International Journal of Scientific & Engineering Research, Volume 6, Issue 1, January-2015 1359

ISSN 2229-5518

Gravity and Density Relationship

(Forward Modeling)

Mohamed El-Tokhey, Mohamed Elhabiby, Ahmed Ragheb, Mohamed Shebl

Public Works Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt

AbstractThe Gravimetry technique is mainly used to measure variation in gravitational field at specific locations on the earth's surface to determine subsurface densities. The variation in gravitational field is clear in zones that having a greater or lesser density than the surrounding material. However, the problem is the ambiguity in gravity anomaly interpretation However; a unique solution can be obtained by incorporating some a-priori information such as assigning a simple geometry to the causative source.

Direct gravity problem is called forward modeling and it means calculating the gravity anomaly from known density of simple bodies through integration. This integration is computed over the volume of anomalous mass. The gravity anomaly (the vertical component of the gravity

attraction) is directly related to the density contrast.

This paper presents basic formulas for the use of relative gravimetry techniques for gravity anomaly computations in geophysical exploration in general and voids detection in specific. The direct techniques explored in this paper extensively through different mathematical bodies that are

corresponding to expected voids that will be under investigation with real case study in the future. The mathematical shaped bodies are picked up and will be tested to help overcoming the ambiguity of gravity anomaly interpretation.

Index TermsGravity anomaly, voids detection, Density, Gravity interpretation and forward modeling.
—————————— ——————————

1 INTRODUCTION

Edifferences in vertical component of the gravity fields,

𝑧

𝑔 = � 𝐺 ∗ 𝛥𝛥 ∗ 𝑑𝑑
xploration applications always involve measurement of
bodies below the computational surface by the following [3]:

𝑣

which are depended on buried geologic features. In fact,
the attraction of the earth does not only affected by buried

𝑧 𝑟3

(1)
geologic features, but there are other sources that cause
variation in attraction value, such as latitude variation. If all
other sources of attraction considered constant over the surveyed area, the contrast density (between certain body and surrounding material), which is known by anomalous density is the main source for vertical component of attraction. Consequently, it can be measured by any type of gravimeters, which are designed for measuring vertical force alone [1].
For indirect interpretation approach, the causative body of a gravity anomaly is simulated by a model whose theoretical anomaly can be computed, and the shape of the model is altered until the computed anomaly closely matches the observed anomaly [2], Although the inverse problem will not be a unique interpretation, but the ambiguity can be decreased by using other constraints on the nature and form of the anomalous body. Indirect methods are either based on the application of simple analytic formulas for elementary source
Where 𝛥𝛥 is the density contrast,𝐺 is the Newton’s
gravitational constant, 𝑧 is the depth to the mass point and 𝑟is
the distance between the mass point and the observation
point.
In general, it’s preferable in the vertical component of
attraction computation to deal with bodies with simplest
geometry or that are symmetrical with respect to vertical axis passing through the center of gravity like (sphere, rectangular prism, horizontal cylinder…)[4].

3 FORWARD GRAVITY MODELING OF SIMPLE-SHAPED BODIES

3.1 The sphere

The attraction at any external point of a homogeneous solid sphere to the attraction of a point mass located at its center and can be computed from simple form [5].
𝐺 ∗ ∆𝛥 ∗ 𝑉 ∗ 𝑧
bodies like vertical and dipping faults, spheres, cuboids,
horizontal and vertical cylinders. Although, that indirect
methods are used extensively, but the inverse problem is not
robust and its stability is an issue and needs extensive
Where:
𝑔𝑧 (𝑚𝑔𝑚𝑚) =

3

(𝑥2 + 𝑧2 )2 (2)
information to help with the regularization of the inverse mathematical problem. Consequently, in this research paper, the main objective will be on building a model and procedure for detecting different voids with different geometrical shapes for void detection using simulated environment will be used.

2 METHODOLOGY

The computation carried out by integrating process for the vertical component of attraction force from three-dimensional
𝑔𝑧 : is the vertical component of attraction force.
G: is universal gravity constant = 6.67 * 10−11 m3Kg−1s−2
∆𝛥(𝑘𝑔/𝑚3): is the density contrast of the sphere
V(𝑚3): is the volume of the sphere
𝑧(𝑚): is the depth of the center of sphere
𝑥(𝑚): is the location of calculation point
For example let: a sphere with radius a= 50 m and z=100 m
and ∆ρ = 2000 Kg. m−3

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ISSN 2229-5518

0, the maximum attraction is

𝑚𝑎𝑥

𝐺 ∗ ∆𝛥 ∗ 𝑉

Surface x p

𝑔𝑧

= 𝑧2 (3)

z g(z)

a

3.2 The horizontal cylinder

The attraction at any external point of a homogeneous solid horizontal cylinder can be calculated from the following equation [1]:

Sphere

Where:
𝑔𝑧 (𝑚𝑔𝑚𝑚) =

2 ∗ 𝐺 ∗ 𝜋 ∗ 𝑚2 ∗ 𝑧 ∗ ∆𝛥
(𝑥2 + 𝑧2 ) (4)

Fig. 1. subsurface sphere with radius (a) and depth (z).

Vertical gravity of sphere in mGal

150

100

0.6

0.55

0.5

𝑔𝑧 : is the vertical component of attraction force.
𝐺: is universal gravity constant = 6.67 * 10−11𝑚3 𝐾𝑔−1𝑠−2
∆𝛥(𝑘𝑔/𝑚3) : is the density contrast of the horizontal cylinder
𝑚(𝑚): is the radius of cylinder
𝑧(𝑚): is the depth of the center of cylinder

𝑥(𝑚): is the location of calculation point

0.45

50

0.4

0.35

0

0.3

Surface x p

-50

-100

0.25

0.2

0.15

z g(z)

a

-150

-150 -100 -50 0 50 100 150

X in meter

0.1

Hz-Cylinder

mGal

Fig. 2. contour map for vertical gravity attraction of subsurface sphere.

Fig. 4. subsurface Hz-cylinder with radius (a) and depth (z)

For example let: a = 50 m and z=100 m and ∆ρ = 2000 Kg. m−3

0.8

0.7

0.6

0.5

X: 0

g: 0.6985

200

150

100

50

0

Vertical gravity of HZ cylinder in mGal

2

1.8

1.6

1.4

0.4

0.3

0.2

0.1

-150 -100 -50 0 50 100 150

-50

-100

-150

-200

-150 -100 -50 0 50 100 150

X in meter

1.2

1

0.8

mGal

X in meter

Fig. 3. profile for vertical gravity attraction of subsurface sphere

Fig. 5. contour map for vertical gravity attraction of subsurface

Hz-cylinder.


𝑔𝑧 has its maximum value directly above the sphere where x =

𝑔𝑧 has its maximum value directly above the cylinder where x
= 0, the maximum attraction is

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𝑚𝑎𝑥

2 ∗ 𝐺 ∗ 𝜋 ∗ 𝑚2
∗ ∆𝛥

3.3 The right rectangular prism

The attraction at any external point of a homogeneous solid
𝑔𝑧

= 𝑧 (5)
right rectangular prism can be calculated from the following equation [6]:

𝑔𝑧 (𝑚𝑔𝑚𝑚) = 𝐺 ∗ ∆𝛥 ∗ ∑2

2

𝑗=1

2

𝑘=1

𝜇𝑖𝑗𝑘 ∗ �∆𝑧𝑘

2.2

2

1.8

X: 0

g: 2.095

𝑚𝑟𝑎𝑎𝑚𝑎 �
Where:

∆𝑥𝑖 ∗∆𝑦𝑗

∆𝑧𝑘∗𝑅𝑖𝑗𝑘

� − ∆𝑥𝑖 ∗ 𝑚𝑙𝑔�𝑅𝑖𝑗𝑘 + ∆𝑦𝑗 � − ∆𝑦𝑗
𝑚𝑙𝑔�𝑅𝑖𝑗𝑘 + ∆𝑥𝑖 ��
(6)

1.6

1.4

1.2

1

0.8

-150 -100 -50 0 50 100 150

X in meter

Fig. 6. profile for vertical gravity attraction of subsurface Hz cylinder.

𝑔𝑧 : is vertical component of attraction force.
𝐺: is the universal gravity constant = 6.67 * 10−11𝑚3𝐾𝑔−1𝑠−2
∆𝛥(𝑘𝑔/𝑚3): is the density contrast of the rectangular prism
𝜇𝑖𝑗𝑘 = (−1)𝑖 (−1)𝑗 (−1)𝑘
∆𝑥𝑖 = �𝑥𝑖 − 𝑥𝑝 �, ∆𝑦𝑖 = �𝑦𝑗 − 𝑦𝑝 �, 𝑚𝑎𝑑 ∆𝑧𝑘 = �𝑧𝑘 − 𝑧𝑝 �The

distances from each corner to calculation point (p).
𝑅𝑖𝑗𝑘 = �∆𝑥𝑖 2 + ∆𝑦𝑗 2 + ∆𝑧𝑘 2 (7)
For example using the following parameters:
𝑥1= -100 m, 𝑥2= 100 m, 𝑦1= -100 m, 𝑦2= 100 m, 𝑧1 = -100 m,

𝑧2= -200 m, ∆𝛥 = 2000 Kg. 𝑚−3
The attraction because of the Hz-cylinder is not as sharp as that from a sphere at the same depth. Since the term involving
the horizontal distance𝐱, which is in the denominator and
taken to a lower power for the cylinder. It is seen by
comparing the both equations 2 and 4 that a horizontal cylinder will have a maximum gravitational attraction about
(1.5 𝐳/ 𝐚) times as great as a sphere of the same radius, depth,
and density. This should be expected in view of the much
greater mass contained in the cylinder.

Fig. 8. right rectangular prism with dimensions(x, y, z)1:2 .

2.5

2

Sphere

Hz-Cylinder

150

100

Vertical gravity of rectangular prism in mGal

1.6

1.4

1.5

50

1.2

0

1 -50

1

0.8

0.5

-100

0.6

0

-200 -150 -100 -50 0 50 100 150 200

X in meter

Fig. 7. comparison between vertical gravity attraction of sphere and Hz-cylinder with the same depth and radius.

-150

-150 -100 -50 0 50 100 150

X in meter

mGal

Fig. 9. contour map for vertical gravity attraction of subsurface prism.

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degree and the vertical gravity attraction will be [7]:

2

1.8

X: 0

g: 1.756


𝑔𝑧 (𝑚𝑔𝑚𝑚) = 2 ∗ 𝐺 ∗ ∆𝛥 ∗ 𝑇 ∗ �𝑎𝑚𝑎−1

𝑙−𝑥


� + 𝑎𝑚𝑎−1 𝑥 �� (12)

1.6 12

1.4

1.2

Dip angle=0 deg.

10 Dip angle=45 deg.

Dip angle=90 deg.

8

1

0.8 6

0.6

4

0.4

-200 -150 -100 -50 0 50 100 150 200

X in meter 2

Fig. 10. profile for vertical gravity attraction of subsurface prism.

3.4 The dipping thin sheet with finite length

The vertical gravitational attraction g zof a dipping thin sheet with an inclination (α) equal (90+dipping angle) with horizontal plan and has finite length (l) and thickness (T) can
be computed from the following form [7]:
𝑔𝑧 (𝑚𝑔𝑚𝑚) = 2 ∗ 𝐺 ∗ ∆𝛥 ∗ 𝑇 ∗ �𝑠𝑠𝑎 ∝ ∗

0

-2000 -1500 -1000 -500 0 500 1000 1500 2000

X in meter

Fig. 12. profile for vertical gravity attraction of three dipping thin sheet with variable dipping angle (0, 45, 90) degrees.

3.5 The semi-infinite horizontal sheet

If the horizontal sheet has infinity in one direction, the vertical
gravity attraction g z will be as follows [1]:
𝑚𝑎 ( 𝑟2 ) − (𝜃
(8)
+ 𝜃 ) ∗ 𝑎𝑙𝑠 𝛼 �
𝑔 (𝑚𝑔𝑚𝑚) = 2 ∗ 𝐺 ∗ ∆𝛥 ∗ 𝑇 ∗ �𝜋 − 𝑎𝑚𝑎−1 𝑥�� (13)
Where:

𝑟1 1 2


𝑧

Where:

2 𝑧

𝑔𝑧 : is the vertical component of attraction force.
𝐺: is the universal gravity constant = 6.67 * 10−11𝑚3𝐾𝑔−1𝑠−2
∆𝛥(𝑘𝑔/𝑚3): is the density contrast

𝑇(𝑚): is the thickness

𝑟1 = �𝑥2 + ℎ2 (9)
𝑟2 = �(𝑥 + 𝑚 ∗ 𝑎𝑙𝑠 ∝)2 + (ℎ + 𝑚 ∗ 𝑠𝑠𝑎 ∝)2 (10)
𝑔𝑧 : is vertical component of attraction force.
𝐺: is the universal gravity constant = 6.67 * 10−11𝑚3𝐾𝑔−1𝑠−2
∆𝛥(𝑘𝑔/𝑚3): is the density contrast

𝑇(𝑚): is the thickness

Fig. 13. semi-infinite horizontal sheet with thickness (T).

Fig. 11. thin Sheet with finite length (l) and (α = 90+dip angle).

When the dipping angle equals 0 degree, the thin sheet will be vertical and the vertical gravity equation is simplified to [7]:
𝑔𝑧 (𝑚𝑔𝑚𝑚) = 2 ∗ 𝐺 ∗ ∆𝛥 ∗ 𝑇 ∗ 𝑚𝑎 �

(ℎ+𝑙)2 +𝑥 2

2 +𝑥2


� (11)

14

The sheet will be horizontal when the dipping angle equals 90

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4 CONCLUSION

Building a library of identified shapes for direct gravimetry void detection technique for geophysical exploration in small areas. It is clear the differentiation between the shape in case of fixing the size, depth, and density. A complete geometrical simulation was done and in the future a complete library of shapes will be tested for direct techniques in archeological applications with constrained environment for void detection.

REFERENCES

[1] Milton, B. D. (1983). “Introduction to Geophysical prospecting”. New York, McGraw-Hill Book Company.
[2] Le Mével, H. (2009). “Relationship between seismic and gravity anomalies at Krafla volcano, North Iceland“. University of Nantes, France.
[3] Corchete, V., Chourak, M., and Khattach, D. (2009). “A Methodology for Filtering and Inversion of Gravity Data: An Example of Application to the Determination of the Moho Undulation in Morocco“. USA: Scientific Research, Engineering.
[4] Sazhina, N. (1971). “Gravity Prospecting“. Moscow: MIR, Amazon.
[5] Telford, W. M., Geldart, L., Sheriff, R., and Keys, D. (1981). “Applied geophysics“. Cambridge, New York, U.S.A.
[6] Nagy. (1966). “The gravitational attraction of a right rectangular prism“. Geophysics, Vol. 31, pp. 362-371.
[7] Patrick, W. (2004). "Gravity Anomalies Produced by
Bodies of Simple Shape".

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