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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Dept. of Mathematics
One of the main inputs of the economic development are the mineral resources.
We define such useful information as “The resolution of potential fields”.
M(s) = M[f(x);s] =
where s is a real number .
(0<Re s <1)
and so on..
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
The gravity effect of the sphere is given by Dobrin(1976), Figure 1(a).
is its density
G universal gravitation constant
R is the radius
Z is the depth to the centre
The Mellin transform of a function g(x) is defined as Sneddon (1979)
using x=ztan , equation(3) reduces to
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)
Table : comparison of assumed values of Z and m used in stimulated models with evaluated values obtained using the Mellin transform (In arbitrary units).
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.
The Mellin transform method as 4976.97m and Nettleton 1976 as 4968.23m.
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