Infrared absorber sensitivity kernel ($\delta L=\int K_p(p)\delta x(p)dp$)

The derivative of outgoing spectral radiance with respect to a pressure-resolved absorber abundance defines a kernel once the vertical measure is fixed. In an optically thin column with $k\propto p$ and $B/B_0=p/p_0$, $K_p=[k_0{B}_0/(p_{0}gM)]p(p/p_0-1)$. It is negative inside the atmosphere and zero at both endpoints. Its largest magnitude lies at $p_0/2$, whereas the logarithmic-pressure kernel $p{K}_p$ peaks in magnitude at $2p_0/3$. A uniformly perturbed abundance instead has one integrated derivative, not a pressure location.