Improvement of the first and second vertical gradients calculated using cosine transform

Summary: First and second vertical gradients are widely used in the interpretation of gravity data. Vertical gradients are sensitive to noise. Accuracy of vertical gradient calculation directly effects the accuracy of interoperations. Therefore, accurate and without noise calculation of vertical gradient is vital. The most common method to calculate vertical gradients is to use Fourier transform. Low noise in the gravity data causes that the vertical gradients, calculated by Fourier transform, have severe noise. In this research, we have used discrete cosine transform (DCT) to calculate vertical gradients. Results of DCT and Fourier transform are completely equal when the gravity data are noise free, but in the case of noisy data, DCT has better performance than FFT. This improvement is investigated by using signal to noise ratio (SNR). The SNR of the results of DCT compared to Fourier transform is larger, therefore less noise enters in the calculation of vertical gradients by using DCT. We have tested these two transforms on the synthetic data containing Gaussian noise. First and second vertical gradients are calculated by DCT and Fourier transform. The results have shown that DCT in comparison with Fourier transform is less sensitive to noise. Moreover, these two transforms are used for calculating first and second vertical gradients of gravity data obtained from Safo manganese mine. The results have shown that less noise enters in vertical gradient map obtained using DCT. Edge detection of anomalies is one of the usage of gradients in the interpretation of gravity data. Analytic Signal has been used for edge detection of anomalies in the cases of real and synthetic gravity data. Vertical gradient of analytic signal calculated by DCT, compared to Fourier transform, has less noise and better quality. Introduction: The first and second vertical gradients are used to distinguish the difference between two adjacent anomalous bodies, reduce the effects of interference of the amplitudes of anomalies, separate the local field superimposed on the background determine, and to determine the location and dimensions of the anomalies. The gradient data are used in direct interpretation and inversion and inputs of many interpretations. Vertical gradients are more sensitive to noise than bouguer map, and second vertical gradients is more sensitive to noise than first vertical gradients. Methodology and Approaches: Generally, the relation between cosine and sine transforms with Fourier transform is equal to ‘F=C + iS’. Where F, denotes Fourier transform and C and S denote cosine and sine transforms, respectively. If the function is positive or constant, imaginary part of the Fourier transform is equal zero. In other words, Fourier transform and DCT will be equal. It can be assumed that noise is a function that is added to gravity signal. Because of the nature of noise, Fourier transform and DCT of noise will not be equal. Then, the signal to noise ratios for DCT and Fourier transform of noisy gravity signals are not the same. Hence, less noise enters the gradient map when calculating vertical gradients using DCT. Results and Conclusions: In this research, we have used DCT to calculate vertical gradients. We analytically have proven that DCT in noise-free data has equal results with Fourier transform results. However, the DCT of noisy data have shown less noise enters the data compared to Fourier transform. We have examined this issue by using synthetic data containing Gaussian noise. A comparison between vertical gradients obtained by DCT and Fourier transform indicates that DCT is less sensitive to noise. We have obtained similar results by using real data. The results of analytic signal calculated by DCT and Fourier transform in both cases of real and synthetic data have been compared. Edge detection calculated by using DCT has shown less noise due to less noise in the DCT and also has better quality compared to using Fourier transform.