A CONTRIBUTION TO THE COMPUTATION OF THE “SECOND DERIVATIVE” FROM GRAVITY DATA

Geophysics ◽  
1953 ◽  
Vol 18 (4) ◽  
pp. 894-907 ◽  
Author(s):  
Otto Rosenbach
Keyword(s):  
Gravity Data ◽  
Second Derivative ◽  
Practical Application ◽  
Arithmetic Means ◽  
Interesting Paper ◽  
Approximation Formulas ◽  
Series Development ◽  
Concentric Circles ◽  
Gravity Interpretation ◽  
Second Derivative Method

The theory and practical application of the second derivative method of gravity interpretation have been discussed by Elkins (1951) in a very interesting paper based partly on an earlier paper by Peters (1949). In this paper, Elkins shows how the second derivative may be computed at the center of a series of concentric circles using the arithmetic means of the gravity values, assumed to be continuous, around each circle. A method is here proposed which does not use a continuum of gravity values but instead requires only a series development. The approximation formulas needed for the routine calculations can be derived by a method different from that of Elkins and the least squares adjustments he used can be dispensed with. Two hypothetical examples using the formulas derived by the series method are given and the results are compared with those computed by Elkins’ formula.

NOTE ON THE EFFECT OF RANDOM ERRORS IN GRAVITY SECOND DERIVATIVE VALUES

Geophysics ◽  
1953 ◽  
Vol 18 (3) ◽  
pp. 720-724
Author(s):  
G. Ramaswamy
Keyword(s):  
Gravity Data ◽  
Prima Facie ◽  
Resolving Power ◽  
Magnetic Data ◽  
Second Derivative ◽  
Random Component ◽  
Random Errors ◽  
Prima Facie Case ◽  
Interesting Paper ◽  
Second Derivative Method

In an interesting paper “The Effect of Random Errors in Gravity Data on Second Derivative Values,” Thomas A. Elkins (1952) points out the need for eliminating the random component of gravity data before proceeding to interpret them with the aid of second derivative maps. There is a prima facie case for the elimination of random errors, if only to ensure reliability in results. The need, however, becomes more apparent if it is remembered: a. that the Second Derivative method of interpreting gravity (or magnetic) data is one of high resolving power, and b. that, therefore, errors creeping into those data may considerably vitiate the interpretation of those data.

THE SECOND DERIVATIVE METHOD OF GRAVITY INTERPRETATION

Geophysics ◽  
1951 ◽  
Vol 16 (1) ◽  
pp. 29-50 ◽  
Author(s):  
Thomas A. Elkins
Keyword(s):  
Horizontal Plane ◽  
Graphical Method ◽  
Gravity Data ◽  
Three Dimensional ◽  
Horizontal Cylinder ◽  
Resolving Power ◽  
Theoretical Formula ◽  
Second Derivative ◽  
Derivative Method ◽  
Second Derivative Method

The second derivative method of interpreting gravity data, although its use is justifiable only on data of high accuracy, offers a simple routine method of locating some types of geologic anomalies of importance in oil and mineral reconnaissance. The theoretical formula by which it is possible to compute the second (vertical) derivative of any harmonic function from its values in a horizontal plane is derived for both the two‐dimensional and the three‐dimensional cases. The graphical method of computing the second derivative is discussed, especially as to the sources of error. A numerical coefficient equivalent of the graphical method is also presented. Formulas and graphs for the second derivative of the gravity effect of such geometrically simple shapes as the sphere, the infinite horizontal cylinder, the semi‐infinite horizontal plane, and the vertical fault, are presented with discussions of their value in the interpretation of practical data. Finally, the gravity and second derivative maps of portions of some important oil provinces are presented and compared to show the higher resolving power of the second derivative.

AMBIGUITY IN GRAVITY INTERPRETATION

Geophysics ◽  
1947 ◽  
Vol 12 (1) ◽  
pp. 43-56 ◽  
Author(s):  
D. C. Skeels
Keyword(s):  
Gravity Data ◽  
Magnetic Data ◽  
Second Derivative ◽  
Two Dimensional ◽  
Wide Range ◽  
Geometrical Shapes ◽  
Unique Interpretation ◽  
Analytical Proof ◽  
Minimum Depth ◽  
Gravity Interpretation

It is shown that contrary to what is stated and implied in much of the literature, gravity data cannot, of themselves, be interpreted uniquely. It is shown by means of a two‐dimensional example that for a given anomaly and a given density contrast a wide range of possible interpretations can be made, at various depths, and that whereas there is a maximum depth for the solution the minimum depth is zero. Other examples are given to show that depth rules based upon the assumption of geometrical shapes may give results very much in error when applied to actual anomalies. Nor does the method of interpretation by vertical gradients allow us to make an unique interpretation, or to distinguish deep from shallow anomalies as has been claimed. It is shown that we do not escape the ambiguity by using second derivative quantities such as gradient and curvature, and that, in fact, gravity and its derivatives are related by a corollary of Green’s theorem. This theorem provides an analytical proof of ambiguity not only for the case of gravity data but for magnetic data as well.

TWO‐DIMENSIONAL FILTERING AND THE SECOND DERIVATIVE METHOD

Geophysics ◽  
1966 ◽  
Vol 31 (3) ◽  
pp. 606-617 ◽  
Author(s):  
C. A. Meskó
Keyword(s):  
Second Derivative ◽  
Two Dimensional ◽  
Frequency Responses ◽  
Center Point ◽  
Average Accuracy ◽  
Two Dimensional Filtering ◽  
Derivative Formulas ◽  
Ring Method ◽  
Second Derivative Method ◽  
Accuracy Of Approximation

The data‐processing operations of second derivative formulas are equivalent to two‐dimensional digital filtering operations. Therefore any coefficient set can be described unambiguously by its two‐dimensional frequency response. The frequency responses can be represented by surfaces over the two‐dimensional frequency plane. We have simplified the representation by giving some curves intersected from these surfaces by a) planes, b) cylindrical surfaces perpendicular to the frequency plane. The coefficient sets given by Elkins (1951), Henderson and Zietz (1949), Rosenbach (1953), and the “center‐point‐and‐one‐ring” method are analysed. These formulas, in order of increasing “average” accuracy of approximation, are Henderson and Zietz, Rosenbach, “center‐point‐and‐one‐ring” method, Elkins. Henderson and Zietz’s formula and the “center‐point‐and‐one‐ring” method have depended significantly on direction, while Rosenbach’s formula is nearly nondirectional. Elkins’ formula lies between them.

Gravity interpretation using Fourier transforms and simple geometrical models with exponential density contrast

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1074-1083 ◽  
Author(s):  
D. Bhaskara Rao ◽  
M. J. Prakash ◽  
N. Ramesh Babu
Keyword(s):  
Los Angeles ◽  
Fourier Transforms ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Sedimentary Basins ◽  
Density Contrast ◽  
Model Parameters ◽  
Exponential Density ◽  
Exponential Density Contrast ◽  
Gravity Interpretation

The decrease of density contrast in sedimentary basins can often be approximated by an exponential function. Theoretical Fourier transforms are derived for symmetric trapezoidal, vertical fault, vertical prism, syncline, and anticline models. This is desirable because there are no equivalent closed form solutions in the space domain for these models combined with an exponential density contrast. These transforms exhibit characteristic minima, maxima, and zero values, and hence graphical methods have been developed for interpretation of model parameters. After applying end corrections to improve the discrete transforms of observed gravity data, the transforms are interpreted for model parameters. This method is first tested on two synthetic models, then applied to gravity anomalies over the San Jacinto graben and Los Angeles basin.

Different Spectrophotometric and Chromatographic Methods for Determination of Mepivacaine and Its Toxic Impurity

2017 ◽  
Vol 100 (5) ◽  
pp. 1392-1399 ◽  
Author(s):  
Nada S Abdelwahab ◽  
Nehal F Fared ◽  
Mohamed Elagawany ◽  
Esraa H Abdelmomen
Keyword(s):  
Particle Size ◽  
Stationary Phase ◽  
Second Derivative ◽  
Spectrophotometric Measurement ◽  
Spectrophotometric Methods ◽  
Derivative Method ◽  
Tlc Plates ◽  
Second Derivative Method ◽  
Developing System

Abstract Stability-indicating spectrophotometric, TLC-densitometric, and ultra-performance LC (UPLC) methods were developed for the determination of mepivacaine HCl (MEP) in the presence of its toxic impurity, 2,6-dimethylanaline (DMA). Different spectrophotometric methods were developed for the determination of MEP and DMA. In a dual-wavelength method combined with direct spectrophotometric measurement, the absorbancedifference between 221.4 and 240 nm was used for MEPmeasurements, whereas the absorbance at 283 nm was used for measuring DMA in the binary mixture. In the second-derivative method, amplitudes at 272.2 and 232.6 nm were recorded and used for the determination of MEP and DMA, respectively. The developed TLC-densitometric method depended on chromatographic separation using silica gel 60 F254 TLC plates as a stationary phase and methanol–water–acetic acid (9 + 1 + 0.1, v/v/v) as a developing system, with UV scanning at 230 nm. The developed UPLC method depended on separation using a C18 column (250 × 4.6 mm id, 5 μm particle size) as a stationary phase and acetonitrile–water (40 + 60, v/v; pH 4 with phosphoric acid) as a mobilephase at a flow rate of 0.4 mL/min, with UV detection at 215 nm. The chromatographic run time was approximately 1 min. The proposed methods were validated with respect to International Conference on Harmonization guidelines regarding precision, accuracy, ruggedness, robustness, and specificity.

scholarly journals Trigonometrically-fitted second derivative method for oscillatory problems

SpringerPlus ◽  
2014 ◽  
Vol 3 (1) ◽  
Author(s):  
Fidele Fouogang Ngwane ◽  
Samuel Nemsefor Jator
Keyword(s):  
Second Derivative ◽  
Derivative Method ◽  
Second Derivative Method

Usefulness of a Curve Fitting Method in the Analysis of Overlapping Overtones and Combinations of CH Stretching Modes

2002 ◽  
Vol 10 (1) ◽  
pp. 85-91 ◽  
Author(s):  
Yukiteru Katsumoto ◽  
Daisuke Adachi ◽  
Harumi Sato ◽  
Yukihiro Ozaki
Keyword(s):  
Curve Fitting ◽  
Second Derivative ◽  
Downward Shift ◽  
Fitting Method ◽  
Derivative Method ◽  
Structural Differences ◽  
Alcohol Mixtures ◽  
Curve Fitting Method ◽  
Second Derivative Method ◽  
Water Alcohol

This paper reports the usefulness of a curve fitting method in the analysis of NIR spectra. NIR spectra in the 7500–5500 cm−1 (1333–1818 nm) region were measured for water–methanol, water–ethanol and water–1-propanol mixtures with alcohol concentrations of 0–100 wt% at 25°C. The 6000–5600 cm−1 (1667–1786 nm) region, where the overtones and combinations of CH3 and CH2 stretching modes are expected to appear, shows significant band shifts with the increase in the alcohol content. To analyse the concentration-dependent spectral changes, a curve fitting method was utilised, and the results were compared with those obtained previously by a second derivative method. It was found that the first overtones of CH3 asymmetric and symmetric stretching modes of alcohols show a downward shift by about 15–30 cm−1 with the increase in the concentration of alcohols. The shifts are much larger for water–methanol mixtures than for water–ethanol and water–1-propanol mixtures. The first overtones and combinations of CH2 stretching modes of ethanol and 1-propanol also show a small downward shift. These shifts support our previous conclusion that there is an intermolecular “CH⃛O” interaction between the methyl group and water in the water–alcohol mixtures. The curve fitting method provided more feasible results for the band shifts than the second derivative method. It was revealed from the curve fitting method that the first overtone of the CH3 asymmetric stretching mode of water–methanol, water–ethanol and water–1-propanol mixtures shows different concentration-dependent plots. The first overtone of CH3 asymmetric stretching mode of water–methanol mixtures shifts more rapidly in the high methanol concentration range while that of water–1-propanol concentration shifts more markedly in the low 1-propanol concentration range. That of water–ethanol mixtures shows an intermediate trend. Based upon these differences structural differences among the three kinds of water–alcohol mixtures are discussed.

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