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.

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.

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.

A fast interpretation of self-potential data using the depth from extreme points method

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. E107-E116 ◽  
Author(s):  
Maurizio Fedi ◽  
Mahmoud Ahmed Abbas
Keyword(s):  
Scaling Law ◽  
Extreme Points ◽  
Real Data ◽  
High Order ◽  
The Self ◽  
Source Distribution ◽  
Imaging Method ◽  
Fast Method ◽  
Structural Index ◽  
Self Potential

We used a fast method to interpret self-potential data: the depth from extreme points (DEXP) method. This is an imaging method transforming self-potential data, or their derivatives, into a quantity proportional to the source distribution. It is based on upward continuing of the field to a number of altitudes and then multiplying the continued data with a scaling law of those altitudes. The scaling law is in the form of a power law of the altitudes, with an exponent equal to half of the structural index, a source parameter related to the type of source. The method is autoconsistent because the structural index is basically determined by analyzing the scaling function, which is defined as the derivative of the logarithm of the self-potential (or of its [Formula: see text]th derivative) with respect to the logarithm of the altitudes. So, the DEXP method does not need a priori information on the self-potential sources and yields effective information about their depth and shape/typology. Important features of the DEXP method are its high-resolution power and stability, resulting from the combined effect of a stable operator (upward continuation) and a high-order differentiation operator. We tested how to estimate the depth to the source in two ways: (1) at the positions of the extreme points in the DEXP transformed map and (2) at the intersection of the lines of the absolute values of the potential or of its derivative (geometrical method). The method was demonstrated using synthetic data of isolated sources and using a multisource model. The method is particularly suited to handle noisy data, because it is stable even using high-order derivatives of the self-potential. We discussed some real data sets: Malachite Mine, Colorado (USA), the Sariyer area (Turkey), and the Bender area (India). The estimated depths and structural indices agree well with the known information.

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 RECIPROCITY AND THE EMERGENCE OF POWER LAWS IN SOCIAL NETWORKS

2006 ◽  
Vol 17 (07) ◽  
pp. 1067-1076 ◽  
Author(s):  
MICHAEL SCHNEGG
Keyword(s):  
Social Networks ◽  
Power Laws ◽  
Scaling Exponent ◽  
Policy Makers ◽  
Public Health Programs ◽  
Scale Free ◽  
The Past ◽  
Naturally Occurring ◽  
Growing Networks ◽  
Law Degree

Research in network science has shown that many naturally occurring and technologically constructed networks are scale free, that means a power law degree distribution emerges from a growth model in which each new node attaches to the existing network with a probability proportional to its number of links (= degree). Little is known about whether the same principles of local attachment and global properties apply to societies as well. Empirical evidence from six ethnographic case studies shows that complex social networks have significantly lower scaling exponents γ ~ 1 than have been assumed in the past. Apparently humans do not only look for the most prominent players to play with. Moreover cooperation in humans is characterized through reciprocity, the tendency to give to those from whom one has received in the past. Both variables — reciprocity and the scaling exponent — are negatively correlated (r = -0.767, sig = 0.075). If we include this effect in simulations of growing networks, degree distributions emerge that are much closer to those empirically observed. While the proportion of nodes with small degrees decreases drastically as we introduce reciprocity, the scaling exponent is more robust and changes only when a relatively large proportion of attachment decisions follow this rule. If social networks are less scale free than previously assumed this has far reaching implications for policy makers, public health programs and marketing alike.

scholarly journals THE HIERARCHY PROBLEM, RADION MASS, LOCALIZATION OF GRAVITY AND 4D EFFECTIVE NEWTONIAN POTENTIAL IN STRING THEORY ON S1/Z2

2010 ◽  
Vol 25 (08) ◽  
pp. 1661-1698 ◽  
Author(s):  
ANZHONG WANG ◽  
N. O. SANTOS
Keyword(s):  
String Theory ◽  
Matter Field ◽  
Newtonian Potential ◽  
Field Equations ◽  
Hierarchy Problem ◽  
Large Extra Dimension ◽  
Gravitational Field Equations ◽  
Spatial Dimensions ◽  
Derivatives Of ◽  
Kaluza Klein

In this paper, we present a systematical study of braneworlds of string theory on S1/Z2. In particular, starting with the toroidal compactification of the Neveu–Schwarz/Neveu–Schwarz sector in D + d dimensions, we first obtain an effective D-dimensional action, and then compactify one of the D - 1 spatial dimensions by introducing two orbifold branes as its boundaries. We divide the whole set of the gravitational and matter field equations into two groups, one holds outside the two branes, and the other holds on them. By combining the Gauss–Codacci and Lanczos equations, we write down explicitly the general gravitational field equations on each of the two branes, while using distribution theory we express the matter field equations on the branes in terms of the discontinuities of the first derivatives of the matter fields. Afterwards, we address three important issues: (i) the hierarchy problem; (ii) the radion mass; and (iii) the localization of gravity, the four-dimensional Newtonian effective potential and the Yukawa corrections due to the gravitational high-order Kaluza–Klein (KK) modes. The mechanism of solving the hierarchy problem is essentially the combination of the large extra dimension and warped factor mechanisms together with the tension coupling scenario. With very conservative arguments, we find that the radion mass is of the order of 10-2 GeV. The gravity is localized on the visible brane, and the spectrum of the gravitational KK modes is discrete and can be of the order of TeV. The corrections to the four-dimensional Newtonian potential from the higher order of gravitational KK modes are exponentially suppressed and can be safely neglected in current experiments. In an appendix, we also present a systematical and pedagogical study of the Gauss–Codacci equations and Israel's junction conditions across a (D - 1)-dimensional hypersurface, which can be either spacelike or timelike.

Sign in / Sign up

Export Citation Format

Share Document