Weighted Least-squares Solution Research Articles

A simple method for estimating the horizontal velocity field in wide zones of active deformation-I. Description, with an example from California

SUMMARY The velocity field is the most complete description of the active kinematics of a wide zone of active deformation. A simple method is described here for estimating the horizontal velocity field in a network of traverses that span the zone. The sum of the relative velocities in such a zone must equal the relative motion of the boundaries, which can usually be determined independently from sea-floor magnetic anomalies. Also, if traverses intersect each other, the velocity of the point of intersection relative to one of the boundaries of the zone, must be the same regardless of the traverse segment along which relative velocities have been summed. In addition, rates of deformation within parts of the deforming zone can be estimated from field observations. All these constraints can be expressed as linear equations. The aim is to maximize the difference between the number of equations (N) and unknowns (M), with the formed exceeding the latter. The problem can be expressed in matrix form: Am = d, where A is an N X M model matrix of constraints, d is a column vector of data, and m is the solution vector of unknowns. If N > M, a solution for m can be determined by minimizing a function of I(Am - dll, using standard techniques and taking into account any uncertainty in the data vector d. This method is used to estimate the velocity field in the Californian plate-boundary zone. A weighted least-squares solution is found that predicts tectonic rotations. Simple tests illustrate the effect on the derived velocity field of changing features of the velocity model.

A simple method for estimating the horizontal velocity field in wide zones of active deformation-II. Examples from New Zealand, Central Asia and Chile

SUMMARY Lamb (1994) has described a simple method for estimating the horizontal velocity field in a wide zone of active deformation. This method is now used to investigate the horizontal velocity or displacement field in part of the New Zealand plate-boundary zone, Central Asia and southern central Chile. In each region, a range of weighted least-squares solutions have been found, depending on assumptions about aspects of the internal deformation. In the New Zealand example, observed tectonic rotations are successfully predicted. Solutions for Central Asia place constraints on the manner in which the northward convergence of India into Eurasia is accommodated. Horizontal surface displacement fields are determined for the elastic rebound during the 1960 Chilean great earthquake, based on available retriangulation data and assumptions about the slip vector on the fault. These solutions suggest, but do not prove, a nearly pure thrust event for this earthquake.

Approximation in nonhierarchic system optimization

This paper reports on the effectiveness of a nonhierarchic system optimization algorithm in application to complex coupled systems problems. A second-order nonhierarchic system optimization algorithm developed in earlier studies is modified in this study to provide for individual constraint/state modeling. A cumulative constraint formulation was used in previous implementation studies. The test problems in this study are each complex coupled systems. Complex coupled systems require an iterative solution strategy to evaluate system states. Nonhierarchic algorithm development is driven by these types of problems, and their study is imperative. The algorithm successfully optimizes each of the complex coupled systems. A significant reduction in the number of system analyses required for optimization is observed as compared with conventional optimization using the generalized reduced-gradient method. in the design database. The design database stores design site information generated during the subspace optimizations. A quadratic polynomial approximation to the design is formed using the strategy of Vanderplaats. 6 A weighted least-squares solution strategy is employed to solve for the second-order terms in Vanderplaats' strategy. Exact data in the design data- base are more heavily weighted in the least-squares solution procedure. The resulting quadratic polynomial forms the basis function of accumulated approximation replacing the linear basis used in the original formulation. In Renaud and Gabriele5 improved convergence is observed for the welded beam test problem. The improved convergence is attributed to the im- proved accuracy of cumulative constraint approximations when using second-order-based approximating functions. Additional studies using the second-order-based coordina- tion procedure of system approximation indicated that replac- ing the cumulative constraints with their component con- straints may improve algorithm performance. Implementation of the second-order-based coordination procedure of system approximation was less effective in reducing cycling when applied to the Golinski speed reducer problem. The speed reducer cumulative constraints were composed of a large num- ber of individual constraints as compared with the cumulative constraints in the welded beam test problem. With a larger number of individual constraints assigned to a cumulative constraint, it will more likely undergo a change in its active set during the coordination procedure. It is difficult to approxi- mate these changes in the cumulative constraints. Inaccurate cumulative constraint approximations reduce algorithm per- formance and delay convergence. Approximating individual constraints/states in the coor-

Image restoration in computed tomography: the spatially invariant point spread function

Image restoration to deblur smoothing caused by the finite-size X-ray beam profile for a simulated computed tomography (CT) system is presented. Three simple image restoration methods are compared when the point-spread-function (PSF) is spatially invariant. In the first restoration method, an iterative least squares solution, regularized with the image norm and constrained by the boundary of the object, is obtained from the projection data. In the second method, a Wiener filter, designed using the power spectrum of CT noise, is applied to the reconstructed CT image. The third method obtains a weighted least-squares solution, by iteration, from the reconstructed CT image; the solution is regularized with the weighted image norm. Restored images were compared with the image obtained using filtered backprojection method. Differences between these images were evaluated qualitatively.

Weighted and unweighted least-squares solutions of linear matrix equations

Some results from the theory of minimization of vector quadratic forms (subjected to linear restrictions) are used to obtain particular solutions of the usual types of linear matrix equations. Following the practice in the mathematical literature, unweighted least-squares solutions are obtained. Also a computational procedure for obtaining weighted least-squares solutions is indicated. An answer to a question raised by Greville is supplied.

Constrained Power System State Estimation

Abstract The structure of electrical power networks is such that there are a number of nodes for which the power injections are known to be exactly zero. These zero-power injections can be treated as equality constraints in the power system state estimation problem. This paper reviews the methods for handling equality constraints and presents a novel technique based on a direct elmination of the dependent variables. A general algorithm is presented which can be adapted to any Weighted Least-Squares (WLS) solution technique. A method based on Givens rotations is particularly suitable, and is described in detail in the paper. Since this method tends to be time-consuming, a hybrid method is also presented which can substantially improve the solution time and reduce storage requirements. The paper also presents some results on the use of the commonly employed weighting method. The conventional and proposed algorithms are tested using an IEEE power system and a New Brunswick Electric Commission transmission network.

A BASIC program for orthogonal polynomials and retrieval of regression coefficients for original model

A fast, short method for calculating orthogonal polynomials has been programmed in BASIC. This is combined with the retrieval of b coefficients for the original, non-orthogonal model in, for example, a curvilinear regression. The program may be used alone, or in conjunction with a package of BASIC programs for weighted least-squares solutions, to which it forms an update. Transformation of the columns of an X, or design matrix, to orthogonal vectors can detect linear dependence among the columns, as well as supplying orthogonal polynomials for regression situations with unequal increments in x values, the values of the independent variable. Retrieval of the b coefficients for the original X matrix prior to transformation allows the user to insert any x values to obtain the corresponding predicted y (dependent variable) values.

Parameter estimation via the kalman filter

A method for parameter estimation is presented using the Kalman filter with appropriate initial conditions. The filter solution is shown to approximate the minimum-norm weighted least-squares solution to any desired accuracy during all phases of estimation. Furthermore, the computations are identical for each measurement, irrespective of whether a minimal observable data set has been established. This procedure contrasts with other techniques for parameter estimation that require additional computation when the process is unobservable.

Gravity field of Jupiter and its satellite from Pioneer 10 and Pioneer 11 tracking data

Accurate two-way Doppler tracking data obtained from Pioneers 10 and 11 during their Jupiter encounters are analyzed to yield significantly improved values for the masses of the Galilean satellites, the harmonic coefficients of Jupiter, and the mass of the planet. The spacecraft trajectories relative to Jupiter are discussed, and nongravitational spacecraft accelerations are taken into account. Gravity results are derived from a simultaneous iterative weighted least-squares solution for the orbital elements of seven bodies, the masses of Jupiter and the Galilean satellites, the right ascension and declination of Jupiter's instantaneous pole relative to the 1950.0 mean earth equator and equinox, the Jovian gravity harmonic coefficients, the mass of a hypothetical mascon at the Great Red Spot, and the nongravitational-acceleration parameters. Four separate solutions are determined, and the best numerical values are given for the ratios of the masses of the four Galilean satellites to the mass of Jupiter; the ratio of the sun's mass to that of the Jupiter system; the second, third, fourth, fifth, and sixth zonal harmonic coefficients of Jupiter; the sectoral harmonics; and the ratio of the hypothetical mascon's mass to that of Jupiter.

Overdetermined photoelastic solutions using least squares

The least-squares method is presented for obtaining an overdetermined solution from the various relationships available in photoelastic stress analysis. Equations are presented for incorporating the various relations into a weighted least-squares solution and advantages of this method are illustrated in two example problems. In addition to the greater accuracy possible in the overdetermined solution, the method permits weighting of the varied information and eliminates the need to separately determine the stress-optical constant. Variations of the method are discussed.