Impedance Contrast Amplification

Seismic impedance is density times shear velocity, $Z = \rho V_s$, and it plays the same role for seismic waves that acoustic impedance plays for sound or optical index plays for light. When a wave crosses a boundary between two impedances, part reflects and part transmits, and the part that transmits changes amplitude. The approximate amplification for a wave passing from stiff material 1 into soft material 2 is $A = \sqrt{Z_1/Z_2}$.

The square root comes from energy bookkeeping. Energy flux through the boundary is conserved, and flux is proportional to $Z$ times amplitude squared. If the impedance on the far side is lower, the amplitude has to rise to carry the same energy through — and because it is the SQUARE of amplitude that appears, the amplitude rises as the square root of the impedance drop. A tenfold impedance contrast, which is entirely ordinary between rock and soft clay, gives roughly a threefold amplification.

This is a different mechanism from the resonance on the site-period page, and the two are frequently conflated. Impedance amplification is broadband and happens at the boundary; resonance is narrowband and happens because the layer has a thickness. In a real soil column they act together, which is why the observed amplification is largest at the site period rather than uniform across the spectrum. The number this equation gives is closer to that peak than to the average.

Three things push the real answer below this estimate. Soil is not elastic: it dissipates energy as it deforms, and material damping alone can cut the amplification substantially in a thick or lossy profile. Strong shaking softens the soil, which reduces $G$ and therefore the contrast. And at very high input levels the soil's shear strength simply caps how much stress it can transmit, so amplification falls away exactly when it would matter most — an effect first documented at Loma Prieta and now built into modern site-factor tables as a reduction at high rock accelerations. Treat this equation as an upper bound and a way of thinking, not as a design number.