The complete $\grave{P}\!\acute{P}(\bar{\theta})$ estimate

To complete the final estimation of $\grave{P}\!\acute{P}(\bar{\theta})$, we make a simple set of mappings from the least-squares (or l₁) estimates derived in the previous sections. A map of $\bar{\theta}({\bf x};{\bf x}_h)$ can be obtained directly from the result of (102) or (103):

 

$${\cal M}_1 \;\; : \;\; \cos\ 2\bar{\theta}({\bf x};{\bf x}_h) \rightarrow \bar{\theta}({\bf x};{\bf x}_h) \;.$$ (104)

Then, since there is a one-to-one mapping of $\bar{\theta}$ to any point $({\bf x};{\bf x}_h)$, and $\grave{P}\!\acute{P}$ to the same point $({\bf x};{\bf x}_h)$, there is a unique map of $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ and $\bar{\theta}({\bf x};{\bf x}_h)$ to $\grave{P}\!\acute{P}(\bar{\theta}({\bf x}))$ such that:

 

$${\cal M}_2 \;\; : \;\; \{ \grave{P}\!\acute{P}({\bf x};{\bf ... ...\} \rightarrow \grave{P}\!\acute{P}(\bar{\theta}({\bf x})) \;,$$ (105)

which is the desired result. This completes the angle-dependent reflectivity estimation process.