The complete data fusion (CDF) method is applied to ozone profiles obtained
from simulated measurements in the ultraviolet and in the thermal infrared in
the framework of the Sentinel 4 mission of the Copernicus programme. We
observe that the quality of the fused products is degraded when the fusing
profiles are either retrieved on different vertical grids or referred to
different true profiles. To address this shortcoming, a generalization of the
complete data fusion method, which takes into account interpolation and
coincidence errors, is presented. This upgrade overcomes the encountered
problems and provides products of good quality when the fusing profiles are
both retrieved on different vertical grids and referred to different true
profiles. The impact of the interpolation and coincidence errors on number of
degrees of freedom and errors of the fused profile is also analysed. The
approach developed here to account for the interpolation and coincidence
errors can also be followed to include other error components, such as
forward model errors.
Introduction
Many remote sensing observations of vertical profiles of atmospheric
variables are obtained with instruments operating on space-borne and
airborne platforms, as well as from ground-based stations. Recently, the
complete data fusion (CDF) method (Ceccherini et al., 2015) was proposed for
use in the combination of independent measurements of the same profile in
order to exploit all the available information and obtain a comprehensive
and concise description of the atmospheric state. This is an a posteriori
method that uses standard retrieval products. With simple implementation
requirements, the CDF products are equivalent to those from a simultaneous
retrieval, considered to be the most comprehensive way of exploiting
different observations of the same quantity (Aires et al., 2012), in spite
of a greater computational complexity. However, so far, the data fusion
method was mainly applied to measurements performed by the same instrument
while sounding the same air sample.
Limited tests were conducted on measurements performed by different
instruments when inconsistencies due to differences in the observed true
profiles (because of the non-perfect coincidence of the space–time location
of the measurements) could degrade the optimal performances of the
simultaneous retrieval. About the fusion of data provided by different
instruments, it has been proved (Ceccherini, 2016) that the CDF method is
completely equivalent to the measurement space solution (MSS) data fusion
method (Ceccherini et al., 2009). The latter was successfully applied to the
data fusion of MIPAS-ENVISAT and IASI-METOP measurements (Ceccherini et al.,
2010a, b) and of MIPAS-STR and MARSCHALS measurements (Cortesi et al., 2016).
However, since in these cases the measurements to be fused (referred to as
fusing profiles hereafter) carried information about basically complementary
altitude ranges, their possible inconsistency did not result in unrealistic
fused profiles.
The first applications of data fusion were made with profiles retrieved on
the same vertical grid. A first analysis of the effect of different grids on
the quality of the fused products was performed and presented by Ceccherini
et al. (2016). In this case, the individual profiles were first obtained on
grids optimally defined according to the information content of the
individual observations. Then, the CDF method was performed using averaging
kernel matrices (AKMs) interpolated to a common grid optimized for the data
fusion product. Compared to the case in which the individual retrievals are
obtained directly on the grid optimized for the data fusion, the number of
degrees of freedom (DOF) is reduced by about a quarter with this approach.
Thus, in data fusion applications the choice of the retrieval grid can lead
to an information content loss that cannot be restored with interpolation.
Here, we consider the general problem posed by the application of the CDF
method to measurements performed by different instruments that are retrieved
on different vertical grids and refer to different true profiles (which
correspond to the case of fusing profiles measured in different
geolocations). The analysis of this problem suggests a modification of the
CDF method, taking into account interpolation and coincidence errors. We
determine the expressions of these errors and show how they enter in the CDF
formula. The study is performed using simulated measurements of ozone
profiles obtained in the ultraviolet and in the thermal infrared in the
framework of the Sentinel 4 (S4) mission (ESA, 2017) of the Copernicus
programme (http://www.copernicus.eu/main/sentinels). The
advantages in using a multispectral approach for observing ozone profiles
from space have been studied, using simulated measurements, by Landgraf and
Hasekamp (2007), Worden et al. (2007), Natraj et al. (2011), Hache et al. (2014) and Costantino et al. (2017), and, using real measurements, by Fu et
al. (2013) and Cuesta et al. (2013). Two review papers on this subject are
Lahoz et al. (2012) and Timmermans et al. (2015).
The paper is organized as follows: Section 2 presents an account of the
problems that occur when the CDF method is applied to vertical profiles
retrieved on different vertical grids and referring to different true
profiles. In Sect. 3, we theoretically analyse the problems discussed in
Sect. 2 and show how the CDF method can be modified to overcome them. In
Sect. 4, we show how the solution proposed in Sect. 3 solves the
problems discussed in Sect. 2. In Sect. 5, we describe how to deal with
forward model errors. Conclusions are drawn in Sect. 6.
Application of the CDF method to profiles retrieved on different vertical
grids and related to different true profiles
The future atmospheric Sentinel missions of the Copernicus programme
(http://www.copernicus.eu/main/sentinels) will provide great
scope and a real test bed for data fusion applications. The wealth of data
that will become available from these missions will likely present technical
challenges to many applications. With the use of data fusion, the number of
products can be reduced while maintaining the information content of the
original datasets. For this reason, we test the CDF method on simulated data
of the S4. We simulate two S4 ozone vertical profile measurements as they
could be obtained from the Infrared Sounder (IRS) in the thermal infrared
and from the Ultraviolet, Visible and Near-Infrared Sounding (UVN)
spectrometer in the ultraviolet (http://www.eumetsat.int/website/home/Satellites/FutureSatellites/MeteosatThirdGeneration/MTGDesign/index.html)
on board the MTG (Meteosat Third Generation) satellite. We refer to these two
simulated measurements as TIR measurement and UV measurement, respectively.
In order to evaluate the effect of the variability of vertical grids and of
true profiles, three cases are considered:
The simulated measurements refer to the same true profile and are retrieved
on the same vertical grid.
The simulated measurements refer to the same true profile but are retrieved
on different vertical grids.
The simulated measurements refer to different true profiles and are
retrieved on the same vertical grid.
In all three cases, the true profile and the vertical grid of the UV
measurement are kept fixed and, when pertinent, are changed for the TIR
measurement. For simplicity, we define the vertical grid of the data fusion
product to coincide with the fixed grid of the UV measurement. In the
following, the vertical grid of the fusion product is referred to as the
fusion grid.
For a meaningful comparison of the quality of fusing and fused profiles, it
is necessary to have common a priori profiles and common a priori covariance
matrices (CMs). Therefore, the a priori of the fusing profiles, which are
produced with individual a priori assumptions, have been modified using the
method described in Ceccherini et al. (2014). In the comparisons, the same a
priori profiles provided by the McPeters and Labow climatology (McPeters and
Labow, 2012) are used for all fusing and fused profiles. The a priori CMs
are obtained using the standard deviation of the McPeters and Labow
climatology when its value is larger than 20 % of the a priori profile and
a value of 20 % of the a priori profile in the other cases. The off-diagonal elements are calculated considering a correlation length of 6 km.
The correlation length is used to reduce oscillations in the retrieved
profile and the value of 6 km is typically used for nadir ozone profile
retrieval (Liu et al., 2010; Kroon et al., 2011; Miles et al., 2015).
The results obtained in the three test cases are reported in Figs. 1–3.
These figures show the true profiles in panel (a), the mean value of the true profiles and the profiles obtained from the
measurements (TIR, UV and data fusion) in panel (b) and the residuals in panel (c),
i.e. the differences between the three estimated profiles and the mean value
of the true profiles.
We observe that, while in case 1 the differences between the profile
obtained from the fusion and the mean of the true profiles are smaller than,
or comparable to, those of the profiles obtained from the TIR and UV
measurements, in cases 2 and 3 these differences are significantly larger.
Therefore, in cases 2 and 3 the fusion provides a product of poorer quality
than that of the single products.
These tests show that the CDF algorithm and the equivalent simultaneous
retrieval work well in case 1, while they have problems in cases 2 and 3,
where the profiles are retrieved on different vertical grids and are
referred to different true profiles, respectively.
(a) True ozone profiles related to TIR (red line) and UV
(blue line) measurements. (b) Ozone profiles obtained from TIR
measurement (red line), from UV measurement (blue line), from the data fusion
(black line) compared with the mean value of the true profiles (green line).
(c) Residual errors obtained as differences of the ozone profiles
obtained from TIR measurement (red line), from UV measurement (blue line) and
from data fusion (black line) from the mean value of true profiles. All the
reported quantities are related to case 1.
As Fig. 1 but for case 2.
As Fig. 1 but for case 3.
The problem encountered in case 2 is due to the fact that the data fusion is
made using estimates of the AKMs on the fusion grid (see Sect. 3.1)
obtained by interpolation of the original AKMs (Ceccherini et al., 2016),
which are only an approximation of the real AKMs on the fusion grid. We
refer to this effect as interpolation error. The problem encountered in case 3 is related to
different true profiles and we refer to this effect as coincidence error because it occurs
when fusing profiles that do not correspond to the same space–time location.
Method
In this section, a theoretical analysis is performed to overcome the
problems highlighted in the previous section. In Sect. 3.1, we recall
the formulas of the CDF method in order to establish the formalism
subsequently used in Sect. 3.2, where an upgrade of the method is
proposed.
CDF
Let us assume to have N independent and simultaneous measurements of the
vertical profile of an atmospheric target referred to the same space–time
location. Performing the retrieval of the N measurements with the optimal
estimation method (Rodgers, 2000), we obtain N vectors x^i (i= 1, 2, ... , N) here assumed to be estimates of the profiles
made on a common vertical grid. The use of a priori information ensures the
possibility of having a common retrieval grid also in the case of
observations with different vertical coverage.
The vectors x^i are characterized by the CMs
Si and the AKMs Ai (Ceccherini et al., 2003;
Ceccherini and Ridolfi, 2010; Rodgers, 2000):
Si≡σiσiT=KiTSyi-1Ki+Sai-1-1KiTSyi-1Ki1KiTSyi-1Ki+Sai-1-1,2Ai≡∂x^i∂x=KiTSyi-1Ki+Sai-1-1KiTSyi-1Ki,
where σi are the errors on x^i obtained by
propagating the errors of the observations through the retrieval processes
(noise errors), Ki are the Jacobians of the forward models,
Syi are the CMs of the observations, Sai are the
CMs of the a priori profiles and x is the true profile.
The CDF solution for the considered profiles is given by (see Ceccherini et
al., 2015)
xf=∑i=1NAiTSi-1Ai+Sa-1-13∑i=1NAiTSi-1αi+Sa-1xa,
where
αi≡x^i-I-Aixai=Aix+σi,xai is the a priori profile used in the ith retrieval and
xa and Sa are the a priori profile and its CM
used to constrain the data fusion.
We note that the vector αi, which can be calculated from the
available retrieval products, is a measurement of the vector x, made
using the rows of the AKM Ai, and no longer depends on the a priori profile
xai. Furthermore, it has the same errors σi as the
retrieved profile x^i; therefore, it is characterized by the
CM Si.
The fused profile has a CM, obtained by propagating the errors of
αi into xf, equal to
Sf=∑i=1NAiTSi-1Ai+Sa-1-1∑i=1NAiTSi-1Ai5∑i=1NAiTSi-1Ai+Sa-1-1
and an AKM, obtained performing the derivative of xf with
respect to the true profile, equal to
Af=∑i=1NAiTSi-1Ai+Sa-1-1∑i=1NAiTSi-1Ai.
The CDF formula (Eq. 3) involves a summation of AKMs made possible by the
common grid. When the fusing profiles x^i are represented
on different vertical grids, the available AKMs are also defined on
different vertical grids; thus in this case, it is necessary to perform a
resampling of the AKMs (Calisesi et al., 2005), which makes their second
index equal to that of the common fusion grid. Following Ceccherini et al. (2016), we define such a transformation as follows:
Ai′=AiRi,
where Ri are the generalized inverse matrices of the linear
interpolation matrices Hi, which interpolate the profiles on
the fusing grids to the fusion grid. In this case, using Eq. (7), Eq. (3)
becomes
xf=∑i=1NRiTAiTSi-1AiRi+Sa-1-18∑i=1NRiTAiTSi-1αi+Sa-1xa.
We notice that, in the case of different vertical grids, only the AKMs must
be interpolated; neither the CMs nor the αi vectors need to
be interpolated.
Interpolation and coincidence errors
Let us first consider the interpolation error. The vectors αi, defined by Eq. (4), are measurements of the true profile, each made
with the averaging kernels Ai. Let us assume that each
measurement is defined on a different retrieval grid, identified by the same
index that identifies the measurements; thus, Eq. (4) becomes
αi=Aixi(i)+σi,
where xi(i) is the true profile related to the
ith measurement that, by definition, is sampled with the ith grid, as
highlighted by the superscript in parentheses.
Equation (8) shows that in the presence of different vertical grids the CDF
method combines measurements with sensitivity to the true profile expressed
by AiRi. This operation assumes that the
measurements are combined on the common fusion grid, i.e. measurements of
AiRixi(f), with
xi(f) being the true profile related to the
ith measurement represented on the fusion grid. If using αi (Eq. 9), which is the measurement of
Aixi(i), the estimate of the
required measurement
AiRixi(f) is made
with an error equal to
Aixi(i)-AiRixi(f).
We can explicitly introduce this error in the expression of αi by rearranging Eq. (9) in the following way:
αi=AiRixi(f)+Aixi(i)-Rixi(f)+σi.
It is useful to introduce the following notations for xi(i) and
xi(f):
11xi(i)=C(i)xi,12xi(f)=C(f)xi,
where xi is the true profile related to the ith measurement
represented on a very fine grid that includes all the levels of the fusion
grid (f) and of the N grids (i). C(i) and
C(f)are the sampling matrices from the fine grid
to the grids (i) and to the grid (f), respectively.
Substituting Eqs. (11) and (12) in Eq. (10), one obtains
αi=AiRiC(f)xi+AiC(i)-RiC(f)xi+σi.
Let us now also consider the coincidence error. In general, measurements made
in different space–time locations are only fused when they lie within a
given coincidence criterion. These measurements correspond to different true
profiles and the purpose of the data fusion can be the determination of
either the mean value of these true profiles or the true profile in a given
space–time location identified as the central point of the coincidence
intervals. We indicate with x‾ the unknown profile
estimated by the data fusion. If we introduce the quantity σi,coin, which gives the deviation of xi from the
unknown profile x‾,
xi=x‾+σi,coin,
then Equation (13) becomes
αi=AiRiC(f)x‾+AiC(i)-RiC(f)x‾+AiC(i)σi,coin+σi=AiRix‾(f)+AiC(i)-RiC(f)x‾15+AiC(i)σi,coin+σi
after using Eq. (12) for x‾.
An estimate of the quantity AiC(i)-RiC(f)x‾ can be obtained writing
x‾ as the a priori profile plus the deviation
σa from it:
x‾=xa+σa.
Substituting Eq. (16) in Eq. (15) and rearranging the terms of the equation,
we can define a new quantity, α̃i, equal to
α̃i≡αi-AiC(i)-RiC(f)xa=AiRix‾(f)+AiC(i)-RiC(f)σa17+AiC(i)σi,coin+σi.
Each α̃i is a measurement of
x‾(f) made using the rows of the matrix
AiRi and a total error given by the sum of the
noise error σi plus the terms AiC(i)-RiC(f)σa
and AiC(i)σi,coin that
can be interpreted as the interpolation error and the coincidence error,
respectively.
For the estimate of the interpolation error, we use the a priori CM
Sa of σa and, therefore, the interpolation
error is characterized by the CM:
Si,int=AiC(i)-RiC(f)SaC(i)-RiC(f)TAiT.
To characterize the coincidence error, we introduce the CM
Scoin of σi,coin. If
x‾ represents the mean value of the true profiles,
Scoin accounts for the dispersion of the true profiles,
thus it depends on the coincidence criteria and it is the same for all the
measurements to be fused together. If x‾ represents the
true profile in a specific space–time location, Scoin
is zero if the measurement is exactly in that location and it increases going
away from that location. The values of Scoin as a
function of space–time location should reflect the variability of the true
profile with the location. Then, the coincidence error is characterized by
the CM
Si,coin=AiC(i)ScoinC(i)TAiT.
In conclusion, the CDF formula, given by Eq. (3), can be modified to account
for the interpolation and coincidence errors by replacing αi with
α̃i=αi-AiC(i)-RiC(f)xa
and Si with
S̃i=Si+Si,int+Si,coin.
The CM given by Eq. (21) is also used in place of Si in Eqs. (5) and (6) for the calculation of the CM and AKM of the fused profile.
Tests with the upgraded algorithm: results and discussionThe effect on fused profiles
The test cases of fusion 2 and 3 shown in Sect. 2 are here repeated with
the modified method described in Sect. 3.2.
In Figs. 4 and 5, we report the noise errors, the interpolation errors and
the coincidence errors related, respectively, to case 2 and case 3, for both
TIR and UV measurements. These errors are calculated as the square root of
the diagonal elements of Si, Si,int and
Si,coin, respectively. In case 2, the vertical grids are
different for the two measurements, and since the fusion grid coincides with
the vertical grid of the UV measurement, the interpolation errors are
different from zero for the TIR measurement and equal to zero for the UV
measurement. The coincidence errors are equal to zero in both TIR and UV
measurements because the true profiles are the same. In case 3, the
interpolation errors are equal to zero for both TIR and UV measurements
because the fusion grid coincides with that of the fusing profiles. The
coincidence errors are instead different from zero because the true profiles
are different and their CMs, chosen equal for both TIR and UV measurements,
are obtained considering an error of 5 % of the a priori profile
(consistent with the difference between the true profiles) and a correlation
length of 6 km.
Figures 6 and 7 show the fused profiles and the residuals obtained with the
modified algorithm compared with the same quantities reported in panels (b) and (c) of Figs. 2 and 3, respectively. In both tests, the
modified method provides residuals that are significantly smaller than those
obtained with the original CDF method.
Noise errors (red lines), interpolation errors (green lines) and
coincidence errors (blue lines) in case 2 for TIR and UV measurements.
As Fig. 4 but for case 3.
These tests show that the upgrade of the CDF method proposed in Sect. 3.2 solves the problems observed in Sect. 2 that occur when either the
fusing profiles are retrieved on different vertical grids or they refer to
different true profiles. The modified method is a generalization of the CDF
that allows its application to a wide range of cases.
The fused profile and the residual error obtained with the modified
algorithm (magenta lines) compared with the same quantities of Fig. 2b and
c.
The fused profile and the residual error obtained with the modified
algorithm (magenta lines) compared with the same quantities of Fig. 3b and
c.
The effect on errors and number of DOF
We now look at the effect of the generalized method on the errors and on the
number of DOF. Figures 8 and 9 show the errors of the fused profile when we
use either the original or the modified method for cases 2 and 3,
respectively. These errors are calculated as the square root of the diagonal
elements of Sf given in Eq. (5), where, in the modified
method, Si is replaced by S̃i. For the
three cases described in Sect. 2, Table 1 gives the number of DOF of the
profiles obtained from the individual TIR and UV measurements, and from the
CDF method using both the original and the generalized formulation. The
numbers of DOF are calculated as the trace of the AKMs. For the fused
products the AKM is Af given by Eq. (6), where, in the
generalized formulation, Si is replaced by
S̃i.
The introduction of the interpolation error (case 2) does not significantly
modify the errors and determines a decrease in the number of DOF of the
fused profile of about 1. The introduction of the coincidence error (case 3)
determines a significant increase in the errors and a small decrease in the
number of DOF of the fused profile equal to about 0.5. However, in both
cases the number of DOF of the fused profile obtained with the modified
method is larger than the number of DOF of the individual fusing profiles,
proving the information gain provided by the fusion.
Errors of the fused profile when we use the original (black line)
and the generalized (magenta line) CDF for case 2.
Errors of the fused profile when we use the original (black line)
and the generalized (magenta line) CDF for case 3.
From the analysis of errors and number of DOF we deduce that the
interpolation error has the largest impact on the vertical resolution, while
the coincidence error has the largest impact on the errors. However, these
numerical results depend on the values that interpolation and coincidence
errors have in the single cases.
Number of DOF of the profiles obtained with the TIR measurement, the
UV measurement, the original fusion method and the modified fusion method for
each of the three cases described in Sect. 2.
In this paper, we considered simulated measurements, which generally do not
include all the error components that are present in real measurements. When
real measurements are considered, there are other important error sources
that can cause inconsistency among the fusing profiles, such as forward
model errors, due, for example, to approximations in the model and
uncertainties in atmospheric and instrumental parameters. When performing
data fusion, these errors can also lead to quality loss and show problems
similar to those described in Sect. 2. These problems can be avoided by
accounting for them in the CDF formulation. In particular, Eq. (21) can be
modified to account for an extra CM term, Si,other, as
follows:
S̃i=Si+Si,other+Si,int+Si,coin.
Conclusions
We analysed the problem posed by the application of the CDF method to
vertical profiles obtained with different instruments, which use different
retrieval grids and observe different true profiles. To this purpose, we
studied simulated ozone profile measurements expected from the MTG payload
for the S4 mission of the Copernicus programme: namely, those provided by
the IRS in the thermal infrared and by the UVN spectrometer in the
ultraviolet. The study showed that the CDF algorithm works well when the
fusing profiles are represented on the same vertical grid and refer to the
same true profile; otherwise the algorithm provides unsatisfactory results
because the fused profile differs from the mean of the true profiles
significantly more than the fusing profiles. In the latter case, the CDF
method, which uses all the existing information for the determination of the
best fused profile, is exploiting the differences due
to the inconsistency of the measurements as useful information and provides unrealistic fused
profiles.
In order to overcome this new problem, we performed a theoretical analysis
that led to a generalization of the CDF method to the cases in which
interpolation and coincidence errors occur. The interpolation error is
present when the vertical grids of the fusing profiles differ from the
fusion grid, meaning that an interpolation of the AKMs is necessary. In this
case, the interpolated AKMs are only an approximation of the real AKMs on
the fusion grid. The coincidence error is a consequence of the fact that the
fusing profiles are not generally co-located in space and time, thus
referring to different true profiles.
The generalized algorithm allows for these inconsistencies and provides
fused profiles that are in better agreement with the true profiles than
those obtained with the original CDF algorithm.
With the new algorithm, the fusion generally provides fused profiles that
are also better than the fusing profiles in terms of total error and number
of DOF. However, a more comprehensive error budget, which may even cause the
fused profile to have larger errors than the fusing profiles (coincidence
and interpolation errors do not have to be considered for the individual
fusing profiles), is now considered. If neither of the qualifiers (total
error and number of DOF) is improved, the fusion process is not justified.
An approach similar to that used to account for interpolation and
coincidence errors can also be useful to include other error components,
such as forward model errors, in the fusion process.
Data availability
The data of the simulations presented in the paper are available upon
request to the authors.
Author contributions
SC deduced the expression of the interpolation and coincidence errors and
wrote the draft version of the paper. BC suggested the idea to introduce the
interpolation and coincidence errors and contributed to the interpretation
of the results. NZ wrote the Python code of the complete data fusion. CT and
SDB performed the simulation of the infrared measurements. JK performed the
simulation of the ultraviolet measurements. UC put together the team of
authors and coordinated its activity. RD performed a detailed revision of the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The results presented in this paper arise from research activities
conducted in the framework of the AURORA project
(http://www.aurora-copernicus.eu/) supported by the Horizon 2020 research
and innovation programme of the European Union (call: H2020-EO-2015; topic:
EO-2-2015) under grant agreement no. 687428.
Edited by: Brian Kahn
Reviewed by: two anonymous referees
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