surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

stable explicit depth extrapolation of seismic wavefields stable explicit filters for depth extrapolation of seismic wavefields may be derived through a modification of the conventional taylor series method the modified taylor series method described here yields extrapolators with maximally flat amplitude spectra in their passband while ensuring that no spectral components in the wavefield are amplified by any factor greater than one the price for stability is increased phase error for low normalized frequencies implicit extrapolators are more accurate than the 39 coefficient stable explicit extrapolator described here however the small spatial sampling intervals required to obtain high phase accuracy in implicit extrapolation imply that this accuracy is rarely achieved in practice over the wide range of normalized frequencies likely to be encountered in practice stable explicit extrapolators outperform implicit ones the method presented here for deriving stable explicit extrapolators is in no formal sense optimal it is only guaranteed to yield stable extrapolators in my limited experience with alternative methods for designing stable extrapolators the method presented here produced the least phase error while ensuring stability nevertheless a simple method for designing optimal in some sense stable explicit extrapolators would be preferred over the method presented here 7 refs 15 figs

constant mean curvature surfaces via an integrable dynamical system it is shown that the equation which describes constant mean curvature surfaces via the generalized weierstrass enneper induction has hamiltonian form its simplest finite dimensional reduction is the integrable hamiltonian system with two degrees of freedom this finite dimensional system admits action and classes of equivalence of its trajectories are in one to one correspondence with different helicoidal constant mean curvature surfaces thus the interpretation of well known delaunay and do carmo dajczer surfaces via an integrable finite dimensional hamiltonian system is established

on certain geometric aspects of cpn harmonic maps a weierstrass type system of equations corresponding to cpn harmonic maps is presented it constitutes a generalization of the previously constructed systems for cp1 and cp2 fields from the linear spectral problem for the cpn model a set of conserved quantities is derived and used for a construction of a generalized weierstrass representation for conformally parametrized surfaces immersed in multidimensional euclidean spaces based on this representation a possible geometrical interpretation of cpn harmonic maps is discussed

3 d migration via mcclellan transformations on hexagonal grids two dimensional 2 d spatially varying convolutional filters for three dimensional 3 d depth extrapolation are efficiently designed and implemented by using mcclellan transformations although efficient mcclellan transformations only approximate circularly symmetric depth extrapolation filters the accuracy of the extrapolation filters can be increased only at an increase in the computational cost of implementing those filters by applying mcclellan transformations on a hexagonal sampling grid the accuracy of the extrapolation filters can be increased at a reduction in cost

designing stable extrapolators for explicit depth extrapolation of 2d and 3d wavefields using projections onto convex sets we have developed a robust algorithm for designing explicit depth extrapolation operators using the projections onto convex sets pocs method the operators are optimal in the sense that they satisfy all required extrapolation design characteristics in addition we propose a simple modification of the pocs algorithm modified pocs or mpocs that further enhances the stability of extrapolated wavefields and reduces the number of iterations required to design such operators to approximately 2 of that required for the basic pocs design algorithm various synthetic tests show that 25 coefficient 1d extrapolation operators which have 13 unique coefficients can accommodate dip angles up to 70 we migrated the seg eage salt model data with the operators and compare our results with images obtained via extrapolators based on modified taylor series and with wavefield extrapolation techniques such as phase shift plus interpolation pspi and split step fourier the mpocs algorithm provides practically stable

riemannian wavefield extrapolation riemannian spaces are described by nonorthogonal curvilinear coordinates we generalize one way wavefield extrapolation to semiorthogonal riemannian coordinate systems that include but are not limited to ray coordinate systems we obtain a one way wavefield extrapolation method that can be used for waves propagating in arbitrary directions in contrast to downward continuation which is used for waves propagating mainly in the vertical direction ray coordinate systems can be initiated in many different ways for example from point sources or from plane waves incident at various angles since wavefield propagation happens mostly along the extrapolation direction we can use inexpensive finite difference or mixed domain extrapolators to achieve high angle accuracy the main applications of our method include imaging of steeply dipping or overturning reflections

complete solution of the schrodinger equation of the complex manifold cp2 passing from the cp2 lagrangian in kahler form to the hamiltonian in terms of polar coordinates this paper gives the complete set of solutions of the corresponding schrodinger equation in a manner that makes fully explicit the su 3 description of the energy eigenspaces the solutions of the self adjoint equation for the radial coordinates are derived most easily directly but also related to jacobi polynomials

implementation of 2d explicit depth extrapolation fir digital filters for 3d seismic volumes using singular value decomposition we propose a new scheme for implementing predesigned 2d complex valued wavefield extrapolation finite impulse response fir digital filters which are used for extrapolating 3d seismic wavefields the implementation is based on singular value decomposition svd of quadrantally symmetric 2d fir filters extrapolators to simplify the svd computations for such afilter impulse response structure we apply a special matrix transformation on the extrapolation fir filter impulse responses where we guarantee the retentionoftheirwavenumberphaseresponse unliketheexisting 2d fir filter implementation methods that are used for this geophysical application such as the mcclellan transformationoritsimprovedversion thisimplementationviasvdresults in perfect circularly symmetrical magnitude and phase wavenumber responses in this paper we also demonstrate that the svd method can save depending on the filter size more than 23 of the number of multiplications per output sample and approximately 62 of the number of additions per output sample when compared to direct implementation with quadrantal symmetry via true 2d convolution finally anapplicationtoextrapolationofaseismicimpulseisshown toproveourtheoreticalconclusions

explicit 3 d migration using equiripple polynomial expansion and laplacian synthesis in f x explicit finite difference depth migration schemes wavefield downward extrapolation is carried out through spatial convolution using finite length filters existing methods for computing these filters are based on nonlinear least squares with a high computational cost or on taylor series expansion which is suboptimal in the 3 d case the physics of wavefield extrapolation requires 2 d extrapolation filters with circular symmetry recently mcclellan transformation has been used to design circularly symmetric extrapolation operators but this approach exhibits artifacts when the data are not spatially oversampled we describe an alternative method to take advantage of the circular symmetry the radial response of the filter is expanded as a polynomial in the laplacian which is synthesized as the sum of two 1 d second derivative filters using the laplacian rather than the mcclellan transform yields an artifact free impulse response for wavenumbers much closer to the nyquist wavenumber at the s

one step extrapolation method for reverse time migration we have proposed a new method a one step extrapolation algorithm to solve the acoustic wave equation by introducing a square root operator the two way wave equation can be formulated as a first order partial differential equation in time which is similar to the one way wave equation to solve the new wave equation we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains unlike the conventional explicit finite difference schemes the new method does not suffer from numerical instability or numerical dispersion problems it can be used to design cost effective and high quality reverse time migration or modeling code

wavefield extrapolation by nonstationary phase shift the phase shift method of wavefield extrapolation applies a phase shift in the fourier domain to deduce a scalar wavefield at one depth level given its value at another the phase shift operator varies with frequency and wavenumber and assumes constant velocity across the extrapolation step we use nonstationary filter theory to generalize this method to nonstationary phase shift nsps which allows the phase shift to vary laterally depending upon the local propagation velocity for comparison we derive an analytic form for the popular phase shift plus interpolation pspi method in the limit of an exhaustive set of reference velocities nsps and this limiting form of pspi can be written as generalized fourier integrals which reduce to ordinary phase shift in the constant velocity limit in the x domain these processes are the transpose of each other however only nsps has the physical interpretation of forming the scaled linear superposition of laterally variable impulse responses i e huygen

seismic absorption compensation a least squares inverse scheme the problem of instability plagues conventional inverse q filtering we formulate the deabsorption problem as an inverse problem in terms of least squares and impose regularization by means of bayes theorem the solution is iterative and nonparametric and returns a reflectivity that has been constrained to be sparse the inverse scheme is tested on both synthetic and real data and the results obtained demonstrate the viability of the approach

linear complementarity problem a novel approach to design finite impulse response wavefield extrapolation filters abstractin this paper the problem of complex valued finite impulse response fir wavefield extrapolation filter design was considered as a linear complementarity problem lcp lcp is not an optimization technique because there is no objective function to optimize however quadratic programming one of the applications of lcp can be used to find an optimal solution for the 1d fir wavefield extrapolation filter quadratic programs are an extremely important source of applications of lcp in fact several algorithms for quadratic programs are based on lcp we found that fir wavefield extrapolation filter design problem can be written as a quadratic program and then finally to an equivalent lcp there are two families of algorithms available to solve for lcp 1 direct pivoting based algorithms and 2 indirect iterative algorithms in this study the lcp has been solved using direct and indirect algorithms to show the effectiveness of the proposed method the seg eage salt velocity model data ha

3 d depth migration via mcclellan transformations three dimensional seismic wavefields may be extrapolated in depth one frequency at a time by two dimensional convolution with a circularly symmetric frequency and velocity dependent filter this depth extrapolation performed for each frequency independently lies at the heart of 3 d finite difference depth migration the computational efficiency of 3 d depth migration depends directly on the efficiency of this depth extrapolation mcclellan transformations provide an efficient method for both designing and implementing two dimensional digital filters that have a particular form of symmetry such as the circularly symmetric depth extrapolation filters used in 3 d depth migration given the coefficients of one dimensional frequency and velocity dependent filters used to accomplish 2 d depth migration mcclellan transformations lead to a simple and efficient algorithm for 3 d depth migration 3 d depth migration via mcclellan transformations is simple because the coefficients of two dimensional depth

on a discrete version of the script cp 1 sigma model and surfaces immersed in bbb r3 we present a discretization of the p1 sigma model we show that the discrete p1 sigma model is described by a nonlinear partial second order difference equation with rational nonlinearity to derive discrete surfaces immersed in three dimensional euclidean space a complex lattice is introduced the so obtained surfaces are characterized in terms of the quadrilateral cross ratio of four surface points in this way we prove that all surfaces associated with the discrete p1 sigma model are of constant mean curvature an explicit example of such discrete surfaces is constructed

symmetry properties and explicit solutions of the generalized weierstrass system the method of symmetry reduction is systematically applied to derive several classes of invariant solutions for the generalized weierstrass system inducing constant mean curvature surfaces and to the associated two dimensional nonlinear sigma model a classification of subgroups with generic orbits of codimension one of the lie point symmetry group for these systems provides a tool for introducing symmetry variables and reduces the initial systems to different nonequivalent systems of ordinary differential equations we perform a singularity analysis for them in order to establish whether these ordinary differential equations have the painleve property these ordinary differential equations can then be transformed to standard forms and next solved in terms of elementary and jacobi elliptic functions this results in a large number of new solutions and in some cases new interesting constant mean curvature surfaces are found furthermore this symmetry analysis is extended to include conditional symmetries by subjecting the original system to certain differential constraints in this case several new types of nonsplitting algebraic trigonometric and hyperbolic multisoliton solutions have been obtained in explicit form some physical interpretation of these results in the areas of fluid membranes string theory two dimensionl gravity and cosmology are given

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

the design of stable sparse wavefield extrapolators using projections onto convex sets abstractwe present the results of poststack explicit depth migration of the well known 2d seg eage salt model zero offset seismic data using sparse wavefield extrapolators the extrapolators are designed to be sparse by forcing some of the very small complex valued coefficients magnitude values to be zero the proposed extrapolators design method combines the previously reported modified projections onto convex sets mpocs for designing explicit depth frequency space f x wavefield extrapolation operators with hard thresholding of the small extrapolators coefficients magnitude the real and imaginary parts of the mpocs operators with small magnitudes are replaced by zeros during the mpocs algorithm iterations the migrated result of the seg eage salt model data using such sparse designed operators shows comparable migrated results using the nonsparse version of the mpocs extrapolation operators as well as the image obtained using the well known phase shift plus interpolation pspi migration techn

design of one way wavefield extrapolation operators using smooth functions in wlsq optimization many depth migration methods use one way frequency space depth extrapolation methods these methods are generally considered to be expensive so it is important to find the most efficient way of implementing them this usually means making spatial convolution operators that are as short as possible applying the extrapolation operators in a recursive way using small depth steps also demands that the operators do not amplify the wavefield at every depth step weighted least squares is an appropriate method to use for designing extrapolation operators that are accurate and efficient and that remain stable in a recursive algorithm the extrapolated wavefields calculated with these operators are comparable with the extrapolation results obtained with other known operator design techniques as the remez exchange method and nonlinear optimization in this paper the weighted least squares technique is refined by using different model functions by smoothing the phase and amplitude transition at the evanescent c

the weierstrass representation for surfaces immersed into r8 and cp2 maps a generalization of the weierstrass system of equations corresponding to cp2 harmonic maps is presented this generalization allows us to study two dimensional surfaces immersed in a flat space r8 with euclidean metric we use this system to suggest a possible geometrical interpretation of cp2 harmonic maps