AN INTRODUCTION TO THE SECOND DERIVATIVE CONTOUR METHOD OF INTERPRETING TORSION BALANCE DATA

Geophysics ◽  
1938 ◽  
Vol 3 (3) ◽  
pp. 234-246 ◽  
Author(s):  
H. Klaus
Keyword(s):  
Local Effect ◽  
Torsion Balance ◽  
Resolving Power ◽  
Contour Method ◽  
Second Derivative ◽  
Regional Effects ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Regional Gradient ◽  
Regional Gravity

After auspicious beginnings in the interpretation of torsion balance data, i.e., gradients and curvatures, the balance has been misused as a gravity instrument, the gradients being integrated into gravity, and the curvatures either neglected or not even observed in the field. Gravity was then made the sole basis of interpretation work, the regional effects being determined with more or less luck, subtracted from the total either before or after integration (regional gravity or regional gradient), and the residue held to be “local effect.” This method appears to be now in vogue for most torsion balance and gravity meter work. In contrast to this procedure, the method here described is based on the quantities measured directly by the torsion balance, the gradients and curvatures, or second derivatives, and constitutes a considerable amplification of the original methods of investigating these quantities. Gravity is simply a by‐product of this method, and is not needed at all for its functioning. The essential parts of this method are: 1) the re‐determination of all second derivative components with respect to a new system of rectangular coordinates, one axis of which has been made parallel to the direction of elongation of anomalous features; 2) the contouring of these second derivative components on four separate maps; and 3), the interpretation of the resulting contour patterns. The outstanding advantages of this method over the total gravity methods are the following: 1) full utilization of the two independent aspects of the gravitational field furnished by the gradients and curvatures; 2) virtual independence from regional effects; 3) much greater resolving power when compared to gravity; and 4), complete absence of assumptions, such as are involved in estimating the regional, and in computing gravity from the gradients.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

TORSION BALANCE STEP ANOMALIES IN NORTHERN TILLMAN COUNTY, OKLAHOMA

Geophysics ◽  
1945 ◽  
Vol 10 (4) ◽  
pp. 507-525
Author(s):  
K. Klaus
Keyword(s):  
Cross Sections ◽  
Torsion Balance ◽  
Second Derivative ◽  
Contour Maps ◽  
Probability Rating ◽  
Order Of Magnitude ◽  
Wichita Mountains ◽  
Gravity Map ◽  
High Degree ◽  
Quantitative Calculations

The results of a semi‐detailed areal torsion balance survey in Southwestern Oklahoma are shown by means of a gradient‐curvature map, a gravity map, two second derivative contour maps, and gravity and second derivative profiles. Detailed quantitative calculations were made of a number of geological cross sections, two of which are shown in Figs. 8 and 9. Fig. 9 represents the subsurface situation with the highest probability rating, since it combines a high degree of geological probability with the fact that it will reproduce the gravity and second derivative curves of Figs. 6 and 7 very closely. This interpretation embodies a fault with a throw of the order of magnitude of 10,000 feet. If this interpretation is substantially correct, it implies a thick sedimentary section in the down‐thrown block, which might be of great economic interest in prospecting for oil. A comparison of the gravity and second derivative data may be of interest to the geophysicist. The geologist may find the results of this survey interesting because of their possible bearing on the orogeny of the Wichita Mountains.

FAULTING IN THE BILLINGS OILFIELD, OKLAHOMA, AS INTERPRETED FROM TORSION BALANCE DATA, AND FROM SUBSEQUENT DRILLING

Geophysics ◽  
1943 ◽  
Vol 8 (4) ◽  
pp. 362-378
Author(s):  
H. Klaus
Keyword(s):  
Torsion Balance ◽  
Second Derivative ◽  
Oil Companies ◽  
Contour Maps ◽  
Made In

The results of an experimental torsion balance survey of the Billings Oilfield are shown by means of the conventional maps, and of second derivative contour maps. The latter show good and consistent anomalies, which are interpreted as faulting. The survey was made in July, 1937, some time after discovery of Ordovician production in the field, but long before faulting was clearly defined by drilling, and the results were communicated to several oil companies interested in the area or in the method of interpretation. In the meantime, one of the faults limiting the field has been defined in detail by drilling, and the present subsurface interpretation is compared with the original torsion balance predictions with respect to this fault. From the amount of agreement between the two sets of data, it is concluded that the torsion balance can still be used effectively for specialized purposes, particularly the investigation of faulting.

Use of growth curve derivatives to illustrate acceleration and deceleration of growth in young plantations under variable competition

2006 ◽  
Vol 36 (10) ◽  
pp. 2515-2522 ◽  
Author(s):  
Michael Newton ◽  
Elizabeth C Cole
Keyword(s):  
Growth Period ◽  
Tsuga Heterophylla ◽  
Douglas Fir ◽  
Early Years ◽  
Western Hemlock ◽  
Richards Equation ◽  
Second Derivative ◽  
Red Alder ◽  
Vegetation Control ◽  
Second Derivatives

Deceleration of growth rates can give an indication of competition and the need for thinning in early years but can be difficult to detect. We computed the first and second derivatives of the von Bertalanffy – Richards equation to assess impacts of density and vegetation control in young plantations in western Oregon. The first derivative describes the response in growth and the second derivative describes the change in growth over time. Three sets of density experiments were used: (i) pure Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco), (ii) mixed Douglas-fir and grand fir (Abies grandis (Dougl. ex D. Don) Lindl.), and (iii) mixed western hemlock (Tsuga heterophylla (Raf.) Sarg.) and red alder (Alnus rubra Bong.). Original planting densities ranged from 475 to 85 470 trees·ha–1 (4.6 m × 4.6 m to 0.34 m × 0.34 m spacing); western hemlock and red alder plots were weeded and unweeded. For the highest densities, the second derivative was rarely above zero for any of the time periods, indicating that the planting densities were too high for tree growth to enter an exponential phase. As expected, the lower the density, the greater and later the peak in growth for both the first and second derivatives. Weeding increased the growth peaks, and peaks were reached earlier in weeded than in unweeded plots. Calculations of this sort may help modelers identify when modifiers for competition and density are needed in growth equations. Specific applications help define onset of competition, precise determining of timing of peak growth, period of acceleration of growth, and interaction of spacing and age in determination of peaks of increment or acceleration or deceleration.

scholarly journals Advanced analysis in epidemiological modeling: Detection of wave

2021 ◽  
Author(s):  
Abdon Atangana ◽  
Seda IGRET ARAZ
Keyword(s):  
Simple Model ◽  
Reproductive Number ◽  
Second Derivative ◽  
Mathematical Concepts ◽  
Epidemiological Modeling ◽  
Derivative Analysis ◽  
Second Derivatives ◽  
Advanced Analysis ◽  
The Stability ◽  
Derivatives Of

Some mathematical concepts have been used in the last decades to predict the behavior of spread of infectious diseases. Among them, the reproductive number concept has been used in several published papers for study the stability of the spread. Some conditions were suggested to predict there would be either stability or instability. An analysis was also suggested to determine conditions under which infectious classes will increase or die out. Some authors pointed out limitations of the reproductive number, as they presented its inability to fairly help understand the spread patterns. The concept of strength number and analysis of second derivatives of the mathematical models were suggested as additional tools to help detect waves. In this paper, we aim at applying these additional analyses in a simple model to predict the future. Keywords: Strength number, second derivative analysis, waves, piecewise modeling.

Direct Use of Second Derivatives in Curve-Fitting Procedures

1989 ◽  
Vol 43 (5) ◽  
pp. 877-882 ◽  
Author(s):  
Farida Holler ◽  
David H. Burns ◽  
James B. Callis
Keyword(s):  
Curve Fitting ◽  
A Priori ◽  
Simulated Data ◽  
Relevant Information ◽  
Second Derivative ◽  
Unknown Variable ◽  
Fitting Method ◽  
Second Derivatives ◽  
Fitting Procedures ◽  
Ft Ir

Most curve-fitting procedures deal with an unknown, variable baseline by modeling it with a function involving a number of parameters. In view of the facts that (1) there is often no analytically relevant information in the baseline, and (2) there is usually no functional form known, a priori, for the baseline, we have chosen to eliminate it by means of the second-derivative transformation. The resulting profile is deconvoluted by fitting it with the second derivative of the sum of an appropriate number of component curves. The utility of this procedure is demonstrated on simulated data with typical baselines and noise levels, and on real FT-IR data. Peak parameters (such as position, width, and area) obtained from this technique are comparable to those obtained by fitting the original spectrum with Lorentzian curves and a simple baseline. The major advantage of this procedure is the reduction in the number of parameters that must be optimized in the fitting method. Applications of the technique could eliminate contributions from other complex baseline profiles in the quantitative analysis of spectral components.

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.

Evaluation of multi-wavelength derivative spectra for quantitative applications in clinical chemistry.

1983 ◽  
Vol 29 (9) ◽  
pp. 1673-1677 ◽  
Author(s):  
W E Weiser ◽  
H L Pardue
Keyword(s):  
Light Scattering ◽  
Clinical Chemistry ◽  
Second Derivative ◽  
Background Spectrum ◽  
Original Solution ◽  
Second Derivatives ◽  
Relative Standard ◽  
Multi Wavelength ◽  
Error Bias ◽  
Second Derivative Spectroscopy

Abstract We have used the quantitation of methemoglobin in the presence of a light-scattering suspension as a model to evaluate and compare the relative merits of zeroth-, first-, and second-derivative spectroscopy for problems in clinical chemistry. Data for second derivatives are most effective in nullifying the effect of the background spectrum; data for first derivatives are less effective; and data for absorbance (zeroth derivative) are least effective. In one set of experiments, variable amounts of the light-scattering component were added to a methemoglobin solution to increase the apparent absorbance up to about 230% of the absorbance of the original solution. Whereas computation with data for absorbance would have yielded errors approaching 100%, concentrations computed with second-derivative spectral data yielded a systematic error (bias) of only 0.5% and a relative standard deviation (CV) of 1.6%.

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.

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