## Outline Of The Retrieval Procedure

Fundamentally, the basic physical quantity required by any kind of surface application of remote sensing in the optical domain is the Bidirectional Reflectance Factor (BRF). Indeed, this quantity expresses the probability of radiation coming from one specific direction, for the particular solar direction), to be scattered into another specific direction, normalized by the reflectance of a Lambertian target illuminated and observed under identical conditions. Accordingly, the upwelling radiance field, / at the surface level in the direction can be expressed as follows:

where jj! is the cosine of the radiation incident from direction Q', /?sfc (z0, Q'->Q) represents the BRF of the surface, and Azo is the down-

welling radiance in the direction at the bottom of the atmosphere which is generated when the Sun is illuminating from the direction Q0. All physical quantities in Equation 1 are monochromatic spectral quantities.

The surface BRF is used to estimate various angularly integrated quantities or albedos, including Directional Hemispherical Reflectances (DHRs):

The estimation of surface BRF values from satellite measurements requires solving an inverse problem in the atmosphere to determine the lowest boundary condition. However, the radiance field emerging at the top of the atmosphere depends on a large number of state variables characterizing the absorption and scattering properties of both the atmosphere and the surface. The inverse problem can therefore be solved in a reliable manner only for the most sensitive state variables, and the radiation transfer model simulating the radiance fields measured by a space-borne instrument must be constrained by a sufficient number of independent observations. The cornerstone of the surface albedo algorithm relies on the exploitation of the temporal sampling of Meteosat VIS channel (individual observations for any given location are acquired every 30 minutes) as if it were an instantaneous angular sampling (such observations are accumulated from sunrise to sunset). The VIS channel of the Meteosat sensor series extends from approximately 0.4 jj.m to 1.1 p.m with a maximum response around 0.65 |im. As such, it is affected by all radiation transfer processes involving the ozone and water vapor contents of the atmospheric column. Since this algorithm is implemented in the EUMETSAT re-processing environment, it benefits from estimates of the total vertical content of ozone and water vapor provided by observations from the Total Ozone Mapping Spectrometer (TOMS) and analyses from the European Centre for Medium-range Weather Forecasts (ECMWF). This reliability permits reducing the full radiation transfer problem to a surface-aerosol absorption-scattering problem. An exhaustive description of the algorithm is given in Pinty et al. (2000a). It is assumed that only a finite set of pre-defined types of atmospheres can be considered and that atmospheric functions and radiance fields can be pre-computed for discrete values of the aerosol optical depth and black surface conditions. This was done for a US-62 type of standard atmosphere implementing a continental aerosol model which includes dust-like, water soluble and soot components (see Vermote et al., 1997, for complete information about this aerosol model). To limit the number of entries in the look-up tables (LUTs), the approach implements a simplified atmospheric model where the gas absorbing layers are located on top of the scattering layers. This scheme is similar to the one adopted in the Second Simulation of the Satellite Signal in the Solar Spectrum (6S) code (Vermote et al, 1997).

The surface BRF, is represented by the RPV parametric

BRF model proposed by Rahman et al. (1993):

where and describe the amplitude and the angular variability of the surface BRF, respectively. The solution of the coupled surface-aerosol absorption-scattering problem is obtained dynamically during the retrieval, given the pre-computation of (1) the function /rJSfC(z0,O0—pc, ©, k) for a set of pre-defined © and k parameter values and, (2) all the atmospheric functions required to solve the atmospheric radiation transfer problem for a black surface condition and a set of predefined aerosol models.

After some mathematical manipulations, the modeled Meteosat spectral response to the total BRF emerging at the top of the atmosphere in direction Q.(jU,(/>) when the Sun is illuminating the system from direction can be approximated by:

where and

In Equations 4, 5 and 6, p~atm (zSoa> ~M> <h<fc>', t) represents the contribution of the intrinsic reflectance of the scattering-only-atmosphere (soa) to the total BRF, weighted by the Meteosat spectral response SCk)\ ?Msat8as(~A /4b U03, UmQ) denotes the transmission factor due to gaseous absorption (C/03 and Um0 are the total content in ozone and water vapor, respectively), weighted by S(X); EO(0O) is the spectral extra-terrestrial solar irradiance; Iu'(zsah Q, Q0) is the radiance measured by the Meteosat sensor and Iatmf(zsoa, Q, Q0) is the scattered radiance field emerging at the top of the scattering-only-atmosphere, i.e., without considering the gaseous absorption effects, bounded by a black surface and weighted by the Meteosat spectral response.

This formulation summarizes the set of dependent and independent variables required to simulate the Meteosat observations under a variety of geophysical situations. The four most critical mathematical manipulations concern

1. the decoupling of the gaseous absorption and aerosol absorption and scattering processes,

2. the linearization of the TOA BRF with respect to the parameter describing the amplitude of the surface BRF (p~o),

3. the expansion of scattered radiation as a Fourier series in relative azimuth angles and,

4. the explicit contribution of atmospheric functions related to the radiation transfer regime for a black surface condition.

This strategy allows a straightforward implementation of the forward radiation transfer model since only sums and products of functions are required during the retrieval process. The Fourier expansion in values also avoids creating LUTs with an entry for this coordinate and, therefore, significantly reduces the memory size required by the processing.

For similar reasons, and as suggested by a sensitivity study, the pc value controlling the hot spot function in was fixed at a value equal to 0.15. This strategy follows the approach applied to the MISR instrument for the retrieval of aerosol over dark surfaces (Martonchik et al., 1998).

The estimated values of denoted by for all the pre-defined conditions of the surface-atmosphere scattering model, described by the pre-defined values of the aerosol optical depth and parameters, are determined by the expression:

where the index i designates the slot (image) number in the daily sequence, and W[m(i) is a weighting function. Since the angular variability function soa> /"> fJo, <h<h> p 0) p CI k~, 0~hg) in Equation 4 is a function of p o, an iteration procedure can be applied to solve Equation 7 until the convergence criterion |/cf0(n)- p~0(n+l)|< 10"3 is satisfied. This convergence is generally achieved in as few as 3 iterations.

The selection of acceptable solutions from the ensemble of retrievals, obtained using the pre-defined models, depends on a comparison of a cost function for each retrieval to a threshold value. This x2 metrics (Kahn et al., 1997) is described by:

X Wcos< (OLRMsa, (Zsa, . 0 ~ RM (Zsa, » 'i ^03 . » Po ' )]'

where Wcost(i) is a weighting function, ^Msilt(zsat, /) is the TOA BRF value measured by Meteosat at the current slot i, and a-data(/) is the assumed uncertainty in both the observation simulations and the actual data i?Msat(zsat, /). The uncertainty erdata(z) is difficult to assess precisely using a theoretical approach, since it takes into account the limitations of the instrument, the uncertainties in the calibration, the stability of the instrument and geometrical rectification, as well as the inaccuracies inherent to the modeling of daily series of Meteosat BRF observations. The value of <7data(0 impacts the number of combinations of surface and atmospheric variables which represent acceptable solutions of the inverse problem obtained daily for all the processed pixels: the larger its value, the larger the number of solutions that are considered acceptable from the radiative point of view. The weighting functions, namely Wiav(i) and Wcost(i), can be chosen such as to maximize the impact of the large solar angles and the corresponding increased atmospheric paths on the retrieval; this, in turn, should lead to a better accuracy in the estimation of the downwelling radiance fields and surface BRFs values.

This approach allows us to identify, for each pixel and on a daily basis, a set of radiatively consistent atmospheric and surface conditions, leading to X2 values less than unity. Furthermore, anyone of these sets of conditions is considered accurate enough to interpret the Meteosat "clear-sky" daily time series with an accuracy at least equal to the value of the denominator of Equation 8. This inversion procedure yields the simultaneous estimate of parameter values characterizing (1) the amplitude (p~Q) and the shape (k~ and ©~hg) of the surface scattering function and, (2) an indication of the aerosol load provided as an effective aerosol optical thickness at 550 nm. Since more than one solution can be retrieved for every single day, the selection of the "Likely" solution is based on inspection of the distribution of retrieved values for p 0, their mean, p~Q, and their average deviation, Ap0:

where N is the number of retrieved solutions.

The solution selected as being the "Likely" solution, p 0, is the one minimizing first the quantity |p~o(/)-p~o| from among those that are not further away from p~Q than Ap0, and, second, the associated X2[p~o(/)] values. This criterion selects the solution giving the lowest x2 value in the range p~o ± Ap0. Once the "Likely" solution for p~0 is identified, the associated values of aerosol optical depth, t, and surface anisotropy parameters, k~ and ©~hg are extracted. This procedure gives explicitly more weight to the control of the "Likely" solution by the value of the amplitude factor of the surface BRF field.

The various experiments conducted with synthetic Meteosat data have shown that the ensemble of solutions to the inverse problem that characterize the surface radiative state can be sampled in an appropriate manner with respect to the envisaged applications (Pinty et al., 2000a). The documentation of the state of the atmosphere is currently tentative, due to the intrinsic nature of the radiative effects and the specific spectral sampling of the Meteosat instrument. The potential to extract an indication of the probable aerosol load over relatively dark surfaces exists, however. Since there is no guarantee that the proper aerosol type is applied at any given time and location, the retrieved aerosol optical depth values must be considered as "effective" in the sense that it permits the interpretation ofMeteosat observations at the accuracy prescribed in the inversion scheme. However, this "effective" value allows the accurate reconstruction of the downwelling atmospheric radiance fields at the surface level.

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