Seismic Statistical Prediction for Fracture Azimuth Based on Fourier Series

The azimuth of fractures has long been a subject of interest for geophysicists, and it holds paramount importance in the exploration and development of oil and gas resources. However, traditional fracture azimuth prediction methods heavily rely on seismic data quality and well-logging data, often encountering severe noise interference and 90° ambiguity. This makes fracture azimuth prediction challenging in areas with complex geological structures. A method for seismic statistical prediction of fracture azimuth based on the Fourier series has been proposed to address these issues. First, the Rüger approximation is rewritten into Fourier series form, combining parameters with high linear correlation to mitigate the ill-conditioning of the coefficient matrix. Second, construct a complex representation of fracture azimuth and initially adjust the sign based on the characteristic that the azimuthal period of the fourth-order Fourier coefficient is $\pi $ /2. Third, considering that the fourth-order Fourier coefficients are susceptible to noise, a directional statistical method is introduced to enhance the stability of fracture azimuth prediction. Then, by analyzing the relationship between second- and fourth-order Fourier coefficients under saturated fluid and gas-filled conditions, the Welch t-test, suitable for data with nonhomogeneous variance, is introduced to eliminate the influence of fluid type on fracture azimuth prediction. Numerical experiments and field data demonstrate that the proposed method overcomes the 90° ambiguity inherent in conventional fracture azimuth prediction, proving its stability and effectiveness in areas with severe structural variations.