Bayesian data assimilation in shape registration

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions\ud for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum p0 and the reparameterisation vector field v, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed MCMC methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem

Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups

In the framework of large deformation diffeomorphic metric mapping (LDDMM), we develop a multi-scale theory for the diffeomorphism group based on previous works. The purpose of the paper is (1) to develop in details a variational approach for multi-scale analysis of diffeomorphisms, (2) to generalise to several scales the semidirect product representation and (3) to illustrate the resulting diffeomorphic decomposition on synthetic and real images. We also show that the approaches presented in other papers and the mixture of kernels are equivalent.Comment: 21 pages, revised version without section on evaluatio

A reduced-order strategy for 4D-Var data assimilation

This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EO F analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a mul tivariate background error covariance matrix $\textbf{B}_r$, and an important decrease of the computational burden o f the method, due to the drastic reduction of the dimension of the control space. % An illustration of the feasibility and the effectiveness of this method is given in the academic framework of twin experiments for a model of the equatorial Pacific ocean. It is shown that the multivariate aspect of $\textbf{B}_r$ brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method

RAMA : the Research Moored Array for African–Asian–Australian Monsoon Analysis and Prediction

Author Posting. © American Meteorological Society, 2009. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Bulletin of the American Meteorological Society 90 (2009):459-480, doi:10.1175/2008BAMS2608.1.The Indian Ocean is unique among the three tropical ocean basins in that it is blocked at 25°N by the Asian landmass. Seasonal heating and cooling of the land sets the stage for dramatic monsoon wind reversals, strong ocean–atmosphere interactions, and intense seasonal rains over the Indian subcontinent, Southeast Asia, East Africa, and Australia. Recurrence of these monsoon rains is critical to agricultural production that supports a third of the world's population. The Indian Ocean also remotely influences the evolution of El Niño–Southern Oscillation (ENSO), the North Atlantic Oscillation (NAO), North American weather, and hurricane activity. Despite its importance in the regional and global climate system though, the Indian Ocean is the most poorly observed and least well understood of the three tropical oceans. This article describes the Research Moored Array for African–Asian–Australian Monsoon Analysis and Prediction (RAMA), a new observational network designed to address outstanding scientific questions related to Indian Ocean variability and the monsoons. RAMA is a multinationally supported element of the Indian Ocean Observing System (IndOOS), a combination of complementary satellite and in situ measurement platforms for climate research and forecasting. The article discusses the scientific rationale, design criteria, and implementation of the array. Initial RAMA data are presented to illustrate how they contribute to improved documentation and understanding of phenomena in the region. Applications of the data for societal benefit are also described

Un-reduction

This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally concieved for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.Comment: 25 pages, revised versio

Supplement to RAMA : the Research Moored Array for African-Asian-Australian Monsoon Analysis and Prediction

Author Posting. © American Meteorological Society, 2009. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Bulletin of the American Meteorological Society 90 (2009): ES5-ES8, doi:10.1175/2008BAMS2608.2

Invariant higher-order variational problems II

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome

Cirene : air-sea iInteractions in the Seychelles-Chagos thermocline ridge region

Author Posting. © American Meteorological Society, 2009. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Bulletin of the American Meteorological Society 90 (2009): 1337-1350, doi:10.1175/2008BAMS2499.1.The Vasco—Cirene program ex-plores how strong air—sea inter-actions promoted by the shallow thermocline and high sea surface temperature in the Seychelles—Chagos thermocline ridge results in marked variability at synoptic, intraseasonal, and interannual time scales. The Cirene oceano-graphic cruise collected oceanic, atmospheric, and air—sea flux observations in this region in Jan-uary—February 2007. The contem-poraneous Vasco field experiment complemented these measure-ments with balloon deployments from the Seychelles. Cirene also contributed to the development of the Indian Ocean observing system via deployment of a moor-ing and 12 Argo profilers. Unusual conditions prevailed in the Indian Ocean during Janu-ary and February 2007, following the Indian Ocean dipole climate anomaly of late 2006. Cirene measurements show that the Seychelles—Chagos thermocline ridge had higher-than-usual heat content with subsurface anomalies up to 7°C. The ocean surface was warmer and fresher than average, and unusual eastward currents prevailed down to 800 m. These anomalous conditions had a major impact on tuna fishing in early 2007. Our dataset also sampled the genesis and maturation of Tropical Cyclone Dora, including high surface temperatures and a strong diurnal cycle before the cyclone, followed by a 1.5°C cool-ing over 10 days. Balloonborne instruments sampled the surface and boundary layer dynamics of Dora. We observed small-scale structures like dry-air layers in the atmosphere and diurnal warm layers in the near-surface ocean. The Cirene data will quantify the impact of these finescale features on the upper-ocean heat budget and atmospheric deep convection.CNES funded the Vasco part of the experiment; INSU funded the Cirene part. R/V Suroît is an Ifremer ship. The contributions from ODU, WHOI, and FOI (Sweden) are supported by the National Science Foundation under Grant Number 0525657. The participation of the University of Miami group was funded though NASA (NNG04HZ33C). PMEL participation was supported through NOAA’s Office of Climate Observation