**2. Theory**

The gravity method is governed by Newton's law of universal gravitation and Newton's law of motion [7]. Newton's law of universal gravitation states that "the force of attraction between two bodies of known mass is directly proportional to the product of their masses and inversely proportional to the square of the separation between their centers of mass." This is expressed as.

$$\mathbf{F} = \frac{\mathbf{G} \times \mathbf{M} \times \mathbf{m}}{\mathbf{R}^2} \tag{1}$$

where F = the force of attraction between the masses, G = constant known as universal gravitation, M and m = respectively the masses of particles 1 and 2, and R = distance between the two masses.

Newton's second law of motion states that "the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force."

Newton's second law can be expressed mathematically as follows:

$$\mathbf{F} = \mathbf{M}\mathbf{a} = \mathbf{M}\mathbf{g} \tag{2}$$

*The Principle of Interpretation of Gravity Data Using Second Vertical Derivative Method DOI: http://dx.doi.org/10.5772/intechopen.100443*

where F is the force of attraction between bodies, M is the mass of the body, and g is the acceleration due to gravity.

In differential form, Newton's second law can be stated as:

$$\mathbf{F} = \frac{\mathbf{dp}}{\mathbf{dt}},\tag{3}$$

Therefore,

$$\mathbf{F} = \frac{\mathbf{d}(\mathbf{m}\mathbf{v})}{\mathbf{d}\mathbf{t}}\tag{4}$$

where F = the applied force, dp = change in momentum, and dt = change in time [7].

The necessary characteristics of the gravity method can be explained in terms of mass and acceleration as illustrated in Newton's second law. The mass distribution and shape of an object are linked by the object's center of mass [8].

Equating (1) and (2), it implies that

$$\mathbf{g} = \frac{\mathbf{G} \times \mathbf{M}}{\mathbf{R}^2} \tag{5}$$

The gravitational potential at a point in a particular field is the work done by the attractive force of M on m as it moves from zero to infinity. The concept of the potential helps in simplifying and analyzing certain kinds of force fields such as gravity, magnetic and electric fields.

Eq. (7) represents the force per unit mass, or acceleration, at a distance r from P, and the work necessary to move the unit mass a distance (ds) having a component dr in the direction is given in Eq. (6).

$$\mathbf{v} = \mathbf{Gm} \int\_{\rightsquigarrow}^{\mathbb{R}} \frac{\mathbf{dr}}{\mathbf{r}^2} \tag{6}$$

$$\mathbf{v} = \frac{\mathbf{Gm}}{\mathbf{r}}\tag{7}$$

where ѵ = the work used in moving a unit mass from infinity to the point in question, m = unit mass at point P, and r = distance covered by the masses.

The gravity anomaly is the difference in values of the actual earth gravity (gravity observed in the field) with the value of the theoretical homogeneous gravity model in a particular reference datum [8].

$$
\delta \mathbf{g}\_{\mathsf{B}} = 2\pi \mathbf{Gph} \tag{8}
$$

$$\mathbf{g\_B} = \mathbf{g\_F} - \partial \mathbf{g\_B} \tag{9}$$

Previously, authors determined the density value at research locations based on the statement that the Bouguer anomaly can be expressed as an equation of the form of "y = mx + b" as.

$$\mathbf{g\_{obs}} - \mathbf{g\_N} + \mathbf{0.3086} \,\mathrm{h} = (0.04193 \,\mathrm{h} - \mathrm{TC})\mathbf{p} + \mathbf{BA} \tag{10}$$

The units for g are cm/s<sup>2</sup> in the c.g.s system and are commonly known as Gals, where the average acceleration of gravity at the earth's surface is 980 Gals. Most realistic gravity studies involved variations in the acceleration of gravity ranging from 10�<sup>1</sup> to 10–<sup>3</sup> Gals, so most workers use the term milliGal (mGal). In some

detailed work involving engineering and environmental applications, workers are dealing with microGal (μGal) variations.
