A study on orthogonality sampling.

The goal of this paper is to study and further develop the orthogonality sampling or stationary waves algorithm for the detection of the location and shape of objects from the far field pattern of scattered waves in electromagnetics or acoustics. Orthogonality sampling can be seen as a special beam forming algorithm with some links to the point source method and to the linear sampling method. The basic idea of orthogonality sampling is to sample the space under consideration by calculating scalar products of the measured far field pattern , with a test function for all y in a subset Q of the space , m = 2, 3. The way in which this is carried out is important to extract the information which the scattered fields contain. The theoretical foundation of orthogonality sampling is only partly resolved, and the goal of this work is to initiate further research by numerical demonstration of the high potential of the approach. We implement the method for a two-dimensional setting for the Helmholtz equation, which represents electromagnetic scattering when the setup is independent of the third coordinate. We show reconstructions of the location and shape of objects from measurements of the scattered field for one or several directions of incidence and one or many frequencies or wave numbers, respectively. In particular, we visualize the indicator function both with the Dirichlet and Neumann boundary condition and for complicated inhomogeneous media.

The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis

It is shown that Bretherton's view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home-base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home-bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self-propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home-bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home-bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV-thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super-modal growth rate for a short period. A phase-locking configuration (and normal-mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft.

The representation of conditional expectations of non-Gaussian noise

For Wiener spaces conditional expectations and $L^{2}$-martingales w.r.t. the natural filtration have a natural representation in terms of chaos expansion. In this note an extension to larger classes of processes is discussed. In particular, it is pointed out that orthogonality of the chaos expansion is not required.

Northern hemisphere winter atmospheric climate: modes of natural variability and climate change

Under anthropogenic climate change it is possible that the increased radiative forcing and associated changes in mean climate may affect the “dynamical equilibrium” of the climate system; leading to a change in the relative dominance of different modes of natural variability, the characteristics of their patterns or their behavior in the time domain. Here we use multi-century integrations of version three of the Hadley Centre atmosphere model coupled to a mixed layer ocean to examine potential changes in atmosphere-surface ocean modes of variability. After first evaluating the simulated modes of Northern Hemisphere winter surface temperature and geopotential height against observations, we examine their behavior under an idealized equilibrium doubling of atmospheric CO2. We find no significant changes in the order of dominance, the spatial patterns or the associated time series of the modes. Having established that the dynamic equilibrium is preserved in the model on doubling of CO2, we go on to examine the temperature pattern of mean climate change in terms of the modes of variability; the motivation being that the pattern of change might be explicable in terms of changes in the amount of time the system resides in a particular mode. In addition, if the two are closely related, we might be able to assess the relative credibility of different spatial patterns of climate change from different models (or model versions) by assessing their representation of variability. Significant shifts do appear to occur in the mean position of residence when examining a truncated set of the leading order modes. However, on examining the complete spectrum of modes, it is found that the mean climate change pattern is close to orthogonal to all of the modes and the large shifts are a manifestation of this orthogonality. The results suggest that care should be exercised in using a truncated set of variability EOFs to evaluate climate change signals.

Backward elimination model construction for regression and classification using leave-one-out criteria

A fundamental principle in practical nonlinear data modeling is the parsimonious principle of constructing the minimal model that explains the training data well. Leave-one-out (LOO) cross validation is often used to estimate generalization errors by choosing amongst different network architectures (M. Stone, "Cross validatory choice and assessment of statistical predictions", J. R. Stast. Soc., Ser. B, 36, pp. 117-147, 1974). Based upon the minimization of LOO criteria of either the mean squares of LOO errors or the LOO misclassification rate respectively, we present two backward elimination algorithms as model post-processing procedures for regression and classification problems. The proposed backward elimination procedures exploit an orthogonalization procedure to enable the orthogonality between the subspace as spanned by the pruned model and the deleted regressor. Subsequently, it is shown that the LOO criteria used in both algorithms can be calculated via some analytic recursive formula, as derived in this contribution, without actually splitting the estimation data set so as to reduce computational expense. Compared to most other model construction methods, the proposed algorithms are advantageous in several aspects; (i) There are no tuning parameters to be optimized through an extra validation data set; (ii) The procedure is fully automatic without an additional stopping criteria; and (iii) The model structure selection is directly based on model generalization performance. The illustrative examples on regression and classification are used to demonstrate that the proposed algorithms are viable post-processing methods to prune a model to gain extra sparsity and improved generalization.

Multiband OFDM modulation and demodulation for ultra wideband communications

This chapter considers the Multiband Orthogonal Frequency Division Multiplexing (MB- OFDM) modulation and demodulation with the intention to optimize the Ultra-Wideband (UWB) system performance. OFDM is a type of multicarrier modulation and becomes the most important aspect for the MB-OFDM system performance. It is also a low cost digital signal component efficiently using Fast Fourier Transform (FFT) algorithm to implement the multicarrier orthogonality. Within the MB-OFDM approach, the OFDM modulation is employed in each 528 MHz wide band to transmit the data across the different bands while also using the frequency hopping technique across different bands. Each parallel bit stream can be mapped onto one of the OFDM subcarriers. Quadrature Phase Shift Keying (QPSK) and Dual Carrier Modulation (DCM) are currently used as the modulation schemes for MB-OFDM in the ECMA-368 defined UWB radio platform. A dual QPSK soft-demapper is suitable for ECMA-368 that exploits the inherent Time-Domain Spreading (TDS) and guard symbol subcarrier diversity to improve the receiver performance, yet merges decoding operations together to minimize hardware and power requirements. There are several methods to demap the DCM, which are soft bit demapping, Maximum Likelihood (ML) soft bit demapping, and Log Likelihood Ratio (LLR) demapping. The Channel State Information (CSI) aided scheme coupled with the band hopping information is used as a further technique to improve the DCM demapping performance. ECMA-368 offers up to 480 Mb/s instantaneous bit rate to the Medium Access Control (MAC) layer, but depending on radio channel conditions dropped packets unfortunately result in a lower throughput. An alternative high data rate modulation scheme termed Dual Circular 32-QAM that fits within the configuration of the current standard increasing system throughput thus maintaining the high rate throughput even with a moderate level of dropped packets.

Elastic-net prefiltering for two-class classification

A two-stage linear-in-the-parameter model construction algorithm is proposed aimed at noisy two-class classification problems. The purpose of the first stage is to produce a prefiltered signal that is used as the desired output for the second stage which constructs a sparse linear-in-the-parameter classifier. The prefiltering stage is a two-level process aimed at maximizing a model's generalization capability, in which a new elastic-net model identification algorithm using singular value decomposition is employed at the lower level, and then, two regularization parameters are optimized using a particle-swarm-optimization algorithm at the upper level by minimizing the leave-one-out (LOO) misclassification rate. It is shown that the LOO misclassification rate based on the resultant prefiltered signal can be analytically computed without splitting the data set, and the associated computational cost is minimal due to orthogonality. The second stage of sparse classifier construction is based on orthogonal forward regression with the D-optimality algorithm. Extensive simulations of this approach for noisy data sets illustrate the competitiveness of this approach to classification of noisy data problems.

Scattered power from non-thermal, F-region plasma observed by EISCAT—evidence for coherent echoes?

Three rapid, poleward bursts of plasma flow, observed by the U.K.-POLAR EISCAT experiment, are studied in detail. In all three cases the large ion velocities (> 1 kms−1) are shown to drive the ion velocity distribution into a non-Maxwellian form, identified by the characteristic shape of the observed spectra and the fact that analysis of the spectra with the assumption of a Maxwellian distribution leads to excessive rises in apparent ion temperature, and an anticorrelation of apparent electron and ion temperatures. For all three periods the total scattered power is shown to rise with apparent ion temperature by up to 6 dB more than is expected for an isotropic Maxwellian plasma of constant density and by an even larger factor than that expected for non-thermal plasma. The anomalous increases in power are only observed at the lower altitudes (< 300 km). At greater altitudes the rise in power is roughly consistent with that simulated numerically for homogeneous, anisotropic, non-Maxwellian plasma of constant density, viewed using the U.K.-POLAR aspect angle. The spectra at times of anomalously high power are found to be asymmetric, showing an enhancement near the downward Doppler-shifted ion-acoustic frequency. Although it is not possible to eliminate completely rapid plasma density fluctuations as a cause of these power increases, such effects cannot explain the observed spectra and the correlation of power and apparent ion temperature without an unlikely set of coincidences. The observations are made along a beam direction which is as much as 16.5° from orthogonality with the geomagnetic field. Nevertheless, some form of coherent-like echo contamination of the incoherent scatter spectrum is the most satisfactory explanation of these data.

Evaluation of a modern-analogue methodology for reconstructing Australian palaeoclimate from pollen

Quantitative palaeoclimate reconstructions are widely used to evaluate climatemodel performance. Here, as part of an effort to provide such a data set for Australia, we examine the impact of analytical decisions and sampling assumptions on modern-analogue reconstructions using a continent-wide pollen data set. There is a high degree of correlation between temperature variables in the modern climate of Australia, but there is sufficient orthogonality in the variations of precipitation, summer and winter temperature and plant–available moisture to allow independent reconstructions of these four variables to be made. The method of analogue selection does not affect the reconstructions, although bootstrap resampling provides a more reliable technique for obtaining robust measures of uncertainty. The number of analogues used affects the quality of the reconstructions: the most robust reconstructions are obtained using 5 analogues. The quality of reconstructions based on post-1850 CE pollen samples differ little from those using samples from between 1450 and 1849 CE, showing that European post settlement modification of vegetation has no impact on the fidelity of the reconstructions although it substantially increases the availability of potential analogues. Reconstructions based on core top samples are more realistic than those using surface samples, but only using core top samples would substantially reduce the number of available analogues and therefore increases the uncertainty of the reconstructions. Spatial and/or temporal averaging of pollen assemblages prior to analysis negatively affects the subsequent reconstructions for some variables and increases the associated uncertainties. In addition, the quality of the reconstructions is affected by the degree of spatial smoothing of the original climate data, with the best reconstructions obtained using climate data froma 0.5° resolution grid, which corresponds to the typical size of the pollen catchment. This study provides a methodology that can be used to provide reliable palaeoclimate reconstructions for Australia, which will fill in a major gap in the data sets used to evaluate climate models.

Comparison of terrain following and cut cell grids using a non-hydrostatic model

Terrain following coordinates are widely used in operational models but the cut cell method has been proposed as an alternative that can more accurately represent atmospheric dynamics over steep orography. Because the type of grid is usually chosen during model implementation, it becomes necessary to use different models to compare the accuracy of different grids. In contrast, here a C-grid finite volume model enables a like-for-like comparison of terrain following and cut cell grids. A series of standard two-dimensional tests using idealised terrain are performed: tracer advection in a prescribed horizontal velocity field, a test starting from resting initial conditions, and orographically induced gravity waves described by nonhydrostatic dynamics. In addition, three new tests are formulated: a more challenging resting atmosphere case, and two new advection tests having a velocity field that is everywhere tangential to the terrain following coordinate surfaces. These new tests present a challenge on cut cell grids. The results of the advection tests demonstrate that accuracy depends primarily upon alignment of the flow with the grid rather than grid orthogonality. A resting atmosphere is well-maintained on all grids. In the gravity waves test, results on all grids are in good agreement with existing results from the literature, although terrain following velocity fields lead to errors on cut cell grids. Due to semi-implicit timestepping and an upwind-biased, explicit advection scheme, there are no timestep restrictions associated with small cut cells. We do not find the significant advantages of cut cells or smoothed coordinates that other authors find.