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GEOPHYSICS, VOL. 35, NO. 1 (FEBRUARY 1970), P. 153-157, i FIGS., 2 TABLES

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SHORT NOTE THE ANOMALOUS VERTICAL GRADIENT OF GRAVITY?

SIGMUND

HAMMER*

The behavior of the earth’s vertical gradient of gravity as observed, for example, in tall buildings or boreholes is a matter of considerable geophysical importance. The following elementary considerations may be of interest.
The theoretical “normal” vertical gradient of gravity varies only slightly with location and elevation on the earth’s surface. Derivation from basic principles of potential theory and known data for the earth gives the well-known formula for the normal “free-air” gravity gradient (Hammer, 1938)

ag

- = 0.308550 + 0.000227 cos 2c$

dh

(1)

- 0.145 X lOFh,

in milligals per meter, where 4 is the geocentric latitude (the small difference between geocentric and geographic latitude is negligible in this context) and h is the elevation in meters above sea level of the point of observation. The sign convention has been adopted to be positive in the sense of increasing gravity. In English units the vertical gradient is, in milligals per foot,

- = 0.09406 + 0.0000692 cos 24

d/z

(2)

- 0.135 X 10-‘/z,

level is less than two-tenths of one percent as shown in Table 1.

Table 1. Normal “free-air” vertical gradient of gravity at sea level

Latitude -__-
0”
z

mgal/m
0.3088 0.3086 0.3083

mgal/f t
0.09413 0.09406 0.09399

The change with elevation amounts to about 0.05 percent per kilometer and 0.01 percent per 1000 ft. For all practical purposes the normal vertical gradient of gravity for the earth as a whole can be taken to be constant over die entire earth’s surface (excluding only very high mountains), nameiy 0.3086 mgaljm or 0.094titi m-gaijft.
The variability caused by local gravity anomalies due to nonhomogeneity of rock density in the earth’s crust and below is anotllcr matter. This may be defined as the “anomalous vertical gradient.” Values of gradient anonlaly in the few measurements which have been published range up to about + 5 percent, more than an order of magnitude larger than the variability in the normal gradient (Hammer, 1938; ThyssenBornemisza and Stackler, 1956; Kumagai et al, 1960; Kuo et al, 1969).’ This is an important

where his in feet. In what follows, the normal freeair vertical gradient of equation (1) and equation (2) will be designated F.
The total change from equator to pole in the value of the normal free-air vertical gradient at sea

1Recent unpublisheddata (Personalcommunication from Professor Charles Drake, Columbia University, New York) show a vertical gradient anomaly of -174 percent in the eastern Mediterranean. This is two orders of magnitude larger than the range in values of the normal gradient.

t Manuscript receivedby the Editor July 29, 1969;revisedmanuscriptreceivedOctober27, 196’1 * Department of Geologyand GeophysicsU, niversity of Wisconsin,Madison,Wisconsin53706. Copyright_@1970by the Societyof Exploration Geophysicists.
153

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154

Hammer

factor in using the normal vertical gradient value, for example in the calibration of gravimeters and the reduction of gravity station data in mountainous country. Analysis of the behavior of the anomalous vertical gradient by classical gravitational potential theory is straightforward but tedious (Morelli and Carrozzo, 1963). However, a very simple model calculation serves to illustrate the general magnitude and behavior.
Related analyses of the anomalous vertical gradient on the axis of a vertical cylinder, have been reported (Thyssen-Bornemisza, 1965; Elkins, 1966). The present note considers the problem from a different point of view.
The maximum variability in the vertical gradient which can occur is near a point mass. Let us therefore consider the vertical gradient of a local anomalous mass represented by a sphere with its top at the surface of the ground. The derivation is as follows.
The vertical component of the gravity anomaly of a spherical mass centered at depth Z, at an elevation lz above ground, and radial distance p

from 0, shown in Figure lB, is

.4 = GM(z + h)/qp” + (z + Iz)“]““, (3)

where M=mass anomaly, and (; is the gravitational constant. On the ground surface, the central magnitude of the anomaly (ior h= 0, p = 0) is

il0 = GM/z”.

(1)

Limiting ourselves to the vertical, axial profile, p=O, we get

A = Agz2/(z + /I)‘).

(5)

The value of gravity in free air directly above the anomalous sphere, at elevation h, from the combined effects of the normal earth and the sphere is

g = go - Fh + A &;(‘ 7 + h)“, (6)

where go is the value of gravity at point 0 in Figure 1. Differentiating with rcqect to h, in the adopted sign convention,

dg

- = F + 2.40z2/(z + Jz)~

(7)

d/z

0.2

(A)

I

I

I

L

h/R

I

3

4

I;Ic. 1. (:I) Dimensionless plot of vertical gradient, (B) Indicated gravity anomaly of spherical and horizontal cylindrical masseswith tops at surface. The point of observation (P) of the vertical gradient is at elevation h above ground. The gravity anomaly is on the surface at elevation h=O.

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Anomalous Vertical Gravity Gradient

155

gives the combined free-air vertical gradient along a vertical profile directly above the sphcrical mass.
The anomalow \w?ical gradient, defined as the departure from 111~ theoretical normal value F, is

This shows the direct relationship on the anomaly axis between the sign of the anomalous vertical gradient and the associated gravity anomaly. r\lso, to the extent that a gravity anomaly may be approxitnated I,)- that ior a single sphere, equation (8j gi\.cs the quantitati1.e magnitudes. Of course, on the flanks of an anomaly this simple relationship, anti e\‘en the sign, will IX different.
‘I‘0 procectl, asbutiie that the 1op of the anomaly oub sphere ia at the jirountl xuriacc 1)~ taking li=z in the equations. This \vill give an approximate simulation of the maximum gradient effect I\-hich can occur for a given gravity- anomaly. Ilquation (8) then hecomes

02 = + 1

2.1,,R2/(R /I)“.

(9)

A dimensionless form is

anti for the cylinder

A

= 2aGu [l + (/I R‘ )]“. (1.3)

The density contrast of tile anomalous mass in the postulated models is uniquely determined in terms of assumed Iz,lK and the magnitude of the anomalous vertical gradient.
A hypothetical application of the theory developed above iollons. _Issume that the observed value of the vertical gradient at a point 100 it (30.5 m) above ground \vas 0.0950 mgal,‘it (0.3117 tngal/m). This is an anomaly of + 17;. Assume further that disturbing effects of building masses and terrain are negligible or have been accurately corrected. ‘1’0 analyze this result let us postulate that the gradient anomaly is caused t)J a spherical mass \vith it< top at the ground surface directly beneath the otwrvation point. The “011. served” data give lrA(ag,‘d//) = 100x0.0009~ =O.OO+ mgal. (The bamc value results for the data expressed in metric units) This value with the assumed values of the parameter II/R applied to the curl-e in Figure 1 yield the results listed in Table 2.
Table 2. Interpretation of gradient anomaly

which is plotted in Figure 1.1. .A similar analysis for an infinite horizontal
cylinder with top at the surface gives the result

which is also plotted for comparison in Figure lr\. Aside from quantitative differences, the two tnodels give essentially similar results. Sate that both curves exhibit maxima. The significance of this fact xvi11be discussed below.
The relationship of the vertical gradient and the density contrast ia) is given by expressing the gravity anomaly 11, in terms of radius and mass. The result for the sphere with top at the ground surface is

A 2 0

= 8rrGg 3[1 + (h/R)]$ (12)

Case a

2

Case 11

1

Case c

3

Case tl

,h

.50 100 200 1000

0 031
0.3i6 0.317 0.626

+1.42 +0 ,442 i-O.186 +0.074

Case c, \vith h/‘K= +, is the extreme case. It indicates the smallest gravity anotnaly that can account for the observed anomalous vertical gradent at the given elevation. The corresponding minimum gravity anomaly for the case of the horizontal cylinder (h/R= 1) is 0.376 mgal, \vhich is about 20 percent larger.
Discrimination between the several postulated cases require additional observations. One way is to make a horizontal survey oi the area to define the gravity anomaly L1, as indicated in Figure 2B. .\ second procedure (applicable in tall huildings, mine shafts, boreholes and-not inconceivably with developing gradiometer technology-in the air) is to extend the vertical gradient measurement to define its behavior over a range oi elevations. The curves in Figure 2.1 show these

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156

Hammer

n(z) 5

mgal/meter

x lO3,O

II I

(B) GRAVITY

ANOMALY

/;\ a

surface

P

C

d

P. .

a

C

d

u-t-

(A ) ANOMALOUS VERTICAL GRADIENT

fl (Z$)

mgal/foot x IO3

FIG. 2. (A) Anomalousvertical gradient, A(dg/ah), versus elevation h for postulatedsphericalmasseswith tops at surface. (B) Gravity anomalies of the masseson the surface it=O. Data refer to Table 2. (Case b is intermediate between casesc and d and is not plotted)

results. The vertical profiles of the anomalous vertical gradient are strongly diagnostic.
Interpreting the postulated gradient anomaly in terms of an infinitely-long horizontal cylinder with top at surface gives the curves in Figure 3. The results and conclusions are similar to that for the sphere.
SUMMARY
The model study reported in this paper is easily extended to single masses of other geometrical forms and depth. In such cases, “RR” is any characteristic dimension (size or depth) of the model. For an assumed model and a given value of the axial anomalous vertical gradient at a known elevation, the magnitude of the associated gravity

anomaly and a complete and unique interpretation (both R and a) are derivable for any assumed value of the dimensionless ratio h/R.
The relationships between vertical gradient and area1 gravity are easy to understand from basic principles. Minor exceptions which may occur (Kumagai et al, 1960) do not apply to localized features. Isolated anomalies in the vertical gradient must correlate directly with associated gravity anomalies. If they do not the data in one or the other or both are inadequate.
Nonlinear behavior of a vertical profile of the vertical gradient can occur only near localized (shallow) mass anomalies. JTcrtical gradient effects of strong, broad gravity anomalies tend to be small and vertically linear.

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Anomalous Vertical Gravity Gradient

157

5

IO

I

I

mgol/meter x IO3

- 100

-gradient

is Iirger than far sphere

? 9

\\

B

-S -r

is smaller than for sphere

I

2

3

#I

x IO3 mgal/foot

FIG. 3. Anomalous vertical gradient versus elevation for postulated infinite horizontal cylinders with tops at surface. Assumed parameters (h/R) for cases a, b, c, d, are respectively 2, 1, i, l/10.

Vertical gradient measurements in tall buildings (and also in underground mine shafts and boreholes) should be supplemented with an area1 gravity survey to define the locally anomalous gravity field in the vicinity. To check the reality of nonlinear vertical gradient effects by an area1 gravity survey it is good practice to have the horizontal station spacing in the immediate vicinity closer than the nonlinear elevation intervals in the gradient data.
REFERENCES
Elkins, T. A., 1966, Vertical gradient of gravity on axis for hollow and solid cylinders: Geophysics, v. 31, p. 816-820.

Hammer, S., 1938, Investigation of the vertical gradient

of gravity: Trans. Am. Geophys. Un., Nineteenth

Ann. Meeting, Pt, I, p. 72-82.

Kumagai, N., Abe, E., and Yoshimura, Y., 1960, Mea-

surement of the vertical gradient of gravity and its

significance: Bull. di Geofisica Teorica ed Applicata, v.

8, p. 607-630.

Kuo, J. T., Ottaviani, M., and Singh, S. K., 1969,

Variations of vertical gravity gradient in New York

City and Alpine, New Jersey: Geophysics, v. 34, p.

235-248.

Morelli, C., and Carrozzo, M. T., 1903, Calculation of

the anomalous gravity gradient in elevation from

Bouguer’s anomalies: Bull. di Geofisica Teorica e

Applicata, v. 5? p. 308-336.

Thyssen-Bornemrsza, S., and Stackler, W. F., 1956,

Observation of the vertical gradient of gravity in

the field: Geophysics, v. 21, p. 771 779.

__

1965, The anomalous free-air vertical gradient

in borehole exploration: Geophysics. v. 30, p. 441-443.