2.1. Theory
A radiance signal measured by an airborne or satellite sensor on a flat, homogeneous, and Lambertian surface can be expressed [
41] as:
where
L = at-sensor radiance in W·m−2·sr·m
= atmospheric radiance, without interactions with the ground
= direct part of the solar irradiance
= scattering part of the solar irradiance
= direct part of the atmospheric transmittance
= scattering part of the atmospheric transmittance
S = atmospheric spherical albedo
= surface reflectance of the studied pixel
= surface reflectance from the environment.
In the presence of a homogeneous semi-transparent plume between the sensor and the soil,
,
,
,
, and
are modified. The total solar irradiance is given by
, and the total atmospheric transmittance is given by
. The pixel environment is assumed to be homogeneous, which means that
is equivalent to
. The surface reflectance,
, is estimated by applying homogeneous atmospheric correction to the radiance image. The variational radiance signal due to the plume aerosols
is modelled by:
where
is the radiance signal measured for a single pixel in the same conditions as
L, but with the presence of a plume,
is the atmospheric radiance variation due to the plume aerosol, and
is the variation of the product of the total irradiance with the total atmospheric transmittance. Philippets et al. [
10] and Alakian et al. [
42] have considered the plume as an infinite horizontal homogeneous layer. The plume is assumed to be here a finite horizontal homogeneous layer; this means that the direct part of the solar irradiance seen by a pixel under the plume may or may not pass through the plume. The variation of the scattering part of the solar irradiance seen by a pixel under the plume is a fraction of that seen in the case of an infinite plume layer. The total variation of the solar irradiance due to the plume and seen by a pixel can be expressed as follows:
where
is equal to 0 or 1,
is a scalar between 0 and 1,
is the direct part of the total irradiance variation, and
is the scattering part of the total irradiance variation in the case of an infinite plume layer. In the same way, the surface reflectance variation,
, due to the plume aerosol is:
where
is the surface reflectance impacted by the plume.
2.2. Definition of the Optimal Estimation Method
The retrieval method is based on the optimal estimation method defined by Rodgers [
28]. The OEM problem can be straightforwardly represented by Equation (
5). The measure
y is explained through the forward model function,
F, associated with the state vector
x and a random noise
. In this study, the state vector
x corresponds to atmospheric and surface properties, while the forward model is a radiative transfer model.
In the linear case, the forward model can be expressed as
, where
K is the Jacobian matrix containing the partial derivatives corresponding to the sensitivities of the direct model to the state parameters. In this form, the problem is ill-posed, as there are more unknowns than observations. To reduce the number of unknowns, prior knowledge of the state vector is used. The optimization method consists of minimizing the cost function to a global minimum. According to Rodgers [
28], the cost function is expressed by Equation (
6):
where the first term of the equation represents the difference between the state vector
x and the a priori state vector, given the a priori variance-covariance matrix
; while the second term represents the error between the forward model
and the observations
y, given the variance–covariance matrix of the observations
. The variance–covariance matrices
and
represent the uncertainties of the observation and the a priori state vector, respectively. According to the formalism of Rodgers,
can be decomposed into:
where
is the variance–covariance matrix representing the uncertainties due to the sensor and
represents the effects of unknowns; that is, parameters that have an impact on retrieval uncertainties but were not retrieved during the OEM processing.
is the Jacobian matrix associated with the unknown parameters. Rodgers [
28] theoretically defined an estimator of the state vector of the retrievals, as follows:
The Jacobian matrix is also defined as the pseudo-inverse of the gain matrix,
G, which represents the sensitivity of the estimated state vector
to the observations
y. The gain matrix
G is computed by the following expression:
The elements of
G correspond to the partial derivatives of the estimated state vector
, with respect to the observation
y. The gain matrix gives access to the averaging kernel matrix
. The diagonal elements of
A represent the degrees of freedom (DOF) associated with the state parameters. The trace of
A gives the total DOF of the problem (i.e., the number of independent parameters). Finally, the rows of
A give the sensitivity of the retrieved parameters to the true state. In the case of a perfect estimate of
x, the matrix
A would be the identity matrix. The true error corresponding to a random measurement noise is defined by:
and the posterior distribution of the estimated parameters has a covariance matrix given by:
The uncertainty given by
depends on the resolving error caused by the lack of resolution in the inverse process and measurement errors due to the measurement noise. The measurement error and resolving error covariance matrices are
and
, respectively, which are defined by:
2.3. Forward Model Definition and State Vectors
The forward radiance model,
, is defined by the following equation:
where
L (see Equation (
1)) is the estimated radiance without the plume and
(see Equation (
2)) is the radiance differential due to the aerosol plume. The forward radiance model
is estimated using the radiative transfer model with the following input parameters: Atmospheric background properties (water vapor, background aerosol properties), the finite plume geometry parameters (i.e., the direct and scattering parts of the irradiance differential, defined as
and
in
Section 2.1), the aerosol plume refractive index
, the plume modal radius
r, the log-normal size distribution standard deviation
, and the plume aerosol optical thickness (AOT) at 550 nm
. We assume that
is linear, with AOT in the range of [0, 0.5] [
10].
where
(
) represents the AOT ratio between a reference AOT
at 550 nm and the observed AOT
at 550 nm. The atmosphere modelling was performed with the COMANCHE software [
43], a frontend of MODTRAN model version 5.2 [
44]. The plume was modelled by a homogeneous layer with a defined vertical extent, a height above the ground level, and an aerosol optical thickness
at 550 nm, defined by the user. The aerosol plume optical properties were simulated using Mie theory, considering a mono-modal size distribution for a given modal radius
r, with an associated standard deviation
set to 1.5. The simulations were performed for 3 different refractive indices
, corresponding to sulphate, brown carbon, and soot particles (
Table 1).
The plume reference AOT was set to 0.1 at 550 nm (optically thin plume). The plume had a vertical extent of 100 m and was located 10 m above ground level. The vertical extent of the plume was within the common range of industrial stack plume extension [
8]. Moreover, the plume reference AOT and vertical extent values have no impact on the retrieved plume properties, as long as the AOT value is less than 0.5 [
10].
The state vector
x is associated with the OEM retrieved parameters, with a prior distribution
, and with a prior variance–covariance matrix
. The state vector
x includes: (i) The surface reflectance vector
, (ii) the plume AOT at 550 nm
, and (iii) the aerosol plume modal radius
r. The prior distribution of the retrieved parameters and their variance–covariance matrix are given by:
Pixel-by-pixel estimation of the prior value is defined in the next section. The covariance matrix,
, of the surface reflectance
was computed class-by-class. The covariance matrix
and
were estimated from the standard deviation of the prior values
and
, respectively, and from the forward model sensitivity analysis (see
Section 2.5).
The state vector b corresponds to the forward model inputs that are not retrieved by the OEM. The associated variance–covariance matrix, , represents the variance of the error of those unknown parameters. The state vector b includes the atmospheric water vapor content, background aerosol Ångström coefficient, and background atmospheric visibility. Visibility is related to ground surface aerosol extinction at 550 nm by the Koschmeider equation, and was used to scale the MODTRAN aerosol extinction profile.
2.4. Forward Model Initialization: First Guess
This step aims to initialize
b and
x and their associated uncertainties. Atmospheric background properties were estimated using MSI surface reflectances and the MODTRAN aerosol database. The MODTRAN background aerosol models were of three types: “maritime”, “rural”, or “urban”. The rural aerosol model was composed of a mixture of 70% water soluble substances and 30% dust-like aerosols. The urban aerosol model was a mixture of rural aerosol particles and soot-like aerosols produced by industrial activities. The proportions of rural aerosols and soot-like aerosols in the urban aerosol model were 80% and 20%, respectively. The maritime aerosol model was composed of sea-salt particles created by the evaporation of sea-spray droplets and of continental aerosols, which were almost identical to those of the rural model. The exception was that the largest particles in the rural aerosol model were removed. Each aerosol model was defined by the normalized extinction and absorption coefficients at 550 nm.
Figure 1a,b presents the normalized extinction and absorption coefficients for each model.
The state vector
b and the associated
matrix were initialized at this stage. The water vapor uncertainty (see
Section 4.1) was set to 10% of the initial concentration of the mid-latitude winter profile defined by MODTRAN. Channels above 920 nm that were strongly affected by water vapour were not considered. The visibility error was fixed at 15%, corresponding to an absolute error of 5 km (see
Section 4.1). The error in the background aerosol model was defined using the Ångström coefficient Å [
45]. The Ångström coefficients were 0.48, 1.32, and 1.15 for the “maritime”, “rural”, and “urban” aerosol models, respectively. The standard deviation of the error associated with the Ångström coefficients was set to 10%.
Surface reflectances were estimated by compensating for the atmospheric effects in the same way for the whole image, including the plume area. An extra step was needed to estimate the “off plume” surface reflectance in the plume area. In the case of a hyperspectral image alone, the “off plume” surface reflectance was assumed to be the estimated average spectrum, class-by-class. The images were classified with a random forest algorithm. The classification was performed with six user-defined classes: “water”, “sparse vegetation”, “dense vegetation”, “concrete soils”, “dark soils”, and “bright soils”. The training data sets were polygons drawn by the user using the open source geographical information system QGIS [
46].
In the case of a single hyperspectral image associated with a multi-spectral image, different surface reflectance estimation methods may be considered. Different fusion algorithms exist, such as MAP-SMM [
47] or FUSE [
48], which are based on Bayesian approaches, or methods like Hysure [
49], ICCV’15 [
50], or CNMF [
51], which are based on unmixing analyses. The coupled non-negative matrix factorization (CNMF) algorithm of Yokoya et al. [
51] does not depend on image classification and the prior unmixing is unsupervised. CNMF merges a hyperspectral image with a multi-spectral image to obtain a final image, where each pixel is computed as a linear combination of the end members of the hyperspectral image. Vertex components analysis (VCA) [
52] is used to obtain an initial set of endmembers for the hyperspectral image. The extracted hyperspectral endmembers and their weights are refined by alternating unmixing of hyperspectral and multi-spectral images by non-negative matrix factorization (NMF) [
53,
54]. Then, CNMF calculates the abundance matrix of hyperspectral endmember spectra using the multi-spectral image. The ground reflectance,
(see Equation (
16)), is then estimated, by combining the abundance matrix and the hyperspectral endmembers. The associated covariance matrix,
, is then computed class-by-class.
First guesses of the AOTs and mean radii were estimated with a sequential approach based on the hyperspectral image corrected for the atmosphere. Atmospheres containing different types of plumes were simulated using the forward model. Three types of aerosol were considered (see
Table 1). The standard deviation of the log-normal size distribution was equal to 1.5 and the modal radius varied from 0.025
m to 1
m. The sequential approach was performed using the surface reflectance differential defined in Equation (
4), where
is the prior reflectance estimated with the CNMF method and
is the hyperspectral data corrected for the atmosphere. The reflectance differential was compared to the simulations
, using the cluster-tuned matched filter (CTMF) or the root-mean-square error (RMSE) as a metric. The analysis of the metric scores led to a first pixel-by-pixel estimation of the optical thickness, aerosol type, and modal radius of plume particles.
CTMF was developed by Funk et al. [
55] and is used to retrieve gas thermal signatures (CO
, SO
, N
O, and O
) in the thermal infrared. It was later adapted by Thorpe et al. [
56,
57] and Dennison et al. [
58], in order to detect other gases in the reflective domain. Most recently, Philippets et al. [
10] used CTMF to detect and characterize the aerosol signature of industrial plumes. The CTMF model can be described as a combination of the surface reflectance
with the spectral reflectance signature of the plume aerosols
. Both vectors
and
have the same dimension (i.e., the number of channels
m in the hyperspectral image). After the classification of a hyperspectral image, an optimal filter
for a soil class
j can be defined as:
where
represents the inverse of the correlation matrix of the soil class
j and
represents the mean aerosol spectral signature equivalent to
. From the optimal filter
associated to a modal radius and an aerosol type, the CTMF score,
, describes the correlation between the simulated and the measured aerosol signal, which can be expressed as:
where
is the optical thickness ratio between the true optical thickness
and the reference optical thickness at 550 nm
, which was fixed to 0.1 in the direct model. A CTMF score map was computed for each aerosol radius and type. As the aerosol type was assumed to be homogeneous inside the plume, the aerosol type associated to the most frequent best score inside the plume was chosen for the entire plume. Once the aerosol type was selected, a second application of the CTMF was performed. The radius map was computed by selecting the best CTMF score in the set of CTMF maps for each pixel. The AOT was then deduced, by comparing the estimated surface reflectance differential map
with the simulated variation of the surface reflectance
corresponding to the selected modal radius
r pixel-by-pixel:
The RMSE defined by Equation (
20) was used as an alternative metric to estimate the aerosol type, radius, and AOT.
where
n is the number of channels
of the spectrum and
r is the modal radius.
was computed, by Equation (
4), for each pixel and for the different aerosol properties. Aerosol properties were set pixel-by-pixel to the minimum RMSE value.
2.5. Sensitivity Analysis
Retrieval uncertainties, in terms of modal radius and AOT, were due to: (i) The signal-to-noise ratio (SNR); (ii) the ground reflectance error (retrieved by the OEM); and (iii) parameters not retrieved by the OEM (forward model assumptions). The OEM non-retrieved parameters were the atmospheric background properties (
b state vector), the finite plume geometry parameters (i.e., the direct and scattering parts of the irradiance differential, defined as
and
in
Section 2.1), the aerosol plume refractive index
, and the aerosol plume size distribution standard deviation
.
The instrument noise
is a diagonal matrix with diagonal elements
, representing the square of the noise equivalent delta radiance (NEDL). The NEDL
was computed by
, where
represents the residual noise and
represents the photonic noise. The coefficients
and
were given by the noise model of the HYSPEX sensor.
Figure 2 represents the modelled SNR of the HYSPEX sensor for the mean radiance of water pixels.
The reference simulation state corresponded to seawater reflectance, a background aerosol model set to the MODTRAN rural mode with a visibility of 15 km. The scattering aerosol plume modal radius was set to 0.15 m and the AOT to 0.1, respectively. The coefficients and were set to 0.0 and 0.3, respectively. The reference state was closed to the observation case. From the reference state, we estimated the retrieval uncertainties due to a nominal variation (or uncertainty) of the forward model inputs or assumptions. Estimated retrieval uncertainties corresponding to the HYSPEX noise were around 0.0024 for AOT and 0.001 m for the modal radius.
Table 2 shows the modal radius and AOT retrieval uncertainties due to forward model input uncertainties. Parameter uncertainties were defined consistently with the data and the initialisation step (
Section 2.4). The surface reflectance uncertainty was set to 10% and uncertainties corresponding to the water vapor, the background aerosol Ångström coefficient, and the visibility used in the forward model were set to 10%, 10%, and 15%, respectively. An uncertainty of 10% was set for the standard deviation of the plume aerosol size distribution. Then, an uncertainty of 10% was set for the value of the real part of its refractive index. The finite plume geometry parameter uncertainties were set to 0.2 for
and to 1 for
.
We observed that most of the uncertainties due to the model assumption were higher than the measurement noise for both the AOT and modal radius. The maximum AOT error of 0.012 (i.e., a relative error of 12%) was due to a reflectance offset error: An error of 0.01 corresponds to more than 20% of water mean reflectance in the VIS spectral domain. As expected, the maximum modal radius error of 0.045 (i.e., a relative error of 30%) was due to uncertainty, as the retrieved modal radius is closely linked with the input size distribution standard deviation. The least sensitive parameters (errors with the same range of magnitude as noise error) were water vapor, for AOT, and and visibility for modal radius. In the case of other parameter uncertainties, retrieval error due to model uncertainties were quite similar, with a mean value around 0.005 (10%) for AOT and around 0.015 (10%) for modal radius. The forward model assumption uncertainties defined in this analysis led to acceptable uncertainties, in terms of the modal radius and AOT.