gravity source moment inversion a versatile approach to characterize position and 3 d orientation of anomalous bodies we present a 3 d gravity interpretation method based on the inversion of source moments employing the series derived from the gravity anomaly expansion in multipoles and retaining moments up to second order only it presents the advantages of being essentially linear and allowing a straightforward computer implementation the method requires neither an explicit assumption about the source geometry nor a priori knowledge about the density contrast distribution which may even be nonuniform the method assumes implicitly 1 that the source fits in the interior of an imaginary sphere whose center coincides with the source center of mass and whose radius is smaller than the depth to the source center of mass relative to the measuring plane and 2 that the spatial density distribution presents three ortogonal planes of symmetry intersecting at the source center of mass the first assumption can be met by upward continuing the observed anomaly when both assumptions are met the method produces reliable and stable estimates of the total anomalous mass the coordinates of the center of mass the three principal axes directions of the anomalous body and the relative importance among the axes the method is particularly suited for interpreting compact isolated or disjoint but spatially correlated sources the method is sensitive to an incomplete removal of the regional field presumably superimposed on the residual anomaly of interest because the unremoved part of the regional field may strongly affect the estimates of the residual source moments

computing with words via turing machines a formal approach computing with words cw as a methodology means computing and reasoning by the use of words in place of numbers or symbols which may conform more to humans perception when describing real world problems in this paper as a continuation of a previous paper we aim to develop and deepen a formal aspect of cw according to the previous paper the basic point of departure is that cw treats certain formal modes of computation with strings of fuzzy subsets instead of symbols as their inputs specifically 1 we elaborate on cw via turing machine tm models showing the time complexity is at least exponential if the inputs are strings of words 2 a negative result of 6 not holding is verified which indicates that the extension principle for cw via tms needs to be re examined 3 we discuss cw via context free grammars and regular grammars and the extension principles for cw via these formal grammars are set up 4 some equivalences between fuzzy pushdown automata respectively fuzzy finite state automata fuzzy context free grammars respectively fuzzy regular grammars are demonstrated in the sense that the inputs are instead strings of words 5 some instances are described in detail summarily formal aspect of cw is more systematically established more deeply dealt with while some new problems also emerge

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

design and implementation of two dimensional fully recursive digital filters the two dimensional 2 d fully recursive filter was recently introduced by the authors as a more general 2 d recursive filter in this paper we present a design algorithm and an implementation method for this class of filters the design algorithm incorporates a novel stability test we discuss the boundary condition problem which occurs in the implementation of fully recursive filters we propose a direct fully recursive implementation method which has been verified in software several design examples and comparisons are provided

consistency measures for hesitant fuzzy linguistic preference relations hesitant fuzzy linguistic term sets hfltss are used to deal with situations in which the decision makers dms think of several possible linguistic values or richer expressions than a single term for an indicator alternative variable etc compared with fuzzy linguistic approaches they are more convenient and flexible to reflect the dms preferences in decision making for further applications of hfltss to decision making we develop a concept of hesitant fuzzy linguistic preference relations hflprs as a tool to collect and present the dms preferences due to the importance of the consistency measures using preference relations in decision making we develop some consistency measures for hflprs to ensure that the dms are being neither random nor illogical a consistency index is defined to establish the consistency thresholds of hflprs to measure whether an hflpr is of acceptable consistency for hflprs with unacceptable consistency we develop two optimization methods to improve the consistency until they are acceptable several illustrative examples are given to validate the consistency measures and the optimization methods

chebyshev approximations for the direct design of 2 d quarter plane iir filters the development of various direct design techniques for inherently stable causal recursive two dimensional 2 d digital filters is described and illustrated by examples these techniques center around the development of 2 d z domain functions for use as arguments in the chebyshev function and provide for the design of low pass bandpass and high pass filters the specifications that can be prescribed are passband ripple filter order and normalized passband position parameters in addition to being inherently stable the filters possess an approximately flat group delay in most of the filter s passband a feature necessary for most applications

operators and comparisons of hesitant fuzzy linguistic term sets the theory of hesitant fuzzy linguistic term sets hfltss is very useful in objectively dealing with situations in which people are hesitant in providing linguistic assessments the purpose of this paper is to develop comparison methods and study the aggregation theory for hfltss we first define operations on hfltss and give possibility degree formulas for comparing hfltss we then define two aggregation operators for hfltss a hesitant fuzzy lwa operator and a hesitant fuzzy lowa operator in actual application we use these operators and the comparison methods to deal with multicriteria decision making problems with different situations in which importance weights of criteria or experts are known or unknown

a stable truncated series approximation of the reduction to the pole operator we combine a stabilized reduction to the pole and an upward continuation filter to produce meaningful reduced to the pole fields at low magnetic latitudes the stabilizing procedure is based on the development in taylor s series of the theoretical expression for the reduction to the pole filter in the wavenumber domain the filter instability is caused by the huge filter amplitudes along the magnetization azimuth which are expressed by the infinite sum of terms close to unity the stabilizing procedure reduces to simply truncating the infinite series the upward continuation filter attenuates the high wavenumber component of the noise and allows us to design a stabilized filter closer to the theoretical one besides quantitative interpretations of source depth based on the filtered field are more reliable when using upward continuation as compared with arbitrary low pass filters the proposed filter was applied to synthetic data of a single prism uniformly magnetized along a supposedly known direction

gridding with continuous curvature splines in tension a gridding method commonly called minimum curvature is widely used in the earth sciences the method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature the minimum curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points minimum curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used these extraneous inflection points can be eliminated by adding tension to the elastic plate flexure equation it is straightforward to generalize minimum curvature gridding algorithms to include a tension parameter the same system of equations must be solved in either case and only the relative weights of the coefficients change therefore solutions under tension require no more computational effort than minimum curvature solutions and any algorithm which can solve the minimum curvature equations can solve the more general system we give common geologic examples where minimum curvature gridding produces erroneous results but gridding with tension yields a good solution we also outline how to improve the convergence of an iterative method of solution for the gridding equations

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

benchmarking in linguistic importance weighted aggregations in this work we concentrate on the issue of weighted aggregations and provide a possibilistic approach to the process of importance weighted transformation when both the importances interpreted as benchmarks and the ratings are given by symmetric triangular fuzzy numbers we will show that using the possibilistic approach i small changes in the membership function of the importances can cause only small variations in the weighted aggregate ii the weighted aggregate of fuzzy ratings remains stable under small changes in the nonfuzzy importances iii the weighted aggregate of crisp ratings still remains stable under small changes in the crisp importances whenever we use a continuous implication operator for the importance weighted transformation we illustrate the proposed method by some simple examples

selecting the individual numerical scale and prioritization method in the analytic hierarchy process a 2 tuple fuzzy linguistic approach the validity of the priority vector used in the analytic hierarchy process ahp relies on two factors the selection of a numerical scale and the selection of a prioritization method the traditional ahp selects only one numerical scale e g the saaty scale and one prioritization method e g the eigenvector method for each particular problem for this traditional selection approach there is disagreement on which numerical scale and prioritization method is better in deriving a priority vector in fact the best numerical scale and the best prioritization method both rely on the content of the pairwise comparison data provided by the ahp decision makers by defining a set of concepts regarding the scale function and the linguistic pairwise comparison matrices lpcms of the priority vector and by using lpcms to unify the format of the input and output of ahp this paper extends the ahp prioritization process under the 2 tuple fuzzy linguistic model based on the extended ahp prioritization process we present two performance measure criteria to evaluate the effect of the numerical scales and prioritization methods we also use the performance measure criteria to develop a 2 tuple fuzzy linguistic multicriteria approach to select the best numerical scales and the best prioritization methods for different lpcms in this paper we call this type of selection the individual selection of the numerical scale and prioritization method we also compare this individual selection with traditional selection by using both random and real data and show better results with individual selection

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

a scattered equivalent source method for interpolation and gridding of potential field data in three dimensions potential field geophysical data observed at scattered discrete points in three dimensions can be interpolated gridded for example onto a level surface by relating the point data to a continuous function of equivalent discrete point sources the function used here is the inverse distance newtonian potential the sources located beneath some of the data points at a depth proportional to distance to the nearest neighboring data point are determined iteratively areas of no data are filled by minimum curvature for two dimensional 2 d data all data points at the same elevation grids calculated by minimum curvature and by equivalent sources are similar but the equivalent source method can be tuned to reduce aliasing gravity data in an area of high topographic relief in southwest u s a were gridded by minimum curvature a 2 d algorithm and also by equivalent sources 3 d the minimum curvature grid shows strong correlation with topography as expected because variation in gravity effect due to

generalized sampling a variational approach i theory we consider the problem of reconstructing a multidimensional vector function f sub in spl ropf sup m spl rarr spl ropf sup n from a finite set of linear measures these can be irregularly sampled responses of several linear filters traditional approaches reconstruct in an a priori given space e g the space of bandlimited functions instead we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion j first we present the solution of the generalized interpolation problem later we also consider the approximation problem and we show that both lead to the same class of solutions imposing generally desirable properties on the reconstruction largely limits the choice of the criterion j linearity leads to a quadratic criterion based on bilinear forms specifically we show that the requirements of translation rotation and scale invariance restrict the form of the criterion to essentially a one parameter family we show that the solution can be obtained as a linear combination of generating functions we provide analytical techniques to find these functions and the solution itself

efficient 3d inversion of magnetic data via octree mesh discretization space filling curves and wavelets abstractairborne magnetic survey data sets can contain from hundreds of thousands to millions of observations and typically cover large areas the large number of measurements combined with a model mesh to accommodate the survey extent can render an inversion of these data intractable faced with this challenge we have developed a three step procedure to locally optimize the degree of model discretization and to compress the corresponding sensitivity matrix for the inversion of magnetic data the mesh optimization is achieved through the use of adaptive octree discretization the compression is achieved by first reordering the model cells using the hilbert space filling curve and then applying the one dimensional wavelet transform to the corresponding sensitivities the fractal property of the hilbert curve groups the spatially adjacent cells into algebraically adjacent positions in the reordered model mesh and thereby maximizes the number of zero or near zero coefficients in the one dimensional wavelet

orthogonality from disjoint support in reproducing kernel hilbert spaces we investigate reproducing kernel hilbert spaces rkhs where two functions are orthogonal whenever they have disjoint support necessary and sufficient conditions in terms of feature maps for the reproducing kernel are established we also present concrete examples of finite dimensional rkhs and rkhs with a translation invariant reproducing kernel in particular it is shown that a sobolev space has the orthogonality from disjoint support property if and only if it is of integer index

the owa based consensus operator under linguistic representation models using position indexes when using linguistic approaches to solve decision problems we need linguistic representation models the symbolic model the 2 tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes together with these three linguistic models the corresponding ordered weighted averaging operators such as the linguistic ordered weighted averaging operator the 2 tuple ordered weighted averaging operator and the extended ordered weighted averaging operator have been developed respectively in this paper we analyze the internal relationship among these operators and propose a consensus operator under the continuous linguistic model or the 2 tuple fuzzy linguistic representation model the proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures some desired properties of the consensus operator are also presented in particular the consensus operator provides an alternative consensus model for group decision making this consensus model preserves the original preference information given by the decision makers as much as possible and supports consensus process automatically without moderator

a unified approach to optimal image interpolation problems based on linear partial differential equation models the unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal spatial the unknown image is reconstructed from a finite set of sampled data in such a way that a mean square error is minimized by first expressing the solution in terms of the reproducing kernel of a related hilbert space and then constructing this kernel using the fundamental solution of an induced linear partial differential equation or the green s function of the corresponding self adjoint operator it is proved that in most cases closed form fundamental solutions or green s functions for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first or second order linear partial differential operator an efficient method for obtaining the corresponding closed form fundamental solutions or green s functions of the operators is presented a computer simulation demonstrates the reconstruction procedure

a comparison of methods for approximating the vertical gradient of one dimensional magnetic field data the vertical gradient of a one dimensional magnetic field is known to be a useful aid in interpretation of magnetic data when the vertical gradient is required but has not been measured it is necessary to approximate the gradient using the available total field data an approximation is possible because a relationship between the total field and the vertical gradient can be established using fourier analysis after reviewing the theoretical basis of this relationship a number of methods for approximating the vertical gradient are derived these methods fall into two broad categories methods based on the discrete fourier transform and methods based on discrete convolution filters there are a number of choices necessary in designing such methods each of which will affect the accuracy of the computed values in differing and sometimes conflicting ways a comparison of the spatial and spectral accuracy of the methods derived here shows that it is possible to construct a filter which maintains a reason

combining numerical and linguistic information in group decision making abstract people give information about their personal preferences in many different ways depending on their background this paper deals with group decision making problems in which the solution depends on information of a different nature i e assuming that the experts express their preferences with numerical or linguistic values the aim of this paper is to present a proposal for this problem we introduce a fusion operator for numerical and linguistic information this operator combines linguistic values assessed in the same label set with numerical ones assessed in the interval 0 1 it is based on two transformation methods between numerical and linguistic values which are defined using the concept of the characteristic values proposed in this paper its application to group decision making problems is illustrated by means of a particular fusion operator guided by fuzzy majority considering that the experts express their opinions by means of fuzzy or linguistic preference relations this operator is used to develop a choice process for the alternatives allowing solutions to be obtained in line with the majority of the experts opinions

consistency of hesitant fuzzy linguistic preference relations an interval consistency index abstract the study of hesitant consistency is very important in decision making with hesitant fuzzy linguistic preference relations hflprs and generally the normalization method is used as a tool to measure the consistency degree of a hflpr in this paper we propose a new hesitant consistency measure called interval consistency index to estimate the consistency range of a hflpr the underlying idea of the interval consistency index consists of measuring the worst consistency index and the best consistency index of a hflpr furthermore by comparative study a connection is shown between the interval consistency index and the normalization method demonstrating that the normalization method should be considered as an approximate average consistency index of a hflpr

reconciliation of the discrete and integral fourier transforms in the application of harmonic analysis to potential field geophysical studies relationships derived in terms of the continuous fourier integral transform are evaluated in terms of the discrete fourier transform the discrete transform obtained by transforming a finite number of equispaced samples of the actual aperiodic continuous function is too low at the dc level and increasingly too high in the high frequencies compared with the theoretical integral transform as a consequence overly restrictive limitations must be placed on high frequency amplifying operators such as differentiation and downward continuation also a spurious and troublesome azimuthal distortion occurs in the discrete fourier analysis of three dimensional 3 d map data represented as grids the discrete transform can be made essentially equivalent to the integral transform if before sampling the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi

rapid construction of equivalent sources using wavelets we have developed a fast algorithm for generating an equivalent source by using fast wavelet transforms based on orthonormal compactly supported wavelets we apply a 2d wavelet transform to each row and column of the coefficient matrix and subsequently threshold the transformed matrix to generate a sparse representation in the wavelet domain the algorithm then uses this sparse matrix to construct the the equivalent source directly in the wavelet domain performing an inverse wavelet transform then yields the equivalent source in the space domain using upward continuation of total field magnetic data between uneven surfaces as examples we have compared this approach with the direct solution using the dense matrix in the space domain we have shown that the wavelet approach can reduce the cpu time by as many as two orders of magnitude

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

a framework for linguistic modelling a new framework for linguistic reasoning is proposed based on a random set model of the degree of appropriateness of a label labels are assumed to be chosen from a finite predefined set of labels and the set of appropriate labels for a value is defined as a random set valued function from a population of individuals into the set of subsets of labels appropriateness degrees are then evaluated relative to the distribution on this random set where the appropriateness degree of a label corresponds to the probability that it is contained in the set of appropriate labels this interpretation is referred to as label semantics a natural calculus for appropriateness degrees is described which is weakly functional while taking into account the logical structure of expressions given this framework it is shown that a bayesian approach can be adopted in order to infer probability distributions on the underlying variable given constraints both in the form of linguistic expressions and mass assignments in addition two conditional measures are introduced for evaluating the appropriateness of a linguistic expression given other linguistic information

a new version of 2 tuple fuzzy linguistic representation model for computing with words in this paper we provide a new proportional 2 tuple fuzzy linguistic representation model for computing with words cw which is based on the concept of symbolic proportion this concept motivates us to represent the linguistic information by means of 2 tuples which are composed by two proportional linguistic terms for clarity and generality we first study proportional 2 tuples under ordinal contexts then under linguistic contexts and based on canonical characteristic values ccvs of linguistic labels we define many aggregation operators to handle proportional 2 tuple linguistic information in a computational stage for cw without any loss of information our approach for this proportional 2 tuple fuzzy linguistic representation model deals with linguistic labels which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having equal distance between them moreover this new model not only provides a space to allow a continuous interpolation of a sequence of ordered linguistic labels but also provides an opportunity to describe the initial linguistic information by members of a continuous linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant meanwhile under the assumption of equally informative which is defined by a condition based on the concept of ccv we show that our model reduces to herrera and mart spl inodot spl acute nez s translational 2 tuple fuzzy linguistic representation model

two dimensional iir filter design with magnitude and phase error criteria two dimensional iir filters are designed using an error criterion consisting of the weighted sum of magnitude phase and stability errors thereby extending the spectral factorization based method to complex approximation the weights are adjusted to effectively constrain stability while minimizing a weighted sum of frequency domain magnitude and phase errors we use a derivative free variant of the marquardt optimization algorithm augmented by a one dimensional search we present examples of approximation of both linear and nonlinear phase ideal transfer functions and conduct comparisons across a range of numerator and denominator orders from fir to all pole while keeping the total number of coefficients approximately constant we find that the designed iir filters offer better magnitude response in the case of linear phase ideal functions even considering coefficient reflection symmetry but of course the fir filter has the better zero phase error when the ideal transfer function has nonlinear phase and hence no coefficient reflection symmetry we find that the designed iir filter can perform better than the fir filter with regard to both magnitude and phase error when magnitude error is weighted more heavily than phase error

smooth approximation of potential fields from noisy scattered data geophysical potential field data generally are observed at scattered sampling points and are contaminated by measurement and preprocessing errors analysis and further treatment require a representation of the underlying potential fields on a regular grid in the presence of noise an approximation of the data is more appropriate than an exact interpolation analysis of the smoothness of potential fields indicates an approximation by band limited functions the resulting least squares problem can be formulated as a linear system of equations in which the system matrix is of block toeplitz type this special structure allows a computationally attractive and robust solution via the fast fourier transform and the conjugate gradient algorithm on this basis we present a gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data experimental results for synthetic and real world gravity data demonstrate good performance for highly sc

smooth fitting of geophysical data using continuous global surfaces continuous global surfaces cgs are a general framework for interpolation and smoothing of geophysical data the first of two smoothing techniques we consider in this paper is generalized cross validation gcv which is a bootstrap measure of the predictive error of a surface that requires no prior knowledge of noise levels the second smoothing technique is to define the cgs surface with fewer centers than data points and compute the fit by least squares lsqr the noise levels are implicitly estimated by the number and placement of the centers relative to the data points we show that both smoothing methods can be implemented using extensions to the existing fast framework for interpolation so that it is now possible to construct realistic smooth fits to the very large data sets typically collected in geophysics thin plate spline and kriging surfaces with gcv smoothing appear to produce realistic fits to noisy radiometric data the resulting surfaces are similar yet the thin plate spline required

applications of bidimensional spline functions to geophysics two bidimensional spline functions are applied to the interpolation and automatic contouring of data from a ground magnetic survey of the rhinegraben between karlsruhe and strasbourg these bidimensional spline functions can be applied in the general case i e where a set of nonequispaced data is given these splines define a surface passing through the original points the most promising spline functions are the thin plate plaque mince and the pseudocubic spline studied by atteia 1966 thomann 1970 and duchon 1975 1976 a contouring method based on them has been implemented

on weighted unbalanced linguistic aggregation operators in group decision making we generalize linguistic evaluation values and their weights in group decision making gdm problems based on unbalanced linguistic terms the gdm problems include two types of weights belief degrees of linguistic evaluation values and experts weights due to the time pressure and conflict of multi source of information etc experts may have various evaluations values on alternatives the belief degree is used to represent the confidence level of the values the experts weight is used to represent the differences among experts importance which caused by experts experience or knowledge distinction we propose the weighted unbalanced linguistic aggregation operators to synthesize linguistic evaluation value belief degree and experts weights some desired properties of the operator are then studied these properties show that the operator extends the linguistic weighted averaging operator lwa and the linguistic ordered weighted averaging lowa operator finally an illustrative example of human resource performance appraisal based on the linguistic aggregation operator is provided

connecting the linguistic hierarchy and the numerical scale for the 2 tuple linguistic model and its use to deal with hesitant unbalanced linguistic information the 2 tuple linguistic representation model is widely used as a basis for computing with words cw in linguistic decision making problems two different models based on linguistic 2 tuples i e the model of the use of a linguistic hierarchy and the numerical scale model have been developed to address term sets that are not uniformly and symmetrically distributed i e unbalanced linguistic term sets ultss in this study we provide a connection between these two different models and prove the equivalence of the linguistic computational models to handle ultss further we propose a novel cw methodology where the hesitant fuzzy linguistic term sets hfltss can be constructed based on ultss using a numerical scale in the proposed cw methodology we present several novel possibility degree formulas for comparing hfltss and define novel operators based on the mixed 0 1 linear programming model to aggregate the hesitant unbalanced linguistic information

using an equivalent source with positivity for low latitude reduction to the pole without striation we present a reformulation of reduction to the pole rtp of magnetic data at low latitudes and the equator using equivalent sources the proposed method addresses both the theoretical difficulty of low latitude instability and the practical issue of computational cost we prove that a positive equivalent source exists when the magnetic data are produced by normal induced magnetization and we show that the positivity is sufficient to overcome the low latitude instability in the space domain we further apply a regularization term directly to the recovered rtp field to improve the solution the use of equivalent source also naturally enables the processing of data acquired on uneven surface the result is a practical algorithm that is effective at the equatorial region and can process large scale data sets with uneven observation heights

interpolation of potential field data by equivalent layer and minimum curvature a comparative analysis interpolation using only the observations at discrete points is an ill posed problem because it admits infinite solutions usually to reduce ambiguity a priori information about the sample function is introduced current interpolation methods in mineral exploration introduce only the constraints of continuity and smoothness of the interpolating function in interpolating potential field anomalies the constraint that the sampled function is harmonic may be introduced by the equivalent layer method elm we compare the performance of the elm and the minimum curvature method mcm in interpolating potential field anomalies by applying these methods to synthetic magnetic data simulating an aeromagnetic survey in the case the anomaly flanks and peak are undersampled the elm performs better than the mcm in recovering the anomaly gradients and peak in the case of elongated linear anomalies the elm recovers the exact linear pattern but the mcm introduces spurious oscillations in the linear pattern also

3 d inversion of gravity data we present two methods for inverting surface gravity data to recover a 3 d distribution of density contrast in the first method we transform the gravity data into pseudomagnetic data via poisson s relation and carry out the inversion using a 3 d magnetic inversion algorithm in the second we invert the gravity data directly to recover a minimum structure model in both approaches the earth is modeled by using a large number of rectangular cells of constant density and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data the model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints we apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information applications of the algorithms to synthetic and field data produce density models representative of true structures our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information

nonlinear magnetic inversion using a random search method a robust magnetic interpretation technique is described it is based on the minimization of an objective function by a random search algorithm instead of just one estimate this algorithm yields several points in parameter space all producing objective function values below an assumed noise level the centroid of the search points is in general a better estimate than each individual search point especially when data are corrupted by noise since no derivatives are employed even nondifferentiable objective functions such as the l1 norm of residuals can be used computation of parameter covariance matrices and confidence ellipses is possible using simple formulas the usefulness of r and q mode factor analysis applied to the correlation matrix is demonstrated the r mode factor analysis can be used in parameter variance control and the q mode factor analysis provides means to find models which are extreme in some sense the technique was applied to theoretical data and to field data from three diffe

a linear programming method for multiple criteria decision making with probabilistic linguistic information abstract the hesitant fuzzy linguistic term set hflts is applicable in expressing people s hesitant qualitative preference information however it may lose preference information as all the terms in the hflts are assigned the same weight the probabilistic linguistic term set plts can avoid this drawback by allowing the decision makers dms to give the weight of each linguistic term as a probability due to the flexibility of the plts the multiple criteria decision making mcdm with probabilistic linguistic information is now becoming a hot research topic the aim of this paper is to propose a linear programming method to solve the mcdm problems in which the dms preferences over the alternatives with respect to the given criteria are expressed as the pltss we first give the weighted deviation square measure for the normalized probabilistic linguistic positive ideal solution and each normalized probabilistic linguistic evaluation vector of each alternative then we define the inconsistency index and the consistency index according to the objective and subjective evaluations given by the dms based on which a linear programming model is then built to derive the weight of each criterion furthermore a linear programming method is developed to solve the mcdm problems with probabilistic linguistic information and a case study about the evaluation of hospitals is conducted to illustrate the proposed method finally some comparative analyses with the aggregation operator based method and the probabilistic linguistic preference ranking organization method for enrichment evaluation pl promethee method are made to show the advantages of the proposed method

splines in geophysics modeling and contouring of geophysical data often require distributions of regularly spaced values splines have been shown to be the most accurate methods to obtain such distributions we emphasize the general problem of interpolating random distributions of data on a given surface splines are classified into unidimensional quasi bidimensional and strictly bidimensional based on this classification a systematic derivation of the corresponding interpolating techniques is conducted two approaches are presented to obtain unidimensional splines one based on the continuity of the first and second derivatives of the polynomials involved and the other based on a variational approach quasi bidimensional splines are constructed based on the unidimensional approach while strictly bidimensional splines are generated by minimizing the bidimensional curvature quasi bidimensional splines can be used for processing data distributions along nearly parallel lines linear projections and parameterization are the techniques used in interpolating this type of distribution strictly bidimensional splines minimize curvature through the analytic solution of the euler lagrange equation or by a finite difference algorithm the maximum error mean error and standard deviation between interpolated data and exact field values produced by various prisms show that quasi bidimensional splines are 2 7 percent more accurate in the maximum error than strictly bidimensional splines when both techniques are applied to regularly spaced data however for irregularly spaced data three examples containing 300 600 and 900 random data points show the superiority of the thin plate approach over the quasi bidimensional splines a comparison between various interpolation densities on regular grids starting from a set of 327 randomly distributed magnetic stations illustrates some differences between geophysically meaningful interpolations and interpolations carried out only for contouring purposes

reproducing kernel hilbert space method for optimal interpolation of potential field data the rkhs based optimal image interpolation method presented by chen and de figueiredo 1993 is applied to scattered potential field measurements the rkhs which admits only interpolants consistent with laplace s equation is defined and its kernel derived the algorithm is compared to bicubic spline interpolation and is found to yield vastly superior results

adaptive sampling of potential field data a direct approach to compressive inversion abstractcompressive inversion uses computational algorithms that decrease the time and storage needs of a traditional inverse problem most compression approaches focus on the model domain and very few other than traditional downsampling focus on the data domain for potential field applications to further the compression in the data domain a direct and practical approach to the adaptive downsampling of potential field data for large inversion problems has been developed the approach is formulated to significantly reduce the quantity of data in relatively smooth or quiet regions of the data set while preserving the signal anomalies that contain the relevant target information two major benefits arise from this form of compressive inversion first because the approach compresses the problem in the data domain it can be applied immediately without the addition of or modification to existing inversion software second as most industry software use some form of model or sensitivity compression the

retraction and generalized extension of computing with words fuzzy automata whose input alphabet is a set of numbers or symbols are a formal model of computing with values motivated by zadeh s paradigm of computing with words rather than numbers ying proposed a kind of fuzzy automata whose input alphabet consists of all fuzzy subsets of a set of symbols as a formal model of computing with all words in this paper we introduce a somewhat general formal model of computing with some special words the new features of the model are that the input alphabet only comprises some not necessarily all fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily by employing the methodology of fuzzy control we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs these principles show that computing with values and computing with all words can be respectively implemented by computing with words some algebraic properties of retractions and generalized extensions are addressed as well

algebraic approach to two dimensional recursive digital filter synthesis an effective algorithm is developed for synthesizing two dimensional recursive digital filters which approximate prescribed ideal frequency response specifications the algorithm is based on an algebraic approach that uses the eigenvalue eigenvector decomposition of the ideal filter s excitation response matrix in conjunction with a recently developed signal enhancement method this results in a recursive filter which has a unit impulse that closely approximates the ideal filter s unit impulse response illustrative examples and comparisons to an existing technique are included

3d euler deconvolution theoretical basis for automatically selecting good solutions we derive the analytical estimators for the horizontal and vertical source positions in 3d euler deconvolution as a function of the x y and z derivatives of the magnetic anomaly within a data window from these expressions we show that in the case of noise corrupted data the x y and z coordinate estimates computed at the anomaly borders are biased toward the respective horizontal coordinate of the data window center regardless of the true or presumed structural indices and regardless of the magnetization inclination and declination on the other hand in the central part of the anomaly the x and y coordinate estimates are very close to the respective source horizontal coordinates regardless of the true or presumed structural indices and regardless of the magnetization inclination and declination this contrasting behavior of the horizontal coordinate estimates may be used to automatically delineate the region associated with the best solutions applying the euler deconvolution operator inside this region would decrease the dispersion of all position estimates improving source location precision

3 d magnetic inversion with data compression and image focusing we develop a method of 3 d magnetic anomaly inversion based on traditional tikhonov regularization theory we use a minimum support stabilizing functional to generate a sharp focused inverse image an iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method the weighting functions are selected based on sensitivity analysis to speed up the computations and to decrease the size of memory required we use a compression technique based on cubic interpolation our method is designed for inversion of total magnetic anomalies assuming the anomalous field is caused by induced magnetization only the method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by exxonmobil upstream research company

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

an interval type 2 fuzzy model of computing with words as a methodology computing with words allows the use of words instead of numbers or symbols in the process of computing and reasoning and thus conforms more to humans inference when it is used to describe real world problems motivated by zadeh s paradigm of computing with words the literature has proposed a kind of type 1 fuzzy automata as a formal model of computing with words which takes type 1 fuzzy subsets of symbols as input however type 1 fuzzy representation provides a limited platform for approximating the meaning of words since it is not able to capture linguistic uncertainty in this paper we develop a formal interval type 2 fuzzy model of computing with words by generalizing the existing type 1 fuzzy sets based model to an interval type 2 fuzzy environment concretely we take interval type 2 fuzzy automata i e interval type 2 fuzzy finite automata and interval type 2 fuzzy pushdown automata which combine interval type 2 fuzzy set theory and automaton theory as a computational model of computing with words furthermore we develop the extension principles to extend from computing with values to computing with words and show that computing with words can be implemented with computing with values in the interval type 2 fuzzy setting at an extra computational cost

an adaptive iterative method for downward continuation of potential field data from a horizontal plane we have developed an improved adaptive iterative method based on the nonstationary iterative tikhonov regularization method for performing a downward continuation of the potential field data from a horizontal plane our method uses the tikhonov regularization result as initial value and has an incremental geometric choice of the regularization parameter we compared our method with previous methods tikhonov regularization landweber iteration and integral iteration method the downwardcontinuation performance of these methods in spatial and wavenumber domains were compared with the aspects of their iterative schemes filter functions and downwardcontinuation operators applications to synthetic gravity and real aeromagnetic data showed that our iterative method yields a better downward continuation of the data than other methods our method shows fast computation times and a stable convergence in addition the l curve criterion for choosing the regularization parameter is expressed here in the wavenumber domain and used to speed up computations and to adapt the wavenumber domain iterative method

integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations abstract the purpose of this paper is to study a fuzzy multipurpose decision making problem where the information about the alternatives provided by the experts can be of a diverse nature the information can be represented by means of preference orderings utility functions and fuzzy preference relations and our objective is to establish a general model which cover all possible representations firstly we must make the information uniform using fuzzy preference relations as uniform preference context secondly we present some selection processes for multiple preference relations based on the concept of fuzzy majority fuzzy majority is represented by a fuzzy quantifier and applied in the aggregation by means of an owa operator whose weights are calculated by the fuzzy quantifier we use two quantifier guided choice degrees of alternatives a dominance degree used to quantify the dominance that one alternative has over all the others in a fuzzy majority sense and a non dominance degree that generalises orlovski s non dominated alternative concept the application of the two above choice degrees can be carried out according to two different selection processes a sequential selection process and a conjunction selection process

an approach to computing with words based on canonical characteristic values of linguistic labels herrera and martinez initiated a 2 tuple fuzzy linguistic representation model for computing with words cw which offers a computationally feasible method for aggregating linguistic information that are represented by linguistic variables with equidistant labels through counting indexes of the corresponding linguistic labels lawry introduced an alternative approach to cw based on mass assignment theory that takes into account the underlying definitions of the words recently we provided a new proportional 2 tuple fuzzy linguistic representation model for cw that is an extension of herrera and martinez s model and takes into account the underlying definitions of linguistic labels of linguistic variables in the process of aggregating linguistic information by assigning canonical characteristic values ccvs of the corresponding linguistic labels in this paper we study further into cw based on ccvs of linguistic labels to provide a unifying link between lawry s framework and herrera and martinez s 2 tuple framework as well as allowing for computationally feasible cw our approach is based on a formal definition of ccv functions of a linguistic variable which is introduced under the context of its proportional 2 tuple linguistic representation model as continuation of our earlier works and on a group voting model that is for the probabilistic interpretation of the whole semantics of the ordered linguistic terms set of an arbitrary linguistic variable after the general framework developed in the former part of this paper we focus on a particular linguistic variable probability we show that for a linguistic probability description the expectation of its posterior conditional probability is a canonical characteristic value of the linguistic probability description then a calculus for reasoning with linguistic syllogisms and inference from linguistic information is introduced it is investigated under the context that linguistic quantifiers in such linguistic syllogisms can be arbitrary linguistic probability description and that the related linguistic information can be linguistic facts overfacts or underfacts intrinsically our approach to this calculus is with computing with words based on canonical characteristic values of linguistic labels

linguistic computational model based on 2 tuples and intervals herrera and martinez initiated a 2 tuple fuzzy linguistic representation model for computing with words moreover wang and hao further developed a new 2 tuple fuzzy linguistic representation model to deal with the linguistic term sets that are not uniformly and symmetrically distributed this study proposes another linguistic computational model based on 2 tuples and intervals which we call an interval version of the 2 tuple fuzzy linguistic representation model the proposed model possesses three steps 1 interval numerical scale 2 computation based on interval numbers and 3 a generalized inverse operation of the interval numerical scale the first step transforms linguistic terms into interval numbers based on which the second step is executed with output as an interval number finally this number is then mapped into the interval of the linguistic 2 tuples by the generalized inverse operation this study also generalizes the numerical scale approach presented in the wang and hao model to set the interval numerical scale by considering the context where semantics of linguistic terms are defined by interval type 2 fuzzy sets it2 fss in order to compare the proposed model with the existing linguistic computational model based on it2 fss we have conducted extensive simulations the simulations demonstrate that the results obtained by our proposal are consistent with the results of the linguistic computational model based on it2 fss in some sense in a vast majority of cases

pdl g splines defined by partial differential operators with initial and boundary value conditions as a natural generalization of l g splines and thin plate splines text pdl g splines are introduced in this paper a text pdl g spline is defined as a solution of the optimal scattered data interpolation problem described by a general higher order separable linear partial differential operator which may or may not incorporate time with initial terminal and boundary value conditions over the time domain and an arbitrary bounded spatial domain with a continuous and piecewise smooth boundary a closed form expression for the text pdl g spline is obtained by means of the reproducing kernel of a related hilbert space a feasible constructive method for finding the reproducing kernel via a fundamental solution or green s function of an induced linear and self adjoint partial differential operator is also included this method provides a way to take care of the initial terminal and boundary value conditions moreover an explicit formulation for finding the best approximation i

application of cauchy type integrals in developing effective methods for depth to basement inversion of gravity and gravity gradiometry data one of the most important applications of gravity surveys in regional geophysical studies is determining the depth to basement conventional methods of solving this problem are based on the spectrum and or euler deconvolution analysis of the gravity field and on parameterization of the earth s subsurface into prismatic cells we have developed a new method of solving this problem based on 3d cauchy type integral representation of the potential fields traditionally potential fields have been calculated using volume integrals over the domains occupied by anomalous masses subdivided into prismatic cells this discretization can be computationally expensive especially in a 3d case the technique of cauchy type integrals made it possible to represent the gravity field and its gradients as surface integrals in this approach only the density contrast surface between sediment and basement needed to be discretized for the calculation of gravity field this was especially significant in the modeling and inversion of gravity data for determining the depth to the basement another important result was developing a novel method of inversion of gravity data to recover the depth to basement based on the 3d cauchy type integral representation our numerical studies determined that the new method is much faster than conventional volume discretization method to compute the gravity response our synthetic model studies also showed that the developed inversion algorithm based on cauchy type integral is capable of recovering the geometry and depth of the sedimentary basin effectively with a complex density profile in the vertical direction

an octree multigrid method for quasi static maxwell s equations with highly discontinuous coefficients in this paper we develop an octree discretization for maxwell s equations in the quasi static regime we then use this discretization in order to develop a multigrid method for maxwell s equations with highly discontinuous coefficients we test our algorithms and compare it to other multilevel algorithms

on advanced computing with words using the generalized extension principle for type 1 fuzzy sets in this paper we propose and demonstrate an effective methodology for implementing the generalized extension principle to solve advanced computing with words acww problems such problems involve implicit assignments of linguistic truth probability and possibility to begin we establish the vocabularies of the words involved in the problems and then collect data from subjects about the words after which fuzzy set models for the words are obtained by using the interval approach ia or the enhanced interval approach eia next the solutions of the acww problems which involve the fuzzy set models of the words are formulated using the generalized extension principle because the solutions to those problems involve complicated functional optimization problems that cannot be solved analytically we then develop a numerical method for their solution finally the resulting fuzzy set solutions are decoded into natural language words using jaccard s similarity measure we explain how acww problems can solve some potential prototype engineering problems and connect the methodology of this paper with perceptual computing

a fuzzy linguistic methodology to deal with unbalanced linguistic term sets many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects in most of these problems a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed however there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets i e using term sets that are not uniformly and symmetrically distributed the use of linguistic variables implies processes of computing with words cw different computational approaches can be found in the literature to accomplish those processes the 2 tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed in this paper we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets to do so we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2 tuple fuzzy linguistic computational model to accomplish processes of cw with unbalanced term sets in a precise way and without loss of information

computing the numerical scale of the linguistic term set for the 2 tuple fuzzy linguistic representation model when using linguistic approaches to solve decision problems we need the techniques for computing with words cw together with the 2 tuple fuzzy linguistic representation models i e the herrera and martinez model and the wang and hao model some computational techniques for cw are also developed in this paper we define the concept of numerical scale and extend the 2 tuple fuzzy linguistic representation models under the numerical scale we find that the key of computational techniques based on linguistic 2 tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2 tuples and numerical values by defining the concept of the transitive calibration matrix and its consistent index this paper develops an optimization model to compute the numerical scale of the linguistic term set the desired properties of the optimization model are also presented furthermore we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations the results in this paper are pretty helpful to complete the fuzzy 2 tuple representation models for cw

a mass assignment theory of the probability of fuzzy events abstract mass assignment theory techniques for processing uncertainty in fril are reviewed the notion of the probability of a fuzzy event is introduced together with the t norm definition of conditional probabilities the latter is then shown to be probability possibility inconsistent an alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties in particular the mass assignment theory of conditional probabilities is shown to be probability possibility consistent

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

a design method for two dimensional recursive digital filters a method for designing two dimensional symmetric half plane recursive digital filters is presented a filter is first designed as a parametrized family of one dimensional filters a simple approximation is then used to find a rational stable two dimensional filter some advantages and disadvantages of the method are discussed and several examples are given

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

polynomial equivalent layer we have developed a new cost effective method for processing large potential field data sets via the equivalent layer technique in this approach the equivalent layer is divided into a regular grid of equivalent source windows inside each window the physical property distribution is described by a bivariate polynomial hence the physical property distribution within the equivalent layer is assumed to be a piecewise polynomial function defined on a set of equivalent source windows we perform any linear transformation of a large set of data as follows first we estimate the polynomial coefficients of all equivalentsource windows by using a linear regularized inversion second we transform the estimated polynomial coefficients of all windows into the physical property distribution within the whole equivalent layer finally we premultiply this distribution by the matrix of green s functions associated with the desired transformation to obtain the transformed data the regularized inversion deals with a linear system of equations with dimensions based on the total number of polynomial coefficients within all equivalent source windows this contrasts with the classical approach of directly estimating the physical property distribution within the equivalent layer which leads to a system based on the number of data because the number of data is much larger than the number of polynomial coefficients the proposed polynomial representation of the physical property distribution within an equivalent layer drastically reduces the number of parameters to be estimated by comparing the total number of floating point operations required to estimate an equivalent layer via our method with the classical approach both formulated with cholesky s decomposition we can verify that the computation time required for building the linear system and for solving the linear inverse problem can be reduced by as many as three and four orders of magnitude respectively applications to synthetic and real data show that our method performs the standard linear transformations of potential field data accurately

a formal model of computing with words classical automata are formal models of computing with values fuzzy automata are generalizations of classical automata where the knowledge about the system s next state is vague or uncertain it is worth noting that like classical automata fuzzy automata can only process strings of input symbols therefore such fuzzy automata are still abstract devices for computing with values although a certain vagueness or uncertainty are involved in the process of computation we introduce a new kind of fuzzy automata whose inputs are instead strings of fuzzy subsets of the input alphabet these new fuzzy automata may serve as formal models of computing with words we establish an extension principle from computing with values to computing with words this principle indicates that computing with words can be implemented with computing with values with the price of a big amount of extra computations

continuation of potential fields between arbitrary surfaces an equivalent source algorithm is described for continuing either one or two dimensional potential fields between arbitrary surfaces in the two dimensional case the dipole surface is approximated as a set of plane faces with constant moments over each face in the one dimensional case the plane faces of the dipole surface reduce to straight line segments application of the algorithm to model and field examples of aeromagnetic data shows the method to be effective and accurate even when the terrain has strong topographic relief and is composed of highly magnetic volcanic rocks

the equivalent data concept applied to the interpolation of potential field data the equivalent layer calculation becomes more efficient by first converting the observed potential data set to a much smaller equivalent data set thus saving considerable cpu time this makes the equivalent source method of data interpolation very competitive with other traditional gridding techniques that ignore the fact that potential anomalies are harmonic functions the equivalent data set is obtained by using a least squares iterative algorithm at each iteration that solves an underdetermined system fitting all observations selected from previous iterations and the observation with the greatest residual in the preceding iteration the residuals are obtained by computing a set of predicted observations using the estimated parameters at the current iteration and subtracting them from the observations the use of cholesky s decomposition to implement the algorithm leads to an efficient solution update everytime a new datum is processed in addition when applied to interpolation problems using equiva

discrete linear transformations of potential field data we present a new approach to perform any linear transformation of gridded potential field data using the equivalent layer principle it is particularly efficient for processing areas with a large amount of data an n n data window is inverted using an m m equivalent layer with m greater than n so that the equivalent sources extend beyond the data window only the transformed field at the center of the data window is computed by premultiplying the equivalent source matrix by the row of the green s matrix associated with the desired transformation corresponding to the center of the data window since the inversion and the multiplication by the green s matrix are independent of the data they are performed beforehand and just once for given values of n m and the depth of the equivalent layer as a result a grid operator for the desired transformation is obtained which is applied to the data by a procedure similar to discrete convolution the application of this procedure in reducing synthetic anoma

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

the fully recursive filter a general 2 d recursive filter the conventional 2 d recursive filter can be thought of as performing a linear combination of 1 d filterings on past and present input and output data rows all of these component 1 d filters are fir except for that of the current output row in a fully recursive filter all the component 1 d filters are iir

robust polynomial fitting method for regional gravity estimation standard polynomial fitting methods are inconsistent in their formulation the regional field is approximated by a polynomial fitted to the observed field as a result in addition to the nonuniqueness in the definition of the regional field the fitted polynomial is strongly influenced by the residual field observed field minus regional field we present a regional residual separation method for gravity data which uses a robust procedure to determine the coefficients of a polynomial fitted to the observations under the hypothesis that the regional can be modeled correctly by the polynomial surface the proposed method minimizes the influence of the residual field in the fitted surface the proposed method was applied to real gravity data from ceara state brazil and produced information on zones of possible crustal thickening and the occurrence of lower crustal granulitic rocks thrust into the shallow subsurface

generalized sampling a variational approach ii applications for pt i see ibid vol 50 no 8 p 1965 76 2000 the variational reconstruction theory from a companion paper finds a solution consistent with some linear constraints and minimizing a quadratic plausibility criterion it is suitable for treating vector and multidimensional signals here we apply the theory to a generalized sampling system consisting of a multichannel filterbank followed by a nonuniform sampling we provide ready made formulas which should permit application of the technique directly to problems at hand we comment on the practical aspects of the method such as numerical stability and speed we show the reconstruction formula and apply it to several practical examples including new variational formulation of derivative sampling landmark warping and tomographic reconstruction

fuzzy linguistic pert a model for program evaluation and review technique pert under fuzzy linguistic contexts is introduced in this fuzzy linguistic pert network model each activity duration is represented by a fuzzy linguistic description aggregation and comparison of the estimated linguistic expectations of activity durations are manipulated by the techniques of computing with words cw to provide suitable contexts for this purpose we first introduce several variations of basic linguistic labels of a linguistic variable such as weighted linguistic labels generalized linguistic labels and weighted generalized linguistic labels etc and then based on the notion of canonical characteristic value ccv function of a linguistic variable we develop some related cw techniques for aggregation and comparison of these linguistic labels afterward using a computing technique of linguistic probability introduced by zadeh and based on the new developed cw techniques for weighted generalized linguistic labels we investigate the associated linguistic expectation pert network of a fuzzy linguistic pert network also throughout the paper several examples are used to illustrate related notions and applications

a new technique for the design of two dimensional fir and iir filters the design problem of two dimensional digital filters is the subject of this paper the problem is formulated for both fir and iir filters in such a way that the performance index is a convex function of the optimization vector the design technique is based on newton s method each iteration consists of n easily computed steps where n is the number of parameters to be optimized the coefficients can be obtained in the quantization levels according to the desired number representation illustrative examples and comparisons with existing techniques are also included