scholarly journals Some remarks on the paper by B. G. Marsden “An attempt to reconcile the dynamical and radar determinations of the Astronomical Unit”

1965 ◽  
Vol 21 ◽  
pp. 237-239
Author(s):  
Eugene Rabe
Keyword(s):  
Least Squares ◽  
Infinite Series ◽  
Infinite Number ◽  
Least Squares Method ◽  
Fundamental Principle ◽  
Astronomical Unit ◽  
Number Of Solutions ◽  
Least Squares Solution ◽  
Minimum Value

While Marsden's solution C leaves residuals with the relatively small [vv] of 13.73, it should be realized that this representation of the observations of Eros does not satisfy the fundamental principle of the least squares method, in so far as the associated value of [vv] is not a minimum with respect to small arbitrary deviations from solution C. As a matter of fact, there is an infinite number of “solutions” with [vv] between the 13.73 of Marsden's solution C and the 8.66 of his solution A, each of these being associated with a certain arbitrarily prescribed value of the mass of Mars and with a related mass of Earth + Moon. Of this infinite series of solutions, only solution A is a least squares solution in the true sense, with a minimum value of [vv]. This can be seen and verified as follows.

scholarly journals Robust Estimations of Current Velocities with Four-Beam Broadband ADCPs

2009 ◽  
Vol 26 (12) ◽  
pp. 2642-2654 ◽  
Author(s):  
M. Gilcoto ◽  
Emlyn Jones ◽  
Luis Fariña-Busto
Keyword(s):  
Least Squares ◽  
Covariance Matrix ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Robust Solution ◽  
Least Squares Solution ◽  
Weighted Least Squares Method ◽  
Radial Beam ◽  
Least Squares Approach ◽  
Variance Covariance Matrix

Abstract An extended explanation of the hypothesis and equations traditionally used to transform between four-beam ADCP radial beam velocities and current velocity components is presented. This explanation includes a dissertation about the meaning of the RD Instrument error velocity and a description of the standard beam-to-current components transformation as a least squares solution. Afterward, the variance–covariance matrix associated with the least squares solution is found. Then, a robust solution for transforming radial beam velocities into current components is derived under the formality of a weighted least squares approach. The associated variance–covariance matrix is also formulated and theoretically proves that the modulus of its elements will be generally lower than the corresponding modulus of the variance–covariance matrix associated with the standard least squares solution. Finally, a comparison between the results obtained using the standard least squares solution and the results of the weighted least squares method, using a high-resolution ADCP dataset, is presented. The results show that, in this case, the weighted least squares solution provides variance estimations that are 4% lower over the entire record period (8 days) and 7% lower during a shorter, more energetic period (12 h).

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method

An Augmented Iteratively Re-weighted and Refined Least Squares Method for lp Minimization Problem in Inverse Modeling of Potential Field Magnetic Data

2020 ◽  
Vol 1 (3) ◽  
Author(s):  
Maysam Abedi
Keyword(s):  
Magnetic Field ◽  
Magnetic Susceptibility ◽  
Least Squares ◽  
Minimization Problem ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Porphyry Copper ◽  
Magnetic Data ◽  
Magnetic Field Data

The presented work examines application of an Augmented Iteratively Re-weighted and Refined Least Squares method (AIRRLS) to construct a 3D magnetic susceptibility property from potential field magnetic anomalies. This algorithm replaces an lp minimization problem by a sequence of weighted linear systems in which the retrieved magnetic susceptibility model is successively converged to an optimum solution, while the regularization parameter is the stopping iteration numbers. To avoid the natural tendency of causative magnetic sources to concentrate at shallow depth, a prior depth weighting function is incorporated in the original formulation of the objective function. The speed of lp minimization problem is increased by inserting a pre-conditioner conjugate gradient method (PCCG) to solve the central system of equation in cases of large scale magnetic field data. It is assumed that there is no remanent magnetization since this study focuses on inversion of a geological structure with low magnetic susceptibility property. The method is applied on a multi-source noise-corrupted synthetic magnetic field data to demonstrate its suitability for 3D inversion, and then is applied to a real data pertaining to a geologically plausible porphyry copper unit.  The real case study located in  Semnan province of  Iran  consists  of  an arc-shaped  porphyry  andesite  covered  by  sedimentary  units  which  may  have  potential  of  mineral  occurrences, especially  porphyry copper. It is demonstrated that such structure extends down at depth, and consequently exploratory drilling is highly recommended for acquiring more pieces of information about its potential for ore-bearing mineralization.

Theoretical accuracy of the last squares method in the isotope dilution analysis

1984 ◽  
Vol 49 (4) ◽  
pp. 805-820
Author(s):  
Ján Klas
Keyword(s):  
Least Squares ◽  
Isotope Dilution ◽  
Least Squares Method ◽  
Isotope Dilution Analysis ◽  
Straight Line ◽  
The One ◽  
Two Parameter ◽  
Theoretical Accuracy

The accuracy of the least squares method in the isotope dilution analysis is studied using two models, viz a model of a two-parameter straight line and a model of a one-parameter straight line.The equations for the direct and the inverse isotope dilution methods are transformed into linear coordinates, and the intercept and slope of the two-parameter straight line and the slope of the one-parameter straight line are evaluated and treated.

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