We developed a novel technique of robust estimation of the discrete Hilbert transform (DHT) of noisy geophysical data. The technique used the sinc method, in which the data were transformed via conformal mapping and the sinc bases were determined by solving a linear matrix equation. A transformation rule was presented for selecting a suitable conformal mapping function that would transform the class of geophysical data set in an appropriate interval range. A novel regularization technique was designed to obtain a robust solution of sinc bases when the data contained noise, in which an optimal regularization parameter was obtained in an automated way using a 1D optimization scheme. The technique of selecting the optimal value of the regularization parameter required no a priori knowledge about the level of noise contamination in the data. Numerical experiments were conducted on synthetically generated and published field data sets with a varying level of noise contamination to test the performance of the scheme. The results obtained using the proposed technique of DHT and those obtained by a standard Fourier domain technique were compared, and it was established that the proposed scheme of discrete Hilbert transformation performed better than that of the standard Fourier domain technique, for noise free and noisy data. The scheme was applied successfully on potential field and infrasound waveform data and also in estimating instantaneous frequency of nonstationary ultrasonic waveform data, which suggested applicability of the scheme to a wide class of geophysical data.