Total (σ T ) and reaction (σ R ) cross sections are derived self consistently from the attenuation cross sections measured in transmission experiments at the AGS for K + on 6 Li, C, Si and Ca in the momentum range of 500-700 MeV/c by using a V opt = t ef f (ρ)ρ optical potential. Self consistency requires, for the KN in-medium t matrix, that Im t ef f (ρ) increases linearly with the average nuclear density in excess of a threshold value of 0.088±0.004 fm −3 . The density dependence of Re t ef f (ρ) is studied phenomenologically, and also applying a relativistic mean field approach, by fitting to the σ T and σ R values. The real part of the optical potential is found to be systematically less repulsive with increasing energy than expected from the free-space repulsive KN interaction.When the elastic scatttering data for 6 Li and C at 715 MeV/c are included in the analysis, a tendency of Re V opt to generate an attractive pocket at the nuclear surface is observed.P ACS: 25.80. Nv; 13.85.Lg; 21.30.Fe Keywords: K + nucleus total and reaction cross sections 500-700 MeV/c on 6 Li, C, Si, Ca; Self consistent analysis; Medium effects. * Dedicated to the memory of Carl Dover, a friend, colleague and scholar. 0