**THE FALLACY OF BOUGUER ANOMALY IN GEOPHYSICAL EXPLORATION AND**

**THE NEW CONCEPT OF THE THEORY OF GRAVITY ANOMALY**

M.KESAVAMANI, C.RAMACHANDRAN, M.V.R.KRISHNA RAO,

R.M.C.PRASAD & P.K.RAMMOHAN

GSI COMPLEX, BANDLAGUDA, HYDERABAD -500068 A. P

Email: manikesava@rediffmail.com

**kesavamanimangalampalli@gmail.com**

**ABSTRACT**

The Bouguer anomaly, in vogue, is really an enigma because it is believed to indicate both the gravity field/mass and the density variations. The theory of gravity states that gravity field is proportional to the mass distribution irrespective of the density of the sources. However, the vertical gradient of gravity field indicates the negative density contrast or the density of the causative source. Hence, if the Bouguer anomaly brings out the mass distribution, it cannot indicate the density variations. On the other hand, if it indicates density variations, it cannot reveal the mass distribution and the measured field should be proportional to vertical gradient of gravity. The primary object is to relate the difference of, obtained observed gravity (g

_{obs}) on the surface of the earth and the theoretical value of the force of gravity on the spheroid (g_{Th}), on to the geoid equi-potential surface for a comparison. In contrast to the conventional gravity anomaly with conceptual corrections, we define the normal gravity anomaly as vertical gravity anomaly (VG) which is the vertical difference

[VG= (g

_{obs}–g_{Th}.)], between the observed gravity on the surface of the earth and the theoretical gravity on the geoid. This is proportional to vertical gradient of gravity and indicates the apparent density variations. This is similar to the calculation of Vertical gradient of gravity in free air correction.The Bouguer anomaly defined by the equation:

(BA)= [g

_{obs}+ (Free-air Correction (FC) - Bouguer correction (BC) + Terrain correction (TC) –g_{Th}] can be written for simple Bouguer correction as [BA= VG + 0.3086* h - 0.04190* h*s ] or

[BA/h=VG/h+0.3086-0.04190*s]. This is the actual obtained anomaly after corrections at station level. This is the difference between the observed gravity and the theoretical gravity with variable ‶h ″. We show that the Free- air anomaly (FA) and Vertical gravity anomaly (VG) are the extremes of Bouguer anomalies (BA) for the assumed densities of Bouguer slab. When density of the Bouguer slab (σ) = 0; the BA = FA= [VG+ 0.3086*h] or [BA/h= FA/h= VG/h+0.3086]. Also, when σ= 7365kg/m

^{3}, the BA= [VG,] or [BA/h = VG/h]. These anomalies indicate different signatures for the same causative source at station levels. However, When ‶h ″ is constant i.e., on a horizontal surface, the VG, FA and BA show similar anomalies as that observed in plain areas. These anomalies are proportional to non-normalized vertical gradient of gravity with change in background levels separated by the correction factor. So, the gravity anomalies indicate only the density variations and not the gravity field.Integrating VG with ‶h ″gives the actual gravity field, called here, as Simple gravity anomaly (SG) at station level which indicates the mass variations. In this case where the VG is not normalized, it is numerically equal to negative of VG. Hence, Simple gravity [SG= - VG]. As the measured anomaly is the vertical differences, the station level VG and SG anomalies can easily be transferred on to the geoid surface without any corrections. By logic, the anomalies obtained on uneven surface should be a mirror reflection of the anomalies when transferred on to the even surface. That is, the observed data on the uneven ground surface equals to negative of the measured field on the even surface. The VG and SG at geoid level are respectively equal to negative of VG and SG on the ground surface. Therefore, station level VG (SLVG) is equal to geoid level SG (GLSG) which is the gravity field on the geoid. Similarly, station level SG (SLSG) is equal to geoid level VG (GLVG) which indicates the density variations on the geoid. Also, for a comparison on a horizontal datum, the free air correction factor (FC) may be used to project the station anomalies on a common datum in free air. If FC is used as a correction factor for the increase in height and added to the station anomalies, all the anomalies show similar signatures as that obtained in VG. However, if FC is used for upward continuation and the correction is subtracted, because of the decrease of natural vertical gradient with height, all the anomalies show similar signatures as obtained in SG, revealing the mass distribution.

The Bouguer anomalies indicate the negative density contrast areas as “highs” and positive contrast areas as “lows”. Therefore, the Bouguer anomaly with conceptual corrections, referred to as the gravity field, is a fallacy in geophysical exploration because it is proportional to Vertical gradient of gravity. So, the difference in the gravitational forces at two levels is proportional to vertical gradient of gravity and hence decreases with height. Consequently, the gravity field/ mass increases with height. Our differing fundamental perception of the concept of theory of gravity anomaly delineates the density and mass variations at both the station and geoid levels, and minimizes the ambiguity in geophysical exploration.

**Introduction**

Gravitational theory states that gravity field is directly proportional to mass distribution irrespective of the density of the sources. If it were true, the gravitational field should be positive and at the geoid level must be less as compared to the gravity field at higher elevations, if the masses are removed. Besides, the vertical gradient of gravity field indicates the negative density contrast or the density of the causative source. In contrast to the existing nomenclature such as gravity highs and lows related to the density highs and density lows respectively, the gravity highs and gravity lows should indicate the large and small masses respectively. The gravity anomalies, in vogue, are really an enigma because they are believed to indicate the density variations, besides representing the gravity field.

Classical geophysical exploration was restricted to relatively flat or gently undulating terrain, that is nearer to theoretical even datum, there by minimizing the ambiguity. With the extension of these surveys to hilly undulating areas, the efforts to minimize the effects of topography and to increase the signal to noise ratio of the causative source, various interpretational techniques have been developed. Although extensive theoretical work in gravity has been carried out over the years, the concept of the theory of gravity anomalies is not completely understood and is elusive with many ambiguities.

However, one of the major problems in gravity is that all the station anomalies are assumed to be on a common datum and interpreted, but very accurate leveling corrections are strictly applied. Normally, there should not be any variation in the signatures of the anomalies for different assumed densities of the Bouguer slab for the same source. Thus, fictitious anomalies are observed in undulating topographic areas for the assumed density of the Bouguer slab for the same source. Any correlation of gravity lows with elevated areas is considered as topographic effects and it appears the terrain correction is introduced at this stage.

Besides, there has been considerable confusion and debate on the subjects right from the geophysical definition of the gravity field (Hualin Zeng and Tianfeng Wan, 2004) to standardization of gravity reduction and corrections (LaFehr, 1991; William Hinze et.,al, 2005) as well as geodetic versus geophysical perspectives of the “gravity anomaly ” (Hackney and Featherstone, 2003). The free-air and Bouguer corrections incorporated in the Bouguer anomaly have been a subject of criticism over the years (Gulatee, 1941; LaFehr, 1991; Chapin, 1996; Kesavamani, 1999). Nine out of 15 English language major standard text books imply with Dobrin (1976) that our intent is data reduction (LaFehr, 1991). The residuals obtained by hand smoothing, ring analysis, trend surface development and fast Fourier transform methods etc., generally do not correspond with known structural configuration from seismic, wells and geological surface maps (Gummert, 1997). These ambiguities raise many basic questions like: (a) Is the concept of Bouguer gravity anomaly, originally conceived in the study of isostasy, relevant in geophysical exploration? (b) Does the Bouguer anomaly represent gravity field or vertical gradient of gravity field? If the Bouguer anomaly represents the gravity field, how are the relative density variations are brought out? (c) Why the Bouguer anomalies vary with height and show different signatures for the same source for assumed density?, (d) Also, the Bouguer anomalies suppress the desired signals and create a false gravitational field due to artificial addition or removal of mass that is proportional to height or depth at the point of observation, why? (e) Are the corrections applied to the observed gravity adequate or are they necessary at all? (f) Why in general the Bouguer anomalies are negative in continental areas and positive in oceanic areas? (Kesavamani, 2001; 2002 and Sarma, 2002) (g) Are the topographic effects removed effectively in Bouguer gravity data? (h) Why along the coast steep gradients with a positive bias towards sea are observed?

In this article, we discuss at length about the concepts of gravity anomalies and the applied corrections in geophysical exploration. We also make an attempt to address the above questions and to bring out some differing fundamental perceptions on the concept of the gravity anomalies (Kesavamani et.al, 2005). We propose the concept of Vertical Gravity (VG) and show that the measured field is proportional to vertical gradient of gravity and not the gravity field.

**A review of the theory of the Bouguer anomaly:**

The gravity anomalies are borne out of the idea to obtain the part of the observed values relating to density variations (Kesavamani et.al, 2005). The observed values are conveniently divided into anomalies that reflect the internal structure of the upper layers of the earth. This is achieved by subtracting the systematically varying part given by the standard formulae. The formulae are to calculate gravity force at any point on the ideal spheroid with slight flattening and a certain regular distribution of mass respectively. Strictly speaking, gravity anomaly should be the difference between the observed and theoretical gravity with reference to the same surface. Such an anomaly will depend on the normal gravity formula used and the difference between the actual and assumed mass distributions (Gulatee, 1941). In the geodetic sense, the anomaly is a single numerical value for any individual observation with reference to the spheroid surface. Relative gravity measurements on the surface of the earth and their anomalies determine variations from ideal theoretical gravity.

In practice, the observations made on the surface of the earth and the theoretical gravity on reference geoid, are subjected to corrections. The corrections enable the value of gravity observed on the surface of the earth to a certain standard surface for comparison. Anomalies are named according to the methods of correction applied for calculation. Those mostly in use are the Free-air gravity anomaly and the Bouguer gravity anomaly. The term Bouguer anomaly describes a field intended to be free of all non-geologic effects not modified by a partial geologic interpretation (LaFehr, 1991). Thus, evolved two concepts, for the theory of Bouguer anomaly associated with conceptual corrections depending upon the point of observation reckoned at the geoid level (Dobrin, 1976) or at station level (Ervin, 1977). What ever be the conceptual ideas about the theory of the Bouguer gravity anomaly, its mathematical equation consists of six factors namely:

1. Observed gravity on the surface of the earth (g

_{obs}.) at station level. This is the observed gravitational force at station level and dependent upon the elevation.2. The theoretical gravity (g

_{Th}.) on the reference spheroid; the theoretical gravitational force, is based on the formulae to calculate gravity force at any point on the ideal spheroid with slight flattening and a certain regular distribution of mass. But, the theoretical values are at the spheroid level.From the above, it is obvious that the observed and theoretical forces are at different levels. These two, on different surfaces are to be brought on to one surface by suitable corrections to relate the difference as gravity field. Thus, the conventional gravity anomaly is defined as the difference between the observed gravity (g

_{obs}.) and the theoretical gravity (g_{Th}.) obtained with reference to the same surface of geoid/spheroid.3. Free-air correction (FC), normally computed by application of the vertical gradient of gravity in free-air as [dg/dh = 0.3086 * h mGal / metre]. This correction is added if the observation station is above the datum (geoid) and subtracted if it is below. Thus, the free-air anomaly (FA) is defined by the equation [FA= g

_{obs}+ 0.3086*h - g_{Th}]. This correction is considered to correct the observed gravity data to the level of the geoid for a better visualization and comparison of the observed data at different heights.4. Bouguer correction (BC) attempts to calculate the gravity effect of the material between the point of observation and datum (geoid) by using an infinite slab (Bouguer slab) with an assumed average density (Bouguer density) having an attraction [dg=2pgsh], where g is the gravitational constant, s is the Bouguer density and h is the thickness of the Bouguer slab respectively. The correction is subtracted if the gravity station lies above the datum and added if it is below. This is called simple Bouguer correction. The assumption of an average single density, approximated to all formations, necessitated by the lack of precise information, causes an over correction in areas of low density and under correction in areas of higher density. An anomalous gravity field is thus created due to artificial addition or removal of masses which is proportional to the height/depth at the point of observation. This is larger, the broader the regions averaged and it is more if the density of the various areas differs from the mean. Thus, Bouguer gravity field, in high terrain regions closely follows topography when the assumed Bouguer density is less. Those with higher value of assumed density the Bouguer gravity field inverts the topography, where as for the correct value it lies in between. The assumption of an infinite Bouguer slab results in an over correction in the data as the topographic features are rarely one dimensional. However, these are partly achieved by computing terrain correction as complete Bouguer correction.

5. Terrain correction (TC), normally applied in hilly areas, is always positive. When TC is applied, the Bouguer correction is termed as complete Bouguer correction. The complete Bouguer reduction includes in addition to the Bouguer slab correction-Bullard A, both curvature-Bullard B and terrain Bullard C corrections (LaFehr, 1991). The application of terrain correction makes it possible to take into account the effect of the attraction of all forms of external relief and to reduce the gravity value at a given point to that would be obtained if there were uniform flat layer, without rises or depressions below the point concerned.

**Concepts of the theory of Bouguer anomaly:**

The Bouguer gravity anomaly (BA), thus, is defined by the following format of equations with two different concepts:

a) [BA = (g

_{obs}+ FC - BC) –g_{Th}.] : In this concept, the FC and BC are applied to the observed gravity data and the corrected data is considered as reduced to the datum (Dobrin, 1976 and Telford, 1976) suggesting that the reduced gravity values are those that would be observed if the measurements could be made on the datum plane. The free air correction factor is considered to reduce the observed value on the surface to the geoid level. This is the original concept that was used in the study of Isostasy from the regional studies but not for the exploration type of studies concerned with relatively shallow masses. In comparing the gravity measurements at elevated stations with the one at sea level, allowance was made for the extra height for the elevated station and the mass between the station and sea level with the aid of FC and BC. Then, the gravity values at two stations are attributed to anomalous masses below the sea level.(b) BA = [g

_{obs}– (g_{Th}- FC+ BC)]: In this concept, the FC and BC are applied to the theoretical gravity and the corrected data is considered as projected to the surface of observation (Ervin, 1977). The free air correction factor is considered to project the theoretical value at geoid level to the surface level. The significance of the datum is only that all the masses below the datum contribute to the Bouguer anomaly field, while only deviations for the idealized mass distributions are included from above the datum (Ervin, 1977). The Bouguer anomaly values do not lie in a common plane but are located at varying elevations of their respective points of measurements called ‶station anomalies″.It is obvious from the above that the obtained gravity field value remains the same, in both the concepts which indicates that neither the observed data is reduced to the datum or the theoretical gravity projected to the surface of observation.

**The new philosophy of gravity anomalies:**

To accomplish the variations in density, the mass should be divided by volume. Let us see how these variations of densities are brought out from the gravity anomalies.

In contrast to the conventional gravity anomaly, we define the gravity anomaly, called here, normal gravity or Vertical Gravity (VG) anomaly as the vertical difference between observed gravity on the surface of the earth and the theoretical gravity at the geoid level, Thus, [VG/h= (g

_{obs}- g_{Th}.) / h].As g

_{ obs}and g_{Th}are the gravitational forces at different levels, the difference between them should be proportional to difference in the masses. Normally, as mass is positive, to get the difference in masses between the two surfaces, the smaller mass should be subtracted from the larger mass. Conventionally, g_{obs}is taken as the larger value as compared to g_{Th}in view of the extra topographic mass above the geoid/ spheroid level, so that the difference between the two should account for the extra/ deficit mass with reference to the assumed uniform spherical mass, as obtained from the International gravity formula (theoretical gravity), for an ideal spheroid. But, by virtue of the decrease of gravitational force with height, the observed value being at a height, the value is negative. Hence, mass (m) divided by volume (height ‶ h ″), equal to dg/dh, should give density and thus VG indicates the density below the station. If the difference is normalized with height, it gives the vertical gradient of gravity field, which eliminates the regional gradients and results in density at a given station. In other words, this is equal to differentiating gravity field with respect to height which gives the vertical gradient of gravity. Thus, the vertical gradient of gravity field indicates the negative density contrast or the density of the causative source.As the value is not divided by h, here it is proportional to vertical gradient of gravity and called vertical gravity anomaly in milligal. [VG= (g

_{obs}- g_{Th.})]. Actually, the observation point should be at the mid point of the observed and theoretical gravity levels, but at the moment it is considered as station anomaly like BA and FA. It is a single numerical value. This may also be called as density anomaly at station elevation. In the relative gravity measurements, if this value is normalized with a constant height, it would be vertical gradient of gravity and indicates the relative variations in density of the hard shell of the crust. Actually the gravity signal is recorded from the density interface. Thus, the Bouguer anomaly indicates a positive density contrast areas as a “low “and the negative contrast areas as a “high”. So, the vertical gradient of gravity indicates the negative density contrast or the density of the causative source. Obviously any point above the geoid level should produce an extra mass, whereas below the geoid it should give rise to a deficit mass. Hence negative gravity anomalies are observed above geoid level and positive anomalies below geoid level.Hence, Vertical Gravity anomaly (VG/h) = [(g

_{obs}- g_{Th})/h] or VG= [(g_{obs}- g_{Th)}]**Role of free-air correction and Theoretical gravity in Bouguer anomaly:**

One major problem is that all the station anomalies are assumed to be on a common datum and interpreted. In our opinion, the crux of the problem lies in understanding the function of the FC in BA. Unfortunately, this is projected in an isolated way that it would reduce or project to the different elevations. Whereas in free-air, the FC can be used to project or reduce the data to any datum, the same cannot be true in a mass medium. We believe this is an inseparable combination called the combined elevation factor (FC-BC). The FC, as a known value in free-air (with near zero density medium) with height, only helps in comparing and computing the vertical gradient of gravity for the effective topographic mass with an assumed density. The role is similar to that of the theoretical gravity in Bouguer anomaly. In normal circumstances, a simple ratio of the vertical gradient of gravity in free air to the density of the medium should give the vertical gradient of gravity for the assumed density of the topography, which indicates an inverse relation as higher values for lesser density. But it gives wrong values, as VG calculated for 1000 kg/m

^{3}equals to that of free-air or zero density and for a density of 500Kg/m^{3}the value equals twice that of free-air correction factor. Hence, the procedure of computation of the effective mass distribution as a BC and subtracting from FC has to be adopted. Therefore, FC cannot perform the dual role of projecting or reducing the data to any surface in Bouguer anomaly. It only helps for a comparison in free air for computing the vertical gradient of gravity in the assumed density of topographic mass. So, the other and only alternative, left in eliciting the information from the Bouguer anomaly is to rewrite the equation in the following format:Bouguer gravity anomaly [BA] = [(g

_{obs}- g_{Th}) + (FC- BC) + TC]i.e. BA= [(g

_{obs}– g_{Th}) + (0.3086 * h – 0.04190 h *s )]This can be written as [(BA/h) = (g

_{obs}– g_{Th})/h + (0.3086 – 0.04190s)].Thus, in practice, what we actually measure is [BA/h = (g

_{obs}– g_{Th})/h + a constant factor] depending on the assumed density for the Bouguer slab at station levels.Hence, Bouguer anomaly [BA = (g

_{obs}– g_{Th }) + (0.3086 * h – 0.04190 h *s)].When s=0, BA= FA, Free air anomaly (FA): [FA/h= (g

_{obs }– g_{Th}) /h+0.3086] or [FA = (g

_{obs}- g_{Th)}+ (0.3086* h)] and When s = 7365kg/m^{3}BA/h = gravity anomaly (VG) and indicate fictitious anomalies.So, Vertical Gravity anomaly VG/h= [(g

_{obs}- g_{Th})/h]. Or VG= [g_{obs}- g_{Th}].If this normalized vertical gradient of gravity is integrated with height h, the resultant would tantamount to gravity field. i.e. (density * volume) or (VG * h). In this case, as h is implicit in VG the inverse of VG or a mere negative sign of the VG should be the gravity field, called here Simple gravity anomaly (SG). So, Simple gravity SG= [-VG= (g

_{th}- g_{obs})] indicates the actual mass distribution and the gravity field. In the geodetic sense, the anomaly is a single numerical value for any individual observation. Relative gravity measurements on the surface of the earth and their anomalies determine variations from ideal theoretical gravity. Relative gravity measurements at each station reveal the density variations at station level. However, these should be brought to one level for comparison. Thus, the difference in the gravitational forces at two different levels is proportional to vertical gradient of gravity which reveals the density variations. This is inversely proportional to the mass. So, the theoretical gravitational force calculated at mean sea level assuming an ideal spheroid model helps in removing the variation of normal gravity with latitude from equator to poles. In addition, by virtue of its position at geoid level, it helps calculating the density below a station from the difference in height between observed and theoretical gravity values. This should be the actual concept of the gravity anomalies.The theoretical gravitational force is calculated at the geoid level, in a conventional way as the force vector is always opposite. But, the computation of gravity fields of the objects is made from the plane of observation with force vector downwards towards the centre of the earth. This perception from the plane of observation results in confusion. So, the difference in the gravitational forces at two levels is proportional to vertical gradient of gravity and hence decreases with height. Consequently, the gravity field/ mass increases with height.

**Projection of uneven datum observations on to an even datum:**

The primary object is to relate the difference of obtained observed gravity (g

_{obs}) on the surface of the earth and the theoretical value of the force of gravity at the surface of the spheroid (g_{Th}) on to the equipotential geoid surface for a comparison. As the measured field represents the non normalized vertical gradient of gravity field, the free air correction factor (FC) can be used to bring the station anomalies (FA, BA and VG) on to a common datum in free air for comparison. If the FC is used as a correction factor for the increase in height and added to the station anomalies, all the anomalies show similar signatures as that obtained in plains, showing the inverse relationship with height. These are termed as FAV, BAV and VGV as they indicate density variations. On the other hand, if the FC is used for upward continuation and the correction is subtracted, because of the decrease of natural vertical gradient with height, all the station anomalies (FA, BA and VG) show similar signatures as obtained in SG. These are termed as FAG, BAG, and VGG as they reveal the mass distribution. Thus, when ‶h ″is constant, on a horizontal datum, i.e. in plain areas, we have Bouguer anomaly, free air anomaly and gravity anomaly similar in character that are separated by constant factors of addition or subtraction with only a change of level in background values. Thus, the obtained anomalies in the form of lows and highs actually indicate the corresponding conventional gravity highs and lows associated with structural highs and lows of the hard shell of the crust. As the measured anomaly at the station level is the vertical differences, by logic, the data on even datum should be a mirror reflection of the uneven surface anomalies. In the case of VG and SG, the data from uneven ground surface can easily be transferred to the geoid level with a change in sign. That is, the observed data on the uneven ground surface equals to negative of the measured field on the even surface. The VG and SG at geoid level are respectively equal to negative of VG and SG on the ground surface. Therefore, station level VG (SLVG) is equal to geoid level SG (GLSG) which is the gravity field on the geoid. Similarly, station level SG (SLSG) is equal to geoid level VG (GLVG) which indicates the density variations on the geoid. The VG and SG anomalies may also be brought to a common datum in free air, if required. SGV (-VGV) is the mirror reflection of VGV which indicates mass variations and resembles SLSG and VGG but differs from SGG in signatures. All these exercises are only to show that they resemble one another on a horizontal datum. But, in practice it may be appropriate only to consider the station level VG and SG anomalies and transfer them on to the geoid.However, in the case of FA and BA, the data on the uneven ground datum has to be brought on to even datum before transferring on to the geoid in view of their changes with height. The FA and BA anomalies brought on to an even datum in free air shall resemble the VG and SG anomalies without any change in signatures. Thus, for topography above the mean sea level, the difference between the SLVG and GLVG is equal to twice of VG. Similarly the difference between the SLSG and GLSG is twice that of SG. These differences appear to indicate the gravity effect of a slab between the station and the geoid levels.

**Analysis of the VG and SG anomalies in relation to free air and Bouguer gravity anomalies:**

Apparent fictitious FA and BA anomalies are observed in undulating topographic areas, when plotted for a horizontal datum. All the VG, FA and BA can be interpreted at the station levels for the same source. But, VG anomalies consistently show an inverse relationship with topography because they are proportional to the vertical gradient of gravity. The VG anomaly depends upon the height and the geological condition below the point of observation. Correlation of gravity lows with elevated areas is considered as topographic effects and it appears the terrain correction in Bouguer anomaly is introduced at this stage.

[VG = 0], or [SG=0] when [g

_{obs}= g_{Th}.]. VG is positive when g_{obs}> g_{Th }and VG is negative when g

_{obs}< g_{Th}. Similarly, SG is positive when g_{obs}< g_{Th }and SG is negative when g_{obs}> g_{Th}.[FA= VG +0.3086 h] for s = 0 kg/m

^{3}. FA =0, When VG = (- 0.3086 h). Thus, FA can become zero at different elevations. So, [FA = VG + FC = VG + 0.3086 * h] and [BA= VG+ (0.3086*h-0.04190*hs)]. [FA = VG] when h=0, FA is positive when SG is greater than (– 0.3086 h) and FA is negative when FA is lesser than (– 0.3086 h). The relative free-air anomalies follow the topography. Hence, FA is a sum proportional to the vertical gradient of gravity field (VG) and the vertical gradient of gravity field in free-air (zero density) as if the topographic medium above the datum is filled with air.Bouguer anomaly [BA = (g

_{obs}– g_{Th }) + (0.3086 * h – 0.04190 h *s)].[BA = VG + 0.3086 h - 0.04190 hs] or [BA = VG + (0.1967 h)] for s =2670 kg/ m

^{3}. When s = 0, [BA = VG +0.3086 h] = FA and when s = 7635kg/m^{3}, BA = VG. Thus, the equation of BA exhibits a density range varying from 0 to 7365 kg/m^{3}indicating the FA and VG respectively as two extremes. When h = 0, BA = VG and when VG=0, [BA = 0.3086h – 0.0419h *s.]BA = 0, when [VG = - (0.3086 h – 0.04190 hs)] i.e., when FC= BC, BA = 0. BA is positive when VG is greater than – (0.3086 h- 0.04190 h *s) and BA is negative when VG is lesser than – (0.3086 h- 0.04190 h *s). The relative Bouguer anomalies do not show any variation with height/depth when s equals 3682.5 kg/m

^{3}. When s < 3682.5 kg/m^{3,}BA tends to follow topography as in the case of FA. However, BA inverts topography as in the case of VG when s >3682.5 kg/m^{3}. The transition of Bouguer anomalies from positive to negative is around 3682.5 kg/m^{3}, as in the case of borehole measurements, with height/depth. As σ and h are inseparable the combination of h and σ indicates the anomaly variations with height or depth.So, BA is the sum of (i) difference of observed gravity and theoretical gravity values at different levels vertical gradient of gravity (VG), (ii) vertical gradient of gravity for the assumed density of the topography indicating the density of the source or the negative density contrast. In fact, these ‶apparent density″ variations are due to the structural highs or the excess mass represented as lows and structural lows or the deficit mass represented as highs. The BA represents the difference of gravity field at two different levels as non normalized vertical gradient of gravity. Thus lateral variations of non normalized vertical gradients are measured in practice. Hence, Bouguer anomalies do not lie on a common datum but are located at varying elevations of their respective points of measurements. So, BA is the sum of natural vertical gradient of gravity (VG) and vertical gradient of gravity for the assumed density of the topography indicating the density of the source or the negative density contrast. In fact, these ‘apparent density’ variations are due to the structural highs or the excess mass represented as lows and structural lows or the deficit mass represented as highs. It is because of the observed measurements indicate the vertical gradient of gravity field, the lateral density variations are indicated and the regional gradients are minimized. Modeling of Bouguer anomalies indicate that the positive contrast areas are represented as “lows” and negative contrast areas reflected as “highs”. Hence, Bouguer anomaly is proportional to Vertical gradient of gravity and not the gravity field.

It is due to this reason the continental areas and oceanic areas are characterized by negative and positive anomalies respectively. The peaks of mountains corresponding to minimum values and deepest ocean parts corresponding to the maximum values also suggest that topographic effects are not removed from the data, in spite of the corrections. On the contrary, the Bouguer gravity highs and lows are misrepresented and interpreted as structural highs or excess mass and structural lows or deficit mass respectively.

VG, FA and BA would not be equal to zero unless there is an equal and opposite distribution of mass below the spheroid/geoid as compared to the topographic mass. VG, FA and BA would be positive when there is an excess mass below the spheroid or geoid as compared to the topographic mass and indicates a shallow Moho. VG, FA and BA would be negative when there is a deficit mass below the spheroid/geoid as compared to the topographic mass as thickened crust, indicating deeper Moho and thus explains anomalies in the mountainous areas.

**PRACTICAL EXAMPLES:**

We consider two practical examples namely the detailed surveys in Mangampeta area and the other semi regional surveys across part of Cuddapah basin to demonstrate that the Bouguer anomaly is proportional to vertical gradient of gravity and not the gravity field. Station gravity anomalies (FA, BA, VG and SG) along with elevation (h) are presented. As these are station anomalies, these anomalies are projected to a horizontal datum (H) in free-air by applying FC as a correction factor. Adding 0.3086*(H-h) to the station anomaly, for the difference in elevations at station to the horizontal datum the FAV, BAV and VGV(Fig.1A,1B and2) anomalies are obtained indicating the density variations. Similarly, by subtracting 0.3086 *(H-h) from the station anomaly for the difference in elevations at station to the horizontal datum, the FAG, BAG and VGG (Fig.1A,1B and2) anomalies are obtained indicating the mass variations.

**Gravity anomalies across Mangampeta bedded barite deposit**:

The Mangampeta Barite deposit (4290kg/m

^{3}) occurs in a synformal trough as a bed sandwiched between volcanic tuffs(2400kg/m^{3}) underlain by dolomite(2880kg/m^{3}) in Pullampeta formations of Cuddapah basin, Andhra Pradesh. The thickest part of the barite body occupies the trough. The formations are characterized by a low dip of 0 to 20^{0 }ENE, with a general trend of NNW-SSE. The area is generally flat in the surveyed area, with a general rise in elevation towards east by about 20m culminating as a mound. Two gravity profiles (Bose, R.N and Vaidyanathan, N.C.1979), taken for an arbitrary gravity base, are selected for the present discussion. The Bouguer anomaly is calculated for a density of 1800 kg/m^{3}based on the Nettleton’s density profile.Elevation and calculated different gravity anomalies along boreholes MGP 21,23,26,29,31 and 42 trending N79E-S79W, across Mangampeta Barite bedded deposit are presented in fig.1A. Normally a gravity high zone is associated with either a basin structure filled with high density sediments or a high density ridge or dome. In contrast to this, the Bouguer anomaly indicates (fig 1A) a high density zone over the synform filled with low density tuffs, as an inverted picture of the high density rock. The FA anomaly also shows a similar picture correlating the topography. The VG also shows a density high zone but inverting the topography. However, the corresponding SG anomaly shows gravity low zone due to low density tuff, indicating a basin like feature, correlating the combined effect of high density barite bed underlain by the dolomites as basement/bed rock. Whereas the SG shows a positive correlation of the high density bed indicating the actual mass distribution, the FA, BA and VG show an inverted response of the high density zone, indicating the density variation. Thus, the Vertical gradient of the gravity field indicates the negative density contrast or the density of the body. The FAV, BAV, VGV anomalies, obtained on a horizontal datum in free air at 200m elevation show similar character as that of VG with a change in background level proportional to the correction factor for the assumed density of the Bouguer slab. Similarly the FAG, BAG, VGG also show a similar character as that of SG with a change in background level proportional to the correction factor for the assumed density of the Bouguer slab.

Elevation and calculated different gravity anomalies along boreholes MGP 9,1,20,28, and 46 trending W-E, across Mangampeta Barite bedded deposit are presented in fig.1B. In contrast to fig.1A, the barite bed almost exposed in the western part is fairly horizontal over a distance of about 400m and follows the dolomite bed as a synform structure. The VG, FA and BA are characterized by a steep rise in the gravity values corresponding to the western edge of the barites bed indicating the contact between shale’s in the west and barites in the east. In the eastern part the VG, FA and BA show a similar picture as in fig.1A. This is in contrast of the profile in SG, which indicates an inverse picture revealing the basin structure that is normally expected if the measured field is gravity field indicating the mass distribution. Small amplitude anomalies over the background low appear to be variation in thickness of barite bed. The FAV, BAV, VGV anomalies, obtained on a horizontal datum in free air at 200m elevation show similar character as that of VG with a change in background level proportional to the correction factor for the assumed density of the Bouguer slab. Similarly the FAG, BAG, VGG also show a similar character as that of SG with a change in background level proportional to the correction factor for the assumed density of the Bouguer slab. Several models constructed over the geological cross sections indicate a density high zone similar to the profiles obtained in VG, FA and BA of figs 1A and 1B. Actually, the gravity signal is recorded from the density interface. Thus, the Bouguer anomalies indicate the negative density contrast areas as “highs” and positive contrast areas as “lows. Hence, the vertical gradient of gravity anomalies indicates the density of the mass or the negative density contrast.

**Gravity anomalies across Cuddapah basin and Nellore schist belt**:

The elevation and gravity anomalies VG,FA and BA (2670k g/m

^{3}) (Bhaskara Rao et.al. 1979) across, part of Cuddapah Basin from West to East comprising meta-sediments like dolomites, phyllites and quartzites as well as the eastern adjoining Nellore schist belt and gneisses of Dharwar Supergroup, are presented in Figure 2. The elevation profile shows a gradual rise from east to west with superposed isolated peaks and depressions. Excepting the short wavelength anomalies, the long wavelength anomalies reveal the same character in spite the variation of elevation from almost mean sea level to about 700m.The variations of VG, FA and BA with topography and geology are discernible (fig2). The BA appears to be fairly in agreement with the density chosen as there is no variation with topography corresponding to the standing quartzites around station 15km. Whereas the FA and SG indicate a positive correlation with short wave length topographic anomalies, the VG shows a negative correlation. The station level VG (SLVG) is equal to geoid level SG (GLSG). Similarly, station level SG (SLSG) is equal to geoid level VG (GLVG) which is the gravity field on the geoid.

The FAV700, BAV700, VGV700 anomalies, obtained on a horizontal datum in free air at 700m elevation show similar character as that of VG with a change in background level proportional to the correction factor for the assumed density of the Bouguer slab. Similarly the SGV,FAG700, BAG700, VGG700 also show a similar character as that of SG with a change in background level proportional to the correction factor for the assumed density of the Bouguer slab.

The VGG700 and VGG 2000 show similar anomalies with a decrease in background level with height. On the other hand, the SGG700 and SGG 2000 also show similar anomalies but indicate an increase in background levels with height. All the FA, BA, VG, show an increase towards the extreme east, whereas the SG indicates a fall towards east. From the above, it is clear that a gravity low zone actually reveals the low density zone whereas a gravity high zone indicates a low density zone.

Therefore, the VG and SG anomalies which respectively indicate the density and the mass variation require no conceptual correction and show no distortions even in highly elevated regions are the actual vertical gradient of gravity and the gravity fields with respect to the theoretical gravity force on the geoid.

**CONCLUSIONS:**

.The Bouguer anomaly actually indicates the negative density contrast areas as “highs” and the positive density contrast areas as “lows”. The difference in the gravitational forces at two levels is proportional to vertical gradient of gravity and hence decreases with height. Consequently, the gravity field/ mass increases with height. Therefore, the vertical gradient of gravity reveals the density variations. This is in turn inversely proportional to the mass. Unless the VG, FA and BA are proportional to the vertical gradient of gravity field, the lateral density variations can not be obtained. The Free air correction in Bouguer anomaly neither reduces the observed data to geoid level nor projects the theoretical gravity to the surface level. The theoretical gravity at the geoid level not only removes the normal variation of gravity with latitude from equator to poles but also helps in indicating the density variations. The station level anomalies can be transferred on to the geoid surface without any corrections by changing the sign. Thus, station level VG (SLVG) is equal to geoid level SG (GLSG). Similarly, station level SG (SLSG) is equal to geoid level VG (GLVG). No elevation related corrections are required for the gravity anomalies. Actually a gravity high zone indicates a low density zone. Similarly a gravity low zone reveals the high density zone. The combination of VG and SG anomalies may help minimize the ambiguities in gravity methods. These preliminary observations may have far reaching implications in understanding the gravity field.

By analogy, the magnetic anomalies also indicate the vertical gradient of magnetic field as the observation and theoretical fields are at different levels.

**Acknowledgement:**The senior author is greatly indebted to Dr. S. K. Mazumder, Dr. R. N. Mishra, Dr. M. Ramakrishnan, Dr. Das Sarma, Dr. M. S. Rao, and Shri M. R. Madhava Rao for their kind help and encouragement at various stages. The authors are grateful to Dr. S. K. Bhushan, Dy. Director General, Southern Region, Geological Survey of India, Hyderabad for his keen interest, encouragement and permission to present the paper at the 31st annual convention and third International seminar and exhibition on exploration geophysics. They are thankful to the anonymous reviewers for their valuable suggestions.

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Bose, R.N and Vaidyanathan, N.C., 1979. Gravity surveys for barites around Mangampeta, Cuddapah Dt., Andhra Pradesh. Jour. Geol.Soc.India, Vol.20, Nov. pp.540-547.

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**CAPTION TO FIGURES**

**Figure 1A**. Elevation and VG, FA, SG and BA profiles across Mangampeta Baryte deposit along Bore holes MGP21,23,26,29,31 and 42.

VGV200, FAV200and BAV200 indicate density variations at a datum height of 200m in free air-Free air correction is added for the difference of elevations between station level and datum. VGG200, FAG200 and BAG200- indicate mass variation in free air at a datum height of 200m-Free air correction is subtracted for the difference of elevations between datum and station levels.

**Figure 1B.**Elevation and VG, FA, SG and BA profiles across Mangampeta Baryte deposit along Bore holes MGP 9, 1, 20, 28 and 46.

VGV200, FAV200and BAV200 indicate density variations at a datum height of 200m in free air- Free air correction is added for the difference of elevations between station level and datum. VGG200, FAG200 and BAG200 indicate mass variation in free air at a datum height of 200m- Free air correction is subtracted for the difference of elevations between datum and station levels.

**Figure 2.**Elevation and gravity anomalies across part of Cuddapah basin and Nellore schist belt.

SLVG, FA, BA, SLSG, are station level anomalies on the surface. GLSG,GLVG anomalies on the geoid .VGV700,FAV700, BAV700 and SGG700 indicate density variations at a datum height of 700m in free air- Free air correction is added for the difference of elevations between station level and datum. VGG700, FAG700, BAG700 indicate mass variation in free air at a datum height of 700m- Free air correction is subtracted for the difference of elevations between datum and station levels. VGV700 (=-SGV700) are at a datum height of 700m. Similarly VGG2000 and SGG2000 (-VGG2000) at a datum height of 2000m in free air.