Spectral Basis Functions Research Articles

A spectral element shallow water model on spherical geodesic grids

SUMMARY The spectral element method for the two-dimensional shallow water equations on the sphere is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements obtained from the generalized icosahedral grid introduced previously (Giraldo FX. Lagrange‐ Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 2000; 160: 336‐368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear. This paper represents a departure from previously published work on solving the shallow water equations on the sphere in that the equations are all written, discretized, and solved in three-dimensional Cartesian space. Because the equations are written in a three-dimensional Cartesian co-ordinate system, the algorithm simplifies into the integration of surface elements on the sphere from the fully three-dimensional equations. A mapping (Song Ch, Wolf JP. The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion. International Journal for Numerical Methods in Engineering 1999; 45: 1403‐1431) which simplifies these computations is described and is shown to contain the Eulerian version of the method introduced previously by Giraldo (Journal of Computational Physics 2000; 160: 336‐368) for the special case of triangular elements. The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapping takes into account the curvature of the high-order spectral elements, thereby allowing the elements to lie entirely on the surface of the sphere. In addition, using this mapping simplifies all of the three-dimensional spectral-type finite element surface integrals because any of the typical two-dimensional planar finite element or spectral element basis functions found in any textbook (for example, Huebner et al. The Finite Element Method for Engineers. Wiley, New York, 1995; Karniadakis GE, Sherwin SJ. Spectral:hp Element Methods for CFD. Oxford University ( ˘

Spectral representation of neutral landscapes

Pattern in ecological landscapes is often the result of different processes operating at different scales. Neutral landscape models were introduced in landscape ecology to differentiate patterns that are the result of simple random processes from patterns that arise from more complex ecological processes. Recent studies have used increasingly complex neutral models that incorporate contagion and other constraints on random patterns, as well as using neutral landscapes as input to spatial simulation models. Here, I consider a common mathematical framework based on spectral transforms that represents all neutral landscape models in terms of sets of spectral basis functions. Fractal and multi-fractal models are considered, as well as models with multiple scaling regions and anisotropy. All of the models considered are shown to be variations on a basic theme: a scaling relation between frequency and amplitude of spectral components. Two example landscapes examined showed long-range correlations (distances up to 1000 km) consistent with fractal scaling.

Data-driven speech analysis for ASR

A typical large vocabulary automatic speech recognition (ASR) system consists of three main components: (1) the feature extraction, (2) the pattern classification, and (3) the language modeling. Replacing hardwired prior knowledge in the pattern classification and language modeling modules by the knowledge derived from the data turned out to be one of most significant advances in ASR research in the past two decades. However, the speech analysis module so far resisted the recent data-oriented revolution and is typically built on textbook knowledge of speech production and perception. Since it is believed that speech was optimized by millenia of human evolution to fit properties of human speech perception, deriving the speech processing knowledge from speech data may make some sense. The work describes some attempts in this direction. The linear discriminant analysis is used to learn about structure of speech signal to derive optimized spectral basis functions and filters (replacing conventional cosines of the cepstral analysis and conventional delta filters for deriving dynamic features) in processing the time-frequency plane of the speech signal. [Work supported by the Department of Defense and by the National Science Foundation.]

Variational discrete variable representation

In developing a pseudospectral transform between a nondirect product basis of spherical harmonics and a direct product grid, Corey and Lemoine [J. Chem. Phys. 97, 4115 (1992)] generalized the Fourier method of Kosloff and the discrete variable representation (DVR) of Light by introducing more grid points than spectral basis functions. Assuming that the potential energy matrix is diagonal on the grid destroys the variational principle in the Fourier and DVR methods. In the present article we (1) demonstrate that the extra grid points in the generalized discrete variable representation (GDVR) act as dealiasing functions that restore the variational principle and make a pseudospectral calculation equivalent to a purely spectral one, (2) describe the general principles for extending the GDVR to other nondirect product spectral bases of orthogonal polynomials, and (3) establish the relation between the GDVR and the least squares method exploited in the pseudospectral electronic structure and adiabatic pseudospectral bound state calculations of Friesner and collaborators.

Vertical Spectral Representation in Primitive Equation Models of the Atmosphere

Attempts to represent the vertical structure in primitive equation models of the atmosphere with the spectral method have been unsuccessful to date. The linear stability analysis of Francis showed that small time steps were required for computational stability near the upper boundary with a vertical spectral method using Laguerre polynomials. Machenhauer and Daley used Legendre polynomials in their vertical spectral representation and found it necessary to use an artificial constraint to force temperature to zero when pressure was zero to control the upper-level horizontal velocities. This ad hoc correction is undesirable, and an analysis that shows such a correction is unnecessary is presented. By formulating the model in terms of velocity and geopotential and then using the hydrostatic equation to calculate temperature from geopotential, temperature is necessarily zero when pressure is zero. This strategy works provided the multiplicative inverse of the first vertical derivative of the vertical basis functions approaches zero more slowly than pressure. The authors applied this technique to the dry-adiabatic primitive equations on the equatorial β and tropical f planes. Vertical and horizontal normal modes were used as the spectral basis functions. The vertical modes are based on the vertical normal modes of Staniforth et al., and the horizontal modes are normal modes for the primitive equations on a β or f plane. The results show that the upper-level velocities do not necessarily increase, total energy is conserved, and kinetic energy is bounded. The authors found an upper-level temporal oscillation in the horizontal domain integral of the horizontal velocity components that is related to mass and velocity field imbalances in the initial conditions or introduced during the integration. Through nonlinear normal-mode initialization, the authors effectively removed the initial condition imbalance and reduced the amplitude of this oscillation. It is hypothesized that the vertical spectral representation makes the model more sensitive to initial condition imbalances, or it introduces imbalance during the integration through vertical spectral truncation. It is also found slow spectral convergence properties for our vertical basis functions. It is concluded that a desirable vertical basis set should have the following properties: 1) nearly uniform distribution of zeros and rapid spectral convergence; 2) vertical structure functions that are bounded at the upper boundary and a multiplicative inverse of the first derivative that goes to zero more slowly than pressure, and 3) expansions for derivatives of the basis functions that need to converge quickly.

Quantitative Determination of Sulfur-Oxygen Anion Concentrations in Aqueous Solution: Multicomponent Analysis of Attenuated Total Reflectance Infrared Spectra

The multicomponent analysis of FT-IR attenuated total internal reflectance spectra of mixtures of sulfur-oxygen anions provides an accurate noninvasive method for their quantitation. The use of spectral artifact basis functions obtained from the singular value decomposition of a difference spectra matrix results in a marked increase in accuracy. Further improvements in accuracy are attained with the use of multicomponent spectra as basis functions. The analysis of a series of solutions of sulfur-oxygen anions ranging in concentration from 0.0005 to 0.0323 M resulted in a root-mean-square error of 0.0002 M

Variability of high-resolution crop reflectance spectra

Abstract High-spectral-resolution reflectance spectra from ground and helicopter measurements of agricultural crops and soils were analyzed to determine spectral variability in the visible and near-infrared (0 4-2 38 μm), using a procedure previously applied to thermal infrared emittance spectra of the atmosphere and to reflectance spectra of soil samples. Five spectral basis functions were sufficient to describe separately the ground and helicopter data, six or, at most, seven are sufficient to describe the pooled data. Thus, five to seven relatively broad band measurements, together with basis functions developed in this analysis, are sufficient to describe the variability of both data sets, to within differences that are probably associated with the measurement process and instrument noise.

On the convergence of matrix elements in planar and waveguide problems

In many integral equation formulations, impedance matrix elements derived for a moment method solution may not be computable. This happens when the summations involved tend to diverge because of a poor choice of basis and testing functions. In cases amenable to the spectral representation, these phenomena can be predicted and avoided, as shown in the paper. The development for many planar structures is shown in a general manner and guidelines for the selection of basis and testing functions for the generation of convergent matrix elements are deduced. The guidelines are applied to the Galerkin formulation, where the decay rate of the spectral basis functions has a double effect on the integrand. Finally, the analysis is applied to waveguide problems, where the full three-dimensional spectral Green's dyad is used. These principles are worked out for transversal slots in waveguides, where divergence has been observed in the past. A stable formulation is then derived and results are presented.