The subtle difference between gravity anomalies and gravity disturbances has lead to conflicting definitions of what a gravity anomaly actually is. The gravity anomaly is typically defined in one of two ways: 1) as the difference between gravity on the geoid and gravity on the reference spheroid (mostly in geodesy), and 2) as the difference between observed gravity and some theoretical value of gravity predicted at the measurement point (mostly in geophysics). By explaining the reasoning behind each of these definitions, we clarify the definition of a gravity anomaly and suggest that for geophysical applications, gravity anomalies, or more correctly gravity disturbances, should be computed using spheroid-referenced elevations. Failure to do so can lead to errors associated with the geophysical indirect effect .
 

1. Introduction

In geodesy, gravity data are used to define the figure of the Earth. In geophysics, the data are used to constrain subsurface density variations to help to understand problems related to tectonics or commodity exploration. To this end, the geodesist seeks to exploit the differences between the difficult-to-measure real gravity field and a mathematically based model gravity field. The aim in geophysics is to remove global and other large-scale gravity effects that mask the local anomalies that are of direct interest [1 , 2]. In order to fully achieve the goal of removing large-scale gravity effects, the geophysicist should actually compute gravity disturbances, not gravity anomalies. This fact is generally not recognised and, in any case, the geophysicist actually fails to achieve the stated aim of gravity  anomaly  computation. This failure has been recognised as the so-called  geophysical indirect effect [3 , 4], whereby the model gravity value is derived by  correcting  normal gravity on a reference spheroid to the level of the gravity measurement using a sea-level (geoid) referenced elevation, not a spheroid-referenced elevation. Below is a short summary that attempts to clarify the definition of gravity anomalies and gravity disturbances. It is suggested that to avoid the indirect effect in geophysical applications, gravity anomalies should be computed using spheroid-referenced elevations.
 

2. Gravity anomalies and gravity disturbances

In the strictest sense, the gravity anomaly vector, , is defined by the equation

                        (1)

where g is gravity at the geoid, appropriately downward-continued from surface data () whilst preserving the mass of material above the geoid.  is normal gravity on the reference spheroid (Figure 1). The gravity disturbance, , at a point P (at the surface, or any other level) is defined as

            (2)

where  is observed gravity measured at the surface point P, and  is normal gravity at the measurement point (Figure 1).  is determined by applying the free-air and Bouguer corrections to normal gravity on the spheroid, .
 
 

Figure 1 Diagram showing parameters used in the definition of gravity anomalies and gravity disturbances. H is orthometric height measured along the plumb-line, h is spheroidal height measured along the spheroidal normal, and N is the geoid height. These quantities are related by the expression H = h - N. The known parameters are: gravity measured at the surface point P , g_P, and normal gravity on the reference spheroid, . Gravity on the geoid is computed by downward continuing g_P to the geoid, and  is computed from the mathematically defined gradient of normal gravity above the reference spheroid.

3. Reduction, correction or downward continuation?

Much of the geophysical literature describes the process of computing gravity anomalies as a reduction process during which gravity values observed at the surface are reduced to some datum level, usually mean sea-level. Reduction to the geoid requires knowledge of the rate of change of the real gravity field above the geoid, . In practice, this quantity is difficult to determine and downward-continuation is an unstable problem. Therefore,  is generally approximated by the theoretically defined gradient of normal gravity above the reference spheroid, . This quantity is the familiar free-air gradient, normally taken as 3.086 µm.s^-2/m.

In geophysical applications, where the aim is to remove large-scale effects on locally measured gravity (such as those arising from the mass of the Earth) it is logical to think of an  anomaly  as the difference between the observed value of gravity at a point, and some predicted, or theoretical, value for that point [1 , 2]. This theoretical value is determined by correcting normal gravity on the spheroid to the level of the measurement point by applying the free-air and Bouguer corrections.

The indirect effect arises because the geoid and reference spheroid do not coincide. Correct computation of gravity disturbances requires that the free-air and Bouguer corrections incorporate spheroidal height, h. If a geoid-referenced elevation is used instead, then the combined free-air and Bouguer correction will be either an under- or over-correction, depending on the geoid height, N (Figure 2). Globally, the indirect effect on Bouguer anomalies varies between about -200 and 150 µm.s^-2 (Figure 3) and on free-air anomalies it varies between about -270 and 250 µm.s^-2 [3]. Therefore, the indirect effect can have a significant effect on regional gravity studies [5].

Figure 2 Diagram showing that using the orthometric height, H, instead of the ellipsoidal height, h, to compute gravity anomalies is either an over- or an under-correction, depending on the geoid-ellipsoid separation, N.

Figure 3 Global map of the geophysical indirect effect on free-air anomalies.
 

The potential errors that arise from the indirect effect mean that, in geophysical applications, we should update the long-held tradition of using geoid-referenced elevations for anomaly computation by computing proper gravity disturbances using spheroid-referenced elevations. Historically, spheroid-referenced elevations have not been readily available, but with the now widespread use of GPS and the availability of high resolution geoid models, these elevations are easily obtainable.

References

[1] T. R. LaFehr. Standardization in gravity reduction. Geophysics, 56(8):1170--1178, 1991.

[2] D. A. Chapin. The theory of the Bouguer gravity anomaly: A tutorial. The Leading Edge, May: 361--363, 1996.

[3] M. E. Chapman & J. H. Bodine. Considerations of the indirect effect in marine gravity modelling. Journal of Geophysical Research, 84(B8):3889--3892, 1979.

[4] M. Talwani. Short note: Errors in the total Bouguer reduction. Geophysics, 63(4):1125--1130, 1998.

[5] W-.Y. Jung & P. D. Rabinowitz. Application of the indirect effect on regional gravity fields in the North Atlantic Ocean. Marine Geodesy, 12: 127--133, 1988.

Further reading:

Hackney, R.I., Featherstone, W.E. and Holmes, S.A. (in press). Geodetic versus geophysical perspectives of the gravity anomaly, Geophysical Journal International.

Li, X. and Götze, H.-J., (2001). Ellipsoid, geoid, gravity, geodesy and geophysics, Geophysics, 66, 1660-1668.

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