Lagrangian Data Assimilation Research Articles

Observing System Simulation Experiments of cross-layer Lagrangian data assimilation

We conduct Observing System Simulation Experiments (OSSEs) with Lagrangian data assimilation (LaDA) in two-layer point-vortex systems, where the trajectories of passive tracers (drifters or floats) are observed on one layer that is coupled to another layer with different dynamics. Depending on the initial position of the observed tracers, the model studied here can exhibit nonlinear features that cause the standard Kalman filter and its variants to fail. For this reason, we adopt a Monte Carlo approach known as particle filtering, which takes the nonlinear dynamics into account. The main objective of this paper is to understand the effects of drifter placement and layer coupling on the precision skill of assimilating Lagrangian data into multi-layered models. Therefore, we analyze the quality of the assimilated vortex estimates by assimilating path data from passive tracers launched at different locations, on different layers and in systems with various coupling strengths between layers. We consider two cases: vortices placed on different layers (heton) and on the same layer (non-heton). In both cases we find that launch location, launch layer and coupling strength all play a significant role in assimilation precision skill. However, the specifics of the interplay of these three factors are quite different for the heton case versus the non-heton case.

The impact of nonlinearity in Lagrangian data assimilation

Abstract. 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.

The impact of nonlinearity in Lagrangian data assimilation

The impact of nonlinearity in Lagrangian data assimilation

Investigating transport pathways in the ocean

The ocean is a very complex medium with scales of motion that range from thousands of kilometers to the dissipation scales. Transport by ocean currents plays an important role in many practical applications ranging from climatic problems to coastal management and accident mitigation at sea. Understanding transport is challenging because of the chaotic nature of particle motion. In the last decade, new methods have been put forth to improve our understanding of transport. Powerful tools are provided by dynamical system theory, that allow the identification of the barriers to transport and their time variability for a given flow. A shortcoming of this approach, though, is that it is based on the assumption that the velocity field is known with good accuracy, which is not always the case in practical applications. Improving model performance in terms of transport can be addressed using another important methodology that has been recently developed, namely the assimilation of Lagrangian data provided by floating buoys. The two methodologies are technically different but in many ways complementary. In this paper, we review examples of applications of both methodologies performed by the authors in the last few years, considering flows at different scales and in various ocean basins. The results are among the very first examples of applications of the methodologies to the real ocean including testing with Lagrangian in-situ data. The results are discussed in the general framework of the extended fields related to these methodologies, pointing out to open questions and potential for improvements, with an outlook toward future strategies.

Variational assimilation of Lagrangian trajectories in the Mediterranean ocean Forecasting System

Abstract. A novel method for three-dimensional variational assimilation of Lagrangian data with a primitive-equation ocean model is proposed. The assimilation scheme was implemented in the Mediterranean ocean Forecasting System and evaluated for a 4-month period. Four experiments were designed to assess the impact of trajectory assimilation on the model output, i.e. the sea-surface height, velocity, temperature and salinity fields. It was found from the drifter and Argo trajectory assimilation experiment that the forecast skill of surface-drifter trajectories improved by 15 %, that of intermediate-depth float trajectories by 20 %, and moreover, that the forecasted sea-surface height fields improved locally by 5 % compared to satellite data, while the quality of the temperature and salinity fields remained at previous levels. In conclusion, the addition of Lagrangian trajectory assimilation proved to reduce the uncertainties in the model fields, thus yielding a higher accuracy of the ocean forecasts.

Capturing eddy shedding in the Gulf of Mexico from Lagrangian observations

The nonlinear process of eddy shedding is studied in the context of the Gulf of Mexico. We show that model runs which do not include eddy detachment can reproduce such an event with the assimilation of suitable data obtained from a control run with eddy detachment. This works surprisingly well and with small amounts of data provided the data originates from instruments that are carried by the flow, i.e. Lagrangian. This is compared with analogous assimilation of data from fixed stations which capture the eddy poorly. The remarkable efficacy of Lagrangian data assimilation in this context is explained by considering the structure of the correlation functions and their associated regions of influence.

Identification of equivalent topography in an open channel flow using Lagrangian data assimilation

We present a Lagrangian data assimilation experiment in an open channel flow above a broad-crested weir. The observations consist of trajectories of particles transported by the flow and extracted from a video film, in addition to classical water level measurements. However, the presence of vertical recirculations on both sides of the weir actually conducts to the identification of an equivalent topography corresponding to the lower limit of a surface jet. In addition, results on the identification of the Manning coefficient may allow to detect the presence of bottom recirculations.

The Diffusion Kernel Filter Applied to Lagrangian Data Assimilation

Abstract The diffusion kernel filter is a sequential particle-method approach to data assimilation of time series data and evolutionary models. The method is applicable to nonlinear/non-Gaussian problems. Within branches of prediction it parameterizes small fluctuations of Brownian-driven paths about deterministic paths. Its implementation is relatively straightforward, provided a tangent linear model is available. A by-product of the parameterization is a bound on the infinity norm of the covariance matrix of such fluctuations (divided by the grid model dimension). As such it can be used to define a notion of “prediction” itself. It can also be used to assess the short time sensitivity of the deterministic history to Brownian noise or Gaussian initial perturbations. In pure oceanic Lagrangian data assimilation, the dynamics and the statistics are nonlinear and non-Gaussian, respectively. Both of these characteristics challenge conventional methods, such as the extended Kalman filter and the popular ensemble Kalman filter. The diffusion kernel filter is proposed as an alternative and is evaluated here on a problem that is often used as a test bed for Lagrangian data assimilation: it consists of tracking point vortices and passive drifters, using a dynamical model and data, both of which have known error statistics. It is found that the diffusion kernel filter captures the first few moments of the random dynamics, with a computational cost that is competitive with a particle filter estimation strategy. The authors also introduce a clustered version of the diffusion kernel filter (cDKF), which is shown to be significantly more efficient with regard to computational cost, at the expense of a slight degradation in the description of the statistics of the dynamical history. Upon parallelizing branches of prediction, cDKF can be computationally competitive with EKF.

A hybrid grid/particle filter for Lagrangian data assimilation. II: Application to a model vortex flow

AbstractWe apply our hybrid filter to a regularised vortex model of a co‐rotating vortex pair. To illustrate the main advantages of our formulation over existing filters, we compare our method to the perturbed observation Ensemble Kalman filter and a particle filter with Gaussian resampling. Our numerical simulations show that both the hybrid and particle filters can track the true vortex positions even when tracer position data is assimilated infrequently into our model. In contrast, the Ensemble Kalman filter diverges in this parameter range as was recently observed in the more realistic shallow‐water model simulations of Salman et al. We have found that our hybrid method can track the true system with as few as 20 members for the vortex model flow. The particle filter on the other hand requires an ensemble comprising in excess of 160 members. The hybrid filter, therefore, provides one solution to the filter divergence problem that has been identified in recent work on Lagrangian data assimilation. Copyright © 2008 Royal Meteorological Society

A hybrid grid/particle filter for Lagrangian data assimilation. I: Formulating the passive scalar approximation

AbstractIn this two‐part paper, we present a new nonlinear method for the assimilation of Lagrangian data. In part I, we formulate the method as a generalization of other particle filters. A particularly novel feature of the formulation is the use of a hybrid discretisation of the probability density function (PDF) in physical/phase space. Moreover, we show that, under the assumption that the drifters are uncorrelated, the projection of the Fokker–Planck equation onto the observation space associated with the drifter positions reduces to a set of passive scalar equations. This property allows us to efficiently compute the transitional PDF. To compute the analysis states, we present a grid/particle filter specifically formulated for use with the hybrid representation of our PDF. In common with other particle filters, our filter can suffer from sample impoverishment. To remedy this problem, we extend the Gaussian resampling procedure of Xiong et al. to produce a very efficient filter. This produces a fully functional scheme for Lagrangian data assimilation when combined with our forecasts of the prior. Copyright © 2008 Royal Meteorological Society

Lagrangian data assimilation for river hydraulics simulations

We present a method to use Lagrangian data from remote sensing observation in a variational data assimilation process for a river hydraulics model based on the bidimensional shallow water equations. The trajectories of particles advected by the flow can be extracted from video images and are used in addition to classical Eulerian observations. This Lagrangian data brings information on the surface velocity thanks to an appropriate transport model. Numerical twin data assimilation experiments in an academic flow configuration demonstrate that this method makes it possible to significantly improve the identification of local bed elevation and initial conditions.

Modified particle filter methods for assimilating Lagrangian data into a point-vortex model

The process of assimilating Lagrangian (particle trajectory) data into fluid models can fail with a standard linear-based method, such as the Kalman filter. We implement a particle filtering approach that affords a nonlinear estimation and does not impose Gaussianity on either the prior or the posterior distributions at the update step. Several schemes for reinitializing the particle filter, specifically tailored to the Lagrangian data assimilation problem, are applied to a point-vortex system. A comparison with the Extended Kalman Filter (EKF) for the same system demonstrates the effectiveness of particle filters for the assimilation of complex, nonlinear Lagrangian data.

Using flow geometry for drifter deployment in Lagrangian data assimilation

Using flow geometry for drifter deployment in Lagrangian data assimilation

A Bayesian approach to Lagrangian data assimilation

A Bayesian approach to Lagrangian data assimilation

Using flow geometry for drifter deployment in Lagrangian data assimilation

Methods of Lagrangian data assimilation (LaDA) require carefully chosen sites for optimal drifter deployments. In this work, we investigate a directed drifter deployment strategy with a recently developed LaDA method employing an augmented state vector formulation for an Ensemble Kalman filter. We test our directed drifter deployment strategy by targeting Lagrangian coherent flow structures of an unsteady double gyre flow to analyse how different release sites influence the performance of the method.We consider four different launch methods; a uniform launch, a saddle launch in which hyperbolic trajectories are targeted, a vortex centre launch, and a mixed launch targeting both saddles and centres. We show that global errors in the flow field require good dispersion of the drifters which can be realized with the saddle launch. Local errors on the other hand are effectively reduced by targeting specific flow features. In general, we conclude that it is best to target the strongest hyperbolic trajectories for shorter forecasts although vortex centres can produce good drifter dispersion upon bifurcating on longer time-scales.

Using flow geometry for drifter deployment in Lagrangian data assimilation

Using flow geometry for drifter deployment in Lagrangian data assimilation

A Bayesian approach to Lagrangian data assimilation

A Bayesian approach to Lagrangian data assimilation

A Bayesian approach to Lagrangian data assimilation

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.

Assimilation of Lagrangian data in oceanography

AbstractWithin the framework of Global Ocean Data Assimilation Experiment (GODAE), an increasing amount of data are available. A crucial issue for oceanographers is to exploit at best these observations, in order to improve models, climatology, forecasts, etc. Thanks to the international program Argo and to more localized experiments, a new type of data is now available: positions of floats drifting at depth in the ocean. Unlike other data, mainly Eulerian, these ones are Lagrangian: the measuring instrument move in the flow. I will briefly described methods about 4D‐Var assimilation of Lagrangian data in the OPAVAR ocean model, and show some results assessing the complementarity with temperature data. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Sampling the posterior: An approach to non-Gaussian data assimilation

The viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a Bayesian framework. In the absence of model error, the correct solution to the data assimilation problem is to find the posterior distribution implied by this Bayesian setting. Methods for dealing with data assimilation should then be judged by their ability to probe this distribution. In this paper we propose a range of techniques for probing the posterior distribution, based around the Langevin equation; and we compare these new techniques with existing methods. When the underlying dynamics is deterministic, the posterior distribution is on the space of initial conditions leading to a sampling problem over this space. When the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths. By writing down a density, and conditioning on observations, it is possible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired posterior distribution, and thereby solve the data assimilation problem. The basic building-blocks for the MCMC methods that we concentrate on in this paper are Langevin equations which are ergodic and whose invariant measures give the desired distribution; in the case of path space sampling these are stochastic partial differential equations (SPDEs). Two examples are given to show how data assimilation can be formulated in a Bayesian fashion. The first is weather prediction, and the second is Lagrangian data assimilation for oceanic velocity fields. Furthermore the relationship between the Bayesian approach outlined here and the commonly used Kalman filter based techniques, prevalent in practice, is discussed. Two simple pedagogical examples are studied to illustrate the application of Bayesian sampling to data assimilation concretely. Finally a range of open mathematical and computational issues, arising from the Bayesian approach, are outlined.