Showing papers on "Gaussian published in 1988"

A diffraction beam field expressed as the superposition of Gaussian beams

Abstract: The diffraction field of a Gaussian planar velocity distribution is a Gaussian beam function under the condition (ka)2≫1. This property makes a series of Gaussian functions attractive as a possible base function set. The new approach presented enables one to express any axisymmetric beam field in a simple analytical form—the superposition of Gaussian beams about the same axis but with beam waists of different sizes located at different positions along the axis. A computer optimization is used to evaluate the coefficients, as well as the beam waists and their positions. The extreme case of a piston radiator is used to test the approach. Good agreement between a ten‐term Gaussian beam solution and the results of numerical integration (or analytical solution on axis) is obtained throughout the beam field: in the farfield, the transition region, and the nearfield. Discrepancies exist only in the extreme nearfield (<0.1 times the Fresnel distance). For surface velocity distributions that are less discontinuous...

Digital Generation of Non‐Gaussian Stochastic Fields

Abstract: A method by which sample fields of a multidimensional non‐Gaussian homogeneous stochastic field can be generated is developed. The method first generates Gaussian sample fields and then maps them into non‐Gaussian sample fields with the aid of an iterative procedure. Numerical examples indicate that the procedure is very efficient and generated sample fields satisfy the target spectral density and probability distribution function accurately. The proposed method has a wide range of applicability to engineering problems involving stochastic fields where the Gaussian assumption is not appropriate.

Comparison of Bessel and Gaussian beams

Abstract: A comparison of beam divergence and power-transport efficiency is made between Gaussian and Bessel beams when both beams have the same initial total power and the same initial full width at half-maximum.

Identifying nonlinear difference equation and functional expansion representations : the fast orthogonal algorithm

Abstract: A method is presented for identifying functional expansion and difference equation representations for nonlinear systems. The method relies on an orthogonal approach which does not require explicit creation of orthogonal functions. This greatly reduces computing time, so that 15-fold increases in speed of estimating kernels or difference equation coefficients are readily obtainable, compared with a previous orthogonal technique. In addition, storage requirements are considerably diminished. A wide variety of input excitation, both random and deterministic, can be used, and the method is not limited to inputs which are Gaussian, white or lengthy. A model of the peripheral auditory system is simulated to show kernel measurement is free of artifacts using the present method, in contrast to the crosscorrelation approach.

Testing of the constrained regularization method of inverting Laplace transform on simulated very wide quasielastic light scattering autocorrelation functions

Abstract: Provencher's constrained regularization method of inverting the Laplace transform was tested on 7 decades wide simulated quasielastic light scattering (QELS) data. The standard method with integration and logarithmic grid was shown to undersmooth seriously theG(Γ) distribution in the region of largeΓ (small relaxation timeτ). The regularization can be considerably improved by switching the integration off. Then, smooth distributions of relaxation timeτ of the generalized exponential type are reproduced essentially correctly with a tendency to replace asymmetric peaks by more symmetric ones with shoulders (in the Gaussian distribution ofτ) or side peaks (in the Gaussian distribution of 1/τ) on the slow decrease sides. In distributions with singularities such as edges of histogram bins or delta functions, the coarse shape of the distribution is recovered essentially correctly, but smoothing of singularities causes a distortion of wide regions of the relaxation spectrum usually in the form of sinusoidal waves. The bias introduced by taking the square root of theg2 function was shown to worsen sometimes the CONTIN results considerably. Thus, the use of Provencher's CONTIN program with logarithmic grid and integration switched off is recommended for the analysis of very wide QELS autocorrelation curves.

A global numerical weather prediction model with variable resolution: Application to the shallow‐water equations

Abstract: We follow the approach suggested by F. Schmidt to implement a spectral global shallow-water model with variable resolution. A conformal mapping is built between the earth and a computational sphere and the equations are discretized on the latter using the standard spectral technique associated with a collocation (Gaussian) grid. We prove that the only non-trivial conformal mapping which exists between the two spheres is based on the transformation introduced by Schmidt, but the pole of the collocation grid has no longer to coincide with the pole of dilatation. We implement the technique in an explicit model, where only minor modifications to a uniform resolution model are needed. The semi-implicit scheme and the nonlinear normal mode initialization are proved to work satisfactorily. 24-hour forecasts show that the method is successful in dealing with the shallow-water equations and allow us to discuss the potential of the approach.

Non-Gaussian models for the statistics of scattered waves

Abstract: This paper addresses problems associated with the development of widely applicable non-Gaussian noise models, particularly with reference to the statistical properties of scattered waves. A combination of phenomenological arguments and exact solutions of specific scattering problems are used to elucidate the significance of one model—K-distributed noise—which has several attractive features and has already found many applications. A full statistical-mechanical formulation is developed for non-Gaussian compound Markov processes, with this model as a special case. The implications for numerical simulation of correlated non-Gaussian noise are explored and comparisons made with experimental data. A brief review of current applications of the K-distribution model is given.

Time delay estimation in unknown Gaussian spatially correlated noise

Abstract: A novel class of methods that estimate the difference in arrival time between signals corrupted by spatially correlated Gaussian noise sources of unknown cross correlation is presented. The methods are based on the idea of comparing the similarities between the two sensor measurements in higher-order spectrum domains (bispectrum) rather than in the cross-correlation domain. It is demonstrated that estimation techniques based on higher-order cumulants suppress the effect of correlated Gaussian noise sources and therefore exhibit improved performance over generalized cross-correlation methods. Results are reported for different types of signals, lengths of data records, and signal-to-noise ratios. >

A time-frequency formulation of optimum detection

Abstract: A time-frequency formulation is proposed for the optimum detection of Gaussian signals in white Gaussian noise. By choosing the Wigner-Ville distribution as the basic time-frequency tool, it is shown that the corresponding receivers generally take the form of a correlation between time-frequency structures, matching mathematical optimality with a physically meaningful interpretation. The case of low SNR is examined and various examples are considered: deterministic signal, Rayleigh fading signal, random jitter, and random time-varying channel. A general class of time-frequency receivers is proposed which admits as limiting cases different known structures, and its suboptimum performance is evaluated. Possible extensions to more elaborate situations (including parameter estimation) are mentioned. >

Objective quality control of observations using Bayesian methods. Theory, and a practical implementation

Abstract: This work attempts to provide a theoretical framework for the quality control of data from a large variety of types of observations, with different accuracies and reliabilities. Bayes' theorem is introduced, and is used in a simple example with Gaussian error distributions to derive the well-known formula for the combination of data with errors. A simple model is proposed whereby the error in each datum is either from a known Gaussian distribution, or a gross error, in which case the observation gives no useful information. Bayes' theorem is applied to this, and it is shown that usual operational practice, which is to reject outlying data and to treat the rest as if their errors are Gaussian, is a reasonable approximation to the correct Bayesian analysis. Appropriate rejection criteria are derived in terms of the observational error and the prior probability of a gross error. These ideas have been implemented in a computer program to check pressure, wind, temperature and position data from ships, weather ships, buoys and coastal synoptic reports. Historical information on the accuracies and reliabilities of various classifications of observation is used to provide prior estimates of observational errors and the prior probabilities of gross error. The latter are then updated in the light of information from a current forecast, and from nearby observations (allowing for the inaccuracies and possible gross errors in these) to give new estimates. The final probabilities can be used to reject or accept the data in an objective analysis. Results from trials of this system are given. It is shown to be possible using an archive generated by the system to update the prior error statistics necessary to make the method truly objective. Some practical case studies are shown, and compared with careful human quality control.

Adaptive segmentation of speckled images using a hierarchical random field model

Abstract: A two-level hierarchical random-field model developed for speckled images and, in particular, for synthetic-aperture radar (SAR) imagery, is described. At the higher level is a Gibbs random field governing the grouping of image pixels into regions, while at the lower level are speckle processes for the different regions, which are also modeled as random fields. In accordance with the physical phenomena that cause speckle, the speckle processes are modeled as circularly symmetric complex Gaussian random fields. As with real Gaussian fields, certain forms of autocovariance of complex Gaussian field lead to Markovianity properties. With the assumption of a separable autocovariance for the complex Gaussian random fields, local joint statistics of the resulting speckle-intensity fields (magnitude-squared of the complex field) are determined. The speckle model concurs with the known marginal statistics and also accounts for the autocorrelation observed in actual speckled images. A MAP segmentation using simulated annealing and based on the hierarchical model is presented. The algorithm is adaptive in that it recursively segments the image and estimates the model parameters necessary for the segmentation. >

Imaging of Gaussian Schell-model sources

Abstract: Imaging of Gaussian Schell-model sources by general lossless systems is analyzed with an extended ray-transfermatrix method. Algebraic expressions are derived for the location, size, and coherence area of the image waist and for the depth of focus and the far-field diffraction angle. These results are shown to provide a continuous transformation between laser-beam optics and geometrical optics. They also lead naturally to several equivalence and invariance relations pertaining to isotropic and anisotropic Gaussian Schell-model sources. As an application, the importance of effects due to partial spatial coherence in beam focusing is examined.

Auto‐Regressive Model for Nonstationary Stochastic Processes

Abstract: An autoregressive model for univariate, one‐dimensional, nonstationary, Gaussian random processes with evolutionary power spectra is introduced. At the same time, an efficient technique for numerically generating sample functions of such nonstationary processes is developed. The technique uses a recursive equation which: (1) Reflects the nature of the nonstationarity of the process whose sample functions are to be generated; and (2) involves a normalized univariate, one‐dimensional white noise sequence. The coefficients of the recursive equation are determined using the autocorrelation function of the process, which in turn is calculated from the evolutionary power spectrum at every time instant. Using the recursive equation with those coefficients, sample functions over a specified domain can be generated with substantial computational ease. Univariate, one‐dimensional, nonstationary processes with three different forms of the evolutionary power spectrum are modeled, and their sample functions are genera...

Gaussian pure states in quantum mechanics and the symplectic group

Abstract: Gaussian pure states of systems with n degrees of freedom and their evolution under quadratic Hamiltonians are studied. The Wigner-Moyal technique together with the symplectic group Sp(2n,openR) is shown to give a convenient framework for handling these problems. By mapping these states to the set of n\ifmmode\times\else\texttimes\fi{}n complex symmetric matrices with a positive-definite real part, it is shown that their evolution under quadratic Hamiltonians is compactly described by a matrix generalization of the M\"obius transformation; the connection between this result and the ``abcd law'' of Kogelnik in the context of laser beams is brought out. An equivalent Poisson-bracket description over a special orbit in the Lie algebra of Sp(2n,openR) is derived. Transformation properties of a special class of partially coherent anisotropic Gaussian Schell-model optical fields under the action of Sp(4, openR) first-order systems are worked out as an example, and a generalization of the ``abcd law'' to the partially coherent case is derived. The relevance of these results to the problem of squeezing in multimode systems is noted.

Statistical aspects of dissipation by Landau-Zener transitions

Abstract: Considers the effect of slowly varying the parameters Xi of a finite-sized quantum mechanical system. The system is excited to higher energies by Landau-Zener transitions at avoided crossing; since this increases the energy of the system, it has the effect of dissipation of the driving motion. The rate of dissipation depends on the level spacing distribution of the system. When the spectral statistics are those of the Gaussian unitary ensemble, the rate of dissipation is proportional to Xi2, i.e. there is viscous or ohmic damping. When the spectral statistics are of those of the Gaussian orthogonal ensemble, the rate of dissipation is proportional to Xi32/.

Optimal linear-quadratic systems for detection and estimation

Abstract: The problem of linear-quadratic systems for detection has long been solved by assuming the deflection criterion and Gaussian noise. It is shown here that the Gaussian assumption can be removed, and a complete solution is presented for an arbitrary probability distribution with finite fourth-order moments. The optimal solution can always be obtained by solving a linear system of equations. Some properties of the optimal systems are developed for particular examples of nonGaussian noise. It is shown that there is a strong relationship between linear-quadratic optimal detection and optimal estimation, which extends results known for the purely linear case. >

Altimetry for non‐Gaussian oceans: Height biases and estimation of parameters

Abstract: The effects of a non-Gaussian ocean on satellite altimetry parameter estimation are discussed. The first part of this paper shows how non-Gaussian ocean parameters affect height estimation for satellites of the Seasat/Geosat/TOPEX type. In the second part, the estimation of the altimeter tracker bias and the non-Gaussian ocean parameters from the altimeter return signal is studied. A new convolution model that facilitates the deconvolution of the ocean surface specular point probability density function is introduced. Next, it is argued that it is not feasible in practice to estimate the electromagnetic bias from noisy altimeter returns. It is then shown that the least squares estimation of the surface parameters in the log-frequency domain has several advantages overestimating parameters in the time domain. The maximum likelihood estimation equations for the estimation of the waveform parameters are derived, and their solution is discussed.

Gaussian likelihood of continuous-time armax models when data are stocks and flows at different frequencies

Abstract: For purposes of maximum likelihood estimation, we show how to compute the Gaussian likelihood function when the data are generated by a higher-order continuous-time vector ARMAX model and are observed as stocks and flows at different frequencies. Continuous-time ARMAX models are analogous to discrete-time autoregressive moving-average models with distributed-lag exogenous variables. Stocks are variables observed at points in time and flows are variables observed as integrals over sampling intervals. We derive the implied state-space model of the discrete-time data and show how to use it to compute the Gaussian likelihood function with Kalman-filtering, predictionerror, decomposition of the data. In this paper, for the purposes of maximum likelihood estimation, we show how to compute the Gaussian likelihood function when the data are generated by a higher-order continuous-time vector ARMAX model and are observed as stocks and flows at different frequencies. Continuous-time ARMAX models are rational spectrum models which are analogous to discrete-time autoregressive moving-average (ARMA) models with distributed-lag exogenous variables (X). By definition, stocks are variables observed at points in continuous time and flows are variables observed as integrals over discrete sampling intervals. We derive the implied state-space model of the discretetime data and show how to use it to compute the Gaussian likelihood function with Kalman-filtering, prediction-error, decomposition of the data. The first contribution of the paper is to extend to multiple frequencies Harvey and Stock's [15] treatment of stocks and flows at the same freThis work was begun during my tenure on the ASA/NSF/Census Fellowship at the U.S. Bureau of the Census in 1983-1984 and has greatly benefitted from conversations with William R. Bell, David F. Findley, the editorial comments of Peter C.B. Phillips, and anonymous referees. The views expressed herein do not necessarily reflect the official policies of ASA, NSF, or the Census.

Calculus on Gaussian and Poisson white noises

Abstract: Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc ([8], [9]), analogously to the works of T Hida ([3], [4], [5]) Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf [10], [11], [12], [13]) Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc in a way completely parallel with the Gaussian case The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case, as will be stated in Section 5 Conversely, those formulae characterize the types of white noises

Estimation via compressed information

Abstract: Some results from classical estimation theory are extended to the case in which data must be communicated from several places where observations are made to the place where the estimate is generated. Particular emphasis is placed on determining how the variance of an unbiased estimator depends on the communication rates. Explicit result are given for Gaussian sources. >

The Effect of Non-Gaussian Statistics on the Mass Multiplicity of Cosmic Structures

Abstract: The mass function of cosmic structures is computed in the framework of the hierarchical clustering picture for a general statistics of density perturbations. Hierarchical distributions are extensively analyzed; it is found that the multiplicity function preserves the Press-Schechter functional form with enhanced power on large scales compared to the Gaussian case. A class of scale-invariant non-Gaussian statistics, among which are a model due to Peebles and the lognormal distribution, are also analyzed. All these predict a mass function which is a decreasing power law at low mass followed by an exponential decay at high mass; none of them, however, yields a mass function of the Press-Schechter type. The effect of a statistical bias on the origin of condensations is also discussed. The comparison of these theoretical formulae with the observed mass multiplicity of galaxies, groups, and clusters may represent a powerful tool to test the statistics of cosmological perturbations. 78 references.

Exact orthogonal kernel estimation from finite data records: extending Wiener's identification of nonlinear systems.

Abstract: A technique is described for exact estimation of kernels in functional expansions for nonlinear systems. The technique operates by orthogonalizing over the data record and in so doing permits a wide variety of input excitation. In particular, the excitation is not limited to inputs that are white, Gaussian, or lengthy. Diagonal kernel values can be estimated, without modification, as accurately as off-diagonal values. Simulations are provided to demonstrate that the technique is more accurate than the Lee-Schetzen method with a white Gaussian input of limited duration, retaining its superiority when the system output is corrupted by noise.

Connection between Gaussian periods and cyclic units

Abstract: This paper finds that all known parametric families of units in real quadratic, cubic, quartic and sextic fields with prime conductor are linear combinations of Gaussian periods and exhibits these combinations. This approach is used to find new units in the real quintic field for prime conductors p n4 + 5n3 + 15n2 + 25n + 25.

Optimization of a distributed Gaussian basis set using simulated annealing: Application to the adiabatic dynamics of the solvated electron

Abstract: A molecular dynamics technique is introduced for the simulation of the adiabatic dynamics of an excess electron coupled to a classical many‐body system. The instantaneous ground state wave function of the electron is represented by a superposition of distributed Gaussian basis functions, each with equal amplitude. We present generalized equations of motion for the coupled system, which optimize the positions and widths of the Gaussians by simulated annealing. The condition of equal amplitude ensures the aggregation of the Gaussians in regions of finite electron probability density and hence yields a particularly efficient representation of localized ground states. The method is applied to an electron solvated in liquid ammonia and results for equilibrium properties are compared to quantum path integral calculations. New results for the dynamics are discussed in the light of mobility measurements.

Computation of body-wave seismograms in absorbing 2-D media using the Gaussian beam method: comparison with exact methods

Abstract: SUMMARY The accuracy and applicability of the Gaussian beam method for the computation of synthetic seismograms in absorbing 2-D laterally inhomogeneous media is discussed in this paper. A computer program was developed for the computation of P-SV- and SH-waves along horizontal and vertical seismic profiles in absorbing 2-D inhomogeneous media with first-order discontinuities in density and velocities. Subdividing the model into triangles with linear density and velocity laws not only allows flexible modelling of complicated structures but also results in fast kinematic and dynamic ray tracing. With this program it is possible to compute phases which are specified by single or multiple reflections, refractions and/or conversions at the different discontinuities in the medium. The width and phase-front curvature of each beam is controlled by its complex beam parameter e. Along with several e-options proposed by other authors, a newly developed option to select the beam parameter was tested for a large set of models. The new option correlates the width of each beam to the size of the triangles the beam has passed through and thus to the structure of the medium. Slowly varying media represented by large triangles will give broad Gaussian beams whereas complicated and rapidly changing media which have to be described by many small triangles will produce narrow Gaussian beams. For the computation of reference seismograms the reflectivity method, the frequency-domain finite-difference method and the finite-difference method were used. The conclusion from these tests is, that the newly developed e-option is the optimum choice, whereas the other options sometimes fail and do not give accurate seismograms for critical incidence and at caustics.

Gaussian basis sets for high-quality ab initio calculations

Abstract: Different types of Gaussian basis sets for accurate LCAO calculations are discussed. For calculations designed to recover a substantial portion of the correlation energy, we suggest the use of basis sets comprising natural orbitals from correlated calculations on the atoms. These basis sets have proven to be very efficient in accounting for large fractions of the molecular correlation energy. For cases in which an SCF or MCSCF treatment is adequate the use of a floating basis provides a rapid convergence to the large-basis limit.