tutorial on seismic interferometry part 2 underlying theory and new advances in the 1990s the method of time reversed acoustics was developed this method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal when ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers they focus at the position of the original source even when the medium is very complex in seismic interferometry the time reversed responses are not physically sent into the earth but they are convolved with other measured responses the effect is essentially the same the time reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers a mathematical derivation based on reciprocity theory formalizes this principle the crosscorrelation of responses at two receivers integrated over different sources gives the green s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver this green s function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving recent developments circumventing these assumptions include interferometric representations for attenuating and or moving media as well as unified representations for waves and diffusion phenomena bending waves quantum mechanical scattering potential fields elastodynamic electromagnetic poroelastic and electroseismic waves significant improvements in the quality of the retrieved green s functions have been obtained with interferometry by deconvolution a trace by trace deconvolution process compensates for complex source functions and the attenuation of the medium interferometry by multidimensional deconvolution also compensates for the effects of one sided and or irregular illumination

robust computational algorithms for dynamic interface tracking in three dimensions front tracking provides sharp resolution of wave fronts through the active tracking of interfaces between distinct materials a major challenge to this method is to handle changes in the interface topology we describe two algorithms implemented in the front tracking code frontier to model dynamic changes in three dimensional interfaces the two methods can be combined to give a hybrid method that is superior to each individual method the success of these algorithms is shown by simulations of rayleigh taylor instability which is an interfacial instability driven by an acceleration directed across a material interface our numerical results are validated by comparing the numerical computation of the velocity of a single rising bubble with an analytic model for the bubble velocity

an unfitted finite element method for the numerical approximation of void electromigration microelectronic circuits usually contain small voids or cracks and if those defects are large enough to sever the line they cause an open circuit we present a numerical method for investigating the migration of voids in the presence of both surface diffusion and electric loading our mathematical model involves a bulk interface coupled system with a moving interface governed by a fourth order geometric evolution equation and a bulk where the electric potential is computed thanks to a novel approximation of the interface equidistribution of its vertices is guaranteed and no remeshing is necessary in addition the used curvature approximation means that our method is unconditionally stable for the evolution by surface diffusion only various examples are performed to demonstrate the accuracy of the method

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

on seismic interferometry the generalized optical theorem and the scattering matrix of a point scatterer we have analyzed the far field approximation of the green s function representation for seismic interferometry by writing each of the green s functions involved in the correlation process as a superposition of a direct wave and a scattered wave the green s function representation is rewritten as a superposition of four terms when the scattered waves are modeled with the born approximation it appears that a three term approximation of the green s function representation omitting the term containing the crosscorrelation of the scattered waves yields a nearly exact retrieval whereas the full four term expression leads to a significant nonphysical event this is because the born approximation does not conserve energy and therefore is an insufficient model to explain all aspects of seismic interferometry we use the full four term expression of the green s function representation to derive the generalized optical theorem unlike other recent derivations which use stationary phase analysis our derivation uses reciprocity theory from the generalized optical theorem we derive the nonlinear scattering matrix of a point scatterer this nonlinear model accounts for primary and multiple scattering at the point scatterer and conforms with well established scattering theory of classical waves the model is essential to explain fully the results of seismic interferometry even when it is applied to the response of a single point scatterer the nonlinear scattering matrix also has implications for modeling inversion and migration

topology adaptive mesh deformation for surface evolution morphing and multiview reconstruction triangulated meshes have become ubiquitous discrete surface representations in this paper we address the problem of how to maintain the manifold properties of a surface while it undergoes strong deformations that may cause topological changes we introduce a new self intersection removal algorithm transformesh and propose a mesh evolution framework based on this algorithm numerous shape modeling applications use surface evolution in order to improve shape properties such as appearance or accuracy both explicit and implicit representations can be considered for that purpose however explicit mesh representations while allowing for accurate surface modeling suffer from the inherent difficulty of reliably dealing with self intersections and topological changes such as merges and splits as a consequence a majority of methods rely on implicit representations of surfaces e g level sets that naturally overcome these issues nevertheless these methods are based on volumetric discretizations which introduce an unwanted precision complexity trade off the method that we propose handles topological changes in a robust manner and removes self intersections thus overcoming the traditional limitations of mesh based approaches to illustrate the effectiveness of transformesh we describe several challenging applications surface morphing and 3d reconstruction

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

robust topological operations for dynamic explicit surfaces we present a solution to the mesh tangling problem in surface tracking using an explicit triangle mesh to track the location of a surface as it moves in three dimensions has many potential advantages for accuracy and efficiency compared to implicit capturing methods such as level sets however particularly when mesh surgery is required for topological changes this approach is prone to tangling the mesh may self intersect or otherwise no longer represent a physical interface our new approach uses robust collision testing to determine when a mesh operation such as motion adaptive refinement coarsening or topological change will lead to an invalid state we then either roll back noncritical operations or apply robust collision response algorithms minimally perturbing the mesh to guarantee validity we present numerical examples demonstrating the robustness and accuracy of the method

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

a simple and exact acoustic wavefield modeling code for data processing imaging and interferometry applications improvements in industrial seismic seismological acoustic or interferometric theory and applications often result in quite subtle changes in sound quality seismic images or information which are nevertheless crucial for improved interpretation or experience when evaluating new theories and algorithms using synthetic data an important aspect of related researchisthereforethat numerical errors due towavefield modeling are reduced to a minimum we present a new matlab code based on the foldy method that models theoretically exact direct and scattered parts of a wavefield its main advantage lies in the fact that while all multiple scattering interactions aretakenintoaccount unlikefinite difference or finite element methods numerical dispersion errors are avoided the method is therefore ideal for testing new theory in industrial seismics seismology acoustics and in wavefield interferometry in particular because the latter is particularly sensitive to the dynamics of scattering interactions we present the theory behind the foldy acoustic modeling method and provide examples of its implementation we also benchmark the code against a good finite difference code because our foldy code was written and optimized to test new theory in seismic interferometry examples of its application to seismicinterferometryarealsopresented showing its validity and importance when exact modeling results are needed

representation theorems and green s function retrieval for scattering in acoustic media reciprocity theorems for perturbed acoustic media are provided in the form of convolution and correlation type theorems these reciprocity relations are particularly useful in the general treatment of both forward and inverse scattering problems using green s functions to describe perturbed and unperturbed waves in two distinct wave states representation theorems for scattered waves are derived from the reciprocity relations while the convolution type theorems can be manipulated to obtain scattering integrals that are analogous to the lippmann schwinger equation the correlation type theorems can be used to retrieve the scattering response of the medium by cross correlations unlike previous formulations of green s function retrieval the extraction of scattered wave responses by cross correlations does not require energy equipartitioning allowing for uneven energy radiation brings experimental advantages to the retrieval of fields scattered by remote lossless and or attenuative scatterers these concepts are illustrated with a number of examples including analytic solutions to a one dimensional scattering problem and a numerical example in the context of seismic waves recorded on the ocean bottom

a hybrid variational front tracking level set mesh generator for problems exhibiting large deformations and topological changes we present a method for generating 2 d unstructured triangular meshes that undergo large deformations and topological changes in an automatic way we employ a method for detecting when topological changes are imminent via distance functions and shape skeletons when a change occurs we use a level set method to guide the change of topology of the domain mesh this is followed by an optimization procedure using a variational formulation of active contours that seeks to improve boundary mesh conformity to the zero level contour of the level set function our method is advantageous for arbitrary lagrangian eulerian ale type methods and directly allows for using a variational formulation of the physics being modeled and simulated including the ability to account for important geometric information in the model such as for surface tension driven flow furthermore the meshing procedure is not required at every time step and the level set update is only needed during a topological change hence our method does not significantly affect computational cost

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

relating source receiver interferometry to an inverse scattering series to derive a new method to estimate internal multiples we have evaluated an explicit relationship between the representations of internal multiples by source receiver interferometry and an inverse scattering series this provides a new insight into the interaction of different terms in each of these internal multiple prediction equations and explains why amplitudes of estimated multiples are typically incorrect a downside of the existing representations is that their computational cost is extremely high which can be a precluding factor especially in 3d applications using our insight from source receiver interferometry we have developed an alternative computationally more efficient way to predict internal multiples the new formula is based on crosscorrelation and convolution two operations that are computationally cheap and routinely used in interferometric methods we have compared the results of the standard and the alternative formulas qualitatively in terms of the constructed wavefields and quantitatively in terms of the computational cost using examples from a synthetic data set

retrieval of green s function and generalized optical theorem for the scattering of complete dyadic fields green s function retrieval has been widely used in different research fields due to the fact that the green s function can be extracted by cross correlating the records at two receivers in this paper the retrieval of the dyadic green s function is studied by investigating the representation theorem the generalized optical theorem for the dyadic fields is derived based on the elastic dynamic interferometric equation by addressing the cross correlation recorded at two receivers the important role of the generalized optical theorem and energy equipartition in retrieving the exact green s function is shown the presented derivation also shows the newton marchenko equation holdsif the condition of equipartition is not satisfied

on unitary elements defined on lorentz cone and their applications abstract in this paper we illustrate a new concept regarding unitary elements defined on lorentz cone and establish some basic properties under the so called unitary transformation associated with lorentz cone as an application of unitary transformation we achieve a weaker version of the triangle inequality and several weak majorizations defined on lorentz cone

nonlinear scattering based imaging in elastic media theory theorems and imaging conditions with the more widespread introduction of multicomponent recording devices in land and marine ocean bottom seismic acquisition elastic imaging may become mainstream in coming years we have derived new nonlinear elastic imaging conditions a correlation type representation theorem for perturbed elastic media commonly used in seismic interferometry to explain how a scattered wave response between two receivers sources may be predicted given a boundary of sources receivers can be considered as a starting point for the derivation here we use this theorem to derive and interpret imaging conditions for elastic migration by wavefield extrapolation e g elastic reverse time migration some approximations lead to a known heuristically derived imaging condition that crosscorrelates p and s wave potentials that are separated in the subsurface after full wavefield extrapolation this formal connection reveals that the nonapproximated correlation type representation theorem can be interpreted as a nonlinear imaging condition that accounts also for multiply scattered and multiply converted waves properly focusing such energy at each image point we present a synthetic data example using either an ideal acquisition on a full closed boundary or a real partial boundary seismic exploration survey and we demonstrate the importance of nonlinearities in pure and converted mode imaging in pp imaging they result in better illumination and artifact reduction whereas in ps imaging they show how zero time lag and zero space lag crosscorrelation imaging conditions are not ideal for imaging of converted mode waves because no conversion arises from zero offset experiments

numerical simulation of interface waves by high order spectral modeling techniques few problems in elastodynamics have a closed form analytical solution the others can be investigated with semianalytical methods but in general one is not sure whether these methods give reliable solutions the same happens with numerical techniques for instance finite difference methods solve in principle any complex problem including those with arbitrary inhomogeneities and boundary conditions however there is no way to verify the quantitative correctness of the solutions the major problems are stability with respect to material properties numerical dispersion and the treatment of boundary conditions in practice these problems may produce inaccurate solutions in this paper the study of complex problems with two different numerical grid techniques in order to cross check the solutions is proposed interface waves in particular are emphasized since they pose the major difficulties due to the need to implement boundary conditions the first method is based on global differential operators where the solution is expanded in terms of the fourier basis and chebyshev polynomials while the second is the spectral element method an extension of the finite element method that uses chebyshev polynomials as interpolating functions both methods have spectral accuracy up to approximately the nyquist wave number of the grid moreover both methods implement the boundary conditions in a natural way particularly the spectral element algorithm we first solve lamb s problem and compare numerical and analytical solutions then the problem of dispersed rayleigh waves and finally the two quarter space problem we show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist the combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics

cancellation of spurious arrivals in green s function extraction and the generalized optical theorem the extraction of the green s function by cross correlation of waves recorded at two receivers nowadays finds much application we show that for an arbitrary small scatterer the cross terms of scattered waves give an unphysical wave with an arrival time that is independent of the source position this constitutes an apparent inconsistency because theory predicts that such spurious arrivals do not arise after integration over a complete source aperture this puzzling inconsistency can be resolved for an arbitrary scatterer by integrating the contribution of all sources in the stationary phase approximation to show that the stationary phase contributions to the source integral cancel the spurious arrival by virtue of the generalized optical theorem this work constitutes an alternative derivation of this theorem when the source aperture is incomplete the spurious arrival is not canceled and could be misinterpreted to be part of the green s function we give an example of how spurious arrivals provide information about the medium complementary to that given by the direct and scattered waves the spurious waves can thus potentially be used to better constrain the medium

analysis of nonsmooth vector valued functions associated with second order cones let open image in new window be the lorentz second order cone in open image in new window for any function f from open image in new window to open image in new window one can define a corresponding function fsoc x on open image in new window by applying f to the spectral values of the spectral decomposition of x open image in new window with respect to open image in new window we show that this vector valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as order semismoothness these results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second order cone programs and complementarity problems

the reciprocity theorem for the scattered field is the progenitor of the generalized optical theorem by analyzing correlation type reciprocity theorems for wavefields in perturbed media it is shown that the correlation type reciprocity theorem for the scattered field is the progenitor of the generalized optical theorem this reciprocity theorem in contrast to the generalized optical theorem allows for inhomogeneous background properties and does not make use of a far field condition this theorem specializes to the generalized optical theorem when considering a finite size scatterer embedded in a homogeneous background medium and when utilizing the far field condition moreover it is shown that the reciprocity theorem for the scattered field is responsible for the cancellation of non physical spurious arrivals in seismic interferometry and as such provides the mathematical description of such arrivals even though here only acoustic waves are treated the presented treatment is not limited to such wavefields and can be generalized to general wavefields therefore this work provides the framework for deriving equivalents of the generalized optical theorem for general wavefields

source receiver wave field interferometry correlation or convolution of recordings of diffuse fields at a pair of locations have been shown to result in estimates of the green s function between the two locations variously referred to as wave field or seismic interferometry in different fields of research green s functions can thus be constructed between either pairs of receivers or pairs of energy sources proofs of these results rely on representation theorems we show how to derive three acoustic and elastic representation theorems that unify existing correlational and convolutional approaches we thus derive three forms of interferometry that provide green s functions on source to receiver paths using only energy that has propagated from surrounding sources or to surrounding receivers the three forms correspond to three possible canonical geometries we thus allow interferometric theory and methods to be applied to commonly used source receiver configurations

source receiver reverse time imaging of dual source vector acoustic seismic data abstractnovel technologies in seismic data acquisition allow for recording full vector acoustic va data pointwise recordings of pressure and its multicomponent gradient excited by pressure only as well as dipole gradient sources building on recent connections between imaging and seismic interferometry we present a wave equation based nonlinear reverse time imaging approach that takes full advantage of dual source multicomponent data the method s formulation relies on source receiver scattering reciprocity thus making proper use of va fields in the wavefield extrapolation and imaging condition steps in a self consistent manner the va imaging method is capable of simultaneously focusing energy from all in and outgoing waves the receiver side up and downgoing receiver ghosts fields are handled by the va receiver extrapolation whereas source side in and outgoing source ghosts arrivals are accounted for when combining dual source data at the imaging condition additionally va imaging handl

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions