  For a rational function $\phi$ on $\mathbb{P}^1$ defined over an
algebraically closed and complete non-archimedean non-trivially valued field
$K$, we introduce the intrinsic reduction at each point in the Berkovich
projective line and the directionwise intrinsic depth of it. After discussing
the relationship between such an intrinsic reduction of $\phi$ and the
classical reduction of $\phi$ modulo $K^{\circ\circ}$, we give as applications
an intrinsic proof of Rumely's moduli theoretic characterization of the minimum
resultant locus and a direct proof of the degenerating limit theorem of
DeMarco--Faber for a meromorphic family of complex rational functions.
