Amplitude Uncertainty Research Articles

Statistical analysis of acoustic emission signals generated during turning of a metal matrix composite

Acoustic emission technique (AET) has been used to monitor the progress of tool wear during turning of silicon carbide (20 wt.%) dispersed Al alloy metal matrix composite. Acoustic emission (AE) signals generated beyond a specific cutting distance increase abruptly. Statistical analysis based on assumed β distribution of AE energy showed that skewness and kurtosis vary with cutting time. Comparison of these results with bparameter of amplitude distribution of AE hits presented in an earlier investigation has shown that while b-parameter is useful for monitoring tool wear up to 0.4 mm, skewness and kurtosis can better monitor the wear beyond that. Uncertainty measurement of AE energy for different cutting distances was determined as per ISO GUM. The combined uncertainty for the measurement of AE energy lies in the range of 0.38 to 1.69, with higher values for the cutting distance between 213.8 mm and 454.5 mm. Different parameters such as skewness and kurtosis of the statistical distribution, b-parameter of amplitude distribution and uncertainties can be used in a complimentary manner for comprehensive evaluation of tool wear.

Fourier analysis of blurred images for the measurement of the in-plane dynamics of MEMS

The goal of this paper is to use FFT imaging techniques to measure in-plane resonances of MEMS devices from blurred microphotographs where the presence of resonance is not visually discernable. A method is presented for measuring the high-frequency (in the 10s–100s of kHz range) response characteristics of MEMS devices using only standard optical microscope cameras (15–30 Hz frame rate) and applying Fourier analysis of camera images of periodic patterns on the oscillating devices. In the frequency domain, in-plane blurring acts as a low pass filter, attenuating all frequency components, but preferentially attenuating the higher order harmonics. A theoretical formula for the blur-induced attenuation of the harmonics of Fourier series components is derived and it is shown that it follows a Bessel curve. The theoretical predictions were verified experimentally using a series of camera microphotographs of three different variations of an electro-thermally driven pad suspended on springs. The predicted attenuations of harmonics were observed and verified. The analysis of the measured attenuation was able to (1) determine in-plane resonant frequencies, (2) measure submicron motions and (3) characterize the nonlinear dynamics (modeled by the Duffing equation). The amplitude uncertainty of the FFT method for detecting in-plane resonant peaks at 75 kHz and 3.5 V was found to be ±0.027 µm using a single image and ±0.011 µm using an average of 10 images.

Accuracy Evaluation of Ultrawideband Time Domain Systems for Microwave Imaging

We perform a theoretical analysis of the measurement accuracy of ultrawideband time domain systems. The theory is tested on a specific ultrawideband system and the analytical estimates of measurement uncertainty are in good agreements with those obtained by means of simulations. The influence of the antennas and propagation effects on the measurement accuracy of time domain near field microwave imaging systems is discussed. As an interesting application, the required measurement accuracy for a breast cancer detection system is estimated by studying the effect of noise on the image reconstructions. The results suggest that the effects of measurement errors on the reconstructed images are small when the amplitude uncertainty and phase uncertainty of measured data are less than 1.5 dB and 15 degrees, respectively.

Default motor preparation under conditions of response uncertainty

In a choice reaction time (RT) paradigm, providing partial advance information (a precue) about the upcoming response has been shown to decrease RT, presumably due to preprogramming of the precued parameters. When advance information about a particular aspect of a movement is provided (precued), several different strategies might be used to prepare the motor system during the foreperiod. For example, in studies where response preparation time was manipulated, precues were provided specifying the required arm and direction but movement amplitude was left uncertain. In this case it was shown that a default movement was preprogrammed whose amplitude was intermediate between the alternatives (Favilla et al. in Exp Brain Res 75(2):280-294, 1989, Exp Brain Res 79(3):530-538, 1990; Ghez et al. in Exp Brain Res 115(2):217-233, 1997). However, this strategy did not appear to be used in a RT task since there was an absence of online adjustments to movement. Therefore, it appeared movements were not initiated until all parameters had been correctly specified and programmed by the nervous system (Bock and Arnold in Exp Brain Res 90:(1):209-216, 1992). The present study reinvestigated the notion of a default movement preparation strategy in a choice RT paradigm, employing the triggering effect of a startling acoustic stimulus. On control trials (80 dB imperative stimulus), the movements were initiated toward the correct targets. Providing a startle stimulus (124 dB) resulted in the early initiation of a "default" movement whose amplitude fell in between the potential response alternatives. Thus, the current experiment found behavioral evidence of default intermediate-amplitude movement preparation as a strategy under conditions of response amplitude uncertainty.

Dielectric response of particles in flowing media: The effect of shear-induced rotation on the variation in particle polarizability

When particles in liquid suspensions flow through channels and pipes in a laminar fashion, the resulting parabolic velocity profile gives rise to shear, which induces the particles to rotate. If flowing suspensions containing dielectric particles are immersed in an external electric field, the anisotropic polarization induced in rotating nonspherical particles will vary with the orientation of the particle with respect to the external field; what results is an uncertainty in experimental measurements that involve particle polarization. The present study establishes the limits of this uncertainty and shows that departure from spherical symmetry in individual particles can lead to a significant overlap in measurements attempting to discriminate between particle subpopulations in suspensions. For example, the uncertainty in signal amplitude for a population of activated T-lymphocytes can be as high as 20%. Such concerns arise in applications like field-flow fractionation, dielectrophoretic sorting of particles, flow impedance measurements and cytometry, and, most recently, isodielectric separation, all of which are used to separate particles in a flow based on their dielectric response. This paper considers axisymmetric particles as the first departure from the approximation of spherical symmetry, shows how to calculate an estimate of the size of the population overlap, and suggests possible strategies to minimize it.

Abstract 3911: Functional copy number alterations (CNAS) in oral cancer

Abstract Background: Alterations in DNA copy number can functionally affect genes and produce a transformed cancer phenotype. Discoevery of novel cancer genes can be facilitated by identifying genomic regions undergoing frequent, significant alteration. Array comparative genomic hybridization techniques have generated vast amounts of raw copy number data requiring computational analysis to identify aberrations likely to be functional. In this analysis, we present data from a comprehensive computational approach designed to robustly map chromosomal aberrations in oral cavity squamous cell carcinoma (OCSCC). Methods: DNA was extracted from 31 microdissected OCSCC frozen tumors and hybridized with reference DNA on the Agilent 1M copy number array, containing nearly 1 million probes across the genome at high resolution. The RAE computational framework was used to identify CNAs. In this analysis, segmentation of probe data is first performed using paired reference HapMap data and individual-tumor thresholds to account for stromal admixture and tumor heterogeneity. Scores (single-copy gain, amplification, hemizygous loss, homozygous deletion) are then assigned, followed by soft discrimination thresholding and significance testing with false discovery rate correction. Regions of interest (ROIs) are then agnostically identified as regions of significant (FDR<0.01) alteration, expanded to adjacent loci to account for spatial and amplitude uncertainty. Finally, described copy number polymorphisms were excluded. Results: 49 ROIs were identified, 27 amplification regions (chromosomes 3,4,5,7,8,9,11,16,18,20,21) and 22 deletion regions (chromosomes 2,3,4,5,7,8,9,10,13,15,16,18,19). ROIs ranged in prevalence from 6.5-77.4% of samples, and encompassed 0-483 genes (total=2799, median=4), and 0-13 microRNAs (total=77, median=0). Eleven ROIs were focal intragenic CNAs. Four ROIs contained no named genes or microRNAs. Hierarchical clustering identified 3 clusters, based on escalating levels of CNA. Complete gene details of the 49 ROIs will be presented. Conclusions: We report significant regions of CNA in oral cancer, by applying a robust computational framework to identify recurrent chromosomal aberrations in OCSCC, which represent highly significant candidate functional events. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 102nd Annual Meeting of the American Association for Cancer Research; 2011 Apr 2-6; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2011;71(8 Suppl):Abstract nr 3911. doi:10.1158/1538-7445.AM2011-3911

Generalized Bragg scattering for uncertain sea surfaces.

Bragg’s law and related concepts based on acoustic scattering due to surface resonances can provide a framework for computational sea surface scattering. Studies for periodic surfaces [Oba, J. Acoust. Soc. Am. 128, 39 (2010)] demonstrate that variation of acoustic resonance with the surface phase allows Fourier analysis separation of Bragg scattering orders. More generally, consider one-dimensional, undulating, sea surfaces composed of several Fourier components of known wavelengths, but of uncertain amplitude and phasing. A combination of polynomial chaos (PC) and Fourier expansion for the acoustic field consists of sums of products of Laguerre polynomials and complex exponentials, corresponding to parameters for uncertainty in surface amplitude and phase, respectively. Each single PC-Fourier index quantifies the acoustic response to an individual order in Bragg’s law. The Laguerre polynomial terms represent the non-linear acoustic variation with surface amplitude. A particular PC-Fourier expansion coefficient corresponding to several surface wavelength components permits evaluation of cross-component coupling. The PC-Fourier coefficient decay rate indicates the degree of Bragg scattering coupling between components. The robustness of the PC-Fourier coefficients in the presence of a larger set of random rough surface components is explored. [This research is sponsored by the Office of Naval Research.]

Pekeris waveguide comparisons of methods for predicting acoustic field amplitude uncertainty caused by a spatially uniform environmental uncertainty (L)

Acoustic field calculations in underwater environments are often uncertain because the environmental parameters required for such calculations are uncertain. This letter compares the accuracy of direct simulations, the field shifting approximation, and polynomial chaos expansions for predicting acoustic amplitude uncertainty in 100-m-deep Pekeris waveguides having spatially uniform uncertain water-column sound speed. When this sound speed is Gaussian-distributed with a standard deviation of 1 m/s, direct simulations and polynomial chaos expansions, based on 21 field calculations, are more accurate than the field shifting approximation, based on two field calculations. This ranking reverses as the sound-speed standard deviation increases to 20 m/s.

Three-Dimensional Surface in LTCC for a MM-Wave Antenna

Low loss LTCC (Low Temperature Cofired Ceramics) materials have already demonstrated their virtues for high density packaging of mm-wave modules and their capability to handle the inherent requirements of heat dissipation. A 60 GHz substrate integrated waveguide fed steerable antenna array is one example of the driving applications for new challenges in the LTCC process. This antenna is suitable for transmit and receive in the 60 GHz WPAN frequency band [1]. Integrating antenna array and mm-wave front-end in one module is a consequent way to avoid amplitude and phase uncertainties associated with connectors and cables. A compact module also helps to reduce losses in the signal path by providing short interconnects. This is important for mm-Wave systems, particularly for the receiver. The antenna design presented here requires a three-dimensional surface structure with features like small and precise cavities and grooves. Different methods to fabricate cavities in LTCC modules will be discussed and a new method which proved suitable for the demanding application will be presented in detail.

Estimating the detection capability of the International Monitoring System infrasound network

A 59 station infrasound network is being established as part of the verification regime for the Comprehensive Nuclear‐Test‐Ban Treaty. We present a probabilistic model for predicting the network detection threshold (defined as the chemical explosive yield at which two stations will detect signals with 90% probability). We incorporate the influence of stratospheric wind and detection bandwidth while accounting for estimates of noise amplitude and propagation uncertainties. Inclusion of the stratospheric wind, using a state‐of‐the‐art climatological model, tends to reduce the detection threshold. For 66%/88% of Earth's surface area the detection threshold is lower than that expected for the windless case for all/≥90% of the time. However, preferential downwind detection makes location of explosions with yields close to the threshold more difficult due to the restricted azimuthal coverage. Completing the IMS network is also important: the detection threshold for the 39 station network operating in October 2008 is approximately 45% higher than that predicted for the full 59 station network. Moreover, the absolute detection threshold values are highly sensitive to the frequency band in which the noise is evaluated. A detector simulation comprising overlapping two‐octave frequency bands indicates that, when accounting for low noise at frequencies above 0.2 Hz, detection capability at low yields (<0.5 kt) may be considerably better than previously estimated. Our simulations, using a two‐octave bandwidth detector and incorporating stratospheric winds from a climatological model, predict that explosions of >210 t chemical explosive (>420 t nuclear equivalent yield) will be detected over ≥95% of Earth's surface at any time of year (with a 90% probability).

On the inner-outer core density contrast from PKiKP/PcP amplitude ratios and uncertainties caused by seismic noise

The inner core boundary (ICB) of the earth is characterized by a discontinuous change in elastic properties between the liquid outer and solid inner core. In the ray theory approximation, a measure of the density contrast at the ICB is given by the amplitude ratio of P waves reflected from the core—mantle boundary (CMB; PcP waves) and the ICB (PKiKP waves), since that ratio conveniently appears in an explicit form in the transmission/reflection coefficient equations. The results for inner—outer core density contrast derived from direct amplitude picks of these waves in the time domain have varied significantly among different authors. The transmission/reflection coefficients on the liquid—solid and solid—liquid boundaries derived from ground displacements enable a direct comparison between the amplitude measurements on displacement seismograms in the time domain and theoretical values. A new approach is proposed and applied to integrate effects of microseismic and signal-generated noise with the amplitude measurements, thus providing a direct maximal uncertainty measure. To suppress the effects of varying radiation pattern and distinctively different ray paths at longer epicentral distances, this new method was applied to high-quality arrivals of PcP and PKiKP waves from a nuclear explosion observed at epicentral distances 10°–20° from recording stations. The resulting uncertainties are high precluding precise estimates of the ICB density contrast, but provide a robust estimate of an upper bound from body waves of about 1100 kg m−3. Median values of two amplitude ratios observed around 17° epicentral distance indicate a small density contrast of 200–300 kg m−3 and suggest the existence of zones of suppressed density contrast between the inner and the outer core, a density contrast stronger than 5000 kg m−3 at the CMB, or a combination of both.

Improved Statistical Processing for Common-Conversion-Point Stacked Receiver Functions

The interpretation of teleseismic receiver functions (RFs) is typically limited by poor constraints on the uncertainty of amplitudes of converted phases. In continental regions these problems are overcome by stacking large amounts of data. In oceanic regions, however, data quality is notoriously noisy and the number of events are often limited by significantly shorter station deployment times. Estimates of RF pulse amplitude uncertainty are, therefore, necessary to allow an analyst to differentiate between real structure in the Earth and random correlations in ambient noise. Here we combine a common-conversion-point stacking technique with multiple-taper correlation RF estimates, which allow frequency-domain weighting. We then compute jackknife statistics to estimate local uncertainties in RF amplitude. We apply this technique to a continental station in Arabia (RAYN) and to the ocean island station at Raratonga, Cook Islands (RAR) in order to compare our method to results obtained via other techniques. The structure we recover matches previous crustal studies at both stations and provides new interpretations of conversions in the upper mantle. At single stations this technique works well to resolve crust and mantle structure up to a depth of 100 km. Geographical dispersion of ray paths at greater depths decreases the number of events per bin and, therefore, increases the uncertainty in converted amplitude. The ability to stack crossing ray paths implies that this method will be well suited to the analysis of data from seismic arrays.

Autoregressive moving average modeling for spectral parameter estimation from a multigradient echo chemical shift acquisition

The authors investigated the performance of the iterative Steiglitz-McBride (SM) algorithm on an autoregressive moving average (ARMA) model of signals from a fast, sparsely sampled, multiecho, chemical shift imaging (CSI) acquisition using simulation, phantom, ex vivo, and in vivo experiments with a focus on its potential usage in magnetic resonance (MR)-guided interventions. The ARMA signal model facilitated a rapid calculation of the chemical shift, apparent spin-spin relaxation time (T2*), and complex amplitudes of a multipeak system from a limited number of echoes (< or equal 16). Numerical simulations of one- and two-peak systems were used to assess the accuracy and uncertainty in the calculated spectral parameters as a function of acquisition and tissue parameters. The measured uncertainties from simulation were compared to the theoretical Cramer-Rao lower bound (CRLB) for the acquisition. Measurements made in phantoms were used to validate the T2* estimates and to validate uncertainty estimates made from the CRLB. We demonstrated application to real-time MR-guided interventions ex vivo by using the technique to monitor a percutaneous ethanol injection into a bovine liver and in vivo to monitor a laser-induced thermal therapy treatment in a canine brain. Simulation results showed that the chemical shift and amplitude uncertainties reached their respective CRLB at a signal-to-noise ratio (SNR) > or =5 for echo train lengths (ETLs) > or =4 using a fixed echo spacing of 3.3 ms. T2* estimates from the signal model possessed higher uncertainties but reached the CRLB at larger SNRs and/or ETLs. Highly accurate estimates for the chemical shift (<0.01 ppm) and amplitude (<1.0%) were obtained with > or =4 echoes and for T2*(<1.0%) with > or =7 echoes. We conclude that, over a reasonable range of SNR, the SM algorithm is a robust estimator of spectral parameters from fast CSI acquisitions that acquire < or =16 echoes for one- and two-peak systems. Preliminary ex vivo and in vivo experiments corroborated the results from simulation experiments and further indicate the potential of this technique for MR-guided interventional procedures with high spatiotemporal resolution approximately 1.6 x 1.6 x 4 mm3 in < or =5 s.

Nonlocal quantum macroscopic superposition in a high-thermal low-purity state.

Quantum state exchange between light and matter is an important ingredient for future quantum information networks as well as other applications. Photons are the fastest and simplest carriers of information for transmission but in general, it is difficult to localize and store photons, so usually one prefers choosing matter as quantum memory elements. Macroscopic superposition and nonlocal quantum interactions have received considerable interest for this purpose over recent years in fields ranging from quantum computers to cryptography, in addition to providing major insights into physical laws. However, these experiments are generally performed either with equipment or under conditions that are unrealistic for practical applications. Ideally, the two can be combined using conventional equipment and conditions to generate a "quantum teleportation"-like state, particularly with a very small amount of purity existing in an overall highly mixed thermal state (relatively low decoherence at high temperatures). In this study we used an experimental design to demonstrate these principles. We performed optical coherence tomography (OCT) using a thermal source at room temperatures of a specifically designed target in the sample arm. Here, position uncertainty (i.e., dispersion) was induced in the reference arm. In the sample arm (target) we placed two glass plates separated by a different medium while altering position uncertainty in the reference arm. This resulted in a chirped signal between the glass plate reflective surfaces in the combined interferogram. The chirping frequency, as measured by the fast Fourier transform (FFT), varies with the medium between the plates, which is a nonclassical phenomenon. These results are statistically significant and occur from a superposition between the glass surface and the medium with increasing position uncertainty, a true quantum-mechanical phenomenon produced by photon pressure from two-photon interference. The differences in chirping frequency with medium disappears when second-order correlations are removed by dual balanced detection, confirming the proposed mechanism. We demonstrated that increasing position uncertainty at one site leads to position uncertainty (quantum position probability amplitude) nonlocally via second-order correlations (two-photon probability amplitude) from a low coherence thermal source (low purity, high local entropy). The implications, first, are that the phenomenon cannot be explained through classical mechanisms but can be explained within the context of quantum mechanics, particularly relevant to the second-order correlations where controversy exists. More specifically, we provide the theoretical framework that these results indicate a nonlocal macroscopic superposition is occurring through a two-photon probability amplitude-induced increase in the target position probability amplitude uncertainty. In addition, as the experiments were performed with a classical source at room temperature, it supports both the quantum-mechanical properties of second-order correlations and that macroscopic superposition is obtainable in a target not in a single coherent state (mixed state). Future work will focus on generalizing the observations outside the current experimental design and creating embodiments that allow practical application of the phenomenon.

A margin model to account for respiration-induced tumour motion and its variability

In order to reduce the sensitivity of radiotherapy treatments to organ motion, compensation methods are being investigated such as gating of treatment delivery, tracking of tumour position, 4D scanning and planning of the treatment, etc. An outstanding problem that would occur with all these methods is the assumption that breathing motion is reproducible throughout the planning and delivery process of treatment. This is obviously not a realistic assumption and is one that will introduce errors. A dynamic internal margin model (DIM) is presented that is designed to follow the tumour trajectory and account for the variability in respiratory motion. The model statistically describes the variation of the breathing cycle over time, i.e. the uncertainty in motion amplitude and phase reproducibility, in a polar coordinate system from which margins can be derived. This allows accounting for an additional gating window parameter for gated treatment delivery as well as minimizing the area of normal tissue irradiated. The model was illustrated with abdominal motion for a patient with liver cancer and tested with internal 3D lung tumour trajectories. The results confirm that the respiratory phases around exhale are most reproducible and have the smallest variation in motion amplitude and phase (approximately 2 mm). More importantly, the margin area covering normal tissue is significantly reduced by using trajectory-specific margins (as opposed to conventional margins) as the angular component is by far the largest contributor to the margin area. The statistical approach to margin calculation, in addition, offers the possibility for advanced online verification and updating of breathing variation as more data become available.

A heuristic derivation of the uncertainty for frequency determination in time series data

Context. Several approaches to estimating frequency, phase, and amplitude errors in time-series analyse have been reported in the literature, but they are either time-consuming to compute, grossly overestimating the error, or are based on empirically determined criteria.Aims. A simple, but realistic estimate of the frequency uncertainty in time-series analyses is our goal here.Methods. Synthetic data sets with mono- and multi-periodic harmonic signals and with randomly distributed amplitude, frequency, and phase were generated and white noise added. We tried to recover the input parameters with classical Fourier techniques and investigated the error as a function of the relative level of noise, signal, and frequency difference.Results. We present simple formulas for the upper limit of the amplitude, frequency, and phase uncertainties in time-serie analyses. We also demonstrate the possibility of detecting frequencies that are separated by less than the classical frequency resolution and of finding that the realistic frequency error is at least 4 times smaller than the classical frequency resolution.

Dissimilar-dimension interference model of fatigue reliability under uncertain cyclic load

The stress-strength interference model for static strength failure is well developed, while difficulties still exist when use the concept of stress-strength interference to calculate the reliability of time-dependent failure modes such as fatigue, abrasion, corrosion, etc. There is not yet a straight forward and effective model such as stress-strength interference model even for fatigue under constant amplitude cyclic stress with uncertainty in stress amplitude. Traditional stress-strength interference model is interpreted as that to express the statistical average of the probability that strength is greater than stress, or vice versa. Thus, the same model format, which traditionally can only be used for the situation of the parameters (load and stress) of the same unit, can be applied to more general situation. In other words, the traditional model was extended to the situation of any two variables, as long as one of the variables can be expressed as a function of the other. By such a model, fatigue failure probability/reliability can be calculated with known load distribution and fatigue life distribution.

System fatigue reliability modelling under stochastic cyclic load

Based on a mathematically insightful interpretation of the 'statistical average' meaning of the traditional stress-strength interference model for reliability calculation, this paper puts the issue of interference analysis of two random variables into a more general framework. Under such a framework, what is expressed by the traditional stress-strength interference model is a statistical stress-weighted average of the 'reliability conditioning to stress', which is a function of the random stress. Thus, the same form of expression as the load-strength interference equation, which traditionally can be applied only in the case of the same system-of-units parameters (e.g., stress and strength, both are measured in the same unit, MPa), can be applied to more general situations involving parameters of different system-of-units. In this way, the traditional stress-strength interference model was extended to deal with any two random variables, as long as one of the variables is a function of the other. Such an extended interference analysis model has the ability to calculate fatigue reliability under cyclic stress with uncertainty in stress amplitude. Using the order statistic of component lives to describe system life, fatigue reliability models for different types of system were developed based on the statistical average algorithm.

Analysis of sources of uncertainty of tooth enamel EPR signal amplitude

The aim of this paper is to investigate some sources of uncertainty in the EPR dosimetric signal amplitude of irradiated tooth enamel. The paper evaluates the uncertainty induced by steps of sample preparation and by the use of a reference sample that are nowadays considered consolidated and that might have been overlooked. The investigated influence quantities are the grain size of the tooth enamel powder sample, sample moisture, the disinfection methods and the range of variability of the reference sample EPR amplitude. It is shown that some of these features need further investigation.

Computing traveltime and amplitude sensitivity kernels in finite-frequency tomography

The efficient computation of finite-frequency traveltime and amplitude sensitivity kernels for velocity and attenuation perturbations in global seismic tomography poses problems both of numerical precision and of validity of the paraxial approximation used. We investigate these aspects, using a local model parameterization in the form of a tetrahedral grid with linear interpolation in between grid nodes. The matrix coefficients of the linear inverse problem involve a volume integral of the product of the finite-frequency kernel with the basis functions that represent the linear interpolation. We use local and global tests as well as analytical expressions to test the numerical precision of the frequency and spatial quadrature. There is a trade-off between narrowing the bandpass filter and quadrature accuracy and efficiency. Using a minimum step size of 10km for S waves and 30km for SS waves, relative errors in the quadrature are of the order of 1% for direct waves such as S, and a few percent for SS waves, which are below data uncertainties in delay time or amplitude anomaly observations in global seismology. Larger errors may occur wherever the sensitivity extends over a large volume and the paraxial approximation breaks down at large distance from the ray. This is especially noticeable for minimax phases such as SS waves with periods >20s, when kernels become hyperbolic near the reflection point and appreciable sensitivity extends over thousands of km. Errors becomes intolerable at epicentral distance near the antipode when sensitivity extends over all azimuths in the mantle. Effects of such errors may become noticeable at epicentral distances>140°. We conclude that the paraxial approximation offers an efficient method for computing the matrix system for finite-frequency inversions in global tomography, though care should be taken near reflection points, and alternative methods are needed to compute sensitivity near the antipode.