At the outset it was decided that RTSWIFT should be compatible with RTTOV (Saunders et al., 1999). This has the advantage of allowing the model to be easily plugged into existing retrieval or assimilation systems that use it. RTTOV is designed for nadir sounding and not limbviewing measurements. RTMIPAS (Bormann et al., 2005), which is applied to the limbviewing MIPAS measurements, has an RTTOV heritage and is designed to be compatible with RTTOV. Like RTTOV, it has a tangent linear model and its adjoint (Bormann et al., 2005), which are generally required in assimilation or retrieval systems. RTSWIFT is an adaptation of RTMIPAS with a modified set of variable gases (RTSWIFT considers N_{2}O as an additional variable) and an added parametrization for the Doppler effect.
The core of RTSWIFT is an iterative regression model similar to RTMIPAS that constructs a limb emission signal from LOS transmittance and temperature profiles. As with MIPAS, the response function is implicitly imbedded within the transmittance model. The product of the SWIFT instrument is a fourpoint interferogram delivered from each pixel on the detector array. Each point of the interferogram is associated with a particular mirror step and its particular response function, thus RTSWIFT requires four transmittance models in order to simulate an interferogram.
The following sections briefly describe the radiative transfer for SWIFT, the limb geometry and its unique interdependence with the wind field, followed by the transmittance/radiance model itself.
3.1. SWIFT radiative transfer
The polychromatic clearsky thermal infrared radiance for an observer at one end of a limb path—assuming local thermodynamic equilibrium and neglecting scattering (Li, 2002)—is the sum of the attenuated emissions from along the path, plus the attenuated emissions from sources at the far end of the path, i.e.
 (3)
Neglecting the first term of Eq. (3)—negligible provided there are no significant sources within the FOV—and assuming that the variation of B over the spectral response function is small, Eq. (3) can be rewritten as
 (5)
where 〈 〉 is the averaging operator,
 (6)
Upon division of the path into a string of N_{cell} homogenous cells, Eq. (5) can be expressed as:
 (7)
where τ_{j} is the mean transmittance from the j^{th} cell to the satellite (CTS) and B_{j} is the mean Planck function of the j^{th} cell. The models for τ_{j} and B_{j} are discussed in more detail in sections 3.3 and 3.5 respectively
Multiple detector array columns (vertical slices) are used with SWIFT for the purpose of reducing the wind error contribution from measurement noise. In principle a regression model is required for each pixel on the detector array. A complete model applied to actual measurements would encompass all or most columns of both the rear and forward views. As well, since the SWIFT instrument characteristics are expected to change in flight over both short and long timescales, e.g. temporal changes in etalon spectral alignment or in optical path difference, a final FFM would have to correctly reflect the impact of these changes. Beyond this, a gradient model version (and its adjoint) would also be needed for use with linearized solution approaches.
For simplicity, this feasibility study for a useful fast forward model was restricted to the application of a single column of pixels, as measurement noise is not being added to the simulated data. The investigation was also restricted to the recovery of the Doppler wind as opposed to the wind vector, and to temporally constant instrument characteristics. The central column of pixels of only the forwardlooking FOV was arbitrarily chosen. Finally, finite differences were used for approximating gradients.
3.2. Definition of the limb path
The atmosphere is assumed to be a set of homogeneous concentric spherical shells defined by a set of vertical profiles of altitude, pressure, temperature, volume mixing ratios and wind. For the purpose of raytracing the path, a supplemental profile of the atmospheric refractivity is determined from this atmosphere.
A limb path is defined by its tangent height and is approximated by a string of homogeneous cells whose boundaries are defined where the path intersects the vertical layer boundaries. For indexing purposes, the cell boundary furthest from the observer is designated with the index of the cell. The cell boundaries closest and furthest from the observer are designated as the near and far sides of the cell respectively. The space between the observer and top of the atmosphere (TOA) defines the first cell and is regarded as empty. The path through the tangent layer is physically very long compared to the other cells. For this reason the tangent layer is further subdivided into two cells, the division point being at the tangent point. The average properties of each cell are determined by an inhouse raytracing program that determines the path length within a cell, and the absorber amount weighted pressure, temperature and volume mixing ratios of the cell. For convenience, the vertical levels are assigned such that the refracted tangent height coincides with the lower boundary of the tangent layer.
The simulations and regression coefficient determinations described in section 3.4 were applied to each of the four points of the interferogram for a group of limb paths defined by a set of fixed tangent pressures. In order to be compatible with RTTOV, the pressure levels of RTSWIFT are coincident with the 100 pressure levels of RTTOV9 (Saunders et al., 2008). Since the instrument's vertical fieldofview extends from about 18 to 65 km, only the upper 40 levels of the RTTOV9 pressure grid are required. The limb paths are defined such that a refracted tangent point coincides with a pressure level as determined by raytracing. The model limb paths are defined as the tangent pressures which coincide with levels 8 to 40 (about 18 to 55 km) of the RTTOV levels (see Table I). The vertical spacing of the tangent heights defined by the instrument FOVs ranges from 600 to 660 m and is smaller than the spacing of the RTSWIFT tangent heights which range from 520 to 1600 m. Most of the RTSWIFT levels are spaced less than 900 m apart. Although the calculations are done on the RTSWIFT levels, it is instructive to know how well the radiances would interpolate to the pixel tangent heights grid, if required. Experiments with the linebyline radiative transfer model demonstrated that interpolation of intensities between the two grids can be done with errors of order 0.5 µW/m^{2}/(cm^{−1})/sr for altitudes above 32 km and up to 7.5 µW/m^{2}/(cm^{−1})/sr for altitudes below 3265 km (Turner and Rochon, 2008).
Table I. The 40 pressure levels of RTSWIFT (hPa).P_{index}  P_{RTTOV}  P_{index}  P_{RTTOV}  P_{index}  P_{RTTOV}  P_{index}  P_{RTTOV} 


 0.0160  4  1.6872  14  11.0038  24  35.6504 
 0.0380  5  2.1526  15  12.6492  25  39.2566 
 0.0770  6  2.7009  16  14.4559  26  43.1001 
 0.1370  7  3.3398  17  16.4318  27  47.1882 
 0.2244  8  4.0770  18  18.5847  28  51.5278 
 0.3454  9  4.9204  19  20.9224  29  56.1259 
 0.5064  10  5.8776  20  23.4526  30  60.9895 
1  0.7140  11  6.9567  21  26.1829  31  66.1252 
2  0.9753  12  8.1655  22  29.1210  32  71.5398 
3  1.2972  13  9.5119  23  32.2740  33  77.2395 
As the instrument only views the lineofsight component of the Doppler shifts, the horizontal wind field must be remapped into an LOS wind representative of each cell. The remapping of the wind field to the LOS winds is tied to the viewing coordinates defined by the satellite orbit and the instrument's FOV pointing relative to the orbit. The viewing coordinate parameters used in this work are: the tangent point coordinates—latitude, longitude and geometric tangent height—and the look angle, ξ, the angle between a limb path's LOS orientation at the tangent point and north. The LOS wind, w^{LOS}, of each cell is the average of its near and far boundary values of the zonal, u, and meridional, v, components of the atmospheric wind projected along the path's lineofsight, i.e.
 (8)
In addition to the Doppler shifts due to the atmospheric winds, the instrument also views the additional shifts due to the forward motion of the satellite and the Earth's rotational velocity. The LOS value of these motions—designated as w^{sr}–is assumed to be constant along the path and is assigned the value determined at the tangent point.
The look angle for an RTSWIFT refracted tangent height is determined by interpolating between the look angles and their corresponding refracted tangent heights associated with the detector array pixels, as determined by the viewing coordinate parameters. Equation (8) uses the interpolated look angle to calculate the appropriate to the RTSWIFT tangent pressure. A value for w^{sr} is determined in a similar manner.
Finally, the mapping of the detector pixels, with their associated refracted tangent heights, to the tangent heights of the RTTOV pressure levels is used to interpolate the instrument characteristics to the RTSWIFT limb paths. These interpolated values are required for the transformation of the simulated radiances to the Doppler wind as described in section 4.
3.3. Transmittance parametrization
As with RTTOV and RTMIPAS, the RTSWIFT total transmittance is broken into absorbing groups which are treated independently. The first three of four groups (O_{3}, N_{2}O and H_{2}O) are treated as variables with respect to altitude, whereas the other absorbers (CH_{4}, SO_{2}, NH_{3}, CFC12 and HCFC22) are treated as a single invariant group, designated as ‘fg’ for fixed gases. The total transmittance is the product of the transmittances of these four groups,
 (9)
Equation (9) assumes that the product of convolved transmittances follows Beer's Law. However, unless there is no significant overlap of spectral lines, Eq. (9) does not generally hold true. Some of the errors inherent with this assumption can be diminished by reformulating the left hand side of Eq. (9) as (McMillin et al., 1995)
 (10)
where is the effective CTS optical depth due to gaseous absorption.
where is the cell's effective optical depth and is defined, for example, as
 (13)
Each component of the cell effective optical depths in Eq. (12) is represented by a recursive polynomial of the form
 (14)
where x_{jm} are profiledependent predictors characterizing the atmosphere state for cell j along the path, M is the number of predictors and a_{jm} are the regression coefficients for each cell.
The regression starts at the cell nearest to the observer and indexes outward. The predictors for O_{3}, H_{2}O and fg (Tables II and III) are similar to those used in the RTMIPAS model (Bormann et al., 2005). RTMIPAS does not have N_{2}O predictors, thus a set similar to the O_{3} predictors is assigned where the volume mixing ratio of N_{2}O replaces the O_{3} mixing ratio.
It should be noted that the order of the ratioing (Eq. 10) is different in RTSWIFT than in RTTOV or RTMIPAS where the predictor sets have been optimized. Ozone is the most significant absorber and it was thought that it should lead the ratioing sequence instead of the fixed gases as in McMillin et al. (1995). However there was no investigation as to whether this is optimal or not. In addition there was no attempt to optimize the N_{2}O predictor set, as the focus is on the wind predictors.
Previously, other regression models have ignored winds since the wind shifts the spectra by extremely small amounts. In this work two multiplicative modifiers are introduced to account for the change in the effective optical depths due to these spectral shifts.
 (15)
In principle, the satellite and Earth's rotational velocities and winds could have been combined into a single velocity. However it is better to model the known and the significantly larger components separately from the smaller unknown atmospheric winds. The choice of the predictor for w^{sr} is based on an examination of how the mean effective optical depth varies with w^{sr} as illustrated in Figure 3. The change in the optical depth for any cell on a path (the vertical line on the plot) appears to be linear. However, closer examination of the variability with respect to w^{sr} for any cell—and other atmosphere/path combinations—indicates that the relation is slightly nonlinear. Hence predictors involving powers of ¡checktext¿ were chosen. In addition, keeping the LOS wind component separate has the advantage of reducing the otherwise larger impact of FFM errors on the recovered LOS winds. This is described and illustrated in sections 4.2 and 5.1.
3.5. Planck function parametrization
The mean Planck function of Eq. (7) follows the model described in Planet (1988) where 〈B(T)〉 is approximated by a monochromatic Planck function at an effective temperature, T^{eff}, and a fixed wave number, :
 (18)
is usually defined as the centroid of the response function.
Generally a linear relationship is assumed between T^{eff} and T. However, a plot of T^{eff} versus T (Figure 4) for four different limb paths and 94 atmospheres over a wide range of T reveals that the relationship is not quite linear. Upon closer examination, the data separate into distinct groups, each corresponding to a limb path (inset of Figure 4). For the region around T = 260 K the spread in T^{eff} across 33 paths is 1 K (inset of Figure 4), indicating that 〈B(T)〉 is path dependent. In addition, each subcurve itself has some error associated with it. These errors are due to the variation of tangent height with atmosphere, and hence T^{eff}, through the response function. These errors are sufficiently small that they can be ignored.
Figure 5 (e.g. path 22) illustrates the residuals between T^{eff} as determined by inverting Eq. (18), and T^{eff} as determined by a fit to polynomials up to order 4. The assumption of linearity does not work well in this case. Consequently a higher order, a cubic, was selected for this study, i.e.
 (19)
a, b, c, d and are collectively known as the band correction coefficients. The band correction coefficients were determined by fitting the data to Eq. (19) for each subcurve of Figure 4.