Fourier Transform Functions for Computing Gravity and Magnetic Anomaly Derivatives in the Wavenumber Domain

In gravity and magnetic anomaly data processing, derivative calculation serves as a common technique to enhance weak anomalies and separate superimposed anomalies. Implementing derivative computation in the wavenumber domain via Fast Fourier Transform (FFT) demonstrates distinct advantages over traditional spatial-domain finite-difference methods.

The fundamental approach for wavenumber-domain differentiation involves transforming gravity/magnetic anomaly data from the spatial domain to the wavenumber domain, multiplying by corresponding wavenumber response functions, and finally applying inverse transformation to obtain derivative results. This method's mathematical foundation lies in the differential property of Fourier transforms - spatial domain differentiation corresponds to multiplication operations in the wavenumber domain. In MATLAB environments, this process can be efficiently implemented using built-in fft and ifft functions, typically involving three key steps: forward FFT, pointwise multiplication with wavenumber operators, and inverse FFT.

For vertical derivative calculations, the wavenumber-domain method significantly improves accuracy compared to conventional Fourier series expansion approaches. This enhancement occurs because vertical derivatives show greater sensitivity to high-frequency noise, while wavenumber-domain methods effectively control high-frequency error amplification through appropriate wavenumber response function design. Horizontal derivative computation exhibits different characteristics, with both methods showing comparable accuracy levels.

Practical applications require attention to several critical aspects: First, proper handling of data periodicity and boundary effects typically necessitates appropriate window functions (e.g., Hanning or Tukey windows). Second, accurate design of wavenumber response functions directly impacts the physical meaning of derivative results, requiring careful consideration of operator formulations. Finally, numerical stability control during computation, particularly strategies for handling high-frequency components, demands implementation of regularization techniques or high-frequency filters.

In a potash exploration project's gravity data processing, wavenumber-domain derivative computation successfully highlighted deep weak anomalies, providing clearer data foundations for subsequent interpretation work. This method proves particularly suitable for high-precision measurement data processing, effectively preserving detailed characteristics of original data while enhancing anomaly visibility through optimized derivative operations.