A LEAST‐SQUARES METHOD OF INTERPRETING MAGNETIC ANOMALIES CAUSED BY TWO‐DIMENSIONAL STRUCTURES

Geophysics ◽  
1969 ◽  
Vol 34 (1) ◽  
pp. 65-74 ◽  
Author(s):  
William W. Johnson
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Magnetic Anomalies ◽  
Descent Method ◽  
The Body ◽  
Steepest Descent Method ◽  
Two Dimensional ◽  
Gauss Method ◽  
Least Squares Estimates

The equations relating the magnetic anomalies to the shape and susceptibility of a body are nonlinear with respect to the coordinates describing the shape. Therefore, iterative procedures must be used to obtain least‐squares estimates of the body coordinates. One method in general use for obtaining nonlinear least‐squares estimates is the Gauss method. This method often fails when the initial values for the structures and susceptibilities do not adequately account for the magnetic anomalies. Another method known as the steepest descent method generally converges to a solution; however, a large number of iterations are required. A method suggested by Marquardt (1963) incorporates the best features of the previous methods. In this paper the Marquardt method is applied to the interpretation of magnetic anomalies. For this purpose the two‐dimensional formulas derived by Talwani and Heirtzler (1964) are used to relate the geometry of a body to the resulting magnetic anomalies. The procedure efficiently controls the amount of change made to an interpreted structure at each iteration, assuring rapid convergence to a solution which satisfies the observed data better in the least‐squares sense than does the initial solution. The method is applied to representative problems.

The solution of nonlinear inverse problems and the Levenberg-Marquardt method

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. W1-W16 ◽  
Author(s):  
Jose Pujol
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Gravity Data ◽  
Nonlinear Problems ◽  
Ordinary Least Squares ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Levenberg Marquardt ◽  
Damped Least Squares

Although the Levenberg-Marquardt damped least-squares method is an extremely powerful tool for the iterative solution of nonlinear problems, its theoretical basis has not been described adequately in the literature. This is unfortunate, because Levenberg and Marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method. The idea of damping the solution was introduced by Levenberg, who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively. This result is not obvious, although it is taken for granted. Moreover, Levenberg derived a solution more general than the one currently used. Marquardt started with the current equation and showed that it interpolates between the ordinary least-squares-method and the steepest-descent method. In this tutorial, the two papers are combined into a unified presentation, which will help the reader gain a better understanding of what happens when solving nonlinear problems. Because the damped least-squares and steepest-descent methods are intimately related, the latter is also discussed, in particular in its relation to the gradient. When the inversion parameters have the same dimensions (and units), the direction of steepest descent is equal to the direction of minus the gradient. In other cases, it is necessary to introduce a metric (i.e., a definition of distance) in the parameter space to establish a relation between the two directions. Although neither Levenberg nor Marquardt discussed these matters, their results imply the introduction of a metric. Some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth. Finally, the work done by early researchers that rediscovered the damped least-squares method is put into a historical context.

Frequency domain least‐squares inversion of thick dike magnetic anomalies using Marquardt algorithm

Geophysics ◽  
1981 ◽  
Vol 46 (12) ◽  
pp. 1745-1748 ◽  
Author(s):  
K. K. Khurana ◽  
S. V. Seshagiri Rao ◽  
P. C. Pal
Keyword(s):  
Least Squares ◽  
Frequency Domain ◽  
Least Squares Method ◽  
Source Parameters ◽  
Transformation Method ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Estimation Of Parameters ◽  
Space Domain ◽  
Marquardt Algorithm

The elegance and ease with which the Fourier transformation method can be employed in the interpretation of potential field geophysical data (Dean, 1958; Gudmundsson, 1966) has resulted in the formulation of a number of frequency‐domain approaches for interpreting the magnetic effects of specific source geometries, e.g., Bhattacharyya (1966), Sengupta (1974), Sengupta and Das (1975), Bhattacharyya and Leu (1975, 1977), etc. In these techniques, the parameters of interest are estimated from the characteristics of the amplitude and phase spectra, a process that is often beset with several problems which adversely affect the estimation of parameters. These problems arise partly from the ill‐defined nature of characteristic points in the spectra, either due to spectral noise or due to inadequate digitization spacing, and partly from the fact that the errors in the estimation of certain parameters which are used for subsequent estimation of the other parameters usually cause added spectral distortions. Rao et al. (1978) advocated the use of end corrections to overcome these problems. An effective alternative would be to incorporate the least‐squares method in the interpretational process. This necessitates iteration owing to the nonlinear nature of equations relating the source parameters and the Fourier spectra of corresponding anomalies. While it is tempting to employ the iteration scheme formulated by Gauss in 1821 for this purpose, as has indeed been done in a number of space‐domain approaches (e.g., Hall, 1958; Bosum, 1968; Al‐Chalabi, 1970; Rao et al., 1973; Won, 1981), the convergence in this approach requires the initially assumed values of the parameters to be very close to the final solution. The steepest descent method, an alternative to this, has an extremely slow convergence. It is shown here, with the example of a two‐dimensional (2-D) structure, that Marquardt’s algorithm (Marquardt, 1963), which offers a compromise between these two iteration schemes and has been effectively employed by Johnson (1969) and Pedersen (1977) in their space‐domain interpretation schemes, provides an efficient approach to accomplish the iteration in the case of frequency‐domain magnetic interpretation.

A modified tangent-gas approximation for two-dimensional steady flow

1958 ◽  
Vol 4 (6) ◽  
pp. 600-606 ◽  
Author(s):  
G. Power ◽  
P. Smith
Keyword(s):  
Least Squares ◽  
Steady Flow ◽  
Variational Approach ◽  
Least Squares Method ◽  
Two Dimensional ◽  
Subsonic Flows ◽  
Von Karman ◽  
Straight Line ◽  
Volume Relationship ◽  
Pressure Volume Relationship

A set of two-dimensional subsonic flows past certain cylinders is obtained using hodograph methods, in which the true pressure-volume relationship is replaced by various straight-line approximations. It is found that the approximation obtained by a least-squares method possibly gives best results. Comparison is made with values obtained by using the von Kármán-Tsien approximation and also with results obtained by the variational approach of Lush & Cherry (1956).

scholarly journals MIM and nonlinear least‐squares inversions of AEM data in Barataria basin, Louisiana

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1126-1131 ◽  
Author(s):  
Melissa Whitten Bryan ◽  
Kenneth W. Holladay ◽  
Clyde J. Bergeron ◽  
Juliette W. Ioup ◽  
George E. Ioup
Keyword(s):  
Least Squares ◽  
Water Depth ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Image Method ◽  
Water Conductivity ◽  
Near Surface ◽  
Barataria Basin ◽  
Airborne Electromagnetic Survey ◽  
Nonlinear Least Squares Method

An airborne electromagnetic survey was performed over the marsh and estuarine waters of the Barataria basin of Louisiana. Two inversion methods were applied to the measured data to calculate layer thicknesses and conductivities: the modified image method (MIM) and a nonlinear least‐squares method of inversion using two two‐layer forward models and one three‐layer forward model, with results generally in good agreement. Uniform horizontal water layers in the near‐shore Gulf of Mexico with the fresher (less saline, less conductive) water above the saltier (more saline, more conductive) water can be seen clearly. More complex near‐surface layering showing decreasing salinity/conductivity with depth can be seen in the marshes and inland areas. The first‐layer water depth is calculated to be 1–2 m, with the second‐layer water depth around 4 m. The first‐layer marsh and beach depths are computed to be 0–3 m, and the second‐layer marsh and beach depths vary from 2 to 9 m. The first‐layer water conductivity is calculated to be 2–3 S/m, with the second‐layer water conductivity around 3 to 4 S/m and the third‐layer water conductivity 4–5 S/m. The first‐layer marsh conductivity is computed to be mainly 1–2 S/m, and the second‐ and third‐layer marsh conductivities vary from 0.5 to 1.5 S/m, with the conductivities decreasing as depth increases except on the beach, where layer three has a much higher conductivity, ranging up to 3 S/m.

The Strength Evaluation and Reliability Analysis of Periodic Symmetric Struts Support

2011 ◽  
Vol 462-463 ◽  
pp. 1164-1169
Author(s):  
Jing Xiang Yang ◽  
Ya Xin Zhang ◽  
Mamtimin Gheni ◽  
Ping Ping Chang ◽  
Kai Yin Chen ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Stress Distribution ◽  
Least Squares ◽  
Reliability Analysis ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Distribution Model ◽  
Different Types ◽  
Nonlinear Least Squares Method

In this paper, strength evaluations and reliability analysis are conducted for different types of PSSS(Periodically Symmetric Struts Supports) based on the FEA(Finite Element Analysis). The numerical models are established at first, and the PMA(Prestressed Modal Analysis) is conducted. The nodal stress value of all of the gauss points in elements are extracted out and the stress distributions are evaluated for each type of PSSS. Then using nonlinear least squares method, curve fitting is carried out, and the stress probability distribution function is obtained. The results show that although using different number of struts, the stress distribution function obeys the exponential distribution. By using nonlinear least squares method again for the distribution parameters a and b of different exponential functions, the relationship between number of struts and distribution function is obtained, and the mathematical models of the stress probability distribution functions for different supports are established. Finally, the new stress distribution model is introduced by considering the DSSI(Damaged Stress-Strength Interference), and the reliability evaluation for different types of periodically symmetric struts supports is carried out.

A NONLINEAR LEAST‐SQUARES METHOD FOR SEISMIC REFRACTION MAPPING—PART I: ALGORITHM AND PROCEDURE

Geophysics ◽  
1972 ◽  
Vol 37 (2) ◽  
pp. 260-272 ◽  
Author(s):  
Leonidas C. Ocola
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Seismic Refraction ◽  
Vertical Plane ◽  
Nonlinear Least Squares ◽  
Inversion Method ◽  
Almost Everywhere ◽  
Isotropic Media ◽  
Time Term ◽  
Nonlinear Least Squares Method

An iterative inversion method (Reframap) based on the kinematic properties of critically refracted waves is developed. The method is based on ray tracing and assumes homogeneous and isotropic media and ray paths confined to a vertical plane through each source‐detector pair. Unlike the earlier Profile or Time‐Term Methods, no restrictions are imposed on interface topography except that it be continuous almost everywhere (in the mathematical sense). As in the preexisting methods, more observations than unknowns are assumed. The algorithm and procedure, on which the Reframap Method is based, generate apparent dips for each source detector pair at the noncritical interfaces from the slope of a least‐squares line approximation to the interface functional in the neighborhood of each refraction point. In turn, the dip and path along the critical refractor is, at every iteration, pairwise approximated by a line through the critical refracting points. The incidence angles are computed recursively by Snell’s law. The solution of the overdetermined, nonlinear multiple refractor time‐distance system of simultaneous equations is sought by Marquardt’s algorithm for least‐squares estimation of critical refractor velocity and vertical thickness under each element.

scholarly journals Methods for Fitting the Limit State Function of the Residual Strength of Damaged Ships

2022 ◽  
Vol 10 (1) ◽  
pp. 102
Author(s):  
Zhiyao Zhu ◽  
Huilong Ren ◽  
Xiuhuan Wang ◽  
Nan Zhao ◽  
Chenfeng Li
Keyword(s):  
Least Squares ◽  
Residual Strength ◽  
Least Squares Method ◽  
Limit State ◽  
Nonlinear Least Squares ◽  
State Function ◽  
Limit State Function ◽  
Distribution Models ◽  
Sample Distribution ◽  
Fitting Method

The limit state function is important for the assessment of the longitudinal strength of damaged ships under combined bending moments in severe waves. As the limit state function cannot be obtained directly, the common approach is to calculate the results for the residual strength and approximate the limit state function by fitting, for which various methods have been proposed. In this study, four commonly used fitting methods are investigated: namely, the least-squares method, the moving least-squares method, the radial basis function neural network method, and the weighted piecewise fitting method. These fitting methods are adopted to fit the limit state functions of four typically sample distribution models as well as a damaged tanker and damaged bulk carrier. The residual strength of a damaged ship is obtained by an improved Smith method that accounts for the rotation of the neutral axis. Analysis of the results shows the accuracy of the linear least-squares method and nonlinear least-squares method, which are most commonly used by researchers, is relatively poor, while the weighted piecewise fitting method is the better choice for all investigated combined-bending conditions.

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