Magnetic interpretation using a least-squares, depth-shape curves method

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. L23-L30 ◽  
Author(s):  
El-Sayed M. Abdelrahman ◽  
Khalid S. Essa
Keyword(s):  
Least Squares ◽  
Magnetic Anomaly ◽  
Least Squares Method ◽  
Random Noise ◽  
Synthetic Data ◽  
Theoretical Data ◽  
Magnetic Anomalies ◽  
Shape Factors ◽  
Characteristic Points ◽  
Horizontal Cylinders

We have developed a least-squares approach to depth determination from residual magnetic anomalies caused by simple geologic structures. By normalizing the residual magnetic anomaly using three characteristic points and their corresponding distances on the anomaly profile, the problem of determining depth from residual magnetic anomalies has been transformed into finding a solution to a nonlinear equation of the form z = f(z). Formulas have been derived for spheres, horizontal cylinders, thin dikes, and contacts. The method is applied to synthetic data with and without random noise. We have also developed a method using depth-shape curves to simultaneously define the shape and depth of a buried structure from a residual magnetic anomaly profile. The method is based on determining the depth from the normalized residual anomaly for each shape factor using the least-squares method mentioned above. The computed depths are plotted against the shape factors on a graph. The solution for the shape and depth of the buried structure is read at the common intersection of the depth-shape curves. The depth-shape curves method was successfully tested on theoretical data with and without random noise and applied to a known field example from Ontario.

Comparison of the least‐squares criterion and the Cauchy criterion in frequency‐wavenumber inversion

Geophysics ◽  
1991 ◽  
Vol 56 (12) ◽  
pp. 2027-2035 ◽  
Author(s):  
Lasse Amundsen
Keyword(s):  
Lower Bound ◽  
Least Squares ◽  
Scale Parameter ◽  
Least Squares Method ◽  
Random Noise ◽  
Synthetic Data ◽  
Error Criterion ◽  
Numerical Examples ◽  
Wavenumber Domain ◽  
Weather Noise

One alternative to the least‐squares inversion technique is the use of a Cauchy error criterion. We show how inversion algorithms of the Gauss‐Newton type based on the least‐squares method can be modified to handle the Cauchy norm. A criterion for the lower bound of the scale parameter in the Cauchy norm is given. We compare the least‐squares and Cauchy error criteria by inverting synthetic data corrupted by random noise and weather noise. The data are transformed to the frequency‐wavenumber domain before the inversion starts. The numerical examples show that the algorithm based on the Cauchy criterion is more robust in the presence of the noise tested here. Per iteration, the computer costs of the two algorithms are approximately the same.

New methods for shape and depth determinations from SP data

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1202-1210 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby ◽  
Abdel‐Rady G. Hassaneen ◽  
Mahfooz A. Hafez
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Least Squares Method ◽  
Random Noise ◽  
Theoretical Data ◽  
Derivative Analysis ◽  
Higher Derivatives ◽  
Optimum Order ◽  
Simple Linear Equation ◽  
Sp Anomaly

We have extended our earlier derivative analysis method to higher derivatives to estimate the depth and shape (shape factor) of a buried structure from self‐potential (SP) data. We show that numerical second, third, and fourth horizontal‐derivative anomalies obtained from SP data using filters of successive window lengths can be used to simultaneously determine the depth and the shape of a buried structure. The depths and shapes obtained from the higher derivatives anomaly values can be used to determine simultaneously the actual depth and shape of the buried structure and the optimum order of the regional SP anomaly along the profile. The method is semi‐automatic and it can be applied to residuals as well as to observed SP data. We have also developed a method (based on a least‐squares minimization approach) to determine, successively, the depth and the shape of a buried structure from the residual SP anomaly profile. By defining the zero anomaly distance and the anomaly value at the origin, the problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of form f(z) = 0. Knowing the depth and applying the least‐squares method, the shape factor is determined using a simple linear equation. Finally, we apply these methods to theoretical data with and without random noise and on a known field example from Germany. In all cases, the depth and shape solutions obtained are in good agreement with the actual ones.

scholarly journals Effective fault detection in structural health monitoring systems

2019 ◽  
Vol 11 (9) ◽  
pp. 168781401987323 ◽  
Author(s):  
Marwa Chaabane ◽  
Majdi Mansouri ◽  
Kamaleldin Abodayeh ◽  
Ahmed Ben Hamida ◽  
Hazem Nounou ◽  
...  
Keyword(s):  
Fault Detection ◽  
Least Squares ◽  
Partial Least Squares ◽  
Least Squares Method ◽  
Random Noise ◽  
Ratio Test ◽  
Multiscale Representation ◽  
Kernel Partial Least Squares ◽  
Partial Least Squares Model ◽  
Partial Least Squares Method

A new fault detection technique is considered in this article. It is based on kernel partial least squares, exponentially weighted moving average, and generalized likelihood ratio test. The developed approach aims to improve monitoring the structural systems. It consists of computing an optimal statistic that merges the current information and the previous one and gives more weight to the most recent information. To improve the performances of the developed kernel partial least squares model even further, multiscale representation of data will be used to develop a multiscale extension of this method. Multiscale representation is a powerful data analysis way that presents efficient separation of deterministic characteristics from random noise. Thus, multiscale kernel partial least squares method that combines the advantages of the kernel partial least squares method with those of multiscale representation will be developed to enhance the structural modeling performance. The effectiveness of the proposed approach is assessed using two examples: synthetic data and benchmark structure. The simulation study proves the efficiency of the developed technique over the classical detection approaches in terms of false alarm rate, missed detection rate, and detection speed.

scholarly journals Calibration for Parameter Estimation of Signals with Complex Noise via Nonstationarity Measure

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Zhiming Zhou ◽  
Zhengyun Zhou ◽  
Liang Wu
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Least Squares Method ◽  
Random Noise ◽  
Calibration Method ◽  
Numerical Studies ◽  
Chaotic Signals ◽  
Inaccurate Estimation ◽  
Non Gaussian ◽  
First Time

The signals in numerous complex systems of engineering can be regarded as nonlinear parameter trend with noise which is identically distributed random signals or deterministic stationary chaotic signals. The commonly used methods for parameter estimation of nonlinear trend in signals are mainly based on least squares. It can cause inaccurate estimation results when the noise is complex (such as non-Gaussian noise, strong noise, and chaotic noise). This paper proposes a calibration method for this issue in the case of single parameter via nonstationarity measure from the perspective of the stationarity of residual sequence. Some numerical studies are conducted for validation. Results of numerical studies show that the proposed calibration method performs well for various models with different noise strengths and types (including random noise and chaotic noise) and can significantly improve the accuracy of initial estimates obtained by least squares method. This is the first time that the nonstationarity measure is applied to the parameter calibration. All these results will be a guide for future studies of other parameter calibrations.

A least‐squares approach to depth determination from gravity data

Geophysics ◽  
1983 ◽  
Vol 48 (3) ◽  
pp. 357-360 ◽  
Author(s):  
O. P. Gupta
Keyword(s):  
Nonlinear Equation ◽  
Least Squares ◽  
Gravity Data ◽  
Synthetic Data ◽  
Numerical Approach ◽  
Random Errors ◽  
Depth Determination ◽  
Vertical Fault ◽  
Horizontal Cylinders ◽  
Least Squares Approach

The present paper deals with a numerical approach to determine the depth of a buried structure from the residual anomaly. The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of the form [Formula: see text]. Formulas have been derived for a sphere, vertical and horizontal cylinders, and for a vertical fault (thin plate approximation). The procedure is applied to synthetic data with and without random errors. Finally, a field example is presented in which the depth to a fault is estimated at 3.8 km and verified from drilling results.

Derivative analysis of SP anomalies

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 890-897 ◽  
Author(s):  
El‐Sayed Mohamed Abdelrahman ◽  
Ahmed Abu Baker Ammar ◽  
Hamdy Ismail Hassanein ◽  
Mahfooz Abdelmottaleb Hafez
Keyword(s):  
Shape Factor ◽  
Random Noise ◽  
Theoretical Data ◽  
Window Length ◽  
Continuous Curve ◽  
Shape Factors ◽  
Derivative Analysis ◽  
Common Intersection ◽  
The Common ◽  
Self Potential

Numerical second horizontal derivative self‐potential (SP) anomalies obtained from SP data using filters of successive window lengths (graticule spacings) can be used to determine the shape and depth of a buried structure. For a fixed window length, the depth is determined using a simple formula for each shape factor. The computed depths are plotted against the shape factors on a graph. All points for a fixed window length are connected by a continuous curve (window curve). The solution for the shape and depth of the buried structure is read at the common intersection of the window curves. The method is applied to theoretical data with and without random noise and tested on a field example from Turkey.

scholarly journals Depth and shape solutions from residual gravity anomalies due to simple geometric structures using a statistical approach

2017 ◽  
Vol 47 (2) ◽  
pp. 113-132 ◽  
Author(s):  
El-Sayed Abdelrahman ◽  
Mohamed Gobashy
Keyword(s):  
Gravity Data ◽  
Random Noise ◽  
Synthetic Data ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
Horizontal Cylinder ◽  
Random Errors ◽  
Residual Gravity ◽  
Characteristic Points ◽  
The Usa

AbstractWe have developed a simple and fast quantitative method for depth and shape determination from residual gravity anomalies due to simple geometrical bodies (semi-infinite vertical cylinder, horizontal cylinder, and sphere). The method is based on defining the anomaly value at two characteristic points and their corresponding distances on the anomaly profile. Using all possible combinations of the two characteristic points and their corresponding distances, a statistical procedure is developed for automated determination of the best shape and depth parameters of the buried structure from gravity data. A least-squares procedure is also formulated to estimate the amplitude coefficient which is related to the radius and density contrast of the buried structure. The method is applied to synthetic data with and without random errors and tested on two field examples from the USA and Germany. In all cases examined, the estimated depths and shapes are found to be in good agreement with actual values. The present method has the capability of minimizing the effect of random noise in data points to enhance the interpretation of results.

On: “Nomogram for the direct interpretation of magnetic anomalies due to long horizontal cylinders,” by T. K. S. Prakasa Rao, M. Subrahmanyam, and A. Srikrishna Murty, (GEOPHYSICS, 51, 2156–2159, November 1986.)

Geophysics ◽  
1987 ◽  
Vol 52 (10) ◽  
pp. 1442-1443
Author(s):  
Ronald Green
Keyword(s):  
Magnetic Anomaly ◽  
Magnetic Anomalies ◽  
Horizontal Cylinder ◽  
Direct Interpretation ◽  
Horizontal Cylinders

In the article by T. K. S. Prakasa Rao, M. Subrahmanyam, and A. Srikrishna Murty, the trivial problem of interpreting the magnetic anomaly over a horizontal cylinder was examined and a set of nomograms to assist with the interpretation was presented. Prakasa Rao et al. begin their discussion with equation (1) from Gay (1965).

3D forward and inverse modeling of total-field magnetic anomalies caused by a uniformly magnetized layer defined by a linear combination of 2D Gaussian functions

Geophysics ◽  
2008 ◽  
Vol 73 (1) ◽  
pp. L11-L18 ◽  
Author(s):  
Juan García-Abdeslem
Keyword(s):  
Linear Combination ◽  
Magnetic Anomaly ◽  
Synthetic Data ◽  
Magnetic Anomalies ◽  
High Amplitude ◽  
Nonlinear Inversion ◽  
Total Field ◽  
Gaussian Functions ◽  
Estimated Parameters ◽  
Forward And Inverse Modeling

I develop a method for 3D forward modeling and nonlinear inversion of the total-field magnetic anomaly caused by a uniformly magnetized layer with its top and bottom surfaces represented by a linear combination of 2D Gaussian functions. The solution of the forward problem is found through both analytic and numerical methods of integration to calculate the theoretical magnetic anomaly. The magnetic anomalies computed by the present numerical method compare well with the ones calculated by using an analytic solution. To test the robustness of the algorithm, the inversion is performed with noisy synthetic data. The estimated parameters in the case of a synthetic model were found to deviate only modestly from the true parameters in the presence of noise. The algorithm is used to interpret a dipolar magnetic anomaly of high amplitude attributable to a laccolith of intermediate composition in northern Mexico.

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