Degrees of homogeneity of potential fields and structural indices of Euler deconvolution

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. L1-L12 ◽  
Author(s):  
Petar Stavrev ◽  
Alan Reid
Keyword(s):  
Potential Field ◽  
Variable Density ◽  
Point Sources ◽  
Index Formula ◽  
Potential Fields ◽  
Positive Zero ◽  
Euler Deconvolution ◽  
Structural Index ◽  
Field Source ◽  
Definition Of

Homogeneity is a well-known property of the potential fields of simple point sources used in field inversion. We find that the analytical expressions of potential fields created by sources of complicated shape and constant or variable density or magnetization also show this property. This is true if all variables of length dimension are involved in the test of homogeneity. The coordinates of observation points and the source coordinates and sizes form an extended set of variables, in relation to which the field expression is homogeneous. In this case, the principal definition of homogeneity applied to a potential field can be treated as an operator of a space transform of similarity. The ratio between the transformed and original fields determines the value and sign of the degree of homogeneity [Formula: see text]. The latter may take on positive, zero, or negative values. The degree of homogeneity depends on the type of field and on the assumed physical parameter of the field source, and can be nonunique for a given field element. We analyze the potential field of one singular point as the simplest case of homogeneity. Thus, we deduce results for the structural index, [Formula: see text], in Euler deconvolution. The structural index can also be positive, zero, or negative, but it has a unique value. Analytical considerations, as well as numerical tests on the gravity contact model, confirm the proposed physical interpretation of [Formula: see text], and lead to an extended version of Euler’s differential equation for potential fields.

DEXP: A fast method to determine the depth and the structural index of potential fields sources

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. I1-I11 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
Potential Field ◽  
Extreme Points ◽  
Power Laws ◽  
Potential Fields ◽  
Newtonian Potential ◽  
Specific Power ◽  
Scaling Exponent ◽  
Fast Method ◽  
Structural Index ◽  
Derivatives Of

We show that potential fields enjoy valuable properties when they are scaled by specific power laws of the altitude. We describe the theory for the gravity field, the magnetic field, and their derivatives of any order and propose a method, called here Depth from Extreme Points (DEXP), to interpret any potential field. The DEXP method allows estimates of source depths, density, and structural index from the extreme points of a 3D field scaled according to specific power laws of the altitude. Depths to sources are obtained from the position of the extreme points of the scaled field, and the excess mass (or dipole moment) is obtained from the scaled field values. Although the scaling laws are theoretically derived for sources such as poles, dipoles, lines of poles, and lines of dipoles, we give also criteria to estimate the correct scaling law directly from the data. The scaling exponent of such laws is shown to be related to the structural index involved in Euler Deconvolution theory. The method is fast and stable because it takes advantage of the regular behavior of potential field data versus the altitude [Formula: see text]. As a result of stability, the DEXP method may be applied to anomalies with rather low SNRs. Also stable are DEXP applications to vertical and horizontal derivatives of a Newtonian potential of various orders in which we use theoretically determined scaling functions for each order of a derivative. This helps to reduce mutual interference effects and to obtain meaningful representations of the distribution of sources versus depth, with no prefiltering. The DEXP method does not require that magnetic anomalies to be reduced to the pole, and meaningful results are obtained by processing its analytical signal. Application to different cases of either synthetic or real data shows its applicability to any type of potential field investigation, including geological, petroleum, mining, archeological, and environmental studies.

Euler deconvolution of gravity anomalies from thick contact/fault structures with extended negative structural index

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. I51-I58 ◽  
Author(s):  
Petar Stavrev ◽  
Alan Reid
Keyword(s):  
Field Data ◽  
Gravity Anomalies ◽  
Potential Fields ◽  
Contact Structures ◽  
Euler Deconvolution ◽  
Structural Index ◽  
Infinite Slab ◽  
Fault Structures ◽  
Structural Indices ◽  
Analytical Expressions

The concept of extended Euler homogeneity of potential fields is examined with respect to all variables of length dimension in their analytical expressions. This reveals the possible existence of positive degrees of homogeneity or corresponding negative structural indices considered as extensions of the Thompson’s structural indices in Euler deconvolution. This approach is implemented for a contact gravity model, represented by a 2D semi-infinite slab with large thickness relative to its depth. Applying Euler deconvolution on synthetic and field data indicates that the positive degree of homogeneity, i.e., the extended negative structural index, is the appropriate one for the inversion of gravity anomalies from contact structures.

Multiridge analysis of potential fields: Geometric method and reduced Euler deconvolution

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. L53-L65 ◽  
Author(s):  
Maurizio Fedi ◽  
Giovanni Florio ◽  
Tatiana A. Quarta
Keyword(s):  
Potential Field ◽  
Extreme Points ◽  
Potential Fields ◽  
Magnetic Data ◽  
Type I ◽  
Tyrrhenian Sea ◽  
Source Position ◽  
Structural Index ◽  
Type Iii ◽  
Homogeneous Potential

A new method based on 3D multiridge analysis of potential fields assumes a 3D subset in the harmonic region and studies the behavior of potential field ridges, which are built by joining extreme points of the analyzed field computed at different altitudes. Three types of ridges are formed by searching for the zeros of the first horizontal and first vertical derivatives of the potential field (types I and II, respectively) and the zeros of the potential field itself (type III). This method uses a redundant set of ridges, called a multiridge set, to determine source type and location. For homogeneous potential fields generated by simple sources, all of the ridges are straight lines converging to the source position. This method analyzes the multiridges by using a geometric criterion to find the source position at the intersection of the multiridge set and by solving the three reduced Euler equations associated with ridge types I, II, and III. The reduced Euler type I and II equations are used to obtain the structural index and the vertical and horizontal source positions; equation type III estimates the horizontal and vertical source positions. Tests on synthetic as well as the Bishop model field yield good results even with noise-corrupted data. Results obtained using magnetic data collected over the wreck of a military ship in the Tyrrhenian Sea successfully determine its vertical and horizontal positions and the structural index.

On the application of Euler deconvolution to the analytic signal

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. L87-L93 ◽  
Author(s):  
G. Florio ◽  
M. Fedi ◽  
R. Pasteka
Keyword(s):  
Potential Field ◽  
Harmonic Functions ◽  
Horizontal Plane ◽  
Homogeneous Function ◽  
Source Parameters ◽  
Analytic Signal ◽  
Euler Deconvolution ◽  
Structural Index ◽  
Vertical Derivative ◽  
First Vertical Derivative

Standard Euler deconvolution is applied to potential-field functions that are homogeneous and harmonic. Homogeneity is necessary to satisfy the Euler deconvolution equation itself, whereas harmonicity is required to compute the vertical derivative from data collected on a horizontal plane, according to potential-field theory. The analytic signal modulus of a potential field is a homogeneous function but is not a harmonic function. Hence, the vertical derivative of the analytic signal is incorrect when computed by the usual techniques for harmonic functions and so also is the consequent Euler deconvolution. We show that the resulting errors primarily affect the structural index and that the estimated values are always notably lower than the correct ones. The consequences of this error in the structural index are equally important whether the structural index is given as input (as in standard Euler deconvolution) or represents an unknown to be solved for. The analysis of a case history confirms serious errors in the estimation of structural index if the vertical derivative of the analytic signal is computed as for harmonic functions. We suggest computing the first vertical derivative of the analytic signal modulus, taking into account its nonharmonicity, by using a simple finite-difference algorithm. When the vertical derivative of the analytic signal is computed by finite differences, the depth to source and the structural index consistent with known source parameters are, in fact, obtained.

CHARACTERIZATION OF THE POTIGUAR RIFT STRUCTURE BASED ON EULER DECONVOLUTION

2014 ◽  
Vol 32 (1) ◽  
pp. 109 ◽  
Author(s):  
Rafael Saraiva Rodrigues ◽  
David Lopes de Castro ◽  
João Andrade dos Reis Júnior
Keyword(s):  
Potential Field ◽  
Window Size ◽  
Gravity Anomalies ◽  
Euler Deconvolution ◽  
Structural Index ◽  
Potential Field Data ◽  
3D Euler Deconvolution ◽  
Maximum Tolerance ◽  
Para Determinar ◽  
Spatial Window

ABSTRACT. The Euler deconvolution is a semi-automatic interpretation method of potential field data that can provide accurate estimates of horizontal position and depth of causative sources. In this work we show the application of 3D Euler Deconvolution in gravity and magnetic maps to characterize the rift structures of the Potiguar Basin (Rio Grande do Norte and Ceará States, Brazil) using the structural index as a main parameter, which represents an indicator of the geometric form of the anomalous sources. The best results were obtained with a structural index equal to zero (for residual gravity anomalies) and 0.5 (for magnetic anomalies reduced to the pole), a spatial window size of 10 km, which is used to determine the area that should be used in the Euler Deconvolution calculation, and maximum tolerance of error ranging from 0 to 7%. This parameter determines which solutions are acceptable. The clouds of Euler solutions allowed us to characterize the main faulted limits of the Potiguar rift, as well as its depth, dip and structural relations with the Precambrian basement. Keywords: Euler deconvolution, potential field, structural index, Potiguar rift.    RESUMO. A deconvolução de Euler é um método de interpretação semiautomático de dados de métodos potenciais, capaz de fornecer uma estimativa da posição horizontal e da profundidade de fontes anômalas. Neste trabalho, mostraremos a aplicação da deconvolução de Euler 3D em mapas gravimétricos e magnéticos para caracterizar as estruturas rifte da Bacia Potiguar (RN/CE), utilizando como principal parâmetro o índice estrutural, que representa um indicador da forma geométrica da fonte anômala. Os melhores resultados foram obtidos com um índice estrutural igual a zero (para as anomalias gravimétricas residuais) e 0,5 (para as anomalias magnéticas reduzidas ao polo), tamanho da janela espacial igual a 10 km, que ´e utilizada para determinar a área que deve ser usada para o cálculo da deconvolução de Euler, e tolerância máxima do erro variando de 0 a 7%, que determina quais soluções são aceitáveis. As nuvens de soluções de Euler nos permitiram caracterizar os principais limites falhados do rifte Potiguar, bem como suas profundidades, mergulho e relações estruturais com o embasamento Pré-cambriano. Palavras-chave: deconvolução de Euler, métodos potenciais, índice estrutural, rifte Potiguar.

scholarly journals Virtual Electric Dipole Field Applied to Autonomous Formation Flight Control of Unmanned Aerial Vehicles

Sensors ◽  
2021 ◽  
Vol 21 (13) ◽  
pp. 4540
Author(s):  
Leszek Ambroziak ◽  
Maciej Ciężkowski
Keyword(s):  
Control System ◽  
Unmanned Aerial Vehicles ◽  
Flight Control ◽  
Electric Dipole ◽  
Potential Field ◽  
Control Algorithm ◽  
Formation Flight ◽  
Potential Fields ◽  
Dipole Potential ◽  
Aerial Vehicles

The following paper presents a method for the use of a virtual electric dipole potential field to control a leader-follower formation of autonomous Unmanned Aerial Vehicles (UAVs). The proposed control algorithm uses a virtual electric dipole potential field to determine the desired heading for a UAV follower. This method’s greatest advantage is the ability to rapidly change the potential field function depending on the position of the independent leader. Another advantage is that it ensures formation flight safety regardless of the positions of the initial leader or follower. Moreover, it is also possible to generate additional potential fields which guarantee obstacle and vehicle collision avoidance. The considered control system can easily be adapted to vehicles with different dynamics without the need to retune heading control channel gains and parameters. The paper closely describes and presents in detail the synthesis of the control algorithm based on vector fields obtained using scalar virtual electric dipole potential fields. The proposed control system was tested and its operation was verified through simulations. Generated potential fields as well as leader-follower flight parameters have been presented and thoroughly discussed within the paper. The obtained research results validate the effectiveness of this formation flight control method as well as prove that the described algorithm improves flight formation organization and helps ensure collision-free conditions.

scholarly journals On the Roots of the Modified Orbit Polynomial of a Graph

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 972
Author(s):  
Modjtaba Ghorbani ◽  
Matthias Dehmer
Keyword(s):  
Positive Root ◽  
Current Work ◽  
Positive Zero ◽  
Unique Positive Root ◽  
Definition Of

The definition of orbit polynomial is based on the size of orbits of a graph which is OG(x)=∑ix|Oi|, where O1,…,Ok are all orbits of graph G. It is a well-known fact that according to Descartes’ rule of signs, the new polynomial 1−OG(x) has a positive root in (0,1), which is unique and it is a relevant measure of the symmetry of a graph. In the current work, several bounds for the unique and positive zero of modified orbit polynomial 1−OG(x) are investigated. Besides, the relation between the unique positive root of OG in terms of the structure of G is presented.

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