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

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

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

optimized finite difference operator for broadband seismic wave modeling high resolution image and waveform inversion of smallscale targets requires the handling of high frequency seismic wavefields however conventional finite difference fd methods have strong numerical dispersions in the presence of high frequency components to reduce these numerical dispersions we optimized the constant coefficients of the fd operator by maximizing the wavenumber coverage within a given error limitation we set up three general criteria to enhance the convergence of the algorithm and reduce the optimization effort we selected the error limitation to be 0 0001 this being the smallest in the literature which led to perfect agreement between theoretical analyses and numerical experiments the accuracy of our optimized fd methods can even reach that of much higher order unoptimized fd methods which means great savings of computational efforts and memory demand these advantages become even more apparent with 3d modeling especially for saving memory demand

globally optimized finite difference extrapolator for strongly vti media implicit finite difference fd migration is unconditionally stable and is popular in handling strong velocity variations but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult traditional local optimizations generate the optimized coefficients for each pair of thomsen anisotropy parameters independently which can degrade results substantially for large anisotropy variations and lead to a huge table we developed an implicit fd method using the analytic taylor series expansion and used a global optimization scheme to improve its accuracy for wide phase angles we first extended the number of the constant coefficients then we relaxed the coefficient of the time delay extrapolation term by tuning a small factor such that error is less than 0 1 finally we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0 5 simultaneously the extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range compared with traditional local optimization our scheme does not need any table and table lookup for each order of the fd method only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy more importantly our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters no matter how strong the anisotropy is for the globally optimized second order fd method the accurate phase angle is up to 58 and the increase is about 18 22 for the globally optimized fourth order fd method the accurate phase angle is up to 77 and the increase is about 22 27

time domain explicit finite difference method based on the mixed domain function approximation for acoustic wave equation abstractexplicit finite difference fd methods with high accuracy and efficiency are preferred in full waveform inversion and reverse time migration the taylor series expansion te based fd methods can only obtain high accuracy on a small wavenumber zone we have developed a new explicit fd method with spatial arbitrary even order accuracy based on the mixed k wavenumber space domain function approximation for the acoustic wave equation and we derived the fd coefficients by minimizing the approximation error in a least squares ls sense the weighted pseudoinverse of mixed k space matrix is introduced into the ls optimization problem to improve the accuracy the new method has an exact temporal derivatives discretization in homogeneous media and also has higher temporal and spatial accuracy in heterogeneous media approximation errors and numerical dispersion analysis demonstrate that the new fd method has a higher numerical accuracy than conventional te based fd and te based time space domain disp

determining the optimal coefficients of the explicit finite difference scheme using the remez exchange algorithm explicit finite difference fd schemes are widely used in the seismic exploration field due to their simplicity in implementation and low computational cost however they suffer from strong artifacts caused by using coarse grids for high frequency applications the optimization of constant coefficients is popular in reducing spatial dispersions but current methods could not guarantee that the bandwidth of the tolerable dispersion error is the widest we have applied the remez exchange algorithm to optimize the constant coefficients of the explicit fd schemes for conventional and staggered grids the resulting dispersion errors are distributed alternately between the maxima and minima in the passband of the filter which is consistent with the most important equal ripple property of the error magnitude for the optimal solution according to the chebyshev criterion the remez exchange algorithm can determine the optimal coefficients of the fd method with only a few iterations and the resulting operator has a wider bandwidth compared with previous solutions it can handle arbitrary orders without the influence of local minima its computational cost for solving the objective function is comparable to that of the least squares method but its bandwidth is wider its accuracy is also higher than that of the maximum norm solved by the simulated annealing algorithm but its computational cost is much lower theoretically the equal ripple error can offer the widest bandwidth for suppressing numerical dispersions among all solutions obtained by the constant coefficient optimization in other words we can obtain a smaller error limitation than traditional methods under the same bandwidth this superiority over traditional methods is essential for reducing the total error accumulation which is helpful to avoid rapid error accumulations especially for large scale models and long term problems

a schur based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite toeplitz matrix abstract recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite toeplitz matrices several algorithms have been proposed in the literature they compute the smallest eigenvalue in an iterative fashion many of them relying on the levinson durbin solution of sequences of yule walker systems exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices respectively an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite toeplitz matrix is derived the algorithm relies on the computation of the r factor of the qr factorization of the toeplitz matrix and the inverse of r the simultaneous computation of r and r 1 is efficiently accomplished by the generalized schur algorithm

seismic modeling by optimizing regularized staggered grid finite difference operators using a time space domain dispersion relationship preserving method the staggered grid finite difference fd method is widely used in numerical simulation of the wave equation with stability conditions grid dispersion often exists because of the discretization of the time and the spatial derivatives in thewave equation therefore suppressing grid dispersion is a key problem for the staggered grid fd schemes to reduce the grid dispersion the traditional method uses high order staggered grid schemes in the space domain however the wave is propagated in the time and space domain simultaneously therefore some researchers proposed to derive staggered grid fd schemes based on the time space domain dispersion relationship however such methods were restricted to low frequencies and special angles of propagation we have developed a regularizing technique to tackle the ill conditioned property of the symmetric linear system and to stably provide approximate solutions of the fd coefficients for acoustic wave equations dispersion analysis and seismic numerical simulations determined that the proposed method satisfies the dispersion relationship over a much wider range of frequencies and angles of propagation and can ensure fd coefficients being solved via a well posed linear system and hence improve the forward modeling precision

on a spatial smoothing technique for multiple source location this correspondence presents an alternate proof for the spatial smoothing technique recently proposed for multiple source location specifically it is proved that the number of subarrays required to ensure a nonsingular covariance matrix is m 1 where m is the nullity of the source covariance matrix also it is shown that this number can be reduced if a specific basis can be found for the null eigenspace of the covariance matrix satisfying certain properties

higher order triangular spectral element method with optimized cubature points for seismic wavefield modeling the mass lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points known as cubature points to date such points are only known analytically in tensor domains such as quadrilateral or hexahedral elements thus the diagonal mass matrix spectral element method sem in non tensor domains always relies on numerically computed interpolation points or quadrature points however only the cubature points for degrees 1 to 6 are known which is the reason that we have developed a p norm based optimization algorithm to obtain higher order cubature points in this way we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9 the dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration simultaneously the lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties as a result such points provide both good interpolation properties and integration accuracy the courantfriedrichslewy cfl numbers are tabulated for the conventional fekete based triangular spectral element tsem the tsem with exact integration and the optimized cubature based tsem otsem a complementary study demonstrates the spectral convergence of the otsem a numerical example conducted on a half space model demonstrates that the otsem improves the accuracy by approximately one order of magnitude compared to the conventional fekete based tsem in particular the accuracy of the 7th order otsem is even higher than that of the 14th order fekete based tsem furthermore the otsem produces a result that can compete in accuracy with the quadrilateral sem qsem the high accuracy of the otsem is also tested with a non flat topography model in terms of computational efficiency the otsem is more efficient than the fekete based tsem although it is slightly costlier than the qsem when a comparable numerical accuracy is required higher order cubature points for degrees 7 to 9 are developed the effects of quadrature rule on the mass and stiffness matrices has been conducted the cubature points have always positive integration weights freeing from the inversion of a wide bandwidth mass matrix the accuracy of the tsem has been improved in about one order of magnitude

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

time space domain implicit finite difference methods for modeling acoustic wave equations it was proved that the implicit spatial finite difference fd method can obtain a higher accuracy than the explicit fd by using even a smaller operator length however when only second order fd in time is used the combined fd is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used the time space domain fd can suppress the temporal dispersion however because the spatial derivatives are solved explicitly the method suffers from spatial dispersion and a large spatial operator length has to be adopted we propose two effective time space domain implicit fd methods for modeling 2d and 3d acoustic wave equations the high order fd is first incorporated into the discretization for the second order temporal derivative and is combined with the implicit spatial fd the plane wave analysis method is used to derive the time space domain dispersion relations and two novel methods are proposed to determine the spatial and temporal fd coefficients in the joint time

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

antenna array design for covariance based direction finding methods angle of arrival estimates based on the analysis of a spatial covariance matrix are particularly vulnerable to error in the presence of coherent signal interference when this condition exists the wave interference field is statistically nonstationary and estimation techniques that are based upon an assumption of stationarity tend to produce poor results a method is proposed for the design of antenna arrays that exploits the effects of time averaging and spatial smoothing to approximate the condition of statistical stationarity and thereby enhance the performance of bearing estimation methods

dispersion relation preserving stereo modeling method beyond nyquist frequency for acoustic wave equation abstractaccording to nyquist shannon sampling theory two sampling points of the displacements per minimal wavelength are required to reconstruct a wavefield however high frequency components above the nyquist frequency can be theoretically recovered by introducing spatial derivatives of the displacements we have developed a new finite difference scheme for solving the acoustic equation named the dispersion relation preserving stereo modeling method which is practical for seismic forward modeling beyond the nyquist frequency the key idea in this model is to approximate the high order spatial derivatives with wavenumber domain optimization in wavefields represented by the wave displacements and their gradients we investigated the theoretical properties of this method including the absolute error of spatial operators numerical error stability criterion and computational efficiency our results indicated that the method can effectively suppress the numerical dispersion on extremely coarse grids wi

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

a staggered grid finite difference scheme optimized in the time space domain for modeling scalar wave propagation in geophysical problems abstract for modeling scalar wave propagation in geophysical problems using finite difference schemes optimizing the coefficients of the finite difference operators can reduce numerical dispersion most optimized finite difference schemes for modeling seismic wave propagation suppress only spatial but not temporal dispersion errors we develop a novel optimized finite difference scheme for numerical scalar wave modeling to control dispersion errors not only in space but also in time our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil we design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers dispersion analysis and numerical examples demonstrate that our optimized finite difference scheme is computationally up to 2 5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2d or 3d problem compared with the high order finite difference scheme using the same new stencil our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy this new optimized finite difference scheme is particularly useful for large scale 3d scalar wave modeling and inversion

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

global optimization of generalized nonhyperbolic moveout approximation for long offset normal moveout the approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media the generalized nonhyperbolic moveout approximation gma brings a considerable improvement in accuracy compared to known analytical approximations however this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters which would degrade the inversion accuracy due to error accumulation in velocity analysis we optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset to depth ratio within practical ranges theoretical analyses and numerical experiments show that the traditional optimization scheme using 2 norm solved by the least squares method could not provide an error constrained result in addition a direct optimization without constant coefficient extension would not lead to a satisfactory accuracy improvement we construct the objective function using the maximum norm and so

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin

determination of finite difference weights using scaled binomial windows the finite difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes conventional finite difference weights can be calculated either from taylor series expansions or by lagrange interpolation polynomials the finite difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain for this reason we also can derive finite difference operators by truncating the convolution series of the pseudospectral method various truncation windows can be employed for this purpose and they result in finite difference operators with different dispersion properties we found that there exists two families of scaled binomial windows that can be used to derive conventional finite difference operators analytically with a minor change these scaled binomial windows can also be used to derive optimized finite difference operators with enhanced dispersion properties

multiple source location a matrix decomposition approach in this paper we present a new method without using the eigenstructure of the sensor covariance matrix for estimating the number of sources d and the direction of arrival of sources based on the matrix decomposition this method uniformly handles coherent and noncoherent sources alike in our procedure a number p is calculated and used for determination of the maximum number of distinguishable sources by a given linear array system