The Generalized Bouguer Anomaly

Earth Planets Space, 58, 287–303, 2006

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 density 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. Therefore, in deriving the concept of tween the observed gravity reduced onto the datum level at generalized Bouguer anomaly, we will use this height sys- an elevation Hd and the reference gravity reduced onto the tem. In Fig. 2(a), we show the correspondence between the same datum level at Hd . Namely, the generalized Bouguer height system used in this paper based on the orthometric anomaly is defined by the form height and the standard height system based on the normal = − γ , height. gp,Hd : gp,Hd Hd (1) Here, we define the generalized Bouguer anomaly of the , γ observed gravity gp. Let the elevation of an arbitrary datum where, gp Hd denotes the reduced observed gravity, and Hd level be Hp (orthometric height from the geoid). The eleva- denotes the reference gravity. The detailed equations are tion of the geoid is zero in this height system. The elevation described in Section 3.2. In Fig. 4, we show a schematic of the datum level of gravity reduction is Hd . The elevation view of the observed gravity reduced onto an arbitrary da- of the surface of the reference ellipsoid is H0. The vertical tum level of elevation Hd and the reference gravity within the mass distribution. This approach of defining the generalized Bouguer anomaly at the same datum level looks classical. How- ever, in the followings, we will find a new relation between the generalized Bouguer anomaly and the ‘gravity anomaly’ defined in the physical geodesy (Heiskanen and Moritz, 1967). In spite of the difference at the same datum level, we use the notation instead of δ. This is because we do not see the terminology ‘Bouguer disturbance’ in the literature. 3.2 Formulation In this section, we discuss the generalized Bouguer anomaly based on the defining equation, Eq. (1). In the formulation, we express the generalized Bouguer anomaly < > gp,Hd for the cases of Hd Hp and Hd Hp separately to make their physical meanings clear, even though the both expressions are equivalent. Symbols used in the formulation are summarized as follows:

Fig. 2. Height systems. (a) The orthometric height system. H: the gp: observed gravity at a station P, orthometric height, N: the geoid height. Symbols H : the elevation p H : elevation of the gravity station, (the orthometric height) of P, H0: the elevation of the ellipsoid, and p Hd : the elevation of the datum level of gravity reduction used in the text. (b) Normal height system. H N : the normal height (Torge, 2001, Hd : elevation of the datum level of the gravity reduction, equation (3.107)), ζ : the height anomaly, h: the geometrical height (Heiskanen and Moritz, 1967) or the ellipsoidal height (Torge, 1989, H0: elevation of the normal ellipsoid surface, equations (2.70a) and (2.71a); h = H N + ζ = H + N).

Fig. 3. Schematic illustration of the truncated spherical shell system of gravity correction. (a) For the case of Hd < Hp. (b) For the case of Hd > Hp. Hatching indicates the areas of mass-redistribution accompanied by the terrain and Bouguer corrections. Hp denotes the elevation of the gravity station P, Hd the elevation of the datum level of the gravity reduction, H0 the elevation of the surface of the normal ellipsoid, and ψ the truncation angle of spherical gravity correction. 290 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

Table 1. Deﬁnition of ‘correction’ and ‘reduction’.

the actual VGG after terrain and Bouguer corrections, and the normal VGG, which is compared at a point in the free-air space where the topographic masses of the density ρB are moved or removed by the terrain and Bouguer corrections,

r: the geocentric radial coordinate (positive upwards),

H ±(1,ψ): sphericity factor for the spherical terrain and Bouguer corrections,

ψ: truncation angle of the spherical terrain and Bouguer corrections. In the VGG anomaly ∂g/∂r, the gravitational effect of the Fig. 4. A conceptual illustration explaining the definition of the gener- near surface density anomaly, which represents the inho- γ ρ alized Bouguer anomaly upon Hd . gp,Hd and Hd denote the reduced mogeneity of the topographic mass-density field from B , observed gravity and the reference gravity at the datum level Hd , re- is included together with the VGG anomaly in the free-air , + − spectively. The generalized Bouguer anomaly ( gp Hd ) is defined by ( ,ψ) ( ,ψ) = − γ ρ space. The functions H 1 and H 1 , which gov- gp,Hd gp,Hd Hd . B denotes the Bouguer reduction density. The reference gravity field is within the Earth’s mass distribution. ern the gravitational behaviour of a thin spherical cap with a truncation angle ψ (Nozaki, 1999), are given as ± 1 − cos ψ γ0: the normal gravity (upon the normal ellipsoid), H (1,ψ)= ± 1, (2) 2 TC (1): the quantity of the terrain correction at the station p (hereafter, double signs should be taken in the same order). P for the unit density, The derivation and the physical properties of these functions

BCp(1): the quantity of the Bouguer correction at the sta- are shown in Appendix A. It is clear that an identical tion P for the unit density, equation H +(1,ψ)− H −(1,ψ)≡ 2 (3) ρB : Bouguer reduction density, holds for any truncation angle ψ. Concerning the spherical TCp: value of the terrain correction (= ρB TCp(1)), gravity corrections, see Nozaki (1981). Corresponding to the functions H +(1,ψ) and H −(1,ψ), we distinguish the BCp: value of the Bouguer correction (= ρB BCp(1)), notation of the Bouguer correction BCp for Hd < Hp from > + FA: gravity anomaly in the physical geodesy or free-air that for Hd Hp: BCp denotes the Bouguer correction for < − > anomaly in the Molodensky sense, Hd Hp and BCp does that for Hd Hp, respectively. In this paper, we distinguish the terminologies between f : sum of the terrain and Bouguer corrections (= TCp + ‘reduction’ and ‘correction’ (after Nettlelton, 1940, Chap- BCp), ter 4). For example, the Bouguer ‘reduction’ is performed by the terrain ‘correction’, Bouguer ‘correction’ and free- G: Newtonian gravitational constant, air ‘correction’. The terminology ‘reduction’ is used for the ∂γ/∂r: VGG of the normal gravity ﬁeld, level-transformation of the gravity value from one elevation level to another. Such technical terms used in this paper are ∂g/∂r: VGG anomaly deﬁned by the difference between summarized in Table 1. K. NOZAKI: GENERALIZED BOUGUER ANOMALY 291

Then, in the case of Hd < Hp, the Bouguer-reduced ob-

served gravity gp,Hd upon the datum level of the gravity reduction Hd can be expressed as Hd + ∂γ ∂g g , = (g + f ) + + dr, (4) p Hd p ∂ ∂ Hp r r

where, f + denotes the sum of the terrain and Bouguer corrections for the datum level of Hd < Hp:

+ + f = TCp + BC p Hd + = ρB TCp(1) + 2πGρB H (1,ψ)dr. (5) Hp The Bouguer reduction (see Table 1) is made up of the spherical terrain correction (TCp), the spherical Bouguer + correction (BCp ), and the free-air correction over the interval [Hp, Hd ]. Notice that, in the free-air correction, the term of the VGG anomaly is added to the integrand on the right- hand side of Eq. (4). A schematic view of the Bouguer- < reduced observed gravity gp,Hd for the case of Hd Hp is illustrated in Fig. 5(a). (2) The case Hd > Hp In this case, as shown in Fig. 5(b), the observed gravity gp at the elevation Hp is reduced to the elevation Hd (> Hp) by the Prey reduction (e.g. Heiskanen and Moritz, 1967) after terrain and Bouguer corrections. When the elevation of the datum level of the gravity reduction Hd is higher than that of the gravity station Hp, the Prey reduction should be applied over the interval [Hp, Hd ] to the observed gravity gp (see also Fig. 3(b)). This is because we have to ﬁll up the open space above the Earth’s surface Hp by the Bouguer correction with the Earth’s materials whose density is ρB . Accordingly, we introduce the spherical Bouguer correction for Hd > Hp denoted by Hd − = π ρ −( ,ψ) . Fig. 5. Schematic illustration of the process to compute the reduced BCp 2 G B H 1 dr Hp observed gravity gp,Hd . (a): Bouguer-reduced observed gravity for the case of Hd < Hp; the observed gravity gp is corrected by the terrain + Thus, in the case of H > H , we have the Prey-reduced and Bouguer corrections (TCp and BCp ), then, reduced by the free-air d p , reduction over the interval [Hp Hd ] as indicated by the arrow. (b): observed gravity, gp,Hd , Prey-reduced observed gravity for the case of Hd > Hp; the observed gravity g is corrected by the terrain and Bouguer corrections (TC and − p p gp,H = (gp + f ) BC−), then, reduced by the Prey reduction over the interval [H , H ]as d p p d H indicated by the arrow. Mass-density distribution after the reduction for d + 2πGρ [H +(1,ψ)− H −(1,ψ)] each case is shown in the ﬁgure. ρB is the Bouguer reduction density. B Hp ∂γ ∂g + + dr, (6) 3.2.1 Reduced observed gravity ∂r ∂r (1) The case Hd < Hp where, f − denotes the sum of terrain and Bouguer correc- In this case, the observed gravity gp at the elevation Hp tions for the datum level of Hd > Hp: is reduced to the Bouguer-reduced observed gravity gpHd at the elevation Hd (< Hp) as shown in Fig. 5(a). − − f = TCp + BC When the elevation of the datum level of gravity reduc- p Hd tion Hd is lower than that of the gravity station Hp (see − = ρB TCp(1) + 2πGρB H (1,ψ)dr. (7) Fig. 3(a)), the Bouguer correction is to remove the Earth’s Hp materials above the datum level of gravity reduction. Ac- cordingly, we introduce the spherical Bouguer correction Notice that the term of the Prey correction as well as that of for Hd < Hp denoted by the VGG anomaly are added to the integrand on the right- hand side of Eq. (6). A schematic view of the Prey-reduced Hd + + observed gravity after terrain and Bouguer corrections g , BC = 2πGρ H (1,ψ)dr. p Hd p b for the case of H > H is illustrated in Fig. 5(b). Although Hp d p 292 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

Fig. 6. Correspondence of the reductions.; (a) the reduced observed gravity, (b) the Prey-reduced reference gravity, and (c) the normal gravity. Upper panels: the case Hd < Hp; lower panels: the case Hd > Hp. (a) The observed gravity at the station level Hp is reduced onto the datum level Hd . (b) The reduction of the reference gravity is done within the earth’s materials whose density is ρB (Prey-reduction). (c) The reduction in the normal gravity ﬁeld is done in the open or null space. The generalized Bouguer anomaly is deﬁned by the difference between the Bouguer- or Prey-reduced observed gravity, and the Prey-reduced reference gravity.

+ − the term 2πGρB [H (1,ψ)− H (1,ψ)] on the right-hand and even the case Hd < H0. We call the newly intro- side of Eq. (6) is 4πGρB regardless of ψ (see Eq. (3)), we duced reference gravity the Prey-reduced reference gravity. shall retain this form in the following sections to show ex- In Fig. 6, the correspondence between the Prey-reduced ref- γ plicitly the gravitational contribution of the Bouguer spheri- erence gravity Hd , the reduced observed gravity gp,Hd , and cal cap. We notice here, the difference between Eqs. (5) and the normal gravity is schematically illustrated. The detailed (7) is that the factor of the integrand on the right-hand side explanation of the reference ﬁeld is added in Appendix C. of Eq. (5) is H +(1,ψ), while in Eq. (7), it is H −(1,ψ). 3.2.3 Formula of the generalized Bouguer anomaly 3.2.2 Prey-reduced reference gravity The reference (1) The case Hd < Hp γ gravity Hd , at the datum level Hd , is deﬁned in this paper When the elevation of the datum level Hd is lower than by the equation that of the gravity station Hp, the formula of the Bouguer- reduced observed gravity is given by Eq. (4). Substituting Hd + Eqs. (4) and (8) into Eq. (1) and arranging the terms with γH = γ0 + 2πGρB [H (1,ψ) d respect to the Bouguer reduction density ρ , we obtain the H0 B , − ∂γ formula of the generalized Bouguer anomaly gp Hd , for − H (1,ψ)] + dr. (8) < ∂r the case of Hd Hp,as γ The reference gravity Hd is reduced by the Prey reduction γ Hp ∂γ Hd ∂g from the normal gravity 0, e.g. from the level H0 to the = − γ − + gp,Hd gp 0 dr dr level Hd . Here we applied the Prey reduction over the inter- H ∂r H ∂r , 0 p val [H0 Hd ], instead of the Bouguer or free-air reduction. Hd + This is because the reduction of the reference gravity from + ρB TCp(1) + 2πGH (1,ψ)dr H the level H0 to another level Hd should be done within the p Earth’s materials whose mass-density is ρ . Thus, one can Hd B − 2πG[H +(1,ψ)− H −(1,ψ)]dr . (9) apply Eq. (8) both for the cases Hd > Hp and Hd < Hp, H0 K. NOZAKI: GENERALIZED BOUGUER ANOMALY 293

The first and second terms in the braces of Eq. (9) are the duction rate’. Equation (12) implies that, when we ignore contribution of the terrain and Bouguer corrections to the the term ∂g/∂r as is usually the case, the upward trans- observed gravity for Hd < Hp, while the third term in formation of the datum level Hd brings the decrease of the the braces is essentially that of the Prey correction to the generalized Bouguer anomaly gp,Hd at the reduction rate − reference gravity (see Eqs. (5) and (4)). of 2πGρB H (1,ψ), and vice versa. Notice that the reduc- (2) The case Hd > Hp tion rate of Eq. (12) does not include the terms of the normal When the elevation of the datum level Hd is higher than gravity field. that of the gravity station Hp, the formula of the Prey- reduced observed gravity is given by Eq. (6). Substituting 4. Generalized Bouguer Anomaly at Some Spe- Eqs. (6) and (8) into Eq. (1) and arranging the terms with cific Datum Level of the Gravity Reduction respect to the Bouguer reduction density ρB , we obtain the In this section, we define the specific datum levels of the formula of the generalized Bouguer anomaly gp,Hd , for gravity reduction so that the generalized Bouguer anoma- the case of Hd > Hp,as lies are not affected by the topographic effects. Also we discuss the physical properties of the generalized Bouguer Hp ∂γ Hd ∂g g , = g − γ − dr + dr anomalies upon the specific datum levels. p Hd p 0 ∂ ∂ H r H r 4.1 Specific datum levels of the gravity reduction 0 p Hd 4.1.1 Specific datum level Hd0 The condition of the + ρ ( ) + π −( ,ψ) B TCp 1 2 GH 1 dr specific datum level Hd0 of gravity reduction, upon which Hp the generalized Bouguer anomaly gp,Hd becomes inde- Hp ρ − 2πG[H +(1,ψ)− H −(1,ψ)]dr . pendent of any B (see Section 2 and Fig. 1), is given by H0 the equation ∂g , (10) p Hd = 0. (13) ∂ρB In the braces of Eq. (10), the first and second terms are the contribution of the terrain and Bouguer corrections to By this condition, one can obtain the defining equation of Hd0 from Eq. (9) or equivalently from Eq. (10) as the observed gravity for Hd > Hp, while the third term is essentially that of the Prey correction to the reference 2(H − H ) TC (1) H = p 0 + H − p . (14) gravity. Notice, that the interval of integration of the Prey d0 H −(1,ψ) p 2πGH−(1,ψ) correction is not [H0, Hd ] but [H0, Hp]. This is because, when Hd > Hp, the term of the Prey reduction over the In the following we shall call this specific datum level of interval [Hp, Hd ] of the reduced observed gravity in Eq. (6) gravity reduction Hd0 ‘ρB -free specific datum level’orin is canceled out by subtracting that of the reference gravity short ‘ρB -free datum level’. in Eq. (8). At the same time, this corresponds to the fact that The meaning of the ρB -free datum level Hd0 can be un- the mass-density above the station height Hp is zero for the derstood as follows. The condition expressed by Eq. (13) case of Hd > Hp. corresponds to that the sum of the terms in the braces on 3.3 Remarks about the generalized Bouguer anomaly the right-hand side of Eq. (9) or Eq. (10) is zero, i.e. inde- 3.3.1 Unified expression of the formula of the gener- pendent of the Bouguer reduction density ρB . alized Bouguer anomaly The unified expression of the 4.1.2 Specific datum levels Hd1 and Hd2 Another generalized Bouguer anomaly gp,Hd can be written in the condition for defining the specific datum levels Hd1 and same form both for Hd < Hp and for Hd > Hp. By arrang- Hd2, upon which the topographic gravitational effects are ing Eqs. (4) and (5) for the case of Hd < Hp, and Eqs. (6) eliminated, is given by Eqs. (5) and (7): > and (7) for the case of Hd Hp, it can be shown that we ± TCp + BC = ρB TCp(1) have p Hd Hd + ± = + ρ ( ) + π ρ ( ,ψ) + 2πGρB H (1,ψ)dr = 0. (15) gp,Hd gp B TCp 1 2 G B H 1 dr H Hp p Hd ∂γ ∂g This is the condition that the sum of the terrain correction + + dr. ∂ ∂ (11) Hp r r TCp and the Bouguer correction BCp is always zero regardless of ρB . This condition leads to the definition of the This means that Eqs. (9) and (10) are equivalent each other. additional specific datum levels (H and H ): 3.3.2 Effect of the datum level change on the gen- d1 d2 eralized Bouguer anomaly When we regard the eleva- TCp(1) Hd1 = Hp − (16) tion of the datum level Hd as an independent variable, the 2πGH+(1,ψ) changing rate of the generalized Bouguer anomaly gp,Hd with respect to the elevation of the datum level H can be and d TC (1) expressed by the equation H = H − p . (17) d2 p 2πGH−(1,ψ) ∂g , ∂g p Hd = πGρ H −( ,ψ)+ . Since the terrain correction TC (1) is almost everywhere ∂ 2 B 1 ∂ (12) p Hd r positive for a small truncation angle ψ (say ψ<3 de- This is directly derived from any one of Eqs. (9) and (10). grees), the elevation of the specific datum level Hd1 is al- ∂ /∂ In this paper, we will call this rate gp,Hd Hd the ‘re- most everywhere lower than that of the gravity station Hp 294 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

In the same manner, from Eqs. (14) and (17), the speciﬁc datum levels Hd0 and Hd2 have a relation 2(H − H ) H = p 0 + H . (20) d0 H −(1,ψ) d2

It is interesting that the relation between Hd0 for H0 and Hd2 for Hp is reciprocal: 2(H − H ) H = 0 p + H . (21) d2 H −(1,ψ) d0 From Eqs. (18) and (19), we have the relation equivalent to Eq. (20) or Eq. (21): − 2(Hp − H0) =−H (1,ψ)(Hd2 − Hd0). Particularly in a flat Earth approximation of the gravity correction, i.e., H +(1,ψ)→+1 and H −(1,ψ)→−1 [for ψ ∼ 0, refer to Eq. (2)], the specific datum levels H and H locate at the same Fig. 7. Geometric relation between the specific datum levels Hd0, Hd1 and d0 d1 Hd2. Each specific datum level spreads the surface with undulation as a distance of lower and upper positions with respect to H0, ( , ) ( , ) function of the horizontal coordinates x and y: Hd0 x y , Hd1 x y or respectively (see Eq. (18)), resulting in H (x, y). For the flat Earth approximation, the specific datum levels d2 + Hd1 and Hd2 are located at the mirror-imaged positions with respect Hd0 Hd1 H0 = , (for ψ ∼ 0). (22) to the elevation of the gravity station Hp (see Eq. (23)); and so the 2 specific datum levels Hd1 and Hd0 with respect to the elevation of the normal ellipsoid H0 (see Eq. (22)). The elevation of the point Q (HQ ) Also from Eq. (19), Hp takes the algebraic mean value of − is HQ = 2(Hp − H0)/H (1,ψ)+ Hp. Hd1 and Hd2: H + H H = d1 d2 , (for ψ ∼ 0). (23) p 2 (Hd1 < Hp), and the elevation of the specific datum level Furthermore, if the topography is very gentle and hence the Hd2 is almost everywhere higher than that of the gravity value of the terrain correction TC (1) is negligibly small, station Hp (Hd2 > Hp). p H degenerates into the elevation H of the point Q, that Notice that Hd1 and Hd2 can be computed from the d0 Q known quantity of the terrain correction for the unit density is, H = H − 2(H − H )/H −(1,ψ) TCp(1) for each gravity station. Also Hd0 is computable Q p p 0 if H0 is given. Remember that H0 is the elevation of the as shown in Fig. 7 (see Eq. (14)). Also, Eq. (22) represents reference ellipsoid in the orthometric height system of this that Hd0 and Hd1 are at the mirror-image position of the paper, and its magnitude is equal to the geoid height N (i.e. reference ellipsoid level H0. Also, Eq. (23) represents that =− H0 N). Hd1 and Hd2 are at the mirror-image position of Hp,and 4.1.3 Relation between the specific datum levels degenerate into Hp when TCp(1) is negligibly small, (see Since the specific datum levels Hd0, Hd1 and Hd2 can be Eqs. (16) and (17)). defined for each gravity station, these specific datum levels On the other hand, particularly in the spherical shell sys- form surfaces as functions of the horizontal position (x, y): tem of gravity correction, i.e., +( ,ψ)→+ −( ,ψ)→− Hd0 = Hd0(x, y), Hd1 = Hd1(x, y) and H 1 2 and H 1 0 ψ = π , Hd2 = Hd2(x, y). [for , see Eq. (2)] the datum levels H and H take the infinite values (see Each of the three surfaces of Hd0(x, y), Hd1(x, y) and d0 d2 Eqs. (14) and (17)), and lose their physical meanings, while Hd2(x, y) has undulation like the Molodensky telluroid (Heiskanen and Moritz, 1967). Hd1 still takes a definite value of The geometric relation among the surfaces of the specific ( ) = − TCp 1 , ψ = π ( , ) ( , ) ( , ) Hd1 Hp (for ) (24) datum levels Hd0 x y , Hd1 x y and Hd2 x y is shown 4πG in Fig. 7. From Eqs. (14) and (16), the specific datum levels (see Eq. (16) for ψ = π). At first sight, this seems to be Hd0 and Hd1 are related to H0 in the following way inconsistent with Eq. (18). However, substituting Eq. (14) − −( ,ψ) + +( ,ψ) into Eq. (18), we get = H 1 Hd0 H 1 Hd1 . H0 (18) + − 2 H (1,ψ)Hd1 = 2H0 + 2(Hp − H0) + H (1,ψ)Hp Also, from Eqs. (16) and (17), the specific datum levels H TC (1) d1 − p , (forr ψ = π). and Hd2 have a relation against Hp as 2πG + − +( ,ψ) = H (1,ψ)H − H (1,ψ)H This relation is consistent with Eq. (24) since H 1 = d1 d2 . − Hp (19) 2 and H ( ,ψ)= ψ = π 2 1 0 for . K. NOZAKI: GENERALIZED BOUGUER ANOMALY 295

4.2 Introduction of the new notation FA 4.3 Representation of the generalized Bouguer Let FAdenote anomaly at the specific datum level 0 ∂γ The generalized Bouguer anomaly upon an arbitrary da- FA = g + dr − γ , (25-1) tum level of Hd is given by any one of Eqs. (9) and (10). p ∂r 0 Hp By substituting FA as defined by Eq. (25) into Eq. (9), we where, have 0 ∂γ Hd ∂g gp: observed gravity at the elevation Hp, = − + gp,Hd FA dr dr H ∂r H ∂r 0 ∂γ 0 p H ∂ dr: free-air correction from the level of elevation Hp Hd p r + to the level of the geoid, + ρB TCp(1) + 2πGH (1,ψ)dr H p γ0: normal gravity upon the normal ellipsoid. Hd − 2πG[H +(1,ψ)− H −(1,ψ)]dr . , Note that the interval of integration [Hp 0] in Eq. (25-1) H0 and the interval [0, H0] in the interval [H0, Hp] in Eq. (9) (28) or (10) are complementary to each other with respect to the whole interval of the free-air correction. The integra- The first and second terms in the braces of Eq. (28) are essentially the terrain and Bouguer corrections for the ob- tion over the interval [H0, 0] plays an important role in the geodetic interpretation of FA (as is described in Sec- served gravity, while the third term in the braces is essen- tion 4.5). tially the Prey correction for the reference gravity. As was Ignoring the VGG anomaly, FA is the approximation previously mentioned, one can derive the same results by using Eq. (10) instead of Eq. (9). of the free-air anomaly gF (e.g. Heiskanen and Moritz, 1967, equation (3-62), p. 146). Alternatively, FA can be Substituting Hd0, Hd1 and Hd2 (Eqs. (14), (16) and (17)) rewritten as into Hd in Eq. (28), we have, respectively, the representation formulae of the generalized Bouguer anomalies at the Hp ∂γ FA = g − γ + dr , specific datum levels of gravity reduction Hd0, Hd1 and Hd2 p 0 ∂ (25-2) 0 r as follows: and also as 0 H ∂γ d0 ∂g H −ς g , = FA− dr + dr p ∂γ p Hd0 ∂r ∂r FA ≈ g − γ + dr , H0 Hp p 0 ∂ (26) −ς r Hd0 + ρ ( ) + π +( ,ψ) where ζ is the height anomaly. This can be easily un- B TCp 1 2 GH 1 dr Hp derstood by changing the interval of integration [0, H ] p Hd0 , + + − in Eq. (25-2) to [H0 Hp H0], and assuming N (geoid − 2πG[H (1,ψ)− H (1,ψ)]dr , height) = ζ (height anomaly), and H (orthometric H0 N height) = H (normal height): (for Hd0 < Hp), (29) + − Hp ∂γ Hp H0 ∂γ Hp N ∂γ ⇒ = dr dr dr 0 ∂γ Hd1 ∂g 0 ∂r H ∂r −N ∂r = − + 0 gp,Hd1 FA dr dr −ς ∂r ∂r Hp ∂γ H0 Hp ≈ . dr (27) Hd1 −ς ∂r + + ρB TCp(1) + 2πGH (1,ψ)dr ∂γ/∂ Hp In this case, it is not necessarily required that r is constant. Importantly, in this case, the integration (free- Hd1 − ρ 2πG[H +(1,ψ)− H −(1,ψ)]dr, air correction) over the interval [Hp − ζ, Hp], which is B H0 complementary to the interval of integration [−ζ, Hp − ζ ] (for Hd1 < Hp), (30) in Eq. (26) for the whole interval [−ζ, Hp] ≈ [H0, Hp]in Eq. (9) or (10), plays essentially the same role as that over and the interval [H0, 0] as mentioned above. 0 ∂γ Hd2 ∂g Thus, FAof Eq. (26) represents the new gravity anomaly g , = FA− dr + dr p Hd2 ∂ ∂ (Heiskanen and Moritz, 1967, equation (8-7), p. 293), or the H0 r Hp r point free-air anomaly (e.g. Torge, 1989, equation (3-7a), Hd2 − p. 54), that is the difference between the measured gravity + ρB TCp(1) + 2πGH (1,ψ)dr H at the ground and the normal gravity at the telluroid. In this p sense, FA is the free-air anomalies in the Molodensky’s Hp − ρ πG H +( ,ψ)− H −( ,ψ) dr sense, although FA of Eq. (25-1) was firstly defined for B 2 [ 1 1 ] H0 the free-air corrected observed gravity on the geoid as the (for Hd2 > Hp), (31) gravity anomaly (Heiskanen and Moritz, 1967, equation (2- 139), p. 83). For the sake of the later calculations, in Eqs. (29)–(31), Using FA, hereafter, we will proceed to formulate the we retain the terms that vanish under the conditions of the ρB -free generalized Bouguer anomaly. specific datum levels. 296 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

Notice that, in Eq. (29), the sum of the terrain and anomaly at the specific datum level ( gp,Hd0 , gp,Hd1 ,or Bouguer corrections (the first and second terms in the gp,Hd2 ), respectively. braces) is canceled out by the term of the Prey correction 4.4 The meaning of the generalized Bouguer anomaly (the third term in the braces), resulting in all the terms con- at the ρB -free datum level Hd0 cerning ρB in the braces on the right-hand side vanish by In the simple case when the term of the VGG anomaly setting the datum level at Hd0. This is because the specific ∂g/∂r is sufficiently small, Eq. (32) of the generalized ρ datum level Hd0 is so defined as to satisfy Eq. (13). Also, Bouguer anomaly at the B -free datum level Hd0 ( gp,Hd0 ) in Eqs. (30) and (31), the sum of the terrain and Bouguer is approximated to corrections, which corresponds to the first and the second 0 terms in the braces on the right-hand sides, vanish by set- ∂γ g , = FA− dr. (35) p Hd0 ∂ ting the datum levels at Hd1 and Hd2, respectively. This is H0 r because the specific datum levels Hd1 and Hd2 are so defined as to satisfy Eq. (15). Notice, that the interval of inte- Rewriting Eq. (35) by using Eq. (25), we obtain the follow- gration of the fifth term on the right-hand side of Eq. (31) is ing approximate representations not [H0, Hd2] but [H0, Hp], because the mass-density ρB is H0 ∂γ zero over the interval [Hp, Hd2]. g , = g + dr − γ : p Hd0 p ∂r 0 When these vanishing terms in Eqs. (29), (30) and (31) Hp are set to zero, we have the final equations gravity disturbance on the ellipsoid 0 ∂γ 0 ∂γ 0 ∂γ = + − γ + , = − gp dr 0 dr : gp Hd0 FA dr ∂r ∂r ∂r Hp H0 H0 Hd0 ∂ gravity disturbance on the geoid g + dr, (for Hd0 < Hp), (32) ∂r Hdc ∂γ Hdc ∂γ Hp = g + dr − γ + dr : p ∂r 0 ∂r Hp H0 0 ∂γ H = − gravity disturbance at any datum level dc. (36) gp,Hd1 FA dr H ∂r 0 Thus, we conclude that the generalized Bouguer anomaly Hd1 + − at the ρ -free datum level H , that is, the ρ -free Bouguer − 2πGρB [H (1,ψ)− H (1,ψ)]dr B d0 B H0 anomaly g , , is the gravity disturbance. Also, Eq. (36) p Hd0 Hd1 ∂ represents that the gravity disturbance is invariant for the + g , < , dr (for Hd1 Hp) (33) level transformation in the free-air space. H ∂r p Equation (35) represents the relation between the gravity disturbance (gp,H ) and FA (the Molodensky’s free-air 0 d0 ∂γ anomaly). Although the details will be described in the gp,H = FA− dr d2 ∂r next section, this fact suggests that Eq. (35) has a tie to the H0 Hp fundamental equation of physical geodesy. Also, Eq. (36) − π ρ +( ,ψ)− −( ,ψ) 2 G B [H 1 H 1 ]dr implies that the gravity disturbance in the free-air space can H0 be defined not only at the elevation of the geoid but also Hd2 ∂ g H g , + dr, H > H , at any level dc. The gravity disturbance ( p Hd0 ) is not ∂ (for d2 p) (34) Hp r defined by the right-hand side of Eq. (36) but results in the right-hand side of Eq. (36). Such a view of the gravity respectively. disturbance (g , ) is schematically illustrated in Fig. 8. Equation (32) is a representation of the condition that the p Hd0 Particularly, when g , is upward-continued in the free- generalized Bouguer anomaly (left-hand side of Eq. (9) or p Hd0 air space to the station level at P, it is interesting that the Eq. (29)) is independent of the Bouguer reduction density gravity disturbance (g , at P) does not change the value ρ . Equations (33) and (34) are representations of the con- p Hd0 B even though the removed or moved topographic masses are dition that the sum of the terrain and Bouguer corrections is ρ completely restored (cf. Eqs. (29) and (32)). zero regardless of B . Namely, by setting the datum level From Eqs. (32), (33) and (34), the relations of gp,Hd1 at the ρB -free datum level Hd0, the generalized Bouguer and gp,H to the ρB -free Bouguer anomaly gp,H are anomaly g , results in being free from the Bouguer re- d2 d0 p Hd0 given as duction density ρB . In this paper, we call this generalized Bouguer anomaly g , the ρ -free Bouguer anomaly. p Hd0 B gp,H = gp,H Also, by setting the datum levels H and H , the gen- d1 d0 d1 d2 Hd1 + − , , eralized Bouguer anomalies gp Hd1 and gp Hd2 result in − 2πGρB [H (1,ψ)− H (1,ψ)]dr being free from the terrain and Bouguer corrections. The H0 + − H integrand of 2πGρB [H (1,ψ)− H (1,ψ)] in Eq. (33), as d1 ∂g + dr well as that in Eq. (34), corresponds to the Prey correction ∂ (37) Hd r for the reference field. 0 Here we shall pay special attention to that Eqs. (32)–(34) yield the relation between FAand the generalized Bouguer K. NOZAKI: GENERALIZED BOUGUER ANOMALY 297

and Eq. (40) is written as ∂γ g , = g , p Hd2 p Hd0 g = δg + N , (41) Hp ∂r − 2πGρ [H +(1,ψ)− H −(1,ψ)]dr B which is equivalent to the fundamental equation of physical H0 Hd2 ∂g geodesy. Comparing Eqs. (39) and (41), it is clear that these + dr, (38) two equations are similar to each other, identifying FAwith ∂r Hd0 δ − g, gp,Hd0 with g, and H0 with N. respectively. 4.5 Relation to the fundamental equation of physical 5. Estimation of the Bouguer Reduction Density geodesy We will show in this section that the Bouguer reduction density ρ is estimated by the plot of FA against the spe- Since H0 =−N, it is shown below that Eq. (35) has a tie B to the fundamental equation of physical geodesy (Heiska- cific datum levels, and also H0 is estimated on the same nen and Moritz, 1967, equation (2-148), p. 86). plot. Regarding VGG of the normal gravity field as constant, 5.1 Derivation of the equation for estimating the Eq. (35) yields Bouguer reduction density As was explained previously, each of the specific da- ∂γ tum levels Hd0, Hd1 and Hd2, upon which the general- = , + (− ) . FA gp Hd0 H0 (39) ∂r ized Bouguer anomalies gp,Hd0 , gp,Hd1 , gp,Hd2 are defined, forms a surface as a function of the horizontal co- On the other hand, the fundamental equation of physical ordinates (x, y): Hd0 = Hd0(x, y), Hd1 = Hd1(x, y) and geodesy is written as Hd2 = Hd2(x, y). ∂T T ∂γ Here, we shall notice that Eqs. (32), (33) and (34), which g =− + , (40) represent the generalized Bouguer anomalies at the spe- ∂r γ ∂r cific datum levels, hold for every point of horizontal coor- where, g denotes the (geodetic) gravity anomaly, and T dinates (x, y). Therefore, one can consider the differential , denotes the gravity disturbing potential. Then, using the quantities of the generalized Bouguer anomalies gp Hd0 , , , relation gp Hd1 , and gp Hd2 with respect to the specific datum lev- ∂T els H , H and H in the neighbourhood of (x, y), re- δg =− d0 d1 d2 ∂r spectively. In this case, it is necessary to differentiate not only g , , g , , g , and FA, but also H and and the Bruns’ formula p Hd0 p Hd1 p Hd2 p TCp(1), since they are functions of Hd0, Hd1 and Hd2 that T = Nγ, are functions of x and y. Differentiating Eqs. (29), (30) and (31) with respect to Hd0, Hd1 and Hd2, respectively, we have dg , dFA dH ∂g p Hd0 = + 1 − p , (42) dHd0 dHd0 dHd0 ∂r

dg , dFA dH ∂g p Hd1 = + 1 − p dHd1 dHd1 dHd1 ∂r + − − 2πGρB [H (1,ψ)− H (1,ψ)], (43)

and dg , dFA dH ∂g p Hd2 = + − p 1 ∂ dHd2 dHd2 dHd2 r dHp + − 2πGρB [H (1,ψ) dHd2 − H −(1,ψ)]. (44)

As for the derivation of these equations, refer to Ap- pendix B. In the above calculation, dHp/dHd0, dHp/dHd1 and dHp/dHd2 are taken into account because Hd0, Hd1 and Fig. 8. Schematic illustration that represents the equivalence of the gen- Hd2 are not independent variables but functions of Hp.Be- eralized Bouguer anomaly at the ρB -free datum level Hd0 to the gravity sides, the elevation of a gravity station Hp is a function of disturbance at any datum level. The VGG anomaly (∂g/∂r)isne- the horizontal coordinates (x, y). Therefore, the total dif- =− glected here. N denotes the geoid height (N H0). ferential of the generalized Bouguer anomaly at the speciﬁc 298 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

= , , datum level (d gp,Hdi , for i 0 1 2) can be written as 5.2 FAvs. Hd0, Hd1 and Hd2 diagram Each free-air anomaly FA in Eqs. (32), (33) and (34), ∂g , ∂g , = p Hdi + p Hdi which is a computable quantity, is a function of each spe- d gp,Hdi dx dy ∂x ∂y cific datum level Hd0, Hd1 and Hd2, respectively. Here, ∂g , we plot the free-air anomaly (FA) against the datum level + p Hdi , = , , dHdi (for i 0 1 2) (45) = ∂ Hdi Hd (e.g. Hd Hd0, Hd1 or Hd2) for every gravity station. Then, we obtain generally a set of plots as schematically as a function of the horizontal coordinates (x, y) and Hdi, shown in Fig. 9. In this paper, we call the diagram of these = , , = (i 0 1 2). When we assume that each gp,Hdi ,(i plots ‘FAvs. Hd diagram’. , , ( , ) 0 1 2) takes a constant value in the neighborhood of x y , The characteristics found in the FA vs. Hd diagram are i.e. the lateral variation of gp,Hdi is sufficiently small, as follows: Eq. (45) yields (1) One can measure the gradient of the regression lines dg , ∂g , on the FA vs. Hd diagram. Namely, the gradi- p Hdi ≈ p Hdi , (for i = 0, 1, 2). (46) ent dFA/dHd0 of the regression line for the FA vs. dHdi ∂ Hdi Hd0 plot. Similarly, the gradients dFA/dHd1 and Since the partial differential coefficients satisfy the equation dFA/dHd2 of the regression lines for the FAvs. Hd1, and Hd2 plots respectively. ∂g , ∂g , ∂g , p Hd0 = p Hd1 = p Hd2 , (2) There exists, in general, an intersection point C of the ∂ Hd0 ∂ Hd1 ∂ Hd2 regression lines Hd1-line and Hd2-line. The intersec- Eq. (46) leads directly to the following approximate relation point C does not generally coincide with the ori- tion: gin (0, 0). Notice, that the position of the intersection dg , dg , dg , p Hd0 = p Hd1 = p Hd2 . (47) point C is definite because the Hd1 and the Hd2 are dHd0 dHd1 dHd2 given by Eqs. (16) and (17). Thus, subtracting Eq. (42) from Eq. (43) and using Eq. (47), (3) Also, one can draw a regression line Hd0-line on the and assuming constant VGG anomaly, (∂g/∂r) = β, FA vs. Hd0 plot so that it passes through the intersec- we have tion point C (H = H , FA = γ ). Notice that d d 0 H -line in Fig. 9 has a degree of freedom for parallel + − dHp dHp d0 2πGρB [H (1,ψ)− H (1,ψ)] + − β translation along the axis H , because the equation of dH dH d d1 d0 H (Eq. (14)) contains H as an unknown parameter. dFA dFA d0 0 = − . (48) / dHd1 dHd0 5.2.1 Evaluation of the gradients dFA dHd0 and dFA/dHd1 Based on the characteristics (1) in the above From Eqs. (43) and (44), one can also derive the result section, Eq. (51-1) shows that one can estimate the Bouguer equivalent to Eq. (48). We shall notice again that all quan- reduction density ρB from the gradients dFA/dHd0 of the ρ β tities in Eq. (48) but for B and are known. Differential Hd0-line and dFA/dHd1 of the Hd1-line. Particularly when coefficients in Eq. (48) have the following relations: ψ ∼ 0, β = 0 and H0 = 0, which is the case for the flat Earth, the relation between the gradient dFA/dHd1 of dH H +(1,ψ) dH p = p , the Hd1-line and the gradient dFA/dHd0 of the Hd0-line is − (49) dHd1 H (1,ψ)dHd0 and +( ,ψ) dFA = H 1 dFA. − (50) dHd1 H (1,ψ)dHd0 These relations of Eqs. (49) and (50) can be derived from Eq. (18). When the VGG anomaly β is sufficiently small, Eq. (48) gives the following equation dFA dFA ρB ≈ − 4πG. (51-1) dHd1 dHd0

Thus, we have the final approximate equation, Eq. (51-1), for estimating the Bouguer reduction density ρB . Equation ,γ (51-1) means that ρB is calculated from the gradients of FA Fig. 9. Typical FAvs. Hd diagram. The intersection point C ( Hd 0) with respect to the specific datum levels. Concrete method is definite because the regression lines Hd1-line for FA vs. Hd1 plot and H -line for FA vs. H plot are definite. One can draw the dFA/dH dFA/dH d2 d2 for evaluating the gradients d0 and d1 regression line Hd0-line for FA vs. Hd0 plot so that it passes through will be described in the next section. The effect of the VGG the definite intersection point C. In case of the flat Earth approximation anomaly (β) on the Bouguer reduction density estimation of the gravity corrections, FAvs. Hd0 plot and FAvs. Hd1 plot become = can be evaluated by Eq. (48). symmetric with respect to the vertical line Hd Hd . K. NOZAKI: GENERALIZED BOUGUER ANOMALY 299 dFA/dHd1 =−dFA/dHd0. Hence, Eq. (51-1) yields dFA ρB ≈ 2πG (51-2) dHd1 which is essentially equivalent to the result given by Hagi- wara et al. (1986) for estimating the Bouguer reduction density. This is a kind of the Nettleton’s method (Nettleton, 1939) for density determination. 5.2.2 Estimation of H0 On the FA vs. Hd diagram (Fig. 9), let the position of the intersection point C between the regression lines Hd1-line and Hd2-line be Hd = Hd , and FA = γ0. Then, the intersecting condition Hd1 = Hd2 (i.e. Eq. (16) =Eq. (17)) leads to

TCp(1) = 0 and Hp = Hd (at the point C). (52)

Next, based on the above characteristics (3), we can ad- just the Hd0-line so as to pass through the intersection point C. Then, the intersecting condition Hd0 = Hd1, (i.e. = Eq. (14) Eq. (16)), leads to Fig. 10. Schematic illustration of computing the Bouguer anomaly on the geoid surface (gp,0). For each gravity station, gp,0 can be computed H0 = Hp. (53) by the level transformation from the generalized Bouguer anomaly at the ρ B -free datum level Hdo ( gp,Hd0 ). Each arrow indicates the amount ρ Finally we have, from Eqs. (53) and (52), at the intersection of level transformation from the B -free datum level to the level of the point C geoid. gp,0 is equivalent to the classical Bouguer anomaly. H0 = Hd . (54)

Of course, the geoid height N is given as N =−H0 = Hd0 to the geoid surface. When such a reduction is done, −Hd . one can study the subsurface density structure by Tsuboi’s In addition to the above results, it can be shown that double Fourier method (Tsuboi, 1938; Tsuboi and Fuchida, the fundamental equation of physical geodesy, equivalently 1938). Eq. (41), is satisfied at the intersection point C (Hd = Hd , Now we will show how the ρB -free Bouguer anomaly = γ = γ = = FA 0). Substituting FA 0 and H0 Hd0 gp,Hd0 is reduced to the geoid. The generalized Bouguer Hd into Eq. (35), we have anomaly at the geoid surface gp,0 is calculated by the level transformation of gp,Hd0 from the datum level Hd0 to the 0 ∂γ = = γ − . level of the geoid (Hd 0). Then, we have gp,Hd0 0 dr ∂r Hd 0 ∂g , = + p Hd . gp,0 gp,Hd0 dHd (55) Because of the correspondence between Eqs. (39)–(41) (i.e. ∂ Hd = δ γ = = = Hd0 gp,Hd0 g, and 0 FA g, and Hd Hd0 = H0 =−N), this equation yields Eq. (41), which is Therefore, by using the reduction rate of Eq. (12), we can equivalent to the fundamental equation of physical geodesy rewrite Eq. (55) as (Heiskanen and Moritz, 1967). This equation shows that , = , gp 0 gp Hd0 gp,Hd0 can be computed from the known quantities Hd 0 and γ . − ∂g 0 + πGρ H ( ,ψ)+ dH . 2 B 1 ∂ d (56) Hd0 r 6. Derivation of the Bouguer Anomaly at the When we ignore the VGG anomaly, Eq. (56) yields Geoid from the ρB -free Bouguer Anomaly at Hd0 The Bouguer anomaly has been used for estimating sub- − g , ≈ g , − 2πGρ H (1,ψ)H . (57) surface structure. The primary purpose of defining the gen- p 0 p Hd0 B d0 eralized Bouguer anomaly is to obtain the Bouguer anomaly This is the final approximate representation of gp,0 that which is free from the density assumption used in the is represented upon an equi-potential surface of the geoid. Bouguer correction as well as in the terrain correction. Such a reduction of the generalized Bouguer anomaly So far, we have found that the ρB -free Bouguer anomaly gp,Hd0 from the specific datum level Hd0 to the level of , ρ gp Hd0 is realized upon the B -free specific datum level of the geoid is schematically illustrated in Fig. 10. It should gravity reduction H , which is not restricted to the geoid d0 be noted that we require the values of ρB and Hd0 (or H0) surface. Furthermore, the ρ -free Bouguer anomaly at H , B d0 for computing gp,0. , gp Hd0 , is nothing but the gravity disturbance in the the- Alternatively, substituting Eq. (35) into Eq. (57), we have ory of physical geodesy. Thus, in order to estimate sub- 0 surface density structure, we have to reduce the generalized ∂γ − gp,0 ≈ FA− dr − 2πGρB H (1,ψ)Hd0. (58) Bouguer anomaly from the ρB -free specific datum level of ∂ H0 r 300 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

Equation (58) gives the relation between the gravity dis- (Eq. (35)). FAis defined as the difference of the free- turbance on the geoid (gp,0), that is, the gravity anomaly air corrected observed gravity upon the geoid from the in the Molodensky sense (FA), and the Bouguer reduction normal gravity γ0 upon the reference ellipsoid (see density ρB . Furthermore, substituting Eq. (51-1) into the Eq. (25)). We found that FA is equal to the ‘gravity density ρB in Eq. (58), we have another expression of the anomaly’ in the Molodensky’s sense. generalized Bouguer anomaly at the geoid (gp,0) in the Relation between the ρB -free Bouguer anomaly form (g , ) and the fundamental equation of physical p Hd0 0 ∂γ geodesy is discussed (Eqs. (39), (40) and (41)). ≈ − gp,0 FA dr (5) A method for estimating the Bouguer reduction den- H ∂r 0 sity ρ is found. The Bouguer reduction density is H −(1,ψ) dFA dFA B − − H . (59) given by the difference between the gradients of FA 2 dH dH d0 d1 d0 with respect to Hd0 and Hd1, respectively (Eq. (51)). In Eq. (59), the Bouguer reduction density ρB is represented Also a condition equation to be satisfied by the in terms of FA. Notice that all the terms including H0 on Bouguer reduction density ρB and the VGG anomaly the right-hand side of Eq. (59) are computable quantities β is derived (Eq. (48)). It can be used for evaluating (see Eqs. (51) and (54)). The distribution of gp,0 calcu- the influence of β on the ρB estimation. lated by Eq. (58) or (59) is nothing but the Bouguer anomaly (6) The generalized Bouguer anomaly upon the geoid distribution, which has been used to study the subsurface (gp,0) is obtained from the ρB -free Bouguer anomaly density structure (e.g. Tsuboi and Fuchida, 1938). gp,Hd0 . It is done by the level transformation of the gravity value from the specific datum level Hd0 7. Conclusions to the geoid (see Eq. (55)), using the reduction rate − (1) We defined a new concept of the generalized Bouguer of 2πGρB H (1,ψ) (Eq. (12)). The generalized anomaly upon an arbitrary datum level whose eleva- Bouguer anomaly distribution upon the geoid, gp,0, tion from the geoid is Hd (see Eq. (1)). will be used for estimating the subsurface structure. (2) Three specific datum levels Hd0, Hd1 and Hd2 are defined for every gravity station (see Eqs. (14), (16) and ρ Acknowledgments. The author expresses sincere thanks to Pro- (17)). The specific datum level Hd0, so-called the B - fessor Emeritus Shozaburo Nagumo of the University of Tokyo, free datum level, is defined by a condition that the the chief technical adviser of OYO Corporation, for his encourage- generalized Bouguer anomaly is independent of the ment and frequent discussions throughout the research. Also the author wishes to express his grateful thanks to Professor Shuhei Bouguer reduction density ρB (see Eq. (13)). The specific datum levels of H and H are defined by a con- Okubo of the University of Tokyo for his hearty guidance and crit- d1 d2 ical comments, and to Professor Emeritus Yoshio Fukao of the dition that the sum of the terrain and the Bouguer cor- University of Tokyo for his stimulating discussions and continuous rections is zero (see Eq. (15)). encouragement during the research. The author’s thanks are also The specific datum levels Hd1 and Hd2 can be com- extended to Drs. Masaru Kaidzu and Yuki Kuroishi of the Geo- puted in practice for each gravity station from the value graphical Survey Institute for their helpful discussions on the basic of terrain correction for the unit density TC (1). H concepts of the physical geodesy, and to Professor Emeritus Yukio p d0 Hagiwara of the University of Tokyo for valuable comments on the can be computed if the elevation of the reference el- latest geodesy. The contribution of Dr. Gabriel Strykowski, Na- lipsoid H0 is known. A method to evaluate H0 is dis- tional Survey and Cadastre, Denmark, who gave insightful and in- cussed (Eq. (54)). structive comments on the paper, is gratefully acknowledged. The (3) Three specific generalized Bouguer anomalies author also expresses his sincere thanks to Professor W. E. Feath- erstone, Curtin University of Technology, Australia, for his critical g , , g , and g , are derived for every p Hd0 p Hd1 p Hd2 comments with his recent publications on the ‘gravity anomaly’. gravity station at their specific datum levels Hd0, Hd1 Finally, the author gratefully acknowledges to Dr. Petr Holota, Re- and Hd2, respectively (see Eqs. (32), (33) and (34)). search Institute of Geodesy, Topography and Cartography, Czech Republic, for his helpful comments to improve the manuscript. The specific generalized Bouguer anomaly gp,Hd0 , the ρB -free Bouguer anomaly, does not include the Bouguer reduction density ρ (see Eq. (32)) and B Appendix A. is therefore free from the assumption of ρB . The ρ -free Bouguer anomaly g , is essentially equal B p Hd0 Effects of the Earth’s sphericity and the truncation an- to the ‘gravity disturbance’ in the physical geodesy gle of the spherical shell (Eq. (36)). Figure A1 shows a schematic illustration of a thin spher- The specific generalized Bouguer anomalies g , p Hd1 ical cap of axial symmetry. Let h be the thickness of the and g , , as well as g , , do not include the p Hd2 p Hd0 thin spherical cap, ρ the density, ψ the truncation angle, and terrain correction explicitly and are not affected by t = r/r ± the normalized geocentric distance of the spheri- the topographic gravitational effects (see Eqs. (33) and cal cap. Then, the gravity (g±) due to the thin spherical cap (34)). at a station P± on the symmetry axis can be written as (4) When the terms of the VGG anomaly are sufficiently small i.e. ∂g/∂r = 0, we found that the gen- t+t ψ 2π 2 ρ ± ± t sin ψ(1 − t cos ψ) eralized Bouguer anomaly at Hd0 (i.e. the B -free g = Gρr dαdψdt, ( 2 + − ψ)3/2 Bouguer anomaly gp,Hd0 ) is equal to FAminus free- t 0 0 t 1 2t cos air correction from the reference ellipsoid to the geoid (A.1) K. NOZAKI: GENERALIZED BOUGUER ANOMALY 301

Clearly, from Eq. (A.3), H ±(t,ψ) defined by Eq. (A.4) is a characteristic function that governs the gravitational behaviour of the spherical cap. A graph of H ±(t,ψ) is shown in Fig. A2 as a function of angular distance (or truncation angle) ψ with an argument t. Especially, when t takes a limit value t = 1, Eq. (A.4) yields 1 − cosψ H ±(1,ψ)= ± 1. (A.5) 2 H ±(1,ψ) means H ±(1 ∓ ε, ψ), where ε denotes an in- finitesimally small positive number. The first term on the right-hand side of Eq. (A.5) corresponds to the term of sphericity, and the second term does to that of an infinite plate or a Bouguer slab. When ψ becomes small enough (i.e. ψ ≈ 0), Eq. (A.3) agrees with gravitational attraction of an infinite plate, i.e. ±2πGρh. On the other hand, when ψ ≈ π, H +(1,ψ) = 2 and Eq. (A.3) becomes π ρ Fig. A1. Schematic illustration of a thin spherical cap with a small 4 G h on the outer surface of a thin spherical shell of thickness h and a truncation angle ψ. r denotes the radial distance of thickness h. On the inner surface of the spherical shell, the spherical cap. P+ and P− denote the computation points of gravity −( ,ψ) = + − H 1 0 and Eq. (A.3) yield zero. From Eq. (A.5), at the radial distances r and r , respectively. clearly holds for the following identical equation:

H +(1,ψ)− H −(1,ψ)≡ 2. (A.6) (double signs should be taken in the same order; the same as below), where, α is the azimuthal angle and t = h/r ±. This relation can be confirmed on the Fig. A2 that the lines Executing the integration of Eq. (A.1) with respect to α for H +(1−ε, ψ) and H −(1+ε, ψ) are parallel to each other and ψ,wehave with the distance of 2. Referring to Eq. (A.3), Eq. (A.6) implies that the gravity difference between the immediately t+t t3 − t2 cos ψ g± = 2πGρr ± ± t2 dt. upper and lower points on the spherical cap with a small t t2 + 1 − 2t cos ψ thickness h is always equal to 4πGρh regardless of the (A.2) truncation angle ψ. By the Taylor expansion of Eq. (A.2) in the neighbourhood of t = t, and neglecting the higher order terms of more than Appendix B. or equal to (t)2 under the condition of t t, Eq. (A.2) yields Differentiation of equations of the generalized Bouguer anomaly with respect to the specific datum levels Hd0, ± ≈ π ρ ±( ,ψ) ,(= ± ) g 2 G H t h h r t (A.3) Hd1 and Hd2 In this appendix, we demonstrate that the differentiation (Nozaki, 1999), where, of Eqs. (29), (30) and (31) in the text with respect to the t3 − t2 cos ψ specific datum levels H , H and H results in Eqs. (42), H ±(t,ψ):= ± t2. (A.4) d0 d1 d2 t2 + 1 − 2t cos ψ (43) and (44) in the text, respectively.

Fig. A2. A graph of the characteristic function H ±(1,ψ)as a function of the truncation angle ψ with an argument t (after Nozaki, 1999). The immediately upper and lower points on the spherical cap correspond to t = 1 − ε and t = 1 + ε, respectively, where ε is an inﬁnitesimally small positive number. Simpliﬁed notations H +(1,ψ)= H +(1 − ε, ψ) and H −(1,ψ)= H −(1 + ε, ψ) are used in the text. 302 K. NOZAKI: GENERALIZED BOUGUER ANOMALY

Here, we shall notice that Eqs. (32), (33) and (33) in the Appendix C. text hold corresponding to every point of horizontal coordinates (x, y). Therefore, one can consider the differen- A Note on the Reference Gravity tial quantities in the neighbourhood of (x, y). In this case, C.1 A comparison of the Prey-reduced reference gravity it is necessary to differentiate not only gp,Hd0 , gp,Hd1 , field and the normal gravity field ( ) gp,Hd2 and FA but also Hp and TCp 1 , since they are It is stated in an old text book by Garland (1965, p. 50) functions of Hd0, Hd1 and Hd2 that are functions of x and that an anomaly of gravity is defined by the difference be- y. tween the observed (or reduced) gravity at some point and Differentiating Eqs. (29), (30) and (31) in the text with the theoretical value predicted for the same point. There- respect to Hd0, Hd1 and Hd2, respectively, we have fore, for defining the difference of gravity at the geoid, the normal gravity will not be the only one theoretical value. dg , dFA dH ∂g p Hd0 = + − p Other theoretical values are possible. Selection of an appro- 1 ∂ dHd0 dHd0 dHd0 r priate value depends on the purpose of the anomaly (Gar- ( ) dTCp 1 land, 1965, p. 59). The Prey-reduced reference gravity field + ρB dHd0 introduced newly in this paper is one of such theoretical ones, and is not the normal gravity field. + − dHp π +( ,ψ) 1 2 GH 1 C.2 The normal gravity field (a review) (see Fig. C1) dHd0 The normal gravity at the ellipsoid is defined by (Heiska- − 2πG[H +(1,ψ)− H −(1,ψ)] , nen and Moritz, 1967, equation (2-78))

< , N (for Hd0 Hp) (B.1) γ = γ ≡ γ (C.1) 0 H0 ∂ N d gp,Hd1 dFA dHp g where the right side term γ is a new notation introduced = + 1 − H0 ∂ N dHd1 dHd1 dHd1 r here. The normal gravity at the normal height H (notation ( ) after Torge, 1989), γ N , is defined by the first order ap- + ρ dTCp 1 H N B proximation (Heiskanen and Moritz, 1967, equation (8-8), dHd1 p. 293) dH + 1 − p 2πGH+(1,ψ) dH ∂γ d1 γ = γ( N ) ≡ γ N ≈ γ + N . + − at H H N 0 H (C.2) − 2πGρB [H (1,ψ)− H (1,ψ)], ∂h (for H < H ), (B.2) d1 p The normal gravity is defined upon and outside the refer- and ence ellipsoid, and not inside the reference ellipsoid. ∂ C.3 Mass density distribution in the normal gravity field d gp,Hd2 dFA dHp g = + 1 − Outside the reference ellipsoid, the space is free-air, i.e., dH dH dH ∂r d2 d2 d2 the mass density is zero. Inside the reference ellipsoid, dTCp(1) + ρB the mass density distribution does not need to be known dHd2 (Heiskanen and Moritz, 1967, Sec. 2–7, p. 64). However, dHp − the amount of the total mass inside the reference ellipsoid + 1 − 2πGH (1,ψ) is constant. The simplest mathematical model of such a dH d2 dHp + − 2πGρB [H (1,ψ) dHd2 − − H (1,ψ)], (for Hd2 > Hp). (B.3) On the other hand, differentiating Eqs. (14), (16) and (17) in the text with respect to Hd0, Hd1 and Hd2, respectively, we have dTC (1) dH p = p 2πGH+(1,ψ) dHd0 dHd0 − 2πGH−(1,ψ), (B.4) dTC (1) dH p =− 1 − p 2πGH+(1,ψ), (B.5) dHd1 dHd1 and dTC (1) dH p =− 1 − p 2πGH−(1,ψ). (B.6) dHd2 dHd2 Fig. C1. Relation between the Prey-reduced reference gravity field and Substituing Eqs. (B.4), (B.5) and (B.6) into Eqs. (B.1), the normal gravity field. The reference ellipsoid connects γ Ref and γ N H0 0 (B.2) and (B.3), respectively, and arranging the terms, we N Ref (or γ ). γ ≡ γH in the text. have Eqs. (42), (43) and (44) in the text. H0 Hd d K. NOZAKI: GENERALIZED BOUGUER ANOMALY 303 mass density distribution is free-air with the centralized to- C.5 Mass density distribution in the reference gravity field tal mass. Another mathematical model is such that a mass Mass density distribution in the Prey-reduced reference density distribution inside the reference ellipsoid is con- gravity field is massive inside the model earth, as stated strained by the Clairaut’s equation (Moritz, 1990, equations above. The characteristics are (1) the mass density of the (2-114) and (4-4)). surface layer is ρB , and (2) the total mass inside the refer- C.4 Reference gravity field introduced in this paper (see ence ellipsoid is equal to the total mass of the normal gravity Fig. C1) field. The model earth of the reference gravity field introduced in this paper is massive, having a surface layer above and References below the reference ellipsoid. The reference gravity field Garland, G. D., The Earth’s Shape and Gravity, 183 pp., Pergamon Press, is defined only inside of the model earth, and not outside. Oxford, 1965. Hackney, R. I. and W. E. Featherstone, Geodetic versus geophysical per- The purpose of defining such a reference gravity field is to spectives of the ‘gravity anomaly’, Geophys. J. Int., 154, 35–43, 2003. introduce the generalized Bouguer anomaly. The surface Hagiwara, Y., I. Murata, H. Tajima, K. Nagasawa, S. Izutuya, and S. of the reference ellipsoid is common in both the normal Okubo, Gravity study of active faults—Detection of the 1931 Nishi- gravity field and the reference gravity field. Saitama Earthquake Fault—, Bull. Earthq. Res. Inst., 61, 563–586, 1986 (in Japanese with English abstract). The reference gravity of this paper at the reference ellip- Heiland, C. A., Geophysical Exploration, 2nd Printing, 1013 pp., Prentice- Ref soid, γ , (i.e. notation γ in this paper) is taken as the Hall, New York, 1946. H0 H0 same as the normal gravity γ0, as shown in Fig. C1, Heiskanen, W. A. and H. Moritz, Physical Geodesy, 364 pp., Freeman and Company, San Francisco, 1967. γ Ref ≡ γ ≡ γ N . Moritz, H., The Figure of the Earth, 279 pp., Wichmann, Karlsruhe, 1990. H 0 H (C.3) 0 0 Nettleton, L. L., Determination of density for reduction of gravimeter The reference gravity at an arbitrary orthometric height observations, Geophysics, 4, 176–183, 1939. Ref Nettleton, L. L., Geophysical Prospecting for Oil, 444 pp., McGraw-Hill, Hd , γ , within the near surface layer, in which the mass Hd New York, 1940. density is ρB , is defined by the next equation Nozaki, K., A computer program for spherical terrain correction, J. Geod. Hd Soc. Japan, 27, 23–32, 1981 (in Japanese with English abstract and γ Ref = γ + 2πGρ [H +(1,ψ)− H −(1,ψ)]dr figure captions). Hd 0 B H0 Nozaki, K., The latest microgravity survey and its applications, OYO Tech- nical Report, No. 19, 35–60, 1997 (in Japanese with English abstract Hd ∂γ + dr. (C.4) and figure captions). ∂ Nozaki, K., A note on the basic characteristics of spherical gravity correc- H0 r tions: gravitational behavior of a thin spherical cap of axial symmetry, J. On the right-hand side of Eq. (C.4), the second and the third Geod. Soc. Japan, 45, 215–226, 1999 (in Japanese with English abstract terms are the Prey reduction. The use of the Prey reduction and figure captions). is due to the level transformation of the gravity value within Torge, W., Gravimetry, 465 pp., Walter de Gruyter, Berlin, 1989. Torge, W., Geodesy, 3rd edition, 416 pp., Walter de Gruyter, Berlin, 2001. the mass. Thus, we call the reference gravity field the Prey- Tsuboi, C., Gravity anomalies and the corresponding subterranean mass reduced reference gravity field. distribution, Japan Acad. Proc., 14, 170–175, 1938. We remember that the normal gravity inside the reference Tsuboi, C. and T. Fuchida, Relation between gravity anomalies and the ellipsoid needs not to be known. Likewise, we remark that corresponding subterranean mass distribution (II), Bull. Earthq. Res. Inst., Univ. of Tokyo, 16, 273–284, 1938. the reference gravity outside the surface layer needs not to be known. The Prey-reduced reference gravity needs not to be defined outside the model earth. Because, the Prey- K. Nozaki (e-mail: [email protected]) reduced reference gravity field is defined only inside the earth.