Mode Conversion

Four Waves from One

At normal incidence a P-wave meeting a boundary makes only two waves: a reflected P and a transmitted P, both still compressional. This is the acoustic world the earlier parts lived in. Tilt the incidence off vertical and the elastic world opens up: part of the P energy converts to shear, adding a reflected S-wave and a transmitted S-wave. One incident wave becomes four.

The Angle Runs the Split

The ray fan shows the geometry. Each outgoing wave leaves at its own Snell angle, and the shear rays bend less than the compressional ones because shear travels more slowly. Crucially, the converted S-rays are invisible at normal incidence and grow as the angle opens: mode conversion is fundamentally an angle effect. As the energy redistributes among four waves, the amplitude of the reflected P-wave, the one a normal survey records, changes with angle.

That change is drawn exactly on the right, the Zoeppritz reflected-P amplitude as a function of incidence angle. Zoeppritz is the full, exact solution of the elastic boundary conditions, and this amplitude-versus-angle variation is the entire signal of AVO, amplitude versus offset. A convolutional model, using a single normal-incidence coefficient, is blind to it. The elastic reflection is a curve, not a number, and reading that curve is how a synthetic can distinguish a gas sand from a wet one. The next section replaces the exact but unwieldy Zoeppritz with the simple Shuey approximation and shows exactly where it holds and where it breaks.