Gravity data interpretation using the algorithm fourth horizontal derivatives and s- curves method

The gravity method is one of the first geophysical techniques used in oil and gas exploration. An algorithm is developed for a fast quantitative interpretation of gravity data generated by geometrically simple but also the estimated depths and other model parameters of a buried structure.
Following Abdelrahman et al (1989). The general gravity anomaly expression produced by a sphere, an infinite long horizontal cylinder and a semi- infinite vertical cylinder can be represented by the following equation
                                                                                                          (1)
where
 
and z is the depth of the body, x_i is the horizontal position coordinate, σ is the density contrast, G is the universal gravitational constant and R is the radius and q is factor related to the shape of the buried structure and is equal to 0.5,1.0,and 1.5 for the semi- infinite vertical cylinder, horizontal cylinder and the sphere respectively.
Consider nine observation point (x_i  -4s),  (x_i  -3s),  (x_i  -2s),  (x_i  -s),  (x_i  ),  (x_i  +s),  (x_i  +2s),  (x_i  +3s),  (xi  + 4s),  along the anomaly profile where s=1,2,3,M spacing units and is called the window length.
Using equation (1) the simplest first numerical horizontal gravity gradient (dg/dx₎
                                                                       (2)
the second horizontal derivative gravity anomaly is obtainedfrom equation (2) as
                            (3)
the third horizontal gradient is(3)
                   (4)
Similarly, the fourth horizontal gradient is (4)
5)
Which yields;
 
 
Where
                                                                                                             (7)
Equation (5) can also be solved using a simple iteration method.
Equations (5) can be used to determine the depth and the shape of a buried structure using the window curves method. The validity of the method is tested on synthetic data white and without random errors.
The method was applied to a gravity anomaly from the Abade of Iran .The results shows that the s-curves intersect each other in a narrow region where  7.220 <z<7995 m and 1.40<q<1.51 ;
The central point of this region occurs at the location z= 7.6900 m and q= 1.43.
The aim of the present study is to develop a simple method (s-curves method) for analysis of gravity anomalies due to derivative calculations that can be used to estimate the depth and the shape of the causative bodies. In all cases examined, the estimated depths are found to be in good agreement with the actual values.