Weighting Functions

SASKTRAN-TIR supports the analytic computation of weighting functions for concentrations of gas species and temperature. It is possible to compute weighting functions for a single parameter or for multiple parameters simultaneously. Weighting functions computed for a single parameter have the form:

\[\mathrm{wf[i, j, k]} = \frac{\partial I(\lambda_\mathrm{i}, \mathrm{L_j})}{\partial \mathrm{S_k}}\]

while those computed for multiple species are given by:

\[\mathrm{wf[i, j, n, k]} = \frac{\partial I(\lambda_\mathrm{i}, \mathrm{L_j})}{\partial \mathrm{S_{n, k}}}\]

where

Symbol	Meaning
\(\lambda_\mathrm{i}\)	Wavelength \(\mathrm{i}\)
\(\mathrm{L_\mathrm{j}}\)	Line of sight \(\mathrm{j}\)
\(\mathrm{S_\mathrm{n, k}}\)	Species \(\mathrm{n}\) at altitude \(\mathrm{k}\)

To calculate weighting functions for a gas species the sasktran.Atmosphere.wf_species must be set. To compute temperature weighting functions set the sasktran_tir_engine.EngineTIR.do_temperature_wf option to True.

Weighting Function Units

By definition, SASKTRAN-TIR weighting functions have units of \(\mathrm{radiance / species}\). The radiance portion has units of \(\mathrm{photon}/(\mathrm{s \: cm^2 \: nm \: sr})\). The units of the weighting function species are configurable. By default, weighting functions are given for gas number density, in \(\mathrm{cm}^{-3}\). To calculate gas species weighting functions with respect to Volume Mixing Ratio (VMR) set the sasktran_tir.engine.EngineTIR.do_vmr_wf option to True.

Multiple Species Calculations

When weighting functions are computed for more than one species, an extra dimension is added to the weighting function array. The index \(n\) of each species are assigned in the same order as the species were listed when setting sasktran.Atmosphere.wf_species. If temperature weighting functions are also being computed, they are assigned the highest species index.

For example, consider a SASKTRAN-TIR engine object with the weighting function species configured as follows:

engine.atmosphere.wf_species = ['CO2', 'O3']
engine.do_temperature_wf = True
rad, wf = engine.calculate_radiance()

The CO2 weighting functions will be given by wf[:, :, 0, :], the O3 weighting functions by wf[:, :, 1, :], and the temperature weighting functions by wf[:, :, 2, :].