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1.
Increasing numerical efficiency of iterative solution for total least-squares in datum transformations 总被引：1，自引：0，他引：1
Cartesian coordinate transformation between two erroneous coordinate systems is considered within the Errors-In-Variables (EIV) model. The adjustment of this model is usually called the total Least-Squares (LS). There are many iterative algorithms given in geodetic literature for this adjustment. They give equivalent results for the same example and for the same user-defined convergence error tolerance. However, their convergence speed and stability are affected adversely if the coefficient matrix of the normal equations in the iterative solution is ill-conditioned. The well-known numerical techniques, such as regularization, shifting-scaling of the variables in the model, etc., for fixing this problem are not applied easily to the complicated equations of these algorithms. The EIV model for coordinate transformations can be considered as the nonlinear Gauss-Helmert (GH) model. The (weighted) standard LS adjustment of the iteratively linearized GH model yields the (weighted) total LS solution. It is uncomplicated to use the above-mentioned numerical techniques in this LS adjustment procedure. In this contribution, it is shown how properly diminished coordinate systems can be used in the iterative solution of this adjustment. Although its equations are mainly studied herein for 3D similarity transformation with differential rotations, they can be derived for other kinds of coordinate transformations as shown in the study. The convergence properties of the algorithms established based on the LS adjustment of the GH model are studied considering numerical examples. These examples show that using the diminished coordinates for both systems increases the numerical efficiency of the iterative solution for total LS in geodetic datum transformation: the corresponding algorithm working with the diminished coordinates converges much faster with an error of at least 10-5 times smaller than the one working with the original coordinates. 相似文献
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Summary In the Part 1 and in a subsequent Part 2 to be published two methods of adjusting a spatial terrestrial network in tri-dimensional space are described. Care has been taken that the nature of the equations used, as well as of the adjustment, correspond to the same in adjustment satellite networks. The adjustment was carried out by the least-squares method according to conditioned observations. Various types of condition equations have been constructed according to the various types of adjusted quantities, and the various alternatives of the introduced errors (changes of input values) and weights. An effort was made to eliminate the ellipsoid of reference to the largest extent. The theory was applied numerically to a model of a smaller network which corresponds in position and height to usual triangulation networks with side lengths of about 30 km.Dedicated to 90th Birthday of Professor Frantiek Fiala 相似文献
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Vahid Mahboub Mohammad Saadatseresht Alireza A. Ardalan 《Studia Geophysica et Geodaetica》2017,61(1):19-34
An applicable algorithm for Total Kalman Filter (TKF) approach is proposed. Meanwhile, we extend it to the case in which we can consider arbitrary weight matrixes for the observation vector, the random design matrix and possible correlation between them. Also the updated dispersion matrix of the predicted unknown is given. This approach makes use of condition equations and straightforward variance propagation rules. It is applicable to data fusion within a dynamic errors-in-variables (DEIV) model, which usually appears in the determination of the position and attitude of mobile sensors. Then, we apply for the first time the TKF algorithm and its extended version named WTKF to a DEIV model and compare the results. The results show the efficiency of the proposed WTKF algorithm. In particular in the case of large weights, WTKF shows approximately 25% improvement in contrast to TKF approach. 相似文献
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The necessary condition for the seismic refraction method to succeed is that the refracted first arrivals from each layer in a multilayered earth system should be detected on a seismogram as first arrivals, and this is possible only when velocities of all underlying layers are successively greater. The usual procedure to interpret the refraction travel times is to fit such a data set with several intersecting straight lines by employing a visual technique which may lead to errors of subjective judgment, as the velocity model depends on the selection of various line segments through the data. To remove the visual fit we propose here a layer stripping method based on minimum intercept time, apparent velocity, rms residual, and maximum data points by least-squares fitting to yield several intersecting straight lines. Once data are segmented out, the conventional equations can be used to determine the velocity structure. 相似文献
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对三维电阻率反演问题进行了深入研究,提供了一种利用地表观测数据实现三维反演的实用算法.该方法应用有限差分求正演解,并通过对粗糙度矩阵元素进行适当改进,使之适用于各种情况下粗糙度矩阵的求取,进而建立在模型的总粗糙度极小条件下的反演方程.对反演方程采用收敛速度快且稳定的最小二乘正交分解（LSQR）法进行迭代求解,在迭代求解过程中只需利用偏导数矩阵和其转置矩阵乘以一个向量的结果,回避了直接求偏导数矩阵的繁琐计算,节省了内存,加快了反演的计算速度.不同的计算实例表明上述方法是求解大规模三维电阻率反演问题的有效方法. 相似文献
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地震勘探方法在深部固体矿产资源勘探中发展潜力巨大,同时也面临挑战.由于固体矿产资源地下分布呈现陡峭构造、尺度小,物性差异小的特点,常规偏移方法对小尺度矿体成像的分辨率提高有限.本文研究了一种基于稀疏促进约束的最小二乘逆时偏移方法.首先,将非均匀分布的矿体等效为随机介质,建立小尺度扰动的矿体模型;其次,改进现有最小二乘偏移方法,以稀疏模型为先验信息约束成像结果,并通过Curvelet变换压缩成像空间,经过多次迭代计算,可以提高小尺度散射体的成像分辨率;再次,对炮域记录进行随机震源编码,减少成像所需的炮集个数,通过稀疏促进约束条件,降低串扰噪声引起的成像误差.通过庐枞金属矿模型数值计算,验证本文方法可以较好的成像包含小尺度散射体的金属矿地质模型. 相似文献
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Elastic least-squares reverse time migration has been applied to multi-component seismic data to obtain high-quality images. However, the final images may suffer from artefacts caused by P- and S-wave crosstalk and severe spurious diffractions caused by complex topographic surface conditions. To suppress these crosstalk artefacts and spurious diffractions, we have developed a topographic separated-wavefield elastic least-squares reverse time migration algorithm. In this method, we apply P- and S-wave separated elastic velocity–stress wave equations in the curvilinear coordinates to derive demigration equations and gradient formulas with respect to P- and S-velocity. For the implementation of topographic separated-wavefield elastic least-squares reverse time migration, the wavefields, gradient directions and step lengths are all calculated in the curvilinear coordinates. Numerical experiments conducted with the two-component data synthetized by a three-topographic-layer with anomalies model and the Canadian Foothills model are considered to verify our method. The results reveal that compared with the conventional method, our method promises imaging results with higher resolution and has a faster residual convergence speed. Finally, we carry out numerical examples on noisy data, imperfect migration velocity and inaccurate surface elevation to analyse its sensitivity to noise, migration velocity and surface elevation error. The results prove that our method is less sensitive to noise compared with the conventional elastic least-squares reverse time migration and needs good migration velocities as other least-squares reverse time migration methods. In addition, when implementing the proposed method, an accurate surface elevation should be obtained by global positioning system to yield high-quality images. 相似文献
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Indrajit G Roy 《Geophysical Prospecting》1999,47(4):527-550
Non-linear least-squares inversion operates iteratively by updating the model parameters in each step by a correction vector which is the solution of a set of normal equations. Inversion of geoelectrical data is an ill-posed problem. This and the ensuing suboptimality restrict the initial model to being in the near vicinity of the true model. The problem may be reduced by introducing damping into the system of equations. It is shown that an appropriate choice of the damping parameter obtained adaptively and the use of a conjugate-gradient algorithm to solve the normal equations make the 1D inversion scheme efficient and robust. The scheme uses an optimal damping parameter that is dependent on the noise in the data, in each iterative step. The changes in the damping and relative residual error with iteration number are illustrated. A comparison of its efficacy over the conventional Marquardt and simulated annealing methods, tested on Inman's model, is made. Inversion of induced polarization (IP) sounding is obtained by inverting twice (true and modified) DC apparent resistivity data. The inversion of IP data presented here is generic and can be applied to any of the IP observables, such as chargeability, frequency effect, phase, etc., as long as these observables are explicitly related to the DC apparent resistivity. The scheme is used successfully in inverting noise-free and noisy synthetic data and field data taken from the published literature. 相似文献
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Xing Fang 《Studia Geophysica et Geodaetica》2014,58(1):1-16
A new proof is presented of the desirable property of the weighted total least-squares (WTLS) approach in preserving the structure of the coefficient matrix in terms of the functional independent elements. The WTLS considers the full covariance matrix of observed quantities in the observation vector and in the coefficient matrix; possible correlation between entries in the observation vector and the coefficient matrix are also considered. The WTLS approach is then equipped with constraints in order to produce the constrained structured TLS (CSTLS) solution. The proposed approach considers the correlation between the observation vector and the coefficient matrix of an Error-In-Variables model, which is not considered in other, recently proposed approaches. A rigid transformation problem is done by preservation of the structure and satisfying the constraints simultaneously. 相似文献
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Summary A new computer-oriented algorithm GSO is presented for solving overdetermined systems of linear observation equations according to the principle of the least-squares method. The matrix of the system of observation equations may be of deficient rank. In this case the algorithm leads to the vector of unknowns with a minimum Euclidean norm. Alternatively, it is possible to minimize the norm of a subvector formed by a selected group of unknowns. The weight coefficient matrix, corresponding to the vector (subvector) of unknows, has the least possible trace. The algorithm GSO is based on the Gram-Schmidt Orthogonalization of suitably defined augmented matrices. The establishing and solving of normal equations is not necessary. Apart from the unknowns and residuals, GSO also determines the factorized weight coefficient matrices of the adjusted values.Presented at the I.A.G. International Symposium on Optimization of Design and Computation of Control Networks, Sopron, Hungary 1977. 相似文献
13.
We have developed a least-squares minimization approach to determine the depth and the amplitude coefficient of a buried structure
from residual gravity anomaly profile. This approach is basically based on application of Werner deconvolution method to gravity
formulas due to spheres and cylinders, and solving a set of algebraic linear equations to estimate the two-model parameters.
The validity of this new method is demonstrated through studying and analyzing two synthetic gravity anomalies, using simulated
data generated from a known model with different random error components and a known statistical distribution. After being
theoretically proven, this approach was applied on two real field gravity anomalies from Cuba and Sweden. The agreement between
the results obtained by the proposed method and those obtained by other interpretation methods is good and comparable. Moreover,
the depth obtained by the proposed approach is found to be in very good agreement with that obtained from drilling information. 相似文献
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E. M. Abdelrahman T. M. El-Araby A. A. Ammar H. I. Hassanein 《Pure and Applied Geophysics》1997,150(1):121-128
—We have developed a least-squares minimization approach to determine the shape (shape-factor) of a buried polarized body from a residual self-potential anomaly profile. By defining the zero anomaly distance and the anomaly value at the origin on the profile, the problem of the shape-factor determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(q) = 0. Procedures are also formulated to estimate the depth of polarization angle, and the electric dipole moment. The method is applied to synthetic data with and without random noise. The obtained shape-factor agrees very well with the model shape-factor when using synthetic data. After adding ± 2 percent random error in the synthetic data, the shape factor obtained is within ± 4 percent. Finally the validity of the method is tested on a field example from the Ergani copper district, Turkey. 相似文献
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The procedure of polynomial fitting by least-squares technique to obtain the regional field from potential field data is commonly used in processing geophysical data. However, it has been found that the unguarded use of this procedure with measured field data may lead to erroneous regional fields. This is due to the fact that the least-squares technique pre-supposes a random variation of the residual or anomaly field. This condition is rarely fulfilled in measured field data where there is the common occurrence of ‘deviatory field points’ in the residual field data. In this paper, a procedure to overcome the problem using robust statistics is described with the straight-forward Downhill Simplex method used to obtain the optimum coefficients for the polynomial. The method is illustrated with aeromagnetic field data from the Mamfe basin of Nigeria and Cameroon. The result shows that the trend of the regional field obtained from robust statistics is consistent with the trend in the aeromagnetic field obtained by using the method of cross-correlation trend analysis. 相似文献
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This study aims at evaluating the optimal properties of friction pendulum bearings to be employed for the seismic protection of elastic isolated structural systems under earthquake excitations with different characteristics in terms of frequency content. A two-degree-of-freedom model is considered to describe the isolated system behavior while accounting for the superstructure flexibility and a non-dimensional formulation of the governing equations of motion is employed to relate the characteristic parameters describing the isolator and structure properties to the response parameters of interest for the performance assessment. Seismic excitations are modeled as time-modulated filtered Gaussian white noise random processes of different intensity within the power spectral density method. The filter parameters control the frequency content of the random excitations and are calibrated to describe stiff, medium and soft soil conditions, respectively. Finally, multi-variate regression expressions are obtained for the optimum values of the friction coefficient that minimize the superstructure displacements relative to the base mass as a function of the structural system properties, of the seismic input intensity and of the soil condition. 相似文献
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Summary In the present paper the adjustment of the Hradilek's spatial network[3] was carried out in space using the least-squares method according to conditional observations. Triangle, side and base equations of condition are given (see also[1]). In the first method of adjustment (Alternative A) the corrections are assigned to oblique (position) angles and the lengths of sides. In the second method of adjustment (Alternative B) the corrections are assigned to horizontal directions, zenith distances and lengths of sides. The refraction coefficients in both alternatives are introduced as unknown parameters. Neither method of adjustment depends on the directions of the verticals. Theoretically, Alternative B is more correct. However, for practical purposes the results yielded by Alternative A are little better than those yielded by Alternative B. As regards the economical aspect Alternative A is considerably more convenient. Both methods seem suitable for computing the rectangular spatial co-ordinates, less so for determining the refraction coefficients. 相似文献
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M. AL-CHALABI 《Geophysical Prospecting》1992,40(3):359-378
There is a general lack of awareness among ‘lay’ professionals (geophysicists included) regarding the limitations in the use of least-squares. Using a simple numerical model under simulated conditions of observational errors, the performance of least-squares and other goodness-of-fit criteria under various error conditions are investigated. The results are presented in a simplified manner that can be readily understood by the lay earth scientist. It is shown that the use of least-squares is, strictly, only valid either when the errors pertain to a normal probability distribution or under certain fortuitous conditions. The correct power to use (e.g. square, cube, square root, etc.) depends on the form of error distribution. In many fairly typical practical situations, least-squares is one of the worst criteria to use. In such cases, data treatment, ‘robust statistics’ or similar processes provide an alternative approach. 相似文献
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This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator. 相似文献
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We propose a wave scattering approach to the problem of deconvolution by the inversion of the reflection seismogram. Rather than using the least-squares approach, we study the full wave solution of the one-dimensional wave equation for deconvolution. Randomness of the reflectivity is not a necessary assumption in this method. Both the reflectivity and the section multiple train can be predicted from the boundary data (the reflection seismogram). This is in contrast to the usual statistical approach in which reflectivity is unpredictable and random, and the section multiple train is the only predictable component of the seismogram. The proposed scattering approach also differs from Claerbout's method based on the Kunetz equation. The coupled first-order hyperbolic wave equations have been obtained from the equation of motion and the law of elasticity. These equations have been transformed in terms of characteristics. A finite-difference numerical scheme for the downward continuation of the free-surface reflection seismogram has been developed. The discrete causal solutions for forward and inverse problems have been obtained. The computer algorithm recursively solves for the pressure and particle velocity response and the impedance log. The method accomplishes deconvolution and impedance log reconstruction. We have tested the method by computer model experiments and obtained satisfactory results using noise-free synthetic data. Further study is recommended for the method's application to real data. 相似文献