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.

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.

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.

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.

On the least‐squares residual anomaly determination

Geophysics ◽  
1985 ◽  
Vol 50 (3) ◽  
pp. 473-480 ◽  
Author(s):  
E. M. Abdelrahman ◽  
S. Riad ◽  
E. Refai ◽  
Y. Amin
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Vertical Cylinder ◽  
Correlation Factor ◽  
Western Desert ◽  
Bouguer Gravity ◽  
Residual Component ◽  
Optimum Order ◽  
Gravity Interpretation ◽  
Residual Maps

This paper discusses an approach to determine the least‐squares optimum order of the regional surface which, when subtracted from the Bouguer gravity anomaly data, minimizes distortion of the residual component of the field. The least‐squares method was applied to theoretical composite gravity fields each consisting of a constant residual component (sphere or vertical cylinder) and a regional component of different order using successively increasing orders of polynomial regionals for residual determination. The overall similarity between each two successive residual maps was determined by computing the correlation factor between the mapped variables. Similarity between residual maps of the lowest orders, verified by good correlation, may generally be considered a criterion for determining the optimum order of the regional surface and consequently the least distorted residual component. The residual map of the lower order in this well‐correlated doublet is considered the most plausible one and may be used for gravity interpretation. This approach was successfully applied to the Bouguer gravity of Abu Roash dome, located west of Cairo in the Western Desert of Egypt.

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.

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.

Higher derivatives analysis of 2-D magnetic data

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 205-212 ◽  
Author(s):  
E. M. Abdelrahman ◽  
E. R. Abo‐Ezz
Keyword(s):  
Theoretical Data ◽  
Horizontal Cylinder ◽  
Magnetic Data ◽  
Random Errors ◽  
New Approach ◽  
Depth Determination ◽  
Higher Derivatives ◽  
Optimum Order ◽  
Fourth Derivative ◽  
Actual Depth

This paper presents a new approach for determining the depth of a buried structure from numerical second‐, third‐, and fourth‐horizontal‐derivative anomalies obtained from 2-D magnetic data using filters of successive graticule spacings. The problem of depth determination has been transformed into the problem of finding a solution to a nonlinear equation of the form z = f(z). Formulas have been derived for a horizontal cylinder and a dike. The depths obtained from the second‐, third‐, and fourth‐derivative anomaly values can be used to determine simultaneously the actual depth to the buried structure and the optimum order of the regional magnetic field along the profile. This powerful technique can solve two major potential field problems: regional residual separation and depth determination. The method is applied to theoretical data with and without random errors and is tested on a field example from Arizona.

A least‐squares derivatives analysis of gravity anomalies due to faulted thin slabs

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 535-543 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby ◽  
Tarek M. El‐Araby ◽  
Eid Ragab Abo‐Ezz
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Gravity Data ◽  
Bouguer Anomaly ◽  
Theoretical Data ◽  
Gravity Anomalies ◽  
Random Errors ◽  
Fault Structure ◽  
Amplitude Coefficient ◽  
Buried Fault

This paper presents two different least‐squares approaches for determining the depth and amplitude coefficient (related to the density contrast and the thickness of a buried faulted thin slab from numerical first‐, second‐, third‐, and fourth‐horizontal derivative anomalies obtained from 2D gravity data using filters of successive graticule spacings. The problem of depth determination has been transformed into the problem of finding a solution to a nonlinear equation of the form f(z) = 0. Knowing the depth and applying the least‐squares method, the amplitude coefficient is determined using a simple linear equation. In this way, the depth and amplitude coefficient are determined individually from all observed gravity data. The depths and the amplitude coefficients obtained from the first‐, second‐, third‐, and fourth‐ derivative anomaly values can be used to determine simultaneously the actual depth and amplitude coefficient of the buried fault structure and the optimum order of the regional gravity field along the profile. The method can be applied not only to residuals but also to the Bouguer anomaly profile consisting of the combined effect of a residual component due to a purely local fault structure (shallow or deep) and a regional component represented by a polynomial of any order. The method is applied to theoretical data with and without random errors and is tested on a field example from Egypt.

Three least‐squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data

Geophysics ◽  
2001 ◽  
Vol 66 (4) ◽  
pp. 1105-1109 ◽  
Author(s):  
E. M. Abdelrahman ◽  
H. M. El‐Araby ◽  
T. M. El‐Araby ◽  
E. R. Abo‐Ezz
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Least Squares Method ◽  
Gravity Data ◽  
Synthetic Data ◽  
Gravity Anomalies ◽  
Maximum Error ◽  
The United States ◽  
Random Errors ◽  
Amplitude Coefficient

Three different least‐squares approaches are developed to determine, successively, the depth, shape (shape factor), and amplitude coefficient related to the radius and density contrast of a buried structure from the residual gravity anomaly. By defining the anomaly value g(max) at the origin on the profile, the problem of depth determination is transformed into the problem of solving a nonlinear equation, [Formula: see text]. Formulas are derived for spheres and cylinders. Knowing the depth and applying the least‐squares method, the shape factor and the amplitude coefficient are determined using two simple linear equations. In this way, the depth, shape, and amplitude coefficient are determined individually from all observed gravity data. A procedure is developed for automated interpretation of gravity anomalies attributable to simple geometrical causative sources. The method is applied to synthetic data with and without random errors. In all the cases examined, the maximum error in depth, shape, and amplitude coefficient is 3%, 1.5%, and 7%, respectively. Finally, the method is tested on a field example from the United States, and the depth and shape obtained by the present method are compared with those obtained from drilling and seismic information and with those published in the literature.

Least‐squares deconvolution of apparent resistivity pseudosections

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1682-1690 ◽  
Author(s):  
M. H. Loke ◽  
R. D. Barker
Keyword(s):  
Least Squares ◽  
Apparent Resistivity ◽  
Least Squares Method ◽  
Random Noise ◽  
Electrode Array ◽  
Earth Model ◽  
Damping Factor ◽  
Data Set ◽  
Single Data ◽  
Starting Model

A fast technique for the inversion of data from resistivity tomography surveys has been developed. This technique is based on the smoothness‐constrained, least‐squares method, and it produces a 2-D subsurface model that is free of distortions in the apparent resistivity pseudosection caused by the electrode array geometry used. A homogeneous earth model is used as the starting model for which the apparent resistivity partial derivative values can be calculated analytically. Tests with a variety of models and data from field surveys show that this technique is insensitive to random noise, provided a sufficiently large damping factor is used, and that it can resolve structures that cause overlapping anomalies in the pseudosection. On a 33 MHz 80486DX microcomputer, it takes about 5 s to process a single data set.

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