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.

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.

On: “The normal vertical gradient of gravity” by J. H. Karl (GEOPHYSICS, 48, 1011–1013, July 1983).

Geophysics ◽  
1986 ◽  
Vol 51 (7) ◽  
pp. 1505-1508 ◽  
Author(s):  
T. R. LaFehr ◽  
Kwok C. Chan
Keyword(s):  
Data Reduction ◽  
Normal Gravity ◽  
Traditional Approach ◽  
Vertical Gradient ◽  
Gravity Gradient ◽  
Average Value ◽  
Terrain Model ◽  
Vertical Gradient Of Gravity

In his reply to C. J. Swain’s (1984) discussion Karl states that no one has disagreed with his proposed (0.265 mGal/m) “average value” for the normal gravity gradient and that his global terrain model can be used to challenge the validity of the traditional approach to data reduction. Our investigations show that Karl is in error on both counts, and we hope that the following analyses will help toward a clearer understanding of this question.

Reply to Discussion by J. R. Hearst and R. C. Carlson

Geophysics ◽  
1979 ◽  
Vol 44 (8) ◽  
pp. 1464-1464
Author(s):  
J. R. Hearst ◽  
R. C. Carlson
Keyword(s):  
Outer Radius ◽  
The Other ◽  
Gradient Term ◽  
The Earth ◽  
Other Hand ◽  
Free Air ◽  
Inner Radius ◽  
The Mean ◽  
The Difference

Our equations (3) and (4) are correct. They represent the difference between the attraction of the shell viewed from [Formula: see text], the outer radius of the shell, and [Formula: see text], its inner radius. (The attraction of the shell viewed from [Formula: see text] is zero.) On the other hand, equations (5) and (6) of Fahlquist and Carlson represent the difference in attraction of the entire earth from the same viewpoints and thus, as they say, include a free‐air gradient term. However, their equation (5) would be correct only if the mean density of the earth were equal to that of the shell, and the free‐air gradient obtained by their equation (10) is correct only under these circumstances.

Optimising the acceleration due to gravity on a planet's surface

2010 ◽  
Vol 94 (530) ◽  
pp. 203-215
Author(s):  
Michael Jewess
Keyword(s):  
Sea Level ◽  
Free Fall ◽  
Gravitational Effect ◽  
Oblate Spheroid ◽  
Practical Significance ◽  
Axis Of Rotation ◽  
The Earth ◽  
Equatorial Diameter ◽  
Ellipsoid Of Revolution ◽  
Rotation Of The Earth

The Earth (more precisely, the ‘geoid’ thereof) is known to approximate closely to a slightly oblate spheroid whose unique axis coincides with the Earth's axis of rotation [1,2]. (By ‘spheroid’ is meant is an ellipsoid of revolution, i.e. one with two semi-axes equal; a slightly oblate one has these two semi-axes slightly longer than the unique one.) To the nearest km, the diameter of the ‘geoid’ pole-to-pole is 43 km less than the equatorial diameter of 12756 km. There is a reduction of practical significance (0.527%) in the acceleration of free fall" at sea level between the poles and the equator, and therefore in the weight of objects. Of this, 0.345% derives directly from the rotation of the Earth; the balance of 0.182% results from the purely gravitational effect of the Earth's deviation from sphericity.

The normal vertical gradient of gravity

Geophysics ◽  
1983 ◽  
Vol 48 (7) ◽  
pp. 1011-1013 ◽  
Author(s):  
John H. Karl
Keyword(s):  
Gravity Field ◽  
Potential Field ◽  
Large Scale ◽  
Normal Gravity ◽  
Vertical Gradient ◽  
Original Model ◽  
Field Methods ◽  
The Earth ◽  
Vertical Gradient Of Gravity

Most gravity surveys are conducted to estimate subsurface density contrasts for one application or another. From large‐scale crustal studies to relatively small exploration surveys, it is necessary to determine in some way what the normal gravity field should be in order to identify anomalous features. The anomalies then represent deviations to be interpreted in light of the original model. It is a central limitation of potential field methods that this model, sometimes representing a so‐called “regional” field, is not unique. In the case of gravity, this model has traditionally involved geometrical approximations. It is generally assumed that variations in station elevations are small compared with the radius of the earth—an obviously excellent approximation, but one needs to be mathematically consistent.

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 anomalous vertical gradient of gravity” by Sigmund Hammer (GEOPHYSICS, February 1970, p. 153–157).

Geophysics ◽  
1971 ◽  
Vol 36 (1) ◽  
pp. 214-216 ◽  
Author(s):  
Stephen Thyssen‐Bornemisza
Keyword(s):  
Horizontal Plane ◽  
Vertical Gradient ◽  
Observation Point ◽  
Gravity Gradient ◽  
Vertical Gravity Gradient ◽  
Free Air ◽  
Plane Passing ◽  
Vertical Gravity ◽  
Vertical Gradient Of Gravity

The interesting analysis by Hammer seems to be founded partly on the approach of Heiskanen and Moritz (1967), in which a real gravity and a vertical gravity gradient are correlated according to the relation [Formula: see text] Here Δg denotes the free‐air anomalies on a horizontal plane passing through observation point p and [Formula: see text] is the anomaly at p, the latter representing the center of the (x, y) coordinates.

scholarly journals XIII. On the constitution of the solid crust of the Earth

1871 ◽  
Vol 161 ◽  
pp. 335-357 ◽  
Keyword(s):  
Sea Level ◽  
Plumb Line ◽  
Level Surface ◽  
Considerable Influence ◽  
The South ◽  
Vertical Column ◽  
The North ◽  
The Earth ◽  
Figure Of The Earth ◽  
The Mean

A few years ago I proposed the following hypothesis regarding the Constitution of the Earth’s Solid Crust, viz.: — that the variety we see in the elevation and depression of the earth’s surface, in mountains and plains and ocean-beds, has arisen from the mass having contracted unequally in becoming solid from a fluid or semifluid condition: and that below the sea-level under m ountains and plains there is a deficiency of m atter, approximately equal in amount to the mass above the sea-level; and th at below ocean-beds there is an excess of matter, approximately equal to the deficiency in the ocean when compared with rock; so that the amount of matter in any vertical column drawn from the surface to a level surface below the crust is now, and ever has been, approximately the same in every part of the earth. 2. The process by which I arrived at this hypothesis I will explain. In the Philosophical Transactions for 1855 and 1858 I showed that the Himalayas and the Ocean must have a considerable influence in producing deflection of the plumb-line in India. But by a calculation of the mean figure of the earth, taking into account the effect of local attraction, it appeared that no where on the Indian Arc of meridian through Cape Comorin is the resultant local attraction, arising from all causes, of great importance*. This result at once indicated that in the crust below there must be such variations of density as nearly to compensate for the large effects which would have resulted from the attraction of the mountains on the north of India and the vast ocean on the south, if they were the sole causes of disturbance, — and that, as this near compensation takes place all down the arc, nearly 1500 miles in length, the simplest hypothesis is, that beneath the mountains and plains there is a deficiency of matter nearly equal to the mass above the sea-level, and beneath ocean-beds an excess of matter nearly equal to the deficiency in the ocean itself.

scholarly journals Refined prediction of vertical gradient of gravity at Etna volcano gravity network (Italy)

2018 ◽  
Vol 48 (4) ◽  
pp. 299-317 ◽  
Author(s):  
Pavol Zahorec ◽  
Juraj Papčo ◽  
Peter Vajda ◽  
Filippo Greco ◽  
Massimo Cantarero ◽  
...  
Keyword(s):  
Gravitational Effect ◽  
Vertical Gradient ◽  
Etna Volcano ◽  
Free Air ◽  
Digital Elevation ◽  
Gravity Network ◽  
Predicted Values ◽  
Elevation Model ◽  
Vertical Gradient Of Gravity

Abstract Predicted values of the vertical gradient of gravity (VGG) on benchmarks of Etna’s monitoring system, based on calculation of the topographic contribution to the theoretical free-air gradient, are compared with VGG values observed in situ. The verification campaign indicated that improvements are required when predicting the VGGs at such networks. Our work identified the following factors to be resolved: (a) accuracy of the benchmark position; (b) gravitational effect of buildings and roadside walls adjacent to benchmarks; (c) accuracy of the digital elevation model (DEM) in the proximity of benchmarks. Benchmark positions were refined using precise geodetic methods. The gravitational effects of the benchmark-adjacent walls and buildings were modeled and accounted for in the prediction. New high-resolution DEMs were produced in the innermost zone at some benchmarks based on drone-flown photogrammetry to improve the VGG prediction at those benchmarks. The three described refinements in the VGG prediction improved the match between predicted and in situ observed VGGs at the network considerably. The standard deviation of differences between the measured and predicted VGG values decreased from 36 to 13 μGal/m.

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