Gravity Interpretation using

the Mellin Transform

Prof.L.Anand Babu Dept. of MathematicsOsmania UniversityHyderabad-500007

One of the main inputs of the economic development are the mineral resources.

They constitute the bulk of raw materials in core industries.

Petroleum and mineral deposits are associated with the subsurface structures.

Hence the major task is geophysical engineering is the estimating of those structure i.e., determining the location and size.

What is the resolution of a potential field ?

It is possible to measure the distribution of a potential field on the surface of the earth at equal intervals along a transverse would cover an area.

These recordings, termed as “Discrete data” naturally convey useful information about the subsurface.

We define such useful information as “The resolution of potential fields”.

The objective of the present discussion is to involve the Mellin transform for resolving the potential field data, both gravity and magnetic due to bodies of common geometries.

The oil and natural gas are generally accumulated in structures of the form like domes, anticlines and synclines.

The domes are approximate as spheres.

The anticlines and synclines are approximate as horizontal circular cylinders.

Here we discuss gravity interpretation using the Mellin transform.

Definition of Mellin transform of a function f(x).

The Mellin transform a function f(x) is defined as

M(s) = M[f(x);s] = where s is a real number . Some properties of Mellin transform

are Multiplying x by a Multiplying f(x) by

dxxfx s )(0

1

)(]);([ sMasaxfM sax

)(]);([ asMsxfxM a

If a>o ; changing x as

(0<Re s <1)

ax

)/(]);([ 1 asMasxfM a

)1(]);([ 11 sMsxfxM

)(/]);([log sdsMdsxxfM )2/1cos()(];[cos sssxM

)(];[ sseM x

)(/)()(];)1[( asassxM a

)2/1sin()(];[sin sssxM (0<Re s <1)

In particular case if we take a=1

and so on.. Discrete Mellin transform is defined as

where N=total number of the observed values. ∆x=station interval of the observed values, and ∆s=interval of the discrete Mellin transform . In the case of sphere it may be noted that 0<n.∆s<3

)1()(];)1[( 1 sssxM

)(/))/()/((];)1[( baasbassxM ba

1

0

1.).)(.().(N

l

snxlxlfsnM

Gravity effect due to sphere The gravity effect of the sphere is given by

Dobrin(1976), Figure 1(a).

where where is its density G universal gravitation constant R is the radius Z is the depth to the centre

)1()/(.)( 2/322 zxzmxg

33/4 GRm

The Mellin transform of a function g(x) is defined as Sneddon (1979)

or

using x=ztan , equation(3) reduces

to

0

1 )2()()( dxxgxsM s

)3(])/(.[)( 2/3221

0

1

dxzxzmxsMs

s

)4()(cos)(sin.)(2/

0

212

dzmsM sss

Evaluation of (4) gives

(0<s<3)

)5(2/)3()2/()/()( 2 ssmzsM s

Analysis From equation (5) the Mellin

transform of the gravity effect of a sphere at two specific values of s are obtained as

From (6) and(7) Where Thus and

)6()1()2/1(]/[)1( zmM

7)2/1()2/3(]/.[)2( 0 hmM

)2(/)1( MMZ

)2()]2/1()2/3(/[)2( MMm )1(/)2( MMZ )2(Mm

Simulated models The theoretical Mellin transform of

the gravity effect is a continuous function in the interval (0,3)

The computed Mellin transform of the simulated models are shown in the following table and figure 1(b).

The Discrete Mellin transform of the gravity effect of the sphere is presented in figure 1(c).

Table : comparison of assumed values of Z and m used in stimulated models with evaluated values obtained using the Mellin transform (In arbitrary units).

MODELS A A B B C C

Assumed Evaluated Assumed Evaluated Assumed Evaluated

Sphere Z 0.50 0.51 0.75 0.77 2.00 1.99

m 0.03 0.03 0.52 0.52 4.19 4.10

Figure - 1

Field exampleHumble Dome Anomaly A profile line AA’ of the gravity map of the humble

dome, near Houston USA (Nettleton 1976 fig 8.17) is analyzed using the residual gravity curve shown in fig 2(a).The anomaly is digitized at an interval of 132.52m.Using these digitized values the Discrete Mellin transform is calculated and shown in 2(b).Because the asymptotic regions are not considered for parametric evaluation the depth to the centre of the sphere is evaluated from the values of the Discrete Mellin transform of the residual gravity effect.

The value of Z is obtained according to The Mellin transform method as 4976.97m

and Nettleton 1976 as 4968.23m.Figure: 2

Discussions The similarity of the curves of the

transformed anomalies and the gamma function curves is expected since the Mellin transform is the generalized form of the gamma function.

It is also expected that the inherent advantage of the gamma function would be present in the transformed anomalies. This is observed since the transform anomalies are bounded by the two asymptotes (equation 5). Further note that the advantage of the Mellin transform method over graphical techniques are

dxexs xs

0

1)(

1. All the observed values are used,2. Only a few transformed values are

required for computation,3. The interpretation procedure can be

computerized, and4. The Mellin transformation method can be

extended to other models in gravity and magnetic interpretation.

References: Dobrin, M. B., 1976, Introduction to Geophysical Prospecting;

McGraw-Hill Book Co. Nettleton, L.L., 1976, Gravity and Magnetics in oil prospecting:

McGraw-Hill Book Co. Sneddon, I. N., 1979, The use of integral transforms: McGraw-Hill

Book Co.References for General Reading: Abramowitz, M., and Stegun. I. A., 1970, Hand Book of

Mathematical functions; Dover Publications, Inc. Bracewell . R., 1965, The Fourier Transform and its Application;

McGraw-Hill Book Co. Gradshteyn. I. S., and Ryzhik. I. M., 1965, Tables of Integral series

and Products; Academic Press, Inc.

Thank You