This paper presents a new algorithm for the joint retrieval of surface reflectance and aerosol properties with continuous variations of the state variables in the solution space. This algorithm, named CISAR (Combined Inversion of Surface and AeRosol), relies on a simple atmospheric vertical structure composed of two layers and an underlying surface. Surface anisotropic reflectance effects are taken into account and radiatively coupled with atmospheric scattering. For this purpose, a fast radiative transfer model has been explicitly developed, which includes acceleration techniques to solve the radiative transfer equation and to calculate the Jacobians. The inversion is performed within an optimal estimation framework including prior information on the state variable magnitude and regularisation constraints on their spectral and temporal variability. In each processed wavelength, the algorithm retrieves the parameters of the surface reflectance model, the aerosol total column optical thickness and single-scattering properties. The CISAR algorithm functioning is illustrated with a series of simple experiments.

Radiative coupling between atmospheric scattering and surface reflectance
processes prevents the use of linear relationships for the retrieval of
aerosol properties over land surfaces. The discrimination between the
contribution of the signal reflected by the surface and that scattered by
aerosols represents one of the major issues when retrieving aerosol
properties using space-borne passive optical observations over land surfaces.
Conceptually, this problem can be modelled to solve a radiative system
composed of at least two sets of layers, where the upper layers include
aerosols and the bottom ones represent the soil–vegetation strata. The
problem can be further complicated by the intrinsic anisotropic radiative
behaviour of natural surfaces due to the mutual shadowing of the scattering
elements, which is also affected by the amount of incident radiation

The strengths and weaknesses of the algorithm proposed by

A new joint surface reflectance–aerosol properties retrieval approach is presented here that overcomes the limitations resulting from precomputed RTE solutions stored in LUTs.

The advantages of a continuous variation of the aerosol properties in the
solution space against a LUT-based approach is discussed in Sect.

The GSA algorithm has been further improved for the processing of SEVIRI data
on board MSG for the retrieval of the total column AOT from observations
acquired in three solar bands centred at 0.6, 0.8
and 1.6

Aerosol dual-mode models based on

Aerosol single-scattering properties include the single-scattering albedo

Example of sensitivity of aerosol single-scattering properties to
particle median radius (green arrows) and imaginary part of the refractive
index (red arrows) at 0.44 and 0.87

A visual inspection of Fig.

To illustrate the dependence of

Example of a region (light-blue area) in the

The actual extent of solutions in the

The position of these vertices is critical as they should
encompass the most likely aerosol single-scattering properties that could be
observed at a given time and location. Different approaches could be used to
define the position of these vertices. The positions can be derived from the
analysis of typical aerosol single-scattering properties available in
databases such as the Optical Properties of Aerosols and Clouds (OPAC)

The forward model, named FASTRE, simulates the TOA bidirectional reflectance
factor (BRF)

The model simulates observations acquired within spectral bands

Atmospheric vertical structure of the FASTRE model. The surface is
at level

The surface reflectance

Aerosol single-scattering properties in the layer

The FASTRE model expresses the TOA BRF in a given spectral band

The multiple-scattering contribution,

The layer

The phase function

The layer total optical thickness,

It is assumed that only molecular absorption takes place in layer

Relative bias and root mean square error in percentage between FASTRE and the reference RTM in various spectral bands.

The simple atmospheric vertical structure composed of two layers is the
principal assumption of the FASTRE model. In order to evaluate the accuracy
of FASTRE, a similar procedure to that in

Surface reflectance characterisation requires multi-angular observations

The retrieved state variables in each spectral band

The fundamental principle of OE is to maximise the
probability

In the same way, the submatrix

Maximising the probability function in Eq. (

Notice that the cost function

The retrieval uncertainty is based on the OE theory, assuming linear
behaviour of

The minimisation of Eq. (

When solving the RTE, the estimation of the multiple scattering term is by
far the most time-consuming step. Hence, during the iterative optimisation
process, when the change

List of aerosol properties used for the simulations. The parameters

A simple experimental set-up based on simulated data has been defined to
illustrate the performance of the CISAR algorithm as a function of the
chosen solution space. More specifically, the capability of CISAR to continuously
sample the

Microphysical parameter values for the four vertices (FA, FN, CS and CL) in the selected spectral bands. Radius are given in

The CISAR algorithm performance evaluation is based on a series of
experiments corresponding to different selections of aerosol properties, both
for the forward simulation of the observations and their inversion. Three
different aerosol models are used in the forward simulations: F0, which only
contains fine-mode; F1, which contains a dual-mode particle size distribution
dominated by small particles; and F2, composed of a dual-mode distribution
dominated by the coarse particles. Table

List of experiments: the names are provided in the first
column. The active vertices in each experiment are indicated with the

Values of the true and retrieved surface RPV parameters for experiment F00. Wavelengths are given in

Retrieved AOT error and uncertainties for the six experiments. The
error

Values used for the RPV parameters in the four selected bands are indicated
in Table

Same as Fig.

The purpose of the first experiment (F00) is to demonstrate that the CISAR
algorithm can accurately retrieve aerosol properties in a simple situation,
showing therefore that the inversion process works correctly. The F0 aerosol
model used to simulate the observations is only composed of fine particles
with a median radius of 0.08

Results are shown in Fig.

A comparison between the true and retrieved values in Table

Let us now examine the case in which both

The results in Fig.

Same as Fig.

Same as Fig.

Same as Fig.

In order to improve the retrieval of the F1 aerosol model properties, the
additional aerosol CS vertex with

Same as Fig.

Same as Fig.

Same as Fig.

For experiment F12, the CS vertex is substituted by vertex CL which has a
median radius of 1.0

Finally, in experiment F13 the inversion was performed using all four
vertices (Fig.

The retrieval of aerosol model F2, a dual-mode particle size distribution
dominated by coarse particles, is now examined. This model is composed of a
fine mode with radius

This paper describes the CISAR algorithm designed for the joint retrieval of
surface reflectance and aerosol properties. Previous attempts to perform a
joint retrieval have been reviewed, discussing their advantages and
weaknesses. The limitations due to a combined used of discrete and continuous
state variables in retrieval methods based on OE as in

A fast-forward radiative transfer model has been designed which solves the radiative transfer equation without relying on precomputed look-up tables. This model considers two atmospheric layers. The upper layer only hosts molecular absorption. The lower layer accounts for both absorption and scattering processes due to aerosols and molecules and is radiatively coupled with the surface represented with the RPV BRF model. Single-scattering aerosol properties in this layer are expressed as a linear combination of the properties of vertices enclosing part of the solution space.

A series of different experiments has been devised to analyse the behaviour
of the CISAR algorithm and its capability to retrieve aerosol single-scattering properties as well as the optical thickness. This discussion
focuses on the retrieval of aerosol models dominated by the fine mode or
coarse mode. These two models have pretty different spectral behaviour in the

These experiments illustrate the possibility of using Eqs. (

The choice of vertices outlining the

This set of experiments represents ideal conditions, i.e. noise-free observations in the principal plane with no bias on the surface prior. This choice is motivated by the need to keep the result interpretation simple to illustrate how the new retrieval concept developed in this paper works. These experiments show the possibility of retrieving aerosol single-scattering properties within the solution space provided it is correctly bounded by the vertices. It is clear that adding noise to the observations will degrade the quality of the retrieval. Similar conclusions can hold if the observations are taking place far from the principal plane, where most of the angular variations occur. Part 2 addresses the performance of CISAR when applied to actual satellite data.

Results presented in Sect.

It includes the plots of case F22, adding a 3 % Gaussian noise to
the simulated TOA BRF for AOT

YG developed the FASTRE model, conceived the experimental set-up and wrote the paper. ML contributed to the development of the inversion method.

The authors declare that they have no conflict of interest.

The authors would like to thank the reviewers for their fruitful suggestions. Edited by: Andrew Sayer Reviewed by: three anonymous referees