## Abstract

This chapter is aimed at demonstrating how the second vertical derivative method is been applied to gravity data for subsurface delineation. Satellite gravity data of some part of Northern Nigeria that lie between latitude 11°–13°E and longitude 8°–14°N obtained from Bureau Gravimetrique International (BGI) were used for demonstration. The Bouguer graph was plotted using surfer software. The second vertical derivative graph was also plotted. Very low gravity anomalies are observed on the Bouguer map, which recommends the presence of sedimentary rocks which have low density. The result of the second vertical derivative method has improved weaker local anomalies, defined the edges of geologically anomalous density distributions, and identified geologic units. This is a clear implication that the second vertical derivative is very important in subsurface delineation.

### Keywords

- gravity data
- second vertical derivative
- interpretation

## 1. Introduction

Gravity method is one of the geophysical methods widely used in environmental, engineering, archeological, and other subsurface investigations. The gravity method measures the difference in earth’s gravitational field at different locations by tools known as gravimeters. Recent developments in observation, processing, and data analysis due to technological improvement added the efficiency and sensitivity of the gravity method, which made it more applicable to a wider range of problems. Airborne gravimeters have the potential to recover a precise gravity field at any place. Recent advancements in detailed aircraft positioning with global positioning system (GPS) carrier phase data have stretched the use of airborne measurement practice to land as well as overwater surveys. To date, large areas of the earth remain unmapped because of the limitations of land and marine surveys. BGI contributes to the recognition of derived gravity products with the aim of providing significant information about the Earth’s gravity field at worldwide or provincial scales. Their products used mostly by scientists are the World Gravity Maps and Grids (WGM), which signify the first gravity anomalies computed in spherical geometry considering a realistic Earth model. With long-range aircraft, nearly all the Earth is accessible to airborne surveying. Improvements in hardware, software, and survey methodology continue to lower the overall error budget for airborne gravity. Even though there is no method universally agreed for evaluating data accuracy and anomaly resolution being a function of the speed of aircraft, reported RMS error and resolution indicates the technique’s accuracy [1]. The gravitational field strength is directly proportional to the density of subsurface materials. This gives gravity anomalies that correlate with source body density variations. Positive gravity anomalies are connected to shallow high-density bodies and negative anomalies are connected to low-density bodies. Potential field anomaly maps present the effects of shallow (residual) and deeper (regional) geological sources. Therefore, the main issue in potential field data interpretation is the separation of anomalies into two components [2]. Consequently, in order to produce meaningful results, potential field datasets generally need many processing techniques that are in accordance with the nature of the geology of the study area. Numerous commercial software packages (e.g. Geosoft Oasis Montaj, GeosystemWinGlink, MagPick, and IGMAS) are commonly used for the analysis of potential field datasets by employing some of the methods stated above.

Quite a lot of graphical and empirical methods have been established for the interpretation of gravity anomalies that are caused by simple bodies [3]. The derivatives have a tendency to expand near-surface structures by increasing the power of the linear dimension in the denominator. This is because the gravity effect differs inversely as the square of the distance, and the first and second derivatives vary as the inverse of the third and the fourth power, respectively, for three-dimensional structures.

The second vertical derivative is frequently employed in gravity interpretation for isolating anomalies and for upward and downward continuation. This chapter aims to emphasize the advantage of using the second vertical derivative on gravity data for subsurface delineation. The second derivative is very important for gravity interpretation due to the fact that the double differentiation with respect to depth tends to emphasize the smaller, shallower geologic anomalies at the expense of larger, regional features [4]. Micro-gravimetric and gravity gradient surveying methods can be applied for the detection and delineation of shallow subsurface cavities and tunnels [5]. The second vertical derivative method is very important in edge location and edges are thought to contain most of the two mineralized ore deposits [6].

## 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.

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:

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:

Therefore,

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

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).

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].

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.

The units for g are cm/s^{2} 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^{−1} to 10^{–3} 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.

## 3. Second vertical derivative

Many researchers [4, 8, 9] have given a thorough picture of the second derivative method of interpretation of gravity and have shown how this method is in effect for the detection of small irregularities in gravity anomalies and thus useful for supposing minute underground mass distribution that cannot be overlooked by the ordinary method. It is fascinating to note that the method is justified only on data of high accuracy [10].

The SVD of the vertical component of gravity, g_{z}, can be calculated in the spatial domain from the horizontal gradients by using Laplace’s equation [9].

For an anomaly extended along the y-axis, the SVD can be approximated by the second horizontal derivative of the gravity data along the x-axis in Eq. (13).

In the wave number domain or Fourier domain, the SVD is usually calculated by using the following Eq. (14) [11].

The second vertical derivatives are the measure of curvature where large curvature is connected to shallow anomalies. It is frequently used to enhance localized subsurface features, that is, weak anomalies due to the sources that are shallow and limited in-depth and lateral extent [10].

## 4. Materials and methods

Qualitative interpretation of the gravity data was performed by applying second vertical derivative methods as filters to find edges of the source of gravity anomaly around the study area.

The materials used for the research include the following:

Work station (computer)

Surfer software

Satellite gravity data

## 5. Source of data

The data used were obtained from Bureau Gravimetrique International (BGI). The survey was carried out in conjunction with IAG international Gravity field service. The main job of BGI is to gather all gravity measurements (relative or absolute) and pertinent information about the gravity field of the Earth on a worldwide basis, compile and authenticate them, and store them in a computerized database in order to redistribute them on request to multiple users for scientific applications. BGI produces the most precise information available on the Earth’s gravity field at short wavelengths today and is very complementary to airborne and satellite gravity measurements. The satellite gravity data were recorded in digital layout (X, Y, and Z). X, Y, and Z represent the longitude, latitude, and Bouguer anomaly of the study area, respectively. Corrections such as drift, earth-tide, elevation and terrain, latitude and were applied on the gravity data by the Bureau Gravimetrique International (BGI).

## 6. The study area

The area is underlain by rock and younger sediments of the Chad formation. The Chad Basin lies within a vast area of Central and West Africa at an elevation between 200 and 500 m above sea level and covers 230,000 km^{2}. The basin is referred to as an interior sag basin, due to a sagging episode that has affected it before the onset of continental separation during which a rift system junction was formed providing an appropriate site for sedimentation. Therefore, it lies at the junction of basins, (comprising the West African rift), which becomes active in the early Cretaceous when Gondwana started to split up into component plates. Sediments are mainly continental, sparsely fossiliferous, poorly sorted, and medium-to-coarse grained, feldspathic sandstones called the Bima Sandstone. Both geophysical and geological interpretations of data suggest a complex series of Cretaceous grabens extending from the Benue Trough to the southwest (Figure 1) [12].

## 7. Methodology

The target of the gravity method is the determination of important facts about the earth’s subsurface. One can just study the grid of gravity values for the determination of the lateral location of any gravity variations or perform a more thorough analysis to calculate the nature (depth, geometry, density) of the subsurface structure that caused the gravity variations. To determine the latter, it is usually necessary to distinguish the anomaly of interest (residual) from the remaining background anomaly (regional).

Gravity data were analyzed and interpreted using the second vertical derivative method. Performance of horizontal and vertical derivatives was evaluated using synthetic data. SVD calculation using 2D was applied to identify fault structure. The first derivatives of the horizontal component (dg/dx and dg/dy) were calculated in the excel software. Then, the second derivatives (

These are used to find the second vertical derivative (

The second vertical derivative map is expected to remove the effect of regional trends. The edge of the residual anomaly was then observed, which is seen on zero contours. This will help to predict the anomaly in the map with its position.

## 8. Results and discussion

The application of the second vertical derivative method in this research has yielded results. Figures 2 and 3 show the results of the Bouguer anomaly distribution, application of second vertical derivative to the Bouguer values. The data obtained from Bureau Gravimetric International (BGI) were used throughout the work. The data had undergone Bouguer correction already. While the Bouguer result shows the density variation across the study area, the second vertical derivative graph shows the major fault zone. These results are presented as follows:

### 8.1 Bouguer graph

The data were converted from excel to data file format using grapher software. Its grid and contour maps were obtained using surfer software. The Bouguer map in Figure 2 indicates that the Bouguer anomaly of the study area varies from −60 to −4 mGal. Low gravity anomalies are observed in the map with its minimum value appearing in around 11.69°N–11.85°N, 11.2°E–11.33°E, and 11.0°N–11.2°N, 11.5°E–11.7°E. Meanwhile, Bouguer gravity anomalies are maximum around 11.70°N–11.80°N, 12.73°E–12.84°E.

### 8.2 Second vertical derivative

The Bouguer data were then filtered using a second vertical derivative on surfer software to produce a new grid and subsequently, the contour map was plotted as shown in Figure 3. The second vertical derivative map shows the distribution of the second-order vertical derivative in mGal/m^{2}. The value ranges from −320,000,000 mGal/m^{2} to 100,000,000 mGal/m^{2}. The zero values show the edges of the deeper feature. As defined earlier, the portion with less density is considered to contain more sediment. The map shows that density distribution is decreasing inward between 8°N to 10.5°N and 11°E to 13°E. The same thing happens between 12.5°N to 13°N and 11.2°E to 12°E. The zero contours can be observed throughout the map, which means that there are so many boundaries or edges in the area.

The second vertical derivative map in Figure 3 shows that the “polarity” of the anomaly can still be recognized, that is, the low density in the Central part relative to its surroundings. The second vertical derivative method of gravity anomaly illustrates the amplitude of gravity anomaly that is triggered by fault structure that gives the impression of residual anomaly. The map of the second vertical derivative method in this study shows that the method is useful in enhancing weaker local anomalies, edges of geologically anomalous density distributions were defined, and geologic units are identified. The second vertical derivative is interested in near-surface anomalous effect at the expense of the effects that are of deep origin. This study embraced the use of the second vertical derivative because of its tendency to emphasize local anomalies and isolate them from the local background, which can be seen in Figure 3. When compared to the Bouguer map in Figure 2, it can be observed that the calculation of gradients has boosted refined features of gravity data that else cannot be noticed visually from the original data. High gradients observed in the middle of the map can be connected to the high contrast of the subsurface physical properties and vice versa [11].

Gradients, and also their magnitude, are commonly engaged to delineate boundaries of anomalous sources. The map produced extended zero contours, which corresponds to the edges of local geologically anomalous density distribution structures. The quantity zero mGal/m^{2} coincides with most of the lithological boundaries, when compared with the major geologic contacts.

## 9. Conclusion

In this chapter, the use of the second vertical derivative is described as one of the efficient methods used for enhancing weaker local anomalies, defining the edges of geologically anomalous density distributions, and identification of geologic units. Satellite gravity data of a particular place in Nigeria were acquired from Bureau Gravimetrique International (BGI). The data that had already undergone Bouguer correction were used to plot the Bouguer map of the study area, which shows that the place is a semidentary basin because negative gravity anomalies are observed throughout the area. The second vertical derivative map was then plotted to emphasize local anomalies and isolate them from the local background, which can be seen in Figure 3. The map has shown areas that have lower and higher anomalies of deeper sources. Boundaries of the anomalies are also observed. On the second vertical derivative maps, the “polarity” of the anomaly can still be identified, that is, the low density in the Central part relative to its surroundings. The second vertical derivative method in this study shows that it is useful in enhancing weaker local anomalies, defining the edges of geologically anomalous density distributions, and identification of geologic units. Boundaries are better delineated by the second vertical derivative method with oscillations between the minimum and maximum (extremum) values through each density contrast transition (Figure 3). It is important to note that the second vertical derivative method is justified only on data that has high accuracy [10].

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