Preamble 1 Variational assimilation 2 Interpretation 3 Implementation 4 The varieties of linear and nonlinear estimation 5 The ocean and the atmosphere 6 Ill-posed forecasting problems References Appendix A Computing exercises Appendix B Euler-Lagrange equations for a numerical weather prediction model Index

/pdf/inverse-modeling-of-the-ocean-and-atmosphere-x50aix99i6.pdf

Inverse Modeling of the Ocean and Atmosphere

Summary Image warping is a transformation which maps all positions in one image plane to positions in a second plane. It arises in many image analysis problems, whether in order to remove optical distortions introduced by a camera or a particular viewing perspective, to register an image with a map or template, or to align two or more images. The choice of warp is a compromise between a smooth distortion and one which achieves a good match. Smoothness can be ensured by assuming a parametric form for the warp or by constraining it using differential equations. Matching can be specified by points to be brought into alignment, by local measures of correlation between images, or by the coincidence of edges. Parametric and non-parametric approaches to warping, and matching criteria, are reviewed.

/pdf/a-review-of-image-warping-methods-2zebe4xgl2.pdf

A review of image-warping methods

The cloud, aerosol and precipitation spectrometer: a new instrument for cloud investigations

Variational approaches to image motion segmentation has been an active field of study in image processing and computer vision for two decades. We present a short overview over basic estimation schemes and report in more detail recent modifications and applications to fluid flow estimation. Key properties of these approaches are illustrated by numerical examples. We outline promising research directions and point out the potential of variational techniques in combination with correlation-based PIV methods, for improving the consistency of fluid flow estimation and simulation.

/pdf/variational-fluid-flow-measurements-from-image-sequences-2s3ciknw7s.pdf

Variational fluid flow measurements from image sequences: synopsis and perspectives

During the last 20 years data assimilation has gradually reached a mature center stage position at both Numerical Weather Prediction centers as well as being at the center of activities at many federal research institutes as well as at many universities.

/pdf/data-assimilation-for-numerical-weather-prediction-a-review-7zi7vp7cna.pdf

Data Assimilation for Numerical Weather Prediction: A Review

A vector spline approximation

We consider two Riemannian geometries for the manifold $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) of all $$m\times n$$ m × n matrices of rank $$p$$ p . The geometries are induced on $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) by viewing it as the base manifold of the submersion $$\pi :(M,N)\mapsto MN^\mathrm{T}$$ ? : ( M , N ) ? M N T , selecting an adequate Riemannian metric on the total space, and turning $$\pi $$ ? into a Riemannian submersion. The theory of Riemannian submersions, an important tool in Riemannian geometry, makes it possible to obtain expressions for fundamental geometric objects on $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) and to formulate the Riemannian Newton methods on $${\mathcal{M }(p,m\times n)}$$ M ( p , m × n ) induced by these two geometries. The Riemannian Newton methods admit a stronger and more streamlined convergence analysis than the Euclidean counterpart, and the computational overhead due to the Riemannian geometric machinery is shown to be mild. Potential applications include low-rank matrix completion and other low-rank matrix approximation problems.

/pdf/two-newton-methods-on-the-manifold-of-fixed-rank-matrices-2diy68da3b.pdf

Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries

Counting loss due to coincidences limit the efficiency of the retriggerable particle counters. When the particle rate increases, the counted rate rapidly reaches a maximum so that it becomes impossible to precisely estimate the actual value. On the contrary, the activity time of the probe increases continuously and is a more efficient parameter for high concentration measurements. Exact relationships between these parameters and the actual particle rate are given here for particles randomly dispersed according to a Poisson distribution. Relations between activity and counted particle rate are also useful for the evaluation of the heterogeneities in the spatial distribution of the particles.

Coincidence and Dead-Time Corrections for Particle Counters. Part I: A General Mathematical Formalism

We consider the general meteorological data assimilation problem including model error estimate. For a linear model, the solution is obtained in the space of observations using a gradient method. The gradient is given by a backward integration of the adjoint equations and a forward integration of the direct equations forced by the adjoint variable. With a computational cost of the same order as for the exact model case, the algorithm may apply for large-dimension systems. The method is illustrated by a numerical example and some indications on the condition number of the system are given.

Solution approchée pour un problème d'assimilation de données météorologiques avec prise en compte de l'erreur de modèle

We present a new vector approximation based on general spline function theory. The smoothing functional involves the divergence ( divV = ∂ x u + ∂ y v ) and the rotational ( rotV = ∂ x v – ∂ y u ) of the approximated vector field V = ( u, v ). The method can be applied for the restitution of velocity fields from observed data, in fluid mechanics and especially in geophysical fluid flows (horizontal wind fields in meteorology, oceanic currents, etc.).

A vector spline approximation with application to meteorology

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