4. Estimation of Retrieval Sensitivity to Instrument Line Shape

The aim of the work described in this chapter is to provide estimates of the degree to which common aberrations in the instrument line shape may affect vertical profile retrievals, and of the degree of accuracy required of any measurement of the actual line shape used in profile retrieval work.

When retrieving total column amounts or vertical distributions of absorbing gases it is common practice to assume that the response of the spectrometer either conforms fully to the mathematical theory (chapter 3) or deviates from it only in some simple way, such as assuming trapezoidal apodisation¹ of the interferogram rather than the boxcar shape of the unapodised interferogram (section 3.4.1). It is well known that this assumption, although useful, is seldom accurate as it is believed that the true instrument line shape may vary between successive measurements (Park [1983], NPL internal report).

There already exist a number sophisticated of retrieval programmes in the community; one such is SFIT, developed by Curtis Rinsland at the NASA Langley Research Centre. This is a non-linear least squares spectral fitting routine which retrieves total column values from measured infrared spectra. The programme is based on a twenty-nine layer forward model and generates synthetic spectra including up to fifteen gases in the spectral region of interest.

The spectrum from the forward model is convolved with a calculated instrument line shape which includes the effects of finite optical path difference, field of view and apodisation function. To cope with any otherwise unaccounted for loss of resolution an additional parameter, "effective apodisation function", may be fitted. This is, in effect, a trapezoidal apodisation function applied to the interferogram. In addition to the above, other parameters such as the 100% transmission level, wavelength shift, zero signal level and channel spectra, may be fitted or constrained. The retrieval process itself involves simulating spectra using the above parameters and scaling the a priori profile for the gas to be fitted by a constant amount at all altitudes.

To improve on such as this is beyond the scope of this work (in view of its well established nature) and to develop a similar retrieval scheme would be of little benefit. A more simple approach has therefore been adopted, tailored to the declared aims of this thesis.

4.1 Forward Model

4.1.1 Theory and Approximations

In order to investigate the importance of accurate knowledge of the instrument line shape it is necessary to remove other potential sources of uncertainty wherever possible and so it was decided to carry out the investigation using wholly synthetic data. In order produce a synthetic spectrum it is necessary to devise a scheme to simulate all the significant optical properties of the Earth's atmosphere. Such a scheme is known as a 'forward model'.

4.1.2 Radiative Transfer

As sunlight passes through the Earth's atmosphere it is be subject to a variety of effects, such as refraction and scattering, in addition to the absorption that we are interested in; these shall now be considered. As it is only possible to make stratospheric measurements in clear conditions, the effects of cloud on radiative transfer shall not be considered.

4.1.3 Refraction and Geometry

As the density of the atmosphere varies as a function of height so will its refractive index. The effect of this is that radiation entering the atmosphere at any non-zero zenith angle will be continually refracted and hence follow a curved path until either leaving the atmosphere again or reaching the ground (assuming no absorption or scattering). For the purposes of this work we consider only the case of radiation normally incident on the atmosphere and so may ignore the effects of refraction. The insistence on normal incidence also serves to avoid the more involved geometry associated with tracking the path of radiation through a spherical atmosphere, since it has the effect of making it totally accurate to treat the atmosphere as planar.

4.1.4 Vertical Structure

It is obvious that the properties of the atmosphere change with height and that these changes are gradual. The most obvious way to accurately represent this numerically would be as a large number of very thin homogeneous layers where the value assigned to each property of interest in each layer is equal to the value of that property in the real atmosphere at the height of the mid point of that layer. While such a scheme could be sufficiently accurate for most purposes, the large number of layers necessary to achieve the required accuracy would carry an unacceptably high computational burden.

If the number of layers is to be reduced sufficiently then their thickness must be increased and, because of the variations in the atmosphere with height, care must then be taken in the manner in which property values for each layer are chosen. The Curtis-Godson approximation (Kyle [1993]) provides values for atmospheric parameters such that the effects of transmission through an inhomogeneous atmosphere may be approximated by an atmosphere consisting of a small number of homogeneous layers (Houghton [1986]). It may be expressed mathematically as

(4.1)

where q_z is the parameter of interest at altitude z, r_z is the density of the atmosphere, and q_eff is the effective value of q over the height range z₁ to z₂ (Liou [1980]).

4.1.5 Scattering Processes

Radiation passing through the atmosphere may be scattered by its constituents; this may take one of two forms.

When the size of atmospheric particles with which the incoming radiation interacts is close to, or greater than, the wavelength of the radiation Mie scattering becomes significant. The equations describing Mie scattering may be derived by the solution of Maxwell's equations for the interaction of electromagnetic radiation with a spherical particle (Kyle [1993]), although the theory may be extended to non-spherical particles. It may be shown that Mie scattering will show very little variation with wavelength when, as is usual in the atmosphere, there is a large distribution of sizes amongst the scattering particles. Since we are interested only in relative absorption in a very narrow spectral window around the R1 line of hydrogen chloride it follows that such broad-band effects, which serve only to shift the base line uniformly, are not significant.

The theory of Rayleigh scattering describes the interaction of light with particles whose sizes are small compared to the wavelength of the radiation. Although it is usually considered as a separate entity, and indeed was proposed earlier, Rayleigh scattering may be considered as the small particle (or long wavelength) limit of Mie scattering (Kyle [1993]).

The scattering coefficient was expressed by Rayleigh (J. W. Strutt, 3rd baron) as

(4.2)

where k is the scattering coefficient, n the refractive index of the medium and N is the number density of the scattering centres (Ditchburn [1953]). From this l^-4 dependence it follows that Rayleigh scattering is most significant at short wave lengths. For the purposes of this study at 2926 cm^-1, it is negligible.

4.1.6 Absorption and Emission

Considering a single homogeneous layer of atmosphere: at any relevant altitude the absorption line will be both Doppler broadened and Lorentz (pressure) broadened. These combine to produce the measured Voigt line shape (Armstrong & Nichols [1972]). The Voigt absorption line may be calculated as follows (after Liou [1980]).

The Lorentz broadened component of the line may be described by

(4.3)

where k_sL denotes the normalised absorption coefficient for a Lorentz broadened line, s₀ the frequency of the line's centre, a_L the half width at half maximum of the line and S the line strength as defined by

(4.4)

The Lorentz half width a_L is quoted in the literature (Rothmann et al., [1996]) under standard conditions (in the case of the HITRAN 96 data: 296 K and 1013.3 hPa) but this may be converted to a value more appropriate for any given temperature and pressure (T, P) using

(4.5)

where a_L0 denotes a_L at temperature T₀ and pressure P₀. It should be noted that the temperature dependence stated here (T^-0.5) is that predicted by kinetic theory and hence is correct only for ideal gases. Although more rigorous theories show that the temperature dependence is actually dependent on the properties of the molecules involved, this approximation was deemed adequate for the work described in this text.

The Doppler broadened component of the line may be described by

(4.6)

where k_sD denotes the normalised absorption coefficient for a Doppler broadened line, S the line strength and a_D is a measure of the width of the line defined by

(4.7)

where c denotes the velocity of light in a vacuum and k Boltzmann's constant their usual meanings and m denotes the mass of the HCl molecule. Note that aD is not the half width at half maximum of the line, this is given by

The Lorentz and Doppler profiles for the layer are calculated according to the above and then the Voigt profile is formed by convolving the two and normalising such that

The normalised absorption coefficient for the Voigt profile, k_s(z), of the layer, the path length, x(z), through that layer (recall that we assume zero zenith angle) and the appropriate number density of HCl in the layer, n(z), are inserted into the Beer/Lambert/Bouger law:

(4.8)

where I_0s denotes the radiation spectrum prior to any atmospheric absorption taking place and I_s describes the spectrum resulting from absorption by HCl.

The foregoing is a simplification in that there is a small frequency shift in the line centre, s₀, associated with pressure (Bernath [1995]) which has been neglected, as has the fact that the line strength, S, shows some variation with temperature (Bernath [1995]). The former may be ignored as the effect is small and the latter has not been included in the forward model as it would merely add complexity to the sensitivity study without affecting the results.

It should also be noted that any absorber will also be emitting radiation according to its emissivity and temperature. This, however, is negligible in this case as the absorption measurements are taken using the sun as a source. It can be shown that if the Sun and the Earth's atmosphere are approximated by black bodies at 5800 K and 300 K, respectively, then at 2926 cm^-1 the radiance of the Sun is some six orders of magnitude greater than that of the atmosphere.

4.1.7 Implementation

A forward model was constructed to simulate the R1 absorption line of hydrogen chloride in the atmosphere. The model assumes that hydrogen chloride is the only atmospheric absorber and uses as the basis of its standard profile one measured as part of the Halogen Occultation Experiment². Standard temperature and pressure profiles, taken from the U.S. Standard Atmosphere 1962 were used in addition to the HalOE HCl profile.

The mixing ratio was assumed to tend towards zero below the lowest measurement of the HalOE profile (about fifteen kilometres), and to be equal to zero below eight kilometres. Above the highest measurement (about sixty kilometres), the volume mixing ratio was assumed to remain constant at the value of that highest measurement up to an altitude of 100 km; we assume that the atmosphere ends here. The resulting vertical profile is shown in figure 4-1.

Figure 4-1. Vertical volume mixing ratio profile used in forward model. The mixing ratio tends to zero below about 15 km, is equal to zero below 8 km and remains constant between about 60 and 100 km (section 4.1.7).

Although these profiles, taken together, do not necessarily give an accurate representation of the real atmosphere at any particular time, they are sufficiently close to that which one might expect to measure to permit this work to proceed.

The solar radiation outside the atmosphere was assumed to have a 'white' (having equal intensity at all frequencies) spectrum across the relevant spectral window and so the spectrum at ground level, in the absence of hydrogen chloride, will likewise be 'white'. Since the R1 line of HCl lies in the wings of a saturated methane absorption band this assumption is not correct but was deemed reasonable on the grounds that the model is only used over 0.05 cm^-1, and since the baseline, although sloping, is effectively linear over this range, the correction is trivial.

For the forward model, the atmosphere defined by the three aforementioned profiles was divided into one hundred layers each of 1 km in thickness. The unattenuated solar radiation is inserted into the Beer's law equation (4.8) along with the relevant parameters for the top layer of the model. The result (I_s) of this represents the effect on the solar spectrum of the absorption occurring in the top 1 km of the atmosphere and is used as the input radiation (I_s0) in the next layer down. The result of continuing this process through the whole profile is to produce the required simulation of the absorption spectrum that might be measured by a ground based instrument. This can then be convolved with a suitable instrument line shape, either theoretical or measured, to provide a simulation of a FTS measurement.

The values of pressure, temperature and number density used in the forward model for each of the layers are calculated from the standard profiles using the Curtis-Godson approximation as described above.

4.2 Retrieval Scheme

For the purposes of estimating the sensitivity of profile retrieval to variations in the instrument line shape away from the assumed value, the following scheme was adopted.

The aforementioned standard profile was divided into four layers with boundaries at fifteen, thirty, and fifty kilometres. The top and bottom layers were constrained and only the middle two (henceforth referred to as the 'upper' and 'lower' layers) permitted to vary. In addition to this, the total column amount, which is known to vary little outside of the polar vortices (Paton-Walsh, private communication), was constrained to that of the standard profile (4.5x10¹⁵ molecules/cm²). It is therefore obvious that the only information being retrieved in this scheme refers to the exchange of column amount between the upper and lower layers.

The actual retrieval was carried out by using this four layer atmosphere in conjunction with the forward model (section 4.1) in a non linear least squares fitting routine written in the Matlab scripting language (using the 'leastsq' non linear least squares minimisation function from the 'Optimisation Toolbox'). This would repeatedly compare the simulated measurement to the line produced by the forward model using the four layer atmosphere at each frequency grid point (0.00167 cm^-1 spacing, to match that of the Bruker 120M), varying amounts of HCl in the upper and lower layers at each iteration, until a 'best fit' of the retrieval line to the measurement was achieved. The best fit was determined by finding the profile which corresponded to the difference (in a least squares sense) between the two absorption lines being a minimum³.

Since, as shall be shown in subsequent sections, common deviations of the instrument line shape from the assumed value give the appearance of HCl moving from the upper to the lower layer, the figure quoted in what follows shall refer to '% subsidence', by which we mean that percentage of the partial column of hydrogen chloride present in the upper layer which has been transferred to the lower layer.

Although this scheme is more simplistic than any which might be considered useful in the field, for current purposes it has the distinct advantage of being totally constrained, and the results obtained are therefore not susceptible to uncertainties which might be caused by the use of a priori information (Mellqvist [1997], Liu [1996], Rodgers [1976]), nor does the scheme itself become unstable in the presence of realistic noise levels (section 4.3).

4.3 Effects of Noise

In order to establish the effects of noise on the retrieval, the scheme was run one thousand times at each of forty-one noise levels ranging from zero up to 0.01 times the background intensity; the results are shown in figures 4-2 and 4-3. In each case the same instrument line shape was used in the retrieval as was used in the simulation of the line, each corresponding to a perfectly aligned instrument operating at 180 cm maximum optical path difference and 0.65 mm aperture diameter, with boxcar apodisation.

Figure 4-2 shows a plot of the mean subsidence obtained from the retrievals at each noise level. The data points can be seen to be spread randomly about zero as might be expected. The solid line in the plot is a least squares best fit (constrained to pass through the origin) to the data and can be seen to lie almost directly on top of the x-axis (dotted).

Figure 4-2. Mean retrieved subsidences as a function of r.m.s. noise level on simulated measurement (section 4.3).

Figure 4-3 shows a plot of the standard deviations associated with the mean subsidence shown in the previous figure. They can be seen to increase linearly with the noise level and the best fitting straight line passes though the origin, as might be expected.

Figure 4-3. Standard deviations of retrieved subsidences as a function of r.m.s. noise level on simulated measurement (section 4.3).

These serve to confirm that the retrieval scheme is indeed stable in the presence of noise.

Summary
Acknowlegements
Contents
Chapters: 1, 2, 3, 4, 5, 6, 7, 8
Appendices
References