At the outset it was decided that RT-SWIFT 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 limb-viewing measurements. RT-MIPAS (Bormann et al., 2005), which is applied to the limb-viewing 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. RT-SWIFT is an adaptation of RT-MIPAS with a modified set of variable gases (RT-SWIFT considers N2O as an additional variable) and an added parametrization for the Doppler effect.
The core of RT-SWIFT is an iterative regression model similar to RT-MIPAS 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 four-point 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 RT-SWIFT 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 clear-sky 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.
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
where 〈 〉 is the averaging operator,
Upon division of the path into a string of Ncell homogenous cells, Eq. (5) can be expressed as:
where τj is the mean transmittance from the jth cell to the satellite (CTS) and Bj is the mean Planck function of the jth cell. The models for τj and Bj 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 time-scales, 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 forward-looking 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 ray-tracing 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 in-house ray-tracing 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 RT-SWIFT are coincident with the 100 pressure levels of RTTOV-9 (Saunders et al., 2008). Since the instrument's vertical field-of-view extends from about 18 to 65 km, only the upper 40 levels of the RTTOV-9 pressure grid are required. The limb paths are defined such that a refracted tangent point coincides with a pressure level as determined by ray-tracing. 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 RT-SWIFT tangent heights which range from 520 to 1600 m. Most of the RT-SWIFT levels are spaced less than 900 m apart. Although the calculations are done on the RT-SWIFT levels, it is instructive to know how well the radiances would interpolate to the pixel tangent heights grid, if required. Experiments with the line-by-line radiative transfer model demonstrated that interpolation of intensities between the two grids can be done with errors of order 0.5 µW/m2/(cm−1)/sr for altitudes above 32 km and up to 7.5 µW/m2/(cm−1)/sr for altitudes below 3265 km (Turner and Rochon, 2008).
Table I. The 40 pressure levels of RT-SWIFT (hPa).
As the instrument only views the line-of-sight 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, wLOS, 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 line-of-sight, i.e.
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 wsr–is assumed to be constant along the path and is assigned the value determined at the tangent point.
The look angle for an RT-SWIFT 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 RT-SWIFT tangent pressure. A value for wsr 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 RT-SWIFT 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 RT-MIPAS, the RT-SWIFT total transmittance is broken into absorbing groups which are treated independently. The first three of four groups (O3, N2O and H2O) are treated as variables with respect to altitude, whereas the other absorbers (CH4, SO2, NH3, CFC-12 and HCFC-22) 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,
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)
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
Each component of the cell effective optical depths in Eq. (12) is represented by a recursive polynomial of the form
where xjm are profile-dependent predictors characterizing the atmosphere state for cell j along the path, M is the number of predictors and ajm are the regression coefficients for each cell.
The regression starts at the cell nearest to the observer and indexes outward. The predictors for O3, H2O and fg (Tables II and III) are similar to those used in the RT-MIPAS model (Bormann et al., 2005). RT-MIPAS does not have N2O predictors, thus a set similar to the O3 predictors is assigned where the volume mixing ratio of N2O replaces the O3 mixing ratio.
It should be noted that the order of the ratioing (Eq. 10) is different in RT-SWIFT than in RT-TOV or RT-MIPAS 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 N2O 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.
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 wsr is based on an examination of how the mean effective optical depth varies with wsr 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 wsr 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, Teff, and a fixed wave number, :
is usually defined as the centroid of the response function.
Generally a linear relationship is assumed between Teff and T. However, a plot of Teff 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 Teff across 33 paths is 1 K (inset of Figure 4), indicating that 〈B(T)〉 is path dependent. In addition, each sub-curve itself has some error associated with it. These errors are due to the variation of tangent height with atmosphere, and hence Teff, through the response function. These errors are sufficiently small that they can be ignored.
Figure 4. Teff vs. T for four different paths Pindex (Table I) and 94 atmospheres. The inset illustrates the separation of limb paths.
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Figure 5 (e.g. path 22) illustrates the residuals between Teff as determined by inverting Eq. (18), and Teff 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.
Figure 5. The residuals for path 22 of Figure 4 of the Planck function assuming Teff is linear (dashed line), quadratic (dotted line), or cubic (solid line), in T. The residuals of the quadratic and cubic are magnified by 10 and 100 times respectively.
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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 sub-curve of Figure 4.