Design of Seismic Wavelets and Reflection Coefficient Sequences with Implementation Approaches

In seismic data processing, a fundamental procedure involves designing appropriate seismic wavelets and reflection coefficient sequences. The implementation typically begins with defining wavelet parameters (such as Ricker wavelet frequency or Ormsby wavelet bandwidth) using functions like `rickerwavelet()` in MATLAB or custom Python implementations with NumPy/SciPy. Reflection coefficients can be generated statistically or based on geological models using random number generators with specific probability distributions. Subsequently, synthetic single-channel seismic records are created through convolutional modeling: seismic_trace = convolution(wavelet, reflection_coefficients) + noise. This process can be efficiently implemented using MATLAB's `conv` function or Python's `numpy.convolve` with optional additive noise simulation. The logarithmic decomposition method (implementable via eigenvalue decomposition or homomorphic filtering techniques) is then applied to extract the wavelet component from the seismic record. This involves using matrix decomposition algorithms like SVD (`numpy.linalg.svd`) or specialized signal processing toolboxes. Finally, deconvolution algorithms (such as Wiener deconvolution, predictive deconvolution, or sparse spike deconvolution) are employed to recover the reflection coefficient sequence. These algorithms can be implemented using inverse filtering approaches with regularization parameters to handle noise sensitivity. The entire workflow often involves iterative refinement cycles where different wavelet types (zero-phase/minimum-phase) and deconvolution parameters are tested to evaluate result stability. This comprehensive process requires careful parameter selection and quality control through metrics like signal-to-noise ratio calculation and residual analysis.