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Launching drifter observations in the presence of uncertainty
... In both cases, the averaged information gain within a time interval is calculated. Although the focus of this paper is on the real-time estate estimation scenario, it is helpful to summarize the reanalysis situation as well [23] to help understand the additional challenge in the real-time case. ...
... Notably, the uncertainty in the inferred or forecast states brings about multiple possible realizations of the flow field. In the reanalysis situation, different realizations of the flow field can be effectively sampled from the posterior distribution of the Lagrangian DA [23]. In the real-time forecast scenario, the forecast ensemble members serve as possible realizations. ...
... . These empirical arguments have been justified using simple analytically solvable examples in [23] for the reanalysis scenario. They will be further illustrated in Section 5. ...
Deploying Lagrangian drifters that facilitate the state estimation of the underlying flow field within a future time interval is practically important. However, the uncertainty in estimating the flow field prevents using standard deterministic approaches for designing strategies and applying trajectory-wise skill scores to evaluate performance. In this paper an information measurement is developed to quantitatively assess the information gain in the estimated flow field by deploying an additional set of drifters. This information measurement is derived by exploiting causal inference. It is characterized by the inferred probability density function of the flow field, which naturally considers the uncertainty. Although the information measurement is an ideal theoretical metric, using it as the direct cost makes the optimization problem computationally expensive. To this end, an effective surrogate cost function is developed. It is highly efficient to compute while capturing the essential features of the information measurement when solving the optimization problem. Based upon these properties, a practical strategy for deploying drifter observations to improve future state estimation is designed. Due to the forecast uncertainty, the approach exploits the expected value of spatial maps of the surrogate cost associated with different forecast realiza-tions to seek the optimal solution. Numerical experiments justify the effectiveness of the surrogate cost. The proposed strategy significantly outperforms the method by randomly deploying the drifters. It is also shown that, under certain conditions, the drifters determined by the expected surrogate cost remain skillful for the state estimation of a single forecast realization of the flow field as in reality.
... Real-time path planning using the LD approach resulted in a significant speed increase for the AUV, leading to unprecedented battery savings and paving the way for routine transoceanic long-duration missions. Furthermore, recent studies such as [107,108] have utilized LD to optimally deploy Lagrangian drifters, facilitating state estimation of the underlying flow field over future time intervals. The importance of uncertainty estimation in these applications has been highlighted by several authors, including [109][110][111]. ...
This survey focuses on the application of Lagrangian descriptors to reveal the geometry of phase space structures that determine transport in dynamical systems. We present diverse formulations of the method and examine various applications of Lagrangian descriptors in geophysical fluids, such as atmospheric flows and oceanic currents. The method of Lagrangian Descriptors has proven to be a powerful tool for characterizing transport and mixing in these contexts, demonstrating how these tools have enhanced our understanding of complex fluid dynamics in critical environments.
... But beyond this, the CGNSs have been utilized to develop an exact nonlinear Lagrangian DA algorithm that advances rigorous analysis to understand model errors and uncertainties in state recovery [75,95,96]. This includes the usage of the framework not just for high-dimensional state estimation of ocean flows, but also in determining the optimal launching locations for Lagrangian tracers under uncertainty [97,98]. Furthermore, the analytically solvable DA schemes have been applied for state estimation and the prediction of intermittent time series associated with monsoons and other climatic events [76,99]. ...
The conditional Gaussian nonlinear system (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise.
... Another interesting topic is the optimal selection of a few Lagrangian trajectories to supplement the Eulerian observations that maximize the uncertainty reduction in the estimated flow field from the hybrid Eulerian-Lagrangian observations. Information criteria [26] are useful cost functions to resolve such an optimization problem. Furthermore, this work assumes the particles appear almost everywhere in the flow field. ...
Lagrangian trajectories are widely used as observations for recovering the underlying flow field via Lagrangian data assimilation (DA). However, the strong nonlinearity in the observational process and the high dimensionality of the problems often cause challenges in applying standard Lagrangian DA. In this paper, a Lagrangian‐Eulerian multiscale DA (LEMDA) framework is developed. It starts with exploiting the Boltzmann kinetic description of the particle dynamics to derive a set of continuum equations, which characterize the statistical quantities of particle motions at fixed grids and serve as Eulerian observations. Despite the nonlinearity in the continuum equations and the processes of Lagrangian observations, the time evolution of the posterior distribution from LEMDA can be written down using closed analytic formulas after applying the stochastic surrogate model to describe the flow field. This offers an exact and efficient way of carrying out DA, which avoids using ensemble approximations and the associated tunings. The analytically solvable properties also facilitate the derivation of an effective reduced‐order Lagrangian DA scheme that further enhances computational efficiency. The Lagrangian DA part within the framework has advantages when a moderate number of particles is used, while the Eulerian DA part can effectively save computational costs when the number of particle observations becomes large. The Eulerian DA part is also valuable when particles collide, such as using sea ice floe trajectories as observations. LEMDA naturally applies to multiscale turbulent flow fields, where the Eulerian DA part recovers the large‐scale structures, and the Lagrangian DA part efficiently resolves the small‐scale features in each grid cell via parallel computing. Numerical experiments demonstrate the skillful results of LEMDA and its two components.
Reconstructing high-resolution flow fields from sparse measurements is a major challenge in fluid dynamics. Existing methods often vectorize the flow by stacking different spatial directions on top of each other, hence confounding the information encoded in different dimensions. Here, we introduce a tensor-based sensor placement and flow reconstruction method which retains and exploits the inherent multidimensionality of the flow. We derive estimates for the flow reconstruction error, storage requirements and computational cost of our method. We show, with examples, that our tensor-based method is significantly more accurate than similar vectorized methods. Furthermore, the variance of the error is smaller when using our tensor-based method. While the computational cost of our method is comparable to similar vectorized methods, it reduces the storage cost by several orders of magnitude. The reduced storage cost becomes even more pronounced as the dimension of the flow increases. We demonstrate the efficacy of our method on three examples: a chaotic Kolmogorov flow, in situ and satellite measurements of the global sea surface temperature and three-dimensional unsteady simulated flow around a marine research vessel.
Plain Language Summary
Tracking individual ice floes is a unique measurement of areas of the Arctic where the ice cover interacts with the open ocean. Unfortunately, optical satellite images of these areas are frequently obscured by clouds, leading to missing observations of the ice floes. Traditional methods of filling in these gaps in the data set have issues. Linear interpolation, which averages between available observations to fill in missing ones, fails to recover the curvature of the floes. Dynamical interpolation methods, which take into account the physical properties of the ice floes, are very computationally expensive. This paper presents a nonlinear dynamical interpolation framework for recovering missing floe observations, which is both computationally efficient and statistically accurate. The framework incorporates a model of the atmosphere, ocean, and sea ice and systematically develops data‐driven reduced‐order stochastic models, which significantly accelerate the dynamical interpolation while retaining accuracy. In addition, the framework estimates key physical parameters, such the floe thickness. This new method succeeds in recovering the locations, curvatures, angular displacements, and strong non‐Gaussian statistics of the missing floes in a data set of ice floes in the Beaufort Sea. These results can provide complete data sets that advance our understanding of Arctic climate.
The chaotic nature of ocean motion is a major challenge that hinders the discovery of spatio-temporal current routes that govern the transport of material. Certain material, such as oil spills, pose significant environmental threats and these are enhanced by the fact that they evolve in a chaotic sea, in a way which still nowadays is far from being systematically anticipated. Recently such an oil spill event has affected the Mediterranean coast of several Middle Eastern countries. No accidents were reported for these spills previous to their arrival at the coast, and therefore there was no hint of their origin. Modelling such an event, in which uncertainties are increased due to the lack of information on where and when the spills was produced, stretches available technologies to their limits, and requires the use of novel ideas that help to understand the essential features of oil and tar transport by ocean currents. In this regard Lagrangian Coherent Structures enable us to find order within ocean chaos and provide powerful insights into chaotic events and their relationships over different locations and times like the one addressed. Using the observed locations of the oil impacting the coast at specific times, we seek to determine its original location and the time it was released in the open ocean. We have determined both using a combination of earlier satellite observations and computational modelling of the time evolution. The observed agreement between modeled cases and satellite observations highlights the power of these ideas.
In many applications, it is important to reconstruct a fluid flow field, or some other high-dimensional state, from limited measurements and limited data. In this work, we propose a shallow neural network-based learning methodology for such fluid flow reconstruction. Our approach learns an end-to-end mapping between the sensor measurements and the high-dimensional fluid flow field, without any heavy preprocessing on the raw data. No prior knowledge is assumed to be available, and the estimation method is purely data-driven. We demonstrate the performance on three examples in fluid mechanics and oceanography, showing that this modern data-driven approach outperforms traditional modal approximation techniques which are commonly used for flow reconstruction. Not only does the proposed method show superior performance characteristics, it can also produce a comparable level of performance to traditional methods in the area, using significantly fewer sensors. Thus, the mathematical architecture is ideal for emerging global monitoring technologies where measurement data are often limited.
For tropical rainfall, there are several potential sources of predictability, including synoptic‐scale convectively coupled equatorial waves (CCEWs) and intraseasonal oscillations such as the Madden–Julian Oscillation (MJO). In prior work, predictability of these waves and rainfall has mainly been explored using forecast model data. Here, the goal is to estimate the intrinsic predictability using, instead, mainly observational data. To accomplish this, Tropical Rainfall Measuring Mission (TRMM) data are decomposed into different wave types using spectral/Fourier filtering. The predictability of MJO rainfall is estimated to be 22–31 days, depending on longitude, as measured by the lead time when the pattern correlation skill drops below 0.5. The predictability of rainfall associated with convectively coupled equatorial Rossby waves, Kelvin waves, and a background spectrum or nonwave component is estimated to be 8–12, 2–3, and 0–3 days, respectively. Combining all wave types, the overall predictability of tropical rainfall is estimated to be 3–6 days over the Indian and Pacific Ocean regions and on equatorial synoptic and planetary length‐scales. For comparison, outgoing longwave radiation (OLR) was more predictable than rainfall by 5–10 days over these regions. Wave‐removal tests were also conducted to estimate the contribution of each wave type to the overall predictability of rainfall. In summary, no single wave type dominates the predictability of tropical rainfall; each of the types (MJO, CCEWs, and nonwave component) has an appreciable contribution, due to the variance contribution, length of decorrelation time, or a combination of these factors.
Complex multiscale systems are ubiquitous in many areas. This research expository article discusses the development and applications of a recent information-theoretic framework as well as novel reduced-order nonlinear modeling strategies for understanding and predicting complex multiscale systems. The topics include the basic mathematical properties and qualitative features of complex multiscale systems, statistical prediction and uncertainty quantification, state estimation or data assimilation, and coping with the inevitable model errors in approximating such complex systems. Here, the information-theoretic framework is applied to rigorously quantify the model fidelity, model sensitivity and information barriers arising from different approximation strategies. It also succeeds in assessing the skill of filtering and predicting complex dynamical systems and overcomes the shortcomings in traditional path-wise measurements such as the failure in measuring extreme events. In addition, information theory is incorporated into a systematic data-driven nonlinear stochastic modeling framework that allows effective predictions of nonlinear intermittent time series. Finally, new efficient reduced-order nonlinear modeling strategies combined with information theory for model calibration provide skillful predictions of intermittent extreme events in spatially-extended complex dynamical systems. The contents here include the general mathematical theories, effective numerical procedures, instructive qualitative models, and concrete models from climate, atmosphere and ocean science.
A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.
This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigorous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for n-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as n-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.
The last decade has seen the success of stochastic parameterizations in
short-term, medium-range and seasonal ensembles: operational weather centers
now routinely use stochastic parameterization schemes to better represent model
inadequacy and improve the quantification of forecast uncertainty. Developed
initially for numerical weather prediction, the inclusion of stochastic
parameterizations not only provides more skillful estimates of uncertainty, but
is also extremely promising for reducing longstanding climate biases and
relevant for determining the climate response to forcings such as e.g., an
increase of CO2. This article highlights recent results from different research
groups which show that the stochastic representation of unresolved processes in
the atmosphere, oceans, land surface and cryosphere of comprehensive weather
and climate models a) gives rise to more reliable probabilistic forecasts of
weather and climate and b) reduces systematic model bias. We make a case that
the use of mathematically stringent methods for derivation of stochastic
dynamic equations will lead to substantial improvements in our ability to
accurately simulate weather and climate at all scales. Recent work in
mathematics, statistical mechanics and turbulence is reviewed and its relevance
for the climate problem demonstrated as well as future research directions
outlined.
We present a sparse sensing and Dynamic Mode Decomposition (DMD) based
framework to identify flow regimes and bifurcations in complex thermo-fluid
systems. Motivated by real time sensing and control of thermal fluid flows in
buildings and equipment, we apply this method to a Direct Numerical Simulation
(DNS) data set of a 2D laterally heated cavity, spanning several flow regimes
ranging from steady to chaotic flow. We exploit the incoherence exhibited among
the data generated by different regimes, which persists even if the number of
measurements is very small compared to the dimension of the DNS data. We
demonstrate that the DMD modes and eigenvalues capture the main temporal and
spatial scales in the dynamics belonging to different regimes, and use this
information to improve the classification performance of our algorithm. The
coarse state reconstruction obtained by our regime identification algorithm can
enable robust performance of low order models of flows for state estimation and
control.
Optimal sensor placement problems are considered for a thermoelastic solid body. The main motivation is the real-time capable prediction of the displacement of the tool center point (TCP) in a machine tool by temperature measurements. A reduced order model based on proper orthogonal decomposition is used to describe the temperature field. The quality of the TCP displacement estimation is measured in terms of the associated covariance matrix. Based on this criterion, a sequential placement algorithm is described which stops when a certain prediction quality is reached. Numerical tests are provided.
The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed high-dimensional turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines with a suite of physically relevant and progressively more complex test models which are mathematically tractable while possessing such important features as the two-way coupling between the resolved dynamics and the turbulent fluxes, intermittency and positive Lyapunov exponents, eddy diffusivity parameterization and turbulent spectra. A large number of new theoretical and computational phenomena which arise in the emerging statistical-stochastic framework for quantifying and mitigating model error in imperfect predictions, such as the existence of information barriers to model improvement, are developed and reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to these remarkable emerging topics with increasing practical importance.
The Argo profiling float project will enable, for the first time, continuous global observations of the temperature, salinity, and velocity of the upper ocean in near-real time.This new capability will improve our understanding of the ocean's role in climate, as well as spawn an enormous range of valuable ocean applications. Because over 90% of the observed increase in heat content of the air/land/sea climate system over the past 50 years occurred in the ocean [Leuitus et al., 2001], Argo will effectively monitor the pulse of the global heat balance.The end of 2003 was marked by two significant events for Argo. In mid-November 2003, over 200 scientists from 22 countries met at Argo's first science workshop to discuss early results from the floats. Two weeks later, Argo had 1000 profiling floats—one-third of the target total—delivering data. As of 7 May that total was 1171.
An important practical problem is the recovery of a turbulent velocity field from Lagrangian tracers that move with the fluid flow. Here, the filtering skill of L moving Lagrangian tracers in recovering random incompressible flow fields defined through a finite number of random Fourier modes is studied with full mathematical rigour. Despite the inherent nonlinearity in measuring noisy Lagrangian tracers, it is shown below that there are exact closed analytic formulas for the optimal filter for the velocity field involving Riccati equations with random coefficients for the covariance matrix. This mathematical structure allows a detailed asymptotic analysis of filter performance, both as time goes to infinity and as the number of noisy Lagrangian tracers, L, increases. In particular, the asymptotic gain of information from L-tracers grows only like ln L in a precise fashion; i.e., an exponential increase in the number of tracers is needed to reduce the uncertainty by a fixed amount; in other words, there is a practical information barrier. The proofs proceed through a rigourous mean field approximation of the random Ricatti equation. Also, as an intermediate step, geometric ergodicity with respect to the uniform measure on the period domain is proved for any fixed number L of noisy Lagrangian tracers. All of the above claims are confirmed by detailed numerical experiments presented here.
The focus of this paper is on how two main manifestations of
nonlinearity in low-dimensional systems - shear around a center fixed
point (nonlinear center) and the differential divergence of trajectories
passing by a saddle (nonlinear saddle) - strongly affect data
assimilation. The impact is felt through their leading to non-Gaussian
distribution functions. The major factors that control the strength of
these effects is time between observations, and covariance of the prior
relative to covariance of the observational noise. Both these factors -
less frequent observations and larger prior covariance - allow the
nonlinearity to take hold. To expose these nonlinear effects, we use the
comparison between exact posterior distributions conditioned on
observations and the ensemble Kalman filter (EnKF) approximation of
these posteriors. We discuss the serious limitations of the EnKF in
handling these effects.
This paper discusses a range of important mathematical issues arising in applications of a newly emerging stochastic-statistical framework for quantifying and mitigating uncertainties associated with prediction of partially observed and imperfectly modelled complex turbulent dynamical systems. The need for such a framework is particularly severe in climate science where the true climate system is vastly more complicated than any conceivable model; however, applications in other areas, such as neural networks and materials science, are just as important. The mathematical tools employed here rely on empirical information theory and fluctuation–dissipation theorems (FDTs) and it is shown that they seamlessly combine into a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect models. Here, we utilize a simple statistically exactly solvable ‘perfect’ system with intermittent hidden instabilities and with time-periodic features to address a number of important issues encountered in prediction of much more complex dynamical systems. These problems include the role and mitigation of model error due to coarse-graining, moment closure approximations, and the memory of initial conditions in producing short, medium and long-range predictions. Importantly, based on a suite of increasingly complex imperfect models of the perfect test system, we show that the predictive skill of the imperfect models and their sensitivity to external perturbations is improved by ensuring their consistency on the statistical attractor (i.e. the climate) with the perfect system. Furthermore, the discussed link between climate fidelity and sensitivity via the FDT opens up an enticing prospect of developing techniques for improving imperfect model sensitivity based on specific tests carried out in the training phase of the unperturbed statistical equilibrium/climate.
In this paper, a series of observing system simulation experiments (OSSEs) are used to study the design of a proposed array of instrumented moorings in the Indian Ocean (IO) outlined by the IO panel of the Climate Variability and Predictability (CLIVAR) Project. Fields of the Ocean Topography Experiment (TOPEX)/Poseidon (T/P) and Jason sea surface height (SSH) and sea surface temperature (SST) are subsampled to simulate dynamic height and SST data from the proposed array. Two different reduced-order versions of the Kalman filter are used to reconstruct the original fields from the simulated observations with the objective of determining the optimal deployment of moored platforms and to address the issue of redundancy and array simplification. The experiments indicate that, in terms of the reconstruction of SSH and SST, the location of the subjectively proposed array compareS favorably with the optimally defined one. The only significant difference between the proposed IO array and the optimal array is the lack of justification for increasing the latitudinal resolution near the equator (i.e., moorings 1.5°S and 1.5°N). An analysis of the redundancy also identifies the equatorial region as the one with the largest amount of redundant information. Thus, in the context of these fields, these results may help define the prioritization of its deployment or redefine the array to extend its latitudinal extent while maintaining the same amount of stations.
A basin-scale, reduced-gravity model is used to study how drifter launch strategies affect the accuracy of Eulerian velocity fields reconstructed from limited Lagrangian data. Optimal dispersion launch sites are found by tracking strongly hyperbolic singular points in the flow field. Lagrangian data from drifters launched from such locations are found to provide significant improvement in the reconstruction accuracy over similar but randomly located initial deployments. The eigenvalues of the hyperbolic singular points in the flow field determine the intensity of the local particle dispersion and thereby provide a natural timescale for initializing subsequent launches. Aligning the initial drifter launch in each site along an outflowing manifold ensures both high initial particle dispersion and the eventual sampling of regions of high kinetic energy, two factors that substantially affect the accuracy of the Eulerian reconstruction. Reconstruction error is reduced by a factor of ~2.5 by using a continual launch strategy based on both the local stretching rates and the outflowing directions of two strong saddles located in the dynamically active region south of the central jet. Notably, a majority of those randomly chosen launch sites that produced the most accurate reconstructions also sampled the local manifold structure.
An experiment design problem–-that of drifter cast strategy–-is discussed. Different optimization techniques used as part of preparations for the Semaphore-93 air-sea experiment, during which drifters were deployed, are examined. The oceanographic experiment objective was to sample a 500-km-square zone cantered at 33°N, 22°W in the Azores current area, using an average of 25 surface drifters for at least one month. We investigate different “orders of merit” for determining the performance of a particular cast strategy, as well as the method of genetic algorithms for optimizing the strategy. Our technique uses dynamic reference knowledge of the area where the simulation takes place. Two reference sets were used: a steady-state field calculated with data collected from the Kiel University April 1982 hydrographic experiment, and data output from a regional quasigeostrophic model assimilating two years of Geosat altimetric data. The strategies obtained via the genetic algorithm method were compared w...
Much of the debris in the near-surface ocean collects in so-called garbage patches where, due to convergence of the surface flow, the debris is trapped for decades to millennia. Until now, studies modelling the pathways of surface marine debris have not included release from coasts or factored in the possibilities that release concentrations vary with region or that pathways may include seasonal cycles. Here, we use observational data from the Global Drifter Program in a particle-trajectory tracer approach that includes the seasonal cycle to study the fate of marine debris in the open ocean from coastal regions around the world on interannual to centennial timescales. We find that six major garbage patches emerge, one in each of the five subtropical basins and one previously unreported patch in the Barents Sea. The evolution of each of the six patches is markedly different. With the exception of the North Pacific, all patches are much more dispersive than expected from linear ocean circulation theory, suggesting that on centennial timescales the different basins are much better connected than previously thought and that inter-ocean exchanges play a large role in the spreading of marine debris. This study suggests that, over multi-millennial timescales, a significant amount of the debris released outside of the North Atlantic will eventually end up in the North Pacific patch, the main attractor of global marine debris.
We determine the optimal location in North America and Fennoscandia for uplift rate or tangential velocity data that will be useful for addressing ice sheet thickness, lithospheric thickness, lateral viscosity variation and background viscosity profile in the lower mantle.
An optimal location is defined by where sensitivity lies above the current accuracy of GPS
measurements. The approach here is different from previous studies that compute sensitivity kernels for viscosity perturbations within a small volume of the mantle. The advantage of the current approach is that the total effect of 3-D lateral heterogeneity in the mantle related to seismic tomography of the whole mantle can be studied. The sensitivity of ice sheet models and lateral lithospheric thickness variations are also studied. Our results show that in North America more permanent GPS stations are needed in northern Canada especially in a region west of the Hudson Bay until the Rocky Mountains. In Fennoscandia, the GPS network is almost adequate, but it should be extended to the last known GIA-affected areas in the Russian part of East Europe and to Central Europe. In addition, we show locations of prospective GPS sites that are sensitive to all four parameters (ice sheet thickness, lithospheric thickness, lateral viscosity variation and background viscosity profile in the lower mantle) and locations that are sensitive to only one, two or three parameters. Thus, the results are useful for the inversion of one individual parameter or for the separation of the effects of two or more parameters in inversions.
Lagrangian coherent structures (LCSs) are time-varying entities which capture the most influential transport features of a flow. These can for example identify groups of particles which have greatest stretching, or which maintain a coherent jet or vortical structure. While many different LCS methods have been developed, the impact of the inevitable measurement uncertainty in realistic Eulerian velocity data has not been studied in detail. This article systematically addresses whether LCS methods are self-consistent in their conclusions under such uncertainty for nine different methods: the finite time Lyapunov exponent, hyperbolic variational LCSs, Lagrangian averaged vorticity deviation, Lagrangian descriptors, stochastic sensitivity, the transfer operator, the dynamic Laplacian operator, fuzzy c–means clustering and coherent structure colouring. The investigations are performed for two different realistic data sets: a computational fluid dynamics simulation of a Kelvin–Helmholtz instability, and oceanographic data of the Gulf Stream region. Using statistics gleaned from stochastic simulations, it is shown that the methods which detect full-dimensional coherent flow regions are significantly more robust than methods which detect lower-dimensional flow barriers. Additional insights into which aspects of each method are self-consistent, and which are not, are provided.
In this paper we bring together the method of Lagrangian descriptors and the principle of least action, or more precisely, of stationary action, in both deterministic and stochastic settings. In particular, we show how the action can be used as a Lagrangian descriptor. This provides a direct connection between Lagrangian descriptors and Hamiltonian mechanics, and we illustrate this connection with benchmark examples.
Lagrangian data assimilation of complex nonlinear turbulent flows is an important but computationally challenging topic. In this article, an efficient data-driven statistically accurate reduced-order modeling algorithm is developed that significantly accelerates the computational efficiency of Lagrangian data assimilation. The algorithm starts with a Fourier transform of the high-dimensional flow field, which is followed by an effective model reduction that retains only a small subset of the Fourier coefficients corresponding to the energetic modes. Then a linear stochastic model is developed to approximate the nonlinear dynamics of each Fourier coefficient. Effective additive and multiplicative noise processes are incorporated to characterize the modes that exhibit Gaussian and non-Gaussian statistics, respectively. All the parameters in the reduced order system, including the multiplicative noise coefficients, are determined systematically via closed analytic formulae. These linear stochastic models succeed in forecasting the uncertainty and facilitate an extremely rapid data assimilation scheme. The new Lagrangian data assimilation is then applied to observations of sea ice floe trajectories that are driven by atmospheric winds and turbulent ocean currents. It is shown that observing only about 30 non-interacting floes in a 200 km×200 km domain is sufficient to recover the key multi-scale features of the ocean currents. The additional observations of the floe angular displacements are found to be suitable supplements to the center-of-mass positions for improving the data assimilation analysis. In addition, the observed large and small floes are more useful in recovering the large- and small-scale features of the ocean, respectively. The Fourier domain data assimilation also succeeds in recovering the ocean features in the areas where cloud cover obscures the observations. Finally, the multiplicative noise is shown to be crucial in recovering extreme events.
We develop a data-driven method, based on semi-supervised classification, to predict the asymptotic state of multistable systems when only sparse spatial measurements of the system are feasible. Our method predicts the asymptotic behavior of an observed state by quantifying its proximity to the states in a precomputed library of data. To quantify this proximity, we introduce a sparsity-promoting metric-learning (SPML) optimization, which learns a metric directly from the precomputed data. The optimization problem is designed so that the resulting optimal metric satisfies two important properties: (i) it is compatible with the precomputed library and (ii) it is computable from sparse measurements. We prove that the proposed SPML optimization is convex, its minimizer is non-degenerate, and it is equivariant with respect to the scaling of the constraints. We demonstrate the application of this method on two multistable systems: a reaction–diffusion equation, arising in pattern formation, which has four asymptotically stable steady states, and a FitzHugh–Nagumo model with two asymptotically stable steady states. Classifications of the multistable reaction–diffusion equation based on SPML predict the asymptotic behavior of initial conditions based on two-point measurements with 95 % accuracy when a moderate number of labeled data are used. For the FitzHugh–Nagumo, SPML predicts the asymptotic behavior of initial conditions from one-point measurements with 90 % accuracy. The learned optimal metric also determines where the measurements need to be made to ensure accurate predictions.
This paper is concerned with real-time background flow field estimation using distributed pressure sensor measurements for autonomous underwater vehicles (AUVs). The goal of this study is to enhance environmental perception of AUVs especially in dynamic environments with changing flow patterns. Dynamic mode decomposition (DMD), a data-driven model reduction approach, is adopted to model the dynamic flow field using spatial basis modes and their corresponding temporal coefficients. This paper computes the DMD modes offline and applies a Bayesian filter to assimilate distributed pressure sensor measurements to estimate the DMD coefficients in real time. Further, fast Fourier transform (FFT) analysis is used to determine the flow pattern/model represented by DMD modes. Aiming to address the flow sensing problem in flow pattern changing environments, the proposed approach is expected to greatly improve environmental perception of AUVs in dynamic and complex flows. Both simulation and experimental results of flow sensing using three testing prototypes are presented to validate the proposed method.
When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. In the present study, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.
Learning nonlinear turbulent dynamics from partial observations is an important and challenging topic. In this article, an efficient learning algorithm based on the expectation-maximization approach is developed for a rich class of complex nonlinear turbulent dynamics using short training data. Despite the significant nonlinear and non-Gaussian features in these models, the analytically solvable conditional statistics allows the development of an exact and accurate nonlinear optimal smoother for recovering the hidden variables, which facilitates an efficient learning of these fully nonlinear models with extreme events. Then three additional ingredients are incorporated into the basic algorithm for improving the learning process. First, the physics constraint that requires the conservation of energy in the quadratic nonlinear terms is taken into account. It plays an important role in preventing the finite-time blowup of the solution and various pathological behavior of the recovered model. Second, a judicious block decomposition is applied to many large-dimensional nonlinear systems. It greatly accelerates the calculation of high-dimensional conditional covariance matrix and provides an extremely cheap parallel computation for learning the model parameters. Third, sparse identification of the complex turbulent models is combined with the learning algorithm that leads to parsimonious models. Numerical tests show the skill of the algorithm in learning the nonlinear dynamics and non-Gaussian statistics with extreme events in both perfect model and model error scenarios. It is also shown that in the presence of noise and partial observations, the model is not uniquely identified. Different nonlinear models all perfectly capture the key non-Gaussian features and obtain the same ensemble forecast skill of the observed variables as the perfect model, but they may have distinct model responses to external perturbations.
Data I/O poses a significant bottleneck in large-scale CFD simulations; thus, practitioners would like to significantly reduce the number of times the solution is saved to disk, yet retain the ability to recover any field quantity (at any time instance) a posteriori. The objective of this work is therefore to accurately recover missing CFD data a posteriori at any time instance, given that the solution has been written to disk at only a relatively small number of time instances. We consider in particular high-order discretizations (e.g., discontinuous Galerkin), as such techniques are becoming increasingly popular for the simulation of highly separated flows. To satisfy this objective, this work proposes a methodology consisting of two stages: 1) dimensionality reduction and 2) dynamics learning. For dimensionality reduction, we propose a novel hierarchical approach. First, the method reduces the number of degrees of freedom within each element of the high-order discretization by applying autoencoders from deep learning. Second, the methodology applies principal component analysis to compress the global vector of encodings. This leads to a low-dimensional state, which associates with a nonlinear embedding of the original CFD data. For dynamics learning, we propose to apply regression techniques (e.g., kernel methods) to learn the discrete-time velocity characterizing the time evolution of this low-dimensional state. A numerical example on a large-scale CFD example characterized by nearly 13 million degrees of freedom illustrates the suitability of the proposed method in an industrial setting.
We use machine learning to perform super-resolution analysis of grossly under-resolved turbulent flow field data to reconstruct the high-resolution flow field. Two machine learning models are developed, namely, the convolutional neural network (CNN) and the hybrid downsampled skip-connection/multi-scale (DSC/MS) models. These machine learning models are applied to a two-dimensional cylinder wake as a preliminary test and show remarkable ability to reconstruct laminar flow from low-resolution flow field data. We further assess the performance of these models for two-dimensional homogeneous turbulence. The CNN and DSC/MS models are found to reconstruct turbulent flows from extremely coarse flow field images with remarkable accuracy. For the turbulent flow problem, the machine-leaning-based super-resolution analysis can greatly enhance the spatial resolution with as little as 50 training snapshot data, holding great potential to reveal subgrid-scale physics of complex turbulent flows. With the growing availability of flow field data from high-fidelity simulations and experiments, the present approach motivates the development of effective super-resolution models for a variety of fluid flows.
Bayesian state estimation of a dynamical system from a stream of noisy measurements is important in many geophysical and engineering applications where high dimensionality of the state space, sparse observations, and model error pose key challenges. Here, three computationally feasible, approximate Gaussian data assimilation/filtering algorithms are considered in various regimes of turbulent 2D Navier-Stokes dynamics in the presence of model error. The first source of error arises from the necessary use of reduced models for the forward dynamics of the filters, while a particular type of representation error arises from the finite resolution of observations which mix up information about resolved and unresolved dynamics. Two stochastically parameterized filtering algorithms, referred to as cSPEKF and GCF, are compared with 3DVAR-a prototypical time-sequential algorithm known to be accurate for filtering dissipative systems for a suitably inflated "background" covariance. We provide the first evidence that the stochastically parameterized algorithms, which do not rely on detailed knowledge of the underlying dynamics and do not require covariance inflation, can compete with or outperform an optimally tuned 3DVAR algorithm, and they can overcome competing sources of error in a range of dynamical scenarios.
Exploiting the complementary character of CryoSat-2 and Soil Moisture and Ocean Salinity (SMOS) satellite sea ice thickness products, daily Arctic sea ice thickness estimates from October 2010 to December 2016 are generated by an Arctic regional ice-ocean model with satellite thickness assimilated. The assimilation is performed by a Local Error Subspace Transform Kalman filter (LESTKF) coded in the Parallel Data Assimilation Framework (PDAF). The new estimates can be generally thought of as combined model and satellite thickness (CMST). It combines the skill of satellite thickness assimilation in the freezing season with the model skill in the melting season, when neither CryoSat-2 nor SMOS sea ice thickness is available. Comparisons with in-situ observations from the Beaufort Gyre Exploration Project (BGEP), Ice Mass Balance (IMB) Buoys and the NASA Operation IceBridge demonstrate that CMST reproduces most of the observed temporal and spatial variations. Results also show that CMST compares favorably to the Pan-Arctic Ice Ocean Modeling and Assimilation System (PIOMAS) product, and even appears to correct known thickness biases in PIOMAS. Due to imperfect parameterizations in the sea ice model and satellite thickness retrievals, CMST does not reproduce the heavily deformed and ridged sea ice along the northern coast of the Canadian Arctic Archipelago (CAA) and Greenland. With the new Arctic sea ice thickness estimates sea ice volume changes in recent years can be further assessed.
Since 1994 the U.S. Global Drifter Program (GDP) and its international partners cooperating within the Data Buoy Cooperation Panel (DBCP) of the World Meteorological Organization (WMO) and the United Nations Education, Scientific and Cultural Organization (UNESCO) have been deploying drifters equipped with barometers primarily in the extratropical regions of the world’s oceans in support of operational weather forecasting. To date, the impact of the drifter data isolated from other sources has never been studied. This essay quantifies and discusses the effect and the impact of in situ sea level atmospheric pressure (SLP) data from the global drifter array on numerical weather prediction using observing system experiments and forecast sensitivity observation impact studies. The in situ drifter SLP observations are extremely valuable for anchoring the global surface pressure field and significantly contributing to accurate marine weather forecasts, especially in regions where no other in situ observations are available, for example, the Southern Ocean. Furthermore, the forecast sensitivity observation impact analysis indicates that the SLP drifter data are the most valuable per-observation contributor of the Global Observing System (GOS). All these results give evidence that surface pressure observations of drifting buoys are essential ingredients of the GOS and that their quantity, quality, and distribution should be preserved as much as possible in order to avoid any analysis and forecast degradations. The barometer upgrade program offered by the GDP, under which GDP-funded drifters can be equipped with partner-funded accurate air pressure sensors, is a practical example of how the DBCP collaboration is executed. Interested parties are encouraged to contact the GDP to discuss upgrade opportunities.
Lagrangian tracers are drifters and floaters that collect real-time information of fluid flows. This paper studies the model error in filtering multiscale random rotating compressible flow fields utilizing noisy Lagrangian tracers. The random flow fields are defined through random amplitudes of Fourier eigenmodes of the rotating shallow-water equations that contain both incompressible geostrophically balanced (GB) flows and rotating compressible gravity waves, where filtering the slow-varying GB flows is of primary concern. Despite the inherent nonlinearity in the observations with mixed GB and gravity modes, there are closed analytical formulas for filtering the underlying flows. Besides the full optimal filter, two practical imperfect filters are proposed. An information-theoretic framework is developed for assessing the model error in the imperfect filters, which can apply to a single realization of the observations. All the filters are comparably skillful in a fast rotation regime (Rossby number ε = 1 ). In a moderate rotation regime (ε = 1), significant model errors are found in the reduced filter containing only GB forecast model, while the computationally efficient 3D-Var filter with a diagonal covariance matrix remains skillful. First linear then nonlinear coupling of GB and gravity modes is introduced in the random Fourier amplitudes, while linear forecast models are retained to ensure the filter estimates have closed analytical expressions. All the filters remain skillful in the ε = 0.1 regime. In the ε = 1 regime, the full filter with a linear forecast model has an acceptable filtering skill, while large model errors are shown in the other two imperfect filters.
This paper introduces a new framework for constructing the Discrete Empirical
Interpolation Method (DEIM) projection operator. The interpolation nodes
selection procedure is formulated using a QR factorization with column
pivoting. This selection strategy leads to a sharper error bound for the DEIM
projection error and works on a given orthonormal frame U as a point on the
Stiefel manifold, i.e., the selection operator does not change if U is
replaced by with arbitrary unitary matrix Q. The new approach allows
modifications that, in the case of gargantuan dimensions, use only randomly
sampled rows of U but are capable of producing equally good approximations.
The recovery of a random turbulent velocity field using Lagrangian tracers that move with the fluid flow is a practically important problem. This paper studies the filtering skill of L -noisy Lagrangian tracers in recovering random rotating compressible flows that are a linear combination of random incompressible geostrophically balanced (GB) flow and random rotating compressible gravity waves. The idealized random fields are defined through forced damped random amplitudes of Fourier eigenmodes of the rotating shallow-water equations with the rotation rate measured by the Rossby number . In many realistic geophysical flows, there is fast rotation so satisfies and the random rotating shallow-water equations become a slow–fast system where often the primary practical objective is the recovery of the GB component from the Lagrangian tracer observations. Unfortunately, the L -noisy Lagrangian tracer observations are highly nonlinear and mix the slow GB modes and the fast gravity modes. Despite this inherent nonlinearity, it is shown here that there are closed analytical formulas for the optimal filter for recovering these random rotating compressible flows for any involving Ricatti equations with random coefficients. The performance of the optimal filter is compared and contrasted through mathematical theorems and concise numerical experiments with the performance of the optimal filter for the incompressible GB random flow with L -noisy Lagrangian tracers involving only the GB part of the flow. In addition, a sub-optimal filter is defined for recovering the GB flow alone through observing the L -noisy random compressible Lagrangian trajectories, so the effect of the gravity wave dynamics is unresolved but effects the tracer observations. Rigorous theorems proved below through suitable stochastic fast-wave averaging techniques and explicit formulas rigorously demonstrate that all these filters have comparable skill in recovering the slow GB flow in the limit for any bounded time interval. Concise numerical experiments confirm the mathematical theory and elucidate various new features of filter performance as the Rossby number , the number of tracers L and the tracer noise variance change.
Incomplete knowledge of the true dynamics and its partial observability pose a notoriously difficult problem in many scientific applications which require predictions of high-dimensional dynamical systems with instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this high-dimensional setting are analyzed through appropriate information measures which naturally extend the common path-wise estimates of filter performance, like the mean-square error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. Particular emphasis is on the notion of practically achievable filter skill which requires trade-offs between different facets of filter performance; a new information criterion is introduced in this context. This information-theoretic framework for assessment of filter performance has natural generalizations to Kalman filtering with non-Gaussian statistically exactly solvable forecast models. Here, this approach is utilized to study the performance of imperfect, reduced-order filters involving Gaussian forecast models which use various spatio-temporal discretizations to approximate the dynamics of the stochastically forced advection-diffusion equation; important examples in this configuration include effects of biases due to model error in the filter estimates for the mean dynamics which are quantified through appropriate information measures.
The filtering/data assimilation and prediction of moisture‐coupled tropical waves is a contemporary topic with significant implications for extended‐range forecasting. The development of efficient algorithms to capture such waves is limited by the unstable multiscale features of tropical convection which can organize large‐scale circulations and the sparse observations of the moisture‐coupled wave in both the horizontal and vertical. The approach proposed here is to address these difficult issues of data assimilation and prediction through a suite of analogue models which, despite their simplicity, capture key features of the observational record and physical processes in moisture‐coupled tropical waves. The analogue models emphasized here involve the multicloud convective parametrization based on three cloud types (congestus, deep, and stratiform) above the boundary layer. Two test examples involving an MJO‐like turbulent travelling wave and the initiation of a convectively coupled wave train are introduced to illustrate the approach. A suite of reduced filters with judicious model errors for data assimilation of sparse observations of tropical waves, based on linear stochastic models in a moisture‐coupled eigenmode basis is developed here and applied to the two test problems. Both the reduced filter and 3D‐Var with a full moist background covariance matrix can recover the unobserved troposphere humidity and precipitation rate; on the other hand, 3D‐Var with a dry background covariance matrix fails to recover these unobserved variables. The skill of the reduced filtering methods in recovering the unobserved precipitation, congestus, and stratiform heating rates as well as the front‐to‐rear tilt of the convectively coupled waves exhibits a subtle dependence on the sparse observation network and the observation time. Copyright © 2012 Royal Meteorological Society
Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.
Fundamental barriers in practical filtering of nonlinear spatio-temporal chaotic systems are model errors attributed to the stiffness in resolving multiscale features. Recently, reduced stochastic filters based on linear stochastic models have been introduced to overcome such stiffness; one of them is the Mean Stochastic Model (MSM) based on a diagonal Ornstein–Uhlenbeck process in Fourier space. Despite model errors, the MSM shows very encouraging filtering skill, especially when the hidden signal of interest is strongly chaotic. In this regime, the dynamical system statistical properties resemble to those of the energy-conserving equilibrium statistical mechanics with Gaussian invariant measure; therefore, the Ornstein–Uhlenbeck process with appropriate parameters is sufficient to produce reasonable statistical estimates for the filter model.
When the climate system experiences time-dependent external forcing (e.g., from increases in greenhouse gas and aerosol concentrations), there are two inherent limits on the gain in skill of decadal climate predictions that can be attained from initializing with the observed ocean state. One is the classical initial-value predictability limit that is a consequence of the system being chaotic, and the other corresponds to the forecast range at which information from the initial conditions is overcome by the forced response. These limits are not caused by model errors; they correspond to limits on the range of useful forecasts that would exist even if nature behaved exactly as the model behaves. In this paper these two limits are quantified for the Community Climate System Model, version 3 (CCSM3), with several 40-member climate change scenario experiments. Predictability of the upper-300-m ocean temperature, on basin and global scales, is estimated by relative entropy from information theory. Despite some regional variations, overall, information from the ocean initial conditions exceeds that from the forced response for about 7 yr. After about a decade the classical initial-value predictability limit is reached, at which point the initial conditions have no remaining impact. Initial-value predictability receives a larger contribution from ensemble mean signals than from the distribution about the mean. Based on the two quantified limits, the conclusion is drawn that, to the extent that predictive skill relies solely on upper-ocean heat content, in CCSM3 decadal prediction beyond a range of about 10 yr is a boundary condition problem rather than an initial-value problem. Factors that the results of this study are sensitive and insensitive to are also discussed.
The history of constant-level balloons as observational platforms for atmospheric research is reviewed. Recent experience in using simple tetroons with Global Positioning System transponders for long-range tracking during the Atlantic Stratocumulus Transition Experiment (ASTEX) is presented, along with an overview of the results of the ASTEX/Marine Aerosol and Gas Exchange Lagrangian strategy experiments. Progress in balloon materials and tracking capabilities is discussed, and a design is presented for an economical, lightweight "smart balloon" for use in future Lagrangian strategy, atmospheric chemistry experiments.
The application of proper orthogonal decomposition for incomplete (gappy) data for compressible external aerodynamic problems has been demonstrated successfully in this paper for the first time. Using this approach, it is possible to construct entire aerodynamic flowfields from the knowledge of computed aerodynamic flow data or measured flow data specified on the aerodynamic surface, thereby demonstrating a means to effectively combine experimental and computational data. The sensitivity of flow reconstruction results to available measurements and to experimental error is analyzed. Another new extension of this approach allows one to cast the problem of inverse airfoil design as a gappy data problem. The gappy methodology demonstrates a great simplification for the inverse airfoil design problem and is found to work well on a range of examples, including both subsonic and transonic cases.
This paper gives an introduction to the connection between predictability and information theory, and derives new connections between these concepts. A system is said to be unpredictable if the forecast distribution, which gives the most complete description of the future state based on all available knowledge, is identical to the climatological distribution, which describes the state in the absence of time lag information. It follows that a necessary condition for predictability is for the forecast and climatological distributions to differ. Information theory provides a powerful framework for quantifying the difference between two distributions that agrees with intuition about predictability. Three information theoretic measures have been proposed in the literature: predictive information, relative entropy, and mutual information. These metrics are discussed with the aim of clarifying their similarities and differences. All three metrics have attractive properties for defining predictability, including the fact that they are invariant with respect to nonsingular linear transformations, decrease monotonically in stationary Markov systems in some sense, and are easily decomposed into components that optimize them (in certain cases). Relative entropy and predictive information have the same average value, which in turn equals the mutual information. Optimization of mutual information leads naturally to canonical correlation analysis, when the variables are joint normally distributed. Closed form expressions of these metrics for finite dimensional, stationary, Gaussian, Markov systems are derived. Relative entropy and predictive information differ most significantly in that the former depends on the ``signal to noise ratio'' of a single forecast distribution, whereas the latter does not. Part II of this paper discusses the extension of these concepts to imperfect forecast models.
A new method for directly assimilating Lagrangian tracer observations for flow state estimation is presented. It is developed in the context of point vortex systems. With tracer advection equations augmenting the point vortex model, the correlations between the vortex and tracer positions allow one to use the observed tracer positions to update the non-observed vortex positions. The method works efficiently when the observations are accurate and frequent enough. Low-quality data and large intervals between observations can lead to divergence of the scheme. Nonlinear effects, responsible for the failure of the extended Kalman filter, are triggered by the exponential rate of separation of tracer trajectories in the neighbourhoods of the saddle points of the velocity field.
This article was chosen from Selected Proceedings of the 4th International Workshop on Vortex Flows and Related Numerical Methods (UC Santa-Barbara, 17-20 March 2002) ed E Meiburg, G H Cottet, A Ghoniem and P Koumoutsakos.
Lagrangian data arise from instruments that are carried by the flow in a fluid field. Assimilation of such data into ocean models presents a challenge due to the potential complexity of Lagrangian trajectories in relatively simple flow fields. We adopt a Bayesian perspective on this problem and thereby take account of the fully non-linear features of the underlying model.In the perfect model scenario, the posterior distribution for the initial state of the system contains all the information that can be extracted from a given realization of observations and the model dynamics. We work in the smoothing context in which the posterior on the initial conditions is determined by future observations. This posterior distribution gives the optimal ensemble to be used in data assimilation. The issue then is sampling this distribution. We develop, implement, and test sampling methods, based on Markov-chain Monte Carlo (MCMC), which are particularly well suited to the low-dimensional, but highly non-linear, nature of Lagrangian data. We compare these methods to the well-established ensemble Kalman filter (EnKF) approach. It is seen that the MCMC based methods correctly sample the desired posterior distribution whereas the EnKF may fail due to infrequent observations or non-linear structures in the underlying flow.