Probabilistic physically-based cloud screening of satellite infra-red imagery for operational sea surface temperature retrieval

We propose and demonstrate a fully probabilistic (Bayesian) approach to the detection of cloudy pixels in thermal infrared (TIR) imagery observed from satellite over oceans. Using this approach, we show how to exploit the prior information and the fast forward modelling capability that are typically available in the operational context to obtain improved cloud detection. The probability of clear sky for each pixel is estimated by applying Bayes' theorem, and we describe how to apply Bayes' theorem to this problem in general terms. Joint probability density functions (PDFs) of the observations in the TIR channels are needed; the PDFs for clear conditions are calculable from forward modelling and those for cloudy conditions have been obtained empirically. Using analysis fields from numerical weather prediction as prior information, we apply the approach to imagery representative of imagers on polar-orbiting platforms. In comparison with the established cloud-screening scheme, the new technique decreases both the rate of failure to detect cloud contamination and the false-alarm rate by one quarter. The rate of occurrence of cloud-screening-related errors of >1 K in area-averaged SSTs is reduced by 83%. Copyright © 2005 Royal Meteorological Society.

Estimating interchannel observation-error correlations for IASI radiance data in the Met Office system

The optimal utilisation of hyper-spectral satellite observations in numerical weather prediction is often inhibited by incorrectly assuming independent interchannel observation errors. However, in order to represent these observation-error covariance structures, an accurate knowledge of the true variances and correlations is needed. This structure is likely to vary with observation type and assimilation system. The work in this article presents the initial results for the estimation of IASI interchannel observation-error correlations when the data are processed in the Met Office one-dimensional (1D-Var) and four-dimensional (4D-Var) variational assimilation systems. The method used to calculate the observation errors is a post-analysis diagnostic which utilises the background and analysis departures from the two systems. The results show significant differences in the source and structure of the observation errors when processed in the two different assimilation systems, but also highlight some common features. When the observations are processed in 1D-Var, the diagnosed error variances are approximately half the size of the error variances used in the current operational system and are very close in size to the instrument noise, suggesting that this is the main source of error. The errors contain no consistent correlations, with the exception of a handful of spectrally close channels. When the observations are processed in 4D-Var, we again find that the observation errors are being overestimated operationally, but the overestimation is significantly larger for many channels. In contrast to 1D-Var, the diagnosed error variances are often larger than the instrument noise in 4D-Var. It is postulated that horizontal errors of representation, not seen in 1D-Var, are a significant contributor to the overall error here. Finally, observation errors diagnosed from 4D-Var are found to contain strong, consistent correlation structures for channels sensitive to water vapour and surface properties.

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

Modelling the reorientation of sea-ice faults as the wind changes direction

A discrete-element model of sea ice is used to study how a 90° change in wind direction alters the pattern of faults generated through mechanical failure of the ice. The sea-ice domain is 400km in size and consists of polygonal floes obtained through a Voronoi tessellation. Initially the floes are frozen together through viscous–elastic joints that can break under sufficient compressive, tensile and shear deformation. A constant wind-stress gradient is applied until the initially frozen ice pack is broken into roughly diamond-shaped aggregates, with crack angles determined by wing-crack formation. Then partial refreezing of the cracks delineating the aggregates is modelled through reduction of their length by a particular fraction, the ice pack deformation is neglected and the wind stress is rotated by 90°. New cracks form, delineating aggregates with a different orientation. Our results show the new crack orientation depends on the refrozen fraction of the initial faults: as this fraction increases, the new cracks gradually rotate to the new wind direction, reaching 90° for fully refrozen faults. Such reorientation is determined by a competition between new cracks forming at a preferential angle determined by the wing-crack theory and at old cracks oriented at a less favourable angle but having higher stresses due to shorter contacts across the joints

Integrated visual vocabulary in latent Dirichlet allocation-based scene classification for IKONOS image

Scene classification based on latent Dirichlet allocation (LDA) is a more general modeling method known as a bag of visual words, in which the construction of a visual vocabulary is a crucial quantization process to ensure success of the classification. A framework is developed using the following new aspects: Gaussian mixture clustering for the quantization process, the use of an integrated visual vocabulary (IVV), which is built as the union of all centroids obtained from the separate quantization process of each class, and the usage of some features, including edge orientation histogram, CIELab color moments, and gray-level co-occurrence matrix (GLCM). The experiments are conducted on IKONOS images with six semantic classes (tree, grassland, residential, commercial/industrial, road, and water). The results show that the use of an IVV increases the overall accuracy (OA) by 11 to 12% and 6% when it is implemented on the selected and all features, respectively. The selected features of CIELab color moments and GLCM provide a better OA than the implementation over CIELab color moment or GLCM as individuals. The latter increases the OA by only ∼2 to 3%. Moreover, the results show that the OA of LDA outperforms the OA of C4.5 and naive Bayes tree by ∼20%. © 2014 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.JRS.8.083690]

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

Comparing hybrid data assimilation methods on the Lorenz 1963 model with increasing nonlinearity

We systematically compare the performance of ETKF-4DVAR, 4DVAR-BEN and 4DENVAR with respect to two traditional methods (4DVAR and ETKF) and an ensemble transform Kalman smoother (ETKS) on the Lorenz 1963 model. We specifically investigated this performance with increasing nonlinearity and using a quasi-static variational assimilation algorithm as a comparison. Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering (1) assimilation window length and observation interval size and (2) ensemble size to investigate the influence of hybrid background error covariance matrices and nonlinearity on the performance of the methods. For short assimilation windows with close to linear dynamics, it has been shown that all hybrid methods show an improvement in RMSE compared to the traditional methods. For long assimilation window lengths in which nonlinear dynamics are substantial, the variational framework can have diffculties fnding the global minimum of the cost function, so we explore a quasi-static variational assimilation (QSVA) framework. Of the hybrid methods, it is seen that under certain parameters, hybrid methods which do not use a climatological background error covariance do not need QSVA to perform accurately. Generally, results show that the ETKS and hybrid methods that do not use a climatological background error covariance matrix with QSVA outperform all other methods due to the full flow dependency of the background error covariance matrix which also allows for the most nonlinearity.

Exploring strategies for coupled 4D-Var data assimilation using an idealised atmosphere-ocean model

Operational forecasting centres are currently developing data assimilation systems for coupled atmosphere-ocean models. Strongly coupled assimilation, in which a single assimilation system is applied to a coupled model, presents significant technical and scientific challenges. Hence weakly coupled assimilation systems are being developed as a first step, in which the coupled model is used to compare the current state estimate with observations, but corrections to the atmosphere and ocean initial conditions are then calculated independently. In this paper we provide a comprehensive description of the different coupled assimilation methodologies in the context of four dimensional variational assimilation (4D-Var) and use an idealised framework to assess the expected benefits of moving towards coupled data assimilation. We implement an incremental 4D-Var system within an idealised single column atmosphere-ocean model. The system has the capability to run both strongly and weakly coupled assimilations as well as uncoupled atmosphere or ocean only assimilations, thus allowing a systematic comparison of the different strategies for treating the coupled data assimilation problem. We present results from a series of identical twin experiments devised to investigate the behaviour and sensitivities of the different approaches. Overall, our study demonstrates the potential benefits that may be expected from coupled data assimilation. When compared to uncoupled initialisation, coupled assimilation is able to produce more balanced initial analysis fields, thus reducing initialisation shock and its impact on the subsequent forecast. Single observation experiments demonstrate how coupled assimilation systems are able to pass information between the atmosphere and ocean and therefore use near-surface data to greater effect. We show that much of this benefit may also be gained from a weakly coupled assimilation system, but that this can be sensitive to the parameters used in the assimilation.

Asset liability modelling and pension schemes: the application of robust optimization to USS

This paper uses a novel numerical optimization technique - robust optimization - that is well suited to solving the asset-liability management (ALM) problem for pension schemes. It requires the estimation of fewer stochastic parameters, reduces estimation risk and adopts a prudent approach to asset allocation. This study is the first to apply it to a real-world pension scheme, and the first ALM model of a pension scheme to maximise the Sharpe ratio. We disaggregate pension liabilities into three components - active members, deferred members and pensioners, and transform the optimal asset allocation into the scheme’s projected contribution rate. The robust optimization model is extended to include liabilities and used to derive optimal investment policies for the Universities Superannuation Scheme (USS), benchmarked against the Sharpe and Tint, Bayes-Stein, and Black-Litterman models as well as the actual USS investment decisions. Over a 144 month out-of-sample period robust optimization is superior to the four benchmarks across 20 performance criteria, and has a remarkably stable asset allocation – essentially fix-mix. These conclusions are supported by six robustness checks.

On the structure of infinity-harmonic maps

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

An Ensemble Smoother with Error Estimates

A smoother introduced earlier by van Leeuwen and Evensen is applied to a problem in which real obser vations are used in an area with strongly nonlinear dynamics. The derivation is new , but it resembles an earlier derivation by van Leeuwen and Evensen. Again a Bayesian view is taken in which the prior probability density of the model and the probability density of the obser vations are combined to for m a posterior density . The mean and the covariance of this density give the variance-minimizing model evolution and its errors. The assumption is made that the prior probability density is a Gaussian, leading to a linear update equation. Critical evaluation shows when the assumption is justified. This also sheds light on why Kalman filters, in which the same ap- proximation is made, work for nonlinear models. By reference to the derivation, the impact of model and obser vational biases on the equations is discussed, and it is shown that Bayes’ s for mulation can still be used. A practical advantage of the ensemble smoother is that no adjoint equations have to be integrated and that error estimates are easily obtained. The present application shows that for process studies a smoother will give superior results compared to a filter , not only owing to the smooth transitions at obser vation points, but also because the origin of features can be followed back in time. Also its preference over a strong-constraint method is highlighted. Further more, it is argued that the proposed smoother is more efficient than gradient descent methods or than the representer method when error estimates are taken into account

A joint state and parameter estimation scheme for nonlinear dynamical systems

We present a novel algorithm for concurrent model state and parameter estimation in nonlinear dynamical systems. The new scheme uses ideas from three dimensional variational data assimilation (3D-Var) and the extended Kalman filter (EKF) together with the technique of state augmentation to estimate uncertain model parameters alongside the model state variables in a sequential filtering system. The method is relatively simple to implement and computationally inexpensive to run for large systems with relatively few parameters. We demonstrate the efficacy of the method via a series of identical twin experiments with three simple dynamical system models. The scheme is able to recover the parameter values to a good level of accuracy, even when observational data are noisy. We expect this new technique to be easily transferable to much larger models.

An LDA and probability-based classifier for the diagnosis of Alzheimer's Disease from structural MRI

In this paper a custom classification algorithm based on linear discriminant analysis and probability-based weights is implemented and applied to the hippocampus measurements of structural magnetic resonance images from healthy subjects and Alzheimer’s Disease sufferers; and then attempts to diagnose them as accurately as possible. The classifier works by classifying each measurement of a hippocampal volume as healthy controlsized or Alzheimer’s Disease-sized, these new features are then weighted and used to classify the subject as a healthy control or suffering from Alzheimer’s Disease. The preliminary results obtained reach an accuracy of 85.8% and this is a similar accuracy to state-of-the-art methods such as a Naive Bayes classifier and a Support Vector Machine. An advantage of the method proposed in this paper over the aforementioned state of the art classifiers is the descriptive ability of the classifications it produces. The descriptive model can be of great help to aid a doctor in the diagnosis of Alzheimer’s Disease, or even further the understand of how Alzheimer’s Disease affects the hippocampus.

Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalizing constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes’ factors for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.