4.4 Effects of Broadening

The effects of the true instrument line shape being broader than that assumed in the retrieval were investigated in the following way. The forward model was used to simulate the R1 absorption line of hydrogen chloride as before but this time it was convolved with a variety of symmetrical instrument line shapes of varying widths; such line shapes will be encountered in the field if there is a drop in the detector signal strength, or fringe contrast towards the end of a scan, as might arise through misalignment. These line shapes were calculated by allowing the aperture diameter to take values greater than 0.65 mm, while keeping the optical path difference at 180 cm. In this way the line shape is broadened and slightly apodised but remains perfectly symmetrical (it is also possible for the aperture to have a smaller diameter than its nominal value; this would result in a narrowing of the line shape). All retrievals were carried out using an unperturbed line shape corresponding to a 180 cm optical path difference and an aperture diameter of 0.65 mm.

Figure 4-4 is an overlay plot of the correct instrument line shape (dashed line) and the 4% broadened⁴ line shape (solid line).

Figure 4-4. Overlay plot of correct (dashed line) and 4% broadened (solid line) instrument line shapes (section 4.4).

Figure 4-5 shows the artefactual subsidence resulting from the true instrument line shape being broader than that assumed in the retrieval. The effect is shown to increase linearly with broadening of the true line shape, as might be intuitively expected. Since any misalignment of the instrument is likely to result in a loss of resolution due to a broadening of the line shape, and since perfect alignment is unlikely to be realised, it follows that any narrowing of the instrument line shape caused by the aperture being smaller than nominal will be at least partly offset by the effects of these misalignments. For this reason the situation where the true line shape is narrower than the assumed one has not been investigated in detail but it follows that an artefactual raising of the the profile might be expected.

Figure 4-5. Relationship between the broadening of the true instrument line shape and resulting artefactual subsidence in the retrieved profile (section 4.4).

These results are put in context in figure 4-6 which combines the data presented in figures 4-3 and 4-5. The percentage by which the supposed true instrument line shape, compared to the theoretical line shape, was broadened, is plotted against the standard deviations of the retrieved subsidences for unbroadened, symmetrical line shapes at various noise levels. The plot shows the point at which the subsidence resulting from a broadening of the true instrument line shape equals the standard deviation of retrievals and therefore gives an estimate of the accuracy with which the width of the line shape must be determined if errors due to random noise on the measurement are to dominate over systematic errors in the results. For example, for retrievals carried out using spectra with a signal-to-noise ratio of 300, say, we can see that the width of the instrument line shape must be known to better than 0.6 %.

This study is useful in two ways: firstly it is useful in the general sense since a variety of common misalignments and defects in the instrument may cause a broadening of the line shape, but more specifically it addresses the problem posed by the manufacturing tolerance of the apertures which, in the case of the Bruker 120M, is quoted as +50 mm in the diameters (Jewkes (private communication)). Figure 4-7 shows a plot of percentage subsidence as a function of aperture diameter where this deviates from the nominal value of 0.65 mm. It is assumed in this that there are no defects in the instrument that could cause a broadening of the instrument line shape other than the diameter of the aperture being greater than the nominal value. It can be seen that if the true aperture diameter is 0.7 mm (+50 mm) when it is assumed to be the stated 0.65 mm, then an error of some 10% subsidence might be expected in the retrieved profile. More significantly, the previous result that the width of the true line shape be known to better than 0.6% to avoid systematic errors, corresponds to a requirement that the aperture diameter be measured to approximately 10 mm - five times better than the manufacturing tolerance.

Figure 4-6. Plot showing the point at which the errors due to broadening of the true instrument line shape equal those due to random noise on the simulated measurement (section 4.4).

A number of retrievals were carried out on noisy simulations to ascertain whether the effects of noise were in anyway altered by the use of an incorrect line shape; it was found that they were unaltered.

Figure 4-7. Relationship between aperture diameter and resulting artefactual subsidence in the retrieved profile (section 4.4).

4.5 Effects of Asymmetry

The effects of using a symmetrical instrument line shape for retrieval when the true one is asymmetric were investigated in a similar manner to the effects of a broadened line shape. A number of line shapes were constructed with different amounts of asymmetry but with their widths constrained to the correct value for 180 cm optical path difference and 0.65 mm aperture. These asymmetric lines were convolved with the calculated absorption line to produce an asymmetric simulated measurement. The retrievals were then carried out using the theoretical unbroadened, symmetrical line shape.

Figure 4-8. Plot illustrating the function used to create asymmetric line shapes (section 4.4).

The asymmetric instrument line shapes were produced by adding a multiple of the solid curve in figure 4-8 to the symmetrical line shape (dotted curve) and then sampling the resultant onto the coarser frequency grid used throughout these studies. The solid curve, which is essentially sextic in nature, was chosen empirically to provide a computationally inexpensive means of obtaining a realistic asymmetrical line shape. The part to the right of the discontinuity is the negative of a mirror image (about the centre line of the sinc function) of the equivalent part to the left; the zero line is a horizontal asymptote to the solid curve right and left of the discontinuity. The discontinuity is positioned away from line centre as this causes the central peak to lean towards the 'low' side of the distortion (as seen in practice). It would obviously be preferable to construct the asymmetric line shape by simulating the effect in the instrument thought to cause asymmetry in the field but since a variety of misalignments and combinations of misalignments can result in similar asymmetric line shapes this simpler, if less rigorous approach was adopted. It was found to produce an adequate approximation to the measured asymmetric line shapes. Figure 4-9 is an overlay plot of the correct instrument line shape (dashed line) and the 4% asymmetric⁵ line shape (solid line).

Figure 4-9. Overlay plot of correct (dashed line) and 4% asymmetric (solid line) instrument line shapes (section 4.5).

The results of this study are shown in figure 4-10. The plot shows the artefactual subsidence plotted as a function of asymmetry. This time the relationship is not linear, as was the case with a broadened line shape, but it does follow the pattern that might be expected intuitively. Since the retrieval scheme merely minimises the difference (in a least-squares sense) between the simulated measurement and its attempt to match that line, it has no means of detecting asymmetry in either the absorption feature or the line shape and so the direction in which the simulation is distorted can make no difference to the result. It follows from this that the gradient of the subsidence/asymmetry curve must be both equal to zero at the origin and be symmetrical about the subsidence axis, and that this curve will apply equally to true line shapes which are asymmetric in either sense. At higher asymmetries the relationship is approximately linear, while near the origin the curve was found to be approximately cubic but no attempt has been made to develop a theory to explain this behaviour. If an asymmetric line shape is used in the retrieval, the effects are likely to be similar to those shown, with the exception that the point of zero subsidence will move away from that of zero asymmetry according to the line shape chosen. The coincidence of the points of zero subsidence and zero asymmetry occurs only because a symmetrical line shape has been used in the retrieval. If an asymmetric line shape is used in the retrieval then the point of zero subsidence will occur when the asymmetry of the assumed line shape (used in the retrieval) matches the true one. This effect has not been investigated in any detail.

Figure 4-10. Relationship between the asymmetry of the true instrument line shape and resulting artefactual subsidence in the retrieved profile (section 4.5).

These results are put in context in figure 4-11 which combines the data presented in figures 4-3 and 4-10 in the same manner as figure 4-6 does for a broadened line shape. The plot shows the point at which the subsidence resulting from asymmetry in the true instrument line shape equals the standard deviation of retrievals. We can calculate the maximum tolerable asymmetry for a signal-to-noise ratio of 300 to be approximately 2.4%. The appearance of the curve not passing through the origin is misleading. The axes have been chosen to best represent that region of the curve of most interest to those making stratospheric observations, which results in some loss of detail at very low noise levels. The curve does, in fact pass through the origin of the graph but approaches the asymmetry axis in a similar to the curve in figure 4-9.

Figure 4-11. Plot showing the point at which the errors due to asymmetry of the true instrument line shape equal those due to random noise on the simulated measurement (section 4.5).

4.6 Conclusions: Implications for Line Shape Measurements

A forward model has been developed to model the R1 absorption line of hydrogen chloride in the atmosphere. A fully constrained retrieval scheme produced to assess the effect of the use of an incorrect line shape during profile retrievals has been developed and proved stable in the presence of noise. It has been found that the use of an incorrect line shape gives rise to subsidence in the retrieved profile that is not present in the atmosphere. The above results have more significance if considered with respect to a particular instrument, such as the Bruker IFS 120M, or the attempt to measure the true instrument line shape.

4.6.1 Broadening

From the above we see that the manufacturing tolerance of +50 mm in the diameter of the apertures of the Bruker permits the 0.65 mm aperture to be up to 0.70 mm in diameter, giving rise to almost 10% subsidence in the retrieved profile. Indeed, even a 0.6% broadening of the instrument line shape (consistent with the aperture diameter being 0.66 mm rather than the nominal 0.65 mm) could have significant effects on profile retrieval. Since this degree of change in aperture diameter is well within the stated manufacturing tolerance, this work suggests that it is essential that the true line shape be measured if accurate profile retrievals are to be carried out, and that this measurement be accurate to significantly better than 0.6%, with respect to width unless systematic effects are to dominate over measurement noise in the error budget for the retrieval.

4.6.2 Asymmetry

It is well known that asymmetry in the instrument line shape can arise from a number of sources, such as misalignments in the instrument (Goorvitch [1975], Kauppinen & Saarinen [1992]) or of the instrument to the radiation source (Saarinen & Kauppinen [1992]), or incorrect calculation of the phase correction (Mertz [1967], Park [1983]) (section 5.4.1). Although these are difficult to quantify in practice, such misalignments as a displacement of the aperture by 10% of its diameter are likely to have a significant effect (Delbouille (private communication)).

If the issue of asymmetry is considered with respect to measurements of the instrument line shape, such as those discussed in chapter 5, we see that if we require the laser spectrum to have less than 2.4% asymmetry due to the laser itself, then the frequency drift rate must not exceed 7.7x10^-5 cm^-1/min during the course of the measurement. This assumes that the frequency of the counting laser does not drift, but since it may do so at up to 3.3x10^-5 cm^-1/min, it would seem prudent to reduce the maximum acceptable drift rate of the infrared laser to less than 4.4x10^-5 cm^-1/min.

4.6.3 Noise

In the foregoing it has been assumed that the retrieval process is constrained only by a noise level of 0.0033 on the measurements but in practice this may well not be the case.

Firstly, in the case of realistic retrieval schemes, rather than the simplistic, totally constrained scheme employed here, random noise on the measurements is likely to be much more of a problem, and hence the targets quoted above for broadening and asymmetry could be relaxed.

Secondly, if we consider the constraining factor to be the requirements of atmospheric models then we arrive at a maximum error limit of ~10% (Chipperfield (private communication)) which relaxes the previous limit on maximum tolerable broadening to about 2.5% and that for asymmetry to 5.6%. In the same way we could say that there is little point in attempting to reduce the errors below 5% ⁶, corresponding to 1.3% broadening and 3.6% asymmetry (Chipperfield (private communication)).

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