A production pitfall: weak instability from v(x)

Some quality problems cannot be understood in the Fourier domain. Unless carefully handled, lateral velocity variation can create instability.

The existence of lateral velocity jumps causes reflections from steep faults. A more serious problem is that the extrapolation equations themselves have not yet been carefully stated. The most accurate derivation of extrapolation equations included in this book so far was done from dispersion relations, which themselves imply velocity constant in x. The question of how a dispersion relation containing a $v\,k_x^2$ term should be represented was never answered. It might be represented by $v(x,z) \partial_{xx}$,$\partial_x v(x,z) \partial_x$,$\partial_{xx} v(x,z)$ or any combination of these. Each of these expressions, however, implies a different numerical value for the internal reflection coefficient. Worse still, by the time all the axes are discretized, it turns out that one of the most sensible representations leads to reflection coefficients greater than unity and to numerical instability.

A weak instability is worse than a strong one. A strong instability will be noticed immediately, but a weak instability might escape notice and later lead to incorrect geophysical conclusions. Fortunately, a stability analysis leads to a bulletproof method at the end of chapter .