Interpretation of magnetic data using tilt-angle derivatives

Geophysics ◽  
2008 ◽  
Vol 73 (1) ◽  
pp. L1-L10 ◽  
Author(s):  
Ahmed Salem ◽  
Simon Williams ◽  
Derek Fairhead ◽  
Richard Smith ◽  
Dhananjay Ravat
Keyword(s):  
Tilt Angle ◽  
Magnetic Data ◽  
Structural Index ◽  
Spectral Content ◽  
Free Data ◽  
Second Derivatives ◽  
Simple Linear Equation ◽  
Theoretical Simulations ◽  
Derivatives Of ◽  
Method Show

We have developed a new method for interpretation of gridded magnetic data which, based on derivatives of the tilt angle, provides a simple linear equation, similar to the 3D Euler equation. Our method estimates both the horizontal location and the depth of magnetic bodies, but without specifying prior information about the nature of the sources (structural index). Using source-position estimates, the nature of the source can then be inferred. Theoretical simulations over simple and complex magnetic sources that give rise to noise-corrupted and noise-free data, illustrate the ability of the method to provide source locations and index values characterizing the nature of the source bodies. Our method uses second derivatives of the magnetic anomaly, which are sensitive to noise (high-wavenumber spectral content) in the data. Thus, an upward continuation of the anomaly may lead to reduce the noise effect. We demonstrate the practical utility of the method using a field example from Namibia, where the results of the proposed method show broad cor-relation with previous results using interactive forward modeling.

Automatic 3-D interpretation of potential field data using analytic signal derivatives

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Nicole Debeglia ◽  
Jacques Corpel
Keyword(s):  
Signal Amplitude ◽  
Vertical Gradient ◽  
Analytic Signal ◽  
Magnetic Data ◽  
Practical Implementation ◽  
Potential Field Data ◽  
Gravity And Magnetic ◽  
Second Derivatives ◽  
Gravity And Magnetic Data ◽  
Derivatives Of

A new method has been developed for the automatic and general interpretation of gravity and magnetic data. This technique, based on the analysis of 3-D analytic signal derivatives, involves as few assumptions as possible on the magnetization or density properties and on the geometry of the structures. It is therefore particularly well suited to preliminary interpretation and model initialization. Processing the derivatives of the analytic signal amplitude, instead of the original analytic signal amplitude, gives a more efficient separation of anomalies caused by close structures. Moreover, gravity and magnetic data can be taken into account by the same procedure merely through using the gravity vertical gradient. The main advantage of derivatives, however, is that any source geometry can be considered as the sum of only two types of model: contact and thin‐dike models. In a first step, depths are estimated using a double interpretation of the analytic signal amplitude function for these two basic models. Second, the most suitable solution is defined at each estimation location through analysis of the vertical and horizontal gradients. Practical implementation of the method involves accurate frequency‐domain algorithms for computing derivatives with an automatic control of noise effects by appropriate filtering and upward continuation operations. Tests on theoretical magnetic fields give good depth evaluations for derivative orders ranging from 0 to 3. For actual magnetic data with borehole controls, the first and second derivatives seem to provide the most satisfactory depth estimations.

Enhancement of the total horizontal gradient of magnetic anomalies using the tilt angle

Geophysics ◽  
2013 ◽  
Vol 78 (3) ◽  
pp. J33-J41 ◽  
Author(s):  
Francisco J. F. Ferreira ◽  
Jeferson de Souza ◽  
Alessandra de B. e S. Bongiolo ◽  
Luís G. de Castro
Keyword(s):  
Tilt Angle ◽  
Real Data ◽  
Magnetic Anomalies ◽  
Detection Methods ◽  
Magnetic Data ◽  
Horizontal Gradient ◽  
Geometrical Properties ◽  
Different Depths ◽  
Deep Sources ◽  
Derivatives Of

Magnetic anomaly maps reflect the spatial distribution of magnetic sources, which may be located at different depths and have significantly different physical and geometrical properties, complicating the identification of the corresponding geologic structures. Filtering techniques are frequently used to balance anomalies from shallow and deep sources, and to enhance certain features of interest, such as the edges of the causative bodies. Most methods used for enhancing magnetic data are based on vertical or horizontal derivatives of the magnetic anomalies or combinations of them, and the edges or centers of the sources are identified by maxima, minima, or null values in the transformed data. Normalized derivatives methods are used to equalize signals from sources buried at different depths. We present an edge detector method for the enhancement of magnetic anomalies, which is based on the tilt angle of the total horizontal gradient. The notable features of this method are that it produces amplitude maxima over the source edges and that it equalizes signals from shallow and deep sources. The method is applied to synthetic and real data. The effectiveness of the method is evaluated by comparing it with other edge detection methods that have been previously reported in the literature and that make use of derivatives. The results show that our method is less sensitive to variations in the depth of the sources and that it indicates the position of the edges of causative bodies in a more accurate fashion, when compared with previous methods, even for anomalies due to multiple interfering sources. These results demonstrate that the proposed method is a useful tool for the qualitative interpretation of magnetic data.

The automatic determination of the location, depth, and dip of contacts from aeromagnetic data

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. J35-J41 ◽  
Author(s):  
Gordon R. J. Cooper
Keyword(s):  
Magnetic Field ◽  
Tilt Angle ◽  
Synthetic Data ◽  
Magnetic Data ◽  
Aeromagnetic Data ◽  
Third Order ◽  
Contour Lines ◽  
Derivatives Of ◽  
The Magnetic Field

A simple new method (termed the contact-depth method) for the determination of the depth, location, and dip of contacts from pole reduced magnetic data was evaluated. The depth was obtained by computing the horizontal derivative of the tangent of the tilt angle of the magnetic field over the contact. Although it is based upon the tilt-depth method, it does not require the distance between contour lines to be measured, and it additionally allows the dip of the contact to be estimated from the gradient of the depth estimates. The horizontal location of the contact is that of the zero value of the tilt angle. The method uses second- and third-order derivatives of the magnetic field to obtain the parameters of the contact, so it is sensitive to noise. When tested on synthetic data and on aeromagnetic data from southern Africa, the method gave plausible results.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

1990 ◽  
Vol 112 (1) ◽  
pp. 83-87 ◽  
Author(s):  
R. H. Fries ◽  
B. M. Coffey
Keyword(s):  
Numerical Simulation ◽  
White Noise ◽  
State Space ◽  
Spectral Characteristics ◽  
The State ◽  
Second Derivatives ◽  
Time Histories ◽  
Sample Interval ◽  
Rail Irregularities ◽  
Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

Investigation of a Technique for the Differentiation of Empirical Data

1983 ◽  
Vol 105 (3) ◽  
pp. 200-202 ◽  
Author(s):  
D. M. Trujillo ◽  
H. R. Busby
Keyword(s):  
Dynamic Programming ◽  
Differentiable Function ◽  
Empirical Data ◽  
Numerical Experiments ◽  
Random Noise ◽  
Second Derivatives ◽  
Derivatives Of

A dynamic programming filter is derived to estimate the first and second derivatives of empirical data. A series of numerical experiments are conducted using a known differentiable function with various amounts of added random noise.

scholarly journals Spectral Derivatives of Optical Depth for Partitioning Aerosol Type and Loading

2021 ◽  
Vol 13 (8) ◽  
pp. 1544
Author(s):  
Tang-Huang Lin ◽  
Si-Chee Tsay ◽  
Wei-Hung Lien ◽  
Neng-Huei Lin ◽  
Ta-Chih Hsiao
Keyword(s):  
Optical Depth ◽  
Complex Refractive Index ◽  
Solar Spectrum ◽  
Collective Models ◽  
Anthropogenic Pollutants ◽  
Satellite Signal ◽  
Aerosol Compositions ◽  
Theoretical Simulations ◽  
Derivatives Of ◽  
Aerosol Index

Quantifying aerosol compositions (e.g., type, loading) from remotely sensed measurements by spaceborne, suborbital and ground-based platforms is a challenging task. In this study, the first and second-order spectral derivatives of aerosol optical depth (AOD) with respect to wavelength are explored to determine the partitions of the major components of aerosols based on the spectral dependence of their particle optical size and complex refractive index. With theoretical simulations from the Second Simulation of a Satellite Signal in the Solar Spectrum (6S) model, AOD spectral derivatives are characterized for collective models of aerosol types, such as mineral dust (DS) particles, biomass-burning (BB) aerosols and anthropogenic pollutants (AP), as well as stretching out to the mixtures among them. Based on the intrinsic values from normalized spectral derivatives, referenced as the Normalized Derivative Aerosol Index (NDAI), a unique pattern is clearly exhibited for bounding the major aerosol components; in turn, fractions of the total AOD (fAOD) for major aerosol components can be extracted. The subtlety of this NDAI method is examined by using measurements of typical aerosol cases identified carefully by the ground-based Aerosol Robotic Network (AERONET) sun–sky spectroradiometer. The results may be highly practicable for quantifying fAOD among mixed-type aerosols by means of the normalized AOD spectral derivatives.

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