Absence Of Attenuation Research Articles

Plane-wave and cylindrical-wave acoustic reflection from a marine sediment with layering representative of the New England Mud Patch.

An analysis is presented of reflection from a marine sediment consisting of a homogeneous mud layer overlying a sand-mud basement, the latter with an upward-refracting, inverse-square sound speed profile. Such layering is representative of the sediment at the New England Mud Patch (NEMP). By applying appropriate integral transforms and their inverses to the Helmholtz equations for the ocean and the two sediment layers, along with the boundary conditions, a Sommerfeld-Weyl type of wavenumber integral is obtained for the cylindrical-wave reflection coefficient of the sediment, R. A stationary phase evaluation of this integral yields a closed-form expression for the plane-wave reflection coefficient, R0. In the absence of attenuation, the plane-wave solution exhibits total reflection up to a critical grazing angle, ac, but when attenuation in the sediment is introduced, the region of total reflection in |R0| is replaced by a sequence of contiguous peaks. With realistic levels of sediment attenuation, the cylindrical-wave solution, |R|, exhibits a quasi-critical grazing angle, less than ac, which is strongly dependent on the source-plus-receiver height above the seabed, which is mildly dependent on the depth of the mud layer but is essentially independent of frequency. Such behavior is consistent with independent experimental observations at the NEMP.

Generation of time-independent torque by ultrasonic guided waves

The excitation of acoustic waves by a unidirectional transducer, integrated in a piezoelectric cylindrical tube or disk, can lead to a time-independent torque. This phenomenon, demonstrated earlier in experiments and analyzed with coupling-of mode theory, is explained in detail, starting on the level of lattice dynamics of a piezoelectric crystal. Expressions are derived for the stationary torque in the form of integrals over the volume or surface of the piezoelectric, involving the electric potential and displacement field associated with the acoustic waves generated by the transducer.Simulations have been carried out with the help of the finite element method for a tube made of PZT for two cases: A pre-defined potential on the surface of the tube and metal electrodes buried in the piezoelectric. The displacement field and electric potential of the high-frequency acoustic waves (between 200 and 300 kHz) were computed and used in the evaluation of the integrals. The attenuation due to various loss channels of the acoustic waves in the system has been analyzed in detail, as this plays a crucial role for the efficiency of torque generation. It is conjectured that time-reversal symmetry, present in the absence of attenuation, prohibits the generation of a static torque at least in the linear limit.A qualitative comparison is made between the simulations and earlier experiments. Discrepancies are attributed to lack of knowledge of the relevant material constants of the piezoelectric and to a simplified modeling of the electrode geometry in the cylindrical tube, which was necessary for reasons of numerical accuracy.

Color-resolved Cherenkov imaging allows for differential signal detection in blood and melanin content.

High-energy x-ray delivery from a linear accelerator results in the production of spectrally continuous broadband Cherenkov light inside tissue. In the absence of attenuation, there is a linear relationship between Cherenkov emission and deposited dose; however, scattering and absorption result in the distortion of this linear relationship. As Cherenkov emission exits the absorption by tissue dominates the observed Cherenkov emission spectrum. Spectroscopic interpretation of this effects may help to better relate Cherenkov emission to ionizing radiation dose delivered during radiotherapy. In this study, we examined how color Cherenkov imaging intensity variations are caused by absorption from both melanin and hemoglobin level variations, so that future Cherenkov emission imaging might be corrected for linearity to delivered dose. A custom, time-gated, three-channel intensified camera was used to image the red, green, and blue wavelengths of Cherenkov emission from tissue phantoms with synthetic melanin layers and varying blood concentrations. Our hypothesis was that spectroscopic separation of Cherenkov emission would allow for the identification of attenuated signals that varied in response to changes in blood content versus melanin content, because of their different characteristic absorption spectra. Cherenkov emission scaled with dose linearly in all channels. Absorption in the blue and green channels increased with increasing oxy-hemoglobin in the blood to a greater extent than in the red channel. Melanin was found to absorb with only slight differences between all channels. These spectral differences can be used to derive dose from measured Cherenkov emission. Color Cherenkov emission imaging may be used to improve the optical measurement and determination of dose delivered in tissues. Calibration for these factors to minimize the influence of the tissue types and skin tones may be possible using color camera system information based upon the linearity of the observed signals.

Joint Estimation of Metal Density and Attenuation Maps with Pencil Beam XFET.

X-ray fluorescence emission tomography (XFET) is an emerging imaging modality that images the spatial distribution of metal without requiring biochemical modification or radioactivity. This work investigates the joint estimation of metal and attenuation maps with a pencil-beam XFET system that allows for direct metal measurement in the absence of attenuation. Using singular value decomposition on a simplified imaging model, we show that reconstructing metal and attenuation voxels far from the detector is an ill-conditioned problem. Using simulated data, we develop and compare two image reconstruction methods for joint estimation. The first method alternates between updating the attenuation map with a separable paraboloidal surrogates algorithm and updating the metal map with a closed-form solution. The second method performs simultaneous joint estimation with conjugate gradients based on a linearized imaging model. The alternating approach outperforms the linearized approach for iron and gold numerical phantom reconstructions. Reconstructing an (8 cm)3 object containing gold concentrations of 5 mg/cm3 and an unknown beam attenuation map using the alternating approach yields an accurate gold map (NRMSE = 0.19) and attenuation map (NRMSE = 0.14). This simulation demonstrates an accurate joint reconstruction of metal and attenuation maps, from emission data, without previous knowledge of any attenuation map.

The ‘ramp filter’ artifact in filtered back projection is due to attenuation not ramp filtering

Artifactually decreased counts in regions of the left ventricle adjacent to intense splanchnic activity have been described in single photon emission computed tomography imaging and attributed to the side lobes of ramp filtering, a crucial step in filtered back projection. However, filtered back projection is based on a mathematical proof that makes no assumptions about the distribution of activity implying there is another cause for this artifact. Computer simulations of a left ventricle with varying intensity, proximity and size of adjacent splanchnic activity was created with and without attenuation. Both filtered back projection and an iterative expectation maximization algorithm were performed. Volume weighted bullseyes were generated then the ratio of average activity in regions in the anterior and inferior walls of the bullseye were calculated. In the absence of attenuation, there was no effect on uptake despite intense adjacent splanchnic activity. The effect of attenuation alone was mild but was greatly amplified by adjacent splanchnic uptake. This effect varied with the intensity, proximity and size of adjacent activity. Moving the attenuation to the apex of the left ventricle produced a paradoxical increase in counts to the inferior wall. The effect on iterative reconstruction was less pronounced than with filtered back projection and the paradoxical increase with attenuation at the apex did not occur. The ramp filter artifact in single photon emission computed tomography reconstruction is due to attenuation, not ramp filtering per se. Adjacent splanchnic activity can result in either an increase or decrease in counts. These effects are less pronounced and more consistent with iterative reconstruction.

Some Theoretical and Experimental Extensions Based on the Properties of the Intrinsic Transfer Matrix.

The work approaches new theoretical and experimental studies in the elastic characterization of materials, based on the properties of the intrinsic transfer matrix. The term ‘intrinsic transfer matrix’ was firstly introduced by us in order to characterize the system in standing wave case, when the stationary wave is confined inside the sample. An important property of the intrinsic transfer matrix is that at resonance, and in absence of attenuation, the eigenvalues are real. This property underlies a numerical method which permits to find the phase velocity for the longitudinal wave in a sample. This modal approach is a numerical method which takes into account the eigenvalues, which are analytically estimated for simple elastic systems. Such elastic systems are characterized by a simple distribution of eigenmodes, which may be easily highlighted by experiment. The paper generalizes the intrinsic transfer matrix method by including the attenuation and a study of the influence of inhomogeneity. The condition for real eigenvalues in that case shows that the frequencies of eigenmodes are not affected by attenuation. For the influence of inhomogeneity, we consider a case when the sound speed is varying along the layer’s length in the medium of interest, with an accompanying dispersion. The paper also studies the accuracy of the method in estimating the wave velocity and determines an optimal experimental setup in order to reduce the influence of frequency errors.

On causality of wave motion in nonlocal theories of elasticity: a Kramers–Kronig relations study

In this paper, the subject of the principle of causality and the physical realizability in the wave motion characteristics within a linear nonlocal elastic medium is examined. The principle of primitive causality is examined via Kramers–Kronig (K–K) relations and the principle of relativistic causality or the Einstein causality is examined via wave motion responses. Gradient as well as integral type nonlocality has been considered. Methodology here involves a Fourier frequency domain based spectral analysis and the wave motion characteristics include: wave modes, group speeds and frequency response function. In general, due to existence of atleast one of the non-physical features in the characteristics, violation of the causality is observed. The non-physical features include: existence of infinitesimally small or zero speeds; existence of very large or infinite speeds; existence of negative speeds; and absence of attenuation of waves. Violation to the primitive causality takes place as a disagreement to the K–K relations, either due to existence of negative and/or zero group speeds or due to absence of wave attenuation in the possible wave modes. Violation to Einstein causality is observed due to existence of infinitely large group speeds. Agreement to the primitive causality is achieved due to the presence of both wave dispersion and wave attenuation in the wave modes. Although existence of infinitely large group speeds violates Einstein causality, however, violation of the primitive causality is not observed. Upon considering only the physically realizable wavemodes, it is observed that, a local Neumann type boundary condition may be sufficient to conduct a wave motion study in a class of nonlocal boundary value problems. As an application of the primitive causality to the Fourier domain analysis, the wavenumbers from the K–K relations are utilized to demonstrate a mitigation effect of certain non-physical features in the wave motion responses.

Trade-off of Elastic Structure and Q in Interpretations of Seismic Attenuation

The quality factor Q is an important phenomenological parameter measured from seismic or laboratory seismic data and representing wave-energy dissipation rate. However, depending on the types of measurements and models or assumptions about the elastic structure, several types of Qs exist, such as intrinsic and scattering Qs, coda Q, and apparent Qs observed from wavefield fluctuations. We consider three general types of elastic structures that are commonly encountered in seismology: (1) shapes and dimensions of rock specimens in laboratory studies, (2) geometric spreading or scattering in body-, surface- and coda-wave studies, and (3) reflectivity on fine layering in reflection seismic studies. For each of these types, the measured Q strongly trades off with the (inherently limited) knowledge about the respective elastic structure. For the third of the above types, the trade-off is examined quantitatively in this paper. For a layered sequence of reflectors (e.g., an oil or gas reservoir or a hydrothermal zone), reflection amplitudes and phases vary with frequency, which is analogous to a reflection from a contrast in attenuation. We demonstrate a quantitative equivalence between phase-shifted reflections from anelastic zones and reflections from elastic layering. Reflections from the top of an elastic layer followed by weaker reflections from its bottom can appear as resulting from a low Q within or above this layer. This apparent Q can be frequency-independent or -dependent, according to the pattern of thin layering. Due to the layering, the interpreted Q can be positive or negative, and it can depend on source–receiver offsets. Therefore, estimating Q values from frequency-dependent or phase-shifted reflection amplitudes always requires additional geologic or rock-physics constraints, such as sparseness and/or randomness of reflectors, the absence of attenuation in certain layers, or specific physical mechanisms of attenuation. Similar conclusions about the necessity of extremely detailed models of the elastic structure apply to other types of Q measurements.

Influence of Hyper-Alkaline pH Leachate on Mineral and Porosity Evolution in the Chemically Disturbed Zone Developed in the Near-Field Host Rock for a Nuclear Waste Repository

This paper evaluates the effect of hyper-alkaline (NaOH/KOH) leachate on the mineralogy and porosity of a generic quartzo-feldspathic host rock for intermediate- and low-level nuclear waste disposal following permeation of the cementitious repository barrier by groundwater. The analysis is made with reference to expected fluid compositions that may develop by contact of groundwater with the cementitious barrier to form a chemically disturbed zone (CDZ) in the adjacent host rock, as informed by relevant natural analogue sites. Theoretical analysis and numerical modelling is used to explore the influence of different host rock mineral assemblages on changes in pore fluid chemistry, multiple mineral dissolution and precipitation reactions and matrix porosity within the CDZ under these conditions. The numerical modelling accounts for kinetic and surface area effects on the mineral transformation and porosity development for periods of up to 10,000 years travel time from the repository and ambient temperature of \(20\,^{\circ }\hbox {C}\). The analysis shows that dissolution of quartz, feldspar and muscovite in the host rock, by the hyper-alkaline waste leachate, will create relatively high concentrations of dissolved Si and Al in the pore fluid, which migrates as chemical fronts within the CDZ. Precipitation of secondary mineral phases is predicted to occur under these conditions. The increase in matrix porosity that arises from dissolution of primary aluminosilicate minerals is compensated by a reduction in porosity due to precipitation of the secondary phases, but with a net overall increase in matrix porosity. These coupled physical and geochemical processes are most important for contaminant transport in the near-field zone of the CDZ and are eventually buffered by the host rock within 70 m of the repository for the 10,000 year travel time scenario. The predicted changes in matrix porosity may contribute to increased transport of radionuclides in the host rock, in the absence of attenuation by other mechanisms in the CDZ.

Environmental radioactivity in the UK: the airborne geophysical view of dose rate estimates

This study considers UK airborne gamma-ray data obtained through a series of high spatial resolution, low altitude surveys over the past decade. The ground concentrations of the naturally occurring radionuclides Potassium, Thorium and Uranium are converted to air absorbed dose rates and these are used to assess terrestrial exposure levels from both natural and technologically enhanced sources. The high resolution airborne information is also assessed alongside existing knowledge from soil sampling and ground-based measurements of exposure levels. The surveys have sampled an extensive number of the UK lithological bedrock formations and the statistical information provides examples of low dose rate lithologies (the formations that characterise much of southern England) to the highest sustained values associated with granitic terrains. The maximum dose rates (e.g. >300 nGy h−1) encountered across the sampled granitic terrains are found to vary by a factor of 2. Excluding granitic terrains, the most spatially extensive dose rates (>50 nGy h−1) are found in association with the Mercia Mudstone Group (Triassic argillaceous mudstones) of eastern England. Geological associations between high dose rate and high radon values are also noted. Recent studies of the datasets have revealed the extent of source rock (i.e. bedrock) flux attenuation by soil moisture in conjunction with the density and porosity of the temperate latitude soils found in the UK. The presence or absence of soil cover (and associated presence or absence of attenuation) appears to account for a range of localised variations in the exposure levels encountered. The hypothesis is supported by a study of an extensive combined data set of dose rates obtained from soil sampling and by airborne geophysical survey. With no attenuation factors applied, except those intrinsic to the airborne estimates, a bias to high values of between 10 and 15 nGy h−1 is observed in the soil data. A wide range of technologically enhanced, localised contributions to dose rate values are also apparent in the data sets. Two detailed examples are provided that reveal the detectability of site-scale environmental impacts due to former industrial activities and the high dose values (>500 nGy h−1) that are associated with former, small-scale Uranium mining operations.

Long Wavelength SAR Backscatter Modelling Trends as a Consequence of the Emergent Properties of Tree Populations

This study describes the novel use of a macroecological plant and forest structure model in conjunction with a Radiative Transfer (RT) model to better understand interactions between microwaves and forest canopies. Trends predicted by the RT model, resulting from interactions with mixed age, mono and multi species forests, are analysed in comparison to those predicted using a simplistic structure based scattering model. This model relates backscatter to scatterer cross sectional or volume specifications, dependent on the size. The Spatially Explicit Reiterative Algorithm (SERA) model is used to provide a widely varied tree size distribution while maintaining allometric consistency to produce a natural-like forest representation. The RT model is parameterised using structural information from SERA and microwave backscatter simulations are used to analyse the impact of changes to the forest stand. Results show that the slope of the saturation curve observed in the Synthetic Aperture Radar (SAR) backscatter-biomass relationship is sensitive to thinning and therefore forest basal area. Due to similarities displayed between the results of the RT and simplistic model, it is determined that forest SAR backscatter behaviour at long microwave wavelengths may be described generally using equations related to total stem volume and basal area. The nature of these equations is such that they describe saturating behaviour of forests in the absence of attenuation in comparable fashion to the trends exhibited using the RT model. Both modelled backscatter trends predict a relationship to forest basal area from an early age when forest volume is increasing. When this is not the case, it is assumed to be a result of attenuation of the dominant stem-ground interaction due to the presence of excessive numbers of stems. This work shows how forest growth models can be successfully incorporated into existing independent scattering models and reveals, through the RT comparison with simplistic backscatter calculations, that saturation need not solely be a direct result of attenuation.

Absence of attenuation in oxidative stress response with repeated lipid-rich feedings

Background: The study of postprandial oxidative stress has received considerable attention in recent years. Excess production of reactive oxygen and nitrogen species (RONS) in response to high fat feeding can lead to oxidative stress. However, repeated exposure to the same stressor may lead to attenuation in the oxidative stress response, possibly via an up-regulation in antioxidant enzyme activity. Objective: To determine if repeated exposure to lipid-rich meal ingestion leads to a lower oxidative stress response over time, possibly due to an increase in antioxidant enzyme activity. Methods: Sixteen healthy men consumed 10 high fat milkshakes over a 3+ week period. Blood was taken from subjects on the first day of each week (before and for 4 hours after milkshake ingestion) and analyzed for oxidative stress and antioxidant biomarkers. Results: The oxidative stress response to repeated lipid-rich feeding was not different over time, as indicated by similar changes in plasma malondialdehyde and hydrogen peroxide, as well as superoxide dismutase, catalase and glutathione peroxidase activity. Conclusion: These data indicate that RONS production is likely similar with repeated exposure to lipid-rich meals and does not induce an adaptive response within the blood antioxidant defense system in a way that is protective to cells.

The effect of seawater attenuation on the directionality and spatial coherence of surface-generated ambient noise

Acoustic attenuation in seawater is sufficiently weak that it usually has little effect on the spatial characteristics of ambient noise. However, at frequencies above 10 kHz and depths below 6 km, seawater attenuation may influence the directionality of surface-generated ambient noise. In a semi-infinite, homogeneous ocean, all the noise is downward-traveling, since there are no bottom reflections, and, in the absence of attenuation, the directional density function varies as the cosine of the polar angle measured from the zenith. When attenuation is present, the angular width of this noise lobe becomes narrower, because sound from distant surface sources is attenuated more than acoustic arrivals from overhead. This narrowing effect increases as frequency rises, since the attenuation is a rapidly increasing function of frequency. The spatial coherence of the noise is also modified by the attenuation. For a pair of horizontally (vertically) aligned sensors, the zeroes in the spatial coherence function are shifted to higher (lower) frequencies. These effects of attenuation on the noise field are sufficiently large to be detectable using the recently developed, deep diving instrument platform Deep Sound, which is capable of measuring broadband ambient noise on multiple hydrophones to depths of 11 km. [Research supported by ONR.]

Elastic and anelastic relaxations accompanying magnetic ordering and spin-flop transitions in hematite, Fe2O3

Hematite, Fe2O3, provides in principle a model system for multiferroic (ferromagnetic/ferroelastic) behavior at low levels of strain coupling. The elastic and anelastic behavior associated with magnetic phase transitions in a natural polycrystalline sample have therefore been studied by resonant ultrasound spectroscopy (RUS) in the temperature range from 11 to 1072 K. Small changes in softening and attenuation are interpreted in terms of weak but significant coupling of symmetry-breaking and non-symmetry-breaking strains with magnetic order parameters in the structural sequence . The transition at TN = 946 ± 1 K is an example of a multiferroic transition which has both ferromagnetic (from canting of antiferromagnetically ordered spin moments) and ferroelastic (rhombohedral → monoclinic) character. By analogy with the improper ferroelastic transition in Pb3(PO4)2, W and W′ ferroelastic twin walls which are also 60° and 120° magnetic domain walls should develop. These have been tentatively identified from microstructures reported in the literature. The very low attenuation in the stability field of the C2/c structure in the polycrystalline sample used in the present study, in comparison with the strong acoustic dissipation reported for single crystal samples, implies, however, that the individual grains each consist of a single ferroelastic domain or that the twin walls are strongly pinned by grain boundaries. This absence of attenuation allows an intrinsic loss mechanism associated with the transition point to be seen and interpreted in terms of local coupling of shear strains with fluctuations which have relaxation times in the vicinity of ∼10−8 s. The first order (Morin) transition occurs through a temperature interval of coexisting phases but the absence of an acoustic loss peak suggests that the relaxation time for interface motion is short in comparison with the time scale of the applied stress (at ∼0.1–1 MHz). Below the Morin transition a pattern of attenuation which resembles that seen below ferroelastic transitions has been found, even though the ideal low temperature structure cannot contain ferroelastic twins. This loss behavior is tentatively ascribed to the presence of local ferromagnetically ordered defect regions which are coupled locally to shear strains.

Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers

We reexamine the derivation of the generalized nonlinear Schr\"odinger equation in the case of nonaxially uniform optical fibers, taking into account the longitudinal and spectral evolutions of all pertinent linear parameters. Our theory leads to an improved form of this equation that highlights an additional term, which ensures the conservation of the total photon number in nonuniform optical fibers in the absence of attenuation. Numerical simulations confirm the validity of this theory in the context of a Raman-induced soliton self-frequency shift, emission of Cherenkov radiation, and a soliton blue shift.

Prospects of the WSR-88D Radar for Cloud Studies

AbstractSounding of nonprecipitating clouds with the 10-cm wavelength Weather Surveillance Radar-1988 Doppler (WSR-88D) is discussed. Readily available enhancements to signal processing and volume coverage patterns of the WSR-88D allow observations of a variety of clouds with reflectivities as low as −25 dBZ (at a range of 10 km). The high sensitivity of the WSR-88D, its wide velocity and unambiguous range intervals, and the absence of attenuation allow accurate measurements of the reflectivity factor, Doppler velocity, and spectrum width fields in clouds to ranges of about 50 km. Fields of polarimetric variables in clouds, observed with a research polarimetric WSR-88D, demonstrate an abundance of information and help to resolve Bragg and particulate scatter. The scanning, Doppler, and polarimetric capabilities of the WSR-88D allow real-time, three-dimensional mapping of cloud processes, such as transformations of hydrometeors between liquid and ice phases. The presence of ice particles is revealed by high differential reflectivities and the lack of correlation between reflectivity and differential reflectivity in clouds in contrast to that found for rain. Pockets of high differential reflectivities are frequently observed in clouds; maximal values of differential reflectivity exceed 8 dB, far above the level observed in rain. The establishment of the WSR-88D network consisting of 157 polarimetric radars can be used to collect cloud data at any radar site, making the network a potentially powerful tool for climatic studies.

Influence of multiple scattering on CloudSat measurements in snow: A model study

The effects of multiple scattering on larger precipitating hydrometers have an influence on measurements of the spaceborne W‐band (94 GHz) CloudSat radar. This study presents initial quantitative estimates of these effects in “dry” snow using radiative transfer calculations for appropriate snowfall models. It is shown that these effects become significant (i.e., greater than approximately 1 dB) when snowfall radar reflectivity factors are greater than about 10–15 dBZ. Reflectivity enhancement due to multiple scattering can reach 4–5 dB in heavier stratiform snowfalls. Multiple scattering effects counteract signal attenuation, so the observed CloudSat reflectivity factors in snowfall could be relatively close to the values that would be observed in the case of single scattering and the absence of attenuation.

Conservative initial‐boundary value problems for the Wide‐Angle PE in waveguides with variable bottoms

We consider the third‐order, Claerbout‐type Wide‐Angle Parabolic Equation (PE) in the context of Underwater Acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There are strong indications, that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition on B, may not be well‐posed, for example when B is downsloping. In previous work we proposed an additional bottom boundary condition that, together with the zero field condition on B, yields a well‐posed problem. In the present paper we continue our investigation of additional bottom boundary conditions that yield well‐posed, physically correct problems. Motivated by the fact that the solution of the wide‐angle PE in a domain with horizontal layers conserves its L2 norm in the absence of attenuation, we seek additional boundary conditions on a variable‐topography bottom, that yield L2‐ conservative solutions of the problem. We identify a family of such boundary conditions after a range‐dependent change of the depth variable that makes B horizontal. We discretize the continuous problems by second‐order accurate Crank‐Nicolson type finite difference schemes, and show, by means of numerical experiments, that the new models yield accurate simulations of the acoustic field in standard, wedge‐type domains with upsloping and downsloping bottoms.

The uses of truncated series in describing the propagation of high-intensity broadband noise

The behavior of the high-frequency part of the spectrum of a broadband noise signal propagating through the atmosphere cannot, according to observation, be accounted for solely by linear mechanisms. Truncated series expansions of the initial signal can afford information on how these linear mechanisms, such as atmospheric attenuation and spherical spreading, might interact with weak nonlinearity to produce the effects observed in the evolved spectrum. Through a Taylor expansion we obtain expressions relating the evolved power spectral density solely to the initial power spectral density and other quantities available directly from measurement, regardless of initial shape. The energy cascade characteristic of broadband noise propagation can be explicitly linked to convolution terms in these expressions. It is also shown that attenuation and spherical spreading affect linear and nonlinear terms in the evolution equation in the same way, producing the same rate of decay. Numerical simulation confirms the validity of these results for distances before shock formation. In the absence of attenuation we find that the range of validity of Taylor expansions is greatly increased by construction of Padé approximants, yielding extremely accurate predictions of the evolved waveforms for distances only slightly shorter than the expected shock formation distance.

Texture of the uppermost inner core from forward- and back-scattered seismic waves

Body waves interacting with the boundary of the solid inner core at narrow and wide angles of incidence provide independent constraints on a heterogeneous texture that may originate from the process of solidification. The equatorial, quasi-eastern hemisphere, of the uppermost 50–100 km of the inner core is characterized by a higher isotropic P wave velocity, lower Q α's inferred from PKIKP, and simpler PKiKP pulses compared to adjacent regions in the western hemisphere and polar latitudes. Compared to this region, the adjacent western (primarily Pacific) equatorial region is characterized by higher Q α's and a higher level of coda excitation following PKiKP. Lateral variations in both inner core Q α inferred from transmitted PKIKP and inner core heterogeneity inferred from the coda of reflected PKiKP can be modeled by lateral variations in a solidification fabric. In an actively crystallizing eastern equatorial region, characterized by upwelling flow in the outer core, fabrics that explain strong attenuation and the absence of attenuation and velocity anisotropy in short range (120°–140°) PKIKP and weak PKiKP codas have an anisotropy of scale lengths with longer scale lengths in the vertical direction, perpendicular to the inner core boundary. In less actively solidifying regions in the equatorial western hemisphere, longer scale lengths tend to be more parallel to the inner core boundary, consistent with outer core flow tangent to the inner core boundary or viscous shearing and recrystallization in the horizontal direction away from more actively crystallizing regions in the eastern hemisphere. This texture is less effective in attenuating PKIKP by forward scattering. Lateral variation in the equatorial western hemisphere between vertical versus horizontal oriented plate-like textures may explain lateral variations from weak to strongly back-scattered PKiKP coda and from strong to weak velocity and attenuation anisotropy in short range PKIKP.