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The Generalized Bouguer Anomaly Earth Planets Space, 58, 287–303, 2006 The generalized Bouguer anomaly Kyozo Nozaki Tsukuba Technical Research and Development Center, OYO Corporation, 43 Miyukigaoka, Tsukuba, Ibaraki 305-0841, Japan (Received November 11, 2004; Revised June 28, 2005; Accepted September 20, 2005; Online published March 10, 2006) This paper states on the new concept of the generalized Bouguer anomaly (GBA) that is defined upon the datum level of an arbitrary elevation. Discussions are particularly focused on how to realize the Bouguer anomaly that is free from the assumption of the Bouguer reduction density ρB , namely, the ρB -free Bouguer anomaly, and on what is meant by the ρB -free Bouguer anomaly in relation to the fundamental equation of physical geodesy. By introducing a new concept of the specific datum level so that GBA is not affected by the topographic masses, we show the equations of GBA upon the specific datum levels become free from ρB and/or the terrain correction. Subsequently utilizing these equations, we derive an approximate equation for estimating ρB . Finally, we show how to compute a Bouguer anomaly on the geoid by transforming the datum level of GBA from the specific datum level to the level of the geoid. These procedures yield a new method for obtaining the Bouguer anomaly in the classical sense (say, the Bouguer disturbance), which is free from the assumption of ρB . We remark that GBA upon the ρB -free datum level is the gravity disturbance and that the equation of it has a tie to the fundamental equation of physical geodesy. Key words: Generalized Bouguer anomaly, Poincare-Prey´ reduction, specific datum level of gravity reduction, free-air anomaly, Bouguer reduction density. 1. Introduction als by the Poincare-Prey´ reduction or, in short, the Prey re- In this paper, we present a new concept of the generalized duction (Heiskanen and Moritz, 1967, p. 146, pp. 163–165) Bouguer anomaly, which is defined upon the datum level from the normal gravity at the reference ellipsoid. The main of gravity reduction of an arbitrary elevation. The classi- aim of such a generalization of the Bouguer anomaly is to cal Bouguer anomaly has been defined upon the geoid by study the subsurface structures (e.g. Nozaki, 1997). The fig- the difference between the observed gravity reduced to the ure of the Earth is not the subject of such a generalization geoid and the reference gravity upon the geoid. The refer- of the Bouguer anomaly. ence gravity has been equated to the standard gravity (e.g. One of the most prominent features of the generalized Heiland, 1946). No distinction was made between the refer- Bouguer anomaly lies in the treatment of the reference grav- ence gravity and the standard gravity. In the theory of mod- ity field: the use of the Prey reduction for the reference ern physical geodesy, the normal gravity upon the reference gravity. This means that the level of gravity reduction is ellipsoid was introduced as the reference gravity (Heiska- within the Earth’s mass distribution outside the reference nen and Moritz, 1967, p. 44). However, it is well known ellipsoid as well as inside. that the surface of the reference ellipsoid is different from In Section 2, we explain the motivations of this study. In that of the geoid. Therefore, the reference gravity upon the Section 3, we describe the details of the formulation of the geoid is represented in terms of the normal gravity and the generalized Bouguer anomaly. Also, we explain the phys- geoid height. Hackney and Featherstone (2003) recently ical properties of the new formula thus obtained. In Sec- discussed geodetic and geophysical ‘gravity anomalies’. tion 4, we define the three specific datum levels of grav- In order to perform the formulation, we attempted to gen- ity reduction: the one is the datum level so that the value eralize the Bouguer anomaly upon an arbitrary elevation of the generalized Bouguer anomaly becomes invariant for (the orthometric height). The reduction level laterally varies any Bouguer reduction density (the so-called ‘ρB -free da- in height depending on the position of the gravity station. tum level’), and the other is the datum level so that the Also it does not coincide with a boundary of a Bouguer sum of the terrain and Bouguer corrections becomes zero plate (or a Bouguer spherical cap) above the geoid. As for any Bouguer reduction density. Also, we derive the is written in the text, we define the generalized Bouguer generalized Bouguer anomaly at each specific datum level. anomaly upon the datum level of an arbitrary elevation by Particularly, it will be shown that the generalized Bouguer the difference between the reduced observed gravity and the anomaly upon the ρB -free datum level, namely, the ρB -free reference gravity that is reduced within the Earth’s materi- Bouguer anomaly, is free from the Bouguer reduction den- sity and is equal to the ‘gravity disturbance’ as defined in Copyright c The Society of Geomagnetism and Earth, Planetary and Space Sci- the physical geodesy. It will be also shown that the equa- ences (SGEPSS); The Seismological Society of Japan; The Volcanological Society tion of the ρB -free Bouguer anomaly is the same as the fun- of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sci- ences; TERRAPUB. damental equation of physical geodesy, which defines the 287 288 K. NOZAKI: GENERALIZED BOUGUER ANOMALY Fig. 1. Examples of the variation of the Bouguer anomaly distributions for Bouguer reduction densities. In the range of the Bouguer reduction densities 3 3 3 3 3 3 ρB s from 1.5 × 10 kg/m to 2.5 × 10 kg/m , the interval is 0.1 × 10 kg/m . β denotes the free-air gradient, DL the elevation of the datum level of gravity reduction. Arrows indicate Bouguer anomaly invariant (BA-invariant) points. Upper panels: Bouguer anomaly profiles, lower panels: topography profiles. The Site (a) profile shows an example in which BA-invariant points occur. The site (b) profile shows an example in which BA-invariant point does not occur. ‘gravity anomaly’ (Heiskanen and Moritz, 1967). In Sec- densities ρB . For convenience, we call each of these points tion 5, we describe the details of the derivation of an ap- a‘BA-invariant point’. On Fig. 1(b), no such BA-invariant proximate equation to be satisfied by the Bouguer reduction points occur. What does such a BA-invariant point mean? density and the anomalous vertical gradient of the gravity. At such a point, the Bouguer anomaly is free from the This approximation can be used for the estimation of the surrounding topographic masses. Bouguer reduction density. In Section 6, using such an esti- Why do BA-invariant points exist? The reason is the mated Bouguer reduction density, we describe a method to elevation of the datum level of the gravity reduction. In obtain the generalized Bouguer anomaly distribution upon this case, it is the geoid. If one changes the elevation of the geoid from that upon the ρB -free datum level. We show the datum level upwards and downwards, the location of the that it is nothing but the Bouguer anomaly in the classical BA-invariant points would also change. In other words, any sense (say, the Bouguer disturbance) which has been used gravity station can become a BA-invariant point for each to study the subsurface structures. gravity data by adjusting the elevation of the datum level. If gravity anomalies are mapped by using only such BA- 2. Motivations of the Approach invariant points, it could be the most useful one for studying The most important motivation in this study is the ‘ex- subsurface structures, because the gravity anomalies are istence of the Bouguer anomaly invariant point’. Figure 1 independent of the Bouguer reduction density ρB . From shows variation of Bouguer anomaly distributions due to the this point of view, the elevation of the datum level of the variation of the Bouguer reduction density ρB . The range gravity reduction could have freedom to be selected. 3 3 of ρB variation is between 0.0 kg/m to 5,000 kg/m . The Bouguer anomaly is defined upon the geoid as is done in 3. Formulation of the Generalized Bouguer a classical textbook (e.g. Heiland, 1946). Namely, the el- Anomaly evation of the datum level of gravity reduction is taken at 3.1 Height system and definition the geoid. The adopted free-air gradient is taken as 0.3086 In this paper, we will use the orthometric height system. mGal/m (10−5 m/s2/m). Firstly, we make some comments on the height system. On Fig. 1(a), one can notice three points that are indicated As mentioned above, the main goal of this paper is the by arrows. The Bouguer anomalies (BAs) for these points generalization of the classical Bouguer anomaly, which has are independent of the variation of the Bouguer reduction been referred to the geoid. In the land gravity survey, the K. NOZAKI: GENERALIZED BOUGUER ANOMALY 289 data set are given by the observed gravity with position and gradient of gravity (VGG) anomaly ∂g/∂r is included. elevation. The truncated spherical shell system of gravity correction is Although the exact determination of the orthometric included by a truncation angle ψ (see Fig. 3). Then, since height requires the complete knowledge of the actual grav- the ‘anomaly’ can be defined by the difference between the ity field within the topographic masses, this type of the observed value and the reference value, we define the gen- height system is the most familiar one in the classical eralized Bouguer anomaly ( gp,Hd ) by the difference be- Bouguer anomaly.
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