When ${\mathcal{X}}$ is snc, ${\mathcal{O}}_{\mathcal{X}}(K^{\mathrm{log}}_{{\mathcal{X}}/S})$ coincides with the relative logarithmic dualizing sheaf $\omega_{{\mathcal{X}}^{+}/S^{+}}$ of [NX13, (3.2.2)]. When ${\mathcal{X}}$ is regular, ${\mathcal{O}}_{\mathcal{X}}(K_{\mathcal{X}})$ is described in [dFEM11, Appendix A] as the determinant of the locally free sheaf $\Omega^{\prime}_{{\mathcal{X}}/k}\subset\Omega^{1}_{{\mathcal{X}}/k}$ of special differentials, corresponding to derivations $D$ of ${\mathcal{O}}_{\mathcal{X}}$ such that $D(f)=f^{\prime}({t})d{t}$ for $f\in k[\![{t}]\!]$.