132 lines
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132 lines
10 KiB
Plaintext
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GEOPHYSICS,
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VOL. 48, NO. 7 (JULY 19X3): P. 1011-1013. I FIG
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Downloaded 08/11/16 to 131.156.224.67. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
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Short Note
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The normal vertical gradient of gravity
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John H. Karl*
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INTRODUCTION
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Most gravity surveysare conductedto estimate subsurfacedensity contrasts for one application or another. From large-scale crustal studiesto relatively small exploration surveys, it is necessary to determine in some way what the normal gravity field should be in order to identify anomalousfeatures. 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 assumedthat variationsin stationelevations arc small compared with the radius of the earth-an obviously excellent approximation, but one needs to be mathematically consistent.
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For example, using the traditional infinite slab Bouguer effect as a first-order terrain correction and then applying topographic data as a second correction results in a station-dependenttruncation at arbitrary distances (Danes, 1982). This planar geometry does not include more distant terrain and furthermore is not an approximation to the spherical earth to any order (Karl. 197I ). I show that this distant terrain is, indeed, important.
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Our approachis to avoid infinite planar geometry and to include distant terrain effects via spherical geometry. Locally. planar geometry can still be usedto estimate the effects of nearby terrain.
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THE VERTICAL GRADIENT
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The two equationsdescribing the gravitational field, in general, arc
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V . G = 4nyp(r)
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and
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VxG=O,
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where positive G is radially inward. By introducing the regional field GK and the anomalousfield G,, , the first equationabove reads
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V * (GR + G,,) = 4nyp(r).
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Next I require that the regional field have only a radial component and average this equation over the surface of the earth at some
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reference radius R,, , giving
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a+$+V*G,,=4r;yp(Ri,j, RO
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where the bar indicatesthis averageand gR is the radial component of the regional field. Becausep representsthe total averagedmass density, there is, on the average. no remaining source for the anomalousfield, i.e.,
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V . G,, = 0.
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Thus the normal vertical gradient is
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-7=6_7-
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dr
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2i?R
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-
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+ 47~yp (R,).
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Ro
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The first term isextremely closeto the familiar elevationcorrection. The secondterm arises from the fact that the gravity stationsare partially “inside” the earth becauseof a spherical shell partially occupiedby massfrom the topographicrelief. This term is stationindependent, substantial in effect, and easy to estimate. While realizing that the definition of the regional field is fully arbitrary, 1 argue that the two assumptionsof (I) only radial components, and (2) its source is the total net mass are most suitable. In fact, these arc the same assumptionsmade in the usual calculation of the free-air’ term. An estimate for fi is obtained from hypsometric data. For example, for a reference datum near sea level about 20 percent of the earth’s area has mass above that level (Wyllic, 1971). Thus i g 2.67 X 20percent = 0.534and4Pyp = 0.0447 mgal/m. Thus near sea level, this additional effect producesabout 14 percent lower vertical gradient than expected from the normal free-air term above. Approximately such a reduction has been reported in the Eastern Mcditerrancan by Hammer (1970). For large variations in station elevations. the hypsomctric data may have to be integratedto give this cortcction sufficiently accurately for practical purposes.
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This approachhas made distant terrain correctionsappearmore like a free-air term. The proper terrain correction to use in this approach is the difference between the standard numerically integratedterrain effect within some volume, sayout to Hammer’s
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Manuscriprteceived by the EditorNovember 16, 1982; revised manuscript received December8, 1982. *Departmentof PhysicsandAstronomy,Universityof Wisconsin,Oshkosh. WI 54901. 0 1983 Society of Exploration Geophysicists. All rights reserved.
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1011
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Downloaded 08/11/16 to 131.156.224.67. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
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1012
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Karl
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zone M and the contribution from p within the same volume (Hammer, 1939). Planar geometry can be used for this.
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DISCUSSION
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Becausethe gravitational force has infinite range, this additive p term is substantialand can profoundly affect interpretation both in large-scale crustal studies and smaller exploration surveys on the ground, in the air, and in boreholes. Its contribution is opposite to the normal free-air correction and hence it reducesthe positive correlation between free-air anomalies and topography that have been observed worldwide. For example, Figure 1 shows gravity data from the Continental Divide (Woollard, 1962) along a line from Gypsum in Colorado to Hein in Nebraska. The free-air anomalyusingour approachshowslesscorrelationwith topography and coincides more with the regional trend.
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The term 4nyp also has significant implications for exploration targets in mountainous areas. For example. topography in the Appalachian ridge and valley province would produce 14 mgal from a 300 m relief due to this additional effect. The small anomaliescommonly producedby thrustsof lessdenseOrdovician shales against more dense middle Mississipian units could be easily confused by improper elevation corrections. It would be interesting to compare the interpretation of gravity data from mountainousareas using both the standarddata reduction method and the one suggestedhere. Certainly the results would be quite different. Unfortunately, the ij effect cannot be avoided even by using direct forward modeling to interpret uncorrected observed data.
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Density determinations by borehole gravity measurementsresponddirectly to p modified by local variations causedby forma-
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FIG. 1. Above: Section acrossthe Rocky Mountain Front through Boulder, Colorado showing topographyand stationelevation. Below: The free-air anomaly both computed by the standardmethod and revised to include global terrain effects. This additional effect is a contribution 4ayc to the normal vertical gradient becauseof the spherical shell passingthrough the station location. This shell is partially occupied by massfrom global topographic relief. For demonstrationpurposes,p has been approximatedas 0.534. The original data are from Woollard (1962). The additional elevation effect reducesthe correlation between free-air anomaly values and topography. This revised free-air curve also shows more agreement with regional gravity.
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Normal Vertical Gradlent of Gravity
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1013
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Downloaded 08/11/16 to 131.156.224.67. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
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tion densitiesand terrain. Unless suchmeasurementsare corrected for this distant terrain effect, resulting formation densities will show systematicerrors. Gibbs and Thomas (1980) reported sucha discrepancyfor two setsof measurementsmade in deep gold-mine shafts in the Archean Yellowknife greenstone belt in the Northwest Territories. On the other hand, McCulloh (1965) reported excellent agreement between density measurements made on core samplesfrom a limestone mine in Ohio and valuesdetermined from gravimetry data. In any case, relative changes in densities over reasonable distances in a borehole are valid without any knowledge of the “normal” vertical gradient. The calculation of absolute formation densities from borehole gravity would, however, require a knowledge of ij.
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The new airborne gravity (Hammer, 1982) could be used to shed further light on this 6 effect. In fact, for exploration purposesthe greatest advantage of airborne gravity nrAy turn out to be its ability to produce data from a fairly constant elevation, thereby reducing the c effect to a negligible amount. Direct forward modeling can be used in this case. On the other hand, long traverses made at different elevations might be used to estimate localized contributions to 6.
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The normal vertical gradient of gravity presented here is the result of global effects, and in many problems more localized influences may have to be included. Regular gravity surveys made over areas of low topographic relief have been successful because anomalies have stood out against a regional field. Presumably, the same would be true of vertical gradient measurements. However, in cases where one wishes to use calculated
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vertical gradients as opposed to measured regional values. there is no obvious solution. Certainly, use of the traditional free-air term is difficult to justify.
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ACKNOWLEIXMENTS
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This work was funded in part by the Submarine Geology and Geophysics program of the National Science Foundation under grant number OCE-8026443. 1 also thank Dr. Sigmund Hammer for his valuable criticisms and suggestions.
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REFEREN<:ES
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Danes, Z. F., 1982,An analyticmethodfor the determinationof distant
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terraincorrectionsG: eophysicsv, . 47. p. 1453-1455.
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Gibb. R. A.. andM. D. Thomas,1980.Density determinations of basic
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volcanic rocks of the Yellowknife supcrgroup by gravity measurements
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in mine shafts-Yellowknife, Northwest Territories: Geophysics. v. 45,
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p. 18-31.
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Hammer, S., 1939. Terrain corrections for gravimetric stations: Geo-
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physics, v. 4, p. 184-194.
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~
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1970, The anomalous vertical gradient of gravity: Geophysics,
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v. 35, p. 153-157.
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~
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1982, Airborne gravity is hem!: The Gil and Gas Journal,
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January II, v. 80, p. 113-122.
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Karl. J. H., 1971, The Bouguer correctton for the spherical earth: Geo-
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physics, v. 36. p. 761.
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McCulloh, T. H., 1965, A confirmation by gravity measurements of an
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underground density profile based on cot-e densities: Geophysics, v. 30,
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p. 1108-1132.
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Woollard, G. P.. 1962, The relation of gravity anomalies to surface eleva-
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tion, cruatal structure and geology:(‘;cophysical and Polar Research
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Center rep. no. 62-9, The Univ. of Wisconsin.
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Wyllie, P. J., 1971. The dynamic earth: New York, JohnWiley & Sons,
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Inc.
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