Retrieval Algorithm

This section outlines the method used to retrieve
vertical profiles of O_{3}, NO_{2}, and BrO from measured
ACDs. A more compact notation is now adopted: let
represent the measured ACD at elevation angle i
and
represent the ACD in the reference, initially
an unknown.
Hence the total ACD at elevation angle i is
.
This is related to the vertical column amounts through,

where is the air mass factor weighting matrix (also known as the kernel) at elevation angle i for an absorber in layer j, is the vertical column density in layer j, and the sum is over vertical layers. Hence each ACD is the weighted sum of contributions from all layers. Before being able to solve for , the column abundance in the reference must be determined. As the majority of the ER-2 flights occurred when the sun was fairly high in the sky ( ) and the reference is composed almost entirely of direct sunlight, the slant path enhancement for the reference through all layers above the aircraft is well approximated as . As such,

(11.18) |

where is the reference air mass factor matrix which has the simple form,

and to are the layers below the aircraft and to are the layers above the aircraft. The column abundance in the reference has simply been expressed as the sum of the vertical column in the layers above the aircraft multiplied by the pathlength enhancement factor. Thus, equation (6.17) can be expressed as,

The main advantage of this formulation is that the column density in the reference need not be solved for prior to the inversion. The use of the horizontal flux as the reference over, for example, the top step in the limb scan allows this simple form for . In order for equation (6.19) to be valid, the altitude retrieval grid must possess a level at the altitude of the aircraft.

There are many techniques available to solve for .
See Rodgers (1976) for a discussion of some of these.
If
then a simple matrix inversion can be
performed,

(11.21) |

where, for convenience, matrix notation is now used so that F is the matrix form of F, C is the matrix form of C, and is the matrix form of . However, given noise on the measurements, this is likely to produce a solution which is physically unrealistic and numerically unstable. For an over-constrained system, , least-squares or some other estimator can be used. In this case,

where the superscript

Another common technique is the method of maximum likelihood, or optimal
estimation. This method has the drawback of requiring an extensive
climatology for use as *a priori* profiles as well as the
covariance matrix.

As a third alternative, a non-linear iterative technique, such as Chahine's
method, can be employed (Chahine, 1977). In Chahine's method, an initial
guess for
is made and equation (6.20) is evaluated.
An improved estimate is determined by,

(11.23) |

such that,

(11.24) |

where the superscript m refers to the modelled ACD, from equation (6.20), and `o' refers to the observed ACD. In practice the iterations are stopped once is within the uncertainties in . As with all methods of retrieval, it is essential that for a given elevation angle there is a distinct peak in the weighting factors as a function of height and that for different elevation angles these peaks must occur at different heights. This technique, using a weighted iteration formula, was used by McKenzie

Testing of Chahine's method on synthetic spectra showed that it was
unable to retrieve the initial profiles, even with the weighted
iteration formula of McKenzie *et al*. (1991). Much better results were
obtained using a similar technique but
where the updated vertical column densities are
determined using an iterative least squares. This technique,
suggested by McDade and Llewellyn (1993) for airglow limb tomography and
adapted for CPFM limb scan retrievals,
has been found to be very robust and is well suited
to handling the similar weighting functions. In this method,
the vertical column densities are sought which minimize,

(11.25) |

where is the standard deviation of , a quantity obtained during the DOAS fitting process, and are calculated using equation (6.20). After each iteration, the vertical column densities are calculated using,

(11.26) |

such that always the larger is used,

(11.27) |

and where,

(11.28) |

is a damping factor.

Due to the similarity of the weighting functions for uplooking angles, it is not possible to obtain significant vertical information above the aircraft. However, they differ sufficiently for downlooking angles, especially in the visible, that vertical information below the altitude of the aircraft can be retrieved.

The retrieval method described above is applied as follows.
The vertical column below the aircraft
is obtained first using the nadir field with the horizontal flux
as the reference. Note that from Figure 6.12 and 6.13, the nadir
airmass factor above the aircraft is equal to
(where a value of
was used).
This implies that the absorption in the nadir field above the aircraft and the
absorption in the reference are essentially identical.
Also, as the nadir air mass
factor is nearly constant below the aircraft, the vertical profile of
the absorber below the aircraft is not required. This is less valid
in the near-UV where there is decreased sensitivity near the surface
(from Figure 6.13).
These points can be seen
using equation (6.20). For all layers above the aircraft,
,
and for the layers below the aircraft
is constant (and
).
Hence, equation (6.20) can be rearranged as,

where the left hand side is simply the vertical column below the aircraft. A three-dimensional nadir air mass factor look-up table has been created so that they need not be computed on-line. The dimensions are solar zenith angle, surface albedo, and altitude of the ER-2. Recall from Figures 6.12 and 6.13 that the nadir air mass factor is largely independent of the height of the absorbing layer. The region in which the absorber is perturbed is from the surface up to the altitude of the aircraft. Tables were generated at the relevant wavelengths both with and without ice and water droplets in the PBL and with background stratospheric aerosols.

The second step is to obtain the column above the aircraft by performing
DOAS on the horizontal flux using an extra-terrestrial spectrum as the
reference. In this case, the column above the aircraft can be expressed
simply as,

(11.30) |

The third step is to recover the vertical profile information. Initially only four layers are used to represent the entire atmosphere. Only four layers are used as with the large overlap and small peaks of the weighting functions, it is expected that little height information is contained in a limb scan. The entire atmosphere above the aircraft is taken as a single level and the atmosphere below the aircraft is divided up into three layers. The best choice for these layers is not clear, however, levels of 0-12, 12-16, and 16- km are initially chosen. These unequal divisions are used as the largest gradients are expected between 12 and 20 km and so this will help minimize the impact of the dependence of air mass factor on the (unknown) profile through each layers. Further, most of the information is expected to originate from altitudes close to the aircraft.

As an initial profile,
,
for the layers below
the aircraft, a constant number density is used such that,

(11.31) |

and for the layer above the aircraft, the VCD obtained using the horizontal flux is used.