Gaussian Autocorrelation Function Research Articles

Synthesis of scalar-wave envelopes in two-dimensional weakly anisotropic random media by using the Markov approximation

SUMMARY Due to the Earth's inhomogeneity, seismogram envelopes increase in duration and decrease in amplitude with increasing travel distance. Those features have been explained on the basis of the forward scattering in random media using the Markov approximation. Since the conventional studies assumed isotropic random media, they were not realistic enough to represent the anisotropic lithosphere characterized by long horizontal and short vertical correlation distances. Using the Markov approximation, the present study formulates the envelope synthesis in 2-D weakly anisotropic random media characterized by Gaussian and von Karman-type autocorrelation functions (ACF). Also, wave propagation is numerically simulated with the finite-difference (FD) method; 2 Hz Ricker wavelets propagate through the random media characterized by 4 km s−1 average velocity and the Gaussian ACF with 5 km horizontal correlation distance, 2.5 km vertical correlation distance, and 5 per cent rms fractional velocity fluctuation. We find a good coincidence between the envelopes of the Markov approximation and those of the FD simulations, which supports the reliability of the synthesis method using the Markov approximation. The envelopes of the Markov approximation are scaled by using a characteristic time, which is a function of propagation direction and other model parameters. It predicts that envelopes increase in duration and decrease in maximum amplitude more rapidly in the horizontal propagation than in the vertical when the media are characterized by long horizontal and short vertical correlation distances. In the case of the vertical wave propagation, the envelope of the anisotropic random media has shorter duration and larger maximum amplitude than those of the isotropic random media. It means that the intensity of the inhomogeneity is underestimated when one analyses seismogram envelopes of deep events considering anisotropic lithosphere as isotropic random media.

Response Variability in Balanced Cortical Networks

We study the spike statistics of neurons in a network with dynamically balanced excitation and inhibition. Our model, intended to represent a generic cortical column, comprises randomly connected excitatory and inhibitory leaky integrate-and-fire neurons, driven by excitatory input from an external population. The high connectivity permits a mean field description in which synaptic currents can be treated as gaussian noise, the mean and autocorrelation function of which are calculated self-consistently from the firing statistics of single model neurons. Within this description, a wide range of Fano factors is possible. We find that the irregularity of spike trains is controlled mainly by the strength of the synapses relative to the difference between the firing threshold and the postfiring reset level of the membrane potential. For moderately strong synapses, we find spike statistics very similar to those observed in primary visual cortex.

Response Variability in Balanced Cortical Networks

We study the spike statistics of neurons in a network with dynamically balanced excitation and inhibition. Our model, intended to represent a generic cortical column, comprises randomly connected excitatory and inhibitory leaky integrate-and-fire neurons, driven by excitatory input from an external population. The high connectivity permits a mean field description in which synaptic currents can be treated as gaussian noise, the mean and autocorrelation function of which are calculated self-consistently from the firing statistics of single model neurons. Within this description, a wide range of Fano factors is possible. We find that the irregularity of spike trains is controlled mainly by the strength of the synapses relative to the difference between the firing threshold and the postfiring reset level of the membrane potential. For moderately strong synapses, we find spike statistics very similar to those observed in primary visual cortex.

Estimation of Trabecular Thickness Using Ultrasonic Backcatter

We present a method to estimate trabecular thickness (Tb.Th) in trabecular bones from ultrasound backscatter measurements. The estimation scheme is based on a nonlinear adjustment of predictions from a model to experimental data. The model assumes weak scattering from bone, where scattering is assumed to arise from the elastic solid trabeculae. The fluctuations of acoustical properties between bone tissue and the saturating fluid are assumed to be random and are described by the 3-D spatial autocorrelation function of the medium. In this paper, a Gaussian autocorrelation function is used. The inversion procedure is applied to a set of data measured on 33 femoral bone specimens. Results show that the model can predict both the magnitude and the frequency-dependence of the backscatter coefficient (root mean square error RMSE = 1 dB). The estimated trabecular thickness values are compared to the true trabecular thickness measured on high resolution microcomputed tomography 3-D reconstruction of bones microarchitecture. A close agreement is obtained on average over the group of specimens between predictions and the reference values: true Tb.Th is 132 +/- 12 microm and estimated Tb.Th is 134 +/- 15 microm. However, a moderate correlation between actual and estimated Tb.Th values is found (R2 = 0.44, p<10(-4), RMSE = 8.7 microm) suggesting a modest predictability at the individual level. Sources for the variability of the estimator are studied. Using synthetic rf signals, we demonstrate that the fundamental limitation of the estimator due to speckle noise is approximately 5 microm. Taking into account the measurement errors, the total uncertainty on Tb.Th estimates is of the order of 7 microm. The influence of the attenuation compensation function used to derive the backscatter coefficient is studied. In particular, we demonstrate the necessity of compensating for the effect of the gating time window. The results are discussed with respect to their meaningful clinical value. The requirements to be fulfilled by the performance of the technique change with regard to the question being posed. Two different strategies are examined: 1. characterize trabecular thickness without consideration of bone quantity (or bone mineral density) and 2. estimate trabecular thickness after adjustment for BMD. Considering the first strategy, a comparison between the precision of our estimator and the biological variability leads us to the conclusion that our estimator should only permit to distinguish between micro-architectures characterized by extreme values of trabecular thickness (i.e., very thin or very thick trabecular thickness). In this respect, it would be interesting to test whether the estimator is able to discriminate between rod-like (thin) and plate-like (thick) structures that are known to influence differently bone strength. The second strategy is more demanding in terms of technique performance and our estimator is not able yet to catch small differences in Tb.Th values expected after adjustment to bone density. Progress in the field will require a significant reduction in speckle noise and measurement errors and/or the development of other and more efficient microstructural estimators.

Correlation of ultrasonic scatterer size estimates for the statistical analysis and optimization of angular compounding

Ultrasonic scatterer size estimates generally have large variances due to the inherent noise of spectral estimates used to calculate size. Compounding partially correlated size estimates associated with the same tissue, but produced with data acquired from different angles of incidence, is an effective way to reduce the variance without making dramatic sacrifices in spatial resolution. This work derives theoretical approximations for the correlation between these size estimates, and the coherence between their associated spectral estimates, as functions of ultrasonic system parameters. A Gaussian spatial autocorrelation function is assumed to adequately model scatterer shape. Both approximations compare favorably with simulation results, which consider validation near the focus. Utilization of the correlation/coherence expressions for statistical analysis and optimization is discussed. Approximations, such as the invariance of phase and amplitude terms with angle, are made to obtain closed-form solutions to the derived spectral coherence near the focus and permit analytical optimization analysis. Results indicate that recommended parameter adjustments for performance improvement generally depend upon whether, for the system under consideration, the primary source of change in total coherence with rotation is phase term variation due to the change in the relative position of scattering sites, or field amplitude term variation due to beam movement.

Errors in ultrasonic scatterer size estimates due to phase and amplitude aberration.

Current ultrasonic scatterer size estimation methods assume that acoustic propagation is free of distortion due to large-scale variations in medium attenuation and sound speed. However, it has been demonstrated that under certain conditions in medical applications, medium inhomogeneities can cause significant field aberrations that lead to B-mode image artifacts. These same aberrations may be responsible for errors in size estimates and parametric images of scatterer size. This work derives theoretical expressions for the error in backscatter coefficient and size estimates as a function of statistical parameters that quantify phase and amplitude aberration, assuming a Gaussian spatial autocorrelation function. Results exhibit agreement with simulations for the limited region of parameter space considered. For large values of aberration decorrelation lengths relative to aberration standard deviations, phase aberration errors appear to be minimal, while amplitude aberration errors remain significant. Implications of the results for accurate backscatter and size estimation are discussed. In particular, backscatter filters are suggested as a method for error correction. Limitations of the theory are also addressed. The approach, approximations, and assumptions used in the derivation are most appropriate when the aberrating structures are relatively large, and the region containing the inhomogeneities is offset from the insonifying transducer.

Mapping random lithospheric heterogeneities in northern and central Europe

SUMMARY An energy-flux model (EFM) and a teleseismic fluctuation wavefield method (TFWM) have been applied to interpret the teleseismic P coda observed at three temporary and two permanent networks in northern and central Europe. The aim is to determine the small-scale random structure of the lithosphere below the receivers. Various subsets of these data have been exploited previously with one of the two methods. The main objectives here are: To compare the performance of both methods with synthetic data sets. To map the random structure of the lithosphere in terms of inverse scattering Q (Q−1s), correlation length a, RMS velocity fluctuation σ, thickness L of the scattering layer and autocorrelation function (ACF) using a combination of both methods. With TFWM, the product aσ2 can be reliably determined if L is known. L can be roughly estimated with EFM. Although EFM can, in principle, resolve a and σ separately, a is recovered with a rather large uncertainty. TFWM does not distinguish much between the ACF type, whereas with EFM determination of the ACF type is sometimes possible. By combining the results of both methods we determine improved random medium parameters of the lithosphere for eleven subregions in northern and central Europe. In the Baltic shield, Rhenohercynian belt, Ardenne and Brabant mountains, eastern Rhenish massif, Eifel, Hunsruck mountains, Lorraine, Frankonian Jura and massif Central scattering predominantly occurs in the crust. For the frequency range from 0.5 to 5 Hz correlation lengths of 1–7 km and rms velocity fluctuations of 3–7 per cent are obtained. For the Rhenohercynian belt (RH) and the N-German basin RMS velocity fluctuations and correlation lengths could not be resolved. The data from the N-German basin (NB) cannot be explained by scattering within the crust only. Smallest scattering Q was found in the N German basin (Qs≈ 100 at 1 Hz) and largest scattering Q in the Baltic shield (Qs≈ 450 at 2–3 Hz). For the Frankonian Jura only a Gaussian ACF can fit the Q−1s values. The data from the eastern Rhenish massif also indicate a random medium with Gaussian ACF. For all other subregions we cannot distinguish between exponential or Gaussian ACF for the random medium structure.

Optimization of angular compounding in scatterer size estimation

Ultrasonic scatterer size estimates generally have large variances due to the inherent noise of the spectral estimates used to calculate size. Compounding partially correlated size estimates associated with the same tissue, but produced with data acquired from different angles of incidence, is an effective way to reduce the variance without making dramatic sacrifices in spatial resolution. This work derives theoretical approximations for the correlation between these size estimates, and between their associated spectral estimates, as functions of data acquisition and processing parameters, where a Gaussian spatial autocorrelation function is assumed to adequately model scatterer shape. Size results exhibit a fair degree of agreement with those of simulation experiments, while spectral results compare favorably with simulation outcomes. Utilization of the theoretical correlation expressions for data acquisition and processing optimization is discussed. Further simplifying approximations, such as the invariance of phase and amplitude terms with rotation angle, are made in order to obtain closed-form solutions to the derived spectral correlation, and permit an analytical optimization analysis. Results indicate that recommended parameter adjustments for performance improvement depend upon whether, for the system under consideration, the primary source of estimate decorrelation with rotation is scatterer phase change or field separation. [Work supported by NIH T32CA09206.]

Statistics of ultrasonic scatterer size estimation with a reference phantom.

A theoretical expression for the variance of scatterer size estimates is derived for a modified least squares size estimator used in conjunction with a reference phantom method for backscatter coefficient measurement. A Gaussian spatial autocorrelation function is assumed. Simulations and phantom experiments were performed to verify the results for backscatter and size variances. The dependence of size estimate errors upon free experimental parameters is explored. Implications of the findings for the optimization of scatterer size estimation are discussed. The utility of scatterer size parametric imaging is examined through the signal to noise ratio comparison with standard ultrasonic B-mode imaging.

Intrinsic parameter fluctuations in decananometer mosfets introduced by gate line edge roughness

In this paper, we use statistical three-dimensional (3-D) simulations to study the impact of the gate line edge roughness (LER) on the intrinsic parameters fluctuations in deep decananometer (sub 50 nm) gate MOSFETs. The line edge roughness is introduced using a Fourier synthesis technique based on the power spectrum of a Gaussian autocorrelation function. In carefully designed simulation experiments, we investigate the impact of the rms amplitude /spl Delta/ and the correlation length /spl Lambda/ on the intrinsic parameter fluctuations in well scaled, but simple devices with fixed geometry as well as the channel length and width dependence of the fluctuations at fixed LER parameters. For the first time, we superimpose in the simulations LER and random discrete dopants and investigate their relative contribution to the intrinsic parameter fluctuations in the investigated devices. For particular MOSFET geometries, we were able to identify the regions where each of these two sources of intrinsic parameter fluctuations dominates.

Envelope broadening of spherically outgoing waves in three‐dimensional random media having power law spectra

High‐frequency S wave seismogram envelopes are broadened with increasing travel distance due to diffraction and scattering. The basic mechanism of the broadening has been studied on the basis of the scattering theory with the parabolic approximation for the scalar wave equation in random media. However, conventional models are not realistic enough since the plane wave modeling is too simple and the Gaussian autocorrelation function (ACF) is far from the reality to represent the inhomogeneity in the Earth. Focusing on the early part of envelopes, we formulated the envelope broadening of spherically outgoing scalar waves in three‐dimensional von Kármán‐type random media, of which the spectra decay according to a power law at large wave numbers. Random media are characterized by three parameters: RMS fractional velocity fluctuation ε, correlation distance a, and order κ that controls the gradient of the power law spectra. This model predicts that the envelope duration increases with both travel distance and frequency when short‐wavelength components are rich in random media, while the duration is independent of frequency when short‐wavelength components are poor. Introducing phenomenological attenuation Q, we developed a method for estimating the parameters of inhomogeneity and attenuation from the envelope duration. Applying this method to S wave seismogram envelopes for the frequency range from 2 to 32 Hz in northeastern Honshu Japan, we estimated the random inhomogeneity parameters as κ = 0.6, ε2.2a−1 ≈ 10−3.6 [km−1] and f/Q = 0.0095 [s−1], where f is frequency. The power law portion of the estimated power spectral density function is P(m) ≈ 0.01 m−4.2 [km3], where m is wave number.

Using waveguide scattering of laser radiation for determining the autocorrelation function of statistical surface roughness within a wide range of changes of the roughness correlation interval

An electrodynamic problem of laser radiation scattering in an integrated-optical waveguide containing small statistical irregularities (interface roughness and irregularities of the refractive indices of the waveguide-forming media) is considered. The possibility of using the waveguide scattering of laser radiation for extracting the information on the statistical properties of irregularities from noisy data of the scattering diagram in a far-field zone is shown. An algorithm for reconstructing the autocorrelation function of irregularities for the correlation interval changing within a wide range is described. The possibility of restoring a given Gaussian autocorrelation function that describes statistical irregularities of the waveguide substrate surface for a correlation interval changing between 10 nm and 10 μm and a high-level additive white real noise is shown by computer simulation.

Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms

Mathematical justifications are given for a Monte Carlo simulation technique based on memoryless transformations of Gaussian processes. Different types of convergences are given for the approaching sequence. Moreover an original numerical method is proposed in order to solve the functional equation yielding the underlying Gaussian process autocorrelation function.

Monostatic and bistatic statistical shadowing functions from aone-dimensional stationary randomly rough surface according tothe observation length: I. Single scattering

When solving electromagnetic rough-surface scattering problems, the effect of shadowing by the surface roughness often needs to be considered, especially as the illumination angle approaches grazing incidence. This paper presents the Ricciardi-Sato, as well as the Wagner and the Smith formulations for calculating the monostatic and bistatic statistical shadowing functions from a one-dimensional rough stationary surface, which are valid for an uncorrelated Gaussian process with an infinite surface length. In this paper, these formulations are extended to include a finite surface length and any uncorrelated process. The inclusion of a finite surface length is needed to extend the single-reflection shadowing function to the more general multiple-reflection case, presented in the following companion paper. Comparisons of these shadowing functions with the exact numerical solution for the shadowing (using surfaces with Gaussian and Lorentzian autocorrelation functions for a Gaussian process) shows that the Smith formulation without correlation is a good approximation, and that including correlation only weakly improves the model. This paper also presents a method to include the shadowing effect in the electromagnetic scattering problem.

Miscellanea. Exact Gaussian maximum likelihood and simulation for regularly-spaced observations with Gaussian correlations

SUMMARY For regularly-spaced observations with the Gaussian autocorrelation function, the finite and infinite autoregressive and moving-average representations can be obtained theoretically. This allows exact Gaussian maximum likelihood to be performed very accurately, and gives a simple exact simulation method.

Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

Observations of seismic waves from earthquakes at depths between 100 and 200 km beneath Japan show that the initial portion of the S-wave arrival has greater duration than can be accounted for by earthquake source-time duration. The observed long duration of S waves has been explained as being caused by multiple forward scattering around the ray path between source and receiver. Array obser- vations of Lg waveforms have also shown that multiple forward scattering along the path between the source and receiver are important influences on the character of Lg waveforms and that the scattering cannot be explained only by vertical variations in velocity. Multiple forward scattering in 3D has been modeled using the Markov approximation for the parabolic-wave equation, which allows the calculation of seis- mogram envelopes in statistically characterized random media. To test the range of validity of the Markov approximation-derived solutions, we made 2D numerical calculations of wavefields in random media using approximations to the parabolic wave equation, which only models forward scattering, and by finite-difference so- lution of the scalar-wave equation, which gives complete wavefields. Media of back- ground velocity 4 km/sec are characterized using a Gaussian autocorrelation function with a5k mcorrelation distance and 5% rms fractional fluctuation. To compare with envelopes obtained from the Markov approximation, we calculated wavefields for source-receiver distances ranging from 50 to 200 km for several statistically identical realizations of random media. We obtain ensemble average envelopes by averaging envelopes from the realizations. We find a good agreement between ensemble- average envelopes obtained from the numerical solution of the parabolic-wave equa- tion and envelopes obtained from the Markov approximation. The later portion of the ensemble-average envelopes calculated using finite difference have larger am- plitudes than those from the Markov approximation, which is probably due to the late-arriving energy that has been scattered at wide angles from the global propa- gation direction between the source and the receiver. We observe that the variations among the numerically calculated envelopes of individual realizations of random media are well fit by a Rayleigh distribution, which describes the distribution of envelope amplitude when signals of one frequency but random phase are summed. Our results show that the Markov approximation provides reliable information about envelope shapes for forward-scattered wavefields but that the influence of wide-angle scattering and backscattering have some influences on envelope shapes and should be considered when analyzing data using random media models.

Large fluctuation of wave amplitude produced by small fluctuation of velocity structure

Seismic velocity structure of the real Earth is described by a combination of deterministic large scale structure and small scale fluctuation. When waves propagate in media having the small scale fluctuation, the fluctuation makes the rays bend, introduces focussing and defocusing, and produces fluctuations of wave amplitude. To estimate the size of wave amplitude fluctuation that is caused by velocity fluctuation, the variances of wave amplitudes are evaluated as a function of propagation distance from numerical simulations of scalar wave propagation in 3D random media having Gaussian or exponential autocorrelation functions. The simulations are performed using a phase screen method, which is a numerical procedure for solving the parabolic wave equation. The following conclusions are obtained from the simulations that use two types of incident wave: stationary monochromatic sine wave and transient Ricker wavelet. For velocity fluctuation as small as several percent, the variance of wave amplitude is larger than the square of its average in some cases of high frequency waves. These results show the importance of small error and fine spatial resolution of velocity structure model for proper modeling of wave amplitude. When the fluctuation of the velocity structure is large, the wave amplitude variance increases with the propagation distance until it reaches some peak value after which it decreases with the increase of propagation distance. The variance for the Ricker wavelet incidence is smaller than that for a monochromatic sine wave. Because of the stochastic dispersion due to the small fluctuation of velocity structure, the maximum amplitude decreases with the propagation distance even if there is neither an intrinsic absorption nor back-scattering for the Ricker wavelet incidence.

Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions

The Wagner (1966) and Smith (1967) classical monostatic one-dimensional (1-D) shadowing functions assume that the joint probability density of heights and slopes is uncorrelated, thus inducing an overestimation of the shadowing function. The goal of this article is to quantify this assumption. More recently, Ricciardi and Sato (1983, 1986) proved that the shadowing function is given rigorously by Rice's infinite series of integrals. We observe that the approach proposed by Wagner retains only the first term of this series, whereas the Smith formulation uses the Wagner model by introducing a normalization function. In this article, we first calculate the shadowing function based on the work of Ricciardi and Sato for an uncorrelated process. We see that the uncorrelated results do not have any physical sense. Next, the Wagner and Smith formulations are modified in order to introduce the correlation. Correlated and uncorrelated results are compared with the reference solution, which is determined by generating a surface for a Gaussian autocorrelation function. So, we show that the correlation improves the results for values /spl mu//spl les/2/spl sigma/, where /spl mu/ represents the slope of incident ray and /spl sigma/ the slopes variance of the surface. Finally, our results are compared to those given by Kapp and Brown (1994), determined from the first three terms of Rice's series, but the shadowing function used is not averaged over the slopes.

The Feasibility of Data Whitening to Improve Performance of Weather Radar

The problem of efficient processing of correlated weather radar echoes off precipitation is considered. An approach based on signal whitening was recently proposed that has the potential to significantly improve power estimation at a fixed pulse repetition rate/scan rate, or to allow higher scan rates at a given level of accuracy. However, the previous work has been mostly theoretical and subject to the following restrictions: 1) the autocorrelation function (ACF) of the process must be known precisely and 2) infinite signal-to-noise ratio is assumed. Here a computational feasibility study of the whitening algorithm when the ACF is estimated and in the presence of noise is discussed. In the course of this investigation numerical instability to the ACF behavior at large lags (tails) was encountered. In particular, the commonly made assumption of the Gaussian power spectrum and, therefore, Gaussian ACF yields numerically ill-conditioned covariance matrices. The origin of this difficulty, rooted in the violation of the requirement of positive Fourier transform of the ACF, is discussed. It is found that small departures from the Gaussian form of the covariance matrix result in greatly reduced ill conditioning of the matrices and robustness with respect to noise. The performance of the whitening technique for various meteorologically reasonable scenarios is then examined. The effects of additive noise are also investigated. The approach, which uses time series to estimate the ACF from which the whitener is constructed, shows up to an order of magnitude improvement in the mean-squared error of the estimated power for a range of parameter values corresponding to typical meteorological situations.

Experimental Evidence of a Gaussian Roughness atSi(111)/SiO2Interfaces

With a plan-view transmission electron microscope technique to unambiguously image the {open_quotes}physical{close_quotes} interface position between Si and furnace grown SiO{sub 2} layers, we first show experimental evidence that the height-height autocorrelation function is a Gaussian function at Si(111)/SiO{sub 2} interfaces. With a simple kinetic model, we have found that this Gaussian autocorrelation function is a natural consequence of step motion during silicon oxidation. This result puts interfacial roughness measurements on a firmer foundation in the future. {copyright} {ital 1998} {ital The American Physical Society }