Steps of making comparisons of TES nadir retrievals to your profiles with higher vertical resolution

TES nadir retrievals of tropospheric species profiles (and the integrated total or partial columns) are not the same as the true profiles (or the columns). Papers have been published to describe the differences and the relationship between the two, e.g., Rodgers (2000), Rodger and Connor (2003), and comparison examples related to the TES data, e.g., Worden et al. (2006), and Luo et al. (2007). The MOPITT team has also given advices and examples on the related topic, e.g., Deeter (2002), and Emmons et al. (2003). This webpage gives detailed steps dealing with TES data and your data with higher vertical resolution, assuming that you understand the basics.

(1) TES data

TES standard product file (.he5) consists following data fields per profile (e.g., iObs) that you need for your inter-comparison: Pressure[67, iObs], speciesVMR[67, iObs] (e.g., TATM, O₃), ConstraintVector[67, iObs], AveragingKernel[67,67,iObs], where iObs is your selected TES profile, 67 is the number of TES standard pressure levels.

In most cases, the 1^st a couple (a few) pressure levels are filled with -999 (index = 0, 1, ...). The 1st non -999 pressure value represent the surface. You need to 'reform' (in IDL) the TES vectors (Pressure, speciesVMR and ConstraintVector) and the AveragingKernel to smaller number of levles starting from the surface level (numLevel < 67).

(2) The standard equation for mimicing TES retrievals assuming your profile is the 'true' (see the above listed references)

x_result = x_tes_aPriori + AK_tes##(x_your_profile - x_tes_aPriori)

where the corresponding TES data is

x_tes_aPriori= alog(ConstraintVector), and
AK_tes = AveragingKernel

You probably noticed that x_your_profile in the equation must be on the same pressure grids as the TES pressure grids for the corresponding TES species profile. The next section gives you suggestions on how to map and extend (if necessary) your profile to all the TES pressure levels.

It is important to know that the above x_tes_aPriori and x_your_profile need to be 'alog(VMR)' (IDL notation), so 'exp(x_result)' is what you want (except atmospheric temperature). Again, this is uniqe to TES due to that TES does volume mixing ratio (VMR) retrievals in alog(VMR).

(3) Map your profile at fine levels to TES standard pressure levels (number < 67)

There are a few things that you need to consider before doing the mapping.

1. Your species profile needs to be a function of pressure in hPa.
2. In most cases, your species profile needs to cover entire TES pressure range, surface to 0.1 hPa. If your profile covers only a small portion of the atmosphere and do not have anything above or below handy, one suggestion is to append the shifted TES a priori profile down or upward of your profile. This extension offers reasonably realistic atmosphere where you don't have measurements.
3. Your profile is treated as 'true' profile here so it should have realistic fine vertical structures. In some cases however, e.g., model profiles, your profile can be simply interpolated (e.g., lnVMR in lnPressure or to preserve layer column amounts when interpolating) to the TES pressure levels in the troposphere (~25 levels) and adequately reproduced the structures in your profile. In this case, you can ignore what described below.
4. An interpolation of your profile to finer levels, by adding the TES pressure levels to your original pressure levels, may be necessary.

Now you are ready to map your profile at fine levels to the TES pressure levels.

There is a way described in Rodgers 2000, section 10.3.1.

x = the species profile vector on your fine grids; z = your species profile vector mapped to TES levels.
x = Wz, or z = W^*x, where W is the map matrix.

The following IDL codes show you how to do this with eq (10.7) in Rodgers 2000, W^* = (W^TW)^-1W^T:
The IDL pseudo code for calculating the map, W, and the IDL pseudo code for obtain your maped profile, x_your_profile (or z in the equation above).

Final note: it is important to check your interpolated, extended, and mapped profiles via, for example overlaying with the original.

(4) References

Deeter (2002), Calculation and Application of MOPITT Averaging Kernels, available at http://mopitt.eos.ucar.edu/mopitt/data/avg_krnls_app.pdf .

Emmons et al., (2004), Validation of Measurements of Pollution in the Troposphere (MOPITT) CO retrievals with aircraft in situ profiles, J. Geophys. Res., 109, D03309, doi:10.1029/2003JD004101.

Luo et al., (2007), Comparison of carbon monoxide measurements by TES and MOPITT the influence of a priori data and instrument characteristics on nadir atmospheric species retrievals, J. Geophys. Res., 112, D09303, doi:101029/2006JD007663.

Rodgers, C. D., Inverse Methods for Atmospheric Sounding: Theory and Practice, World Sci., River Edge, N. J., 2000.

Rodgers, C. D., and B. J. Connor (2003), Intercomparisons of remote sounding instruments, J. Geophys. Res., 108, 4116, doi:10.1029/2002JD002299.

Worden et al., (2007), Comparisons of Tropospheric Emission Spectrometer (TES) ozone profiles to ozonesondes: Methods and initial results, J. Geophys. Res., 112, D03309, doi:10.1029/2006JD007258.