gravimagnetic anomaly formulas for polyhedra of spatially linear media formulas for the gravity potential field and field gradient tensor are derived for a polyhedral target body of a spatially linear density medium the formulas also define the magnetic potential and field in the case of a medium of spatially linear magnetization this work generalizes existing solutions for the gravity field of a polyhedral target of linearly varying density the formulas are analyzed for singularities and for numerical error growth error growth with increasing target distance is found to be higher than in the corresponding uniform polyhedral case examination of the error sources reveals that some error reduction is possible on this basis new algorithms of improved error control over existing algorithms are proposed for the linear medium computational results confirm the expected improvement

the algebraic theory of quasicrystals with five fold symmetries an algebraic binary operation is introduced into quasicrystals admitting five fold symmetry in terms of this many quasicrystals displaying full pentagonal or icosahedral symmetry are seen to be finitely generated examples are given in dimensions 1 2 3 and 4 the operation of left quasicrystal addition is affine linear the monoid generated by these operators is discussed and a presentation for it given in the generic case

a two dimensional eshelby problem for two bonded piezoelectric half planes the present paper concerns a two dimensional eshelby problem for an inclusion of arbitrary shape embedded within one of two bonded dissimilar piezoelectric halfplanes the elastic and piezoelectric constants of the inclusion and its surrounding half plane are assumed to be the same a simple explicit solution is derived in terms of some auxiliary functions which can be determined using several related conformal mappings of the inclusion shape the obtained solution is exact provided that the expansions of all conformal mappings include only a finite number of terms on the other hand if an exact conformal mapping includes infinite terms a truncated polynomial mapping function should be used and then the method gives an approximate solution the existing solutions obtained in an earlier work for a homogeneous piezoelectric plane or half plane can be derived from the present solution as special cases in particular the closed form solutions are given for the eshelby problem of an arbitrarily shaped inclusion in a piezoelectric half plane with various mixed surface conditions such as rigid insulating surface or traction free conducting surface these solutions are used to study the effects of various surface electrical conditions on an electro elastic field in a piezoelectric half plane

new scheme for computing the magnetic field resulting from a uniformly magnetized arbitrary polyhedron the magnetic field at any point outside a uniformly magnetized polyhedron of arbitrary shape is obtained by adding the fields resulting from the effective free magnetic poles on each of the polygonal surfaces of the polyhedron for each polygonal surface the components of the field at the point of observation are expressed in terms of new line integrals around the edges of the polygon and the solid angle subtended by the polygon at the point of observation the line integrals are standard elementary forms this new approach makes the numerical evaluation of the magnetic fields for such models much simpler and faster than previously published methods

new parallel sor method by domain partitioning in this paper we propose and analyze a new parallel sor method the psor method formulated by using domain partitioning and interprocessor data communication techniques we prove that the psor method has the same asymptotic rate of convergence as the red black r b sor method for the five point stencil on both strip and block partitions and as the four color r b g o sor method for the nine point stencil on strip partitions we also demonstrate the parallel performance of the psor method on four different mimd multiprocessors a ksr1 an intel delta a paragon and an ibm sp2 finally we compare the parallel performance of psor r b sor and r b g o sor numerical results on the paragon indicate that psor is more efficient than r b sor and r b g o sor in both computation and interprocessor data communication

lie algebras associated with one dimensional aperiodic point sets the set of points of a one dimensional cut and project quasicrystal or model set while not additive is shown to be multiplicative for appropriate choices of acceptance windows this leads to the definition of an associative additive graded composition law and permits the introduction of lie algebras over such aperiodic point sets these infinite dimensional lie algebras are shown to be representatives of a new type of semi direct product induced lie algebras

on the convergence of the symmetric successive overrelaxation method abstract the symmetric successive overrelaxation ssor iterative method is applied to the solution of the system of linear equations a x b where a is an n n nonsingular matrix we give bounds for the spectral radius of the ssor iterative matrix when a is an hermitian positive definite matrix and when a is a nonsingular m matrix then we discuss the convergence of the ssor iterative method associated with a new splitting of the matrix a which extends the results of varga and buoni 1

aperiodic virasoro algebra an aperiodic analog to the virasoro algebra is introduced and its representation theory is investigated in particular highest and lowest weight representations are constructed an analog to the kac determinant formula is derived and the implications for unitarity are discussed

on the singularities of the gravity field of a homogeneous polyhedral body the study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body

magnetic gradients produced by a 2d homogeneous polygonal source we use a set of analytic equations for calculating horizontal and vertical gradients of the magnetic anomaly produced by a 2d homogeneous magnetic source with a polygonal cross section the magnetization vector of the source is assumed to be constant and known and the anomalous field is assumed to be much smaller than the ambient geomagnetic field in magnitude the derivation begins from an expression of the magnetic potential then we evaluate clockwise integrations to establish the equations a comparison between this analytic approach and a numerical approach using the finite difference method to compute gradients indicates the reliability of the equations the proposed method is numerically efficient and does not present singularity on the edges of the polygon prism except at the vertices where the gradients approach to infinity

analytic solution of the gravity anomaly of irregular 2d masses with density contrast varying as a 2d polynomial function the analytic solution of the gravity anomaly caused by a 2d irregular mass body with the density contrast varying as a polynomial function in the horizontal and vertical directions is extrapolated from a historical version in which the analytic solution for the gravity anomaly was given only at the origin of the coordinate system to any point for the density function in terms of variables relative to that origin to calculate the gravity anomaly at stations that are not at origins a coordinate transformation is performed in which case the polynomial density contrast function must also be expressed in the transformed coordinates or a transformed solution must be obtained these analytic solutions can be obtained at any station using 1 a solution transformation method in which the density function and boundary of a mass body are kept intact or 2 a coordinate transformation method in which polynomial coefficient and boundary of a mass body are transformed accordingly the issue of singularity and instability of the analytic methods has been related to case studies caution should be exercised in modeling or interpreting the gravity survey data using the analytic methods for large target distance to target size ratios outside the range of numerical stability compared with other published methods the analytic solution results agree very well with other numerical or seminumerical methods indicating the solution is correct and can be applied for any gravity anomaly calculation caused by an irregular 2d mass body with the density contrast approximated as a polynomial function of horizontal position and or vertical position when the observation is within the range of numerical stability

constrained inversion of gravity fields for complex 3 d structures as part of a research program to develop gravity interpretation tools that can be merged with seismic techniques a full 3 d complex structural inversion scheme for possibly multibody polyhedral models has been developed the forward modeling algorithm was adopted from previous work because the inverse problem is generally very ill posed several methods of regularizing the inversion were investigated and a combination of the most useful was adopted the combination includes i a structured matrix formulation for the system equations ii an analytical expression for the jacobian calculation iii first derivative damping iv a choice of damping parameter based on a variation of the trust region method v a weighted scheme for parameter correction and vi complete freezing of degrees of freedom found not to influence the gravity field significantly this combination yields a robust inversion which was successfully demonstrated on data over the galveston island salt dome offshore texas variat

tiles in quasicrystals with cubic irrationality we determine completely all tiles of one dimensional cut and project quasicrystals which have a cubic irrationality as self similarity factor and are given in terms of connected rectangular acceptance windows it is shown that there are at least five and at most nine tiles and they are indicated explicitly depending on the lengths of the two acceptance windows in terms of which the quasicrystals are defined

gravity anomalies of arbitrary 3d polyhedral bodies with horizontal and vertical mass contrasts up to cubic order abstracta new singularity free analytical formula has been developed for the gravity field of arbitrary 3d polyhedral mass bodies with horizontally and vertically varying density contrast using third order polynomial functions first the observation sites are moved to the origin of the coordinate system then the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies furthermore singularity free closed form solutions are derived for these line integrals over the edges thus the observation sites can be located inside on or outside the 3d distributions a synthetic prismatic mass body is adopted to verify the accuracy and singularity free property of our newly developed analytical expressions excellent agreements are obtained between our solutions and other published closed form solutions with relative errors in the order of 10 12 to 10 13 in a

efficient gravity inversion of basement relief using a versatile modeling algorithm abstractwe have developed a novel approach to compute in an efficient and versatile way the gravity anomaly produced by an arbitrary discrete 3d distribution of density contrast the method allows adjustable precision and is particularly suited for the interpretation of sedimentary basins because the gravity field decays with the square of the distance we use a discrete green s operator that may be made much smaller than the whole study area for irregularly positioned observations this discrete green s operator must be computed just at the first iteration and because each of its horizontal layers presents a center of symmetry only one eighth of its total elements need to be calculated furthermore for gridded data on a plane this operator develops translation symmetry for the whole region of interest and has to be computed just once for a single arbitrary observation position the gravity anomaly is obtained as the product of this small operator by any arbitrary density contrast distribution in

magnetic anomaly fourier spectrum of a 3d gaussian source i obtain a closed form expression for the total field magnetic anomaly generated by a 3d gaussian magnetization distribution the solution is evaluated by a novel contour integration in the complex wavenumber domain the spatial domain solution is then computed by a numerical inverse fourier transform this calculated field can be used to interpret magnetic data because a 3d gaussian function can represent a compact and smooth source the method moreover allows for calculation of the fourier components as a function of the distance between the depth of the source and the height of the observation level the solution is thus useful for the inverse problem spectral analysis or data continuation

fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast abstractprismatic bodies with variable density contrasts are very useful in modeling and inversion of gravity anomalies for sedimentary basins in which some simple mathematical functions offer better approximations for the density depth relationship of the basin fill than a constant density model we have explored fourier domain modeling of vector and tensor gravity anomalies due to 2d and 3d prismatic bodies following a wide class of variable density functions we placed the emphasis on general polynomial models because they provided a flexible way to approximate arbitrary density functions the recently proposed gauss fast fourier transform method was applied to improve the accuracy of inverse fourier transform the numerical performance of the presented method was examined by comparing with space domain analytical seminumerical or completely numerical solutions singularity and numerical stability of the algorithm were also analyzed with ranges of numerical stability for polynomial density models of

gravitational attraction of a rectangular prism with depth dependent density the gravity effect produced by two and three dimensional bodies with nonuniform density contrast has been treated by several authors one of the first attempts in this direction made by cordell 1973 who developed a method to compute the gravity effect due to a two dimensional prism whose density decreases exponentially with depth a different approach was proposed by murthy and rao 1979 they extended the line integral method to obtain the gravity effect for bodies of arbitrary cross sections with density contrast varying linearly with depth chai and hinze 1988 have derived a wavenumber domain approach to compute the gravity effect due to a vertical prism whose density contrast varies exponentially with depth recently rao 1990 has developed a closed expression of the gravity field produced by an asymmetrical trapezoidal body whose density varies with depth following a quadratic polynomial

analysis of the sor iteration for the 9 point laplacian the sor iteration for solving linear systems of equations depends upon an overrelaxation factor omega a theory for determining omega was given by young iterative methods for solving partial differential equations of elliptic types trans amer math soc 76 1954 pp 92 111 for consistently ordered matrices here we determine the optimal omega for the 9 point stencil for the model problem of laplace s equation on a square we consider several orderings of the equations including the natural rowwise and multicolor orderings all of which lead to nonconsistently ordered matrices and find two equivalence classes of orderings with different convergence behavior and optimal omega s we compare our results for the natural rowwise ordering to those of garabedian estimation of the relaxation factor for small mesh size math comp 10 1956 pp 183 185 and explain why both results are in a sense correct even though they differ we also analyze a pseudo sor method for the model p

extensions of ssor abstract kahan in his 1957 thesis was probably the first to use quadratic form arguments to analyze the sor method in 1973 richard varga used quadratic form arguments to extend the sor theory to a more general splitting and to obtain upper bound estimates for the spectral radius of an sor iteration matrix these estimates then led to a suggested value for the overrelaxation factor independent of the theory for consistently ordered matrices the purpose of this paper is to obtain similar results for ssor it presents for ssor a generalization of the ostrowski reich theorem bounds for the spectral radius of the ssor iteration matrix suggestions for the selection of an overrelaxation factor and a numerical computation for the nine point stencil for the model problem of laplace s equation on a square

2 d gravity modeling with analytically defined geometry and quadratic polynomial density functions new methods for 2 d modeling of gravity anomaly data are developed following an approach that uses both analytic and numerical methods of integration the forward model solution developed here is suitable to calculate the gravity effect caused by a 2 d source body bounded either laterally or vertically by continuous functions in our models the density contrast is defined by a second order polynomial function of depth and distance along the profile we present several examples to show that our models are capable of accommodating a broad variety of geologic structures

gravity inversion using open reject and shape of anomaly fill criteria a gravity inversion method is developed by iteratively applying open reject and fill o r f criteria within a model space comprising a great many rectangular prisms each prism is assigned a density contrast the modeling procedure consists of filling some prisms while leaving others empty only one element is filled for each pass generally elements are added only to the periphery of the growing model models can be allowed to grow in any combination of directions or in all directions by application of a shape of anomaly fill criterion the model rapidly attains a form which yields gravity approximating the given gravity scaled down by some constant factor as the model continues to grow this scale factor approaches unity the method readily yields inverse models comprising several thousand individual prisms examples presented here give applications to 2 d problems the method is readily applicable to 212 d and 3 d problems as well overhanging elements are obtained by appropriate use of model

closed form formula of magnetic gradient tensor for a homogeneous polyhedral magnetic target a tetrahedral grid example abstracta closed form formula is developed for the full magnetic gradient tensor of a polyhedral body with a homogeneous magnetization vector it is based on the direct derivative technique on the closed form of the magnetic field these analytical expressions are implemented into an easy to use c package which simultaneously calculates the magnetic potential the magnetic field and the full magnetic gradient tensor for magnetic targets modern unstructured tetrahedral grids are adopted to represent the polyhedral body so that our code can deal with arbitrarily complicated magnetic targets a prismatic body is tested to verify the accuracies of our closed form formula excellent agreements are obtained between our closed form solutions and solutions of a prismatic magnetic body with differences up to machine precision a pipeline model is used to demonstrate its capability to deal with complicated magnetic targets this c code is freely available to the magnetic exploration community

singularity removal a refinement of resistivity modeling techniques modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available a singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth the inability to adequately represent in discrete space a discontinuous function in this case the delta function describing the introduction of current at a point commonly results in numerical error near the source of a modeled singularity inspection of an integrated finite difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin a procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step thus preventing it from contributing to the numerical error using this procedure the average error in apparent resistivity values for a model of a polar dipole traverse over a nonconducting sphere is reduced by 40 percent for a dipole dipole traverse of a two layer model the error decreases by 75 percent and in the case of a wenner profile of a model of a vertically faulted earth the average error is diminished by 90 percent

high precision fourier forward modeling of potential fields we analyzed the numerical forward methods in the fourier domain for potential fields existing fourier domain forward methods applied the standard fast fourier transform fft algorithm to inverse transform a conjugate symmetrical spectrum into a real field it had significant speed advantages over space domain forward methods but suffered from problems including aliasing imposed periodicity and edge effect usually grid expansion was needed to reduce these errors which was equivalent to the numerical evaluation of the oscillatory fourier integral using the trapezoidal rule with smaller steps we tested a high precision fourier domain forward method based on a combined use of shift sampling technique and gaussian quadrature theory the trapezoidal rule applied by the standard fft algorithm to evaluate the continuous fourier transform was modified by introducing a shift parameter by choosing optimum values of as gaussian quadrature nodes we developed a gauss fft method for fourier forward modeling of potential fields no grid expansion was needed the sources can be set near the boundary of the fields or even go beyond the boundary the gauss fft method converged to the space domain solution much faster than the standard fft method with grid expansion forward modeling results almost identical to space domain ones can be obtained in less time numerical examples of both simple and complex 2d and 3d source forward modeling revealed the reliability and adaptability of the method

analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals the analytical computation of the full gravity tensor from a polyhedral source of homogeneous density is presented with emphasis on its algorithmic implementation the theoretical development is based on the subsequent transition of the general expressions from volume to surface and from surface to line integrals defined along the closed polygons building each polyhedral face however the accurate numerical computation of the obtained transcendental expressions is linked with the relative position of the computation point and its corresponding projections on the plane of each face and on the line of each segment with respect to the polygons defining each face depending on this geometric setup the application of the divergence theorem of gauss leads to the appearance of additional correction terms valid only for these boundary conditions and crucial for the correct numerical evaluation of the polyhedral related gravity quantities at those locations of the computation point a program in fortran is su

three dimensional fourier gravity inversion with arbitrary density contrast the use of variable density contrasts in gravity inversion has gained increasing importance in recent years due to the necessity of constructing more realistic models of geophysical structures such as sedimentary basins linear quadratic and exponential variations either in the space or in the frequency domain are the basis of several methods see among others the papers by granser 1987 chai and hinze 1988 reamer and ferguson 1989 and rao et al 1990 guspi 1990 used polynomial density depth functions for inverting gravity anomalies into 2 d polygons in the space domain

forward modeling of gravity data using finite volume and finite element methods on unstructured grids abstractminimum structure inversion is one of the most effective tools for the inversion of gravity data however the standard gauss newton algorithms that are commonly used for the minimization procedure and that employ forward solvers based on analytic formulas require large memory storage for the formation and inversion of the involved matrices an alternative to the analytical solvers are numerical ones that result in sparse matrices this sparsity suits gradient based minimization methods that avoid the explicit formation of the inversion matrices and that solve the system of equations using memory efficient iterative techniques we have developed several numerical schemes for the forward modeling of gravity data using the finite element and finite volume methods on unstructured grids in the finite volume method a delaunay tetrahedral grid and its dual voronoi grid are used to find the primary solution i e gravitational potential at the centers and vertices of the tetrahedra respectively cel

gravity inversion of a discontinuous relief stabilized by weighted smoothness constraints on depth we present a new stable gravity inversion method applied to the mapping of an interface separating two homogeneous media in contrast with previous similar methods it does not impose an overall smoothness on the estimated interface to stabilize the solution the density contrast between the media is assumed to be known the interpretation model for the upper medium consists of rectangular juxtaposed prisms whose thicknesses represent the depths to the interface and are the parameters to be estimated the true interface is assumed to be flat everywhere except at faults to incorporate this attribute into the estimated relief we developed an iterative process in which three kinds of constraints are imposed on parameters 1 proximity between values of adjacent parameters 2 lower and upper bounds to parameters and 3 proximity between the values of parameters and fixed numerical values starting with an initial solution which presents an overall smooth relief the method enhances initially estimated

a two level four color sor method a two level four color sor method is proposed for the nine point discretization of the poisson equation on a square instead of examining the jacobi iteration matrix in the space domain we consider an equivalent but much simpler four color iteration matrix in the frequency domain a two level sor method is introduced to increase the convergence rate for the frequency domain iteration matrix at the first level the red and orange points and then the black and green points are treated as groups and a block sor iteration is performed on these two groups at the second level another sor iteration is used to decouple values at the red and orange points and then at the black and green points the conventional red black sor iteration for a five point stencil is shown to be a degenerate case of the general two level four color sor method for the case of the nine point stencil a closed form expression for the optimal relaxation parameters omega b and omega p at the two iteration levels is

comparison of gravimetric formulas for uniform polyhedra the gravitational anomaly of a polyhedral target of uniform density is amenable to analytical solution and has received wide attention we analyze a number of apparently dissimilar solutions for consistency and comment on their suitability for numeric computation consistency is determined by evaluating a method over a special contour for which all methods must agree numerical performance is assessed through operation counts and algorithm classification into vertex line or surface type each type having a known accuracy bound for a given floating point precision we verify performance predictions through model calculations in particular we demonstrate the effectiveness of revised vertex and line algorithms and of a new analytical surface method our scheme of analysis and classification has general application it provides criteria by which existing and novel solutions may be examined for consistency and numerical performance

domains of divergence of the ussor method applied on p cyclic matrices using a recently derived classical type general functional equation relating the eigenvalues of a weakly cyclic jacobi iteration matrix to the eigenvalues of its associated unsymmetric successive overrelaxation ussor iteration matrix we obtain bounds for the convergence of the ussor method when applied to systems with ap cyclic coefficient matrix

an analytical expression for the gravity field of a polyhedral body with linearly varying density over the past 24 years the gravity fields of homogeneous polyhedra have been studied extensively paul 1974 barnett 1976 okabe 1979 golizdra 1981 strakhov et al 1986a 1986b gotze and lahmeyer 1988 pohanka 1988 ivan 1990 pohanka 1990 holstein and ketteridge 1996 this intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run time library for most computational programming languages

gravimagnetic similarity in anomaly formulas for uniform polyhedra gravitational and magnetic anomalies of an arbitrary target body are linked through poisson s differential relation for a uniform polyhedral target poisson s relation reduces to an algebraic link between gravity and magnetic anomaly formulas the derivation is given in tensor form it identifies for each target facet edge a vector function in terms of which the gravitational and magnetic potential and field anomaly formulas are similarly expressed as appropriately weighted linear combinations this similarity unifies the theory of uniform polyhedral anomalies it benefits analysis and construction of software that naturally embraces all anomalies in a single code the analysis is exemplified by a discussion of singularities and by the adaptation of three gravity field algorithms to the remaining gravitational and magnetic cases while retaining the respective computational advantages of the former gravity field algorithms

inversion modeling of gravity with prismatic mass bodies a combined method for forward and inverse modeling of gravity data is presented based on the fourier transform of poisson s equation the forward modeling is suitable for observation points above within and below causative masses with any prescribed density distribution the inversion is linearized in the spatial domain by superimposing numerous prismatic bodies each having constant but different density and fixed geometry our inversion algorithm adopts a sampling window to reduce memory storage and computations testing with synthetic and field data demonstrates that a successful inversion can be obtained from crudely estimated a priori density distributions and uncertainties lateral variations in density are well resolved but depth resolution often requires better constrained a priori information under various a priori conditions our modeling indicates that sediment density tends to vary exponentially with depth in the san jacinto basin southern california

a versatile solution for the gravity anomaly of 3d prism meshed bodies with depth dependent density contrast abstractwe have developed a generalized solution for computing the gravity anomalies of 3d irregular mass bodies with complicated density contrast variation the 3d irregular shaped bodies can be approximated flexibly by a collection of finite juxtaposed right rectangular prisms the complicated density contrast variation of each prism can be well represented by a depth dependent polynomial function a novel analytic solution of gravity anomalies due to a right rectangular prism with an arbitrary order of polynomial density contrast function of depth is then derived the solution is singularity free in the upper half space over the prism and its singularity in the lower half space containing the prism is resolved by assigning their limit values to the singular terms the numerical stability of the solution is also evaluated through numerical tests hence the solution can be used to compute the gravity anomalies of 3d irregular bodies with variable density contrasts without singularities when computation

new method for fast computation of gravity and magnetic anomalies from arbitrary polyhedra we show that at any point the gravity field from a solid body bounded by plane surfaces and having uniform density can be computed as a field from a fictitious distribution of surface mass density on the same body the surface mass density at every surface element is equal to the product of the volume density of the body and the scalar product of 1 the unit outward vector normal to that surface element and 2 the position vector of the surface element with respect to the point of observation accordingly the contribution to the gravity field from any plane surface of the body vanishes if the observation point lies in the plane of that surface as a result we can compute the gravity field everywhere including points inside on the surface on an edge or at a corner of the body where more than two surfaces meet this new result lets us compute the gravity field using exactly the same simple procedure as for the magnetic field of a uniformly magnetized object computed from an equivalent surface distr

3d vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3d variable density contrast three dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3d mass bodies with spatially variable density contrasts a 3d vector gravity potential is defined for a 3d rectangular prism with density contrast varying in depth and horizontally the vertical component of the gravity anomaly equals the flux of the 3d vector gravity potential through the enclosed surface of the prism thus the 3d integral for the gravity anomaly is reduced to a 2d surface integral in turn a 2d vector gravity potential is defined the vertical component of the gravity anomaly equals the net circulation of the 2d vector gravity potential along the enclosed contour bounding the surfaces of the prism the 3d integral for the gravity anomaly is reduced to 1d line integrals further analytical or numerical solutions can then be obtained from the line integrals depending on the forms of the density contrast functions if an analytical solution cannot be obtained the line integral method is semianalytical requiring numerical quadratures to be carried out at the final stages singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them then the vector potential line integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast varying simply in one direction and sophisticatedly in three directions the advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast

convergence and performance of iterative methods for solving variable coefficient convection diffusion equation with a fourth order compact difference scheme abstract we conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection diffusion equation discretized by a fourth order compact difference scheme several conditions are formulated under which the coefficient matrix is guaranteed to be an m matrix we further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods multigrid method and preconditioned gmres three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions the orderings of the grid points and the magnitude of the convection coefficients

elastic and piezoelectric fields due to polyhedral inclusions abstract we derive explicit closed form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half space and bi materials our analysis is based on the linear elasticity theory and green s function method the method involves evaluation of volume and surface integrals of harmonic and bi harmonic potentials in case of polyhedra such integrals are expressed through algebraic functions our results generalize numerous studies on this subject and they allow to obtain fully analytical solutions for a number of physical and engineering problems in the limiting case of an infinite space our relations have an essentially more compact form than relations obtained by other authors we present solutions to classical mindlin and cherruti problems we describe the elastic relaxation of a misfitting polygonal quantum dot in bi materials assuming isotropic and vertically isotropic properties it is explained how to analyze non hydrostatic and non uniform inclusions we also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices among other results we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in gaas 0 0 1 and gaas 1 1 1 depending on the orientation of the cube this suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

2d vector gravity potential and line integrals for the gravity anomaly caused by a 2d mass of depth dependent density contrast using line integrals lis used to calculate the gravity anomaly caused by a 2d mass of complicated geometry and spatially variable density contrast is a computationally efficientalgorithm thatreducesthecalculationfromtwodimensions to one dimension this work has developed a mechanism for defining lis systematically for different types of density functions two dimensional vector gravity potential is defined as a vector the net circulation of which along the closedcontourboundinga2dmass equalsthegravityanomaly caused by the 2d mass two representative types of lis aredefined anliwithanarctangentkernelforanydepth dependent density contrast function which has been studied historically and an li with a simple algebraic kernel for any integrabledensity contrastfunction thepresentworkoffers 1avectorial basedderivationofformulasthatdonotsuffer from the arbitrary sign conventions found in some historical approaches and2asimplealgebraickernelinlineintegrals as an alternative to the historical arctangent kernel with the possibilityofextensiontomoregeneralcases theconceptof 2dvectorgravitypotentialprovidesausefultoolfordefining lis systematically for any mass density function helping us understand how dimensions can be reduced in a calculating gravity anomaly especially when the density contrast varies with space lis have been tested in case studies the maximumdifferencesincalculatedgravityanomaliesbydifferent lis for the case studies were between 5 9310

polynomial preconditioning on vector computers abstract this paper is concerned with the computation of the minimal eigenpair of the generalized eigenproblem with the preconditioned conjugate gradient method the ssor polynomial preconditioner and the additive polynomial preconditioner have been analyzed the results of an extensive computer experimentation on cray y mp are reported

on the equivalence between the scheduled relaxation jacobi method and richardson s non stationary method abstract the scheduled relaxation jacobi srj method is an extension of the classical jacobi iterative method to solve linear systems of equations a u b associated with elliptic problems it inherits its robustness and accelerates its convergence rate computing a set of p relaxation factors that result from a minimization problem in a typical srj scheme the former set of factors is employed in cycles of m consecutive iterations until a prescribed tolerance is reached we present the analytic form for the optimal set of relaxation factors for the case in which all of them are strictly different and find that the resulting algorithm is equivalent to a non stationary generalized richardson s method where the matrix of the system of equations is preconditioned multiplying it by d diag a our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix a or the corresponding iteration matrix of the underlying weighted jacobi scheme is replaced by the much easier calculation of the maximum and minimum frequencies derived from a von neumann analysis of the continuous elliptic operator this set of weights is also the optimal one for the general problem resulting in the fastest convergence of all possible srj schemes for a given grid structure the amplification factor of the method can be found analytically and allows for the exact estimation of the number of iterations needed to achieve a desired tolerance we also show that with the set of weights computed for the optimal srj scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter in the successive overrelaxation sor method in some cases finally we demonstrate with practical examples that our method also works very well for poisson like problems in which a high order discretization of the laplacian operator is employed e g a 9 or 17 points discretization this is of interest since the former discretizations do not yield consistently ordered a matrices and hence the theory of young cannot be used to predict the optimal value of the sor parameter furthermore the optimal srj schemes deduced here are advantageous over existing sor implementations for high order discretizations of the laplacian operator in as much as they do not need to resort to multi coloring schemes for their parallel implementation

gravity inversion of an interface above which the density contrast varies exponentially with depth mapping of an interface above which the density contrast varies exponentially with depth as is common at the basement surface of sedimentary basins is efficiently achieved by a theoretically precise gravity method which can be applied to either profile data or twodimensional data the contrast in mass above the interface is modeled by an array of vertical rectangular prisms with density contrasts varying exponentially with depth gravity anomalies due to the prisms are calculated in the wavenumber domain and then converted to the space domain the precision of the inverse numerical fourier transform in this procedure is significantly increased by a shift sampling technique based on the discrete fourier deviation equation depth to the interface is determined by iterative adjustment of the vertical extent of the prisms in accordance with observed gravity anomaly data the basement surface of the los angeles basin california calculated by this method closely duplicates the published configuration based

gravity interpretation using fourier transforms and simple geometrical models with exponential density contrast the decrease of density contrast in sedimentary basins can often be approximated by an exponential function theoretical fourier transforms are derived for symmetric trapezoidal vertical fault vertical prism syncline and anticline models this is desirable because there are no equivalent closed form solutions in the space domain for these models combined with an exponential density contrast these transforms exhibit characteristic minima maxima and zero values and hence graphical methods have been developed for interpretation of model parameters after applying end corrections to improve the discrete transforms of observed gravity data the transforms are interpreted for model parameters this method is first tested on two synthetic models then applied to gravity anomalies over the san jacinto graben and los angeles basin

eshelby s problem for two dimensional piezoelectric inclusions of arbitrary shape eshelby s problem for piezoelectric inclusions of arbitrarily shaped crosssection remains a challenging topic in this paper a simple method is presented to obtain an analytic solution for eshelby

an accurate robust and easy to implement method for integration over arbitrary polyhedra application to embedded interface methods we present an accurate method for the numerical integration of polynomials over arbitrary polyhedra using the divergence theorem the method transforms the domain integral into integrals evaluated over the facets of the polyhedra the necessity of performing symbolic computation during such transformation is eliminated by using one dimensional gauss quadrature rule the facet integrals are computed with the help of quadratures available for triangles and quadrilaterals numerical examples in which the proposed method is used to integrate the weak form of the navier stokes equations in an embedded interface method eim are presented the results show that our method is as accurate and generalized as the most widely used volume decomposition based methods moreover since the method involves neither volume decomposition nor symbolic computations it is much easier for computer implementation also the present method is more efficient than other available integration methods based on the divergence theorem efficiency of the method is also compared with the volume decomposition based methods and moment fitting methods to our knowledge this is the first article that compares both accuracy and computational efficiency of methods relying on volume decomposition and those based on the divergence theorem

the gravity anomaly of two dimensional sources with continuous density distribution and bounded by continuous surfaces the gravity anomaly of a complicated two dimensional 2 d source having arbitrary surfaces and density varying in either horizontal or vertical direction is calculated using a combination of closed form solutions and numerical integration the surfaces and density can be defined by continuous or piecewise continuous two dimensional functions in the integration interval for example the anomalies for intrusions or folded sedimentary units having an arbitrary density in the horizontal direction and a polynomial density distribution in the vertical direction can be calculated using surfaces represented by functions of the horizontal dimension when modeling dipping layered intrusions or sedimentary beds the surfaces are represented by functions of the vertical dimension in which case the density can be an arbitrary function of depth and a polynome function of horizontal coordinate the accuracy of the method is defined by the user the method gives simple and general equations to calculate anomalies of c

gravimetric analysis of uniform polyhedra analytical formulas for the gravity anomaly of a uniform polyhedral body are subject to numerical error that increases with distance from the target while the anomaly decreases this leads to a limited range of target distances in which the formulas are operational beyond which the calculations are dominated by rounding error we analyze the sources of error and propose a combination of numerical and analytical procedures that exhibit advantages over existing methods namely 1 errors that diminish with distance 2 enhanced operating range and 3 algorithmic simplicity the latter is achieved by avoiding the need to transform coordinates and the need to discriminate between projected observation points that lie inside on or outside a target facet boundary our error analysis is verified in computations based on a published code and on a code implementing our methods the former requires a numerical precision of one part in 1016 double precision in problems of geophysical interest whereas our c

a bayesian approach to modeling 2d gravity data using polygons abstractwe apply a bayesian markov chain monte carlo formalism to the gravity inversion of a single localized 2d subsurface object the object is modeled as a polygon described by five parameters the number of vertices a density contrast a shape limiting factor and the width and depth of an encompassing container we first constrain these parameters with an interactive forward model and explicit geologic information then we generate an approximate probability distribution of polygons for a given set of parameter values from these we determine statistical distributions such as the variance between the observed and model fields the area the center of area and the occupancy probability the probability that a spatial point lies within the subsurface object we introduce replica exchange to mitigate trapping in local optima and to compute model probabilities and their uncertainties we apply our techniques to synthetic data sets and a natural data set collected across the rio grande gorge bridge i

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa