Densities determined from surface and subsurface gravity measurements

Geophysics ◽  
1981 ◽  
Vol 46 (11) ◽  
pp. 1568-1571 ◽  
Author(s):  
B. A. Sissons
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Average Density ◽  
Laboratory Measurements ◽  
Density Estimates ◽  
Direct Inversion ◽  
Gravity Measurements ◽  
The Difference ◽  
Made In

A least‐squares method for the direct inversion of surface and subsurface gravity measurements to obtain in situ density estimates is presented. The method is applied to a set of measurements made in a tunnel through the flank of an andesitic volcano. Densities obtained are [Formula: see text] for material in the top 100 m increasing to [Formula: see text] at about 200 m depth. The average density for rocks penetrated by the tunnel is, from laboratory measurements, [Formula: see text] i.e., about 4 percent higher. The difference is ascribed to joints and voids present in situ and not sampled in the laboratory specimens.

scholarly journals On the Geometric Mean Method for Incomplete Pairwise Comparisons

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1873
Author(s):  
Konrad Kułakowski
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Geometric Mean ◽  
Computational Method ◽  
Pairwise Comparisons ◽  
Similar Relationship ◽  
Final Solution ◽  
The Difference ◽  
The Relationship ◽  
Theoretical Considerations

One of the most popular methods of calculating priorities based on the pairwise comparisons matrices (PCM) is the geometric mean method (GMM). It is equivalent to the logarithmic least squares method (LLSM), so some use both names interchangeably, treating it as the same approach. The main difference, however, is in the way the calculations are done. It turns out, however, that a similar relationship holds for incomplete matrices. Based on Harker’s method for the incomplete PCM, and using the same substitution for the missing entries, it is possible to construct the geometric mean solution for the incomplete PCM, which is fully compatible with the existing LLSM for the incomplete PCM. Again, both approaches lead to the same results, but the difference is how the final solution is computed. The aim of this work is to present in a concise form, the computational method behind the geometric mean method (GMM) for an incomplete PCM. The computational method is presented to emphasize the relationship between the original GMM and the proposed solution. Hence, everyone who knows the GMM for a complete PCM should easily understand its proposed extension. Theoretical considerations are accompanied by a numerical example, allowing the reader to follow the calculations step by step.

scholarly journals Effectiveness of in situ P-wave measurements in monuments: examples from Greece

2000 ◽  
Vol 22 ◽  
Author(s):  
B. Christaras
Keyword(s):  
Modulus Of Elasticity ◽  
Building Stone ◽  
P Wave ◽  
Mechanical Anisotropy ◽  
Laboratory Measurements ◽  
S Wave ◽  
Wave Velocities ◽  
Wave Measurements ◽  
Made In

P and S wave velocities can be used for both in situ and laboratory measurements of stones. These methods are used for studying such properties as mechanical anisotropy and modulus of elasticity. In this paper, the P-wave velocities were used for the estimation of the depth of weathered or artificially consolidated layers as well as the depth of cracks developed at the surface of the building stone. This estimation was made in relation to the lithology and texture of the materials, given that in many cases different lithological data create similar diagrams. All tests were carried out on representative monuments in Greece.

scholarly journals Comparison of Relaxation Modulus Converted from Frequency- and Time-Dependent Viscoelastic Functions through Numerical Methods

2018 ◽  
Vol 8 (12) ◽  
pp. 2447 ◽  
Author(s):  
Weiguang Zhang ◽  
Bingyan Cui ◽  
Xingyu Gu ◽  
Qiao Dong
Keyword(s):  
Least Squares ◽  
Collocation Method ◽  
Creep Compliance ◽  
Least Squares Method ◽  
Relaxation Modulus ◽  
Prony Series ◽  
Significant Difference ◽  
Time Region ◽  
The Difference ◽  
Reduced Time

Due to the difficulty of obtaining relaxation modulus directly from experiments, many interconversion methods from other viscoelastic functions to relaxation modulus were developed in previous years. The objectives of this paper were to analyze the difference of relaxation modulus converted from dynamic modulus and creep compliance and explore its potential causes. The selected methods were the numerical interconversions based on Prony series representation. For the dynamic to relaxation conversion, the time spectrum was determined by the collocation method. Meanwhile, for the creep to relaxation conversion, both the collocation method and least squares method were adopted to perform the Laplace transform. The results show that these two methods do not present a significant difference in estimating relaxation modulus. Their difference mostly exists in the transient reduced time region. Calculating the average of two methods is suggested to avoid great deviation of single experiment. To predict viscoelastic responses from creep compliance, the collocation method yields comparable results to the least squares method. Thus, simply-calculated collocation method may be preferable in practice. Further, the master curve pattern is sensitive to the Prony series coefficients. The difference in transient reduced time region may be attributed to the indeterminate Prony series coefficients.

scholarly journals Recent Advances in Sire Evaluation Methods: A Review

2021 ◽  
Author(s):  
Amol Jagannath Talokar ◽  
Harshit Kumar ◽  
Arnav Mehrotra ◽  
Kaiho Kaisa ◽  
K.A. Saravanan ◽  
...  
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Error Variance ◽  
Multiple Trait ◽  
Index Method ◽  
Linear Unbiased Prediction ◽  
Sire Evaluation ◽  
Derivative Free ◽  
Best Linear Unbiased ◽  
Made In

This review deals with the various traditional and recent methods of sire evaluation. A sire evaluation is a method of prediction of sire’s next-generation produced by breeding with specified females and creating their records in a specific environment. Breeding worth of progeny tested sires are obtained through sire index method that assigns ranking to each sire based on their genetic merit. Numerous sire indices are broadly classified in two type’s viz., indices which are purely meant for ranking purposes and those which, besides, provide an estimate of the breeding worth of each sire. Sires can be evaluated in single or multiple herds. The statistical equations represent sire evaluation as linear or non-linear methods. Numerous methods can be incorporated as the advances are made in sire evaluation based on data structures, breeding approaches and selection methods. There are various approaches of sire evaluation such as Least Squares Method (LSM), Simple Regressed Least Squares (SRLS), Best Linear Unbiased Prediction (BLUP) and Derivative-Free Restricted Maximum Likelihood (DFREML) for single as well as multiple trait models which can be used to derive genetic worth of an individual. An efficient method of sire evaluation shows minimum within sire variance or error variance. 

scholarly journals IN INTERPOLATING THE VALUES OF FUNCTIONS SPECIFIED BY THE TABLE EFFECTIVENESS OF THE LEAST SQUARES METHOD

2020 ◽  
pp. 68-71
Author(s):  
М.Е. Eskaliyev ◽  
◽  
А.А. Masimgazieva ◽  
N.A. Nurgali ◽  
◽  
...  
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Main Idea ◽  
Gram Matrix ◽  
Method Of Least Squares ◽  
Newton Interpolation ◽  
Wide Range ◽  
Mathematical Formulas ◽  
The Difference ◽  
Interpolation Polynomials

The article provides a general algorithm of the method of least squares (OLS) for the compilation of a program account, taking into account the features of the Gram matrix. In fact, the difference and advantage of the OLS from the long-known Langrange and Newton interpolation polynomials is emphasized. Using the general MNC algorithm, the square version of the MNC is considered. The characteristic of the full algorithm of the square version is given using special selected mathematical formulas. The scope of OLS in household and practical computational problems is indicated. The application of the least squares method has a wide range, especially for geographical forecasts, hydrometeorological control, and dosage of geological resources. Therefore, for applied calculations, DVI is used for more accurate calculation of approximate values of functions that are suitable in some values and are presented in the form of tables. Its main idea is to create a function and correct deviations caused by errors made during measurement.

Monopulse Estimation of Unknown Polarization for Conformal Phased Array with Arbitrarily Oriented Dipoles

2021 ◽  
Author(s):  
Yuhao Deng ◽  
Julan Xie ◽  
Zishu He ◽  
Jun Li
Keyword(s):  
Least Squares ◽  
Phased Array ◽  
Least Squares Method ◽  
Likelihood Function ◽  
Hessian Matrix ◽  
Mean Square ◽  
Similar Technique ◽  
Dipole Orientation ◽  
Initial Direction ◽  
The Difference

Abstract In this paper, a novel monopulse estimator is proposed to surmount obstacles caused by unknown polarized pattern and the difference among each dipole orientation. Polarized pattern often alters the phased array response and we could hardly recover it if we known nothing about polarized parameters. The sum and difference beamforming of conformal phased array is affected due to the difference among each dipole orientation. Therefore the conventional monopulse estimator is dumped in this circumstance. The method proposed in this paper is a remarkable estimator based on maximum likelihood methodology. In this method, polarized parameters are considered as a part of desired signal and the least-squares solution of desired signal is obtained. With the desired signal solution, the likelihood function with respect to direction is derived at first. Then from the above, Jacobian and Hessian matrix of likelihood function is deduced. The boresight is considered as the initial direction value and the estimator of desired signal direction is obtained by Newton's formula. Finally, the polarized parameters are estimated by least-squares method using the direction estimator. The root-mean-square error (RMSE) of angle estimation is acceptable when prior polarized information is completely unknown. Polarized parameters are estimated by similar technique after we find out azimuth and elevation. Our research fills a gap of monopulse estimation with arbitrary polarization and diverse dipole arrangement.

Electromagnetic seismograph constants by least-squares inversion

1969 ◽  
Vol 59 (3) ◽  
pp. 1335-1348
Author(s):  
B. J. Mitchell ◽  
M. Landisman
Keyword(s):  
Least Squares ◽  
Love Wave ◽  
Least Squares Method ◽  
Linear Trend ◽  
Wave Train ◽  
Vertical Component ◽  
Long Period ◽  
Electromagnetic Seismograph ◽  
Before And After

abstract A least-squares method has been developed to determine the free periods and damping constants of an electromagnetic seismograph from its calibration pulse. The resulting values are correct to within a few tenths of one per cent for synthetic calibration pulses, even when moderate levels of microseismic noise are present. The seismograph constants of the long-period vertical component at Dallas, Texas were determined from in situ measurements and compared with those calculated from the calibration pulse. The results agreed to within four per cent or better for the free periods and damping constants, after correcting for the linear trend and the base coordinate system of the observed pulse. The values differed by as much as 30 per cent when the linear trend and base coordinate corrections were ignored. Two sets of instrumental constants from an unmatched pair of horizontal seismographs were determined from their calibration pulses. Directions of particle motion for a Love wave train recorded on the same instruments were computed before and after instrumental corrections; the results differed by as much as 8 degrees.

A DENSITY DETERMINATION BY UNDERGROUND GRAVITY MEASUREMENTS IN MICHIGAN

Geophysics ◽  
1963 ◽  
Vol 28 (4) ◽  
pp. 663-664
Author(s):  
George B. Secor ◽  
Howard J. Meyer ◽  
William J. Hinze
Keyword(s):  
Sedimentary Rocks ◽  
Michigan Basin ◽  
Established Method ◽  
Density Determination ◽  
The Earth ◽  
Gravity Measurements ◽  
Made In

The in situ density of an 1,100‐ft section of Paleozoic sedimentary rocks in the southeastern portion of the Michigan Basin was determined by the established method of observing the change in gravity over a known vertical interval within the earth. The measurements were made in and adjacent to the Detroit Mine (Sec. 33, T2S, R11E) of the International Salt Company in Detroit, Michigan, which is the only deep shaft mine in the Southern Peninsula of Michigan.

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