The outgoing radiance or upward radiance, R^u, for channel i from an atmospheric column is given by

R_i^u = B_i(T(p)) · ( ^u(p)/ ln p) d ln p
The weighting function is defined as

W(p) ^u(p)/ ln p
and the contribution function is the integrand of the outgoing radiance calculation

C(p) B_i(T(p)) · ( ^u(p)/ ln p)
and can be normalized as

C(p) B_i(T(p))/R_i^u · ( ^u(p)/ ln p)
the information content will require derivative w.r.t. temperature of the radiation transfer equation and, therefore, the Kernel function represents the best estimate of the sounding level for temperature retrievals.

K(p) ( B_i(T(p))/ T) · ( ^u(p)/ ln p)