Componentes de ingreso de los hogares según deciles del ingreso per cápita

The SASKTRAN model has been developed using a successive orders of scattering algorithm in a spherical geometry. The calculation of the multiple scatter source term is exact, with respect to the solar geometry, for the first two orders of scattering from the atmosphere and the ground. The integral over the unit sphere of the incoming diffuse radiance field that is required for the estimation of the source term is highly

variable near the local horizon. Careful spacing of the integration nodes, with a high density of rays near and just below the local horizon, provides an accurate solution of the source term to within 1% for a relatively small number of rays. When the sun is relatively high in the sky, i.e. solar zenith angle less than 70^◦, the estimate of the multiple scatter source term at the tangent point of the line of sight alone is sufficient; closer to the solar terminator, the multiple scatter source term must be estimated at several points along the line of sight in order to avoid a systematic error of approximately 10% at tangent altitudes below 20 km.

Comparisons at wavelengths and geometries specified by Loughman et al. (2004) show that the SASKTRAN results compare better with a reference Monte-Carlo model than the approximate spherical models and generally as well as the true spher-ical models. The SASKTRAN results compare well with the OSIRIS measurements at wavelengths from 310–810 nm and at altitudes from 15–40 km for a best fit scene albedo and the retrieved profiles of ozone, stratospheric aerosol and nitrogen diox-ide. The implementation of the algorithm using caching and multi-threading com-putational techniques has resulted in an efficient code that is suitable for improved operational inversion of limb scatter satellite data with modern desktop computers.

Chapter 5

Aerosol Number Density Retrieval

An algorithm for the retrieval of global stratospheric aerosol profiles is presented in this chapter using the OSIRIS limb scatter measurements as an example data set.

The retrieval utilizes a one dimensional version of the MART non-linear relaxation in-version suitable for limb scatter. A height profile of the particle size distribution must be assumed in order to retrieve the aerosol number density. An altitude normalized wavelength ratio measurement vector is employed to minimize effects of upwelling radiation from ground albedo and uncertainties in the neutral density profile. Using a method of numerical perturbation, a formal error analysis is performed that shows that the dominant error term is the measurement noise. Comparison of SAGE II and SAGE III coincident measurements with the OSIRIS result converted to extinction shows agreement with the limb scatter retrievals to within 15% throughout the lower stratosphere for an appropriate choice of particle size distribution. A sample set of the OSIRIS Level 2 aerosol product is presented that highlights the relatively high sampling resolution of the limb scatter technique.

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5.1 Aerosol signature in limb scatter

For spherical droplets of sulphuric acid, the scattering cross section at visible and near infrared wavelengths is several orders of magnitude larger than the absorption cross section. Therefore in a basic sense, where the total atmospheric optical depth is small, stratospheric aerosols enhance the total limb radiance beyond the Rayleigh background level by a small fraction. In regions of high total atmospheric optical depth, addition of aerosols contribute to the extinction of radiation from the scatter-ing point along the line of sight more than they contribute to additional scatterscatter-ing into the line of sight. This results in a reduction of the total limb radiance by a small fraction. Therefore, unlike the limb sensitivity to an absorbing gas species, which is always negative, the addition of aerosol in the atmosphere can increase or decrease the limb signal, depending on the altitude and wavelength range.

Figure 5.1 is a plot of the sensitivity, or kernel matrix (Equation 3.8), of the limb radiance at selected OSIRIS wavelengths to the aerosol number density profile for an assumed log-normal particle size distribution and under normal OSIRIS viewing conditions (solar zenith angle of 72 degrees, scattering angle of 88 degrees) calculated using the SASKTRAN forward model. Each curve in Figure 5.1 represents the sen-sitivity of the radiance at a single tangent altitude calculated using a typical aerosol number density profile that is successively perturbed by +1% at each altitude from 1 to 40 km. The curves tend to peak at the altitude corresponding to the tangent al-titude. This is because the majority of the information in a limb spectra comes from the tangent point of the line of sight and is discussed in more detail in Section 5.3.

In general, the sensitivity to aerosol increases at longer wavelengths. At the shortest measured wavelengths (less than 300 nm), the limb radiance is essentially insensitive to stratospheric aerosols. The sensitivity at the other wavelengths, greater than 300 nm, shows an enhancement of the limb radiance due to an increase in aerosol density for upper altitudes, and an extinction of the signal at lower altitudes where the optical thickness is large. Note that the altitude where the sensitivity changes

-0.025 0 0.02 0.04

Figure 5.1: Sensitivity of the modelled limb radiance, I, at selected OSIRIS wave-lengths to changes in the aerosol number density at each 1 km altitude layer.

from positive to negative is dependent on wavelength and is lower at the longest wavelengths. This point of insensitivity also moves in altitude depending on the shape and magnitude of the aerosol profile. The increased sensitivity of the limb signal to aerosol at longer wavelengths is partly due to the fact that the Mie scattering cross section of the aerosol particles does not decrease as rapidly with wavelength as the Rayleigh scattering cross section of the neutral density.

For these calculations, a height profile of a single mode, log-normal size distribu-tion and a number density profile that are simple parameterizadistribu-tions consistent with SAGE II retrievals as calculated by Bingen et al. (2004) for post-volcanic conditions (September, 1993) are used. Mode radius varies from 0.4 to 0.3 µm from 10 to 40 km and mode width varies from 1.2 to 1.1 over the same range. This size distribution is chosen as it is significantly influenced by the larger volcanic aerosols and repre-sents a worst case scenario for the modelling and sensitivity studies of this inversion technique. It is shown in Section 5.6 that a size distribution of smaller particles that better represents the background aerosol state of more recent years is required in order to retrieve OSIRIS extinction that agrees well with SAGE II/III measure-ments. However, the validity of the technique presented here, which is a retrieval of the number density for an assumed distribution, still applies.

In the forward model, two parameters are required to characterize the effect of the aerosol particles. One of these is the aerosol extinction (see Equation 4.13),

k(λ) = naσ(λ) (5.1)

defined as a product of the aerosol number density, n_a and the scattering cross sec-tion, σ(λ), as calculated by Mie theory based on the particle size distribution. The absorption cross section of stratospheric sulphate aerosols is negligible compared to the scattering cross section. As the extinction is a product, the effect of particle size distribution and number density are intertwined in such a way that, roughly

300 400 500 600 700 800 0

0.5 1 1.5 2 2.5

Wavelength (nm)

LimbRadiance

Background Aerosol Clean Atmosphere

Figure 5.2: Modelled limb radiance spectra (units of 10¹³ photons/s/cm²/sterad) at 25 km tangent altitude for clean Rayleigh/O₃atmosphere and for the same conditions with a typical background stratospheric aerosol load.

speaking, a small number of large particles can produce the same extinction as a large number of small particles. In a retrieval sense, the extinction is not as strongly affected as the number density by incomplete knowledge of the particle size distribu-tion; however, particle size still has an important effect through the second required parameter of the radiative transfer calculation, the scattering phase function, p(Θ).

Conveniently, for stratospheric aerosol particle sizes and visible wavelengths McLin-den et al. (1999) show that the phase function changes slowly with mode radius such that the extinction is not a strong function of particle size.

Figure 5.2 is a plot of the modelled OSIRIS spectrum at 25 km tangent altitude for a clean, Rayleigh/O₃ atmosphere and for the same conditions with a typical stratospheric aerosol load. It is clear that the addition of aerosol enhances the signal at long wavelengths; short wavelengths are relatively insensitive to aerosol loading.