  The Hitchin morphism is a map from the moduli space of Higgs bundles
$\mathscr{M}_X$ to the Hitchin base $\mathscr{B}_X$, where $X$ is a smooth
projective variety. When $X$ has dimension at least two, this morphism is not
surjective in general. Recently, Chen-Ng\^o introduced a closed subscheme
$\mathscr{A}_X$ of $\mathscr{B}_X$, which is called the space of spectral data.
They proved that the Hitchin morphism factors through $\mathscr{A}_X$ and
conjectured that $\mathscr{A}_X$ is the image of the Hitchin morphism. We prove
that when $X$ is a smooth projective surface, this conjecture is true for
vector bundles. Moreover, we show that $\mathscr{A}_X$, for any dimension, is
invariant under proper birational morphisms, and apply the result to study
$\mathscr{A}_X$ for ruled surfaces.
