برآورد عمق بی‌هنجاری‌های گرانی حاصل از شکل‌های هندسی منظم  با استفاده از تبدیل هیلبرت تغییریافته

One of the most important parameters in the interpretation of gravity data is the depth to the center or top of the buried body. In this study, the interpretation of gravity anomalies of spherical and cylindrical models is examined using the modified Hilbert transform.
The Hilbert transform of a real function f(x) is defined as:
                                                                        (1)
where
                                                               (2)
RF(ω) and FI(ω) are the real and imaginary component of the Fourier transform of f(x). The Hilbert transform defined by Eq.(1) is a mathematical operation which shifts the phase of a function by 90° without changing its amplitude.
Modified Hilbert transform is identical in amplitude with the conventional Hilbert transform but differs in phase as it yields a phase shift of 270°. Modified Hilbert transform, MH(x), can be obtained from Hilbert transform H(x) by replacing x with –x in Eq.(1) (Shriniivas 2000; Sundararajan et al 2000), i.e,
                                                          (3)
The general gravity effect caused by simple models such as horizontal circular cylinder and sphere, centered at x=0 and buried at a depth z is given by (Abdelrahman et al 2001):
                                                                                                              (4)
where q is shape factor and depends on the nature of the source, and is 3/2, for a sphere and 1, for a horizontal cylinder. And A is amplitude factor given by: , for a sphere and  for a horizontal cylinder, where ρ is the density contrast, G is the universal gravitational contrast and R is the radius.         
The application of the method is examined using noise free and noise corrupted synthetic gravity data created for spherical and cylindrical models with a density contrast of  1 and 1.5g/cm3 respectively. The gravity anomaly g(x), modified Hilbert transform MH(x), along a profile at an interval of 1m are computed for both data sets. The procedure has been tested for several models at different dephts and radii, for three of which the results are presented here. It is observed that the depth to the origin of the gravity anomaly can be computed as a function of the intersection point of gravity anomaly g(x) and its modified Hilbert transform MH(x).
The effect of random noise on the models shows that even by including up to 16% random noise, interpretational values do not differ significantly from thoes of the noise free case. Hence the effect of noise is negligible on the procedure. 
To illustrate the applicability of the method two field examples from “Abade” in Fars province and “Havasan” in Ilam province, Iran, are also included. A Scintrex CG3 gravimeter with a sensitivity of 5 microGal was used for micro-gravity observations in the selected areas. Station altitudes were measured with a total station model Leica Tc 407 with an accuracy of 1-5mm in horizontal and vertical coordinates. The residual gravity grids of were obtained obtained using Geosoft software. 
 To demonstrate the reliability of the proposed method, the Euler de-convolution method is used to detect the depth of the real gravity anomalies. The results from the interpretation of real data by modified Hilbert transform method are compared to the ones obtained from the Euler de-convolution method and the known depth values from drilling information.