Accurate Solution for Zoeppritz Equations

The Zoeppritz equations serve as a fundamental theoretical basis in seismic exploration, calculating reflection and transmission coefficients of plane waves at elastic medium interfaces. This program implements an exact numerical solution, providing critical support for AVO (Amplitude Versus Offset) inversion and synthetic seismogram generation.

The core mathematical principle relies on boundary conditions for elastic wave propagation, establishing equations for displacement and stress continuity to solve energy distribution of P-waves and S-waves at interfaces. Compared to traditional approximations (such as Aki-Richards or Shuey approximations), direct solving of Zoeppritz equations yields more accurate reflection coefficients, particularly for large incidence angles or strong impedance contrasts. The implementation employs matrix inversion techniques to solve the 4x4 coefficient matrix system, ensuring numerical precision through proper eigenvalue handling.

In AVO analysis, the program generates reflection coefficient curves for various incidence angles, aiding in hydrocarbon reservoir characterization. For synthetic seismogram creation, it accurately simulates seismic responses at different offsets, providing theoretical references for seismic data interpretation. Users must input four parameters for upper and lower media: P-wave velocity, S-wave velocity, and density. The code automatically handles numerical stability issues including complex root selection through robust algorithm design that checks for critical angle conditions and implements proper branch selection for complex-valued solutions.

This implementation accounts for complex propagation angles beyond critical angles, correctly handling wave mode conversions and total reflection phenomena. The output clearly displays amplitude variation characteristics for reflected PP-waves, PS-converted waves, and transmitted waves, providing quantitative basis for reservoir fluid identification and lithology interpretation. The code structure includes modular functions for parameter validation, matrix formulation, and solution verification, ensuring reliable results across diverse geological scenarios.