1. Weighting Functions for Microwave and Infra-Red Satellite Nadir Sounding Remote Sensing I Lecture 7 Summer 2006

2. Radiative Transfer Without Scattering Remember RTE without scattering: (Also known as Schwarzschild equation.) With T(z,z´) the transmission between level z and z´

3. RTE for Satellite Nadir Sounder The RTE without scattering can be solved to give the upward directed radiance at the altitude z´ This is what a satellite nadir sounder in the microwave or infra-red spectral region will observe (for ).

4. Weighting Functions For a nadir looking satellite sensor this can be written as: with the Weighting Functions K(z) defined as: I.e., the measured radiance is determined by the thermal emission from the different levels, weighted by the Weighting Function.

5. Temperature Retrieval If the absorption results from a uniformly mixed compound (such as CO 2 or O 2 ), the absorption coefficient can be assumed to be of the form: with the number density profile n(z) decreasing exponentially with height (scale height H, typically about 7km) Thus:

6. Weighting Functions for Temperature Retrieval If then the transmission will be given by And the Weighting Functions are given by

7. Mathematical Discussion of Weighting Functions The weighting functions have their maximum at z max when At the maximum the weighting function have the value:

8. Idealized Weighting Functions for Temperature Retrieval Scale height H = 7km

9. Idealized Weighting Functions for Temperature Retrieval