Fresnel Volumes Research Articles

Evaluation of the 3D Near-Surface Velocity Structure in an Urban Environment from Ambient Noise Array Tomography: The Case of the City of Thessaloniki (Northern Greece)

Abstract We examine the implementation of ambient noise array tomography in an urban environment to assess the 3D near-surface shear wave velocity (VS) structure at an intermediate spatial scale (∼1 km2, depth range 200–300 m). The application employs cross correlation traces of vertical component ambient noise recordings from a local network installed in Thessaloniki city (Northern Greece), allowing the determination of Rayleigh wave travel times for the frequency range of 1.5–14 Hz. The results confirm the presence of a complex subsurface with strong lateral variations in the geology, with travel times varying up to almost one order of magnitude. A surface wave travel time tomography approach was applied for each frequency to determine the spatial variability of the group velocity, involving the use of approximate Fresnel volumes, as well as damping and spatial smoothing constraints to stabilize the results. We also employed an interfrequency smoothing scheme to obtain smooth but data-compatible dispersion curves at the cost of inverting all travel time data simultaneously. Following the application of several quality cutoff criteria, we reconstructed local group slowness dispersion curves for a predefined tomographic grid in the study area. The final 3D velocity model was determined by a modified Monte Carlo inversion of these dispersion curves and the spatial integration of the obtained 1D VS profiles. Different model parameterizations were tested for the inversion to determine the optimal datafit. The final 3D velocity model is in a very good agreement with the local geology, previous larger scale studies, and other geophysical surveys, providing additional structural constraints (such as hidden fault identification) for the complex sedimentary deposits and bedrock formation in Thessaloniki, up to the depth of ∼250–300 m. The introduction of the aforementioned modifications to the ambient noise array tomography suggests that it can be efficiently adjusted and employed as a reliable tool for imaging the 3D seismic structure in urban environments with complex geology.

3-D Wave-Equation-Based Finite-Frequency Tomography for Ultrasound Computed Tomography.

Ultrasound computed tomography (USCT) has great potential for 3-D quantitative imaging of acoustic breast tissue properties. Typical devices include high-frequency transducers, which makes tomography techniques based on numerical wave propagation simulations computationally challenging, especially in 3-D. Therefore, despite the finite-frequency nature of ultrasonic waves, ray-theoretical approaches to transmission tomography are still widely used. This article introduces a finite-frequency traveltime tomography to medical ultrasound. In addition to being computationally tractable for 3-D imaging at high frequencies, the method has two main advantages: 1) it correctly accounts for the frequency dependence and volumetric sensitivity of traveltime measurements, which are related to off-ray-path scattering and diffraction. 2) It naturally enables out-of-plane imaging and the construction of 3-D images from 2-D slice-by-slice acquisition systems. Our method rests on the availability of calibration data in water, used to linearize the forward problem and to provide analytical expressions of cross correlation traveltime sensitivity. As a consequence of the finite-frequency content, sensitivity is distributed in multiple Fresnel volumes, thereby providing out-of-plane sensitivity. To improve computational efficiency, we develop a memory-efficient implementation by encoding the Jacobian operator with a 1-D parameterization, which allows us to extend the method to large-scale domains. We validate our tomographic approach using laboratory measurements collected with a 2-D setup of transducers and using a cylindrically symmetric phantom. We then demonstrate its applicability for 3-D reconstructions by simulating a slice-by-slice acquisition system using the same data set.

An efficient nonlinear Fresnel volume tomography using an improved scattering-integral algorithm

A new nonlinear Fresnel volume tomography (FVT) method is proposed in which a gradient-guided approach is used to replace the conventional equation-solving method. An improved scattering-integral algorithm is used to compute the objective function gradient with respect to the model parameters and the diagonal Hessian. The key principle of this algorithm is the transformation of the gradient computation into an accumulation of multiple vector-scalar products, where each vector is the corresponding individual Fresnel volume kernel of a source-receiver pair. This implementation has several advantages over conventional equation-solving approaches. Firstly, neither the Fresnel volumes nor the Hessian matrix must be stored in memory in advance, thereby ensuring low memory requirements. Secondly, no large equations must be iteratively solved using a general inverse algorithm such as least-squares QR decomposition (LSQR), thereby ensuring a highly efficient inversion procedure. Thirdly, the steepest-descent direction is used to ensure stable convergence. Finally, and most importantly, the preconditioning of the steepest-descent direction, which is equivalent to achieving illumination compensation, can be conveniently performed. This proposed nonlinear FVT method is applied to both synthetic and field data to invert finite-frequency traveltimes of first-arrival waveforms for the near-surface velocity structure. Shallow accuracies comparable to or better than those obtained using the equation-solving FVT or ray-based tomography are achieved. Furthermore, both the memory requirements and computation time are reduced and deeper inversion results are obtained using the diagonal Hessian.

Frequency-dependent traveltime tomography using fat rays: application to near-surface seismic imaging

Frequency-dependent traveltime tomography does not rely on the high frequency assumption made in classical ray-based tomography. By incorporating the effects of velocity structures in the first Fresnel volume around the central ray, it offers a more realistic and accurate representation of the actual physics of seismic wave propagation and thus, enhanced imaging of near-surface structures is expected. The objective of this work was to apply frequency-dependent first arrival traveltime tomography to surface seismic data that were acquired for exploration scale and near-surface seismic imaging. We adapted a fat ray tomography algorithm from global-earth seismology that calculates the Fresnel volumes based on source and receiver (adjoint source) traveltime fields. The fat ray tomography algorithm was tested on synthetic model data that mimics the dimensions of two field data sets. The field data sets are presented as two case studies where fat ray tomography was applied for near-surface seismic imaging. The data set of the first case study was recorded for high-resolution near-surface imaging of a Quaternary valley (profile length<1km); the second data set was acquired for hydrocarbon search (profile length>10km). All results of fat ray tomography are compared against the results of classical ray-based tomography. We show that fat ray tomography can provide enhanced tomograms and that it is possible to recover more information on the subsurface when compared to ray tomography. However, model assessment based on the column sum of the Jacobian matrix revealed that especially the deep parts of the structure in the fat ray tomograms might not be adequately covered by fat rays. Furthermore, the performance of the fat ray tomography depends on the chosen input frequency in relation to the scale of the seismic survey. Synthetic data testing revealed that the best results were obtained when the frequency was chosen to correspond to an approximate wavelength-to-target depth ratio of 0.1.

2D time-lapse seismic tomography using an active time-constraint (ATC) approach

A 2D seismic time-lapse inversion approach can image the evolution of seismic velocities over time and space. The forward modeling is based on solving the eikonal equation using a second-order fast-marching method. Wave paths are represented by Fresnel volumes rather than by conventional rays. This approach accounts for complex velocity models and has the advantage of considering the effects of wave frequency on velocity resolution. The aim of time-lapse inversion is to find changes in velocities of each cell in the model as a function of time. Each model along the time axis is called a reference space model. This approach can be simplified into an inverse problem that seeks the optimum of several reference space models taken together, using the approximation that the change in seismic velocity varies linearly in time between two subsequent reference models. The method is demonstrated on a synthetic example that includes regularization in time in the cost function and reduces inversion artifacts associated with noise in the data by comparison with independent inversions at each time.

Three‐dimensional shallow structure from high‐frequency ambient noise tomography: New results for the Mygdonia basin‐Euroseistest area, northern Greece

AbstractWe examine the use of ambient noise cross‐correlation tomography for shallow site characterization using a modified two‐step approach. Initially, we extract Rayleigh wave traveltimes from correlation traces of vertical component seismic recordings from a local network installed in Mygdonia basin, northern Greece. The obtained Rayleigh wave traveltimes show significant spatial variability, as well as distance and frequency dependence due to the 3‐D structure of the area, dispersion, and anelastic attenuation effects. The traveltime data sets are inverted through a surface wave tomography approach to determine group velocity maps for each frequency. The proposed tomographic inversion involves the use of approximate Fresnel volumes and interfrequency smoothing constraints to stabilize the results. In the last step, we determine a final 3‐D velocity model using a node‐based Monte Carlo 1‐D dispersion curve inversion. The reliability of the final 3‐D velocity model is examined by spatial and depth resolution analysis, as well as by inversion for different model parameterizations. The obtained results are in very good agreement with previous findings from seismic and other geophysical methods. The new 3‐D VS model provides additional structural constraints for the shallow sediments and bedrock structure of the northern Mygdonia basin up to the depth of ∼200–250 m. Present work results suggest that the migration of ambient tomography techniques from large scales (tens or hundreds of km) to local scales (few hundred meters) is possible but cannot be used as a black box technique for 3‐D modeling and detailed geotechnical site characterization.

A ray model of sound focusing with a balloon lens: An experiment for high school students

A weather balloon filled with carbon dioxide gas is used as a positive spherical acoustic lens. High frequency but audible sound from a circular loudspeaker ensonifies the balloon and produces increased sound pressure levels in a region along the principal axis according to a ray acoustics model. This enhancement was measured experimentally and was found to agree with theory. The possibility that interference from reflected sound off walls or the floor could mask or mimic the expected focusing was countered by calculating and measuring within a "shadow zone" in which only direct rays or rays refracted by the balloon exist by the method of Fresnel volumes. The experiment described in this paper would be a suitable learning experience for junior high and high school students showing how rays and Snell's law apply to sound as well as light and giving them a measurable predicted focal region for enhanced sound pressure levels.

Fresnel volume migration of single-component seismic data

Standard implementations of Kirchhoff prestack depth migration (PSDM) distribute the recorded wavefield along two-way-traveltime isochrons and an image is generated by constructive interference of these isochrons along the actual reflector elements. Beside the recent developments of wave-equation-based approaches, Kirchhoff PSDM is still considered widely as a state-of-the-art technique in obtaining high-quality images of the subsurface, particularly for highly irregular survey layouts and target-oriented imaging tasks. However, for sparse sampling or limited aperture, the resulting image is affected by significant migration noise as a result of limited constructive interference of the back-propagated wavefield. Some modifications have been proposed to reduce these artifacts. These modifications include constructing a specular path of wave propagation, derived from estimates of the emergent angle of coherent phases in the seismogram section, and the mainly heuristic restriction of the imaging operator to the neighborhood of that wavepath. Our approach uses Fresnel volumes to restrict the migration operator in a physically frequency-dependent way. Using the emergent angle at the receiver, determined by a local slowness analysis, a ray is propagated into the subsurface; the back-propagation of the wavefield is restricted to the vicinity of this ray according to its approximated Fresnel volume. This so-called Fresnel volume migration approach enhances image quality significantly compared with standard Kirchhoff PSDM because of the inherent focusing and the restriction of the back-propagation to the region around the actual reflection point.

Steep reflections from the earth's core reveal small-scale heterogeneity in the upper mantle

We investigate arrivals of seismic phases reflected from the core–mantle boundary (PcP waves) and those reflected from the inner-core boundary (PKiKP waves) at subcritical angles, with the goal of measuring their amplitude ratios. This adds invaluable data points required to study the density jump across the inner-core boundary (ICB). One nuclear explosion and one earthquake, both with favourable focal mechanisms are identified as sources that produce an excellent and abundant record of arrivals of both PcP and PKiKP waves at multiple stations. There is only a limited number of detections of both types of waves on the same seismogram, while more frequently, either one or another of the two phases is detected. Thus, for those cases where at least one phase is above a detectable level, we observe a highly significant negative correlation (anti-correlation) of phase appearances on seismograms, where PcP and PKiKP phase-detections are treated as dichotomous, categorical random variables that can take values detected or undetected. The fact that similar anti-correlation in detection is observed for both explosive and tectonic sources makes less likely the possibility that source effects or a specific near source structure is responsible for this phenomenon. Although laterally varying structure near the core–mantle boundary (CMB) can account for the magnitude of observed fluctuations in the amplitude ratio of PKiKP to PcP, the Fresnel volumes surrounding their ray paths are well separated at the CMB at the frequencies of interest. This separation excludes the possibility that complex structure at or near the CMB is the dominant effect responsible for the observed anti-correlation. We demonstrate that the interaction of the wavefield with near-receiver heterogeneities is an important additional source of amplitude fluctuations across arrays of stations, and the likely cause of the anti-correlation between PKiKP and PcP detections and their spatial coherences at regional arrays. The combined effects of heterogeneities near the surface and the CMB have a profound impact on the estimates of the PKiKP/PcP amplitude ratios and the subsequent estimates of the density jump at the ICB.

Crosshole radar velocity tomography with finite-frequency Fresnel volume sensitivities

SUMMARY Crosshole-radar velocity tomography is increasingly being used to characterize the electrical and hydrologic properties of the Earth’s near-surface. Because radar methods are sensitive to the water content of geologic materials, velocity tomography is a good proxy for imaging soil water retention in the vadose zone and porosity in the saturated zone. In many near-surface environments, radar velocity varies over a few orders of magnitude. Common velocity tomography applies ray theory that assumes infinite frequency propagation. The ray approximation may induce velocity modelling artefacts and loss of localization. We propose an alternative method for computing velocity tomogram sensitivities using Fresnel volumes based on first-order scattering. The Fresnel volume sensitivities account for the finite-frequency of the crosshole radar signal and model the physics of radar propagation more accurately than the ray theory approximation. We demonstrate that applying finite-frequency Fresnel volume sensitivities provides improved radar velocity tomograms in low contrast environments. Analysis of the singular value decomposition of the sensitivity matrix demonstrates how the finite-frequency inversion recovers and localizes velocity heterogeneities better than ray theory. The singular value spectrum obtained from the full waveform sensitivities matches well with the Fresnel volume results. Furthermore, these basis functions are smooth and localized because the kernels capture the first order wave propagation effect compared to ray based sensitivity, which is a high frequency approximation. Through forward modelling experiments, we validate the finite-frequency sensitivity for crosshole radar velocity. In the Fresnel volume approach, the traveltime picking is more efficient because the datum is the peak of the first pulse rather than the first arrival, and therefore, data pre-processing is simpler and may be easily automated. The synthetic Fresnel volume inversion results show improvements in the final model and the data fits are better when compared to the ray theoretical inversions.

Acoustic scattering from an obstacle in a homogeneous non-uniform moving medium via the ray method

For an accurate noise prediction for aircraft in forward flight, scattering of engine noise from airframe must be included into the description of the sound field. A realistic model of acoustic scattering by airframe in flight must account for the effects of the background flow. We consider acoustic scattering from an obstacle in the presence of a time-independent background flow at low Mach number and we neglect all other effects of the variation of the medium on sound propagation. The model is based on a convected Helmholtz equation with coefficients dependent on the background flow, which can be given either analytically or by CFD. A high-frequency solution is sought by the ray acoustics with a simple method based on the Fresnel volumes of the rays employed to determine the caustic zones and the field on the caustics. The weaker effects of a tip or smooth-body diffraction are neglected and only the reflection and edge diffraction are taken into account. The source of noise is modeled as a point source. The upstream shielding effect of a boundary layer is investigated as an example and uniform versus non-uniform flow effects are illustrated. Other numerical examples will be considered. [Work supported by NRC.]

JaTS: a fully portable seismic tomography software based on Fresnel wavepaths and a probabilistic reconstruction approach

JaTS, a Java 2D seismic tomography software, is presented. It implements original algorithms achieving optimal accuracy with reasonable computing costs. A second-order Fast Marching Method (FMM) is used for solving the eikonal equation, therefore enabling a fast and robust computation of seismic traveltimes between sources and receivers. The wavepaths are materialized by Fresnel volumes rather than by conventional rays. This approach accounts for complex velocity models and has the advantage of considering the effects of the wave frequency in the velocity model resolution. The model is computed by a Simultaneous Iterative Reconstruction Technique (SIRT) which has been reformulated to integrate Fresnel wavepaths by using a probabilistic approach. In addition, various utilities are implemented, such as a tapering filter, used to decrease artifact effects occurring in the vicinity of the sources and receivers. The software also offers the possibility of reconstructing the velocity field on a grid larger than the one used for the wave propagation computation. This contributes to stabilize the estimated values. All of the seismic processing tools have been integrated with a user-friendly graphical interface. JaTS represents a tightly integrated tool suite that supports the entire process of importing the SG2 field records, first-break picking, forward modeling and velocity-field computing across multiple platforms.

Sensitivity kernels for finite-frequency ocean acoustic observables

Wave theoretic modeling was applied to obtain sensitivity kernels representing the amount by which ocean acoustic observables, such as pressure, intensity, or arrival times, are affected by localized sound-speed variations anywhere in the medium. These kernels can be derived using the first Born approximation, yielding an integral representation of observable perturbations as functionals of sound-speed perturbations. The kernel in each integral represents the spatial sensitivity of the observable to sound-speed perturbations in the medium. The travel-time sensitivity kernel represents a full-wave generalization of the highly localized ray kernel from geometric optics. This work extended recent wave-theoretic results for the travel-time sensitivity kernel for short-range propagation in simple environments to long-range transmissions in more complicated ocean environments relevant to ongoing propagation and inversion experiments. It was shown that the geometry of the wave-theoretic sensitivity kernels is related to the geometry of Fresnel volumes surrounding eigenrays, provided that the effects of refraction are taken into account for the calculation of the latter. [Work supported by ONR.]

The isochron ray in seismic modeling and imaging

The isochron, the name given to a surface of equal two‐way time, has a profound position in seismic imaging. In this paper, I introduce a framework for construction of isochrons for a given velocity model. The basic idea is to let trajectories called isochron rays be associated with iso chrons in an way analogous to the association of conventional rays with wavefronts. In the context of prestack depth migration, an isochron ray based on conventional ray theory represents a simultaneous downward continuation from both source and receiver. The isochron ray is a generalization of the normal ray for poststack map migration. I have organized the process with systems of ordinary differential equations appearing on two levels. The upper level is model‐independent, and the lower level consists of conventional one‐way ray tracing. An advantage of the new method is that interpolation in a ray domain using isochron rays is able to treat triplications (multiarrivals) accurately, as opposed to interpolation in the depth domain based on one‐way traveltime tables. Another nice property is that the Beylkin determinant, an important correction factor in amplitude‐preserving seismic imaging, is closely related to the geometric spreading of isochron rays. For these reasons, the isochron ray has the potential to become a core part of future implementations of prestack depth migration. In addition, isochron rays can be applied in many contexts of forward and inverse seismic modeling, e.g., generation of Fresnel volumes, map migration of prestack traveltime events, and generation of a depth‐domain–based cost function for velocity model updating.

Investigating effects of 3-D ray tracing methods in local earthquake tomography

The evaluation of accuracy and precision of the employed forward solution method (ray tracer) and of its effects on the inverse solution is one of the key issues in quality and reliability assessment of tomographic images. In general, an analytical solution to the forward problem does not exist for arbitrary three-dimensional (3-D) velocity models. To analyze ray tracing performance and to evaluate the effects of numerical inaccuracies and approximations inherent to any such forward calculation method, we implement a 3-D ray shooting algorithm in the widely used SIMULPS software and compare it to its’ standard approximative pseudo-bending method (ART_PB). In a first step the effect of the different parametrizations of the velocity model required by the two ray tracers is assessed. This leads to the definition of specific ‘forward grids’ derived from and numerically representing the same seismic velocity model, thus ensuring that only insignificant differences in the ray tracing results originate from the different forward model representations. Hence, the observed differences in travel times and ray paths are then due to inaccuracies and lack of precision of the ray tracers. Precision of each ray tracer is assessed by exchanging source and receiver coordinate pairs and comparing the results from ray tracing in both directions, whereas accuracy is analyzed by comparing the results from the two ray tracers for the same source-receiver pair. Using realistic heterogeneous synthetic velocity models, our results indicate that both ray tracers are precise within ±10 ms in travel time for ray lengths less than about 60 km, and ray paths are identical with respect to appropriate Fresnel volumes. For longer rays, however, the ray shooting method yields significantly smaller errors than the ART_PB ray bending. Even with regard to optimal resolution in real tomographic images, differences between the two ray tracers are insignificant up to about 80 km raylength. Noteworthy differences, however, appear in the resolution matrix calculated for the same inverse problem with the two different ray tracers. In addition, differences in the take-off angle of the ray from the source are significant with respect to calculations of focal mechanisms.

Fresnel volumes corresponding to PKP waves in the IASP91 model

Fresnel volumes provide a suitable tool for estimating the resolution limits or accuracy of tomographic methods based only on the arrival times of these seismic waves, which penetrate the deep interior of the Earth. The standard definition of the Fresnel volume was modified to include effects of multipathing, i.e., multivalued travel times. Computations were performed in the IASP91 model for three representative epicentral distances and two prevailing signal periods. Preliminary results have been obtained. They place a rather strong restriction on the resolution limits of tomographic methods, depending on the prevailing period, due to the relatively large fatness of PKP Fresnel volumes.

Three-dimensional interface modelling with two-dimensional seismic data: the Alpine crust-mantle boundary

SUMMARY We present a new approach to determine the 3-D topography and lateral continuity of seismic interfaces using 2-D-derived controlled-source seismic reflector data. The aim of the approach is to give the simplest possible structure consistent with all reflector data and error estimates. We define simplicity of seismic interfaces by the degree of interface continuity (i.e. shortest length of oVsets) and by the degree of interface roughness (least surface roughness). The method is applied to structural information of the crust‐mantle boundary (Moho) obtained from over 250 controlled-source seismic reflection and refraction profiles in the greater Alpine region. The reflected and refracted phases from the Moho interface and their interpretation regarding crustal thickness are reviewed and their reliability weighted. Weights assigned to each reflector element are transformed to depth errors considering Fresnel volumes. The 2-D-derived reflector elements are relocated in space (3-D migration) and interpolation is performed between the observed reflector elements to obtain continuity of model parameters. Interface oVsets are introduced only where required according to the principle of simplicity. The resulting 3-D model of the Alpine crust‐mantle boundary shows two oVsets that divide the interface into a European, an Adriatic and a Ligurian Moho, with the European Moho subducting below the Adriatic Moho, and with the Adriatic Moho underthrusting the Ligurian Moho. Each sub-interface depicts the smoothest possible (i.e. simplest) surface, fitting the reflector data within their assigned errors. The results are consistent with previous studies for those regions with dense and reliable controlledsource seismic data. The newly derived Alpine Moho interface, however, surpasses earlier studies by its lateral extent over an area of about 600 km by 600 km, by quantifying reliability estimates along the interface, and by obeying the principle of being consistently as simple as possible.

Discussion of the Features of Synthetic Amplitude-Distance Curves of Teleseismic P Waves for Global Earth Models

The reason why the synthetic amplitude distance curves of P-waves for models IASP91 and PREM, observed for shallow sources up to epicentral distance of 28°, oscillate is illustrated by means of synthetic seismograms. Furthermore the position of the beginning of the diffraction of the P-wave at the CMB, depending on the prevailing signal period, is discussed in connection with the extension of Fresnel volumes.

Analytical expressions for Fresnel volumes and interface fresnel zones of seismic body waves. Part 2: Transmitted and converted waves. Head waves

Fresnel volumes and interface Fresnel zones of transmitted and head waves are studied. The relation derived for transmitted waves may also be used for converted reflected waves. Considerable attention is devoted to the penetration of Fresnel volumes across structural interfaces, particularly for head waves.

Analytical expression for fresnel volumes and interface fresnel zones of seismic body waves. Part 1: Direct and unconverted reflected waves

Fresnel volumes, plane-sectional Fresnel zones and interface Fresnel zones of direct and unconverted reflected waves are studied. Exact analytical expressions for various parameters of Fresnel volumes and Fresnel zones are derived and discussed. Among others, these expressions are related to semi-axes of Fresnel zones, to overshooting and penetration distances, and to the off-ray shifts of Fresnel zones.