  Optical potentials for $K^+$-nucleus interactions are constructed from
$K^+$-nucleon amplitudes using recently developed algorithm based on $K^+$-N
kinematics in the nuclear medium. With the deep penetration of $K^+$ mesons
into the nucleus at momenta below 800~MeV/c it is possible to test this
approach with greater sensitivity than hitherto done with $K^-$ and pions. The
energy-dependence of experimental reaction and total cross sections on nuclei
is better reproduced with this approach compared to fixed-energy amplitudes.
The inclusion of Pauli correlations in the medium also improves the agreement
between calculation and experiment. The absolute scale of the cross sections is
reproduced very well for $^6$Li but for C, Si and Ca calculated cross sections
are (23$\pm4$)\% smaller than experiment, in agreement with earlier analyses.
Two phenomenological models that produce such missing strength suggest that the
imaginary part of the potential needs about 40\% enhancement.
