Topics in data assimilation: Stochastic processes

Stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences. One application is data assimilation, i.e., the combination of model output with observations to form the best picture of the system under study. For any given quantity to be estimated, the relative weights of the model and the data will be adjusted according to estimated model and data error statistics, so implementation of any data assimilation scheme will require some assumption about errors, which are considered to be random. For dynamical models, some assumption about the evolution of errors will be needed. Stochastic models are also applied in studies of predictability.

The formal theory of stochastic processes was well developed in the last half of the twentieth century. One consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification. In some cases the rules of ordinary calculus must be modified.

The formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists. The purpose of this article is to provide an informal introduction to the relevant theory, and to point out those situations in which that theory must be applied in order to model random processes correctly.     

One-dimensional non-nearest-neighbor random walks in the presence of traps

A one-dimensional lattice random walk in the presence ofm equally spaced traps is considered. The step length distribution is a symmetric exponential. An explicit analytic expression is obtained for the probability that the random walk will be trapped at thejth trapping site.     

The Geometry of the Newton Method on Non-Compact Lie Groups

An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is non-compact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. In this paper the invariant structure of a Lie group is exploited to provide a strong interpretation of a Newton iteration on a general Lie group. The paper unifies several previous algorithms proposed in the literature in a single theoretical framework. Local asymptotic quadratic convergence is proved for the algorithms considered.     

Fluid and diffusion limits for transient sojourn times of processor sharing queues with time varying rates

We provide an approximate analysis of the transient sojourn time for a processor sharing queue with time varying arrival and service rates, where the load can vary over time, including periods of overload. Using the same asymptotic technique as uniform acceleration as demonstrated in [12] and [13], we obtain fluid and diffusion limits for the sojourn time of the M_t/M_t/1 processor-sharing queue. Our analysis is enabled by the introduction of a “virtual customer” which differs from the notion of a “tagged customer” in that the former has no effect on the processing time of the other customers in the system. Our analysis generalizes to non-exponential service and interarrival times, when the fluid and diffusion limits for the queueing process are known.     

Nonaqueous diazotization of 4-amino-3-arylisothiazole-5-carboxylate esters

4-Amino-3-arylisothiazole-5-carboxylate esters are converted to the corresponding desamino, chloro, bromo, and iodo esters by processes involving nonaqueous diazotization. These related procedures open up the possibility for the conversion of readily available heterocyclic ortho-amino esters and nitriles to related heterocyclic derivatives, which are not easily obtained by alternate routes.     

A short and efficient synthesis of grisan

A three-step synthesis of grisan (spiro[benzofuran-2(3H)-1′-cyclohexane]), using methoxymethoxybenzene and 1-chloromethylcyclohexene as starting materials, is described.     

Acetylene weak bands at 2.5 μm from intracavity Cr:ZnSe laser absorption observed with time-resolved Fourier transform spectroscopy

The spectral dynamics of a mid-infrared multimode Cr(2+):ZnSe laser located in a vacuum sealed chamber containing acetylene at low pressure is analyzed by a stepping-mode high-resolution time-resolved Fourier transform interferometer. Doppler-limited absorption spectra of C(2)H(2) in natural isotopic abundance are recorded around 4000 cm(-1) with kilometric absorption path lengths and sensitivities better than 3 10(-8) cm(-1). Two cold bands are newly identified and assigned to the ν(1)+ν(4) (1) and ν(3)+ν(5) (1) transitions of (12)C(13)CH(2). The ν(1)+ν(5) (1) band of (12)C(2)HD and fourteen (12)C(2)H(2) bands are observed, among which for the first time ν(2)+2ν(4) (2)+ν(5) (-1).     

Latex particles bearing hydrophilic grafted hairs with controlled chain length and functionality synthesized by reversible addition–fragmentation chain transfer

A method is described for synthesizing latex particles with anchored hairs by the grafting of hydrophilic chains, synthesized by reversible addition–fragmentation chain transfer, onto functionalized latex particles. These have the potential to bind biologically active species. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1188–1195, 2003     

Focused ion beam Moiré method

A focused ion beam (FIB) Moiré method is proposed to measure the in-plane deformation of object in a micrometer scale. The FIB Moiré is generated by the interference between a prepared specimen grating and FIB raster scan lines. The principle of the FIB Moiré is described. The sensitivity and accuracy of deformation measurement are discussed in detail. Several specimen gratings with 0.14 and 0.20 μm spacing are used to generate FIB Moiré patterns. The FIB Moiré method is successfully used to measure the residual deformation in a micro-electro-mechanical system structure after removing the SiO₂ sacrificial layer with a 5000 lines/mm grating. The results demonstrate the feasibility of this method.