DEFINITION OF OPERATORS

Let us denote $\bold H$ as the operator of summation along hyperbolas. The operator $\bold H$ transforms (t,x)-space into $(\tau,m)$-space:

$$\bold u = \bold H \bold d.$$ (2)

An adjoint operator $\bold H^{\bold T}$ transforms $(\tau,m)$-space into (t,x)-space:

$$\bold d = \bold H^{\bold T} \bold u.$$ (3)

We can also define a stacking operator in $(\tau,m)$-space. Because $\bold H$ maps a point onto a parabola (Jedlicka, 1989b), it is natural to define an operator $\bold P$ as an operator of summation along parabolas:

$$\bold d = \bold P \bold u.$$ (4)

The adjoint operator $\bold P^{\bold T}$,

$$\bold u = \bold P^{\bold T} \bold d,$$ (5)

may serve as an alternative operator for velocity analysis. Here each point in (t,x)-space is spread onto a parabola in $(\tau,m)$-space. Both operators $\bold H$ and $\bold P$ are linear. The relationship between them is shown in Appendix A.