BICUBIC SPLINE INTERPOLATION AS A METHOD FOR TREATMENT OF POTENTIAL FIELD DATA

Geophysics ◽  
1969 ◽  
Vol 34 (3) ◽  
pp. 402-423 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Field Data ◽  
Spline Interpolation ◽  
Amplitude Spectrum ◽  
Accurate Method ◽  
Fourier Series Expansion ◽  
Small Scale ◽  
Spline Functions ◽  
Two Dimensional ◽  
Potential Field Data ◽  
Bicubic Spline

A method for the generation of bicubic spline functions is presented in this paper. From this method it becomes apparent that these functions derive their potential strength in accurate and reliable representation of two‐dimensional data by maintaining continuity of the variable and its slope and curvature throughout the area of observation. The results obtained by computing horizontal and vertical derivatives with model and field data illustrate the exceptional accuracy achieved with spline functions. The piecewise cubic polynomial functions expressing observed data analytically in space are used to estimate amplitude and phase spectra of magnetic anomalies. At relatively long wavelengths the amplitude spectrum thus calculated displays remarkable similarity with the true spectrum and is found to be superior to that obtained with two‐dimensional Fourier series expansion. A cubic spline method is also presented for computing values of an observed variable at equispaced points along two orthogonal directions with the help of irregularly distributed data. The interpolation technique applied to field data shows high resolution by maintaining the separation of neighboring anomalies and the small‐scale features. The shapes, peaks, and troughs of both large and small amplitude anomalies are faithfully reproduced. The gradients of the magnetic field do not undergo any appreciable distortion. It can thus be concluded that cubic splines are a reliable and accurate method of interpolation.

DESIGN OF SMALL OPERATORS FOR THE CONTINUATION OF POTENTIAL FIELD DATA

Geophysics ◽  
1972 ◽  
Vol 37 (3) ◽  
pp. 488-506 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Potential Field ◽  
Field Data ◽  
Grid Point ◽  
Frequency Characteristics ◽  
Potential Fields ◽  
Two Dimensional ◽  
Potential Field Data ◽  
Computational Work ◽  
Considerable Loss ◽  
Equivalent Operators

Two‐dimensional continuation of potential fields is commonly achieved by employing a continuation operator which consists of a number of coefficients operating upon uniformly gridded field data. To obtain accurate results, the size of the operator has to be quite large. This not only requires a lot of computational work, but also causes a considerable loss of information due to the reduced size of the field obtained after continuation. Small‐size “equivalent” operators were designed which are free from these drawbacks but yield accurate results. In order to demonstrate the efficiency of these operators, a potential field was continued upward by using 31×31 Tsuboi coefficients. This required 961 multiplications for computing the continued field at each grid point. When procedure was repeated using the equivalent operator, the number of multiplications required for each grid point was reduced to 15, the size of the resulting map was much larger, but the results in both cases were practically identical in accuracy. Frequency characteristics of the equivalent operators and the continuation of data very close to the boundary of the field map are discussed.

The Method of Curvature Attribute applied in the Depth Inversion of the Geological Bodies Edge by Potential Field Data

2020 ◽  
Author(s):  
Jinlan Liu ◽  
Wanyin Wang ◽  
Shengqing Xiong
Keyword(s):  
Objective Function ◽  
Potential Field ◽  
Field Data ◽  
Small Scale ◽  
Inversion Method ◽  
Potential Field Data ◽  
Upward Continuation ◽  
Extraction Algorithm ◽  
Key Technologies ◽  
Key Techniques

<p>It is vital to quickly and effectively determine the extent and depth of geological body by using potential field data in gravity and magnetic survey. In this study, three key techniques studying the extent and depth of geological sources based on curvature attribute are studied: the optimal solutions to the objective function, the edge of geological bodies and picking out solutions. Firstly, the optimal solution to the objective function is studied, that is, the key extraction algorithm about the curvature attribute. The Huber norm is introduced into the extraction algorithm of curvature attribute, which more accurately detect the depth of edge of the geological bodies. Secondly, the normalized vertical derivative of the total horizontal derivative (NVDR-THDR) technique is introduced into curvature attribute, which shows more continuous results about the edge position of the geological bodies and more sensitive to the small-scale tectonic structure. Finally, we study the way to pick out the inversion solution, that is, to solve the multi-solution equations in the inversion. The upward continuation of a certain height with strict physical significance was introduced into the inversion method, which was used to suppress the noise, and the final and actual inversion depth was equal to the inversion depth minus the height of upward continuation. And the average value of threshold limitation technology of the potential fields data was also introduced into this method. Using the two technologies, solutions of non-field source edge positions were eliminated, and make the inversion solutions closer to the actual situation. Through the above three key techniques, the accuracy, continuity and recognition to the small-scale structure of the inversion result are optimized. The theoretical models are used to verify the effectiveness of the above key technologies, the results show that the three key technologies have achieved good results, and the combined models are used to verify the effectiveness of the optimized inversion method. The measured aeromagnetic data were used to inversing the edge depth of the intrusive rock in a mining area, and the inversion results are in good agreement with the rock depth revealed by borehole.</p>

RECURSION FILTERS FOR DIGITAL PROCESSING OF POTENTIAL FIELD DATA

Geophysics ◽  
1976 ◽  
Vol 41 (4) ◽  
pp. 712-726 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Frequency Domain ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Vertical Gradient ◽  
Two Dimensional ◽  
Single Variable ◽  
Potential Field Data ◽  
Recursive Filter ◽  
Rational Expression

Zero‐phase two‐dimensional recursive filters, with a specified frequency domain response, have been designed for processing potential field data. In the case of second vertical derivative filters, it is possible to use the rational approximation of symmetrical functions of a single variable for the design of a short recursive filter. The filter so designed has an excellent response in the frequency domain. For vertical gradient and continuation filters, a method is developed for obtaining, by the least‐squares method, a rational expression for a two‐dimensional symmetrical function. In order to ensure the stability of the recursive filter, the denominator of the rational expression is approximated by a product of two factors, each being a function of a single variable. Finally, to keep the error of the filter response as small as possible, an iterative procedure is used for adjusting the zeros of the denominator and then determining the coefficients of the numerator of the rational expression.

AUTOMATIC TREND ANALYSIS AND INTERPRETATION OF POTENTIAL FIELD DATA

Geophysics ◽  
1971 ◽  
Vol 36 (2) ◽  
pp. 339-348 ◽  
Author(s):  
R. G. Agarwal ◽  
E. R. Kanasewich
Keyword(s):  
Fourier Transform ◽  
Correlation Matrix ◽  
Field Data ◽  
Empirical Method ◽  
Magnetic Data ◽  
Two Dimensional ◽  
Degree Polynomial ◽  
Potential Field Data ◽  
Maximum Coefficient ◽  
Crosscorrelation Function

In this investigation, the fast Fourier transform is applied to the analysis of two‐dimensional data. In particular, a crosscorrelation function has been used to obtain trend directions of magnetic data. The technique is applicable to other geological or geophysical data. An empirical method has been devised to trace the trends by fitting a least squares third degree polynomial to the maximum coefficient in the correlation matrix. An example from Stony Rapids, Northern Saskatchewan illustrates the method.

ADDITIONAL COMMENTS ON THE ANALYTIC SIGNAL OF TWO‐DIMENSIONAL MAGNETIC BODIES WITH POLYGONAL CROSS‐SECTION

Geophysics ◽  
1974 ◽  
Vol 39 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Misac N. Nabighian
Keyword(s):  
Cross Section ◽  
Potential Field ◽  
Input Data ◽  
Field Data ◽  
Field Component ◽  
Analytic Signal ◽  
Two Dimensional ◽  
Field Profile ◽  
Total Field ◽  
Potential Field Data

In a previous paper (Nabighian, 1972), the concept of analytic signal of bodies of polygonal cross‐section was introduced and its applications to the interpretation of potential field data were discussed. The input data for the proposed treatment are the horizontal derivative T(x) of the field profile, whether horizontal, vertical, or total field component. As it is known, this derivative curve can be thought of as being due to thin magnetized sheets surrounding the causative bodies.

scholarly journals Vibration Analysis of Rectangular Plates with One or More Guided Edges via Bicubic B-Spline Method

2005 ◽  
Vol 12 (5) ◽  
pp. 363-376 ◽  
Author(s):  
W.J. Si ◽  
K.Y. Lam ◽  
S.W. Gong
Keyword(s):  
Vibration Analysis ◽  
Plate Theory ◽  
Spline Interpolation ◽  
Accurate Method ◽  
Spline Functions ◽  
Rectangular Plates ◽  
B Spline ◽  
Thin Plate Theory ◽  
Edge Conditions ◽  
New Type

A simple and accurate method is proposed for the vibration analysis of rectangular plates with one or more guided edges, in which bicubic B-spline interpolation in combination with a new type of basis cubic B-spline functions is used to approximate the plate deflection. This type of basis cubic B-spline functions can satisfy simply supported, clamped, free, and guided edge conditions with easy numerical manipulation. The frequency characteristic equation is formulated based on classical thin plate theory by performing Hamilton's principle. The present solutions are verified with the analytical ones. Fast convergence, high accuracy and computational efficiency have been demonstrated from the comparisons. Frequency parameters for 13 cases of rectangular plates with at least one guided edge, which are possible by approximate or numerical methods only, are presented. These results are new in literature.

DESIGN OF SPATIAL FILTERS AND THEIR APPLICATION TO HIGH‐RESOLUTION AEROMAGNETIC DATA

Geophysics ◽  
1972 ◽  
Vol 37 (1) ◽  
pp. 68-91 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
High Resolution ◽  
Field Data ◽  
Vertical Gradient ◽  
Double Fourier Series ◽  
Two Dimensional ◽  
Total Field ◽  
Spatial Filters ◽  
Aeromagnetic Survey ◽  
Potential Field Data ◽  
High Frequency Response

Methods for the design of spatial filters are discussed in this paper. For a given response of a one‐dimensional filter, the weighting coefficients are calculated by solving a set of simultaneous equations with a simple matrix inversion procedure. In the case of a two‐dimensional filter, the method for obtaining the coefficients of a double Fourier series representing a set of given values is used to design the spatial operator. The problems connected with the length of the operator and the choice of a suitable decay in the high‐frequency response are discussed in detail. In order to show the usefulness of these methods, the paper presents several examples of operators designed for computing the vertical gradient, the second vertical derivative, and downward continuation of potential field data. A two‐dimensional vertical gradient filter is applied to the total field data obtained during a high‐resolution aeromagnetic survey over an area in the Precambrian Shield of Northeastern Ontario. The calculated gradient maps are compared with maps showing measured gradient values. The quality of the calculated maps in defining trends, patterns, and detailed features of anomalies shows the feasibility of obtaining very accurate vertical gradient maps from observed total field data.

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