Error from the forward model can be separated into two distinct types. The first type includes all of the systematics that arise from the approximate nature of the model. These are extremely difficult to gauge as an accurate characterization requires comparison with an exact model that is, of course, impossible to attain. Often this type of error is used to describe the systematics that arise when it is possible to use the forward model in an approximate sense. While the SASKTRAN model is certainly not perfect, it is utilized to the fullest extent for the aerosol retrieval.
Fully spherical geometry is always used. The multiple orders of scattering and the resolution of all spatial integrals in the calculation of the ith order diffuse profile are done to a sufficiently high resolution that effectively no change in the retrieved aerosol profile is observed. No attempt is made to quantify the effect of assuming horizontal homogeneity.
The second type of error in this category, often called forward model parameter error, occurs due to uncertainty in the external inputs specified in the radiative transfer. These errors are most certainly systematic in nature. In this approach, the effect of each parameter is investigated independently through numerical perturbation of the simulated retrieval by an amount in the parameter that represents a realistic uncertainty. Total forward model parameter error is the quadrature sum of each term.
Surface albedo and neutral density are the most important inputs to the radiative transfer calculation in terms of their uncertainty affecting the aerosol retrieval. Again,
a note about particle sizes must be made. The assumed particle size distribution is, in fact, the most important forward model parameter that affects the aerosol retrieval.
However, it is such a crucial aspect of the product that it should not be considered an error, but a crucial assumption. The retrieved density is only meaningful with respect to the assumed size distribution and the two must always be considered together. The accuracy of the product as an optical depth or extinction is shown by the analysis accompanying Figure 5.7.
The reflectivity of the earth, or the albedo, represents a significant unknown in the radiative transfer modelling of the limb signal. The signature of the radiation reflected from the ground in the limb radiance is much stronger at red wavelengths than at shorter UV/blue wavelengths because shorter wavelengths suffer greater extinction along the path length to the ground and back up to the scattering point due to the Rayleigh optical depth. The reddening of the spectrum is similar to the effect of an increased aerosol load and results in potential confusion between the signal attributed to aerosol and the signal caused by an error in the assumed albedo. Even though the altitude normalization of the measurement vector tends to cancel out the majority of the effect of the albedo, it is important that the best estimate of the albedo be used in the forward model as an error in the assumed albedo does effect the solution.
Figure 5.11 is a plot of the same simulated retrieval used to investigate the effect of particle sizes but for different values of albedo. This time the forward model of the measurements is calculated using an albedo of 0.4. The retrievals are performed using assumed albedos of 0.3, 0.4, and 0.5. When the correct albedo is used, i.e. 0.4, the solution is accurate to within the same retrieval error discussed in Sections 5.4 and 5.5.1. With an assumed albedo of 0.3, the algorithm attributes some reddening of the spectrum that is actually due to the upwelling radiation to the aerosol and retrieves an aerosol number density that is systematically about 10% too high at all altitudes.
The error is slightly larger at the uppermost and lowermost altitudes. The opposite is true when the assumed albedo is 0.5. That is, some reddening of the spectrum
0 0.5 1
Number Density (cm−3)
Altitude(km)
Figure 5.11: Simulations to show the variation in retrieved number density for error in the assumed albedo. One simulated measurement set with an albedo of 0.4 was used to retrieve number densities assuming an albedo 0.3, 0.4 and 0.5.
actually caused by the aerosol profile is attributed to the modelled upwelling radiation that is too large. The retrieved solution is too low, by approximately 10%, with a slightly larger error at the altitude extremes.
Because of these similar effects on the limb radiance, given a set of measurements in the region of the aerosol layer, it is difficult to estimate the albedo. However, at altitudes above the aerosol layer a much more reliable estimate of the albedo can be obtained. For OSIRIS scans, the reference altitude chosen for the measurement vector, href=40 km, is also a suitable altitude for determining the albedo. For this work, an estimate of the albedo is obtained, as discussed in Section 4.7.2, by fitting the absolute value of the modelled limb radiance at 700 nm and 40 km altitude to the measured value by adjusting the albedo in the forward model. The reflection from the earth is assumed to be Lambertian with no variation in the horizontal direction and the same value of albedo is used at all wavelengths. Even though these sweeping assumptions must be made, it is better to attempt to estimate the albedo from the
measurements directly than to use a climatology of earth reflectivity because of the frequency of clouds that significantly modify the amount of upwelling radiation from that predicted by clear sky conditions and earth albedo.
The main problem with this technique of albedo estimation is that it relies on the absolute calibration of the instrument. Even so, the calibration is believed to be good to within less than 10% for measurements on the long wavelength side of the order sorter. This is based on extensive in-flight radiative transfer testing and the validation of other retrievals that rely on the absolute calibration (Lloyd, personal communication, 2006). This translates into an uncertainty in the estimated albedo of approximately 20%. In order to relate this uncertainty in the albedo to an uncertainty in the retrieved profile, the Rodgers error analysis requires the sensitivity of the forward modelled measurement vector to the albedo parameter, a,
Ka= ∂F(xa, ˜b)
∂a . (5.12)
Note that this is a diagonal matrix as the albedo is a single parameter that affects the entire profile of the measurement vector. It is determined numerically by computing the difference between the forward model simulation of the measurement vector due to a small change in albedo. For an albedo covariance matrix, Sa, the forward model parameter error covariance is
SS= DKaSaKTaDT. (5.13)
Cloud cover is handled in this work as a modification of the ground albedo. By using a tangent altitude near 40 km, the retrieved albedo is effective for the entire scene below and compensates for cloud cover modification to the upwelling radiation. A systematic error arises in the calculation of the multiple scattering component in the forward model because the light path is modified from the cloud free scenario assumed in the model; however this error is small compared to the uncertainty from
0 10 20 30 40 50 10
15 20 25 30 35
Number Density Error (%)
Altitude(km)
Detector Pointing Albedo Neutral Total
Figure 5.12: Percent error in retrieved number density from measurement error due to detector noise and attitude registration and from forward model parameter error due to uncertainty in albedo and neutral density. The total error is the quadrature sum of all terms.
the absolute calibration.
The same error analysis is applied to the sensitivity to the neutral density with an assumed uncertainty of 1% at all altitudes. Figure 5.12 is a summary plot of the error terms discussed here as a percentage of the aerosol number density. The total error is the quadrature sum of all terms. Especially at upper altitudes, the dominant uncertainty arises from detector noise. This error is very large in terms of percentage, but it is quite constant with altitude in terms of number density. The large percentage error reflects the very small aerosol load at high altitude. For this case, the total error is less than 20% between 12 and 23 km and increases rapidly above 25 km altitude.
0 2 4 6
Figure 5.13: OSIRIS limb radiance profiles (units of 1013 photons/s/cm2/sterad) for scan 06432019 at 470 nm and 750 nm and the forward model profiles using an initial guess aerosol density profile and retrieved albedo of 0.57.
5.6 OSIRIS Measurements
This described technique has been applied to the OSIRIS limb scatter measurements.
Figure 5.13 is a plot of the limb radiance at 470 nm and 750 nm for a typical mid-latitude OSIRIS scan (06432019, 73◦ solar zenith angle, 104◦ solar scattering angle).
Also shown on the plot is the limb radiance calculated using SASKTRAN with an initial guess aerosol number density and the particle size distribution used in the simulations. Radiances are shown on an absolute scale and no altitude normalization has been performed. The albedo used in the model for this scan is 0.57 and was determined using the technique described above. The largest difference between the measurements and the model is at 750 nm at altitudes below 15 km. Slight discrepancies are noticeable at both wavelengths for all altitudes up to approximately 25 km where the measurements and the model are in close agreement.
The difference between the measurement and the model is much more apparent
0 0.5 1 1.5
Figure 5.14: The measurement vector, y, constructed using the OSIRIS measure-ments and with the forward model, before and after the retrieval of the aerosol number density.
in the measurement vector (see Equation 5.3) plotted in the left panel of Figure 5.14 for the OSIRIS measurements and for the forward model with an initial guess aerosol profile. Because the measurement vector is constructed in such a way that the ker-nel matrix elements are always positive, the difference between the measured and modelled vector can be simplistically interpreted as an overestimate of the initial guess aerosol load at altitudes above 13 km and an underestimate below. This is not strictly true due to the coupling between altitudes from multiple scattering.
The measurement vector calculated with SASKTRAN after 30 iterations of the MART inversion of the aerosol density is shown in the second panel of this figure.
The forward model vector is now in very good agreement with the measurements.
Convergence to within 2% in the measurement vector, i.e. |αi − 1| < 0.02, at all altitudes is obtained except at the lowest altitudes where a larger difference is still present due to the retrieval error discussed in Section 5.5.1. The retrieved aerosol number density for this OSIRIS scan is shown in Figure 5.15 as an example of a
0 0.2 0.4 0.6 10
15 20 25 30
Number Density, (cm−3)
Altitude(km)
initial guess retrieved
Figure 5.15: The retrieved aerosol number density for scan 06432019 assuming the particle size distribution parameters shown in Figure 5.6.
typical result.
Finally, the modelled calculation of the limb radiance at 470 nm and 750 nm was repeated using the retrieved aerosol density. The results are shown in Figure 5.16.
Even though the fitting of the measurement vector does not require the agreement of the limb radiance with the measurements, at both wavelengths, close agreement is obtained at altitudes above 15 km. Below this altitude, some discrepancy remains;
however, especially at 750 nm, the modelled profile is much closer to the measure-ments than it was before the inversion. Because the modelled radiance profiles using the retrieved profile closely match the observations, systematic error due to aerosol in the further retrieval of trace gases such as ozone is significantly reduced by this solution. The good agreement between the modelled and measured radiances pro-duced by this aerosol retrieval is also demonstrated in the plots of the entire spectral range presented in the previous chapter, Figure 4.23.
0 2 4 6
Figure 5.16: OSIRIS limb radiance profiles (units of 1013 photons/s/cm2/sterad) for scan 06432019 and the forward model profiles using the retrieved aerosol number density and retrieved albedo.