  This is the first in a series of papers on scattering theory for
one-dimensional Schr\"odinger operators with highly singular potentials $q\in
H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive
Schr\"odinger operators that admit a Riccati representation $q=u'+u^2$ for a
unique $u\in L^1(R)\cap L^2(R)$. Such potentials have a well-defined reflection
coefficient $r(k)$ that satisfies $|r(k)|<1$ and determines $u$ uniquely. We
show that the scattering map $S:u\mapsto r$ is real-analytic with real-analytic
inverse. To do so, we exploit a natural complexification of the scattering map
associated with the ZS-AKNS system. In subsequent papers, we will consider
larger classes of potentials including singular potentials with bound states.
