measured grid, suggesting a more robust behaviour of CDA compared to NA.

Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional B\'enard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress free boundary condition on the walls, and fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

The challenge of integrating offshore wind power in the U.S. electric grid. Part I: Wind forecast error

The purpose of this two-part study is to model the effects of large penetrations of offshore wind power into a large electric system using realistic wind power forecast errors and a complete model of unit commitment, economic dispatch, and power flow. The chosen electric system is PJM Interconnection, one of the largest independent system operators in the U.S. with a generation capacity of 186 Gigawatts (GW). The offshore wind resource along the U.S. East Coast is modeled at five build-out levels, varying between 7 and 70 GW of installed capacity, considering exclusion zones and conflicting water uses.This paper, Part I of the study, describes in detail the wind forecast error model; the accompanying Part II describes the modeling of PJM's sequencing of decisions and information, inclusive of day-ahead, hour-ahead, and real-time commitments to energy generators with the Smart-ISO simulator and discusses the results.Wind forecasts are generated with the Weather Research and Forecasting (WRF) model, initialized every day at local noon and run for 48 h to provide midnight-to-midnight forecasts for one month per season. Due to the lack of offshore wind speed observations at hub height along the East Coast, a stochastic forecast error model for the offshore winds is constructed based on forecast errors at 23 existing PJM onshore wind farms. PJM uses an advanced, WRF-based forecast system with continuous wind farm data assimilation. The implicit (and conservative) assumption here is that the future forecast system for offshore winds will have the same performance as the current PJM's forecast system for onshore winds, thus no advances in weather forecasting techniques are assumed.Using an auto-regressive moving-average (ARMA) model, 21 equally-plausible sample paths of wind power forecast errors are generated and calibrated for each season at a control onshore wind farm, chosen because of its horizontally uniform landscape and large size. The spatial correlation between pairs of onshore wind farms is estimated with an exponential function and the matrix of error covariance is obtained. Validation at the control farm and at all other onshore farms is satisfactory. The ARMA model for the wind power forecast error is then applied to the offshore wind farms at the various build-out levels and combined with the matrix of error covariance to generate multiple samples of forecast errors at the offshore farms. The samples of forecast errors are lastly added to the WRF forecasts to generate multiple samples of synthetic, onshore-based, actual offshore wind power for use in Part II.

Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations

We consider the two-dimensional (2D) Navier–Stokes equations (NSE) with space periodic boundary conditions and an algorithm for continuous data assimilation developed by Azouani et al. (2014). The algorithm is based on the observation that existence of finite determining parameters for nonlinear dissipative systems can be exploited as a feedback control mechanism for a companion system into which observables, e.g, modes, nodes, or volume elements, are input directly for the purpose of assimilation. It has been shown that in the case of the 2D NSE, the approximating solution induced by the algorithm synchronizes with the exact solution of the 2D NSE in the topology of the Sobolev space, H1, provided that the number of observed modes, nodes, or volume elements is sufficiently large in terms of the Grashof number. In this article, we adapt a technique, introduced by Grujić and Kukavica (1998) and Kukavica (1999) to obtain good estimates of the analyticity radius for the 2D NSE, and show that one can in fact obtain synchronization in the analytic Gevrey class in the case of modal observables, given sufficiently many, but fixed number of such observables. For these types of observables, we additionally show that synchronization in the uniform norm, L∞, can be achieved by assuming the same number of modal observables (in terms of the order of the Grashof number) as is required for the H1 synchronization.

Improvement of the forecast of convective activity from the AROME‐France system

AROME‐France is a convective‐scale numerical weather prediction system which has been running operationally at Météo‐France since the end of 2008. In order to determine its initial conditions, it uses a 3D‐Var assimilation scheme at the same resolution as the model in a continuous data assimilation cycle. In addition to conventional and satellite observations used in global data assimilation systems, dedicated observations for the mesoscale such as surface observations and radar measurements (radial winds and reflectivities) are assimilated. A major update of this system occurred in April 2015 with, among several improvements, (i) an increase of both horizontal and vertical resolutions (1.3 km and 90 vertical levels versus 2.5 km and 60 levels), and (ii) the reduction of the period of the data assimilation cycle from 3 to 1 h (as a result of the model spin‐up reduction and the tuning of the background‐error covariances). This study presents the preparatory work to these modifications and explores the main impact expected on convective activity forecasts. (i) appears to result in more realistic convective cells and better rainfall and wind gust scores and (ii) allows assimilation of more observations with information at the mesoscale which provides more accurate initial conditions and hence better subsequent rainfall forecasts. The benefits of using both (i) and (ii) in a pre‐operational configuration are shown using objective precipitation scores and illustrated by a case‐study.