direct vector field method to obtain angle domain common image gathers from isotropic acoustic and elastic reverse time migration we have developed an alternative new method to produce common image gathers in the incident angle domain by calculating wavenumbers directly from the p wave polarization rather than using the dominant wavenumber as the normal to the source wavefront in isotropic acoustic media the wave propagation direction can be directly calculated as the spatial gradient direction of the acoustic wavefield which is parallel to the wavenumber direction the normal to the wavefront instantaneous wavenumber obtained via a novel hilbert transform approach is used to calculate the local normal to the reflectors in the migrated image the local incident angle is produced as the difference between the propagation direction and the normal to the reflector by reordering the migrated images over all common source gathers with incident angle common image gathers are produced in the incident angle domain instantaneous wavenumber takes the place of the normal to the reflector in the migrated image p and s wave separ

wave field separation in two dimensional anisotropic media until recently the term elastic usually implied two dimensional 2 d and isotropic in this limited context the divergence and curl operators have found wide use as wave separation operators for example mora 1987 used them in his inversion method to allow separate correlation of p and s arrivals although the separation is buried in the math and not obvious clayton 1981 used them explicitly in several modeling and inversion methods devaney and oristaglio 1986 used closely related operators to separate p and s arrivals in elastic vsp data

isotropic angle domain elastic reverse time migration multicomponent data usually are not processed with specifically designed procedures but with procedures analogous to those used for single component data in isotropic media the vertical and horizontal components of the data commonly are taken as proxies for the p and s wave modes which are imaged independently with the acoustic wave equations this procedure works only if the vertical and horizontal components accurately represent p and s wave modes which generally is not true therefore multicomponent images constructed with this procedure exhibit artifacts caused by incorrect wave mode separation at the surface an alternative procedure for elastic imaging uses the full vector fields for wavefield reconstruction and imaging thewavefieldsarereconstructedusingthemulticomponentdata as a boundary condition for a numerical solution to the elastic wave equation the key component for wavefield migration is theimagingcondition whichevaluatesthematchbetweenwavefields reconstructed from sources and receivers for vector wave fields a simple component by component crosscorrelation between two wavefields leads to artifacts caused by crosstalk between the unseparated wave modes we can separate elastic wavefields after reconstruction in the subsurface and implement theimagingconditionascrosscorrelationofpurewavemodesinstead of the cartesian components of the displacement wavefield thisapproachleadstoimagesthatareeasiertointerpretbecause they describe reflectivity of specified wave modes at interfaces of physical properties as for imaging with acoustic wavefields the elastic imaging condition can be formulated conventionally crosscorrelation with zero lag in space and time and extendedtononzerospaceandtimelags theelasticimagesproduced by an extended imaging condition can be used for angle decomposition of primary pp or ss and converted ps or sp reflectivity angle gathers constructed with this procedure have applicationsformigrationvelocityanalysisandamplitude variation with angleanalysis

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

multidirectional vector based elastic reverse time migration and angle domain common image gathers with approximate wavefield decomposition of p and s waves abstractelastic reverse time migration e rtm has limitations when the migration velocities contain strong contrasts first the traditional scheme of p s wave mode separation is based on helmholtz s equations which ignore the conversion between p and s waves at the current separation time thus it contains an implicit assumption of the constant shear modulus and requires smoothing the heterogeneous model to approximately satisfy a locally constant condition second the vector based imaging condition needs to use the reflection image normal and it also cannot give the correct polarity of the pp image in all possible conditions third the angle domain common image gathers adcigs calculated using the poynting vectors pvs do not consider the wave interferences that happen at each reflector therefore smooth models are often used for e rtm we relax this condition by proposing an improved data flow that involves three new contributions the first contribution is an improved system of p s wave mode

fast algorithms for elastic wave mode separation and vector decomposition using low rank approximation for anisotropic media abstractwave mode separation and vector decomposition are significantly more expensive than wavefield extrapolation and are the computational bottleneck for elastic reverse time migration in heterogeneous anisotropic media we have expressed elastic wave mode separation and vector decomposition for anisotropic media as space wavenumber domain operations in the form of fourier integral operators and developed fast algorithms for their implementation using their low rank approximations synthetic data generated from 2d and 3d models demonstrated that these methods are accurate and efficient

comparison of methods for extracting adcigs from rtm abstractmethods for extracting angle domain common image gathers adcigs during 2d reverse time migration fall into three main categories direction vector based methods local plane wave decomposition methods and local shift imaging condition methods the direction vector based methods which use either amplitude gradients or phase gradients cannot handle overlapping events because of an assumption of one propagation direction per imaging point per imaging time however the adcigs from the direction vector based methods have the highest angle resolution a new direction vector based method using instantaneous phase gradients in space and time gives the same propagation directions and adcigs as those obtained by the poynting vector or polarization vector based methods where amplitudes are large angles calculated by the phase gradients have larger uncertainties at smaller amplitudes but they do not significantly degrade the adcigs because they contribute only small amplitudes the local plane wave d

hopf bifurcation analysis for a model of genetic regulatory system with delay this paper deals with a mathematical model that describe a genetic regulatory system the model has a delay which affects the dynamics of the system we investigate the stability switches when the delay varies and show that hopf bifurcations may occur within certain range of the model parameters by combining the normal form method with the center manifold theorem we are able to determine the direction of the bifurcation and the stability of the bifurcated periodic solutions finally some numerical simulations are carried out to support the analytic results

multiple attenuation in the image space multiples can be suppressed in the angle domain image space after migration for a given velocity model primaries and multiples have different angle domain moveout and therefore can be separated using techniques similar to the ones employed in the data space prior to migration we use radon transforms in the image space to discriminate between primaries and multiples and employ accurate functions describing angledomain moveouts since every individual angle domain common image gather incorporates complex 3d propagation effects our method has the advantage of working with 3d data and complicated geology therefore our method offers an alternative to the more expensive surface related multiple elimination srme approach operating in the data space radon transforms are cheap but not necessarily ideal for separating primaries and multiples particularly at small angles where the moveout discrepancy between the two kinds of events are not large better techniques involving signal noise separation using prediction error filters can be employed as well although such approaches fall outside the scope of this paper we demonstrate using synthetic and real data examples the power of our method in discriminating between primaries and multiples after migration by wavefield extrapolation followed by transformation to the angle domain

elastic wave mode separation for vti media elastic wave propagation in anisotropic media is well represented by elastic wave equations modeling based on elasticwaveequationscharacterizesbothkinematicsanddynamics correctly however because p and s modes are both propagated using elastic wave equations there is a need to separate p and s modes to efficiently apply single mode processing tools in isotropic media wave modes are usually separated using helmholtz decomposition however helmholtz decomposition using conventional divergence and curl operators in anisotropic media does not give satisfactory resultsandleavesthedifferentwavemodesonlypartiallyseparated theseparationofanisotropicwavefieldsrequiresmore sophisticated operators that depend on local material parameters anisotropic wavefield separation operators are constructedusingthepolarizationvectorsevaluatedateachpoint of the medium by solving the christoffel equation for local mediumparameters thesepolarizationvectorscanberepresented in the space domain as localized filtering operators which resemble conventional derivative operators the spatially variable pseudo derivative operators perform well in heterogeneous vti media even at places of rapid velocity densityvariation syntheticresultsindicatethattheoperators can be used to separate wavefields for vti media with an arbitrarydegreeofanisotropy

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

the stability and hopf bifurcation analysis of a gene expression model abstract in this paper we investigate a model for gene expression unlike the models mathematically analyzed previously we have both transcriptional and translational time delays the stability and hopf bifurcation of the equilibrium point are investigated different to previous papers a multiple time scale mts technique is employed to calculate the normal form on the center manifold of system of delay differential equations which is much easier to implement in practice than the conventional method center manifold reduction our results show that when time delay is small the equilibrium is stable when it is at its critical value hopf bifurcation happens and while for very large value of time delay the oscillation sustains which has been confirmed by the published data and proved mathematically by using the global continuity of the hopf bifurcation in this paper

elastic least squares one way wave equation migration abstractleast squares migration seeks a reflectivity model that fits the observed data it is used to compensate for acquisition noise poor sampling of sources and receivers on the surface as well as poor illumination of the subsurface to date least squares migration has been mainly restricted to the imaging of acoustic wavefields we have developed an extension of one way wave equation least squares migration for elastic wavefields in isotropic media least squares migration is an iterative method that requires a forward and an adjoint operator in elastic least squares one way wave equation migration the forward operator generates data components from multiparameter images by recursive wavefield decomposition extrapolation and recomposition conversely the adjoint operator generates multiparameter images from data components by recursively applying the adjoint of the wavefield recomposition extrapolation and wavefield decomposition operators we use an extended imaging condition and regularize

stability and bifurcations of equilibria in a multiple delayed differential equation the influence of multiple negative delayed feedback loops on the stability of a single action mechanism are considered a characteristic equation for the linearized stability of the equilibrium is completely analyzed as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops the bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold in particular the nature of hopf and more degenerate higher codimension bifurcations are explicitly determined

elastic least squares reverse time migration abstractwe use elastic least squares reverse time migration lsrtm to invert for the reflectivity images of p and s wave impedances elastic lsrtm solves the linearized elastic wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration numerical tests on synthetic data and field data reveal the advantages of elastic lsrtm over elastic reverse time migration rtm and acoustic lsrtm for our examples the elastic lsrtm images have better resolution and amplitude balancing fewer artifacts and less crosstalk compared with the elastic rtm images the images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic lsrtm images similar to conventional least squares migration elastic lsrtm also requires an accurate estimation of the p and s wave migration velocity models however the problem remains that when there are moderate errors in the velocity model and strong multiples lsrtm

exact solution of a general two dimensional ising model the partition function abstract the exact partition function per spin is obtained for a two dimensional generalization of the ising model this general model consists of a unitary cell that is repeated throughout the system the unitary cell is made up of spins in t columns and q rows with arbitrary energies between them the exact result is given in terms of certain quantities see section 2 which can be determined for particular cases it is always possible to transform the unit cell in order to obtain a desired given model thus the results for several known models are recovered e g the utiyama and the hexagonal with three different interaction parameters models furthermore explicit results are presented for other models hitherto not reported in the literature as are the general hexagonal with six different interaction parameters the general triangular model with six different interaction parameters and the tetragonal triangular with seven different interaction parameters models

elastic wavefield separation based on the helmholtz decomposition abstractdivergence and curl operators used for the decomposition of p and s wave modes in elastic reverse time migration rtm change the amplitudes units and phases of extrapolated wavefields i separate the p and s waves in elastic media based on the helmholtz decomposition the decomposed wavefields based on this approach have the same amplitudes units and phases as the extrapolated wavefields to avoid expensive multidimensional integrals in the helmholtz decomposition i introduce a fast poisson solver to efficiently solve the vector poisson s equation this fast algorithm allows us to reduce computational complexity from o n2 to o n log n where n is the total number of grid points because the decomposed p and s waves are vector fields i use vector imaging conditions to construct pp ps ss and sp images several 2d numerical examples demonstrate that this approach allows us to accurately and efficiently decompose p and s waves in elastic media in addition elastic rtm images based

3d ps wave imaging with elastic reverse time migration abstractone of the important features of elastic reverse time migration is that it is a depth domain prestack migration method based on a vectorial wavefield pp and ps wave images can be easily generated by implementing an imaging condition to pure wave modes separated with divergence and curl operators in isotropic media in the 3d case however the curl calculation will generate a 3c vector s wave which makes it impossible to generate a scalar ps image with the crosscorrelation imaging condition therefore we first analyzed the polarity distribution of a 3d s wave and concluded that the separated s wave keeps perpendicular to a plane determined by the particle motion direction and the propagation direction of the s wave this specific plane is the raypath plane when the s wave is excited by an incident p wave the normal direction of the raypath plane can be calculated by computing the vector product of the propagation directions of source and receiver wavefields then we used this normal directio

laplace fourier domain elastic full waveform inversion using time domain modeling to obtain subsurface information from onshore seismic exploration data using full waveform inversion fwi based on the acoustic wave equation elastic waves such as ground roll and mode converted waves should be suppressed through heavy preprocessing however such preprocessing deforms not only the elastic waves but also the acoustic waves moreover it is not easy to separate body waves from surface waves in seismic traces for these reasons we need to generate both types of waves in the modeling step to obtain seismic waves that are similar to real seismic waves therefore elastic fwi using the elastic wave equation is necessary to achieve a more accurate fwi in addition elastic fwi can provide better geologic information than acoustic fwi because it inverts the p wavevelocity s wave velocity and density laplace fourier domain elastic fwi is an effective method because it inverts these multiple parameters and can be applied to real seismic data that lack low frequency components however the conventional laplace fourier domain fwi requires a matrix solver with a huge memory cost to perform the modeling in the laplace fourierdomain and memory usage becomes more intensive in the elastic case in the present study we combined time domain wave propagation modeling and laplace fourier domain elastic fwi to invert multiple parameters with less memory cost by using time domain modeling which does not require a matrix solver we obtainedthe forwardand adjoint wavefields withless memory cost the residuals between the recorded and modeled data the virtual sources the hessian matrices and the gradient directions were calculated in the laplace fourier domain to validate the proposed algorithm we performed numerical tests with model 94 synthetic data and benjamin creek real seismic data

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

stability and bifurcation in a two variable delay model for circadian rhythm of neurospora crassa abstract a two variable delay model for circadian rhythm of neurospora crassa is considered in this paper conditions for the global attractivity of the unique positive equilibrium are given moreover hopf bifurcation and the global continuation of the hopf bifurcation branches are addressed through a global hopf bifurcation result

3 d elastic prestack reverse time depth migration by combining and extending previous algorithms for 2 d prestack elastic migration and 3 d prestack acoustic migration a full 3 d elastic prestack depth migration algorithm is developed reverse time extrapolation of the recorded data is by 3 d elastic finite differences computation of the image time for each point in the 3 d volume is by 3 d acoustic finite differences the algorithm operates on three component vector wavefield common source data and produces three component vector reflectivity distributions converted p to s reflections are automatically imaged with the primary p wave reflections there are no dip restrictions as the full wave equation is used the algorithm is illustrated by application to synthetic data from three models a flat reflector a dipping truncated wedge overlying a flat reflector and the classical french double dome and fault model

exact magnetic transitions in the union jack model abstract we present the exact expression for the partition function per spin in the thermodynamic limit for the general union jack model in effect in this model we treat nearest and next nearest neighbors interactions for the particular case in which the interaction parameters within the unit cell have the values e 1 e 2 e 3 ae 4 e 5 ne 3 and e 6 ne 4 with a 1 and n any real number we present explicit phase diagrams and rigorous exact expressions for the loci of the transitions these diagrams split up in several regions which we are able to characterize by their magnetic configurations we do that by calculating the exact correlation functions between the spins in the unit cell

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

elastic imaging with exact wavefield extrapolation for application to ocean bottom 4c seismic data abstracta central component of imaging methods is receiver side wavefield backpropagation or extrapolation in which the wavefield from a physical source scattered at any point in the subsurface is estimated from data recorded by receivers located near or at the earth s surface elastic reverse time migration usually accomplishes wavefield extrapolation by simultaneous reversed time injection of the particle displacements or velocities recorded at each receiver location into a wavefield modeling code here we formulate an exact integral expression based on reciprocity theory that uses a combination of velocity stress recordings and quadrupole dipole backpropagating sources rather than the commonly used approximate formula involving only particle velocity data and dipole backpropagating sources the latter approximation results in two types of nonphysical waves in the scattered wavefield estimate first each arrival contained in the data is injected upward and downward rather than unidirectionally as

improving the efficiency of elastic wave mode separation for heterogeneous tilted transverse isotropic media wave mode separation for ti transversely isotropic models can be carried out by nonstationary filtering the elastic wavefields with localized filters these filters are constructed based on the polarization vectors obtained by solving the christoffel equation using local medium parameters this procedure although accurate is computationally expensive especially in 3d we develop an efficient method for wave mode separation which exploits the same general idea of projecting wavefields onto polarization vectors the method consists of two steps 1 separate wave modes in the wavenumber domain at a number of reference models to obtain the same number of partially separated wavefields then transform all the wavefields to the space domain 2 interpolate the wavefields obtained in step 1 in the space domain using the spatially variable model parameters the new method resembles the phaseshift plus interpolation pspi technique which interpolates the wavefields that are reconstructed at several reference velocities synthetic examples indicate that the separation followed by interpolation is effective for models with complex geology the new technique has the benefits of speed and accuracy

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

reentrant phase transition in a particular 3 3 uc two dimensional ising model abstract using the method presented in an earlier paper we obtain the exact partition function per spin l t corresponding to the particular 3 3 unitary cell two dimensional ising model studied by kitatani et al in contrast with these authors that worked with numerical methods we use algebraic procedures to obtain the analytic expression for l t and the exact equation for the critical temperature t c for other values of the interaction parameters we find results for more extensive ranges of the values of the interaction parameters than obtained by kitatani et al we use these results to discuss physical properties of the systems we exhibit an extended range of values for which double reentrancy is found

prestack scalar reverse time depth migration of 3d elastic seismic data using two independent 3d scalar reverse time depth migrations we migrate the reflected p and s waves in a prestack 3d three component 3 c elastic seismic data volume generated with a p wave source in a 3d model and recorded at the top of the model reflected p and s waves are extracted by divergence a scalar and curl a 3 c vector calculations respectively during shallow downward extrapolation of the elastic seismic data the imaging time for the migrations of both the reflected p and p s converted waves at each point is the one way p wave traveltime from the source to that point the divergence the extracted p waves is reverse time extrapolated using a finite difference solution of the 3d scalar wave equation in a 3d p velocity modeland is imaged to obtain the migrated p image the curl the extracted s waves is first converted into a scalar s wavefield by taking the curl s absolute value as the absolute value of the scalar s wavefield and assigning a positive sign if the curl is counterc

elastic least squares reverse time migration via linearized elastic full waveform inversion with pseudo hessian preconditioning abstracttime domain elastic least squares reverse time migration lsrtm is formulated as a linearized elastic full waveform inversion problem the elastic born approximation and elastic reverse time migration rtm operators are derived from the time domain continuous adjoint state method our approach defines p and s wave impedance perturbations as unknown elastic images our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage optimize before discretize approach the discretized numerical versions of the elastic born operator and its adjoint elastic rtm operator pass the dot product test the conjugate gradient least squares method is used to solve the least squares migration quadratic optimization problem in other words the hessian operator for elastic lsrtm is implicitly inverted via a matrix free algorithm that only requires the action of forward and adjoint operators on vectors the diagonal of the pseudo hessian operator is used t

2d isotropic elastic gaussian beam migration for common shot multicomponent records gaussian beam migration is flexible and adaptable for imaging geologically complicated areas it avoids some limitations of traditional ray based migration approaches such as amplitude singularity in caustic zones and inability to image multiple arrivals previous studies on gaussian beam migration mainly focused on acoustic media we develop a 2d isotropic elastic gaussian beam migration scheme for common shot multicomponent records the proposed method extrapolates p and s mode wavefields simultaneously using the kirchhoff helmholtz integral solution of the isotropic elastodynamic equation we separate the extrapolated wavefields into compressional and shear modes using the helmholtz decomposition and then implement a modified dot product imaging condition this approach enables us to produce clear pp images and avoid polarity reversal issues for ps images in addition based on the theory of wavefield approximation in the effective vicinity of central rays we derive a formula to compute the propaga

elastic reflection based waveform inversion with a nonlinear approach abstractfull waveform inversion fwi is a highly nonlinear problem due to the complex reflectivity of the earth and this nonlinearity only increases under the more expensive elastic assumption in elastic media we need a good initial p wave velocity and even better initial s wave velocity models with accurate representation of the low model wavenumbers for fwi to converge however inverting for the low wavenumber components of p and s wave velocities using reflection waveform inversion rwi with an objective to fit the reflection shape rather than produce reflections may mitigate the limitations of fwi because fwi performing as a migration operator is preferred of the high wavenumber updates along reflectors we have developed an elastic rwi that inverts for the low wavenumber and perturbation components of the p and s wave velocities to generate the full elastic reflection wavefields we derive an equivalent stress source made up by the inverted model perturbations and incident wavefields w

multicomponent kirchhoff migration this paper presents a method for elastic and viscoelastic imaging of multicomponent seismic data the method is based on claerbout s survey sinking concept and the visco elastic kirchhoff integral for the displacement field assuming a multishot and multireceiver experiment the migration process is formulated as a wavefield reconstruction problem using the visco elastic kirchhoff integral twice first the receiver coordinates are downward continued second the source coordinates are downward continued the multicomponent seismic data are treated as a vector wavefield in which the data measurements may be displacement velocity or traction pressure the theoretical formulation is based on the viscoelastic hooke s law and newton s equation of motion as the physical model for seismic wave propagation it is valid for linear viscoelastic media with any anisotropic symmetry when the lowest order ray approximation is introduced the migration equation takes a form similar to conventional kirchhoff migra

phase transitions in regularly diluted 2d ising models with a complicated inherent structure of a lattice cell abstract we analyze the critical point and study the nonuniversal peculiarities occuring in the temperature dependence of the specific heat for a sequence of the exactly solvable highly decorated two dimensional triangular type ferromagnetic ising models with a complicated fractal type inherent structure of elementary cell the lattices are indexed by the decoration parameter n 0 1 2 3 the elementary cell for the n lattice is the n th order lattice approximation of the sierpinski gasket fractal at any finite n there is the second order phase transition indicated by the logarithmic singularity in the specific heat the critical temperature decreases as 4 ln 4 n 1 with increasing decoration the specific heat critical amplitude vanishes as 3 n n ln n 2 as n the basic nonuniversal features observable in the specific heat at strong decoration are the anomalous narrowing of the critical interval and the appearance of a rounded peak above the true t c point these effects are closely related and can both be explained by the above t c local ordering in the local lattice inhomogeneities the mechanism of the above t c local ordering is analyzed the models under discussion can as well be interpreted as the regular bond diluted 2d ising ferromagnets with a hierarchy of the nonmagnetic holes at different scales and varying dilution a comparative discussion on the properties of the regularly and randomly diluted 2d ising ferromagnets is given

sp and ss imaging for 3d elastic reverse time migration three dimensional elastic reverse time migration has been confronted with the problem of generating scalar images with vector s waves the underlying principle for solving this problem is to convert the vector s waves into scalars previous methods were mainly focused on ps imaging but usually cannot work properly on sp and ss cases the complexity of sp and ss imaging arises from the fact that the incident s wave has unpredictable relationship with the raypath plane we suggest that s wave should be treated separately as sv and sh waves which keep predictable relationships with the raypath plane firstly the elastic wavefield is separated into p and s waves using helmholtz decomposition then we evaluate the normal direction of raypath plane at each imaging grid next we separate the vector s wave obtained with curl operator into sh and sv waves both of which are scalars finally correlation imaging conditions are implemented to those scalar wave modes to produce scalar sv p sv sv and sh sh images

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

elastic least squares reverse time migration with hybrid l1 l2 misfit function abstract we have developed the theory and synthetic tests of elastic least squares reverse time migration elsrtm in this method a least squares reverse time migration algorithm is used to image multicomponent seismic data based on the first order elastic velocity stress wave equation in which the linearized elastic modeling equations are used for forward modeling and its adjoint equations are derived based on the adjoint state method for back propagating the data residuals also we have developed another elsrtm scheme based on the wavefield separation technique in which the p wave image is obtained using p wave forward and adjoint wavefields and the s wave image is obtained using p wave forward and s wave adjoint wavefields in this way the crosstalk artifacts can be minimized to a significant extent in general seismic data inevitably contain noise we apply the hybrid l1 l2 misfit function to the elsrtm algorithm to improve the robustness of our elsrtm to noise numerical tests on synthetic dat

elastic least squares reverse time migration abstractleast squares migration can produce images with improved resolution and reduced migration artifacts compared with conventional imaging we have developed a method for elastic least squares reverse time migration lsrtm based on a new perturbation imaging condition that yields scalar images of squared p and s velocity perturbations these perturbation images do not suffer from polarity reversals that are common for more conventional elastic imaging methods we use 2d synthetic and field data examples to demonstrate the proposed lsrtm algorithm using the perturbation imaging condition our results show that elastic lsrtm improves the energy focusing and illumination of the elastic images and it attenuates artifacts resulting for instance from sparseness in the wavefield sampling and crosstalk of the p and s modes compared with rtm images the lsrtm images provide more accurate relative amplitude information that is useful for reservoir characterization

time delay induced instabilities in reaction diffusion systems time delay in the kinetic terms of reaction diffusion systems has been investigated it has been shown that short delay beyond a critical threshold may induce spatiotemporal instabilities for unequal diffusivities and appropriate parameter space delay may induce turing instability resulting in stationary patterns and also interesting turing hopf transition with the formation of spirals the theoretical scheme has been numerically explored in two different prototypical reaction diffusion systems

multidimensional signature deconvolution and free surface multiple elimination of marine multicomponent ocean bottom seismic data this paper presents a wave equation method for multidimensional signature deconvolution designature and elimination of free surface related multiples demultiple in four component 4c ocean bottom seismic data the designature demultiple method has the following characteristics it preserves primary amplitudes while attenuating free surface related multiples it requires no knowledge of the sea floor parameters and the subsurface it requires information only of the local density and acoustic wave propagation velocity just above the sea floor it accommodates source arrays and no information except location of the physical source array its volume and its radiation characteristics wavelet is required designature is an implicit part of the demultiple process hence the method is capable of transforming recorded reflection data excited by any source array below the sea surface into free surface demultipled data that would be recorded from a point source with any desired signature in addition the incident wavefield is not subtracted from the data prior to free surface demultiple hence separation of incident and scattered fields is not an issue as it is for most other free surface demultiple schemes the designature demultiple algorithm can be divided into two major computational steps first a multidimensional deconvolution operator inversely proportional to the time derivative of the downgoing part of the normal component of the particle velocity just above the sea floor is computed second an integral equation is solved to find any component of the designatured free surface demultipled multicomponent field when the geology is horizontally layered the designature and free surface demultiple scheme greatly simplifies and lends itself toward implementation in the p domain or frequency wavenumber domain as deterministic deconvolution of common shot gathers or common receiver gathers when source array variations are negligible