Ground Reflectance Retrieval on Horizontal and Inclined Terrains Using the Software Package REFLECT

Abstract

This paper presents the software package REFLECT for the retrieval of ground reflectance from high and very-high resolution multispectral satellite images. The computation of atmospheric parameters is based on the 6S (Second Simulation of the Satellite Signal in the Solar Spectrum) routines. Aerosol optical properties are calculated using the OPAC (Optical Properties of Aerosols and Clouds) model, while aerosol optical depth is estimated using the dark target method. A new approach is proposed for adjacency effect correction. Topographic effects were also taken into account, and a new model was developed for forest canopies. Validation has shown that ground reflectance estimation with REFLECT is performed with an accuracy of approximately ±0.01 in reflectance units (for the visible, near-infrared, and mid-infrared spectral bands), even for surfaces with varying topography. The validation of the software was performed through many tests. These tests involve the correction of the effects that are associated with sensor calibration, irradiance, and viewing conditions, atmospheric conditions (aerosol optical depth AOD and water vapour), adjacency, and topographic conditions.

Full text

remote sensing
Article
Ground Reflectance Retrieval on Horizontal and
Inclined Terrains Using the Software
Package REFLECT
Yacine Bouroubi 1,* , Wided Batita 1, François Cavayas 2and Nicolas Tremblay 3
1Department of Applied Geomatics, Universitéde Sherbrooke, Sherbrooke, QC J1K 2R1, Canada;
wided.batita@usherbrooke.ca
2Department of Geography, Universitéde Montréal, Montreal, QC H2V 2B8, Canada;
francois.cavayas@umontreal.ca
3Agriculture and Agri-Food Canada, Saint-Jean-sur-Richelieu, QC J3B 3E6, Canada;
nicolas.tremblay@agr.gc.ca
*Correspondence: yacine.bouroubi@usherbrooke.ca; Tel.: +1-819-821-8000 (ext. 62522)
Received: 31 August 2018; Accepted: 12 October 2018; Published: 15 October 2018


Abstract:
This paper presents the software package REFLECT for the retrieval of ground reflectance
from high and very-high resolution multispectral satellite images. The computation of atmospheric
parameters is based on the 6S (Second Simulation of the Satellite Signal in the Solar Spectrum)
routines. Aerosol optical properties are calculated using the OPAC (Optical Properties of Aerosols
and Clouds) model, while aerosol optical depth is estimated using the dark target method. A new
approach is proposed for adjacency effect correction. Topographic effects were also taken into
account, and a new model was developed for forest canopies. Validation has shown that ground
reflectance estimation with REFLECT is performed with an accuracy of approximately
±
0.01 in
reflectance units (for the visible, near-infrared, and mid-infrared spectral bands), even for surfaces
with varying topography. The validation of the software was performed through many tests. These
tests involve the correction of the effects that are associated with sensor calibration, irradiance,
and viewing conditions, atmospheric conditions (aerosol optical depth AOD and water vapour),
adjacency, and topographic conditions.
Keywords:
ground reflectance retrieval; radiometric corrections; atmospheric corrections; topographic
corrections; 6S code; dark target method
1. Introduction
Multispectral satellite imagery, especially at high spatial resolution (30 m or finer), represents an
invaluable source of information for decision making [
1
] in various domains in connection with natural
resources management, environment preservation, or urban planning and management. Common
applications of remotely sensed imagery concern vegetation cover (species, biomass, damages, crop
yields, etc.), surface conditions (land use, types of materials, soil physics, and chemistry, etc.) or
the inventory of natural resources [
2
]. This involves the transition from a relative scale (gray scale)
of measured physical variables (radiances) to a scale of biophysical variables (biomass, leaf cover,
etc.) or a system of classes (e.g., type of land use). This transition is generally accomplished by
combining a series of models, algorithms, equations, and/or classification methods. Depending on the
spatial, spectral, and radiometric resolution of the sensors, it is possible to identify various classes and
conditions of objects. The basic assumption is that the signal recorded by the sensor is directly related
to the state and/or nature of the target objects on the ground.
Remote Sens. 2018,10, 1638; doi:10.3390/rs10101638 www.mdpi.com/journal/remotesensing

Remote Sens. 2018,10, 1638 2 of 34
Variation in object reflectance in the solar spectrum is the key parameter for many optical remote
sensing applications. However, in addition to object reflectance values, sensor measurements are
affected by many factors. Some of them are interrelated, and can be summarised as follows: (1) the
atmosphere: composition and optical properties of gases and aerosols; (2) the sun: its position in
the sky during data acquisition and its spectral irradiance; (3) the terrain: elevation and topography;
(4) ground reflectivity: reflectance anisotropy and reflectivity of the surrounding area; and (5) the
sensor: calibration, spectral sensitivity, viewing angle, and altitude. In this article, the REFLECT
software package is presented, which considers all of these aspects for the radiometric correction of
multispectral satellite images and ground reflectance retrieval.
There are many methods that can be used for ground reflectance retrieval [
3
]. These methods could
be grouped into two main approaches. The first is empirical, and the second is based on the physical
modelling of the signal. Empirical methods require a series of recognisable targets on the images of
invariable and known reflectance. Therefore, it is therefore by regression to establish the relationship
between the signal measured by the sensor and the ground reflectance. Using this relationship,
the reflectance of any pixel on the images is then estimated [
4
–
6
]. The underlying hypothesis here is
that the coefficients of the equations, which link the numerical values to the reflectance, are constants
for the entire image. However, these coefficients are variable over time, and at best can be used only
for one date. The physical modelling approach is by far the most widely used. The atmospheric
effects (additive and multiplicative) and ground irradiances (direct and diffuse) are estimated by
applying approximate solutions of the radiative transfer equation in the earth–atmosphere system.
These solutions are incorporated in atmospheric codes simulating the desired parameters as a function
of a series of inputs.
The purpose of this paper is to present the theoretical and practical bases of the REFLECT
software package for ground reflectance retrieval based on physical modelling (6S atmospheric code).
The first version of this software was developed in the remote sensing laboratory of the Department
of Geography at the University of Montreal in order to estimate the reflectance of water in coastal
environments based on Landsat-7 ETM+ images [
7
]. A second version was developed in collaboration
with Agriculture and Agri-Food Canada’s Horticulture Research and Development Centre (HRDC),
St-Jean-sur-Richelieu, Quebec. The aim of calculating the ground reflectance of crops is based on
Landsat-7 ETM+ images [
8
,
9
]. The current version that is presented in this paper includes several
new features concerning: (a) terrain with variable topography; (b) adjacency effects; (c) atmospheric
conditions; and (d) observation conditions and sensor properties. In order to better present the
theoretical and practical bases of REFLECT, this article is subdivided into eight sections. The “Material
and Methods” are described in Sections 2–4, including a brief overview of the Second Simulation of the
Satellite Signal in the Solar Spectrum (6S) code, basic elements of REFLECT, and the methodology that
was used for validation. The “Results and Discussions” are presented in Sections 5–7, where validation
tests of the ground reflectance retrieval are shown. These tests involve the correction of effects that are
associated with the sensor calibration, irradiance, and viewing conditions, atmospheric conditions
(aerosol optical depth AOD and water vapour), adjacency, and topographic conditions. Section 8
concludes the paper.
2. The 6S Atmospheric Code
The version of the 6S code that was used in REFLECT is the one published in 2006 by Vermote,
E.F. et al. [
10
]. Conversely, 6S can estimate the reflectance of a ground object based on spectral radiance
measured by a sensor under atmospheric conditions that are also defined by the user. The 6S code
assumes a cloud-free atmosphere stratified in parallel layers, in which each one is characterised
by its content in gas molecules and aerosol particles. The atmospheric constituents vary in space
(water vapour and aerosols). A way of addressing this variability is proposed in REFLECT.
The multi-layer structure of the atmosphere in the 6S code divides the atmospheric column into
34 levels between the ground and the top of the atmosphere, allowing the elevation of the target and

Remote Sens. 2018,10, 1638 3 of 34
the altitude of the sensor to be considered. Thus, the code accounts for the reduction in the number of
scattering particles (molecules and aerosols) and the absorbing gases that are above a target at a certain
altitude. The atmospheric pressure and temperature profiles are used to adjust the optical depths of
the atmospheric constituents, which are more concentrated in the lower layers. Standard models for
the vertical distribution of atmospheric gases (US62, MIDSUM, etc.) are used therein.
Gaseous absorption is computed using the Goody model for H
2
O and the Malkmus model
for O
2
, CO
2
, O
3
, N
2
O, and CH
4
. Scattering by atmospheric gases is modelled by the Rayleigh
theory, while aerosol scattering is described by the Lorenz–Mie theory [
11
]. The single scattering
hypothesis is valid solely if the optical path is very thin (
τ
<< 1) and the scattering albedo is very low
(SC << 1). Since these conditions do not always hold [
12
], multiple scattering is considered using
the successive-orders-of-scattering method as an approximation of the radiative transfer equation
solution [13].
The 6S code assumes that the atmosphere is limited by a horizontal ground surface. Three cases
are then accounted for: (a) the target object is part of a Lambertian surface that extends to infinity;
(b) the target object is part of an anisotropic surface (bidirectional properties) that extends to infinity;
and (c) the target object occupies the central portion of a regular surface (a circle) of limited extent with
Lambertian properties, and is surrounded by a Lambertian surface that extends to infinity with a level
of reflectance that is different from that of the target object. Among these three cases, the latter one most
closely approaches the conditions that are usually observed in remote sensing. This configuration was
selected for REFLECT. However, significant modifications were introduced, as described in Section 3.2.
The code formulates the problem of radiative transfer in terms of reflectance factors, and not in
terms of radiance. All of the radiances that are generated in the Earth–atmosphere system are expressed
relative to the Lambertian radiance (
Lsat
Lamb
) of a target with unit reflectance, which was illuminated
and observed under the same conditions as the target, and placed at the top of the atmosphere.
This radiance can be expressed as follows (see symbols in Abbreviations):
Lsat
Lamb (λ) = 1
πE0(λ)d2
scos θs(1)
Hence, the radiance L
sat
, which is measured by the sensor at a spectral band designated by its
central wavelength λ, is described in terms of apparent reflectance factor ρsat by:
ρsat (λ) = Lsat (λ)/Lsat
Lamb (λ)(2)
For the sake of readability, the references to wavelength and the viewing and illumination
geometry are sometimes omitted in the equations.
Although atmospheric scattering may be considered independent from gaseous absorption, water
vapour presents a problem, since it is often concentrated in the lower atmospheric layers where the
aerosols are also concentrated. To take this into account, the code proposes the following formulation
(the case of a Lambertian surface):
(ρsat )j=1,2,3 =ρRTOG
gas +ρA(TOG +H2O
gas )j=1,2,3 +TOG +H2O
gas hT↓
1−Salb ρenv t↑
dir ρtar +t↑
di f f ρenv i (3)
The atmospheric reflectance is divided into two components that were introduced by the gas
molecules (first term on the right) and aerosols (second term on the right). The index OG means
“Other Gases” than water vapour. Equation (3) can lead to three different formulations (index j= 1, 2,
3). The first corresponds to an extreme situation where water vapour has a minimal effect on aerosol
scattering, since it is concentrated at a layer located below the aerosol layer (j= 1). In this case, we find
the formulation usually cited in the literature:

Remote Sens. 2018,10, 1638 4 of 34
ρsat =Tgas 
ρatm +T↓(θs)t↑
dir (θv)ρtar +T↓(θs)t↑
di f f (θv)ρenv
1−Salb ρenv 
(4)
The formulation corresponding to j= 3 in Equation (3) also describes an extreme case where the
water vapour has a maximal effect, since it is mainly concentrated at a layer above the aerosol layer.
Finally, the case j= 2 assumes that half of the water vapour is concentrated at the same layer as the
aerosols, and therefore, the water vapour has an effect that is halfway between the two extreme cases.
In Equations (3) and (4), the term
T↓(θs)t↑
dir (θv)ρtar
contains the useful signal
ρtar
(L
tar
in Figure 1)
that must be estimated from satellite measurements by eliminating the multiplicative effects of the
atmosphere (total transmittance in the descending path and direct transmittance in the ascending
path). The term
T↓(θs)t↑
di f f (θv)ρenv
is due to the radiance reflected by the environment surrounding
the observed target (adjacency effect) that was introduced into the field of view of the sensor by
atmospheric scattering (L
env
in Figure 1). This radiance is also considered as a noise (additive), since it
does not contain information about the target viewed by the sensor; it is significant for low-reflectance
targets. The term in the denominator is added in order to account for multiple reflections between the
ground and the atmosphere, which is characterised by its spherical albedo, S
alb
. This component can
be significant in case of a very hazy atmosphere and a highly reflective environment.
Figure 1.
Solar radiation components received by a remote sensing sensor according to the Second
Simulation of the Satellite Signal in the Solar Spectrum (6S) code.
3. REEFLECT: Principles and Main Developments
This section presents the basic elements of the proposed software package: atmospheric properties,
surface properties, topographic terrain with variable topography, adjacency effect, observation
conditions, and sensor properties.
3.1. Atmospheric Properties
This section shows the modifications made to the atmospheric properties used in radiative transfer
calculations of the 6S code for both gaseous absorption and aerosol scattering.
3.1.1. Gaseous Absorption
Except for water vapour and ozone, the content of gases in the atmosphere is fixed and known.
It is also possible to better represent actual atmospheric conditions by using meteorological data for
the location and date of acquisition [
14
]. Therefore we have introduced this option in REFLECT by

Remote Sens. 2018,10, 1638 5 of 34
incorporating data on ozone content variation according to latitude and month of the year, as well as
the formula of Leckner [14] (Equations (5a) and (5b)) for water vapour content estimation [15].
w=0.493 HrPs/Ta(5a)
where wis the total water vapour content along the length of the atmospheric column in g cm
−2
,
Hr
is
the relative humidity as a fraction of 1,
Ta
is the ambient temperature at ground level in Kelvin, and
Ps
is the partial pressure of water vapour in saturated air given by the equation:
Ps=exp(26.23 −5816/Ta)(5b)
3.1.2. Aerosol Scattering
The knowledge of aerosol properties is essential in remote sensing, since their effect on solar
radiation is present throughout the optical spectrum. These properties are calculated using the Mie
theory based on the microphysical properties of the particles [
16
]. However, the chemical composition
of aerosols is quite variable, and depends both on the geographic distribution of the sources and on
atmospheric dynamics. Determining the proportion of the various types of aerosols at a given location
and at a particular time is therefore not easy. In most cases, standard models are used that assume the
presence of a mixture of typical aerosol particles (dust, water-soluble, soot, etc.) with known and stable
properties. As used in 6S, typical atmospheres (continental, urban, maritime, etc.) are constructed with
different mixtures of these particles.
The atmosphere includes a type of aerosol that is called water-soluble, whose particle size increases
when they are in contact with water vapour molecules. This contributes to modifying the refractive
index and the effective section of the particles, and hence all of the optical properties of this type of
aerosol, including the AOD [
17
–
19
]. Therefore, for the atmosphere types that are rich in water-soluble
aerosols, as continental and urban mixtures that are found in the regions most frequently targeted by
remote sensing, the AOD is expected to be higher for high values of air relative humidity.
We examined the relationship between the AOD measured by the Microtops II Sun photometer
(Solar Light Company, Philadelphia, PA, USA.) at two experimental sites (Saint-Jean-sur-Richelieu
and St-Valentin, Quebec, Canada), and the corresponding relative humidity measured at the closest
weather station (L’Acadie, Quebec, Canada). Figure 2shows that there is a proportional relationship
(
r = 0.62
) between the AOD at 550 nm and the relative humidity. We can assume that this relationship
is due to the proportion of water-soluble aerosols that are present in the continental atmosphere whose
optical depth increases with humidity. Thus, we updated the aerosol model of the 6S code (published in
1983 by the World Meteorological Organization) with a more recent model (OPAC: Optical Properties
of Aerosols and Clouds), which was presented by Hess, M. et al. [
18
], and takes into account the effect
of relative humidity on the properties of aerosols and thus on the values of the AOD.
The OPAC model
The OPAC model [
18
] describes the optical and microphysical properties of 10 aerosols that
are considered the most typical, as well as their mixtures that are usually found in the atmosphere.
The optical properties are: the extinction coefficient (EX), the scattering coefficient (SC) (or scattering
albedo), the asymmetry parameter (ASY), and the phase function (PH). The data are available for
61 wavelengths between 0.25–40
µ
m, and eight relative humidity values (for the water-soluble type).
OPAC proposes a larger number of atmosphere models (aerosol mixtures) than 6S.
The particle types that are considered in the OPAC model are: (1) water-insoluble (INSO), which is
equivalent to dust-like in 6S; (2) several mineral-type aerosols corresponding to the desert dust particles
in 6S; (3) water-soluble (WASO), which is composed of various particles such as sulfates, nitrates, and
other organic matter, whose size and optical properties are a function of the air relative humidity;
the properties of this type of aerosol are considered stable in 6S; and (4) soot, which is identical to
that in 6S, which is made up of very absorbent (black) particles composed of carbon. OPAC proposes

Remote Sens. 2018,10, 1638 6 of 34
the following mixtures: continental clean, continental average, continental polluted, urban, desert,
maritime clean, maritime polluted, maritime tropical, Arctic, and Antarctic. The properties of a mixture
are the averages weighted by the percentages of each aerosol in each mixture.
Figure 2.
Relation between aerosol optical depth (AOD) at 550 nm measured by Microtops-II instrument
in Montérégie, Québec, Canada, and relative humidity given by the closest meteorological station.
A comparison of the OPAC dust-type aerosols (INSO and mineral) with the 6S dust aerosol
(present in the continental and urban mixtures) has shown that the optical properties (EX, SC, and ASY)
of the OPAC particles are more variable than those in the 6S [
20
]. The WASO aerosols in OPAC
are characterised by higher extinction and scattering coefficients than the same type of aerosol
that are used in the 6S code (Figure 3). Also, the higher the relative humidity, the higher these
coefficients are as well, since the particles are larger (and capture more photons by their larger effective
section). The asymmetry parameter of WASO in OPAC is higher for short wavelengths and lower for
wavelengths greater than 1.5
µ
m. The phase functions of INSO in OPAC and of dust in the 6S code are
fairly close and very directional [
20
]. For most wavelengths, the phase function curves of the WASO
aerosols in the 6S code are close to those of OPAC for moderate humidity percentages (Figure 4).
Figure 3.
Comparison between aerosol properties extinction coefficient (EX), scattering coefficient
(SC), and asymmetry parameter (ASY) of the Optical Properties of Aerosols and Clouds (OPAC) and
6S models.

Remote Sens. 2018,10, 1638 7 of 34
Figure 4.
Comparison between phase function (PH) of water-soluble aerosol in 6S (WATE) and OPAC
(WASO).
Adaptation of the OPAC aerosol parameters to the 6S code
The aerosol properties (EX, SC, and ASY) of the OPAC model are defined for 61 wavelengths,
17 of which fall between 0.4–3.0
µ
m (0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.9, 1.0, 1.25, 1.5, 1.75,
2.0, 2.5, and 3.0). Therefore, it has been necessary to calculate by interpolation the properties of the
OPAC aerosols in the 10 reference wavelengths of the 6S code (0.4, 0.488, 0.515, 0.55, 0.633, 0.694,
0.86, 1.536, 2.25, and 3.0). The same problem arises for the phase functions: in OPAC, 112 angles
between –
π
and
π
are considered, while the 6S code divides the circle into 83 angles. Of the 112 OPAC
angles, we therefore chose those that are the closest to the 83 angles of the 6S code, and performed
interpolations according to wavelengths [20].
Calculation of relative humidity profiles
Using the WASO aerosol from the OPAC model requires knowing the relative humidity in the
layer of the atmosphere where the aerosol-scattering properties are calculated. The relative humidity
profile is reconstructed using the formula of Leckner, B. [
14
] (Equations (5a) and (5b)), which is based
on the temperature and pressure profiles according to the atmospheric model used (e.g., MIDSUM).
The parameters of the water-soluble aerosols for the relative humidity values calculated at various
heights are determined by means of a linear interpolation.
Methods for estimating the AOD and choice of an approach for REFLECT
Three approaches are commonly for measuring and characterising aerosols: intensive in situ
measurement campaigns; ground-based networks; and indirect estimates by remote sensing [
21
].
The AOD can also be estimated from the visibility measured in weather stations. Measurement
campaigns allow to characterise the physical, chemical, and optical properties of aerosols [
22
,
23
].
However, since the photoradiometers that are used are very expensive, these measurements are
limited to a certain number of stations. The largest ground-based network is AERONET (AErosol
RObotic NETwork), which was established by NASA [
24
]. This network currently includes more than
450 stations that perform spectral radiometric measurements that are capable of determining the most
important properties of aerosols (single scattering albedo, size distribution, phase function, asymmetry
factor, AOD). Other local networks also exist, such as the Canadian AEROCAN network (which
includes about 10 stations and is part of the AERONET network), the EARLINET network (European
Aerosol Research Lidar Network), and the AD-Net network (Asian Dust Network). These measurement
networks are of insufficient spatial density for specific earth observation applications using high and
very high-resolution satellite images.
Global indirect measurements of the AOD are regularly performed from satellite observations
above dark targets with low and invariable reflectance values [
21
]. The satellite approach is greatly

Remote Sens. 2018,10, 1638 8 of 34
appreciated, since it allows monitoring the great spatial and temporal variability of the AOD.
For example, the AVHRR (Advanced Very High-Resolution Radiometer) sensor provides AOD
estimates in the visible and near infrared bands with a resolution that is close to 1 km [
25
–
28
].
The POLDER (Polarisation and Directionality of the Earth Reflectance) sensor is used to measure
the properties of the aerosols above the oceans and continents with a resolution of approximately
6 km [
29
,
30
]. Since 2000, the MODIS (Moderate Resolution Imaging Spectroradiometer) sensor has
been used to deduce the AOD over the oceans and portions of the ground with daily coverage and
for seven multispectral bands between 0.47–2.13
µ
m, including a blue band for which the ground
reflectance is low and aerosol scattering is high [
31
–
36
]. The AOD estimated from MODIS and MISR
(Multi-angle Imaging Spectroradiometer) sensors has a relative error of about 20% compared to
AERONET measurements [37,38]. The AOD data obtained from the MODIS sensor are provided at a
daily scale for a latitude/longitude grid of 1
◦
[
37
], with a spatial resolution of 10 km. This resolution is
not adequate for the correction of the spatial variability of the AOD in a HR or VHR image (pixels of
30 m or less). Models to estimate AOD at 550 nm from visibility are proposed in the literature, but the
results are very approximate [39,40].
We performed a series of in situ measurements of AOD by a Microtops II Sun photometer in
the vicinity of Montreal, Quebec, Canada. These measurements were compared to data from the
nearest AEROCAN station located in Sherbrooke, Quebec, which was more than 100 km from the
measurement sites. In addition, the data from this station were not available for all of the dates of
interest. The AOD measured by the Microtops was also compared to estimates performed based on
visibility measured by the weather station in St-Hubert, Quebec. For estimating the AOD based on
visibility, we used the empirical formula: AOD
550,VIS
= 3/(VIS
−
3), yielding values intermediate
between the model of So [
39
] and Qiu [
40
]. The comparison showed that the data obtained from the
AEROCAN network (Sherbrooke, Quebec, station), when available, are fairly close to the Microtops
measurements. The AOD estimated based on visibility (AOD
550,VIS
) is often far from the measured
values, which is probably because the visibility data are not precise measurements (e.g.,: 5 km, 15 km,
23 km . . . ).
Given the state of the art concerning the sources of AOD data, the approach adopted in REFLECT
is the dark target method applied to the images to be corrected. This approach is explained in
Section 3.4.
New AOD spectral extrapolation model
For each wavelength of the spectral bands of each sensor, the scattering parameters (
t↓
dir
,
t↓
di f f
,
t↑
dir
,
t↑
di f f
,
ρatm
, or
Salb
) are evaluated by interpolation of their values that were calculated in the 10
reference wavelengths (400 nm, 488 nm, 515 nm, 550 nm, 633 nm, 694 nm, 860 nm, 1536 nm, 2250 nm,
and 3750 nm). In the 6S code, the AOD is estimated at these 10 wavelengths by extrapolation based
on the AOD at 550 nm (AOD
550
) according to the power law:
AOD(λ) = AOD550(λ[nm]/550)−a
.
The coefficient
avaries between 0 and 4, with the standard value being 2. For each spectral band,
these parameters are calculated by integrating the interpolated values for all of the wavelengths
covered by the spectral band, taking into account the spectral sensitivity weighting of the sensor and
the exoatmospheric solar spectral irradiance [41].
Bouroubi, M.Y. et al. [
9
] pointed out that the extrapolation law of AOD
550
that was used in
the 6S code was not adapted to our study area (Montérégie, Quebec), which was possibly because
of a dominance of small particles. The study of this issue led us to propose another law based
on measurements (at 380 nm, 870 nm, 936 nm, and 1020 nm) that were taken by the Microtops in
agricultural (Montérégie, Quebec), urban (Montreal, Quebec), and forest (Laurentians, Quebec) areas.
Figure 5shows that the power law that was used in the 6S code cannot follow the measured values of
AOD across wavelengths. For a= 1, the law is suitable for short wavelengths (around 400 nm), but not
for long wavelengths (800 nm and more). Conversely, the curve with a= 2 is close to the measurements
for long wavelengths, but deviates for short wavelengths. We therefore propose a Gaussian function

Remote Sens. 2018,10, 1638 9 of 34
(Equation (6)) that closely matches the average measurements for the five wavelengths offered by
the Microtops.
AOD(λ) = K1AOD550 exp −K2(λ
550 )
2!(6)
The values of the coefficients are K
1
= 1.91 and K
2
= 0.65. The type of site (rural or urban) does
not affect this relationship [20].
Figure 5. Comparison of average relative AOD to its value at 550 nm with 6S and proposed models.
3.2. Surface Properties
The reflectance of the environment (adjacent pixels) affects the digital value of the viewed pixel
due to scattering within the viewing angle of the sensor. This secondary source of radiance becomes
important if the target object is surrounded by a highly reflective object [
15
]. In this case, the adjacency
effect can be similar in magnitude to the reflectance of the target [
42
]. According to several authors,
the reflectance of the environment is the sum of the reflectances of all of the surfaces surrounding
the target, and is weighted by a function f(r), which depends on their distance rto the target and the
diffusivity of the atmosphere [42–44]:
ρenv =Z∞
0ρ(r)f(r)dr (7)
The environment function f(r) is calculated by taking into account the scattering properties of gases
and aerosols. This formulation was included by Tanré, D. et al. [
42
] in the 5S (Simulation of the Satellite
Signal in the Solar Spectrum) code and by Vermote, E.F. et al. [
43
] in the 6S code. The latter offers an
anisotropic model to better account for the scattering properties of the atmospheric constituents in a
specific direction.
Calculation of the reflectance of the environment according to the rings method proposed by
Vermote, E.F. et al. [
45
] was introduced by Cavayas, F. et al. [
8
] in the first version of the REFLECT
program. This method assumes that the target occupies the central portion of a circular area of
radius r. The tests performed with the rings method led to the conclusion that the phenomenon of
adjacency should be more significant than estimated by Vermote’s formulation [
45
]. An approach
with the individual contribution of each pixel of the environment represents more accurately the
actual conditions of a scene with a heterogeneous spatial reflectivity. Also, the specular reflection of
certain pixels of the environment must be accounted for. These ideas were behind the development of
the method used in REFLECT, which was based on: (1) the anisotropic environment function model
proposed in the 6S code [
45
], (2) the independent contribution of surrounding pixels to the adjacency

Remote Sens. 2018,10, 1638 10 of 34
effect when calculating the weighted average of Equation (7), and (3) the presence of both diffuse
(Lambertian) and specular components in the reflectance of the environment (Figure 6); hence:
ρenv =
∞
Z
0
ρe(r)f(r)g(θs,θv,∆ϕ)dr (8)
where
ρe
is reflectance of each surrounding pixel (individual contribution) at a distance r, and
f
is the
anisotropic environment function. The function
g
, which was inspired from the Phong model [
46
],
takes into account the specularity of each pixel (Figure 7), and it is given by:
g(γsp ) = kd+ks(cos(γsp ))n(9)
Figure 6. Adjacency effect with a specular component of the reflectance of the environment.
Figure 7.
Function
g(θs
,
θv
,
∆ϕ)
with a Lambertian and a specular component of the reflectance of the
environment for different values of θs,kd,ks, and n.
The coefficients k
d
and k
s
are, respectively, the proportions of diffusely and specularly reflected
radiances, with k
d
+k
s
= 1. The calculation window must respect a compromise between two
conditions: (i) accounting for all of the pixels of the environment for which f(r) has a significant
value; and (ii) computation time savings.
In the case of a nadir view (Figure 6), the specular direction γsp is given by:
cos γsp =cos θscos θe(i,j)+sin θssin θe(i,j)cos(ϕs p −ϕe(i,j))(10)

Remote Sens. 2018,10, 1638 11 of 34
The angles
θs
and
ϕs
are respectively solar zenith and azimuth, and
ϕsp =ϕs+
180
◦
. For a
pixel (i,j):
θe(i,j)=tan−1(r(i,j)/Hsat )(11a)
ϕe(i,j)=arccos(−i
pi2+j2).sign(j)(11b)
where r(i,j) is the distance between the environment pixel and the central pixel, and H
sat
is the height
of the satellite. However, since Hsat >> r, it is also possible to consider a single value θe(i,j)=θv.
Observation of the specular effect and estimation of the parameters kd,ks, and n
Figure 8shows a near-infrared (NIR) Landsat-7 ETM+ image of the Hertel Lake on Mount
Saint-Hilaire, Montérégie, Quebec (45
◦
32
0
40”N; 73
◦
09
0
00”W), which is in the middle of a forest. This is
an example of a very dark surface surrounded by very bright surfaces. We can observe that the profiles
of the digital numbers of the NIR band follow a certain slope when they are extracted in the direction
parallel to the solar azimuth. The profiles of the pixels extracted in the direction orthogonal to the solar
azimuth did not show this trend. This phenomenon was observed for other dark surfaces surrounded
by bright surfaces and for other sensors (data not shown) suggesting the presence of a specular effect.
Figure 8.
Specular effect of the reflectance of the environment in the NIR band of a Landsat-7 ETM+
image acquired on 8 June 2001 and covering Hertel lake in Mont St-Hilaire, Quebec, Canada.
The parameters (k
d
,k
s
) of Equation (9) can be deduced from these observations. In Figure 8,
the apparent reflectances (simplified formula) for the dark pixels 1 and 2 are calculated, considering
only the effect of the bright pixels on the dark pixels by:
For point 1: ρs at
dark (1) = Tgas ρatm +TgasT↓t↑
dir ρdar k +Tgas T↓t↑
di f ρlight (kd+ks)
For point 2: ρs at
dark (2) = Tgas ρatm +TgasT↓t↑
dir ρdar k +Tgas T↓t↑
di f ρlight kd

Remote Sens. 2018,10, 1638 12 of 34
Subtracting the apparent reflectances of pixels 1 and 2 gives: ks=ρsat
dark (1)−ρsat
dark (2)
Tgas T↓t↑
di f ρlight
By calculating the apparent reflectances for the example of Figure 9and using the atmospheric
parameters given by REFLECT, we obtain k
s
= 0.14 and k
d
= 0.86. These values are in accordance with
those found by the Columbia-Utrecht Reflectance and Texture Database (CUReT) [
47
]. The exponent
“n” is more difficult to estimate; we took the value n= 50, which is intermediate between the cases
illustrated in Figure 6where n= 20 and n= 100.
Figure 9.
Specular adjacency effect of a bright pixel on a dark pixel according to the position relative to
sun azimuth.
3.3. New Model for Topographic Correction
The formulation used in REFLECT is a generalisation of Equations (3) and (4) to include terrain
with variable topography, as well as non-Lambertian reflectances. We explain here this formulation
with expressions involving irradiances and radiances, as a first step, and then translate all of the values
into reflectance terms, as used in the 6S code. The various sources of irradiance whose intensity is
modulated by an inclined surface are: (1) solar irradiance transmitted directly by the atmosphere;
(2) solar irradiance scattered by the atmosphere, which reaches the surface from any direction of the
sky (hemispherical source); and (3) total solar irradiance (direct and diffuse), which is reflected by the
surrounding area, and which illuminates the surface viewed after multiple reflections between the
Earth and the atmosphere (hemispherical source). These three types of irradiance are given by the
following equations:
-
Direct solar irradiance:
Edir =E0t↓
dir cos is
, in which i
s
is the angle of incidence of the sun’s rays
on an inclined terrain; it is given by:
cos is=cos θscos β+sin θssin βcos(ϕs−α)
. The angles
β
and αare, respectively, the slope and orientation of the inclined surface.
-
Diffuse sky irradiance:
Edi f f ,sky =EHoriz
di f f ,sky gsky
, where the factor g
sky
takes into account the losses
in intensity of the sky irradiance relative to the irradiance incident to a horizontal terrain. These
losses are incurred because parts of the sky are masked by the surface itself, given its inclination.
-
Diffuse irradiance due to multiple Earth–atmosphere reflections:
Edi f f ,env =Etot ρenvSalb
1−ρenvSal b genv
;
the factor g
env
considers the losses because parts of the hemisphere of the sky are masked
due to the inclination of the surface itself and of the surrounding area (environment).
These irradiances are reflected in the direction of the sensor by the surface, which has its own
factors of bidirectional reflectance and hemispherical–directional reflectance. Hence, the radiances can
be written according to their source as follows:
- Ground radiance due to direct radiation: Ldir = (1/π)Edir ρB
- Ground radiance due to diffuse sky radiation: Ld i f f ,sky =(1/π)Edi f f ,sky ρHD
- Ground radiance due to multiple Earth–atmosphere reflections: Ldi f f ,env =(1/π)Edi f f ,env ρHD
Normalising these radiances in accordance with the 6S code (Equations (1) and (2)) gives:
ρdir =t↓
dir
cos is
cos θs
ρB;ρdi f f ,sky =t↓
di f f gskyρH D; andρdi f f ,env =T↓ρenv S
1−ρenvSgenv ρHD
ρB
and
ρHD
are respectively the bidirectional and hemispherical–directional components of the
target reflectance
ρtar
. All of these reflectances are transmitted directly to the sensor with losses due to

Remote Sens. 2018,10, 1638 13 of 34
the atmospheric scattering depending on the transmittance
t↑
dir
. In addition, the adjacency effect can be
written as a first approximation: ρad j =T↓t↑
di f f ρenv.
By homogenising all of these reflectances for the denominator 1
−ρenvS
and ignoring (in keeping
with the 6S code) all of the terms involving the products of two reflectances and the spherical albedo,
because their value approaches zero, we will finally get the equation:
ρsat =Tgas (ρatm +t↓
dir
cos is
cos θsρBt↑
dir +t↓
di f f gsky ρHD t↑
dir +T↓ρenv t↑
di f f
1−Salb ρenv
)(12)
Equation (12) is the REFLECT form of Equation (4), which was used in the 6S code. It allows
separately modulating the direct and diffuse components of solar radiation. For the direct component,
the cosine correction is applied. The definition of the correction factor g
sky
of the diffuse component
requires the use of a diffuse irradiance on the inclined plane model. Loutzenhiser, P.G. et al. [
48
]
presented a review of these models, some of which have already been used in remote sensing
studies [
49
–
53
]. The one developed by Temps and Coulson [
54
] is the most elaborate according
to Mefti, A. et al. [55]. It is given by Equation (13):
gsky =1+cos β
2×1+sin3β
2×1+cos2issin3θs(13)
3.4. Dark Targets Method for AOD Estimation
The dark targets method developed in REFLECT for AOD estimation is based on clear deep water
and very dense forest pixels identification by an automatic (or semi-automatic) histogram thresholding
in the NIR and visible spectral bands. For clear deep-water pixels, REFLECT uses the NIR band to
distinguish water from land. By keeping among the water pixels those having the lowest values in the
available visible bands, turbid or shallow water pixels are avoided. The thresholds in the NIR and in
the visible bands (examples: blue, green, and red) are defined, respectively, as the first local minimum
(MIN
NIR
) and the first local modes (MOD
VIS
=MOD
blue
,MOD
green
, and MOD
red
) of the histograms of
the image digital numbers (Figure 10). However, for an image contaminated by clouds, and to avoid
selecting cloud-shadowed pixels, a stricter condition can be imposed in the NIR band by defining a
lower threshold than MINNIR; for example, the first mode of the histogram MODNIR.
1
Figure 10.
Histogram thresholding of blue, green, red, and near-infrared (NIR) bands (e.g., image
ETM+ of 8 June 2001) for clear deep-water dark targets detection.

Remote Sens. 2018,10, 1638 14 of 34
The red–infrared relationship is used to differentiate vegetation from bare soil and water surfaces
in the extraction of dense forest pixels (Figure 11). Among these pixels, only those that have an
appropriate reflectance in the red band are kept. The condition to select a pixel (with the digital number
DN) as dense forest is based on three thresholds: (
DNN IR −D NRed>T1
) and (
T2≤DNRed<T3
).
To find the T
1
threshold, the digital numbers corresponding to reflectances of 0.02 in the red band
and 0.15 in the NIR band are simulated for the conditions under which the image was acquired.
The atmospheric parameters are calculated using REFLECT atmospheric routines by assuming the
presence of a clear atmosphere (AOD = 0.05). The T
3
threshold is the first mode of the red band
histogram, which was calculated for the pixels that meet the first criterion. The T
2
threshold was
added to avoid selecting the border pixels of lakes and the edges of cloud-shaded areas. It was set at
T2=T3−3.
Figure 11.
Histogram thresholding of digital number (DN) differences of the NIR and red bands (e.g.,
image ETM+ of 8 June 2001) for the detection of dense forest dark targets.
In order to give the user more options to choose the “acceptance” thresholds of dark targets,
we added to the algorithm a manual selection of the “degree of severity” of the darkness conditions.
This option enables the user, for water-type dark targets, to vary the thresholds from MIN
NIR
to
MOD
NIR
for the NIR and some digital numbers around MOD
VIS
for the visible bands. The same
principle is applied for threshold T1when looking for dense forest pixels.
Another improvement was made to the search process for dark targets due to the following
observation: in the case of images with a heterogeneous spatial distribution of the AOD, the histogram
thresholding technique selects only dark targets that are located in the parts of the image where
the atmosphere is the clearest. Thus, REFLECT allows dividing the scene into NxM sub-images
and conducting the search in each of them. This local search enables detecting the spatial variation
of the AOD and correcting it. It should be noted that Ouaidrari, H. and Vermote, E.F. [
56
] also
proposed dividing the image into 4
×
4 quadrants to search for the dark targets in an image with
a non-homogeneous atmosphere. Ahern, F.J. et al. [
57
] and Teillet, P.M. et al. [
58
] also addressed
this issue.
3.5. Sensors Integration
We have incorporated in REFLECT most of the HR and VHR sensors that are currently used
for Earth observation applications. These sensors are Landsat-5 TM, Landsat-7 ETM+, Landsat-8
OLI, Terra ASTER, SPOT-1 to SPOT-7, Sentinel-2, Ikonos, QuickBird, Pleiades, and WorldView-2 and
WorldView-3. New sensors will be incorporated as they come on line. Incorporating a sensor requires:
(1) the calibration coefficients “gain” and “offset” for each spectral band (provided with the image as
metadata) that are used to convert the digital numbers (given in eight or 16 bits) to apparent radiances

Remote Sens. 2018,10, 1638 15 of 34
in W.m
-2
.sr
-1
.
µ
m
-1
according to sensor technical documentation; and (2) the spectral sensitivities SS
k
(
λ
)
used for the averaging of spectral properties (
Lsat
k
,T
gas
,
t↓
dir
,
t↓
di f f
,
t↑
dir
,
t↑
di f f
,
ρatm
, or
Salb
) given at the
wavelengths
λ
(with a step of
∆λ
= 2.5 nm in the 6S code) within the limits of the spectral band k.
Hence, for any parameter p, this integration is performed by the equation:
pk= λ2
∑
λ1
p(λ)SS(λ)E0(λ)∆λ!/ λ2
∑
λ1
SS(λ)E0(λ)∆λ!(14)
E0(λ) is the exoatmospheric solar spectral irradiance.
The calibration coefficients are determined during pre-launch operations, but their values drift
over the years with the age of the optical system. Certain methods are proposed for making post-launch
corrections [
59
,
60
], and these coefficients are regularly updated by the satellite operators. Since the
satellites cover almost the entire planet, the areas observed often have very different reflectances;
consequently, several gains are available on certain sensors to optimise the accuracy of the images
(better contrast without saturation). For example, REFLECT allows the user to choose predefined
calibration coefficients when it is possible (examples: ETM+ and ASTER) [
61
]. Otherwise, values from
the image metadata file should be used.
3.6. Software Implementation
The REFLECT software was programmed in C++ language. It was provided with a graphic
interface and organised in modular form to reduce the computational burden on the main program,
which calculates ground reflectance. A fast computation option based on the generation of internal
look-up tables of the atmospheric parameters for a combination of AOD and terrain altitude values
(which distinguish the various pixels of the image) considerably reduces the computation time. Another
tool is provided in the interface for the calculation of atmospheric parameters.
4. Methodology and Data Used for Validation
The accuracy of ground reflectance retrieval by REFLECT was evaluated by means of several tests
and using various types of images and related data. These tests involved:
a.
Calculation of atmospheric parameters: Temperature and relative humidity data for the image
acquisition dates were obtained from the Environment Canada website [
62
] to estimate the water
vapour content in the atmosphere as well as gaseous transmittances, as explained in Section 3.1.
The AOD that was used for the calculation of the diffusion parameters was calculated from
images using the dark targets method.
b.
Correction of sensor gain effect and atmospheric effects: we used five Landsat-5 TM images,
two Landsat-7 ETM+ images, one SPOT-1 HRV image, and one SPOT-5 HRG image. Images
were acquired over several years between June and August for an area (intersection of the scenes
of these images) of approximately 80 km
×
65 km around Montreal, Quebec, Canada (Figure 12;
Table 1). By comparing the fifth and 95th percentiles of the DNs and ground reflectance values
calculated by REFLECT, we assessed the correction of sensor gain, atmospheric effects (additive
and multiplicative), and conditions of illumination and observation.
c.
Comparison of spectral signatures of selected materials: The ground reflectance values of three
materials (water, asphalt, and vegetation) identified on the images of the scene in Figure 12 were
compared with their simulated values from the USGS ASTER spectral library. The reflectance
values of these materials for the spectral bands of the sensors used are simulated using following
Equation (similar to Equation (14)):
Rband,material = (
λ2
∑
λ1
ρmaterial SSband E0)/(
λ2
∑
λ1
SSbandE0)

Remote Sens. 2018,10, 1638 16 of 34
where
ρmaterial
is the spectral reflectance from the ASTER library for a given material,
SSband
is the
spectral sensitivity of the sensor by band, and E
0
is the exoatmospheric spectral solar irradiance.
d.
Comparison with ASD measurements: The ground reflectance values calculated for four ETM+
images acquired between 2000–2002 and one WorldView-2 image from 2011 were compared
with the spectroradiometer (ASD FieldSpec HH Pro) measurements taken in agricultural fields
in the Montérégie region, Quebec, Canada. The ASD reflectance values, which were measured
concurrently with image acquisition, were incorporated by the spectral band of the images that
was used according to the principle of Equation (16).
e.
Correction of adjacency effects: A series of nine Formosat-2 images (Taiwanese experimental
sensor operated by SPOT Image, a spatial resolution of 8 m, fast revisit time [
63
]) acquired
between 5 June and 3 July 2005 near Montreal were used to show correction of the adjacency
effect (in addition to correction of sensor gain and atmospheric effects) for a vegetated surface
surrounded by highly reflective surfaces in the red band.
f.
Topographic corrections for flat surfaces: One Ikonos image acquired on 29 August 2002 over
the Paulatuk region, Northwest Territories, Canada, was used to test REFLECT’s topographic
effects correction model on flat inclined surfaces (roofs of large buildings).
g.
Topographic corrections for a forest canopy: Three SPOT-1 HR images acquired during the same
period in 1988, 1989, and 1990 over the mountainous region of Tarn, France, as well as a DEM
(digital elevation model) of the region enabled us to validate the adaptation of the topographic
correction model to a forest canopy.
Figure 12.
Scene used to compare DNs and ground-based reflectances for five Landsat-5 TM images,
two Landsat-7 ETM + images, 1 HRV SPOT image and 1 SPOT 5 HRG images.
Validation tests (a) to (c)—where all of the surfaces are considered horizontal—are discussed
together in Section 5. Tests (d) and (e) are presented in Section 6, which is devoted to vegetation targets.
Tests (f) and (g) deal mainly with topographic corrections, and are presented in Section 7.

Remote Sens. 2018,10, 1638 17 of 34
Table 1.
Gaseous transmittances calculated for Landsat and SPOT images used for validation.
Temperature and relative humidity were used for atmospheric water vapour estimation.
MIR: mid-infrared.
Images Temperature
(◦C)
Relative
Humidity (%)
Total Gaseous Transmittances (Tgas)
Blue Green Red NIR MIR
TM
17 June 1984 22 60 0.9931 0.9508 0.9492 0.9159
0.8723
25 July 1992 23 65 0.9932 0.9494 0.9473 0.9081
0.8647
18 June 1996 22 46 0.9929 0.9522 0.9512 0.9268
0.8824
27 August 1998 24 70 0.9929 0.9463 0.9439 0.8991
0.8560
27 June 2005 27 56 0.9933 0.9495 0.9473 0.9061
0.8636
ETM+ 8 June 2001 20 42 0.9946 0.9595 0.9565 0.9470
0.9462
11 August 2001 23 43 0.9948 0.9588 0.9551 0.9391
0.9407
HRV 1 August 1987 21 42 - 0.9725 0.9588 0.9339 -
HRG 29 July 2006 27 60 - 0.9675 0.9478 0.9069
0.9574
5. Reflectance Retrieval for Global Scenes
This section deals with ground reflectance retrieval for a region comprising Montreal, Quebec,
Canada, which is common to nine Landsat and SPOT images (Table 1; Figure 11). This operation
validates the radiometric corrections that are associated with sensor gain, illumination and observation
conditions, and additive and multiplicative atmospheric effects. After an atmospheric parameter
calculation step, analysis of the digital numbers and ground reflectance values was carried out as
follows: (1) by comparing the fifth and 95th percentiles of the digital numbers and ground reflectance
(
ρground
, which is
ρtar
of Equation (3), and includes values of all of the available images; and (2) by
comparing the spectral signatures from raw (DNs) and corrected images (
ρground
) of selected materials
from the USGS ASTER library (water, asphalt, and vegetation).
5.1. Atmospheric Parameters for TM, ETM+, HRV, and HRG Images
5.1.1. Gaseous Transmittances
Gaseous transmittances are calculated by estimating the water vapour content in the air from
the air relative humidity (H
r
) and ambient temperature (T
a
) data obtained from Canada’s National
Climate Archive for the Pierre Elliott Trudeau Airport station, which is located approximately at the
centre of the scene delimited around Montreal (Table 1). The effect of H
r
(lower T
gas
for higher H
r
,
and vice versa) can be seen in the NIR band because of the water vapour absorption lines between
the 800–840 nm wavelengths [
20
]. Gaseous transmittances are close to one in the short wavelengths
(blue band), and decline with increasing proximity to the MIR (mid-infrared) wavelengths, mainly
owing to water vapour absorption.
5.1.2. AOD from Dark Targets Method and Diffusion Parameters
The AOD at 550 nm is estimated using the dark targets method for all of the images (Table 2).
An average value was taken because the AOD spatial distribution did not show any significant
differences. We can see that most of the images have a clear atmosphere. The atmospheric parameters
also show the magnitude of the additive effect in the short wavelengths (blue band). However,
owing to the new interpolation rule AOD(
λ
) = f (AOD550,
λ
) given by Equation (6) of Section 3.1.2,
the diffusion parameters (t
dir↓
,t
dir↑
,t
diff↓
,t
diff↑
,
ρatm
and S
alb
) are no longer overestimated in the blue
band compared to tests carried out with the original rule of the 6S code (data not shown).

Remote Sens. 2018,10, 1638 18 of 34
Table 2.
AOD values calculated with REFLECT dark objects method and corresponding atmospheric parameters (t
dir↓
,t
diff↓
,t
dir↑
,t
diff↑
,
ρatm
, and S
alb
) for Landsat and
SPOT images used for validation.
Band TM TM TM TM TM ETM+ ETM+ HRV HRG
17 June 1984 25 July 1992 18 June 1996 27 August 1998 27 June 2005 8 June 2001 11 August 2001 1 August 1987 29 July 2006
AOD550 0.12 0.11 0.06 0.23 0.11 0.06 0.07 0.10 0.22
tdir↓
Blue 0.7236 0.7114 0.7669 0.5977 0.7292 0.7670 0.7011 - -
Green 0.8124 0.8035 0.8536 0.6993 0.8164 0.8538 0.7962 0.7953 0.7079
Red 0.8697 0.8633 0.9050 0.7751 0.8726 0.9103 0.8645 0.8626 0.7971
NIR 0.9327 0.9293 0.9544 0.8745 0.9342 0.9561 0.9286 0.9302 0.8902
MIR 0.9981 0.9980 0.9984 0.9972 0.9982 0.9983 0.9978 - 0.9971
tdiff↓
Blue 0.1009 0.1037 0.0840 0.1318 0.0996 0.0817 0.1035 - -
Green 0.0831 0.0858 0.0640 0.1199 0.0819 0.0640 0.0882 0.0885 0.1193
Red 0.0627 0.0648 0.0456 0.0964 0.0617 0.0435 0.0645 0.0661 0.0919
NIR 0.0279 0.0287 0.0197 0.0425 0.0275 0.0193 0.0293 0.0301 0.0405
MIR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - 0.0000
tdir↑
Blue 0.7584 0.7584 0.8012 0.6795 0.7584 0.7918 0.7488 - -
Green 0.8372 0.8372 0.8762 0.7645 0.8372 0.8701 0.8306 0.8144 0.7425
Red 0.8875 0.8875 0.9200 0.8259 0.8875 0.9206 0.8882 0.8760 0.8225
NIR 0.9422 0.9422 0.9618 0.9042 0.9422 0.9613 0.9414 0.9372 0.9046
MIR 0.9984 0.9984 0.9986 0.9979 0.9984 0.9985 0.9982 - 0.9975
tdiff↑
Blue 0.0924 0.0924 0.0745 0.1190 0.0924 0.0752 0.0926 - -
Green 0.0751 0.0751 0.0560 0.1051 0.0751 0.0582 0.0774 0.0826 0.1114
Red 0.0563 0.0563 0.0396 0.0834 0.0563 0.0394 0.0561 0.0613 0.0849
NIR 0.0253 0.0253 0.0172 0.0380 0.0253 0.0175 0.0258 0.0281 0.0379
MIR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - 0.0000
ρatm
Blue 0.0758 0.0727 0.0700 0.0859 0.0753 0.0728 0.0770 - -
Green 0.0426 0.0429 0.0378 0.0508 0.0423 0.0398 0.0460 0.0457 0.0535
Red 0.0258 0.0262 0.0219 0.0335 0.0257 0.0215 0.0260 0.0246 0.0306
NIR 0.0125 0.0126 0.0099 0.0183 0.0124 0.0099 0.0129 0.0105 0.0161
MIR 0.0017 0.0017 0.0011 0.0030 0.0017 0.0012 0.0019 - 0.0027
Salb
Blue 0.0863 0.0863 0.0898 0.0809 0.0863 0.0927 0.0887 - -
Green 0.0626 0.0626 0.0611 0.0642 0.0626 0.0637 0.0648 0.0691 0.0686
Red 0.0459 0.0459 0.0416 0.0517 0.0459 0.0413 0.0457 0.0492 0.0524
NIR 0.0277 0.0277 0.0222 0.0361 0.0277 0.0224 0.0280 0.0290 0.0360
MIR 0.0072 0.0072 0.0042 0.0122 0.0072 0.0044 0.0075 - 0.0129

Remote Sens. 2018,10, 1638 19 of 34
5.2. Comparison between Low and High Values of DN and Ground Reflectance
The images listed in Table 1, which were all acquired in the summer, were used to demonstrate
the homogenisation of dynamic ranges by the radiometric corrections. We examined the minima
(
fifth percentiles
) and maxima (95th percentiles) of the DNs and ground reflectance values (
ρground
)
calculated by REFLECT. The minima provide an indication of the correction of the additive atmospheric
effects and sensor offsets, while the maxima show the correction of the multiplicative atmospheric
effects and sensor gain. Table 3shows that the differences between the DNs are significant for the
different dates (atmospheric conditions) and different sensors (gains).
The DNs of the SPOT-1 HRV image of 1 August 1987 are particularly low, and those of the SPOT-5
HRG image of 29 July 2006 are particularly high. The reason for this is that SPOT has variable gain
settings for each spectral band. The gain values in the green, red, and NIR bands for the HRV image of
1 August 1987 were 0.852, 0.794, and 0.888, respectively, versus 2.33, 2.8, and 1.31 in the same bands
for the HRG image of 29 July 2006. The fifth and 95th percentiles of
ρground
are much less variable;
differences are around 0.01 for the low reflectance values (visible bands), and around 0.02 for the high
reflectance values (NIR and MIR).
5.3. Comparison between Values of DN and Ground Reflectance for Selected Materials
Pixels corresponding to clear water (lakes), asphalt (highways), and vegetation (deciduous trees)
surfaces were selected (Table 4) on all of the previously used Landsat and SPOT images. The purpose
of this test was to compare the ground reflectance values calculated by REFLECT for these materials
from the various images and compare these reflectance values with those obtained from the USGS
ASTER library. The latter values are calculated as indicated by Equation (16).
As we can see from Table 5, there is significant variability in the DNs by type of sensor.
The variability by date is relatively less significant for the TM and ETM+ sensors. The reason for
this is that the images were acquired under the same conditions of: (i) sensors gain per spectral band
(unique for TM and fixed to high in visible bands and low in infrared bands for ETM+); (ii) season
and time (similar illumination and observation conditions); and (iii) clear skies. However, the levels of
the DNs from the SPOT HR images are not only different from those of the TM and ETM+ sensors;
they also differ among themselves. As mentioned above, this is due to the difference in gains for the
two acquisitions. Hence, the variation in DNs is affected more by the sensor than by the atmospheric
conditions corresponding to clear skies in our case. The ground reflectance values derived from the
various images show significantly lower variability compared to the DNs (Table 6).
For each material and each spectral band, the ground reflectance values are very similar for all of
the sensors and all of the acquisition dates. In most cases, the variation does not exceed
±
0.01 in the
reflectance units. In the case of vegetation (deciduous trees), the reflectance values derived from the
images are lower than those given in the ASTER library, which was probably because of mixed pixels.
Note that for the different sensors, we assigned unique values for the reflectance values of the
materials from the ASTER library (Table 6). The reason for this is that the differences between the
integrated reflectance values by spectral band of the sensors used (TM, ETM+, HRV, and HRG) were
very small (less than 0.01).

Remote Sens. 2018,10, 1638 20 of 34
Table 3. The fifth and 95th percentiles of DNs and ground reflectances for Landsat and SPOT images.
Band
TM TM TM TM TM ETM+ ETM+ HRV HRG
17 June 1984 25 July 1992 18 June 1996 27 August 1998 27 June 2005 8 June 2001 11 August 2001 1 August 1987 29 July 2006
Fifth percentile of DN
Blue 70 67 65 61 66 66 61 - -
Green 28 26 26 23 26 48 43 30 80
Red 20 20 19 18 20 32 30 16 50
NIR 12 19 13 12 13 17 17 14 17
MIR 7 10 8 7 7 15 13 - 16
Fifth percentile of ρground
Blue 0.039 0.041 0.037 0.031 0.034 0.029 0.031 - -
Green 0.047 0.045 0.046 0.039 0.041 0.033 0.036 0.034 0.039
Red 0.029 0.031 0.030 0.027 0.031 0.027 0.026 0.027 0.028
NIR 0.030 0.045 0.032 0.027 0.029 0.034 0.036 0.041 0.038
MIR 0.008 0.011 0.010 0.007 0.008 0.011 0.009 - 0.010
95th percentile of DN
Blue 118 98 105 85 105 117 97 - -
Green 58 45 52 37 51 106 83 57 205
Red 65 48 61 41 60 123 93 44 175
NIR 132 133 134 118 132 141 122 101 171
MIR 137 103 132 88 131 161 131 - 185
95th percentile de ρground
Blue 0.132 0.115 0.121 0.112 0.111 0.120 0.010 - -
Green 0.146 0.123 0.141 0.118 0.133 0.142 0.119 0.121 0.136
Red 0.151 0.124 0.149 0.119 0.141 0.153 0.135 0.128 0.143
NIR 0.494 0.514 0.495 0.512 0.487 0.511 0.495 0.484 0.501
MIR 0.308 0.283 0.305 0.275 0.299 0.309 0.295 - 0.301

Remote Sens. 2018,10, 1638 21 of 34
Table 4.
Sites used for comparing ground reflectances of selected materials with their values given by the USGS ASTER library. All of these sites are around
Quebec, Canada.
Materials Site Coordinates
Water (Lakes)
Hertel lake, Mont St-Hilaire 45◦32040”N; 73◦09000”W
Seigneurie lake, Mont St-Bruno 45◦32050”N; 73◦19035”W
L’Achigan lake, Laurentides 45◦56030”N; 73◦58000”W
Connelly lake, Laurentides 45◦53050”N; 73◦58000”W
Asphalt (Roads) Highways arround Montreal
Deciduous trees Ste-Thérèse-de-Blainville 45◦43015”N; 73◦49000”W
Verchères 45◦43016”N; 73◦18036”W
Table 5. Averages of DNs of the selected materials on Landsat and SPOT images.
Band
TM TM TM TM TM ETM+ ETM+ HRV HRG
17 June 1984 25 July 1992 18 June 1996 27 August 1998 27 June 2005 8 June 2001 11 August 2001 1 August 1987 29 July 2006
DN of water (lakes)
Blue 66 65 64 59 63 62 60 - -
Green 22 23 21 20 21 36 36 40 101
Red 15 18 16 14 16 24 25 22 63
NIR 13 20 15 14 15 13 12 16 19
MIR 8 9 7 6 7 11 10 - 21
DN of asphalt (roads)
Blue 101 88 90 78 91 94 89 - -
Green 46 40 41 34 41 76 75 50 151
Red 46 40 43 35 42 73 78 37 136
NIR 69 69 69 60 71 74 64 59 68
MIR 88 77 81 69 75 86 94 - 114
DN of vegetation (deciduous trees)
Blue 73 70 67 63 67 68 63 - -
Green 32 29 29 24 28 54 47 32 90
Red 22 22 21 19 21 35 32 17 58
NIR 116 112 122 87 123 124 100 88 118
MIR 82 67 74 57 75 82 71 - 109

Remote Sens. 2018,10, 1638 22 of 34
Table 6. Averages of ground reflectances of the selected materials on Landsat and SPOT images.
Band ASTER Lib.
TM TM TM TM TM ETM+ ETM+ HRV HRG
17 June 1984 25 July 1992 18 June 1996 27 August 1998 27 June 2005 8 June 2001 11 August 2001 1 August 1987 29 July 2006
ρground of water (lakes)
Blue 0.02 0.03 0.03 0.03 0.02 0.02 0.01 0.02 - -
Green 0.04 0.03 0.04 0.03 0.02 0.02 0.02 0.01 0.04 0.05
Red 0.03 0.02 0.03 0.02 0.03 0.03 0.02 0.01 0.04 0.03
NIR 0.00 0.01 0.02 0.02 0.00 0.01 0.02 0.01 0.03 0.02
MIR 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 - 0.01
ρground of asphalt (roads)
Blue 0.15 0.10 0.09 0.09 0.08 0.08 0.08 0.08 - -
Green 0.19 0.12 0.11 0.11 0.10 0.10 0.09 0.10 0.10 0.12
Red 0.22 0.11 0.10 0.10 0.09 0.09 0.09 0.10 0.10 0.11
NIR 0.26 0.25 0.26 0.25 0.25 0.25 0.25 0.24 0.25 0.23
MIR 0.37 0.20 0.19 0.19 0.18 0.17 0.18 0.20 - 0.19
ρground of vegetation (deciduous trees)
Blue 0.06 0.05 0.05 0.04 0.03 0.04 0.03 0.02 - -
Green 0.09 0.07 0.07 0.06 0.05 0.06 0.06 0.05 0.05 0.05
Red 0.06 0.04 0.04 0.04 0.03 0.03 0.04 0.03 0.03 0.03
NIR 0.55 0.42 0.43 0.44 0.40 0.44 0.45 0.40 0.39 0.40
MIR 0.31 0.18 0.16 0.17 0.15 0.17 0.17 0.16 - 0.18

Remote Sens. 2018,10, 1638 23 of 34
6. Reflectance Retrieval on Vegetation Targets
6.1. Comparison with ASD Measurements for Agricultural Targets
A series of reflectance measurements were made with the ASD FieldSpec HH Pro spectroradiometer
in agricultural fields (corn at several growth stages) located in the Montérégie, Quebec, Canada,
concurrently with the acquisition of the Landsat-7 ETM+ images (5 June 2000, 26 July 2001,
11 August 2001
, and 14 August 2002) and WorldView-2 images (5 July 2011) (Figure 15).
These measurements were compared with the ground reflectance values that were obtained
with REFLECT.
- Landsat-7 ETM+ images
The ASD measurements corresponding to the Landsat-7 ETM+ images showed significant variability
by date and measurement point (Figure 13), indicating a certain variability in vegetation density.
Figure 13.
ASD spectroradiometer measurements in agricultural fields located in Montérégie, Quebec,
Canada. The blue, green, red, and NIR spectral bands of ETM+ are indicated by blue, green, red,
and purple lines, respectively.
The comparison of the measured reflectance values and the reflectance values that are estimated
using REFLECT (Figure 14) shows that the estimates are fairly dispersed relative to the measurements
with mean absolute errors (|estimate-measurement|) of 0.012 in the blue band, 0.014 in the green
band, 0.019 in the red band, and 0.046 in the NIR, which corresponds to a mean relative absolute
error (|estimate-measurement|/measurement) of approximately 10%. These deviations seem to be
acceptable, since the images were obtained under variable atmospheric conditions and have spatial
resolutions (30 m) that are very different from those of the ASD measurements (1 circular metre). In the
case of the image of 5 June 2000, a systematic overestimation is observed, which was probably due
to the presence of a fine cloud layer that led to a high additive effect in all of the the spectral bands
(non-selective diffusion). If the AOD is increased to correct the additive effect of the blue and green
bands, the NIR reflectance is increased, since the transmittances (T
↓
and T
↑
) are reduced without
increasing the atmospheric reflectance (ρatm).
- WordView-2 image
The WorldView-2 has eight multispectral bands: coastal blue (400–450 nm), blue (450–510 nm),
green (510–580 nm), yellow (585–625 nm), red (630–690 nm), red-edge (705–745 nm), NIR-1
(770–895 nm), and NIR-2 (860–1040 nm, including water vapour absorption lines between 900–990 nm).

Remote Sens. 2018,10, 1638 24 of 34
Differences between the estimated reflectance from the WorldView-2 image and ASD measurements
are slightly less than those found for ETM+ images (mean relative absolute error of less that 10%)
(Figure 15). The good correction of the coastal blue band indicated that the AOD spectral extrapolation
formulae that was proposed in REFLECT gave correct AOD values for the lower wavelengths (close to
400 nm). The original power law used in 6S gave over estimation of AOD and thus overcorrection in
this band (data not shown). The reflectances that were estimated for NIR-1 and NIR-2 (particularly
affected by water vapour absorption) bands were very close, indicating a good correction of water
vapour effect in the NIR domain.
Figure 14.
Comparison of reflectances measured (by ASD) and estimated (by REFLECT) for Landsat-7
ETM+ images.
Figure 15.
Comparison of reflectances measured (ASD) and estimated (REFLECT) for the World-View-2
image of 5 July 2011.

Remote Sens. 2018,10, 1638 25 of 34
6.2. Formosat-2 Images of a Vegetal Surface: Atmospheric and Adjacency Effects
This test shows the atmospheric corrections—including the adjacency effect—for nine Formosat-2
images that were provided in apparent reflectance values (16-bit format). The AOD was determined
on dense vegetation dark targets only, since water was too affected by specular reflection (Figure 16a).
In order to show the effect of atmospheric conditions as well as the adjacency effect on agricultural
applications, we selected a vegetated surface that was located in the Boucherville area, Quebec, Canada,
with very bright edges in the red band (Figure 16b).
Figure 16.
Example of an agricultural field bordered by glossy surfaces in the red band on the
Formosat-2 image dated 5 June 2005. Location coordinates: 45
◦
33
0
30”N; 73
◦
25
0
30”W; (
a
) global
location; (b) zoom on the field.
For the nine Formosat-2 images used (acquired on 5 June, 19 June, 21 June, 22 June, 25 June,
27 June, 28 June, 2 July, and 3 July 2005), we calculated the apparent NDVIs and extracted the profiles
by field width (Figure 17a). Figure 16b shows that the apparent NDVIs are very different, even for
dates separated by only one or two days; this is due to the difference in atmospheric conditions.
The adjacency effects can also be seen on the NDVI profiles: except for the mixed pixels (containing
soil and vegetation), the pixels at the edge of the field have lower NDVIs than the pixels in the centre
(Figure 17b).
Figure 17.
Profile of apparent NDVI (normalized difference vegetation index) extracted over the width
of the plant surface selected on the Formosat-2 images that were used.

Remote Sens. 2018,10, 1638 26 of 34
Figure 18 shows the red and NIR apparent reflectance values of the study area, as well as the
environment reflectance values that were calculated as explained previously.
Figure 18.
Example of apparent reflectance and environmental reflectance in the red and NIR bands
for the selected farm field on the Formosat-2 images used.
Figure 19 illustrates the NDVIs calculated from ground reflectance values retrieved by REFLECT.
Figure 19.
Corrected NDVI profile extracted on the width of the selected field on the Formosat-2 images.

Remote Sens. 2018,10, 1638 27 of 34
Correction of the atmospheric effects reduces the NDVI underestimation obtained with apparent
reflectances [
64
]. In addition, with correction of the adjacency effect, the edge pixels have values closer
to those of the centre. By excluding the ground pixels (numbered 1 to 4 and 14 to 18) and the mixed
pixels (numbered 5, 6, and 13), the comparison between the apparent and corrected NDVIs of the
field pixels (7 to 12) with the centre pixels (9 and 10) shows that the NDVIs that were calculated from
the corrected images are less affected by the adjacency effect than the apparent NDVI. In addition,
the adjacency effect is greater for pixels 11 and 12, which were located on the side of the field where the
reflectance values in the red band from the neighbouring surface are higher (northeast direction, pixels
14 to 18). Correction of the adjacency effect for pixels 11 and 12 is clearly expressed by the reduction in
the difference between their NDVIs and those of the pixels in the centre of the field.
7. Reflectance Retrieval for Inclined Surfaces
We conducted two validation tests for the topographic corrections that were used in REFLECT:
the first for flat inclined planes (building roofs) in a high northern latitude, and the second for a
high-elevation mountainous terrain covered by forests. In the first case, the choice of high latitudes
results in a high solar zenith angle and very low atmospheric effects because of the clear sky. The choice
of high elevation in the second case also minimises the incidence of aerosol particles, which are fairly
rare at these elevations.
7.1. Flat Inclined Surfaces on an Ikonos Image
On an Ikonos image dating from 29 August 2002 covering an area of the Paulatuk region in
the Northwest Territories, Canada, we identified large buildings with inclined symmetrical roofs.
Since solar elevation was very low, the effect of the difference in irradiance on the two sides of the roof
is quite significant (Figure 20).
Figure 20.
Ikonos image of Paulatuk region de, Nordwest Territories, Canada, showing the illumination
difference of tilted rooftops. Scene center is 69
◦
20
0
30”N, 124
◦
04
0
00”O; sun zenith angle is 60.06
◦
;
sun azimuth
is 182.75
◦
; viewing angle is 25.49
◦
; viewing azimuth is 287.78
◦
; AOD = 0.05 (very clear
sky).
The roof slopes can be approximately estimated (ignoring the difference in azimuth between the
sensor and the surface) from the ratio of the apparent lengths of each of the half roofs by means of a
trigonometric calculation (Figure 21).

Remote Sens. 2018,10, 1638 28 of 34
Figure 21. Calculation of the slope of the rooftop parts from viewing geometry of the Ikonos image.
The ratio between the half roof with a large apparent width and the half roof with a small apparent
width can be expressed as follows: A
B=L+d
L−d=k; therefore: d=Lk−1
k+1.
From the view incidence angle, we obtain: tan(iv) = d
h; therefore: h=d
tan(iv).
The slope of the roof can be expressed as follows: tan(β) = h
L=d
tan(iv)L=L(k−1)/(k+1)
tan(iv)L
Finally: β=arctan((k−1)
(k+1)tan(iv))
Angle i
v
represents the view incidence angle, and is calculated by taking into account the curvature
of the earth; the result is i
v
= 28.44
◦
. For buildings 1, 2, and 3 (Figure 19), the kvalues are 8/6, 7/5,
and 4/2.5 pixels, which gives βslopes of 15◦, 18◦, and 24◦, respectively.
By considering these slopes with the buildings’ orientations, the illumination (solar zenith of
60.06
◦
and solar azimuth of 182.75
◦
) and viewing parameters (incidence angle of 28.44
◦
), the REFLECT
radiometric corrections that were used with and without accounting for the topographic effects gives
the results provided in Table 7.
Table 7.
DNs and ground reflectances without and with topographic corrections for bright and dark
sides of the rooftops in Ikonos Paulatuk.
Bands Roof 1 Roof 2 Roof 3
Bright
Side
Dark
Side
Bright
Side
Dark
Side
Bright
Side
Dark
Side
Orientation (◦)225 45 225 45 135 315
DNs
Blue 35 23 40 26 38 25
Green 26 17 31 18 33 17
Red 16 10 22 10 25 11
Reflectance without
topographic corrections
Blue 0.166 0.065 0.2087
0.0903
0.192 0.082
Green 0.107 0.051 0.1377
0.0576
0.150 0.051
Red 0.097 0.056 0.1377
0.0560
0.158 0.063
Reflectance with
topographic corrections
Blue 0.123 0.109 0.152 0.141 0.144 0.145
Green 0.082 0.081 0.102 0.093 0.109 0.095
Red 0.076 0.081 0.104 0.097 0.118 0.122
From this table, we can see that there are significant differences between the DNs of the light
and dark sides of the three buildings. The same is true for the ground reflectance values without
topographic corrections. We can also see the effectiveness of the topographic corrections in finding
reflectance values that are very close for the two sides of the same building. Hence, the proposed
topographic correction method works well for flat surfaces.

Remote Sens. 2018,10, 1638 29 of 34
7.2. Forest Canopy in Inclined Terrain on SPOT-1 Images
For the validation (and adaptation) of the topographic corrections in forested environments,
we used three SPOT-1 images (5 September 1988, 24 September 1989, and 3 September 1990) of the
Tarn region, France (43
◦
39
0
N; 02
◦
42
0
E), with a highly varied relief (elevations between 300–1200 m).
The result shows that the distribution of the DNs of the NIR band for low slopes (
β
< 8
◦
) is not much
affected by the orientation of the pixels
α
relative to the solar azimuth
ϕs
. On the other hand, for slopes
of more than 18
◦
, surfaces oriented towards the sun (|
α−ϕs
| <20
◦
) have DNs that are clearly higher
than those of surfaces with low slopes. However, surfaces with slopes of more than 18
◦
and oriented
in the direction opposite the sun (|
α−ϕs
| > 160
◦
) have DNs that are barely lower than the surfaces
with a low slope. Hence, in a forest environment, the magnitude of the slope effect varies depending
on the orientation of the surfaces relative to the sun. This observation has also been reported by
Ekstrand, S. [65].
For atmospheric corrections, AOD values were estimated using the dark targets method on the
lakes in the scene. Values obtained were 0.05, 0.05, and 0.07 for the 1988, 1989, and 1990 images,
respectively. First, we used REFLECT to calculate the ground reflectance values without considering
the topographic effects. The overestimation of the reflectance values in inclined terrains oriented
toward the sun is observed. However, surfaces oriented in the direction opposite to the sun are not
significantly underestimated relative to the flat terrains. This could be explained by the geotropic
nature of trees and the amount of shading within the canopy that varies depending on the slope
exposition to the sun [
66
]. Hence, by applying the topographic correction of the model that we
developed for flat inclined surfaces, a very high overcorrection is obtained for the surfaces that are
oriented in the direction opposite the solar azimuth. The model that was developed for inclined planes
is not applicable to a forest canopy. Therefore, one would expect that the equivalent slope of a forest
canopy would be gentler than that of the terrain in which it is located. We therefore propose a slope
correction that depends on the orientation of the pixel relative to the solar azimuth given by:
β0=κ(fs−α)β(15)
The parameters
β
and
β
’ are actual and corrected slopes, respectively. The function
κ(ϕs−α)
,
which is less than or equal to 1, is determined by analysing the data derived from the SPOT images.
In order to find the equivalent slope that adapts our topographic correction model to the forest
canopy, we looked for the multiplicative factor
κ
(by varying it from 1 to 0), which minimises the
RMSE between the ground reflectance of inclined terrains (three slope ranges: 6
◦
<
β
< 12
◦
, 12
◦
<
β
< 18
◦
, and
β
> 18
◦
) and of flat terrains (
β
< 6
◦
) for all of the forested area of the three SPOT-1 images
used, and all of the forest stands combined. The obtained values of
κ
can be smoothed based on the
difference between the solar azimuth
ϕs
and the orientation of the surface
α
, which was noted as
δϕ
,
by the function :
κ(δϕ) = A+B1−exp(−2π
|δϕ|)C
(16)
where A= 0.1, B= 0.9, and Cis an exponent that defines the shape of the curve. All that remains
to determine is whether these values are of the same order of magnitude under other conditions
(other images, other forested environments, etc.).
The application of topographic correction with the equivalent slopes principle yields very good
results for the NIR band (the most affected by topographic effects in our example) of the SPOT-1
images used; the mean error is 0.02 reflectance units. This error is very small compared to that obtained
with the Minnaert model [
67
], which is frequently used for this type of application [
68
,
69
], and yields
an error in the order of 0.1 reflectance units. For example, we can see on the NIR band of the image of
5 September 1988 (Figure 22), in the circled areas, that the ground reflectance values of the inclined
surfaces (steep slope) oriented toward the sun (low |orientation-azimuth|) have been homogenised
relative to the digital numbers.

Remote Sens. 2018,10, 1638 30 of 34
Figure 22.
Example of topographic effects reduction by the equivalent slopes model applied to an NIR
band of the SPOT 1 image acquired on 5 September 1988 in the Tarn region, France.
8. Conclusions
The objective of this research was to develop a tool (REFLECT software package) for
ground reflectance restitution, which considers all of the radiometric effects affecting multispectral
images, including illumination and viewing conditions, sensor properties, atmospheric conditions,
and topography. REFLECT uses the formulation and the routines of the 6S code, and introduces
several new features such as: the incorporation of a more recent aerosol model (OPAC); the proposal
of a law of extrapolation of the AOD at 550 nm to the other wavelengths that are better adapted to
the studied area (Quebec, Canada); accounting for the adjacency effect by means of a model inspired
by the Phong model; improvement of the dark targets method that is used for estimating the AOD
with the possibility of subdividing the image and the semi-automatic choice of the target “darkness”
criterion; topographic corrections for smooth surfaces by means of a model that separates the direct
and diffuse components of solar radiation and for forest canopy by means of the “equivalent slopes”
principle; and the reduction of processing time in the case of large images by generating internal
look-up tables (LUTs) of atmospheric parameters.
Validation tests showed that the ground reflectance is retrieved with a good accuracy:
the variability of this reflectance for the same surfaces was approximately
±
0.01 among the atmospheric
conditions and sensors for all of the spectral bands. The topographic correction that was proposed in
REFLECT allowed reducing the error of ground reflectance retrieval for flat inclined surfaces (roofs)
observed by an Ikonos image from 0.1 without topographic corrections to 0.01. This topographic
correction model adapted to forest canopy reduced the error of NIR reflectance retrieval from three
SPOT images from 0.08 (without topographic correction) to less than 0.02 (with the “equivalent slopes”
principle). These performances are very satisfactory compared to ATCOR (used in PCI Geomatics) or
FLAASH (used in ENVI), according to the comparative study of Fuyi, T. et al. [70].
Author Contributions:
Conceptualization, Y.B. and F.C.; Methodology, Y.B.; Software, Y.B.; Validation, Y.B., F.C.
and N.Y.; Formal Analysis, Y.B.; Investigation, Y.B.; Resources, N.T.; Writing-Original Draft Preparation, Y.B. and
W.B.; Writing-Review & Editing, W.B.; Supervision, F.C.; Funding Acquisition, N.T.

Remote Sens. 2018,10, 1638 31 of 34
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
AOD550 aerosols optical depth at 550 nm
E0(λ) exoatmospheric solar spectral irradiance
Edir direct solar irradiance
Ediff, sky diffuse sky irradiance
Hrrelative humidity
Hsat height of the satellite
Lsat apparent luminance at the satellite level
Pspartial pressure of water vapour in saturated air
Salb spherical albedo of the atmosphere
Tgas total gas transmittance (descending and ascending path)
Taambient temperature
isangle of incidence of the solar radiation on an inclined terrain
t↑dir, t↓dir diffusion transmittances of the direct solar radiation in the ascending and descending paths
t↑diff, t↓diff diffusion transmittances of the diffuse solar radiation in the ascending and descending paths
wtotal water vapour content in the atmospheric column
αorientation of inclined surface
βslope of inclined surface
β0corrected slope (for vegetation)
λwavelength
θssolar zenith angle
θvsensor viewing angle (relative to the nadir)
ρatm atmospheric reflectance
ρsat reflectance of the target at the satellite level
ρenv reflectance of the environment
ρBbidirectional components of the target reflectance ρtar
ρHD hemispherical-directional components of the target reflectance ρtar
τtotal optical thickness of the atmosphere
ϕssolar azimuth angle
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