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TRANSCRIPT
Non-normal distribution of the Top-Of-Atmosphere satellite optical
measurements over calibration sites
Javier Gorroño1,2*, Agnieszka Bialek1,2, Paul D. Green1, Peter Harris1, Tracy Scanlon1,
Nigel P. Fox1 and Craig Underwood2
1National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW
2Surrey Space Centre, University of Surrey, Guildford, Surrey, GU2 7XH, UK
*corresponding author
E-mails: [email protected], [email protected], [email protected]
[email protected], [email protected], [email protected] and
This work was supported by the Centre for Earth Observation and Instrumentation (CEOI), the
National Measurement System (NMS) and European Metrology Research Programme (EMRP).
The EMRP is jointly funded by the EMRP participating countries within EURAMET and the
European Union.
Non-normal distribution of the Top-Of-Atmosphere satellite optical
measurements over calibration sites
This paper studies the distribution associated with the measurement of the
satellite derived Top-Of-Atmosphere (TOA) reflectance on a pixel-to-pixel level,
within a defined spatial region of interest (ROI) within a vicarious calibration
target site. The study analyses the effects of the atmosphere and surface
reflectance distribution spatial shape. The analysis shows that some of the
contributing effects are inherently non-linear, so produce non-normal
distributions. For these non-normal distributions, the use of the mean and
standard deviation alone does not allow sufficient parameterisation of the
distribution to capture all the information associated with the ROI reflectance
measurement. Therefore, additional information concerning the distribution is
required to provide a full site reflectance characterisation. This additional
information can be useful in establishing the sources of change in the distribution
and ultimately improve the site radiometric characterisation, particularly for long
term monitoring. In this study LandSat-8 (L8) Operational Land Imager (OLI)
measurements over the CEOS Libya-4 Pseudo Invariant Calibration Site (PICS)
are used as a demonstration.
Keywords: Libya-4; PICS; non-normal distribution; asymmetry
1. Introduction
Several EO missions currently use desert sites to monitor the temporal evolution of
optical sensors on-board satellites, as well as for the inter-calibration of different optical
sensors in-orbit. Desert sites are chosen because of their relatively stable and uniform
surface conditions and the characteristics of the overlying atmosphere. Such sites were
initially studied and defined in Cosnefroy, Leroy, and Briottet (1996) and further
revisited in Lachérade et al. (2013). In 2010 a subset of six of these sites were
identified by Committee of Earth Observation Satellites (CEOS) to serve as community
Pseudo-Invariant Calibration Sites (PICS) of which Libya 4 historically has had the
most international attention.
The ideal configuration for sensor to sensor inter-calibration using vicarious
targets is that the two instruments should make matched measurements viewing the
same target at the same time; with the same spatial and spectral responses at the same
viewing geometry. Similarly, for the temporal monitoring of an instrument, stable
acquisition conditions and angular geometry are required over time. Since these
idealised conditions rarely occur in reality, there will always be some need for
additional compensatory steps to allow comparison of the two instruments. Therefore,
inter-calibration and temporal monitoring of optical sensors requires many observations
made at different times, different illuminating/viewing geometries, and over different
reference targets, in order to reduce the associated uncertainties statistically (Lachérade
et al. 2013).
Several efforts have been made to characterise and model the atmosphere and
surface reflectance of desert calibration sites. They range from semi-empirical models
as in Bouvet (2014), to empirical models as in Mishra, et al. (2014) and, as in Govaerts
(2015), they study specific effects, such as the dune morphology. In addition, efforts
have been made to provide automatic extraction algorithms and databases, such as
SADE (Cabot 1997; Cabot et al. 1999), that provide a long-term record of instrument
acquisitions over several desert sites.
In parallel to the data management and site characterisation efforts, there has
been a continuous improvement of the satellite instrument uncertainty levels. In the next
decades, proposed missions such as TRUTHS (Traceable Radiometry Underpinning
Terrestrial- and Helio- Studies) (Fox et al. 2011) led by the National Physical
Laboratory (NPL), UK, or its NASA-proposed sister mission CLARREO (Climate
Absolute Radiance and Refractivity Observatory) (Wielicki et al. 2013), could further
reduce the radiometric uncertainty of satellite optical sensors to unprecedented levels.
The continued improvement of the vicarious sites’ characterisation, together
with the upgrading of the satellite optical sensor performance, will eventually require a
reconsideration of the main sources of limitations for the inter-calibration and temporal
monitoring over vicarious sites. Re-evaluating some of the assumptions will ultimately
allow the full exploitation of these new opportunities in the mid- and long-term.
This paper revisits the adequacy of a two-term parameterisation (mean and
standard deviation (SD)) to describe the full distribution of the pixel-to-pixel
radiometric information for a vicarious calibration site. The traditional approach for
inter-calibration and instrument monitoring uses the expectation value (commonly, the
mean) of the pixel-level TOA reflectance/radiance and evaluates the dispersion of
values across the ROI by measuring the SD (Lachérade et al. 2013). However, these two
parameters can only fully describe the distribution if it is normal. In the scenario of a
non-normal distribution, these two parameters are not sufficient to capture all the
relevant information. The additional parameterisation of the distribution can provide a
better understanding of the spatial and temporal dynamics of the measurements. This is
possible by attributing the changes of the distribution shape to different sources
associated with either the vicarious site and/or the on-board optical sensor, as is partially
demonstrated within this paper.
2. The TOA reflectance distribution of a ROI for Libya-4
The CEOS Libya-4 site is considered as one of the most suitable sites for the
radiometric monitoring and cross-calibration due to its excellent climatology. The study
in Cosnefroy, Leroy, and Briottet (1996) reported < 1 mm average monthly
precipitation over the area and a majority of clear-sky days. Furthermore, the same
study also showed a temporal variation —after removal of directional effects — of the
TOA reflectance of 1-2% in the Visible and Near-Infrared (VNIR) region. The
follow-up study in Lachérade et al. (2013) pointed to values < 3 % in radiometric spatial
homogeneity for a ROI of 0.45º × 0.45º latitude and longitude.
The study here does not focus on the temporal and spatial mean and SD
associated to these ROIs but on a better characterisation of the TOA reflectance
distribution. This chapter intends to demonstrate the non-normality of the data
distribution by using Libya-4 site as an example.
To evaluate the TOA reflectance probability distribution, Level 1 L8 scenes over
Libya-4 for the available images closest to the summer (22 June 2014 – day 173) and
winter solstices (31 December 2014 – day 365) of 2014 have been selected. These dates
represent the two extreme positions of the Sun throughout the year. The ROI centre
position is selected at the centre of the Libya-4 site as defined in Lachérade et al. (2013)
— i. e. 28.55º N, long: 23.39º E — and with a size of 50 km × 50 km (approx.
0.45º × 0.45º latitude and longitude). The Figure 1 presents the TOA reflectance images
associated to one of the Landsat-8 OLI products for the Day 173 of Year 2014 and
bands 1, 5, and 7. The crosshair represents the centre of Libya-4 site as defined in
Lachérade et al. (2013).
[Figure 1]
It is noted that the OLI instrument Field Of View (FOV) of approximately
185 km (Irons 2012) permits the selection of a number of different scale ROIs within a
single image, however a 50 km × 50 km ROI provides a sufficient number of pixels
(>106) to provide suitable details of the distribution of reflectance across the site whilst
being within the boundaries defined in Lachérade et al. (2013). It is noted that, for both
images, the cloud cover declared in the image metadata is lower than 10 % and no
pixels within the ROI are marked invalid in the data provider quality assessment mask
band.
The L1 Digital Number (DN) in the L1 product are converted to TOA
reflectance, ρλ', as (USGS 2015):
ρ λ' =M L(DN )+ AL (1)
Where ML refers to the reflectance multiplicative scaling factor for the band and
AL refers to the reflectance additive scaling factor for the band. Both values can be
extracted from the product metadata.
The obtained reflectance values are not yet corrected for the Solar Zenith Angle
(SZA). The SZA for the centre coordinates of the Libya-4 site (noting this is different to
the SZA provided for the centre of the image within the L8 OLI metadata) is calculated
using the image timestamp and lat / long position at the centre of the ROI using the
Pysolar library (Stafford 2015). The SZA is then used to normalise the reflectance
values between the images by 1/cos (θ) where θ refers to the SZA (USGS 2015).
In Figure 2, the TOA reflectance pixel distributions for L8 band 1 (B1; central
wavelength (CW) ≈ 443 nm), band 5 (B5; CW ≈ 865 nm) and band 6 (B7; CW ≈
2201 nm) at the two different days of the year are shown. These particular bands are
chosen since B1 is dominated by atmospheric effects, B5 is dominated by surface
effects and B7 is in the SWIR region. The latter is also dominated by the surface effects
but built on Cadmium Mercury Telluride (CMT) detectors (Knight and Kvaran 2014).
In order to capture the distribution shape, the binning width of B1 has been
selected as 0.001 whereas for B5 and B7 the binning width increases to 0.005, these
values being chosen as a compromise to allow optimal visualisation of the distribution
shape. For all panels in Figure 2, the distribution is normalised to an area equal to 1 and
the theoretical normal distribution — obtained from the mean and SD of the distribution
— is plotted as a reference.
[Figure 2]
A significant level of asymmetry in the reflectance distribution can be seen in all
panels, with a skew towards high reflectance values. The reference normal distribution
disagrees in all the cases to a large extent, with Figure 2 (a) being the only the case
where the level of agreement is considerable.
In order to understand the level of asymmetry from a quantitative point of view,
several metrics are shown in Table 1. The statistics include the traditionally-used
distribution mean and SD but also other parameters that determine the level of
asymmetry and non-normality of the data distribution. These are the median (or 50th
percentile), the first quartile (or 25th percentile) and the third quartile (or 75th percentile).
For the quartile values, the corresponding normalised probability has been provided in
parenthesis. In addition, we provide the interval shifted by subtracting the mean that
contains the value of TOA reflectance with a probability of approximately 68.27 %. For
a normal distribution, the interval of values [-σ, +σ] represents the 68.27 % (or k = 1)
coverage interval of the total probability distribution as specified in BIPM et al. (2008).
[Table 1]
Using the statistics in Table 1, the asymmetry can be quantitatively described by
the comparison of the median and mean parameters of the distribution and by
comparing the first and third quartiles of the distribution. For a symmetric distribution,
the 50th quantile (i.e. the median) coincides with the centre of the distribution (i.e. the
mean). In addition, the first and third quartiles should be equidistant from the median
and their associated normalised probability coincident. However, in none of the cases
presented in Figure 2 and Table 1 do the parameters coincide. The maximum difference
appears during the winter overpass (Figure 2 (d), (e) and (f)) with a minimum difference
for B1 in summer overpasses (Figure 2 (a)). It is noted that, even for B1 in summer,
there is a significant difference between the normalised probability associated to the
first and third quartiles (in Table 1 column (a); 56.40. c.f. 46.47).
The level of non-normality of the distribution can also be enumerated by the
difference between the SD and the approximated 68.27% probability coverage interval
around the mean. In Figure 2, these two dispersion metrics disagree for all the panels.
The most pronounced disagreement occurs for B7 during the winter (in Table 1 column
(f); 7.43 % c.f. 5.01 %).
In addition, there are specific parameters that can be used to describe the shape
of the distribution. The ones introduced here are the skewness as a measure of
asymmetry and the excess kurtosis as an indication of the flatness vs. sharpness of the
distribution. They are defined as follows (Zwillinger and Kokoska 1999):
skewness=m3
m23/2 ; excess kurtosis=
m4
m22
(2)
Where the values of mr refer to the rth moment of the distribution and it is
defined for n observations as:
mr=1n∑i=1
n
( xi− x̄ )r
(3)
A value of skewness close to zero indicates a symmetric distribution. When the
parameter is positive, it tends to indicate the effect of a distribution tail on the right and
vice versa when it is negative. The “excess kurtosis” is calculated here by taking the
standard definition of the kurtosis for each TOA distribution and subtracting three,
which is the kurtosis for a normal distribution (i.e. the excess kurtosis is equal to zero
when the distribution is normal). For positive values, it indicates a sharpness of the
distribution — w. r. t. the normal distribution — with a well-defined peak.
The skewness is positive for all the panels presented here, indicating the larger
tail of the distribution towards the right side. The excess kurtosis is also positive in all
panels meaning that the distribution tends to be sharper than a normal distribution. This
can also be appreciated in Figure 2 since the mode of the distribution is always higher
than the mode of the “reference” normal distribution. Furthermore, the skewness and
excess kurtosis are both higher for B5 and B7 than for B1. This higher level indicates
that B5 and B7 are highly-peaked distributions with a long tail to the right side, as can
be seen from the panels in Figure 2.
As presented in Figure 2, and confirmed with the statistics in Table 1, the TOA
reflectance distribution of L8 values over the Libya-4 calibration site is typically not a
normal distribution, but contains a significant level of asymmetry and sharpness. The
non-normality of the distribution varies depending on the spectral region and the time of
the year and cannot be clearly defined with any specific shape. This implies that:
The SD does not necessarily link the dispersion of the values with a certain
probability in the range [-σ, +σ].
Additional information on the TOA reflectance distribution can be useful in
order to attribute better the sources of variation in the distribution.
3. Analysis of the asymmetry of the TOA reflectance distribution of an ROI
forLibya-4
In this section we aim to identify the contributing factors that lead to the observed
asymmetric distribution of the TOA reflectance values specific to the Libya-4 site for
the date / time under consideration.
It is considered that the asymmetry associated with the instrument calibration is
not a significant contributor to the overall asymmetry shown at the Libya-4 site. The
instrument noise produced by the scanning of the pixels in the Across-track (ACT)
direction is assumed to be normally distributed since it is produced by the temporal
noise of the detectors. In the Along-track (ALT), the residual of pixel-to-pixel non-
uniformity of the L8 OLI images has been evaluated in between 0.2% and 0.3%
(Morfitt, et al 2015). This value represents, in a worst case, less than 10% of the SD
quoted in Table 1. In addition, the analysis has included the B5 and B7 that are both
dominated by the surface effects but the focal plane detectors are technologically
different. The results in Table 1 have shown similar levels of asymmetry which further
indicates the low sensitivity of the asymmetry to the detector noise. Furthermore, the
level of noise of the OLI instrument has been tested pre-flight and further verified in-
flight. The work in Ren et al. (2014) verified using ground sites imagery that the Signal-
to-Noise-Ratio (SNR) is above 160 at typical radiance level. Considering these as the
worst case scenario, the noise would account for a small fraction of the dispersion of
values quoted in Table 1.
Thus, the origin of the asymmetry seen in all panels of Figure 2 is likely to be
predominantly attributed to atmospheric and surface effects, both of which are discussed
in the following sections.
Atmospheric Effects
A potential origin of asymmetry in the TOA reflectance pixel distribution shown in
Figure 2 may be the effect of atmospheric transmission. Atmospheric transmission, Tatm,
can be modelled by the Beer-Lambert law:
T atm=e−mτ(4)
where τ refers to an integrated vertical optical depth (which can be separated for
multiple contributions: aerosols, oxygen, water vapour, etc.) and m is the relative air-
mass factor. The atmospheric optical depth is a consequence of the atmospheric state,
whereas m depends on the SZA and viewing conditions.
For the purpose of this study the air-mass factor is expressed as m = 1 +
1/cos (θ), where both up-welling and down-welling paths are considered. This air-mass
factor is valid for the range of SZA measured by L8 OLI and nadir viewing, represented
by the additional “1” in the equation, (note that at higher values of SZA the curvature of
the Earth would need to be taken into account). The SZA variation across a ROI size of
50 km × 50 km is, at maximum, ±0.2º (calculated using (Stafford 2015)). This small
angular range produces an essentially linear variation across the ROI of ±0.4 % for a
SZA of 60º (winter case) and ±0.1 % for a SZA of 20º (summer case). A similar linear
variation occurs in the reflectance conversion due to the 1/cos (θ) term. The application
of a fixed SZA from the centre of the ROI for the reflectance conversion, rather than a
local pixel-specific SZA, produces a linear spread in the calculated reflectance across
the ROI of ±0.6 % for a SZA of 60º and ±0.1 % for a SZA of 20º.
Therefore, the use of orthorectified data without individual correction for the
variation in SZA does not limit the analysis of the distribution of the atmospheric
effects. The effect of both the relative air-mass factor and reflectance conversion is of
limited effect compared to the total dispersion of values in Figure 2 and it is largely
linearised.
The integrated optical vertical depth, τ, is likely to vary in the ROI as a
consequence of differences in angular conditions for each pixel and due to the different
atmospheric composition along the path length. The differences in angular conditions
are produced by either a change of the solar or viewing angles across the ROI.
For the effect of angular conditions, the comparison of B1 results in the
temporal domain (Figure 2 (a) and (d)) are examined. The difference between the mean
and median indicates a better resistance of the latter against changes in the skew shape
of the distribution (2.9 % change of the mean vs. 2.1 % change of the median). This is
the expected result of a change in skew of a log-normal distribution as expected from
Equation (4). The result indicates that the B1 shape is largely dominated by the
atmospheric variations within the ROI as a consequence of per-pixel angular changes.
No in-situ atmospheric data of the Libya-4 site is available that can provide
reliable information about the variability and distribution shape of the atmospheric
composition across the entire ROI. This information could be obtained by analysing
aerosol products, for example, however, the performance of these products tends to be
limited for high reflectance surfaces. The MODIS deep blue aerosol product, for
example, largely overcomes these limitations, but the applicability of the product in the
Sahara desert tends to be challenging due to the high surface reflectance (Sayer et al.
2013; Shi et al. 2013).
In order to test the later point, a study of the MODIS collection 6 Aqua Level 2
aerosol products at 10 km resolution is performed, centred at the Libya-4 site with a
4º × 4º latitude and longitude scenes, for a period of approximately one year. The
retrieved Aerosol Optical Thickness (AOT) at 550 nm (‘Deep Blue Aerosol Optical
Depth 550 Land’) is selected for a 100 × 100 km (100 pixels) ROI. Only pixels with the
best confidence flag (i.e. QA=3) are selected. Images are discarded where the mean
AOT exceeds 1, since it is assumed that this is an exceptional phenomenon as e.g. a dust
storm. Only images with <= 90 % valid pixels are used. The Figure 3 shows the
distribution (bin size 0.01) of the Aerosol Optical Thickness (AOT) pixel values for the
100 × 100 km ROI in Libya-4 for the period from 12 February 2014 to 9 February 2015.
[Figure 3]
The Figure 3 clearly shows how the distribution shape follows a multi-peaked
pattern. It is possible to distinguish the overlap of (at least) four distributions which are
likely to be the result of different aerosol micro-physical model selection. In Libya-4
site, the aerosol contribution at the TOA becomes small and more sensitive to the
surface reflectance model and/or assumptions in the aerosol microphysical properties
during the retrieval. Nonetheless, the Figure 3 demonstrate the non-normality of spatial-
temporal variations of AOT. This exercise also exemplifies how the use of AOT
products to analyse the distribution shape in a ROI has important limitations.
Despite the absence of reliable information, there is an expected asymmetry in
the TOA reflectance distribution as a consequence of the atmospheric composition
variation. The Figure 3 demonstrates the expected asymmetry of the aerosol variations.
The atmospheric optical depth propagation is exponential w.r.t. air mass factor and
optical depth, as indicated in Equation (4). Only in the case when the variations are
sufficiently small to linearise the process, would the impact on the TOA reflectance
distribution be symmetric.
However, this situation is not true for B5 and B7 in Figure 2 since the median
does not show better resistance to distribution changes in the temporal domain and,
indeed, the TOA reflectance decreases for the winter case. However, as mentioned in
the previous section (and shown in Table 1), the skewness and excess kurtosis for B5
and B7 are higher than for B1 during both the summer and winter; this indicates that
atmospheric changes are not likely to be the dominant cause of the main changes
observed in the distribution shape.
Surface Effects
To determine if the surface is impacting the TOA reflectance over Libya-4, a surface
reflectance model is derived using an elevation model and a Bi-directional Reflectance
Distribution Function (BRDF) model. The distribution of the surface reflectance from
this model is then studied. It is noted that atmospherically-corrected L8 OLI images are
not utilised for this purpose due to the limited atmospheric data available for the Libya-
4 site (as mentioned in the previous section).
The ASTER Global Digital Elevation Model version 2 (GDEMv2) is used to
provide altitude information at 30 m spatial sampling (matching the L8 OLI instrument
spatial resolution) with 1 m altitude resolution. The GDEMv2 validation in Tachikawa
(2011) reported a mean elevation error ranging from +3.34 m to +15.02 m, however,
specific errors are not reported for the Libyan Desert.
The ROI selected here to provide the altitude at Libya-4 is the same as that used
to derive Figure 2 (28.55º N 23.39º E with a size of 50 km × 50 km). The altitude
across the site and the associated distribution are shown in Figure 4 (a) and Figure 4 (b)
respectively. For the altitude distribution, a 2 m binning width has been selected, which
is just above the vertical resolution of the DEM.
[Figure 4]
The altitude image is converted to slope and aspect images using a 9 pixel kernel
gradient function (Liu and Mason 2009). The resulting slope and aspect distributions for
the ROI are shown in Figure 5 (a) and Figure 5 (b) respectively. The images of the
slope and aspect are also provided in Figure 5 (c) and Figure 5 (d). The slope
distribution (binned at 1º intervals) shows that the dunes in the area have low elevation
angles, with few showing a slope > 15º. The aspect distribution (binned at 10º intervals)
shows weak peaks around 0º and 180º which aligns with knowledge that the dune ridges
run North-South (as studied in Govaerts (2015)).
[Figure 5]
The Rahman-Pinty-Verstraete (RPV) model (Rahman, Pinty and Verstraete
1993) is utilised as the BRDF model. The model provides the reflectance, ρ, defined by
four parameters (ρ0, k, Θ and ρc) for the viewing and illumination conditions (SZA ≡ θs,
Viewing Azimuth Angle (VZA) ≡ θV and Relative Azimuth Angle (RAA) ≡ Δϕ) as
follows:
ρ(θS , θV , Δφ , ρ0 , k ,Θ , ρc )=ρ0 M1 (θS , θV , k ) FHG( g ,Θ) H ( ρc ,G)(5)
Where each one of the terms is defined as:
M 1(θS , θV , k )=cosk−1 θScosk−1θV
(cosθS+cosθV )1−k(6)
FHG( g , Θ)= 1−Θ2
(1+2Θ cos g )3
2 (7)
H ( ρc ,G)=1+1−ρc
1+G (8)
cos g=cos (θS)cos (θV )+sin (θS )sin(θV )cos( Δφ ) (9)
G=( tan 2(θS ) tan2(θV )−2 tan (θS ) tan(θV )cos( Δφ ))1
2(10)
The terms described in Equation (6)–(10) represent different features of the
reflectance function (Rahman, Pinty and Verstraete 1993). The amplitude component is
set by ρ0 and then modified by the term M1 which defines the overall shape of the
angular field using the parameter k. FHG is a Henyey-Greenstein function that provides
the balance between forward and backward scattering and is described through the
parameter Θ and g (described in Equation 9). H describes the hotspot effect through the
parameter ρc.
Values for k, Θ and ρc for the Libya-4 site have been extracted from the results
obtained in Bouvet (2014) for a surface BRDF model. The values have been derived
from data pertaining to the whole Libya-4 ROI site and therefore describe the BRDF of
large scale structures at the site such as the dunes. It is recognised that these values are,
therefore, not strictly valid for application to the 30 m spatial resolution model derived
here, however, in the absence of other data, they are recognised as the best available
parameterisation of the BRDF of the site using the RPV model for this application. A
more suitable model for the current application would be the use of the BRDF of the
sand reflectance; it is known this can be well characterised by the RPV model (Boucher
et al 1999), however, values for the model parameters relating to sand are not available.
To provide the overall surface reflectance model using the GDEMv2 and the
BRDF model, the angular configuration (SZA, SAA, VZA and VAA) are required. The
solar angular configuration is taken as an approximate of the summer solstice overpass
of L8 over Libya-4 (SZA= 21º and SAA= 99º) and winter overpass (SZA= 56 º and
SAA= 130). The Viewing Zenith Angle (VZA) is set equal to 0º (i.e. nadir viewing).
This configuration simulates an acquisition of the central pixel in the focal plane of L8
OLI instrument.
The global solar and view angular configuration, together with slope and aspect
information of each pixel, are used to transform to local viewing and solar angles for
each pixel within the ROI using equations provided in Hejmanowska 1992. The local
angular configuration of each pixel is then used as the input of the surface BRDF model
to provide the surface reflectance model.
Figure 6 (a) and Figure 6 (b) shows the surface reflectance distribution at 865
nm (which is the same as the L8 OLI B5 CW), for the summer and winter overpasses
respectively. The Figure 6 (c) and Figure 6 (d) are the simulated surface reflectance
image for the corresponding summer and winter cases. Only 865 nm is studied here as
this is the NIR band which is less prone to atmospheric effects and the study of only the
specular component in the surface reflectance is justified. The SWIR region is not
studied since the available surface BRDF model from Bouvet (2014) does not cover this
range. The selected reflectance binning width is 0.0005 for summer case and 0.002 for
winter case.
[Figure 6]
Figure 6 indicates that the dune shape results an asymmetric distribution for the
ROI surface reflectance. As previously stated, the viewing and solar angular
configuration has been fixed for all the pixels meaning that, the resultant distribution of
reflectance is independent (in this simulated case) of the variations in these across the
ROI (as would be seen in real L8 OLI imagery). Therefore, the source of distribution
shape shown in Figure 6 is inherent to the dune morphology.
The effect of the H term has very limited impact in our simulations, since the
local pixel angular configurations are not close to the hotspot angle (note that a nadir
viewing is set). A sensitivity analysis has shown that the function FHG has an impact on
the reflectance value, but only a very limited effect on the shape of the distribution.
Thus, the distribution shape is mainly dominated by the term M1 from Equation (6).
The parameters associated to the Figure 6 (a) and (b) distributions have been
reported in Table 2. These parameters are the same ones as reported in Table 1 and
previously explained.
[Table 2]
The difference between the dispersion of the values in Figure 2 (b) and (e) and
Figure 6 can be attributed partly due to the BRDF model ability to reproduce the local
angular variations and the low (but present) atmospheric effect in the NIR band of L8
(i.e. B5). It is noted that the small difference, due to the dates used, between the angular
configuration of the observations in Figure 2 and the simulations in Figure 6 is not
expected to be a large contributor to the differences in the dispersions.
The skewness and kurtosis in Table 2 are in both cases in positive values as in
Table 1. However, both skewness and kurtosis are higher for the summer case than the
winter case. This is the opposite of the statistics quoted in Table 1 for Figure 2 (b) and
(e). These differences could be explained at some extent by the combined effect of
surface and atmospheric non-normal effects.
The results in Figure 6 and Table 2 tend to indicate the difficulty to model the
distribution shape due to the dune morphology. Nonetheless, this exercise is considered
a sufficient reference to demonstrate the non-normal distribution of the surface
reflectance as a consequence of the dune morphology.
Thus, for spectral regions with a TOA reflectance dominated by the surface
reflectance and with a spatial resolution lower than the dune size, the dune morphology
will have an important impact in the distribution shape and asymmetry. For example,
this is the case of B5 and B7 in Figure 2 (b), (c), (e) and (f). The atmospheric
transmission for these two bands is higher and, even during the summer solstice, an
important asymmetry persists in the distribution due to the dune morphology.
It could be possible that some variation in reflectance is introduced by the
surface sand shape and variability in its composition. Bouvet (2014) discusses the
variability of spectral properties of sand in the visible to NIR spectral region and how
the composition changes could be explained by the presence of iron oxide coatings on
sand grain surface as indicated in Bullard and White (2002). The deposition of iron
coatings and different grain shapes are expected to vary between the windward and slip
face of the dune due to wind transportation effects. In the absence of qualifying data,
this effect is expected to result in a quasi-normal surface reflectance distribution at each
side of the dune if a random distribution of sand is considered. Nonetheless, future
access to in-situ samples should confirm this assumption.
4. Discussion
L8 OLI data has been used to show the asymmetry and non-normal shape of the TOA
reflectance distribution over the Libya-4 site. The observed non-normality of the
distribution has been analysed in different ways. The first metric parameterised the
asymmetry of the distribution by comparing the mean and median values of the
distribution and the first to the third quartiles of the distribution. The second metric
studied the skewness and kurtosis to evaluate the asymmetry of the distribution and the
sharpness with respect to a normal distribution. Finally, the non-normal shape of the
distribution was evaluated by comparing the SD to the interval of values that represent
around 68.27 % of the total probability distribution (k = 1 coverage interval).
The following section moved on to analyse the individual sources of the TOA
reflectance distribution asymmetry. The analysis assumed the observed asymmetry is
not due to the measurement instrument, but a consequence of the viewed scene.
The effect of both the relative air-mass factor and reflectance conversion have
been studied and concluded to be largely linearised and have limited effect in terms of
asymmetry. Thus, not compensating for the spatial variation of these parameters at a
pixel level does not alter the study of the distribution shape undertaken here. The effect
of angular variations within the ROI and the difference between the summer and winter
solstices has been shown to have a large impact on the distribution asymmetry for bands
dominated by the atmospheric reflectance. Here, no exhaustive modelling of the
atmosphere has been included due to the limited information on spatial changes of the
atmospheric constituents. The study of Aqua MODIS deep blue AOT products over a
year and a ROI has shown the asymmetric nature of these variations but has also
flagged the limitations of these products to reveal a “realistic” spatial distribution shape.
The use of a fix aerosol model for a ROI as proposed in Govaerts et al. (2010) could
reveal better properties of the spatial distribution since it assumes that the aerosol model
does not change abruptly in the study region (100 × 100 km). Nonetheless, it is known
that the atmospheric contribution is intrinsically non-linear since it is modelled by an
exponential decay that directly translates into a non-normal distribution at the TOA
reflectance measurements.
The dune morphology is also established as a key contributor to the observed
non-normal shape of the TOA reflectance distribution over Libya-4. The validity of the
model input data (BRDF and GDEMv2 model), however, is questioned as, for example,
there are limitations in the BRDF model to reproduce the distribution shape. In addition,
the GDEMv2 used may introduce a further limitation to the surface reflectance model,
due to the quantised nature of the GDEMv2 (at 1 m vertical resolution). The error
reported in Tachikawa (2011) does not provide sufficient knowledge to understand the
relative uncertainty of the Libya-4 elevation model and, therefore, these limitations
cannot be quantifiably assessed. The results in Figure 5 (a) approximate well with the
shape and maximum peak of the slope distribution in Govaerts (2015). In addition, the
aspect distribution in Figure 5 (b) indicates the directionality of the dunes as expected.
Both the BRF and DEM used in this exercise are not considered fundamental limitations
when testing the non-normality of the distribution due to the dune morphology.
The next steps for this research should, among others, seek the access to in-situ
spatial data — both atmospheric and surface — from the calibration sites. This will
permit a better modelling and quantification of the sources of non-normal distribution
and will eventually define a TOA reflectance distribution model to be used in practical
EO calibration and performance operations.
One of the elements of the model set up which significantly contributes to the
observed asymmetry is the choice of pixel size. In the current work, a 30 m pixel size
equivalent to the L8 OLI instrument has been used and has shown asymmetry in the
resultant reflectance distribution. This asymmetry is, in part, due to the small spatial
variations in the reflectance of the Libya-4 site being captured by the instrument.
However, if a larger spatial resolution were utilised such that one or more dunes are
entirely encompassed (from east to west, noting that the dunes run approximately the
whole length of the site in the North-South direction) by a single pixel, the resultant
reflectance distribution would be less asymmetric. The resultant image would contain
only pixels which represented the average reflectance of an entire dune structure, all of
which would be similar to one another, hence the distribution would tend towards
normal. A similar situation would be applicable for the atmospheric propagation where
the variations would be averaged out if a larger pixel size were used and vice versa.
Certain corrections could be applied to the data as, for example, the atmospheric
dependence on the sun angle and viewing angular changes across the ROI (Mishra, et al.
2014). These corrections have the potential to reduce the non-normality of the
distribution to a certain extent. In such a situation parameters such as skewness and
kurtosis can be useful in order to test the validity of the correction applied to the
reflectance distribution by comparing the values obtained with and without this
correction. Nonetheless, we have appreciated that there are intrinsic features in the site,
such as the dune morphology or the atmospheric composition variations, which are
inherently non-normal effects and more difficult to cancel out.
The example here is based on L8 OLI images for the Libya-4 site, however, the
developed software allows a similar approach to be followed for other instrument and
alternative sites. Depending on the characteristics of the study, different sources will
dominate the distribution shape. Here we do not (and cannot) study each specific
satellite instrument and site, however, the evolution of space-borne sensors towards
higher spatial resolution and lower noise, indicates that approximating the reflectance
distribution to a normal shape may not always be a valid assumption. That is, if the
TOA reflectance over a calibration site is described by the mean and SD the validity of
its approximation as a normal distribution needs to be demonstrated.
The results have shown that the non-normal distribution of TOA reflectance
over the Libya-4 site is clear and important. Consequently, it means that the systematic
study of, not only the mean and SD of the observations, but of additional parameters,
can enhance and provide extra information about the sensor and site characteristics and
hence enhance trend analysis. Such complementary parameters could include the
median as a measurement of location, the skew and kurtosis for the study of the shape
variations, and the consideration of the quantile information of the distribution.
Recent studies of the Solar and Heliospheric Observatory (SOHO) mission have
shown the utility in the usage of quantile information and complementary parameters to
study the solar radiance distribution. Del Zanna (2010) analyses the SOHO Coronal
Diagnostic Spectrometer (CDS) and the Normal Incidence Spectrograph (NIS) spectral
responsivity variations based on the fitting of the solar measurements to a log-normal
distribution. From that distribution, the mode was selected as the statistical estimate that
minimises the influence from solar cycle radiance variations. Furthermore, the analysis
in Shakeri, Teriaca, and Solanki (2015), uses the ‘contrast ratio’ to characterise the
radiance distribution, defined as the ratio between the lower and higher quantiles, which
is selected either as 5 % or 10 %. The authors argue that this technique cancels out the
effects of correction and calibration factors and permits the study of subtle solar
changes. Both studies (Del Zanna (2010) and Shakeri, Teriaca, and Solanki (2015))
indicate that the incorporation of additional parameters and quantile information of the
TOA reflectance/radiance distribution to understand variations of either the instrument
or vicarious test site is important.
The non-normality of the TOA reflectance distribution showed that the link
between the SD and a certain probability is inadequate. For a non-normal distribution,
the area included in the confidence interval [-σ, σ] does not cover the 68.27 % (k = 1) of
samples. The work initially described in Willink (2005) and refined in Willink (2006)
develops a method to calculate the 90%, 95%, 98% and 99% coverage intervals by
using the first four moments of an approximating distribution. Thus, the use of the third
and four moments of the distribution (skewness and kurtosis) is a useful mechanism to
estimate the variation of the coverage interval independently of changes in the
distribution shape in time or space domains.
5. Conclusion and further work
Historically, the mean and SD have been used ubiquitously to characterise the TOA
reflectance over satellite calibration sites. However, this information provides a full
characterisation of distribution of values only for a normal (or near-normal) distribution.
In this paper, an example of L8 OLI data over the Libya-4 site is used to observe
the non-normality of the distribution and then to study the sources of this non-normality
in the TOA reflectance. In a scenario where the normality assumption of the TOA
reflectance distribution is not applicable, it is proposed that complementary information
is considered to provide a more robust representation of the radiometric information
over a calibration site. The next steps of the research should:
Include in-situ data and/or alternative data sources so that a model can reproduce
the TOA reflectance distribution over a ROI for selected calibration sites.
Explore new techniques inspired in the ones used by the astrophysics and other
fields of research in order to improve the detection of scene and instrument
variations.
The authors would like to thank the reviewer for the helpful suggestions that have improved the
final manuscript. Thanks to the National Aeronautics and Space Administration (NASA) for the
free access of data and inputs used in this study. Also thanks are due to Dr. Marc Bouvet
(European Space Agency) for the access to the Libya-4 surface reflectance model.
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Figure 1. TOA reflectance image corresponding to Landsat-8 OLI product
“LC81810402014173LGN00” — Day 173 of Year 2014, WRS path 181 and row 40 —
for band 1 (a), band 5 (b), and band 7 (c). The crosshair represents the centre of Libya-4
site at 28.55º N, 23.39º E.
Figure 2. TOA reflectance distribution for a ROI (28.55º N, 23.39º E, size: 50 km × 50
km) of L8 OLI for Day of Year 173 (summer) for band 1 (a), band 5 (b), band 7 (c), and
for Day of Year 365 (winter) for band 1 (d), band 5 (e) and band 6 (f).
Figure 3. Aqua MODIS deep blue AOT distribution over Libya-4 100 × 100 km area for
the period from 12 February 2014 to 09 February 2015.
Figure 4. 50 km × 50 km ROI from ASTER GDEMv2 at 28.55º N, 23.39º E (a) and its
altitude histogram (b).
Figure 5. Slope distribution (a) and aspect distribution (b) for a 50 km × 50 km ROI at
28.55º N, 23.39º E and their associated images of slope (c) and aspect (d).
Figure 6. Surface reflectance distribution at 865 nm and nadir viewing for a 50 km × 50
km ROI at 28.55º N, 23.39º E for Day of Year 172 (summer) (a) and Day of Year 355
(winter) (b) and their associated images for summer (c) and winter (d).
Table 1. Statistics relating to Figure 2, showing several statistical parameters for the two
dates considered.
Day of year 173 (summer) Day of year 365 (winter)B1 panel
(a)B5 panel
(b)B7 panel
(c)B1 panel
(d)B5 panel
(e)B7 panel
(f)Mean 0.2274 0.5920 0.6087 0.2339 0.5743 0.5865Median/50th percentile
0.2270 0.5891 0.6052 0.2318 0.5635 0.5769
First quartile/ 25th percentile
0.2232(56.40)
0.5801(24.56)
0.5983(36.50)
0.2288(66.69)
0.5498(16.68)
0.5638(18.71)
Third quartile/ 75th percentile
0.2313(46.47)
0.6012(14.04)
0.6156(10.13)
0.2366(33.99)
0.5855(6.75)
0.5973(5.73)
SD [%] 2.85 3.36 2.69 3.63 7.68 7.43Coverage interval 68.27% 1 [%]
2.68 2.83 2.25% 2.76 5.35 5.01
Skewness 0.2261 0.6808 1.0863 1.8080 2.1185 2.0115Excess kurtosis
0.4729 1.4827 2.0113 4.8398 7.3861 7.5878
1 This probability area show a residual variation around 0.1% from the 68.27% level as a
consequence of the finite binning resolution.
Table 2. Statistics relating to Figure 6 (a) and (b), showing several statistical parameters
for the two dates considered.
Day of year 172 (summer) Day of year 355 (winter)865 nm panel (a) 865 nm panel (b)
Mean 0.5824 0.5965Median/50th percentile
0.5818 0.5954
First quartile/ 25th percentile
0.5803(163.12)
0.5903(49.12)
Third quartile/ 75th percentile
0.5838(87.41)
0.6015(26.89)
SD[%] 0.50 1.53Coverage interval 68.27% 1[%]
0.46 1.42
Skewness 1.5059 0.9721Excess kurtosis
4.5032 3.0423
1 This probability area show a residual variation around 0.1% from the 68.27% level as a
consequence of the finite binning resolution.