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

An infrared absorber sensitivity kernel is a functional derivative of outgoing spectral radiance with respect to a resolved absorber abundance, holding the physical temperature profile fixed. Specify the vertical measure: for pressure $p$ increasing downward from zero to surface value $p_0$,Here $x$ is absorber mole fraction. Under hydrostatic balance, take mean mass $M$ per atmospheric particle and gravitational acceleration $g$. An absorption cross section $\sigma (p)$ gives $d\tau =\sigma (p)x(p)dp/(Mg)$.

Consider a vertical, optically thin non-scattering column in local thermodynamic equilibrium, above a surface of spectral radiance $B_0$. Stipulate $\sigma (p)={\sigma }_{0}p/p_0$ and a thermal source function $B(p)=B_{0}p/p_0$. Expanding the formal transfer solution givesThe kernel is negative inside the column and zero at both endpoints: adding absorber replaces warmer surface light with cooler atmospheric emission. Its magnitude per unit pressure peaks at $p_0/2$. Changing measure to $d\log p=dp/p$ gives $K_{\log p}=p{K}_p$, whose magnitude peaks at $2p_0/3$. A uniform abundance change has the integrated response $dL/dx=\int K_{p}dp$, rather than a unique pressure of action.