A method for merging flow-dependent forecast error statistics from an ensemble with static statistics for use in high resolution variational data assimilation

The background error covariance matrix, B, is often used in variational data assimilation for numerical weather prediction as a static and hence poor approximation to the fully dynamic forecast error covariance matrix, Pf. In this paper the concept of an Ensemble Reduced Rank Kalman Filter (EnRRKF) is outlined. In the EnRRKF the forecast error statistics in a subspace defined by an ensemble of states forecast by the dynamic model are found. These statistics are merged in a formal way with the static statistics, which apply in the remainder of the space. The combined statistics may then be used in a variational data assimilation setting. It is hoped that the nonlinear error growth of small-scale weather systems will be accurately captured by the EnRRKF, to produce accurate analyses and ultimately improved forecasts of extreme events.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Approximations to the scattering of water waves by steep topography

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

The effects of environmental perturbation and measurement error on estimates of the shape parameter in the theta-logistic model of population regulation

The theta-logistic is a widely used generalisation of the logistic model of regulated biological processes which is used in particular to model population regulation. Then the parameter theta gives the shape of the relationship between per-capita population growth rate and population size. Estimation of theta from population counts is however subject to bias, particularly when there are measurement errors. Here we identify factors disposing towards accurate estimation of theta by simulation of populations regulated according to the theta-logistic model. Factors investigated were measurement error, environmental perturbation and length of time series. Large measurement errors bias estimates of theta towards zero. Where estimated theta is close to zero, the estimated annual return rate may help resolve whether this is due to bias. Environmental perturbations help yield unbiased estimates of theta. Where environmental perturbations are large, estimates of theta are likely to be reliable even when measurement errors are also large. By contrast where the environment is relatively constant, unbiased estimates of theta can only be obtained if populations are counted precisely Our results have practical conclusions for the design of long-term population surveys. Estimation of the precision of population counts would be valuable, and could be achieved in practice by repeating counts in at least some years. Increasing the length of time series beyond ten or 20 years yields only small benefits. if populations are measured with appropriate accuracy, given the level of environmental perturbation, unbiased estimates can be obtained from relatively short censuses. These conclusions are optimistic for estimation of theta. (C) 2008 Elsevier B.V All rights reserved.

Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

OFDM joint data detection and phase noise cancellation based on minimum mean square prediction error

This paper proposes a new iterative algorithm for orthogonal frequency division multiplexing (OFDM) joint data detection and phase noise (PHN) cancellation based on minimum mean square prediction error. We particularly highlight the relatively less studied problem of "overfitting" such that the iterative approach may converge to a trivial solution. Specifically, we apply a hard-decision procedure at every iterative step to overcome the overfitting. Moreover, compared with existing algorithms, a more accurate Pade approximation is used to represent the PHN, and finally a more robust and compact fast process based on Givens rotation is proposed to reduce the complexity to a practical level. Numerical Simulations are also given to verify the proposed algorithm. (C) 2008 Elsevier B.V. All rights reserved.

Unit-cell approximation for diblock−copolymer brushes grafted to spherical particles

This paper examines the equilibrium phase behavior of thin diblock-copolymer films tethered to a spherical core, using numerical self-consistent field theory (SCFT). The computational cost of the calculation is greatly reduced by implementing the unit-cell approximation (UCA) routinely used in the study of bulk systems. This provides a tremendous reduction in computational time, permitting us to map out the phase behavior more extensively and allowing us to consider far larger particles. The main consequence of the UCA is that it omits packing frustration, but evidently the effect is minor for large particles. On the other hand, when the particles are small, the UCA calculation can be readily followed up with the full SCFT, the comparison to which conveniently allows one to quantitatively assess the effect of packing frustration.

The inflation hedging characteristics of US and UK investments: a multi-factor error correction approach

Historic analysis of the inflation hedging properties of stocks produced anomalous results, with equities often appearing to offer a perverse hedge against inflation. This has been attributed to the impact of real and monetary shocks to the economy, which influence both inflation and asset returns. It has been argued that real estate should provide a better hedge: however, empirical results have been mixed. This paper explores the relationship between commercial real estate returns (from both private and public markets) and economic, fiscal and monetary factors and inflation for US and UK markets. Comparative analysis of general equity and small capitalisation stock returns in both markets is carried out. Inflation is subdivided into expected and unexpected components using different estimation techniques. The analyses are undertaken using long-run error correction techniques. In the long-run, once real and monetary variables are included, asset returns are positively linked to anticipated inflation but not to inflation shocks. Adjustment processes are, however, gradual and not within period. Real estate returns, particularly direct market returns, exhibit characteristics that differ from equities.

Implications of non-cylindrical flux ropes for magnetic cloud reconstruction techniques and the interpretation of double flux rope events

Magnetic clouds (MCs) are a subset of interplanetary coronal mass ejections (ICMEs) which exhibit signatures consistent with a magnetic flux rope structure. Techniques for reconstructing flux rope orientation from single-point in situ observations typically assume the flux rope is locally cylindrical, e.g., minimum variance analysis (MVA) and force-free flux rope (FFFR) fitting. In this study, we outline a non-cylindrical magnetic flux rope model, in which the flux rope radius and axial curvature can both vary along the length of the axis. This model is not necessarily intended to represent the global structure of MCs, but it can be used to quantify the error in MC reconstruction resulting from the cylindrical approximation. When the local flux rope axis is approximately perpendicular to the heliocentric radial direction, which is also the effective spacecraft trajectory through a magnetic cloud, the error in using cylindrical reconstruction methods is relatively small (≈ 10∘). However, as the local axis orientation becomes increasingly aligned with the radial direction, the spacecraft trajectory may pass close to the axis at two separate locations. This results in a magnetic field time series which deviates significantly from encounters with a force-free flux rope, and consequently the error in the axis orientation derived from cylindrical reconstructions can be as much as 90∘. Such two-axis encounters can result in an apparent ‘double flux rope’ signature in the magnetic field time series, sometimes observed in spacecraft data. Analysing each axis encounter independently produces reasonably accurate axis orientations with MVA, but larger errors with FFFR fitting.

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω 2 u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Nonlinear error dynamics for cycled data assimilation methods

We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ..., with a first guess given by the state propagated via a dynamical system model from time tk − 1 to time tk. In particular, for nonlinear dynamical systems that are Lipschitz continuous with respect to their initial states, we provide deterministic estimates for the development of the error ||ek|| := ||x(a)k − x(t)k|| between the estimated state x(a) and the true state x(t) over time. Clearly, observation error of size δ > 0 leads to an estimation error in every assimilation step. These errors can accumulate, if they are not (a) controlled in the reconstruction and (b) damped by the dynamical system under consideration. A data assimilation method is called stable, if the error in the estimate is bounded in time by some constant C. The key task of this work is to provide estimates for the error ||ek||, depending on the size δ of the observation error, the reconstruction operator Rα, the observation operator H and the Lipschitz constants K(1) and K(2) on the lower and higher modes of controlling the damping behaviour of the dynamics. We show that systems can be stabilized by choosing α sufficiently small, but the bound C will then depend on the data error δ in the form c||Rα||δ with some constant c. Since ||Rα|| → ∞ for α → 0, the constant might be large. Numerical examples for this behaviour in the nonlinear case are provided using a (low-dimensional) Lorenz '63 system.

Growth rate and vertical propagation of the initial error in baroclinic models

The present study investigates the growth of error in baroclinic waves. It is found that stable or neutral waves are particularly sensitive to errors in the initial condition. Short stable waves are mainly sensitive to phase errors and the ultra long waves to amplitude errors. Analysis simulation experiments have indicated that the amplitudes of the very long waves become usually too small in the free atmosphere, due to the sparse and very irregular distribution of upper air observations. This also applies to the four-dimensional data assimilation experiments, since the amplitudes of the very long waves are usually underpredicted. The numerical experiments reported here show that if the very long waves have these kinds of amplitude errors in the upper troposphere or lower stratosphere the error is rapidly propagated (within a day or two) to the surface and to the lower troposphere.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.