Relative precision of vertical and horizontal gravity gradients measured by gravimeter

Geophysics ◽  
1979 ◽  
Vol 44 (1) ◽  
pp. 99-101 ◽  
Author(s):  
Sigmund Hammer
Keyword(s):  
Survey Data ◽  
Hilbert Transform ◽  
Gravity Data ◽  
Gradient Methods ◽  
Vertical Gradient ◽  
Gravity Gradients ◽  
Relative Precision ◽  
The Hilbert Transform ◽  
Vertical Gradients ◽  
Vertical Gradient Of Gravity

Several recent publications advocate the use of the vertical gradient of gravity from gravimeter measurements at two elevations in a portable tower (Thyssen‐Bornemisza, 1976; Fajklewicz, 1976; Mortimer, 1977). Contrary opinions have also been expressed (Hammer and Anzoleaga, 1975; Stanley and Green, 1976; Thysen‐Bornemisza, 1977; Arzi, 1977). The disagreement revolves around the question of practically attainable precision of the vertical gradient tower method. Although it is possible to calculate both horizontal and vertical gradients from conventional gravity survey data by use of the Hilbert transform (Stanley and Green, 1976), it should be noted that highly precise gravity data are required. Also the need for connected elevation and location surveys, the major cost in gravity surveying, is not avoided. This is a significant advantage of the gradient methods. The purpose here is to present a brief consideration of the relative precision of the horizontal and vertical gradients, as measured in the field by special gravimeter observations.

GRAVITY GRADIENTS AND THE INTERPRETATION OF THE TRUNCATED PLATE

Geophysics ◽  
1976 ◽  
Vol 41 (6) ◽  
pp. 1370-1376 ◽  
Author(s):  
John M. Stanley ◽  
Ronald Green
Keyword(s):  
Hilbert Transform ◽  
Theoretical Data ◽  
Vertical Gradient ◽  
Density Contrast ◽  
Horizontal Gradient ◽  
Gravity Gradients ◽  
Gravity Method ◽  
Commercially Important ◽  
Dip Angle ◽  
Vertical Gradients

The truncated plate and geologic contact are commercially important structures which can be located by the gravity method. The interpretation can be improved if both the horizontal and vertical gradients are known. Vertical gradients are difficult to measure precisely, but with modern gravimeters the horizontal gradient can be measured conveniently and accurately. This paper shows how the vertical gradient can be obtained from the horizontal gradient by the use of a Hilbert transform. A procedure is then presented which easily enables the position, dip angle, depth, thickness, and density contrast of a postulated plate to be precisely and unambiguously derived from a plot of the horizontal gradient against the vertical gradient at each point measured. The procedure is demonstrated using theoretical data.

Interval gravity‐gradient determination concepts

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 828-832 ◽  
Author(s):  
Dwain K. Butler
Keyword(s):  
Finite Difference ◽  
Gravity Data ◽  
Vertical Gradient ◽  
Gravity Gradient ◽  
Gravity Gradients ◽  
Tower Structure ◽  
Gravity Measurements ◽  
Vertical Gradients

Considerable attention has been directed recently to applications of gravity gradients, e.g., Hammer and Anzoleaga (1975), Stanley and Green (1976), Fajklewicz (1976), Butler (1979), Hammer (1979), Ager and Liard (1982), and Butler et al. (1982). Gravity‐gradient interpretive procedures are developed from properties of true or differential gradients, while gradients are determined in an interval or finite‐difference sense from field gravity data. The relations of the interval gravity gradients to the true or differential gravity gradients are examined in this paper. Figure 1 illustrates the concepts of finite‐difference procedures for gravity‐gradient determinations. In Figure 1a, a tower structure is illustrated schematically for determining vertical gradients. Gravity measurements are made at two or more elevations on the tower, and various finite‐difference or interval values of vertical gradient can be determined. For measurements at three elevations on the tower, for example, three interval gradient determinations are possible: [Formula: see text]; [Formula: see text]; [Formula: see text]; where [Formula: see text] and [Formula: see text] etc. For a positive downward z-;axis, these definitions for [Formula: see text] and [Formula: see text] will result in positive values for the vertical gradient. Relations of the interval gradients to each other and to the true or differential gradient are examined in this paper.

An alternate derivation of the three‐dimensional Hilbert transform relations from first principles

Geophysics ◽  
1986 ◽  
Vol 51 (4) ◽  
pp. 1014-1015 ◽  
Author(s):  
J. Bradley Nelson
Keyword(s):  
Hilbert Transform ◽  
Magnetic Field Intensity ◽  
Three Dimensional ◽  
Gradient Methods ◽  
Vertical Gradient ◽  
Horizontal Gradient ◽  
Implicit Assumption ◽  
Limited Applicability ◽  
Magnetic Gradients ◽  
The Magnetic Field

Several techniques for determining the location, geometry, and strength of a source are based on a knowledge of the magnetic gradients generated by that source. Hood (1965), Bhattacharyya (1966), and Rao et al. (1981) detailed three of these gradient methods. For many years, geophysicists have used the two‐dimensional (2-D) Hilbert transform to approximate the vertical gradient from measurements of the horizontal gradient in the magnetic‐field intensity (Nabighian, 1972; Stanley and Green, 1976; Stanley, 1977; Mohan et al., 1982). This technique is of limited applicability because of the implicit assumption that the source is a linear, 2-D body oriented at right angles to the profile direction.

Advantages of using the vertical gradient of gravity for 3-D interpretation

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1588-1595 ◽  
Author(s):  
I. Marson ◽  
E. E. Klingele
Keyword(s):  
Gravity Data ◽  
Vertical Gradient ◽  
Three Dimensions ◽  
Gravity Gradient ◽  
Horizontal Gradient ◽  
Euler Deconvolution ◽  
Model Studies ◽  
Combined Use ◽  
Deconvolution Techniques ◽  
Vertical Gradient Of Gravity

Gravity gradiometric data or gravity data transformed into vertical gradient can be efficiently processed in three dimensions for delineating density discontinuities. Model studies, performed with the combined use of maxima of analytic signal and of horizontal gradient and the Euler deconvolution techniques on the gravity field and its vertical gradient, demonstrate the superiority of the latter in locating density contrasts. Particularly in the case of interfering anomalies, where the use of gravity alone fails, the gravity gradient is able to provide useful information with satisfactory accuracy.

On: “The normal vertical gradient of gravity” by J. H. Karl (GEOPHYSICS, 48, 1011–1013).

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1563-1563 ◽  
Author(s):  
C. J. Swain
Keyword(s):  
Sea Level ◽  
Gravity Data ◽  
Vertical Gradient ◽  
Practical Significance ◽  
Average Value ◽  
The Earth ◽  
Free Air ◽  
The Mean ◽  
Local Anomalies ◽  
Vertical Gradient Of Gravity

The implication of the author’s hypothesis, that the conventional free‐air correction factor is difficult to justify and can lead to large errors (e.g., 14 mGal from 300 m of topographic relief), would be very serious indeed for many interpretations of gravity data if it were true. He predicts a normal vertical gradient of 0.264 mGal/m near sea level, 14 percent lower than the conventional theoretical value. However, precise measurements of the free‐air gradient near sea level have been reported (Kuo et al., 1969) which differ by less than [Formula: see text] percent from the theoretical value; moreover these differences correlate with small local (isostatic) anomalies. My own observations at Leicester, England (elevation 100 m) and Nairobi (elevation 1 650 m) (made with students) also differ by less than [Formula: see text] percent from the theoretical values and again the differences correlate with small local anomalies. If these values represent the normal free‐air gradient, it would appear that the author’s analysis must be wrong. The formula he derives gives, correctly, the mean vertical gradient at some level over and within the Earth to a good approximation. This can be seen simply by considering the well‐known formulas for the gradient at a point within the Earth where the density is ρ [Formula: see text] and at a point outside the Earth [Formula: see text] and taking averages at this radius. However, the average value has no practical significance. It does not apply to any point on the Earth’s surface; it is merely a mean.

Model of multicomponent narrow-band periodical non-stationary random signal

2020 ◽  
Vol 2020 (48) ◽  
pp. 17-24
Author(s):  
I.M. Javorskyj ◽  
◽  
R.M. Yuzefovych ◽  
P.R. Kurapov ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Real Time ◽  
Hilbert Transform ◽  
Spectral Properties ◽  
Narrow Band ◽  
Analytic Signal ◽  
Random Signal ◽  
Hilbert Transformation ◽  
The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

A Bayesian View on the Hilbert Transform and the Kramers-Kronig Transform of Electrochemical Impedance Data: Probabilistic Estimates and Quality Scores

2020 ◽  
Author(s):  
Jiapeng Liu ◽  
Ting Hei Wan ◽  
Francesco Ciucci
Keyword(s):  
Power Generation ◽  
Electrochemical Impedance Spectroscopy ◽  
Hilbert Transform ◽  
Electrochemical Impedance ◽  
The Self ◽  
Impedance Data ◽  
The Hilbert Transform ◽  
Validation Tests ◽  
Self Consistency ◽  
Mathematical Relations

<p>Electrochemical impedance spectroscopy (EIS) is one of the most widely used experimental tools in electrochemistry and has applications ranging from energy storage and power generation to medicine. Considering the broad applicability of the EIS technique, it is critical to validate the EIS data against the Hilbert transform (HT) or, equivalently, the Kramers–Kronig relations. These mathematical relations allow one to assess the self-consistency of obtained spectra. However, the use of validation tests is still uncommon. In the present article, we aim at bridging this gap by reformulating the HT under a Bayesian framework. In particular, we developed the Bayesian Hilbert transform (BHT) method that interprets the HT probabilistic. Leveraging the BHT, we proposed several scores that provide quick metrics for the evaluation of the EIS data quality.<br></p>

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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