Sets of equations and unknowns

Our system of equations is constituted by collecting relations (4), (6), (7), (8), and (13):  

$$\left\{ \begin{array} {ccc} \xi(p_g,t_g)\frac{{\bf p}_g}{p... ... p}_s + {\bf p}_g) \cos \theta(p_0,t_0) \end{array} \right. .$$ (14)

Because the ${\bf x}$ and ${\bf p}$ vectors are two-dimensional, the first, second, and sixth relations of system (14) give $3\times2$ equations, yielding a set of nine equations.

The unknowns are ${\bf p}_s$, ${\bf p}_g$, t_s, t_g, t₀, and ${\bf x}_0$. As described in the preamble, the ${\bf p}$ vectors have only two unknown parameters, p_x and p_y (or p and $\phi$). Similarly, ${\bf x}_0$ is a two-dimensional unknown vector. Therefore, system (14) relates nine unknowns with the known parameters ${\bf p}_0$, t_sg, ${\bf x}_s$, ${\bf x}_g$, and the ray tracing tables $\xi(p,t)$, $\tau(p,t)$, and $\theta(p,t)$.