Transformation of nonlinear problems into linear ones applied to the magnetic field of a two‐dimensional prism

Geophysics ◽  
1989 ◽  
Vol 54 (1) ◽  
pp. 114-121 ◽  
Author(s):  
João B. C. Silva
Keyword(s):  
Linear Problem ◽  
Least Squares Method ◽  
Synthetic Data ◽  
Nonlinear Problems ◽  
Parameter Estimate ◽  
Initial Guess ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Magnetization Direction ◽  
Discrete Values

I present a magnetic interpretation method which transforms into a linear problem the nonlinear problem of obtaining the geometric and position parameters of a two‐dimensional vertical, infinite prism. The magnetization, the only linear parameter, becomes nonlinear after the transformation. By assuming a few discrete values over a prescribed interval for the magnetization, I obtain several solutions for the geometric and position parameters. By storing only the extreme solutions, bounds for each parameter are produced. The method was applied to synthetic anomalies due to isolated and interfering sources for which robust alternatives performed better than the least‐squares method. The correlation between the magnetization and the prism width is the most important factor controlling ambiguity of parameters. The horizontal position is the least affected parameter, followed by the depth to the top of the prism. Application to a real anomaly confirmed the results from synthetic data, except for a greater uncertainty in the estimation of the horizontal position. The uncertainty results from the requirement in the present method that the observations be reduced to the pole; an imprecise knowledge of the magnetization direction distorts the position, which is highly correlated with the magnetization inclination. Because the estimation of the position, depth, and width is transformed into a linear problem, the method is simple, fast, and independent of the initial guess. The method might, therefore, be useful in automatic interpretation of basement relief. By producing bounds for each parameter estimate, an analysis of parameter precision and ambiguity is also possible.

Dynamic Response of Surface-Moored Vertical Barrier to Wave Action by Analog and Digital Simulation

1974 ◽  
Vol 96 (1) ◽  
pp. 335-342
Author(s):  
J. R. Fowler ◽  
E. I. Bailey
Keyword(s):  
Theoretical Model ◽  
Digital Simulation ◽  
Nonlinear Problems ◽  
Analog Computer ◽  
Two Dimensional ◽  
Test Program ◽  
Analog Simulation ◽  
Vertical Barrier ◽  
Amplitude Data ◽  
Lab Test

The two-dimensional dynamics of an oil containment barrier, which was designed to have very low tensile loads due to current and waves, were simulated with a theoretical model. The model was solved on both analog and digital computers, and a lab test program conducted to verify the model. For nonlinear problems such as this, for which “exact” solutions do not exist, the analog computer has many advantages, principally rapid parameter studies and convenient plotting output, plus giving the engineer a real time “feel” for the problem. The problem treated here was especially well-suited to analog simulation. Charts and graphs present maximum force and amplitude data, and experimental verification of the solution was obtained from wave tank studies.

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).

Asymptotic analysis of some two-dimensional diffusion problems

1995 ◽  
Vol 448 (1933) ◽  
pp. 213-236 ◽  
Keyword(s):  
Linear Problem ◽  
Vacancy Concentration ◽  
Surface Concentration ◽  
Two Dimensional ◽  
Surface Source ◽  
Dimensional Diffusion ◽  
Equilibrium Vacancy ◽  
Equilibrium Vacancy Concentration ◽  
Dimensional Surface ◽  
Diffusion Problems

In this paper we discuss two-dimensional surface source and implant problems for a substitutional-interstitial diffusion model. We present asymptotic solutions in the limit of the surface concentration of impurity (or peak concentration of the implant) being far greater than the equilibrium vacancy concentration. Using leading order composite solutions we plot contours of constant impurity concentration. Some of these contours differ markedly from those of the corresponding linear problem, having the ‘bird’s beak’ shape which is frequently observed in experiments. We also discuss a two-dimensional surface source problem for a va­cancy model.

scholarly journals A Subspace Preconditioned LSQR Gauss-Newton Method with a Constrained Line Search Path Applied to 3D Biomedical Microwave Imaging

2015 ◽  
Vol 2015 ◽  
pp. 1-21
Author(s):  
Jürgen De Zaeytijd ◽  
Ann Franchois
Keyword(s):  
Complex Permittivity ◽  
Line Search ◽  
Synthetic Data ◽  
Iterative Solution ◽  
Initial Guess ◽  
Microwave Imaging ◽  
Single Frequency ◽  
Problem Solution ◽  
Source Position ◽  
Search Path

Three contributions that can improve the performance of a Newton-type iterative quantitative microwave imaging algorithm in a biomedical context are proposed. (i) To speed up the iterative forward problem solution, we extrapolate the initial guess of the field from a few field solutions corresponding to previous source positions for the same complex permittivity (i.e., “marching on in source position”) as well as from a Born-type approximation that is computed from a field solution corresponding to one previous complex permittivity profile for the same source position. (ii) The regularized Gauss-Newton update system can be ill-conditioned; hence we propose to employ a two-level preconditioned iterative solution method. We apply the subspace preconditioned LSQR algorithm from Jacobsen et al. (2003) and we employ a 3D cosine basis. (iii) We propose a new constrained line search path in the Gauss-Newton optimization, which incorporates in a smooth manner lower and upper bounds on the object permittivity, such that these bounds never can be violated along the search path. Single-frequency reconstructions from bipolarized synthetic data are shown for various three-dimensional numerical biological phantoms, including a realistic breast phantom from the University of Wisconsin-Madison (UWCEM) online repository.

Magnetic quantum oscillations in doped antiferromagnets

2017 ◽  
Vol 31 (25) ◽  
pp. 1745015
Author(s):  
V. V. Kabanov
Keyword(s):  
Magnetic Field ◽  
Polar Angle ◽  
Two Dimensional ◽  
Plane Angle ◽  
Arbitrary Direction ◽  
Magnetization Direction ◽  
Quantum Oscillations ◽  
Out Of Plane ◽  
Plane Direction ◽  
Magnetic Quantum Oscillations

Energy spectrum of electrons (holes) doped into two-dimensional (2D) antiferromagnetic (AF) semiconductors is quantized in an external magnetic field of arbitrary direction. A peculiar dependence of de Haas–van Alphen (dHvA) magneto-oscillation amplitudes on the azimuthal in-plane angle from the magnetization direction and on the polar angle from the out-of-plane direction is found. The angular dependence of the amplitude is different if the measurements are performed in the field above and below of the spin-flop field.

scholarly journals Estimation of the total magnetization direction of approximately spherical bodies

2014 ◽  
Vol 1 (2) ◽  
pp. 1465-1507
Author(s):  
V. C. Oliveira ◽  
D. P. Sales ◽  
V. C. F. Barbosa ◽  
L. Uieda
Keyword(s):  
Synthetic Data ◽  
Magnetization Vector ◽  
Spherical Shape ◽  
Time Interval ◽  
Alkaline Complex ◽  
Multiple Sources ◽  
Geological Time ◽  
Magnetization Direction ◽  
Reduction To The Pole ◽  
Horizontal Grid

Abstract. We have developed a fast total-field anomaly inversion to estimate the magnetization direction of multiple sources with approximately spherical shape and known centres. Our method can be applied to interpret multiple sources with different magnetization directions. It neither requires the prior computation of any transformation like reduction to the pole nor the use of regularly spaced data on a horizontal grid. The method contains flexibility to be implemented as a linear or non-linear inverse problem, which results, respectively, in a least-squares or robust estimate of the components of the magnetization vector of the sources. Applications to synthetic data show the robustness of our method against interfering anomalies and errors in the location of the sources' centre. Besides, we show the feasibility of applying the upward continuation to interpret non-spherical sources. Applications to field data over the Goiás Alkaline Province (GAP), Brazil, show the good performance of our method in estimating geological meaningful magnetization directions. The results obtained for a region of the GAP, near from the alkaline complex of Diorama, suggest the presence of non-outcropping sources marked by strong remanent magnetization with inclination and declination close to -70.35° and -19.81°, respectively. This estimated magnetization direction leads to predominantly positive reduced-to-the-pole anomalies, even for other region of the GAP, in the alkaline complex of Montes Claros de Goiás. These results show that the non-outcropping sources near from the alkaline complex of Diorama have almost the same magnetization direction of that ones in the alkaline complex of Montes Claros de Goiás, strongly suggesting that these sources have emplaced the crust almost within the same geological time interval.

scholarly journals On the Non-Negative Radial Solutions of the Two Dimensional Bratu Equation

2021 ◽  
Vol 45 (02) ◽  
pp. 275-288
Author(s):  
MBE KOUA CHRISTOPHE NDJATCHI ◽  
PANAYOTIS VYRIDIS ◽  
JUAN MARTÍNEZ ◽  
J. JUAN ROSALES
Keyword(s):  
Finite Difference ◽  
Boundary Value Problem ◽  
Theoretical Result ◽  
Theoretical Method ◽  
Parameter Estimate ◽  
Two Dimensional ◽  
Negative Solution ◽  
Radial Solutions ◽  
Seidel Method ◽  
Bratu Equation

In this paper, we study the boundary value problem on the unit circle for the Bratu’s equation depending on the real parameter μ. From the parameter estimate, the existence of non-negative solution is set. A numerical method is suggested to justify the theoretical result. It is a combination of the adaptation of finite difference and Gauss-Seidel method allowing us to obtain a good approximation of μc, with respect to the exact theoretical method μc = λ = 5.7831859629467.

TWO-DIMENSIONAL DC RESISTIVITY MAPPING FOR SUBSURFACE INVESTIGATION USING SOFT COMPUTING APPROACHES

2015 ◽  
Vol 77 (17) ◽  
Author(s):  
Herman Wahid ◽  
Mohd. Hakimi Othman ◽  
Ruzairi Abdul Rahim
Keyword(s):  
Inverse Problem ◽  
Synthetic Data ◽  
Least Square Method ◽  
Least Square ◽  
Dc Resistivity ◽  
Two Dimensional ◽  
Actual Model ◽  
Subsurface Investigation ◽  
Artificial Neural Network Ann ◽  
Mlp Model

In geophysical subsurface surveys, difficulty to interpret measurement of data obtain from the equipment are risen. Data provided by the equipment did not indicate subsurface condition specifically and deviates from the expected standard due to numerous features. Generally, the data that obtained from the laws of physics computation is known as forward problem. And the process of obtaining the data from sets of measurements and reconstruct the model is known as inverse problem. Researchers have proposed multiple estimation techniques to cater the inverse problem and provide estimation that close to actual model. In this work, we investigate the feasibility of using artificial neural network (ANN) in solving two- dimensional (2-D) direct current (DC) resistivity mapping for subsurface investigation, in which the algorithms are based on the radial basis function (RBF) model and the multi-layer perceptron (MLP) model. Conventional approach of least square (LS) method is used as a benchmark and comparative study with the proposed algorithms. In order to train the proposed algorithms, several synthetic data are generated using RES2DMOD software based on hybrid Wenner-Schlumberger configurations. Results are compared between the proposed algorithms and least square method in term of its effectiveness and error variations to the actual values. It is discovered that the proposed algorithms have offered better performance in term minimum error difference to the actual model, as compared to least square method. Simulation results demonstrate that proposed algorithms can solve the inverse problem and it can be illustrated by means of the 2-D graphical mapping.

The IDQ curve: A tool for evaluating the direction of remanent magnetization from magnetic anomalies

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. J85-J98
Author(s):  
Shuang Liu ◽  
Xiangyun Hu ◽  
Dalian Zhang ◽  
Bangshun Wei ◽  
Meixia Geng ◽  
...  
Keyword(s):  
Remanent Magnetization ◽  
Field Data ◽  
Eastern China ◽  
Synthetic Data ◽  
Magnetic Anomalies ◽  
Natural Remanent Magnetization ◽  
Magma Intrusion ◽  
Magnetization Direction ◽  
Ore Bodies ◽  
Using Data

Natural remanent magnetization acts as a record of the previous orientations of the earth’s magnetic field, and it is an important feature when studying geologic phenomena. The so-called IDQ curve is used to describe the relationship between the inclination ( I) and declination ( D) of remanent magnetization and the Köenigsberger ratio ( Q). Here, we construct the IDQ curve using data on ground and airborne magnetic anomalies. The curve is devised using modified approaches for estimating the total magnetization direction, e.g., identifying the maximal position of minimal reduced-to-the-pole fields or identifying correlations between total and vertical reduced-to-the-pole field gradients. The method is tested using synthetic data, and the results indicate that the IDQ curve can provide valuable information on the remanent magnetization direction based on available data on the Köenigsberger ratio. Then, the method is used to interpret field data from the Yeshan region in eastern China, where ground anomalies have been produced by igneous rocks, including diorite and basalt, which occur along with magnetite and hematite ore bodies. The IDQ curves for 24 subanomalies are constructed, and these curves indicate two main distribution clusters of remanent magnetization directions corresponding to different structural units of magma intrusion and help identify the lithologies of the magnetic sources in areas covered by Quaternary sediments. The estimated remanent magnetization directions for Cenozoic basalt are consistent with measurements made in paleomagnetism studies. The synthetic and field data indicate that the IDQ curve can be used to efficiently estimate the remanent magnetization direction from a magnetic anomaly, which could help with our understanding of geologic processes in an area.

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