Heterogeneity of the Attractor of the Lorenz '96 Model: Lyapunov Analysis, Unstable Periodic Orbits, and Shadowing Properties

The predictability of weather and climate is strongly state-dependent: special and extremely relevant atmospheric states like blockings are associated with anomalous instability. Indeed, typically, the instability of a chaotic dynamical system can vary considerably across its attractor. Such an attractor is in general densely populated by unstable periodic orbits that can be used to approximate any forward trajectory through the so-called shadowing. Dynamical heterogeneity can lead to the presence of unstable periodic orbits with different number of unstable dimensions. This phenomenon - unstable dimensions variability - implies a serious breakdown of hyperbolicity and has considerable implications in terms of the structural stability of the system and of the possibility to describe accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, we provide here an extensive numerical study of the dynamical heterogeneity of the Lorenz '96 model in a parametric configuration leading to chaotic dynamics. We show that the detected variability in the number of unstable dimensions is associated with the presence of many finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered. The transition between regions of the attractor with different degrees of instability comes with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing unstable periodic orbits, we can characterize the slow fluctuations of the system between regions featuring, on the average, anomalously high and anomalously low instability. In turn, such regions are associated, respectively, with states of anomalously high and low energy, thus providing a clear link between the microscopic and thermodynamical properties of the system.Comment: 28 pages, 11 figures, final accepted versio

On Variational Data Assimilation in Continuous Time

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalisation of what is known as weakly constrained four dimensional variational assimilation (WC--4DVAR) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two point boundary value problem. For imperfect models, the trade off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. For (nearly) perfect models, this trade off turns out to be (nearly) trivial in some sense, yet allowing for some dynamical error is shown to have positive effects even in this situation. The presented formalism is dynamical in character; no assumptions need to be made about the presence (or absence) of dynamical or observational noise, let alone about their statistics.Comment: 28 Pages, 12 Figure

Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically

Shadowing-based data assimilation method for partially observed models

In this article we develop further an algorithm for data assimilation based upon a shadowing refinement technique [de Leeuw et al., SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2446-2477] to take partial observations into account. Our method is based on a regularized Gauss-Newton method. We prove local convergence to the solution manifold and provide a lower bound on the algorithmic time step. We use numerical experiments with the Lorenz 63 and Lorenz 96 models to illustrate convergence of the algorithm and show that the results compare favorably with a variational technique --- weak-constraint four-dimensional variational method --- and a shadowing technique-pseudo-orbit data assimilation. Numerical experiments show that a preconditioner chosen based on a cost function allows the algorithm to find an orbit of the dynamical system in the vicinity of the true solution

On shadowing methods for data assimilation

Combining orbits from a model of a (chaotic) dynamical system with measured data to arrive at an improved estimate of the state of a physical system is known as data assimilation. This thesis deals with various algorithms for data assimilation. These algorithms are based on shadowing. Shadowing is a concept from the theory of dynamical systems. When a dynamical system has the property that an exact orbit of the dynamical system is located in a neighborhood of each pseudo-orbit, then this exact orbit shadows the pseudo-orbit. Shadowing can be used to show that a numerical solution of a dynamical system is located in a neighborhood of an exact solution. Shadowing refinement is a numerical technique in which an improved approximation to an exact solution is found from a pseudo-orbit. It is possible to use a shadowing refinement technique for data assimilation. Starting from observations, Newton's method is applied to approximate a zero of a cost operator, where the cost operator assigns costs to deviations from model solutions. The algorithms of Chapter 2 are based on a numerical time-dependent split between stable and unstable directions. The algorithm uses time-dependent projections onto the unstable subspace determined by using Lyapunov exponents and Lyapunov vectors. A shadowing algorithm is used in the unstable subspace, while synchronization is used in the stable subspace. The method is further extended to include parameter estimation and to some cases where only partial observations are available. Chapter 3 discusses data assimilation for imperfect models. Through regularization according to the Levenberg-Marquardt method, imperfections in the model are considered. It also describes how the shadowing method compares, both analytically and numerically, with the weak constraint 4DVar method and shows that the shadowing method is consistent with the measurement error distribution, which is not the case for the weak constraint 4DVar method. This effect is particularly evident when there are fewer observations. Moreover, when there are few observations, they have a smaller impact on unobserved variables in the shadowing method than in the weak constraint 4DVar method. Chapter 4 extends the method of Chapter 2 to other cases of partial observations, in a similar way to Chapter 3. Local convergence to a solution manifold is proved and a lower bound on an algorithmic time step is provided. Numerical experiments with the Lorenz-'63 and Lorenz-'96 models show convergence of the algorithm and further show that the method compares favorably with the weak constraint 4DVar method and another shadowing method called pseudo-orbit data assimilation. Chapter 5 further develops the method of the previous chapters. The algorithm is extended to an ensemble of states for estimating uncertainties of the algorithm, based on the concept of indistinguishable states. The chapter also includes some proofs on uniqueness, accuracy and consistency of the algorithm. The algorithm is applied to an imperfect model to show how the unmodeled components of the model can be estimated using the data assimilation algorithm

Tropospheric phase delay in interferometric synthetic aperture radar estimated from meteorological model and multispectral imagery

ENVISAT Medium Resolution Imaging Spectrometer Instrument (MERIS) multispectral data and the mesoscale meteorological model MM5 are used to estimate the tropospheric phase delay in synthetic aperture radar (SAR) interferograms. MERIS images acquired simultaneously with ENVISAT Advanced Synthetic Aperture Radar data provide an estimate of the total water vapor content W limited to cloud-free areas based on spectral bands ratio (accuracy 0.17 g cm^(−2) and ground resolution 300 m). Maps of atmospheric delay, 2 km in ground resolution, are simulated from MM5. A priori pertinent cumulus parameterization and planetary boundary layer options of MM5 yield near-equal phase correction efficiency. Atmospheric delay derived from MM5 is merged with available MERIS W product. Estimates of W measured from MERIS and modeled from MM5 are shown to be consistent and unbiased and differ by ~0.2 g cm^(−2) (RMS). We test the approach on data over the Lebanese ranges where active tectonics might contribute to a measurable SAR signal that is obscured by atmospheric effects. Local low-amplitude (1 rad) atmospheric oscillations with a 2.25 km wavelength on the interferograms are recovered from MERIS with an accuracy of 0.44 rad or 0.03 g cm^(−2). MERIS water product overestimates W in the clouds shadow due to mismodeling of multiple scattering and underestimates W on pixels with undetected semitransparent clouds. The proposed atmospheric filter models dynamic atmospheric signal which cannot be recovered by previous filtering techniques which are based on a static atmospheric correction. Analysis of filter efficiency with spatial wavelength shows that ~43% of the atmospheric signal is removed at all wavelengths

Combining statistical methods with dynamical insight to improve nonlinear estimation

Physical processes such as the weather are usually modelled using nonlinear dynamical systems. Statistical methods are found to be difficult to draw the dynamical information from the observations of nonlinear dynamics. This thesis is focusing on combining statistical methods with dynamical insight to improve the nonlinear estimate of the initial states, parameters and future states. In the perfect model scenario (PMS), method based on the Indistin-guishable States theory is introduced to produce initial conditions that are consistent with both observations and model dynamics. Our meth-ods are demonstrated to outperform the variational method, Four-dimensional Variational Assimilation, and the sequential method, En-semble Kalman Filter. Problem of parameter estimation of deterministic nonlinear models is considered within the perfect model scenario where the mathematical structure of the model equations are correct, but the true parameter values are unknown. Traditional methods like least squares are known to be not optimal as it base on the wrong assumption that the distribu-tion of forecast error is Gaussian IID. We introduce two approaches to address the shortcomings of traditional methods. The first approach forms the cost function based on probabilistic forecasting; the second approach focuses on the geometric properties of trajectories in short term while noting the global behaviour of the model in the long term. Both methods are tested on a variety of nonlinear models, the true parameter values are well identified. Outside perfect model scenario, to estimate the current state of the model one need to account the uncertainty from both observatiOnal noise and model inadequacy. Methods assuming the model is perfect are either inapplicable or unable to produce the optimal results. It is almost certain that no trajectory of the model is consistent with an infinite series of observations. There are pseudo-orbits, however, that are consistent with observations and these can be used to estimate the model states. Applying the Indistinguishable States Gradient De-scent algorithm with certain stopping criteria is introduced to find rel-evant pseudo-orbits. The difference between Weakly Constraint Four-dimensional Variational Assimilation (WC4DVAR) method and Indis-tinguishable States Gradient Descent method is discussed. By testing on two system-model pairs, our method is shown to produce more consistent results than the WC4DVAR method. Ensemble formed from the pseudo-orbit generated by Indistinguishable States Gradient Descent method is shown to outperform the Inverse Noise ensemble in estimating the current states. Outside perfect model scenario, we demonstrate that forecast with relevant adjustment can produce better forecast than ignoring the existence of model error and using the model directly to make fore-casts. Measurement based on probabilistic forecast skill is suggested to measure the predictability outside PMS