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Gravity Interpretation using the Mellin Transform. Prof.L.Anand Babu Dept. of Mathematics Osmania University Hyderabad-500007. One of the main inputs of the economic development are the mineral resources. They constitute the bulk of raw materials in core industries.

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gravity interpretation using the mellin transform

Gravity Interpretation using the Mellin Transform

Prof.L.Anand Babu

Dept. of Mathematics

Osmania University

Hyderabad-500007

slide2

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.
slide3

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.
slide4

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.
slide5

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.
slide6

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
slide9

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

slide10

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

slide11

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

or

using x=ztan , equation(3) reduces to

slide13

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

Thusand

slide14

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).
slide15

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

slide18

Field example

Humble 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.

slide19

The value of Z is obtained according to

The Mellin transform method as 4976.97m and Nettleton 1976 as 4968.23m.

Figure: 2

slide21

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

slide22

All the observed values are used,

  • Only a few transformed values are required for computation,
  • The interpretation procedure can be computerized, and
  • The Mellin transformation method can be extended to other models in gravity and magnetic interpretation.
slide23

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.
slide24

Thank

You