Scatterer Number Density Research Articles

Vegetation characteristics and underlying topography from interferometric radar

This paper formulates and demonstrates methods for extracting vegetation characteristics and underlying ground surface topography from interferometric synthetic aperture radar (INSAR) data. The electromagnetic scattering and radar processing, which produce the INSAR observations, are modeled, vegetation and topographic parameters are identified for estimation, the parameter errors are assessed in terms of INSAR instrumental performance, and the parameter estimation is demonstrated on INSAR data and compared to ground truth. The fundamental observations from which vegetation and surface topographic parameters are estimated are (1) the cross‐correlation amplitude, (2) the cross‐correlation phase, and (3) the synthetic aperture radar (SAR) backscattered power. A calculation based on scattering from vegetation treated as a random medium, including the effects of refractivity and absorption in the vegetation, yields expressions for the complex cross correlation and backscattered power in terms of vegetation characteristics. These expressions lead to the identification of a minimal set of four parameters describing the vegetation and surface topography: (1) the vegetation layer depth, (2) the vegetation extinction coefficient (power loss per unit length), (3) a parameter involving the product of the average backscattering amplitude and scatterer number density, and (4) the height of the underlying ground surface. The accuracy of vegetation and ground surface parameters, as a function of INSAR observation accuracy, is evaluated for aircraft INSAR, which is characterized by a 2.5‐m baseline, an altitude of about 8 km, and a wavelength of 5.6 cm. It is found that for ≈0.5% accuracy in the INSAR normalized cross‐correlation amplitude and ≈5° accuracy in the interferometric phase, few‐meter vegetation layer depths and ground surface heights can be determined from INSAR for many types of vegetation layers. With the same observational accuracies, extinction coefficients can be estimated at the 0.1‐dB/m level. Because the number of parameters exceeds the number of observations for current INSAR data sets, external extinction coefficient data are used to demonstrate the estimation of the vegetation layer depth and ground surface height from INSAR data taken at the Bonanza Creek Experimental Forest in Alaska. This demonstration shows approximately 5‐m average ground truth agreement for vegetation layer depths and ground‐surface heights, with a clear dependence of error on stand height. These errors suggest refinements in INSAR data acquisition and analysis techniques which will potentially yield few‐meter accuracies. The information in the INSAR parameters is applicable to a variety of ecological modeling issues including the successional modeling of forested ecosystems.

Origin of the Doppler ultrasound spectrum from blood.

Backscattering of an incident ultrasound beam by blood gives rise to a Doppler spectrum whose characteristics are affected not only by the velocity distribution, but also by certain basic aspects of the RBC's behavior. Starting from fundamental assumptions, an explicit expression is derived for the Doppler spectrum in terms of the variance in the scatterer number density and the backscattering cross section. This shows that the Doppler power at a given frequency is weighted by the backscattering cross section and the manner in which the red blood cells (RBC's) are packed (packing factor). Since spatial variations in the flow field can result in changes in the variance and backscattering cross section, the mean Doppler frequency will not necessarily be proportional to the mean flow through the sample volume. Experimental results for two different flow fields are used to illustrate these effects. The implications of these findings are discussed in relation to volumetric flow estimation and the power mode display used in some color Doppler flow imaging systems.

Experimental demonstration of the frequency dependence of the effective scatterer number density

In previous papers, a data reduction method for determining a frequency-dependent ‘‘effective scatterer number density’’ in ultrasound was reported [Chen et al., J. Acoust. Soc. Am. 95, 77–95 (1994); 96, 2556–2563 (1994)]. The effective scatterer number density is the actual number density of scatterers multiplied by a frequency-dependent factor that depends on the differential scattering cross sections of the scatterers. Evaluation of this technique based on measurements applied to a phantom with a distribution of scatterer diameters is reported here. This condition yields an effective scatterer number density that depends on the dominant scatterer concentration at the analysis frequency. Measured scatterer number densities at 3.5, 5, and 8 MHz were in agreement with theoretical computations of this parameter for the phantom.

Speckle analysis using signal to noise ratios based on fractional order moments.

The SNR (signal-to-noise ratio) of the echo envelope image is a monotonically-increasing function of scatterer number density. Various SNRs, like amplitude SNR and intensity SNR, can be used to quantify the scatterer density. The problem of using a SNR based on higher order moments like the intensity SNR is that they require large sample sizes to obtain estimates with high confidence (the variance of the estimate becomes large for higher moments). In this paper, we consider SNRs based on fractional order moments (moments of order less than 1), and obtain mathematical analyses of their properties using the K distribution, which has been shown to be a good model for the density function of backscatter echo envelope signal. Statistics of SNRs based on fractional moment are derived and appear to be more robust and useful than the amplitude and intensity SNRs previously studied. The SNRs based on fractional order moments have greater dynamic range and the sample size requirements are smaller than those for integral order moment SNRs, like amplitude SNR or intensity SNR. Thus, SNRs based on fractional order moments could be used to better quantify the variations in scatterer density which can be used for tissue classification problems.

Speckle Analysis Using Signal to Noise Ratios Based on Fractional Order Moments

The SNR (signal-to-noise ratio) of the echo envelope image is a monotonically-increasing function of scatterer number density. Various SNRs, like amplitude SNR and intensity SNR, can be used to quantify the scatterer density. The problem of using a SNR based on higher order moments like the intensity SNR is that they require large sample sizes to obtain estimates with high confidence (the variance of the estimate becomes large for higher moments). In this paper, we consider SNRs based on fractional order moments (moments of order less than 1), and obtain mathematical analyses of their properties using the K distribution, which has been shown to be a good model for the density function of backscatter echo envelope signal. Statistics of SNRs based on fractional moment are derived and appear to be more robust and useful than the amplitude and intensity SNRs previously studied. The SNRs based on fractional order moments have greater dynamic range and the sample size requirements are smaller than those for integral order moment SNRs, like amplitude SNR or intensity SNR. Thus, SNRs based on fractional order moments could be used to better quantify the variations in scatterer density which can be used for tissue classification problems.

Non-Rayleigh statistics of ultrasonic backscattered echo from tissues

The envelope of the backscattered signal from tissues can exhibit non-Rayleigh statistics if the number density of scatterers is small or if the variations in the scattering cross sections are random. The K distribution which has been used extensively in radar, is introduced to model this non-Rayleigh behavior. The generalized K distribution is extremely useful since it encompasses a wide range of distributions such as Rayleigh, Lognormal, and Rician. Computer simulations were conducted using a simple one-dimensional discrete scattering model to investigate the properties of the echo envelope. In addition to cases of low number densities, significant departures from Rayleigh statistics were seen as the scattering cross sections of the scatterers become random. The validity of this model was also tested using data from tissue mimicking phantoms. Results indicate that the density function of the envelope can be modeled by the K distribution and the parameters of the K distribution can provide information on the nature of the scattering region in terms of the number density of the scatterers as well as the scattering cross sections of the scatterers in the range cell. [Work was supported by NSF Grant No. BCS-9207385.]

Berry phases for three-neutrino oscillations in matter

We show that nontrivial Berry phase factors can arise in the evolution operator of the standard, three-flavor-neutrino system when a neutrino beam propagates through a medium whose parameters - number densities of scatterers - vary cyclically with distance. The conditions are established whereby the Berry phases take their extreme (resonance) values independently of the density distribution in the background. This universal “geometric” resonance is capable to change drastically the survival and transition probabilities for neutrinos in matter.

A method for determination of frequency-dependent effective scatterer number density.

Previous researchers have shown that for low scatterer concentrations, the scatterer number density in an ultrasound pulse-echo experiment can be obtained from ratios of the fourth to the second moment of the backscattered signal. In this paper a new method is presented for determining an effective scatterer number density, which is the actual number density multiplied by a frequency-dependent factor that depends on the differential scattering cross sections of all scatterers. The method of data reduction goes beyond the work of previous authors in that, in addition to accounting for the possibility that different sets of scatterers may dominate the echo signal at different frequencies, it also explicitly retains both the temporal nature of the data acquisition and the properties of the ultrasound field in the data reduction. Tests of the method in phantoms yielded good agreement between measured values of the effective scatterer number density and number densities calculated using first principles.

Electromagnetic scattering from turbulent plasmas.

A self-consistent multiple-scattering theory of vector electromagnetic waves scattered from a turbulent plasma is presented. This approach provides a general and systematic treatment to all orders in turbulence of the scattering of electromagnetic waves in terms of the properties of the turbulent structure of the scattering system and is applicable in the full regime from underdense to overdense plasmas. To illustrate the theory, a plasma consisting of a finite number density of discrete scatterers with a simple geometry and statistical properties is chosen. In this approach the exact solution for a single scatterer is obtained first. From it the configuration-dependent solution for N scatterers is constructed. Rather than solving explicitly for this solution and then averaging, the averaging operation will be taken first in order to find an approximate equation obeyed by the mean or coherent field. The coherent and incoherent scattering are then determined in terms of the coherent field and the backscatter is evaluated. The coherent and incoherent scattering, our principal results, are expressed in a plane-wave basis in a form suitable for numerical computation. A number of interesting phenomena which may readily be incorporated into the theory are indicated.

Solution of stochastic integral equation for multiple scattering by method of Taylor series expansion

Using the solution for multidimensional integral equation obtained in the author's earlier paper by the Taylor expansion method, the stochastic integral equation for a multiply scattered field is solved in stochastic form. From this, the mean value and space‐time correlation function of the multiply scattered field are obtained, in terms of the shape, size, number density and distribution region of scatterers. Both the near and far field are obtained. The results show that, for the first‐order approximation, the results agree with those obtained from the Born approximation, for the second‐ and third‐order approximations, the results agree with those obtained from the Neumann series expansion method in [1]. Therefore, these results also show better agreement with experimental results than those obtained from Twersky's formula.

Angularly resolved light scattering from silica glass: Determination of the critical angle and scatterer density.

Light-scattering studies of chemical-vapor-deposition silica glass in the angular range 10\ifmmode^\circ\else\textdegree\fi{}--50\ifmmode^\circ\else\textdegree\fi{} are presented. The patterns obtained are characterized by intensity fluctuations which are slow and very large at small angles while rapid at large angles. The intensity statistics of angularly resolved scattering spectra from glass are analyzed analytically. Numerical calculations are performed in order to localize the critical scattering angle ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$ which marks the transition of the intensity distribution from negative exponential to nonexponential statistics. The crucial role of angular resolution in the determination of ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$ is stressed. The obtained value ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$=35.8\ifmmode^\circ\else\textdegree\fi{} leads to the determination of the number density of scatterers, 4.2\ifmmode\times\else\texttimes\fi{}${10}^{17}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$, which is in agreement with data reported elsewhere. The results of fixed-angle measurements obtained at large aperture area are also discussed. A brief discussion of Fourier transforms of angular scattering patterns is given, though Fourier analysis was found to be an inefficient alternative method for the determination of ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$. The transforms obtained show random, speckle-like fluctuations if the input scattered intensities fluctuate randomly with the angle of observation.

Microwave propagation at X band in cylindrical-shaped forest components: attenuation observations

Experimental results are presented for attenuation measurements performed on cylindrical-shaped forest components. Microwave propagation in simulated canopies composed of bare deciduous twigs and leafy coniferous branches was investigated at 9 GHz. In order to account for the influence of natural component parameters, the experiments were carried out under laboratory conditions where those parameters could be easily and fully controlled. The influence of the physical parameters (number density of scatterers and gravimetric water content) and the geometrical parameters (dimensions and orientation of the scatterers) on the angular and polarization attenuation behaviors is pointed out. These results supplement previous attenuation observations obtained with natural cylindrical elements.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Density effects on high-n molecular Rydberg states: CH3I and C6H6 in H2 and Ar

The absorption spectra of high-n Rydberg states of methyl iodide and benzene perturbed by varying number densities of hydrogen or argon, range 0.9×1020–10.5×1020 cm−3 for H2 and 0.6×1020–7.5×1020 cm−3 for Ar, have been investigated. The high-n molecular states of both absorbers were found to shift linearly with the number density of atomic Ar and molecular H2 scatterers. The Fermi formula modified by the Alekseev–Sobel’man polarization term provides an excellent fit of the shift data. The electron scattering lengths obtained are: 0.93 a0 for H2 and −1.63 a0 for Ar using the CH3I absorber; and 0.99 a0 for H2 and −1.57 a0 for Ar using the C6H6 absorber. The electron scattering lengths for H2 and Ar agree with the results of an empirical model that correlates scattering lengths and the polarizabilities α(spherical) for inert atoms and α2(nonspherical) for H2 molecule.

Tissue characterization based on scatterer number density estimation

The authors propose a robust model for characterizing the statistical nature of signals obtained from ultrasonic backscatter processes. The model can accommodate frequency-dependent attenuation, spatially varying media statistics, arbitrary beam geometries, and arbitrary pulse shapes. On the basis of this model, statistical schemes are proposed for estimating the scatterer number density (SND) of tissues. The algorithm for estimating the scatterer number incorporates measurements of both the statistical moments of the backscattered signals and the point spread function of the acoustic system. The number density algorithm has been applied to waveforms obtained from ultrasonic phantoms with known number densities and in vitro mammalian tissues. There is an excellent agreement among theoretical, histological, and experimental results. The application of this technique for noninvasive clinical tissue characterization is discussed.

A maximum entropy derivation of the integrated direct methods–isomorphous replacement or integrated direct methods–anomalous scattering probability distributions

It is shown that taking the appropriate terms from a series expansion of the Shannon-Jaynes entropy of a density map subject to intensity constraints gives the standard direct methods structure factor probability distribution functions. The use of two maps, one to represent a native structure, the other to represent either heavy atoms or the number density of anomalous scatterers, and the application of a similar expansion to the total entropy of both maps rapidly gives either the integrated direct methods-single isomorphous replacement or the integrated direct methods-anomalous scattering probability densities.

On estimating the number density of random scatterers from backscattered acoustic signals

When an acoustic pulse interacts with an inhomogeneous, attenuating medium, the backscattered signals exhibit random fluctuations which are correlated with the physical properties of the medium. This paper proposes a robust model for characterizing the statistical nature of these backscattered signals. This model takes into account frequency-dependent attenuation, spatially varying media statistics, arbitrary beam geometries, and arbitrary pulse shapes. Based on this model, statistical estimation schemes are proposed for estimating both the attenuation coefficient and scatterer number density of the medium. Using appropriate simplifying assumptions, it is shown that this model is consistent with attenuation estimation algorithms currently used for ultrasonic tissue characterization. A statistical approach for estimating the number density of scatterers is described and its theoretical performance is evaluated. The algorithm for estimating the scatterer number density incorporates measurements of both the statistical moments of the backscattered signals and the point spread function of the acoustic system. The number density algorithm has been applied to simulated waveforms, waveforms obtained from ultrasonic phantoms with known number densities, and in vitro mammalian tissues. There is an excellent agreement between theoretical, simulation, and experimental results. The application of this technique to ultrasonic tissue characterization is also discussed.

Acoustic backscatter from highly dense scattering arrays

Acoustic backscatter signals from several specimens containing high number densities of small spherical scatterers were measured in order to compare the signals obtained with those calculated from the independent scatterer model for the case of high scatterer concentration and long wavelengths. The scatterers were prepared by precipitation of less dense second-phase spherical particles from a glass host by controlling the melt chemistry and cooling rate. Statistics on the size, number, and distribution of the scatterers were obtained from microscopic observation. The results show that the backscatter signal energy measured is directly related to the mean interparticle spacing between scatterers. The reason for this relationship is shown, from the independent scatterer model, to come from the fact that the distribution of interparticle spacings can be represented by a Gaussian probability with a specified mean. Results of the calculation and comparison with experimental results illustrating the effect of the spacing distribution on the scattering structure factor are presented. [Work supported by the Interior Department's Bureau of Mines through Department of Energy.]

A statistical approach to determining the number density of random scatterers from backscattered pulses

This paper describes a method of using the statistical character of the waveform backscattered from random scatterers to estimate the scatterer number density. The moments of the probability density function of the backscattered intensity depend on the scatterer number density. Expressions exist for these moments in terms of the number of scatterers contributing to the echo signal. These expressions can give an estimate for the scatterer number density, provided the resolution cell size of the backscattering configuration is known, and provided the number of scatterers in the resolution cell is small. This latter constraint is equivalent to a requirement that the statistics of the return signal envelope deviate significantly from a Rayleigh distribution. The validity of these expressions is investigated by analyzing the acoustic waveform backscattered from a randomized volume distribution of polystyrene spheres suspended in water when insonified by a short acoustic pulse. Experimentally, the second-order moment of intensity is found to be given the most accurate predictions of number density. A specific application suggested for this work is acoustic fish-stock assessment. Other possible applications are ultrasonic tissue characterization and acoustic ocean bottom identification.

Ultrasonic scattering properties of three random media with implications for tissue characterization.

Random media with different structural properties were used to simulate some of the differences in liver morphology that may occur with disease. First, a reference medium consisting of glass spheres in agar was studied to verify the accuracy and precision of the data obtained with our equipment and processing procedures. Then, studies were conducted on a pair of media comprised of graphite particles in gelatin, one of the pair with twice as many particles as the other. Finally, studies were carried out on a set of media composed of Sephadex particles in water. Three samples were employed, each with a different size of Sephadex. The average differential scattering cross section per unit volume sigma sd (v) of each media was obtained as a function of scattering angle v and frequency. The measured sigma sd were compared with predictions based on models of scattering from the media. The agreement between the measured and predicted sigma sd of the glass sphere medium was excellent. The graphite medium with twice the number of particles as the other was observed to scatter twice as much power as the other. The shape of the angular scattering pattern measured from each size of Sephadex followed the prediction reasonably well. The largest size exhibited marked variations in the shape of sigma sd as a function of frequency, while the absolute magnitude of sigma sd of the smallest size grade was extremely sensitive to frequency. Our results suggest that the dependence of sigma sd (v) on scatterer number density and size, both in absolute magnitude and shape, can provide reliable information which may be useful in the diagnosis of some diseases.

Overall properties of a cracked solid

AbstractThe differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.