A LEAST‐SQUARES PROCEDURE FOR GRAVITY INTERPRETATION

Geophysics ◽  
1965 ◽  
Vol 30 (2) ◽  
pp. 228-233 ◽  
Author(s):  
Charles E. Corbató
Keyword(s):  
Least Squares ◽  
High Speed ◽  
Gravity Anomalies ◽  
The Body ◽  
Two Dimensional ◽  
Partial Derivatives ◽  
Dimensional Gravity ◽  
Gravity Interpretation ◽  
Relationship Of ◽  
Derivatives Of

A procedure suitable for use on high‐speed digital computers is presented for interpreting two‐dimensional gravity anomalies. In order to determine the shape of a disturbing mass with known density contrast, an initial model is assumed and gravity anomalies are calculated and compared with observed values at n points, where n is greater than the number of unknown variables (e.g. depths) of the model. Adjustments are then made to the model by a least‐squares approximation which uses the partial derivatives of the anomalies so that the residuals are reduced to a minimum. In comparison with other iterative techniques, convergence is very rapid. A convenient method to use for both the calculation of the anomalies and the adjustments is the two‐dimensional method of Talwani, Worzel, and Landisman, (1959) in which the outline of the body is polygonized and the anomalies and the partial derivatives of the anomaly with respect to the depth of a vertex on the body can be expressed as functions of the coordinates of the vertex. Not only depths but under certain circumstances regional gravity values may be evaluated; however, the relationship of the disturbing body to the gravity information may impose certain limitations on the application of the procedure.

On: “Gravity Interpretation using the Fourier Integral” by J. W. Berg, Jr., and M. E. Odegard (GEOPHYSICS, vol. 30, no. 3, p. 424–438, June 1965)

Geophysics ◽  
1970 ◽  
Vol 35 (2) ◽  
pp. 358-358 ◽  
Author(s):  
H. A. Meinardus
Keyword(s):  
Fourier Transform ◽  
Gravity Anomalies ◽  
Fourier Integral ◽  
Hankel Transform ◽  
Cylindrical Symmetry ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Gravity ◽  
The One ◽  
Gravity Interpretation

The authors do not state explicitly that equations (1) and (12) express gravity anomalies of the two‐dimensional cylinder and fault respectively. These structures have an infinite extension along the strike, and all the gravity profiles perpendicular to the strike are identical. For this reason it is admissible to apply the one‐dimensional Fourier transform to obtain the one‐dimensional amplitude spectra shown in Figures (1) and (5). The situation, however, is different for the sphere, which does not have a one‐dimensional gravity profile in the above sense, but rather a two‐dimensional field with cylindrical symmetry. Instead of using the one‐dimensional Fourier transform as given by (6) along a straight line, we must apply the two‐dimensional Fourier transform over the whole area. Because of the circular symmetry, one radial variable will suffice in place of the two Cartesian coordinates x and y, and the two‐dimensional Fourier transform can be expressed as Hankel transform, so that equation (6) becomes [Formula: see text]

A COMPREHENSIVE SYSTEM OF AUTOMATIC COMPUTATION IN MAGNETIC AND GRAVITY INTERPRETATION

Geophysics ◽  
1960 ◽  
Vol 25 (3) ◽  
pp. 569-585 ◽  
Author(s):  
Roland G. Henderson
Keyword(s):  
Effective Means ◽  
Gravity Anomalies ◽  
Comprehensive System ◽  
Diagnostic Features ◽  
Special Cases ◽  
Desk Calculator ◽  
Concentric Circles ◽  
Gravity Interpretation ◽  
Magnetic And Gravity Anomalies ◽  
Derivatives Of

In the interpretation of magnetic and gravity anomalies, downward continuation of fields and calculation of first and second vertical derivatives of fields have been recognized as effective means for bringing into focus the latent diagnostic features of the data. A comprehensive system has been devised for the calculation of any or all of these derived fields on modern electronic digital computing equipment. The integral for analytic continuation above the plane is used with a Lagrange extrapolation polynomial to derive a general determinantal expression from which the field at depth and the various derivatives on the surface and at depth can be obtained. It is shown that the general formula includes as special cases some of the formulas appearing in the literature. The process involves a “once for all depths” summing of grid values on a system of concentric circles about each point followed by application of the appropriate one or more of the 19 sets of coefficients derived for the purpose. Theoretical and observed multilevel data are used to illustrate the processes and to discuss the errors. The coefficients can be used for less extensive computations on a desk calculator.

Numerical Analysis of Super-Cavitating Flow Around a Two-Dimensional Cavitator Geometry

2011 ◽  
Author(s):  
Sunho Park ◽  
Shin Hyung Rhee
Keyword(s):  
Experimental Data ◽  
High Speed ◽  
Cavity Length ◽  
Stokes Equations ◽  
Cavitating Flow ◽  
The Body ◽  
Computational Method ◽  
Computational Results ◽  
Two Dimensional ◽  
Cell Counts

Mostly for military purposes, which require high speed and low drag, super-cavitating flows around under-water bodies have been an interesting, yet difficult research subject for many years. In the present study, high speed super-cavitating flow around a two-dimensional symmetric wedge-shaped cavitator was studied using an unsteady Reynolds-averaged Navier-Stokes equations solver based on a cell-centered finite volume method. To verify the computational method, flow over a hemispherical head-form body was simulated and validated against existing experimental data. Through the verification tests, the appropriate selection of domain extents, cell counts, numerical schemes, turbulence models, and cavitation models was studied carefully. A cavitation model based on the two-phase mixture flow modeling was selected with the standard k-epsilon model for turbulence closure. The cavity length, surface pressure distribution, and the flow velocity at the interface were compared with experimental data and analytic solutions. Various computational conditions, such as different wedge angles and caviation numbers, were considered for super-cavitating flow around the wedge-shaped cavitator. Super-cavitation begins to form in the low pressure region and propagates downstream. The computed cavity length and drag on the body were compared with analytic solution and computational results using a potential flow solver. Fairly good agreement was observed in the three-way comparison. The computed velocity on the cavity interface was also predicted quite closely to that derived from the Bernoulli equation. Finally, comparison was made between the computational results and cavitation tunnel test data, along with suggestions for cavitator designs.

scholarly journals EL: Local Image Descriptor Based on Extreme Responses to Partial Derivatives of 2D Gaussian Function

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jasna Maver ◽  
Danijel Skočaj
Keyword(s):  
Image Matching ◽  
Gaussian Function ◽  
Two Dimensional ◽  
Image Descriptor ◽  
Partial Derivatives ◽  
Illumination Changes ◽  
Local Image Descriptor ◽  
Extreme Responses ◽  
Derivatives Of ◽  
Local Image

We propose a two-part local image descriptor EL (Edges and Lines), based on the strongest image responses to the first- and second-order partial derivatives of the two-dimensional Gaussian function. Using the steering theorems, the proposed method finds the filter orientations giving the strongest image responses. The orientations are quantized, and the magnitudes of the image responses are histogrammed. Iterative adaptive thresholding of histogram values is then applied to normalize the histogram, thereby making the descriptor robust to nonlinear illumination changes. The two-part descriptor is empirically evaluated on the HPatches benchmark for three different tasks, namely, patch verification, image matching, and patch retrieval. The proposed EL descriptor outperforms the traditional descriptors such as SIFT and RootSIFT on all three evaluation tasks and the deep-learning-based descriptors DeepCompare, DeepDesc, and TFeat on the tasks of image matching and patch retrieval.

Gravity interpretation using the Mellin transform

Geophysics ◽  
1986 ◽  
Vol 51 (1) ◽  
pp. 114-122 ◽  
Author(s):  
N. L. Mohan ◽  
L. Anandababu ◽  
S. V. Seshagiri Rao
Keyword(s):  
Circular Cylinder ◽  
Gravity Anomaly ◽  
Mellin Transform ◽  
The Body ◽  
Two Dimensional ◽  
Gravity Effect ◽  
Horizontal Derivative ◽  
Gravity Interpretation ◽  
Body Parameters ◽  
Horizontal Circular Cylinder

The Mellin transform of the gravity effect of a buried sphere and two‐dimensional horizontal circular cylinder, and the first horizontal derivative of the gravity effect of a two‐dimensional thin fault layer are derived. The transformed functions are bounded by two asymptotes. They are analyzed and procedures are formulated excluding the asymptotic regions for the extraction of the body parameters. The application of the Mellin transform is tested on simulated models as well as on two field examples: (1) the Humble Dome gravity anomaly near Houston, USA; and (2) the Louga gravity anomaly, USA.

TECHNIQUE FOR IMPROVED CONVERGENCE IN ITERATIVE ANALYSIS OF GRAVITY ANOMALY PROFILES

Geophysics ◽  
1973 ◽  
Vol 38 (3) ◽  
pp. 500-506 ◽  
Author(s):  
K. P. Fournier ◽  
S. F. Krupicka
Keyword(s):  
Bouguer Anomaly ◽  
Gravity Anomalies ◽  
Computer Time ◽  
Two Dimensional ◽  
Profile Data ◽  
Plate Method ◽  
Modeling Process ◽  
Iterative Analysis ◽  
Dimensional Gravity ◽  
Number Of Iterations

Narrow two‐dimensional gravity anomalies are difficult to interpret iteratively by the relatively simple flat‐plate method suggested by Bott (1960) and employed by others. In this report Bott’s formula is modified empirically after a number of iterations have proceeded. The process is repeated until satisfactory convergence is obtained. The results can be applied to other profiles across the anomaly of interest and thus save substantial computer time by requiring fewer iterations. The technique is applied to two Bouguer anomaly profiles. Analysis of adjoining anomaly profile data with similar amplitudes and widths can be used to speed the convergence in the iterative modeling process.

Gravity interpretation using correlation factors between successive least‐squares residual anomalies

Geophysics ◽  
1989 ◽  
Vol 54 (12) ◽  
pp. 1614-1621 ◽  
Author(s):  
E. M. Abdelrahman ◽  
A. I. Bayoumi ◽  
Y. E. Abdelhady ◽  
M. M. Gobashy ◽  
H. M. El‐Araby
Keyword(s):  
Least Squares ◽  
Bouguer Anomaly ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
The United States ◽  
Horizontal Cylinder ◽  
Regional Component ◽  
Gravity Interpretation ◽  
Regional Gravity ◽  
Correlation Factors

The correlation factors between successive least‐squares residual (or regional) gravity anomalies from a buried sphere, a two‐dimensional (2‐D) horizontal cylinder, and a vertical cylinder and the first horizontal derivative of the gravity from a 2‐D thin faulted layer are computed. Correlation values are used to determine the depth to the center of the buried structure, and the radius of the sphere or the cylinder and the thickness of the fault are estimated. 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 structure and a regional component represented by a polynomial of any order. The method is easy to apply and may be automated if desired. It can also be applied to the derivative anomalies of the gravity field. The validity of the method is tested on two field examples from the United States and Denmark.

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