Computers & Geosciences 29 (2003) 801–804

Short Note

Depth determination from residual gravity anomalydata using a simple formula

Ahmed Salema,*,1, Eslam Elawadib, Keisuke Ushijimab

a Airborne Geophysics Department, Nuclear Materials Authority of Egypt, P.O. Box 530, Maadi Kattamiya Road, Maadi, Cairo, Egyptb Earth Resources Department, Faculty of Engineering, Kyushu University 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan

Received 20 September 2002; received in revised form 19 March 2003; accepted 25 March 2003

1. Introduction

The problem of ambiguty in the interpretation of

potential-field data cannot be solved by any processing

or interpretation technique (Roy, 1962). However, a

unique solution may be obtained by incorporating

certain a priori information such as assigning a simple

geometry of the causative source (Roy et al., 2000).

Although simple models may not be geologically

realistic, they usually are sufficient to analyze sources

of many isolated anomalies (Nettelton, 1976; Abdelrah-

man and El-Araby, 1993).

Several numerical methods have been presented for

interpreting gravity anomalies to estimate depths of the

geological structures assuming simple source geometry.

These methods include, for example, Fourier transfor-

mation (Sharma and Geldart, 1968), Euler deconvolu-

tion (Thompson, 1982), Mellin transform (Mohan et al.,

1986), least-squares minimization approaches (Gupta,

1983; Abdelrahman, 1990) and neural network (Elawadi

et al., 2001).

In this note, we present an alternative method for

interpreting residual gravity anomalies caused by simple

models. A linear equation is derived to provide the

depth of buried gravity sources. The proposed technique

is simple and allows depth to be calculated using a

simple calculator. The utility of the method is demon-

strated using three field examples.

2. The method

The general gravity effect at an observation point

(x; z ¼ 0) caused by simple gravity models (such as a

sphere, an infinite horizontal cylinder, and a semi-finite

vertical cylinder) centered at x ¼ 0 and buried at a depth

z is given (Abdelrahman et al., 2001) by

gðxÞ ¼Azm

ðx2 þ z2Þq; ð1Þ

where q is a value characterizing the nature of the source

(shape factor)

q ¼

3=2 for a sphere;

1 for a horizontal cylinder;

1=2 for a vertical cylinder;

8><>:

m ¼

1;

1;

0

8><>:

and A is an amplitude factor given by

A ¼

4pGrR3

3for a sphere;

2pGrR2 for a horizontal cylinder;

pGrR2 for a vertical cylinder;

8>>><>>>:

where r is the density contrast, G is the universal

gravitational constant, and R is the radius. Using the

gravity value (g0) over the source (x ¼ 0), Eq. (1) can be

normalized (Abdelrahman et al., 2001) as

gnðxÞ ¼gðxÞg0

¼z2

x2 þ z2

� �q

: ð2Þ

Rearranging Eq. (2), we get

ðgnðxÞÞ1=qx2 þ ðgnðxÞÞ

1=qz2 ¼ z2: ð3Þ

ARTICLE IN PRESS

*Corresponding author. Fax: +81-92-642-3614.

E-mail addresses: salem@mine.kyushu-u.ac.jp (A. Salem),

eslam@mine.kyushu-u.ac.jp (E. Elawadi).1 Current address: Earth Resources Department, Faculty of

Engineering, Kyushu University, 6-10-1, Higashi-ku, Fukuoka,

Japan.

0098-3004/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0098-3004(03)00106-7

It is obvious that this equation can provide a depth

estimate (z) from even a single normalized residual

gravity value. However, due to several sources of errors,

multiple values are required to obtain a good estimate

of the depth. In a least-squares sense, Eq. (3) can be

solved as

z ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ð1 � ðgnðxiÞÞ

1=qÞðgnðxiÞÞ1=qx2

iPNi¼1ð1 � ðgnðxiÞÞ

1=qÞ2

s; ð4Þ

where N is a number of observations. Eq. (4) can be

implemented using a simple calculator using the follow-

ing procedure:

1. Normalize the residual gravity observations using

Eq. (2).

2. CalculateP

U ¼PN

i¼1ð1 � ðgnðxiÞÞ1=qÞðgnðxiÞÞ

1=qx2i :

3. CalculateP

L ¼PN

i¼1ð1 � ðgnðxiÞÞ1=qÞ2:

4. Then the depth is given by z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

U=P

Lp

:

3. Synthetic examples

The reliability of the proposed method has been tested

using synthetic gravity data created for both cylinder

and sphere models, having a density contrast of 2.5 g/

cm3, and buried at different depths. The gravity

anomaly values were calculated along a 30 m profile at

an interval of 1 m. Then the calculated data were

contaminated with 5% and 10% random noise to

simulate errors that might arise during measurement

and processing of raw gravity data and/or small

interference due to nearby sources. Applying the method

to these test data (Table 1), the errors in the estimated

depths of the cylinder model are less than 4.6% and

8.0% for data corrupted by 5% and 10% noise,

respectively. Meanwhile, the errors in the estimated

depths for the sphere model corrupted by 5% and 10%

noise are less than 3.3% and 5.8%, respectively. We also

tested the effect of background errors that may arise due

to inappropriate separation of the regional field. A

background level, representing 10% of the maximum

anomaly value (g0) was added to the simulated gravity

anomaly data of the above models. The data were also

corrupted by 10% random noise. Then the formula was

applied to estimate the depth. For all cases, depth errors

are found to be less than 20% of the actual depths. Even

though the data were degraded by the inclusion of up to

10% noise and 10% regional background, the method

provided good estimates of the depth. However, these

estimates would be highly improved when the residual

gravity data are less affected by noise and the regional

field is adequately separated.

4. Field examples

We demonstrate the practical utility of the proposed

method to estimate the source depth from three field

examples of gravity data for which source depths are

known. For each example, we repeat the depth

determination using different number of observations,

centered above the horizontal location of the source.

Then an average depth is calculated for each field

example.

4.1. Humble salt dome

A residual gravity profile of the gravity map of the

Humble salt dome, TX, USA (Nettelton, 1976), was

digitized at an interval of 0.76 km (Fig. 1). Several

authors have used a spherical model to interpret this

anomaly (Nettelton, 1976; Mohan et al., 1986; Abdel-

rahman and El-Araby, 1993). Applying the proposed

method to these data, assuming a spherical model

(q ¼ 1:5), the estimated depths (Table 2) are between

4.97 and 5.32 km with an average depth of 5.15 km,

which agrees well with results from drilling and seismic

information (4.97 km, Nettelton, 1976).

4.2. Chromite deposit

Fig. 2 shows a normalized residual gravity anomaly

measured over a chromite deposit in Camaguey

province, Cuba (Davis et al., 1957). The normalized

residual anomaly was digitized at an interval of 4.3 m.

Robinson and Coruh (1988) showed that a sphere model

at a depth of 21.0 m approximates the source of

this anomaly. Application of the method to these data

using a spherical model (q ¼ 1:5), the estimated depths

(Table 2) are between 23.02 and 24.9 m with an average

value of 23.8 m, which is in good agreement with the

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Table 1

Error percentage in calculated depths from gravity data for test

models contaminated by 5% and 10% random noise

Model

depth (m)

Error percentage in the calculated depth

Cylinder Sphere

5% noise 10% noise 5% noise 10% noise

1 2.0 5.0 1.0 3.0

2 2.5 5.5 1.5 3.5

3 3.0 6.6 2.0 4.3

4 3.5 8.0 2.2 4.7

5 4.0 6.6 2.6 5.8

6 4.6 6.0 2.8 5.0

7 4.2 5.7 3.2 4.5

8 3.8 5.6 3.2 4.3

9 3.7 4.1 3.1 4.3

10 3.7 2.6 3.0 4.4

A. Salem et al. / Computers & Geosciences 29 (2003) 801–804802

information obtained from drill-hole information

(21.0 m).

4.3. Medford cavity

Fig. 3 shows a residual gravity anomaly for a cavity

located at the Medford cave site, Florida, USA (Butler,

1984). The residual anomaly data was interpolated at an

interval of 1 m. This anomaly has an elongated shape in

the residual gravity map of the Medford site, which may

indicate that a cylinder model better approximates the

cavity geometry. Applying the proposed method to these

data, assuming a cylinder model (q ¼ 1:0), the estimated

depths (Table 2) are between 3.76 and 4.02 m with an

average value of 3.92 m, which correlates well with the

center of the cavity as obtained from drill-hole

information (3.57 m).

5. Conclusion

This note presents a simple method to estimate the

depth of a buried structure from residual gravity

anomaly data. A simple formula has been derived to

ARTICLE IN PRESS

-7.6 0 7.6Distance (km)

-16

-12

-8

-4

0

Gra

vity

ano

mal

y (m

Gal

)

Fig. 1. Residual gravity profile over Humble salt dome, TX,

USA (Nettelton, 1976).

-21.5 0 21.5Distance (m)

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

gra

vity

ano

mal

y (m

Gal

)

Fig. 2. A normalized residual gravity profile over the chromite

deposit, Cuba (Davis et al., 1957).

-5 0 5Distance (m)

-60

-40

-20

0

Gra

vity

ano

mal

y (µ

Gal

)

Fig. 3. Residual gravity profile over a cavity at Medford test

site (Butler, 1984).

Table 2

Depth estimates from gravity field examples of Humble dome,

USA, chromite deposit, Cuba, and Medford cavity, USA

N Depth to

dome (km)

Depth to

chromite

deposit (m)

Depth to

cavity (m)

5 5.31 23.02 4.02

7 5.32 23.75 4.09

9 5.25 23.29 4.04

11 5.17 23.48 3.89

13 5.10 23.84 3.76

15 5.09 24.32 3.76

17 5.02 24.90

19 4.97

Average

depth

5.15 23.80 3.92

N is number of observations centered above source body.

A. Salem et al. / Computers & Geosciences 29 (2003) 801–804 803

estimate the depth of a sphere and of a vertical and

horizontal cylinder. Depth to the buried sources can be

obtained from a few measurements using a simple

calculator. The method was tested using synthetic data

on spherical and cylindrical models buried at different

depths. Even though the data were degraded by the

inclusion of up to 10% noise, the method provided good

estimates of the depth. The practical utility of the

method is demonstrated using three field examples. For

these examples, the estimated depths are very close to

the known depth values.

Acknowledgements

We greatly appreciate constructive and thoughtful

comments of M.D. Thomas. We are also indebted to

Prof. D. Ravat of Southern Illinois University for his

comments on the manuscript. The authors are thankful

to all the staff of the Exploration Geophysical labora-

tory, Kyushu University for their contribution and

support during this work. The work of the first author

on this paper was made possible by a funding from the

Japan Society of Promotion of Science (JSPS).

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ARTICLE IN PRESSA. Salem et al. / Computers & Geosciences 29 (2003) 801–804804