Gaussian Autocorrelation Function Research Articles

The application of dynamic light scattering to complex plasmas

The dynamic light scattering (DLS) technique is applied to the dust component of a complex (dusty) plasma, revealing a Gaussian intensity autocorrelation function for scattering angles between 4° and 175°. The Gaussian decay form represents free (ballistic) particle motion and allows determination of the one-dimensional squared particle velocity . At scattering angles below 1°, the intensity autocorrelation function is shown to be a combination of a Gaussian and an exponential function. This allows determination of the particle velocity and the diffusion constants at the same time. The dust system is fully described by the two components of motion in the horizontal and vertical directions. The two components are simultaneously measured on two scattering paths using only a single incident laser beam. In contrast to standard imaging techniques, the DLS method can be applied even to the disordered phase state where the dust particles have very high kinetic energies. In the ordered phase state, the assumptions of the DLS approach were verified by the independent Charge Coupled Device technique on the fundamental kinetic level. Furthermore, a careful discussion of the standard deviation of the DLS method proves that it can be used to study phase transitions of complex plasmas in detail.

Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory

Coupled-mode and coupled-power theories are described for multi-core fiber design and analysis. First, in order to satisfy the law of power conservation, mode-coupling coefficients are redefined and then, closed-form power-coupling coefficients are derived based on exponential, Gaussian, and triangular autocorrelation functions. Using the coupled-mode and coupled-power theories, impacts of random phase-offsets and correlation lengths on crosstalk in multi-core fibers are investigated for the first time. The simulation results are in good agreement with the measurement results. Furthermore, from the simulation results obtained by both theories, it is confirmed that the reciprocity is satisfied in multi-core fibers.

Evaluation of a Numerical Modeling Approach Based on the Finite-Element Method for Calculating the Rough Surface Scattering and Emission of a Soil Layer

We evaluate a new 3-D numerical modeling approach for calculating the rough-surface scattering and emission of a soil layer. The approach relies on the use of Ansoft's numerical computation software High-Frequency Structure Simulator, which solves Maxwell's equations directly using the finite-element method. The interest of this approach is that it can be easily extended to studies of heterogeneous media. However, before being applied in this way, it must first be validated for the rough-surface case. In this letter, we perform this validation by comparing the results of rough-surface scattering and emission with the results of the method of moments (MoM) for a range of different roughness and permittivity conditions and with both Gaussian and exponential rough-surface autocorrelation functions. For the scattering case, we obtain results that are in agreement with the MoM to within approximately 1-3 dB for angles up to and including 40° and 2-4 dB for angles from 50° to 70°. Agreement for emissivity is to within 3.2 and 3.6 K for low and high roughness conditions, respectively. We then illustrate the application of the new approach by calculating the emission of a two-layer system with rough surfaces, representing the soil-litter system in forests.

Statistical Model of Line-Edge and Line-Width Roughness for Device Variability Analysis

The authors propose a model of line-edge and line-width roughness (LER and LWR) of actual device patterns, which received some smoothing steps, for accurate estimation of device variability. The model assumes that LER/LWR has originally an exponential autocorrelation function (ACF) and is smoothed using another exponential function. The power spectrum of this ACF almost completely fits the experimental one of polycrystalline silicon lines, which were formed using plasma etching. The authors investigate the effect of LER/LWR on the current factor of metal-oxide-semiconductor field-effect-transistors, comparing this to conventional models. The Gaussian ACF, which is widely used in device simulations, calculates the variation in the current factor with considerable accuracy as long as accurate LER/LWR statistics are used. However, it alone cannot provide the statistics. The exponential ACF underestimates the variation by a nonnegligible amount. From these results, the authors propose to use the aforementioned smoothed exponential ACF in the device simulations. They also alert to the possibility that a little-known long-range correlation exists universally in the LER/LWR even of the present-day devices and is causing an unexpectedly large mismatching between wide-channel devices.

Synthesis of vector wave envelopes on the free surface of a random medium for the vertical incidence of a plane wavelet based on the Markov approximation

In short‐period seismograms of earthquakes, we often observe the broadening of apparent duration of P and/or S waves and the excitation of the transverse component especially for P waves as travel distance increases. Such phenomena are well explained by scattering caused by random velocity fluctuation in the lithosphere. The Markov approximation is known as one of the powerful stochastic methods for the direct synthesis of wave envelopes. We extend the method to synthesize vector wave envelopes on the free surface of a random medium since seismic observation is usually done on the ground surface. We evaluate the mean square (MS) envelope on the free surface by multiplying the amplification factor on the free surface to the angular spectrum in an infinite random medium. We synthesize MS envelopes for the vertical incidence of an impulsive plane P or S wavelet into a 3‐D random medium characterized by a Gaussian autocorrelation function with typical parameters of the lithosphere. As a result, the vertical and horizontal component MS envelopes show different amplification rates on the free surface; however, we may say that “a factor of 4” is a good approximation for the amplification rate for both components. Finally, we numerically confirm the validity of our direct envelope syntheses by the comparison with finite difference simulations of waves in 2‐D random media. The Markov approximation is accurate when the wavelength is shorter than the correlation distance and the fractional fluctuation is much smaller than the ratio of the correlation distance to the propagation distance.

Simulation of non-Gaussian multivariate stationary processes

Mathematical justifications are given for a simulation technique of multivariate non-Gaussian random processes and fields based on Rosenblatt's transformation 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.

Statistical solutions of the Khokhlov–Zabolotskaya–Kuznetsov equation for analyzing mechanisms of image quality enhancement in tissue harmonic imaging.

Statistical solutions of the Khokhlov–Zabolotskaya–Kuznetsov equation were obtained to analyze mechanisms contributing to image quality enhancement in tissue harmonic imaging (THI). The focus is on suppression of image clutter due to phase aberration and reverberation. Tissue heterogeneity is modeled with a random phase screen characterized by its variance and spatial correlation length. Solutions for the mean intensities of the linear (fundamental mode) and second-harmonic fields were derived from a focused Gaussian beam that is transmitted through a phase screen located an arbitrary distance from the source. The random phase variations of the screen are assumed to be small and described by a Gaussian autocorrelation function. The solutions are validated by comparison with ensemble averages of direct numerical simulations. A benefit of the analytical approach is separation of the different contributions to deformation of the beam by the phase screen, and the statistical approach is convenient for quantifying the merits of THI. The degree to which THI reduces beam deformation is assessed using a measure based on signal-to-clutter ratios introduced previously for this purpose by C. E. Bradley [Proceedings of 17th International Symposium on Nonlinear Acoustics (AIP, New York, 2006)]. Statistical solutions will also be presented for backscattering. [Work supported by NIH DK070618.]

Frequency-dependent attenuation of S-waves in the Kanto region, Japan

AbstractApparent, scattering, and intrinsicS-wave attenuations (QS−1,Qscat−1andQint−1) of the upper lithosphere in the Kanto region of Japan were measured in the 1- to 32-Hz frequency range using Multiple Lapse Time Window Analysis (MLTWA) for 115 borehole seismograms of local earthquakes. A new set of time windows for MLTWA, in which multiple isotropic scattering is assumed, was proposed and employed to estimate the frequency dependence ofS-wave attenuation parameters. Scattering attenuation was found to dominate intrinsic attenuation in theS-wave attenuation mechanism at low frequencies (<;2 Hz), whereas the opposite relation was observed at high frequencies. The transition is caused by the different frequency dependences ofQscat−1(∝f−1.5) andQint−1(∝f−0.7) at this frequency. Interestingly,Qscat−1is almost frequency independent at frequencies >8 Hz, which implies the self-similar nature of short-wavelength heterogeneities in the upper lithosphere. In terms of the upper lithosphere of the Kanto region, these results may indicate that the random heterogeneities characterized by the Gaussian autocorrelation function with a fractional fluctuationε≈ 10% and a correlation lengtha≈ 2 km are superimposed on the weak background self-similar heterogeneity.

Impact of Line-Edge Roughness on Double-Gate Schottky-Barrier Field-Effect Transistors

The impact of line-edge roughness (LER) on double-gate (DG) Schottky-barrier field-effect transistors (SBFETs) in the level of device and circuit was investigated by a statistical simulation. The LER sequence is statistically generated by a Fourier analysis of the power spectrum of the Gaussian autocorrelation function. The results show that SBFETs are more sensitive to the LER effect in the high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gs</sub> region and less sensitive in the subthreshold region compared with DG FinFETs. The aggressive fluctuation of drive current can be attributed to the variation of tunneling barrier width. Lowering the Schottky-barrier height and increasing the silicon-body thickness can suppress the parameter fluctuations from the LER effect. The simulation also shows that a 6T SRAM cell consisting of SBFETs is more vulnerable to noise disturbance than its counterpart consisting of FinFETs, particularly for the read operation, which is due to a larger mismatch of drivability of SBFETs within the cell.

The impact of line edge roughness on the stability of a FinFET SRAM

3D mixed-mode device-circuit simulation is presented to investigate the impact of line edge roughness (LER) on the stability of a FinFET SRAM. In this work, LER sequence is statistically generated by a Fourier analysis of the power spectrum of Gaussian autocorrelation function. The sensitivity of 20 nm FinFET SRAM of Read and Write static noise margins (SNM) to fin LER is evaluated. The results show that FinFET SRAM is more tolerant of disturbance in write operation than in read disturbance. The dependence of Read SNM on fin LER's root mean square (RMS) amplitude, fin thickness and supply voltage is also analyzed. Furthermore, methods to reduce the LER effect on the FinFET SRAM's read stability are introduced. Optimization of the cell ratio by a multiple-fin design, control of the access transistor's gate bias voltage and replacement of a 6T cell with an 8T cell are possible solutions to continue the scaling trend of SRAM in the nanoscale CMOS technology.

Radar Bistatic Configurations for Soil Moisture Retrieval: A Simulation Study

The possible contribution of bistatic radar measurements for bare soil moisture retrieval is investigated in this paper. A simulation study based on well-established electromagnetic models of rough surface scattering (both coherent and incoherent components) has been accomplished for this purpose. The retrieval accuracy has been evaluated by using both the Cramer-Rao lower bound and the error variance of a linear regression estimator, thus considering slightly different assumptions on retrieval conditions. Both methods have allowed us to identify the optimal system configurations in terms of observation directions, polarizations, and frequency. This identification has been carried out for single-polarization and multipolarization receivers and for the case in which bistatic measurements are complemented by monostatic ones, which are expected to be available through already-existing spaceborne synthetic aperture radars. The optimal systems have first been singled out by considering a Gaussian autocorrelation function (ACF) and a constant value of correlation length. Successively, the simulations for an exponential ACF and a variable correlation length have been analyzed, demonstrating that the results substantially remain the same. The comparison between the soil moisture estimation accuracy yielded by the optimal configurations and that provided by the standard monostatic radar has shown that a significant improvement in the quality of retrieval can be achieved by complementing bistatic and monostatic measurements.

Synthesis of vector‐wave envelopes in 3‐D random media characterized by a nonisotropic Gaussian ACF based on the Markov approximation

The earthquake source duration is short; however, the apparent duration time of observed seismogram increases with travel distance. The amplitude excitation is observed even on the transverse component for P waves, on the longitudinal component for S waves. These phenomena are well explained by scattering due to random velocity inhomogeneities around the global seismic ray. We directly synthesize vector‐wave envelopes in 3‐D random elastic media statistically characterized by a nonisotropic Gaussian autocorrelation function (ACF). The method uses the Markov approximation in the case that the wavelength is shorter than the correlation distance and the ray direction is parallel to one of the principal axes of the ACF. A spherical outgoing vector wavelet radiated from a point source in the random elastic media is used as a basic model for high‐frequency seismogram envelopes from micro‐earthquakes in the inhomogeneous lithosphere. The stochastic master equation for the two‐frequency mutual coherence function (TFMCF) of the potential field is analytically solved. The Fourier transform of TFMCF gives mean square (MS) envelopes of band‐pass filtered vector‐wave traces. If the ACF is axially symmetric around the ray direction, MS envelopes of vector components are analytically solved. The aspect ratio of the correlation distance in the longitudinal direction to that in the transverse direction is the key parameter for the envelope broadening and the excitation in the orthogonal component. Envelope broadening becomes longer and the transverse (longitudinal) component amplitude increases for a P wavelet (for an S wavelet) when the correlation distance in the transverse plane becomes smaller. When the vertical correlation distance is shorter than the horizontal one as seen in the real Earth, the envelope broadening is larger for horizontal raypaths compared with vertical raypaths.

Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA

In this paper, a fast exact numerical method, based on the method of moments, is developed to calculate the scattering by an object above a rough surface. N. Déchamps et al. have recently developed the PILE (Propagation-Inside-Layer Expansion) method for a stack of two one-dimensional rough interfaces separating homogeneous media. This method allows us to calculate separately and exactly the multiple scattering contributions inside the layer. This is done with a decomposition by block of the impedance matrix (the inverse of the impedance matrix of each interface and two coupling matrices are involved). The purpose of this paper is to extend the PILE method to the more general case of two illuminated surfaces and to apply it to an object located above a rough surface. In addition, to invert a matrix of large size, the Forward–Backward Spectral Acceleration (FB-SA) approach of complexity 𝒪(N) proposed by Chou and Johnson is applied. The new method, extended PILE combined with FB-SA, is tested on Perfectly Conducting (PC) circular and elliptic cylinders located above a rough surface (dielectric or PC) obeying a Gaussian process with Gaussian and exponential height autocorrelation functions.

Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA

A fast, exact numerical method based on the method of moments (MM) is developed to calculate the scattering from an object below a randomly rough surface. Déchamps et al. [J. Opt. Soc. Am. A23, 359 (2006)] have recently developed the PILE (propagation-inside-layer expansion) method for a stack of two one-dimensional rough interfaces separating homogeneous media. From the inversion of the impedance matrix by block (in which two impedance matrices of each interface and two coupling matrices are involved), this method allows one to calculate separately and exactly the multiple-scattering contributions inside the layer in which the inverses of the impedance matrices of each interface are involved. Our purpose here is to apply this method for an object below a rough surface. In addition, to invert a matrix of large size, the forward-backward spectral acceleration (FB-SA) approach of complexity O(N) (N is the number of unknowns on the interface) proposed by Chou and Johnson [Radio Sci.33, 1277 (1998)] is applied. The new method, PILE combined with FB-SA, is tested on perfectly conducting circular and elliptic cylinders located below a dielectric rough interface obeying a Gaussian process with Gaussian and exponential height autocorrelation functions.

Evaluation of strength of heterogeneity in the lithosphere from peak amplitude analyses of teleseismic short-period vectorPwaves

SUMMARY We quantitatively characterize the regional variations in the strength of heterogeneity in the lithosphere of the globe by analysing the observed seismogram envelopes of teleseismic P waves in the frequency band of 0.5‐4 Hz. We apply a theoretical scattering model based on the Markov approximation for a plane P wave propagating through the random medium characterized by a Gaussian autocorrelation function. Since this model presumes the vertical incidence of an impulsive plane wavelet, we first analyse teleseismic P waves from deep earthquakes occurring along the western Pacific regions. We measure the ratios of peak intensity of transverse components to that of the sum of the three components, and determine the quantity of randomness e 2 z/a, where e, a and z are fractional fluctuation, correlation distance and thickness of a heterogeneous structure, respectively. Although source time functions of shallow earthquakes are too complex to directly apply the scattering model, a good correlation between the ratios of peak amplitude and the normalized transverse amplitude, which is the square root of the energy partition of the P-coda waves into the transverse component, enables us to use the shallow earthquakes that occur widely around the world. As a result, the quantity e 2 z/a extends from 1.15 × 10 −4 to 6.34 × 10 −2 at 0.5‐1 Hz, 2.02 × 10 −3 to 1.89 × 10 −1 at 1‐2 Hz and 1.49 × 10 −4 to 1.89 × 10 −1 at 2‐4 Hz, which are in agreement with the results of previous studies using different methods. The spatial distribution of randomness almost agrees with various tectonic settings and roughly correlates with lateral variations of Lg coda Q and shear wave velocity perturbations at 80 km depth, suggesting that lateral heterogeneity extends from the shallow crust to uppermost mantle.

Envelope syntheses of cylindrical vector-waves in 2-D random elastic media based on the Markov approximation

Abstract Seismograms of microearthquakes are complex; however, their envelopes broaden as the travel distance increases. P-waves are recorded in transverse components, S-waves are recorded in the longitudinal component, and waves are observed at sites even in the nodal direction of the source radiation. These phenomena, which are typically found in short-period seismograms, can be interpreted to be the result of scattering due to lithospheric inhomogeneity. We report here our study of a simple statistical model in which the propagation of waves radiated from a point source in two-dimensional (2-D) random elastic media is characterized by a Gaussian autocorrelation function. For the case that the wavelength is shorter than the correlation distance, two methods based on the Markov approximation are introduced for the direct synthesis of vector wave envelopes. One is to analytically solve the stochastic equation for the two-frequency mutual coherence function; the validity of the solution is confirmed by using finite difference simulations. The second is to numerically solve the stochastic equation for the mutual coherence function. The two methods are equivalent, but the latter is applicable to nonisotropic source radiation. For the case of a point shear dislocation source, a peak delay from the onset and a smoothly decaying tail are found to be common to synthesized envelopes in all azimuths. Scattered waves are excited even at receivers in the nodal direction, and amplitudes become independent of the radiation pattern as lapse time increases.

Synthesis of vector wave envelopes in three‐dimensional random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: Spherical wave case

High‐frequency seismograms of microearthquakes show broadened envelopes with travel distance increasing. Scattered P waves appear in the transverse components and scattered S waves also appear in the longitudinal component. These phenomena are clear evidence of scattering due to random inhomogeneities spreading over the lithosphere. There is a statistical method to directly simulate wave envelopes when the medium inhomogeneity is weak and the wavelength is shorter than the medium‐scale length. Applying the Markov approximation to spherically outgoing vector waves radiated from a point source in three‐dimensional random elastic media statistically characterized by a Gaussian autocorrelation function, we synthesize their envelopes. Solving the stochastic master equation for the development of the two‐frequency mutual coherence function of potential with travel distance, and taking the Fourier transform of it, we newly derive analytic representation of mean square (MS) envelopes of three component vector waves by using the elliptic theta function. For P waves, the radial component shows a broadened envelope with a delayed peak and a smoothly decaying tail, where the peak MS amplitude is approximately proportional to the minus fourth power of travel distance. The transverse component shows a broadened envelope with a smaller peak value and a longer peak delay and a longer tail compared with those in the radial component, where the peak MS amplitude is proportional to the minus third power of travel distance. S wave envelopes are also solved. S wave envelope broadening is larger than that of P wave broadening. These features well explain qualitatively the characteristics of seismogram envelopes of microearthquakes.

Influence of the precision of spectral backscatter measurements on the estimation of scatterers size in cancellous bone

The goal of this study is to propose a model for the ultrasonic frequency-dependent backscatter coefficient in femoral cancellous bone. This model has been developed with success to predict backscatter in human calcaneal bone [Jenson, Ultr. Med. Biol. 2003]. A weak scattering model is used and the backscatter coefficient is expressed in terms of a Gaussian autocorrelation function of the medium. The backscatter coefficient is computed and comparison is made with experimental data for 37 specimens and for frequency ranging from 0.4 to 1.2 MHz. An excellent agreement between experimental data and predictions is found for both the magnitude and the frequency-dependence of the backscatter coefficient. Then, a nonlinear regression is performed for each specimen, and the mean trabecular thickness is estimated. Experimental data and theoretical predictions are averaged over the 37 specimens. We also find a close agreement between theoretical predictions obtained using the Gaussian autocorrelation function (scatterer size = 134 ± 15 μm) and the mean trabecular thickness (Tb.Th = 132 ± 12 μm) derived from the analysis of bone 3-D micro-architecture using high-resolution micro-tomography. However, the correlation between individual experimental and estimated Tb.Th values is moderate ( R 2 = 0.44). The performance of the estimator are limited mainly by two factors: interference noise due to random positioning of the scatterers and attenuation. We show that the fundamental limitation of our estimator due to the speckle noise is around 5 μm for trabecular thickness estimation. This limitation is lower than the observed biological variability which is around 30 μm and should not be a limiting factor for individual prediction. A second limitation is the tremendous attenuation encountered in highly scattering media such as cancellous bone, which results in highly damped backscatter signals. The compensation for attenuation is difficult to perform, and it may be a critical point that limits the precision of the estimator.

Velocity shift in two-dimensional anisotropic random media using the Rytov method

SUMMARY When high-frequency waves propagate through a randomly inhomogeneous medium, the apparent wave velocity is larger than the spatial average of the velocity distribution. The difference between the two velocities is referred to as the velocity shift. The present study formulates the velocity shift in 2-D anisotropic random media using the Rytov method. Anisotropic random media in (x, z) coordinates are characterized by autocorrelation functions (ACFs) with a long correlation distance along the x-axis and a short correlation distance along the z-axis, where the velocity structure varies smoothly along the x-axis and rapidly along the z-axis. Note that the spatial average values of the velocity distribution along the x-axis and the z-axis are the same. The formulation gives the velocity shift for general types of ACF. An analytic solution is obtained for the case of a Gaussian ACF. To examine the reliability of the Rytov method, the velocity shift is estimated from numerical simulations of wave propagation using Ricker wavelets with dominant frequencies 80 and 40 Hz. The random media are realized with a spatial average velocity of 2700 m s−1 and an exponential ACF with 5 per cent rms fractional velocity fluctuation, a correlation distance of 80 m along the x-axis and 40 m along the z-axis. Numerical simulations show that waves apparently propagate faster with increasing travel distance, frequency and the angle of incidence measured from the z-axis to the global ray direction. For example, in the case of the 80 Hz Ricker wavelet at a distance of 520 m, the values of the velocity shift are about 0.9 and 0.3 per cent along the x-axis and the z-axis, respectively. The Rytov method quantitatively explains these general tendencies except for short travel distances along the x-axis. The discrepancy at short travel distances could be due to the small-angle scattering approximation and the long travel distance approximation employed in the Rytov method. Observations of P-wave velocity anisotropy have usually been interpreted in terms of preferred orientations of cracks and minerals in past studies. However this study indicates that wave scattering due to anisotropic random media can provide an alternative explanation for those observations.

Synthesis of vector wave envelopes in three‐dimensional random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: Plane wave case

High‐frequency seismograms of microearthquakes observed at long travel distances have longer apparent durations than the source durations caused by scattering due to lithospheric inhomogeneity. Frequency dependence of the peak delay from the onset and the broadening of seismogram envelope well reflect the spectra of random velocity inhomogeneity. Here we propose a formulation for the envelope synthesis of vector waves in three‐dimensional random elastic media in the case that wavelengths are smaller than the characteristic scale of random inhomogeneity. A stochastic master equation for the two‐frequency mutual coherence function (TFMCF) of potential field is derived on the basis of the Markov approximation, which is a stochastic extension of the phase screen method for solving the parabolic wave equation. From the Fourier transform of TFMCF at a given travel distance, we are able to synthesize the mean square traces of three vector wave components. For the incidence of an impulsive plane P wavelet to random elastic media characterized by a Gaussian autocorrelation function, the mean square envelope of each component at a long travel distance is analytically written by using an elliptic theta function. Each synthesized envelope shows peak delay from the onset and broadening with travel distance increasing; however, the peak delay of the transverse component is larger than that of the longitudinal component. For the same fractional velocity fluctuation, the envelope broadening for S waves is larger than that for P waves.