Plot of mean branch point density versus propagation distance for r₀ = 10.28 cm and averaged over twenty values of the inner scale from 0.05 to 1.0 cm. Overlaid are z₀, z_ss and z_ρmax. In standard texts (Sasiela 2007), the Rytov approximation is shown to fail at . Empirically, it begins to deviate from the theoretical curve for , but remains reasonably valid until . These are shown as z_ss and z_ρmax, respectively, and the location at which branch points form as z₀.