The purpose of model testing is to investigate the program performance. As an analytical solution to the radiative transfer equations does not exist under realistic conditions, a standard

80.5 80.75 81 81.25 81.5
υ [cm^{-1}]

1e-04 2e-04 3e-04 4e-04 5e-04 6e-04

d I / d O 3 [erg / (s cm2 sr cm-1 ) / ppmv]

Automatic Diff.: 271.87 s Finite Diff. (+ 0.5%): 27 x 89.57 s Finite Diff. (+ 50%)

*(derivatives w.r.t. 26 altitude levels, 11 spectra)*

Jacobian @ 22 km (Aura/MLS; tangent: 12 km)

### (a)

4 4.5 5 5.5 6

*f*_{IF} [GHz]

0 1e-11 2e-11 3e-11

d *I* / d OH [W / (m2 sr Hz) / ppmv]

Automatic Diff.: 65 s Finite Diff.(+ 0.1%): 24 x 34 s Finite Diff.(+ 50%)

*(derivatives w.r.t. 23 altitude levels, 10 spectra)*

Jacobian @ 21 km (TELIS f_{LO} = 1830.10 GHz; tangent:19 km)

### (b)

Figure 3.3: Results of automatic differentiation versus finite differences. Panel (a): comparison of
partial derivatives with respect to the O_{3}volume mixing ratio in a typical wavenumber microwindow of
Aura/MLS. The plotted derivatives are evaluated at an altitude level of 22 km and for a tangent altitude
of 12 km. Panel (b): comparison of partial derivatives with respect to the OH volume mixing ratio in
a typical frequency microwindow of TELIS. The plotted derivatives are evaluated at an altitude level of
21 km and for a tangent altitude of 19 km.

way of verifying the mathematical/numerical performance of radiative transfer models relies on a cross-checking against similar models. Several intercomparisons between GARLIC and other radiative transfer codes have been (or are currently being) performed, as mentioned in Sect. 3.2.

In addition, an extensive intercomparison of forward calculations in conjunction with TELIS configurations has been carried out. The forward models are the two Level-2 data analysis

Table 3.3: Forward model parameters and input files for the intercomparison of the two Level-2 data processing codes PILS and AdL.

Forward model parameters and input files Description

Sideband ratio 0.7–0.8

Pointing offset −4.5 arcmin

Temperature profile MIPAS-B retrievals

Pressure profile ECMWF

Major gases (O_{3}, HCl, and ClO) MIPAS-B and MLS profiles
Remaining interfering gases AFGL subarctic winter model
Spectroscopic line parameters HITRAN 2004

20.82 20.84 20.86 20.88

υ [cm^{-1}]
0

1 2

Rel. diff. [%]

8e-20 1.2e-19 1.6e-19 2e-19

*k**m* [cm2 ]

PILS AdL

(a)

20.82 20.84 20.86 20.88

υ [cm^{-1}]
0

1 2

Rel. diff. [%]

3e-11 4e-11 5e-11 6e-11 7e-11

α [cm-1 ]

PILS AdL

(b)

Figure 3.4: Comparison of (a) absorption cross sections and (b)absorption coefficient for one HCl line at the altitude level of 10 km. The results correspond to the two Level-2 data processing codes PILS and AdL, and both quantities are expressed as a function of wavenumber.

programs developed by DLR (PILS) and SRON [de Lange et al., 2009, 2012] (hereafter referred to AdL), respectively. For model testing, a frequency microwindow (fLO = 619.1 GHz, fIF = 5–7 GHz) covering HCl lines of both isotopes is selected. The comparison procedure consists of line-by-line and radiative transfer calculations. To avoid any discrepancies stemming from instrumental, atmospheric, and spectroscopic parameters (from external sources), both forward models make use of the identical input parameters (see Table 3.3).

3.4.1 Monochromatic spectra: HCl only

It is essential to start with the simplest possible case, before moving on to more complex cases.

The first exercise is to compare the following quantities by taking into account only one HCl
transition line that is located at 20.8470 cm^{−1}:

• absorption cross sections,

• absorption coefficients,

• monochromatic pencil beam spectra with respect to single- and double sideband modes.

5 5.5 6 6.5 7
*f*IF [GHz]

-0.2 -0.1 0 0.1 0.2

Abs. diff. [K]

0 10 20 30 40

Intensity [K]

PILS
AdL
*f*LO = 619.1 GHz; tangent: 10 km; single sideband (USB)

### (a)

5 5.5 6 6.5 7

-6 -4 -2 0 2 4

Rel. diff. [%]

0 5e-16 1e-15

Radiance [W / (m2 sr Hz)] PILS

AdL
*f*LO = 619.1 GHz; tangent: 10 km; double sideband

### (b)

Figure 3.5: Comparison of monochromatic spectra for one HCl line at a tangent height of 10 km.

Upper panel(a): spectra in brightness temperature units and for the upper sideband. Lower panel(b):

radiance spectra in the double sideband mode.

Figure 3.4 depicts the comparison of absorption cross sections and absorption coefficients as
functions of wavenumber. Overall, the quantities computed by PILS are slightly larger than
that by AdL, with a maximum 1.5 % relative difference. It has been identified that the
differ-ences between the two forward modules are mainly due to the different temperature conversion
schemes of the line strength (values of total partition functions) and the values of the molecular
mass of H^{37}Cl.

Figure 3.5 shows the monochromatic pencil beam spectra evaluated at a tangent height of 10 km computed by the two radiative transfer models. The upper panel (a) shows the spectra in brightness temperature units and for the upper sideband mode, while the lower panel (b) shows the radiance spectra in the double sideband mode. The absolute difference for the single sideband spectra ranges roughly from−0.2 to 0.2 K. For the double sideband spectra, the largest relative difference is about 5 % and corresponds to the peak value of the radiance around the intermediate frequency of approximately 5.9 GHz.

3.4.2 TELIS-like spectra: all absorbers

A complete forward model comprises both atmospheric radiative transfer and sensor character-istics. The second exercise is to compare an entire TELIS-like limb sequence by convolving the monochromatic radiance spectra with the dedicated ILS function for the 480–650 GHz channel.

The spectra covering the tangent heights between 10 and 32.5 km (equidistant spacing: 1.5 km) are plotted in Fig. 3.6. For this exercise, the real ILS function for the TELIS 480–650 GHz chan-nel and all relevant molecules are considered so that the modelled spectra bear resemblance to the actual observations. Furthermore, a pencil beam is assumed (no FoV convolution), while the refraction effect and additional instrumental features (standing waves) are neglected.

At lower tangent heights (10 and 11.5 km), the discrepancies between the two forward models
appear to be almost constant over the frequency range. When the tangent height increases,
the differences between the two models mostly occur at the intermediate frequency points of
approximately 5.9, 6.3, and 6.8 GHz where the emission line centers of HCl and O_{3} are located.

In the line wings where the absorption coefficient is very low, the different continuum models chosen by the two models result in differences. However, the differences in the continuum models may not be crucial for trace gas retrievals, because the continuum absorption can be included in the retrieval. Other factors for causing the differences in the spectra are the different interpolation approaches (atmospheric parameters as a function of altitude) and the integration of the radiative transfer equation.

In this section, we have presented a set of intercomparisons of radiative transfer results computed by the forward modules of two Level-2 retrieval codes (PILS and AdL). The model configurations with respect to the spectral range and the observing geometry are based on a submillimeter microwindow observed by TELIS’s 480–650 GHz channel during the 2010 flight.

For one HCl line, discrepancies in the monochromatic spectra mostly stem from differences in the partition function and the value of the molecular mass. The TELIS-like spectra corre-sponding to an entire limb sequence show that PILS delivers accurate spectra by taking into account the instrument characteristics, which is fundamental to the inversion process.

5 5.5 6 6.5 7
*f*IF [GHz]

-0.9 -0.8 -0.7

Abs. diff. [K]

155 160 165 170

Intensity [K]

PILS AdL

5 5.5 6 6.5 7

*f*IF [GHz]

-1 -0.9 -0.8

Abs. diff. [K]

175 180 185 190

Intensity [K]

PILS AdL

### (a) (b)

*f*LO = 619.1 GHz; tangent: 10 and 11.5 km

5 5.5 6 6.5 7

*f*IF [GHz]

-0.8 -0.7 -0.6

Abs. diff. [K]

110 120 130 140

Intensity [K]

PILS AdL

5 5.5 6 6.5 7

*f*IF [GHz]

-0.9 -0.8 -0.7

Abs. diff. [K]

120 130 140 150

Intensity [K]

PILS AdL

### (c) (d)

*f*LO = 619.1 GHz; tangent: 13 and 14.5 km

5 6 7

*f*IF [GHz]

-1 -0.8 -0.6

Abs. diff. [K]

60 80 100 120

Intensity [K]

PILS AdL

5 6 7

*f*IF [GHz]

-1 -0.8 -0.6

Abs. diff. [K]

80 100 120 140

Intensity [K]

PILS AdL

### (e) (f)

*f*LO = 619.1 GHz; tangent: 16 and 17.5 km

Figure 3.6: Comparison of modelled TELIS radiance spectra for a GHz-channel HCl microwindow.

The results correspond to the two Level-2 processing codes PILS and AdL. The local oscillator frequency fLO is set to 619.1 GHz and fIF ranges from 5 to 7 GHz. For each pair of spectra, the corresponding residual in terms of absolute difference is shown in the lower panel. The plotted spectra are given in equivalent brightness temperature units. The comparisons are done for tangent heights of (a)10, (b) 11.5,(c)13,(d)14.5,(e) 16, and(f )17.5 km.

5 6 7
*f*IF [GHz]

-0.6 -0.4 -0.2

Abs. diff. [K]

20 40 60 80 100

Intensity [K]

PILS AdL

5 6 7

*f*IF [GHz]

-1.2 -0.8 -0.4

Abs. diff. [K]

40 60 80 100 120

Intensity [K]

PILS AdL

### (g) (h)

*f*LO = 619.1 GHz; tangent: 19 and 20.5 km

5 5.5 6 6.5 7

*f*IF [GHz]

0 0.2 0.4 0.6 0.8

Abs. diff. [K]

20 40 60 80 100

Intensity [K]

PILS AdL

5 5.5 6 6.5 7

*f*IF [GHz]

0 0.2 0.4 0.6

Abs. diff. [K]

20 40 60 80 100

Intensity [K]

PILS AdL

### (i) (j)

*f*LO = 619.1 GHz; tangent: 22 and 23.5 km

5 5.5 6 6.5 7

*f*IF [GHz]

0 0.4 0.8 1.2

Abs. diff. [K]

20 40 60 80

Intensity [K]

PILS AdL

5 5.5 6 6.5 7

*f*IF [GHz]

0 0.2 0.4 0.6 0.8

Abs. diff. [K]

20 40 60 80

Intensity [K]

PILS AdL

### (k) (l)

*f*LO = 619.1 GHz; tangent: 25 and 26.5 km

Figure 3.5: Continued from the previous page. The comparisons are done for tangent heights of (g) 19,(h)20.5,(i)22,(j)23.5,(k) 25, and(l)26.5 km.

5 5.5 6 6.5 7
*f*IF [GHz]

0 1 2

Abs. diff. [K]

20 40 60 80

Intensity [K]

PILS AdL

5 5.5 6 6.5 7

*f*IF [GHz]

0 1 2

Abs. diff. [K]

20 40 60 80

Intensity [K]

PILS AdL

### (m) (n)

*f*LO = 619.1 GHz; tangent: 28 and 29.5 km

5 5.5 6 6.5 7

*f*IF [GHz]

0 1 2 3

Abs. diff. [K]

10 20 30 40 50 60

Intensity [K]

PILS AdL

5 5.5 6 6.5 7

*f*IF [GHz]

0 1 2 3

Abs. diff. [K]

10 20 30 40 50 60 70

Intensity [K]

PILS AdL

### (o) (p)

*f*LO = 619.1 GHz; tangent: 31 and 32.5 km

Figure 3.4: Continued from the previous page. The comparisons are done for tangent heights of (m) 28,(n)29.5,(o)31, and(p)32.5 km.

### Chapter 4

## Inversion Methodology

Inverse problems arising in atmospheric remote sensing aim to estimate certain atmospheric state parameters based on indirect measurements (spectra) of these parameters. These problems are nonlinear, and mostly ill-posed in the sense that the noise in the data produces large errors in the state vector. The inversion frequently explores the simultaneous retrieval of several gas concentration profiles and optionally of some auxiliary (instrumental/geophysical) parameters, and works out the underlying multi-component problem by means of regularization.

In this chapter, the theoretical and practical aspects of the direct and iterative regulariza-tion methods for solving nonlinear inverse problems are presented. To assess the numerical performances of the methods, an example using TELIS submillimeter spectra is considered.