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Retrieval of the vertical column of an atmospheric constituent from data fusion of remote sensing measurements
ARTICLE IN PRESS Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 507–514
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Retrieval of the vertical column of an atmospheric constituent from data fusion of remote sensing measurements Simone Ceccherini , Bruno Carli, Ugo Cortesi, Samuele Del Bianco, Piera Raspollini Istituto di Fisica Applicata ‘‘Nello Carrara’’ del Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Firenze, Italy
a r t i c l e in fo
abstract
Article history: Received 10 June 2009 Received in revised form 1 September 2009 Accepted 2 September 2009
Techniques of data fusion are presently being considered with increasing interest for application to atmospheric observations from space because of their capability to optimally exploit the complementary information provided by different instruments operating aboard on-going and future satellite missions. The task of combining measurements of the same target, when carried out at the level of the retrieved state vectors, faces with two major problems: the need to interpolate the products represented on different retrieval grids which determines a loss of information and the presence of a priori information in the products that can determine a bias in the product of the data fusion. The measurement space solution method avoids these problems. Based on this method we present a novel approach to retrieve the vertical column of an atmospheric constituent from data fusion of remote sensing measurements. We apply the method to retrieve the ozone column from the fusion of simulated measurements of the IASI nadir-viewing spectrometer onboard the METOP-A platform and of the MIPAS limb sounder onboard the ENVISAT satellite. The performance of the method is evaluated in terms of retrieval errors and averaging kernels of the products. The results show the evidence of improved retrieval quality when comparing the outcome of data fusion with that of the inversion process applied to spectra from either of the two instruments. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Vertical column Data fusion Atmospheric composition
1. Introduction Many instruments onboard satellites are performing remote sensing measurements in order to improve our knowledge of the chemistry and physics of the atmosphere. In case that two or more instruments sound the same portion of atmosphere there is the problem of how to combine the different measurements in order to exploit all the available information for a more comprehensive and accurate description of the atmospheric state. This problem is generally referred to as data fusion and has been addressed by a variety of techniques which combine information from multiple sources to enhance the quality of the retrieval
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products in terms of precision and accuracy, spatial and temporal coverage and resolution and overall consistency. Comprehensive reviews of the mathematical methods developed for merging atmospheric measurements from different sensors can be found in the papers by Nirala about data fusion of aerosol optical thickness [1] and total column ozone [2]. Data fusion is a complex problem, indeed the quantities retrieved from the different measurements are in general represented on different grids (defined by the different observation and retrieval methods) and their combination implies some interpolation that produces a loss of information. Furthermore, it is possible that the quantities retrieved from the different measurements contain a priori information that may introduce a bias in the fused data. Interpolations and biases are the two main problems encountered in data fusion. Recently a new method [3], called the MSS method, has been proposed for the optimal use of the information
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provided by indirect measurements of atmospheric vertical profiles. This method provides the retrieved profile as the sum of a component belonging to the measurement space (the space generated by the rows of the Jacobian matrix of the forward model), referred to as measurement space solution (MSS), and a component belonging to the null space (the orthogonal complement space to the measurement space). The observations only provide information on the MSS and leave completely undetermined the component belonging to the null space, an estimation of which can only be obtained by the use of some a priori information. The MSS method has the advantage that, without any a priori information, the retrieved profile can be determined on a vertical grid as fine as desirable. In case that two or more independent measurements of the same portion of the atmosphere are available, we can calculate the relative MSSs on the same vertical grid and from these obtain the MSS in the union space of the measurement spaces of the original measurements. This new MSS includes all the information contained in the observations of the measurements to be fused without any bias due to a priori information. In this paper the application of the MSS method is extended to the case of the column retrieval of an atmospheric constituent. We describe how the MSS method can be used to determine the vertical column of the atmospheric constituent and in particular how this result is obtained in the case of an individual remote sensing measurement and in the case of the data fusion of two or more measurements. The method is applied to the retrieval of partial ozone columns from data fusion of simulated measurements of IASI (infrared atmospheric sounding interferometer) [4] and MIPAS (Michelson interferometer for passive atmospheric sounding) [5] instruments that fly aboard Metop-A (metereological operational) and Envisat (ENVIronmental SATellite) satellites, respectively. MIPAS performs limb observations and mainly provides information on the stratospheric ozone, while IASI performs nadir observations and consequently its measurements contain information also on the tropospheric ozone [6]. IASI and MIPAS measurements have, therefore, complementary altitude coverage and provide a conspicuous example of the data fusion advantages. We describe the products of this data fusion and compare their quality, characterized by errors and averaging kernels (AKs), with that obtained when only one of the two measurements is used. In Section 2 the theory to retrieve the column of an atmospheric constituent from data fusion of remote sensing measurements is described and in Section 3 a test case is presented where this procedure is used for the fusion of two simulated measurements.
2. Theory 2.1. Column of an atmospheric constituent The column of an atmospheric constituent between two altitudes z1 and z2 is defined by Z z2 rðzÞxðzÞ dz; ð1Þ c¼ z1
where r(z) and x(z) are, respectively, the air number density and the volume mixing ratio (VMR) of the constituent as functions of altitude. Expressing the air number density as a function of pressure p(z) and temperature T(z) by means of the ideal gas law we obtain Z 1 z2 pðzÞ c¼ xðzÞ dz; ð2Þ k z1 TðzÞ where k is the Boltzmann constant. The discretization of Eq. (2) can be obtained dividing the altitude range between z1 and z2 in n layers of thickness Dzi (i=1,2,y,n), so that pressure, temperature and VMR can be assumed constant inside a single layer. In this case, replacing the continuous functions p(z), T(z) and x(z) with vectors, Eq. (2) becomes c¼
n 1X pi x Dz : k i¼1 Ti i i
ð3Þ
If we define the discrete air column vector d: di ¼
Dzi pi k Ti
with i ¼ 1; 2; . . . ; n;
ð4Þ
whose i-th element provides the air column in the i-th layer, the column c can be written as the scalar product between d and the VMR profile x: T
c ¼ d x;
ð5Þ
where the superscript T denotes transposed vectors (in the following also transposed matrices). Therefore, the column represents the component of x along d. 2.2. Application of the MSS method to the calculation of the column The unknown VMR profile x is a vector of n elements and, according to the MSS method proposed in [3], can be decomposed as the sum of a vector belonging to the measurement space (the space generated by the rows of the Jacobian matrix of the forward model) and a vector belonging to the null space (the orthogonal complement space to the measurement space): x ¼ xa þ xb ;
ð6Þ
where xa and xb belong to the measurement space and to the null space, respectively. These two components can be expressed as xa ¼ Va;
ð7Þ
xb ¼ Wb;
ð8Þ
where V is a matrix whose columns are an orthonormal basis of the measurement space, W is a matrix whose columns are an orthonormal basis of the null space and a and b are the projections of x on these orthonormal bases: a ¼ VT x;
ð9Þ
b ¼ WT x;
ð10Þ
Substituting Eq. (6) in Eq. (5) we obtain the following expression for the column: T
T
c ¼ d xa þ d xb ¼ ca þ cb
ð11Þ
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where T
ca ¼ d xa
ð12Þ
is the contribution to the column from the component of the profile in the measurement space and T
cb ¼ d xb
ð13Þ
is the contribution to the column from the component of the profile in the null space. By means of Eqs. (7)–(10) the components ca and cb can also be written as follows: T
T
T
ca ¼ d Va ¼ d VVT x ¼ da x; T
T
T
cb ¼ d Wb ¼ d WWT x ¼ db x;
db ¼ WW d
ð21Þ
which is characterized by the error qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eb ¼ dTb Sc db :
ð22Þ
ð15Þ
2.5. Solution and its characterization
ð16Þ
From Eqs. (18) and (21) we obtain the estimation c^ for the column:
ð17Þ
is the component of d in the null space. 2.3. Estimation of the contribution to the column from the component of the profile in the measurement space Following the procedure described in [3] the measurements can be used to calculate the matrix V and the estimation a^ of the vector a. From Eq. (14) we obtain the estimation of the contribution to the column from the component of the profile in the measurement space: T ^ c^ a ¼ d Va;
ð18Þ
which is characterized by the error qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ea ¼ dT VSa VT d;
ð19Þ
^ Sa where Sa is the variance-covariance matrix (VCM) of a. is a diagonal matrix [3] because the elements of a^ are independent of each other. The sensitivity of a^ to the true profile x is given by ([3], Eq. (17)) @a^ ¼ VT : @x
T c^ b ¼ db xc ;
ð14Þ
is the component of d in the measurement space and T
with a climatological profile xc (characterized by a VCM Sc):
The use of the climatological profile for the calculation of c^ b introduces an a priori information in the estimation of the column; however, the a priori information is kept separate from the component retrieved from the observations.
where da ¼ VVT d
509
ð20Þ
2.4. Estimation of the contribution to the column from the component of the profile in the null space The contribution cb depends on the component xb of the profile x in the null space for which the observations do not provide any information. Any available a priori information can be used for the estimation of cb. For instance a regularization method, called null-space regularization, has been used in [3] for the determination of the null-space component. However, in the case of column measurements an a priori information based on the climatological knowledge of the profile is typically used in order to fill the observation gaps that may be present in the measurements. Accordingly, in order to calculate an estimation c^ b of this contribution, we replace x in Eq. (15)
T T c^ ¼ c^ a þ c^ b ¼ d Va^ þ db xc :
ð23Þ
Since the two contributions are determined independently, the error of c^ can be estimated by the square root of the quadratic sum of the errors of the two contributions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ e2a þ e2b ¼ dT VSa VT d þ dTb Sc db : ð24Þ Using Eq. (20) and Eq. (23) we can calculate the sensitivity of the estimated column c^ to the true VMR profile: ^ @c^ T @a T T ð25Þ ¼d V ¼ d VVT ¼ da : @x @x The AK A in the partial column space is the vector whose i-th element is the derivative of the retrieved column c^ with respect to the partial column ci=dixi of the ith layer: Ai ¼
@c^ 1 @c^ d ¼ ¼ ai di @ci di @xi
ð26Þ
Since the MSS method provides the solution as sum of the contribution coming from the observations plus the contribution coming from the a priori information, the AK of this solution is independent from the a priori information. 2.6. Data fusion In the case that two or more instruments sound the same portion of atmosphere the information contained in the observations of all the instruments can be exploited to obtain a more precise estimation of the column of an atmospheric constituent with respect to when a single instrument performs the measurement. In this paper we use the approach of data fusion proposed in [3], where, starting from the MSSs of the VMR profile of the individual measurements, a new MSS (data fusion MSS) is calculated which lies in the union space of the original measurement spaces. Once that this new MSS has been calculated (corre^ the same sponding to a new matrix V and a new vector a) procedure described in the previous subsections can be applied to the data fusion MSS in order to determine a
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new estimation of the column that includes the information coming from all the considered measurements.
3. Retrieval of the ozone column in a simulated case 3.1. Introduction In this section we apply the procedure proposed in the previous section to the retrieval of partial ozone columns from data fusion of simulated atmospheric radiance measurements of IASI and MIPAS instruments. Since in the case of simulated measurements the true ozone VMR profile that determines the observed radiances is available, we can evaluate the performances of the proposed procedure comparing the true columns with the retrieved columns. The simulation is made in the case of a Northern midlatitude climatological atmosphere of April [7] with a modified ozone VMR profile. The calculation of the MSSs of the two individual measurements as described in [3] requires the definition of a predefined vertical grid (common for the two instruments) on which to represent the ozone VMR profile and of the linearization points (in general different for the two instruments) close enough to the true profile in such a way that the linear approximation of the forward models is appropriate. The predefined grid can be chosen as fine as wished and, thanks to this freedom, it can be determined on the basis of the application rather than according to the vertical resolution of the measurements. In this work we have chosen a vertical grid of 1 km step between 0 and 80 km. The linearization points have been obtained interpolating at the predefined grid the ozone VMR profiles obtained by the retrieval codes of the two instruments. With these choices we have calculated the MSSs of both the IASI and MIPAS measurements following the procedure described in [3].
in the frame of a project of the European Spatial Agency (ESA). IASI observations, corresponding to a single IFOV measured at nadir (12 12 km ground pixel from an altitude of approximately 817 km), are simulated using the forward model and adding a random noise based on the nominal values of IASI noise equivalent spectral radiance. The full spectral coverage of IASI measurements (8461 channels) is used for the retrieval. 3.3. Simulation of the MIPAS measurement MIPAS [5] is a Fourier-transform spectrometer operating in the middle infrared that observes the atmospheric emission at the limb for the retrieval of the vertical profiles of several minor atmospheric constituents. The code adopted by ESA for the operational retrieval [11,12] (and used in this work for the calculation of the linearization point) uses a non-linear least-squares fit of the observed spectra with forward model simulations to retrieve the vertical profiles of pressure, temperature, water vapor, ozone, nitric acid, methane, nitrous oxide and nitrogen dioxide between 7 and 72 km of altitude. The simulated spectra correspond to the MIPAS measurement mode adopted after January 2005, for which the unapodized spectral resolution is 0.0625 cm1 and the tangent altitudes are with a 1.5 km step between 7 and 22 km, a 2 km step between 22 and 32 km, a 3 km step between 32 and 47 km, a 4 km step between 47 and 63 km and a 4.5 km step between 63 and 72 km (for a total of 27 tangent altitudes). The simulated observations are obtained adding a realistic random noise to the radiances calculated with the forward model. The microwindow approach, described in [13], is adopted and of the 27 spectra only a subset of 4557 spectral points containing the maximum information on ozone profile is used. 3.4. Results of the simulation
3.2. Simulation of the IASI measurement The IASI instrument [4], launched on-board the sunsynchronous polar orbiting satellite METOP-A on 19 October 2006, is a nadir-viewing Fourier transform spectrometer for passive atmospheric sounding in the thermal infrared region (from 645 to 2760 cm1 with unapodized spectral resolution of 0.25 cm1). IASI observations are mainly devoted to the retrieval of accurate information on meteorological parameters of interest for numerical weather prediction applications. Operational products include, along with vertical profiles of temperature and water vapor and surface temperature and emissivity, total and partial columns of ozone and columnar values of CH4, CO and N2O. Calculations of synthetic IASI radiances and retrieval of ozone VMR profiles, used in this work as inputs for the simulation of the proposed approach to data fusion with MIPAS measurements, are based on a version of the MARC retrieval code [8] recently upgraded for the analysis of the REFIR measurements [9,10] and subsequently optimized
In order to evaluate the gain of information obtained performing the data fusion of IASI and MIPAS measurements with respect to when only one of the two measurements is used we have retrieved three ozone partial columns (corresponding to the vertical ranges: 0–16 km, 16–80 km and 0–80 km, 16 km being the tropopause altitude in the considered atmosphere) with their errors and AKs in the following three cases: using only the IASI measurement, using only the MIPAS measurement and performing the data fusion of the IASI and MIPAS measurements. Once that the MSSs for the ozone VMR profile have been calculated as described in [3] for the IASI measurement, the MIPAS measurement and for the union space of the two measurement spaces (obtained considering the 20 largest singular values for the MSS of IASI and the 50 largest singular values for the MSS of MIPAS), we have calculated the estimations for the three partial columns by means of Eq. (23). The vector d has been calculated with Eq. (4) using the climatological pressure and temperature vertical profiles [7] in the vertical ranges to
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which the partial columns refer to and set to zero outside. The ozone climatological profile xc needed for the estimation of c^ b has been taken from the ozone climatology reported in [14]. Since these climatological ozone profiles are given in the altitude range between 0 and 60 km, they have been extended between 60 and 80 km using the climatology reported in [7]. The ozone climatology provided in [14] is given for latitudinal bands of 101 and for each month, while that provided in [7] is given for wider latitudinal bands (the mid latitude band extends between 201 and 651) and for each season. As a consequence the ozone climatological profiles provided in [14] have standard deviations smaller than those of the profiles reported in [7] and, therefore, their use determines a smaller error on the retrieved column. For this reason we have chosen to use the ozone climatological profiles reported in [14] in the altitude range where they are available. In order to characterize the retrieved columns we have calculated their errors and AKs by means of Eqs. (24) and (26), respectively. The VCM Sc of the climatological profile has been calculated using the climatological variances for the diagonal elements and a correlation that decreases
IASI
511
exponentially with a correlation length of 5 km for the offdiagonal elements. As described in [3] the MSS generally includes a large number of components but only a small number of these is determined with an acceptable error. In the retrieval of the column it is preferable not to use components of the MSS with large errors that introduce large uncertainties, but it is better to reduce the dimension p of the measurement space and to consider components affected by a large error as belonging to the null space (consequently they are determined by means of the climatology). In Fig. 1 the percentage total errors (calculated using Eq. (24)) for the three partial columns retrieved using only the IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)) are reported as a function of the number of singular values considered to determine the measurement space. The corresponding percentage difference between the retrieved values and the true values are shown in panels (d), (e) and (f). The total errors on the retrieved columns depend on the number of considered singular values. Indeed increasing the number of considered singular
MIPAS
FUSION
10 20 30 Number of singular values
10 20 30 Number of singular values
20
Error [%]
15
10
5
Difference [%]
10 5 0 -5 0-16 km 16-80 km 0-80 km
-10 -15 0
2 4 6 Number of singular values
40
Fig. 1. Percentage total errors for the three partial columns retrieved using only the IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)) as a function of the number of singular values considered to determine the measurement space. Panels (d), (e) and (f) show the corresponding percentage difference between the retrieved values and the true values. The number of singular values corresponding to the minimum of the total error is emphasized with a triangle for the column between 0 and 16 km, a diamond for the column between 16 and 80 km and a square for the column between 0 and 80 km. For this condition also the error bars are reported in panels (d), (e) and (f).
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values we have a decrease of eb and an increase of ea and the square root of their quadratic sum has a minimum for a particular number of singular values. This minimum can be taken as a criterion for the choice of the number of singular values. The number of singular values corresponding to the minimum of the total error is emphasized in Fig. 1 with a triangle for the column between 0 and 16 km, a diamond for the column between 16 and 80 km and a square for the column between 0 and 80 km. Panels (d), (e), and (f) of Fig. 1 show that the differences between the retrieved columns and the true values are consistent with the errors.
In Table 1 the errors ea, eb, and e (respectively, of the measurement space component, of the null space component and of the total value), calculated from Eqs. (19), (22) and (24) for the number p of singular values that corresponds to the minimum of the total error e, are reported for the three retrieved partial columns. We can see that the fusion between MIPAS and IASI measurements provides a significant reduction of all the three total errors with respect to the single measurements. In Fig. 2 the AKs calculated by means of Eq. (26) are reported for the three columns retrieved using only the
Table 1 Errors ea, eb, and e (in percentage), calculated from Eqs. (19), (22) and (24) for the number p of singular values that corresponds to the minimum of the total error e, of the three retrieved partial columns. Partial column 0–16 km
IASI MIPAS Fusion
Partial column 16–80 km
Total column 0–80 km
p
ea (%)
eb (%)
e (%)
p
ea (%)
eb (%)
e (%)
p
eb (%)
eb (%)
e (%)
2 12 14
2.52 0.80 1.20
7.53 5.94 2.28
7.94 5.99 2.58
4 14 20
0.77 0.16 0.17
0.85 0.17 0.15
1.15 0.23 0.22
1 23 11
0.09 0.19 0.25
2.34 1.33 0.41
2.34 1.35 0.48
The values are reported for when only the IASI measurement is used, only the MIPAS measurement is used and the data fusion of them is performed.
IASI
FUSION
MIPAS
80 0-16 km 16-80 km 0-80 km
Altitude [km]
60
40
20
0 -1
0
1
Averaging kernel
0
1
Averaging kernel
0
1
2
Averaging kernel
Fig. 2. AKs in the partial column space for the three retrieved ozone columns obtained using only the IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)). For each case the number of considered singular values is the one that corresponds to the minimum of the total error e and is reported in Table 1.
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IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)). For each case the number of considered singular values is the one that corresponds to the minimum of the total error e and is reported in Table 1. The ideal AK that we would have in case of a complete measurement space would be 1 in the vertical range where the partial column is calculated and 0 outside. We can see that the IASI AKs are significantly different from the ideal ones, while the MIPAS AKs are much closer to the ideal ones with the only exception that those of the columns in the vertical ranges 0–16 and 0–80 km are 0 instead of 1 below 6 km. This is due to the fact that the vertical range below 6 km is not sounded by the MIPAS measurement. In the case of the data fusion the AK of the partial column between 16 and 80 km is quite comparable to the MIPAS AK as far as the capability of reproducing the ideal AK is concerned. The AKs of the columns in the vertical ranges 0–16 and 0–80 km retrieved with the data fusion show instead deviations from the expected value that at high altitude are larger than those observed for the corresponding MIPAS AKs. However, an improvement of the AKs is still obtained because in the data fusion case a better representation of the 0–6 km contribution is provided by
513
IASI and the oscillating values at high altitude are a small contribution to the measured column. The shape of the AKs depends on the number of considered singular values p that we have chosen on the basis of the criterion of the minimum total error. In order to analyze the dependence of the AKs on p we have reported in Fig. 3 the AKs of the column between 0 and 80 km retrieved from the data fusion of the IASI and MIPAS measurements for several values of p. We can see that by increasing the number of p the AK approaches the ideal shape of a constant equal to 1. Looking at Figs. 1(c) and 3 we can see that the choice of p corresponding to the minimum of the total error determines an AK that in some vertical ranges significantly deviates from the ideal shape. The choice of a larger value of p would determine an improved AK but an increase of the total error. The choice of the optimal value for p is, therefore, a compromise between the error value and the AK shape, and it should be driven by our confidence in the estimation of the climatological profile and of its VCM. If we trust in the estimation of these quantities we can safely choose the condition that corresponds to the minimum of the total error because the not measured components are well represented by the climatological values within their
80 p=5
p = 23
p = 17
p = 11
p = 29
Altitude [km]
60
40
20
0 -1
0
1
0
1
0
1
0
1
0
1
2
Averaging kernel Fig. 3. AKs in the partial column space of the ozone column between 0 and 80 km retrieved by the data fusion of the IASI and MIPAS measurements for several values of the number p of considered singular values.
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errors. Otherwise a larger value of p may be more appropriate because it improves the AK.
and application of KLIMA algorithm to OCO and GOSAT validation’’ under ESA-ESRIN contract N. 21612/08/I-OL.
4. Conclusions
References
We have proposed a procedure to determine the vertical column of an atmospheric constituent using the MSS method and extended this procedure to retrieve the column from data fusion of two or more measurements. The use of the MSS method to perform data fusion permits to avoid any loss of information due to interpolation and the propagation of possible biases present in the quantities retrieved from the measurements to be fused into the product of the data fusion. We have applied the proposed procedure to the calculation of the ozone column from simulated measurements of IASI and MIPAS and from their data fusion. We have characterized the quality of the obtained products in terms of retrieval errors and AKs. The retrieval errors of the partial columns obtained from the fusion of the two measurements are significantly smaller than those obtained when only the measurement of one instrument is used. This result demonstrates the ability of the proposed procedure to exploit the complementary information provided by the two instruments. The analysis of the AKs shows that the sensitivity of the retrieved quantities to the true quantities depends on the number of singular values considered to build the MSS. Increasing this number the sensitivity increases but above a certain value also the error increases. Therefore, the choice of the optimal value for this number is a compromise between the error and the sensitivity to the true value.
Acknowledgments The forward and inverse models, that we used for retrieval simulations from IASI spectra, have been optimized and tested in the frame of the project ‘‘Sensitivity analysis
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Retrieval of the vertical column of an atmospheric constituent from data fusion of remote sensing measurements Simone Ceccherini , Bruno Carli, Ugo Cortesi, Samuele Del Bianco, Piera Raspollini Istituto di Fisica Applicata ‘‘Nello Carrara’’ del Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Firenze, Italy
a r t i c l e in fo
abstract
Article history: Received 10 June 2009 Received in revised form 1 September 2009 Accepted 2 September 2009
Techniques of data fusion are presently being considered with increasing interest for application to atmospheric observations from space because of their capability to optimally exploit the complementary information provided by different instruments operating aboard on-going and future satellite missions. The task of combining measurements of the same target, when carried out at the level of the retrieved state vectors, faces with two major problems: the need to interpolate the products represented on different retrieval grids which determines a loss of information and the presence of a priori information in the products that can determine a bias in the product of the data fusion. The measurement space solution method avoids these problems. Based on this method we present a novel approach to retrieve the vertical column of an atmospheric constituent from data fusion of remote sensing measurements. We apply the method to retrieve the ozone column from the fusion of simulated measurements of the IASI nadir-viewing spectrometer onboard the METOP-A platform and of the MIPAS limb sounder onboard the ENVISAT satellite. The performance of the method is evaluated in terms of retrieval errors and averaging kernels of the products. The results show the evidence of improved retrieval quality when comparing the outcome of data fusion with that of the inversion process applied to spectra from either of the two instruments. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Vertical column Data fusion Atmospheric composition
1. Introduction Many instruments onboard satellites are performing remote sensing measurements in order to improve our knowledge of the chemistry and physics of the atmosphere. In case that two or more instruments sound the same portion of atmosphere there is the problem of how to combine the different measurements in order to exploit all the available information for a more comprehensive and accurate description of the atmospheric state. This problem is generally referred to as data fusion and has been addressed by a variety of techniques which combine information from multiple sources to enhance the quality of the retrieval
Corresponding author. Tel.: þ39 055 5226352; fax: þ39 055 5226348. E-mail address: [email protected] (S. Ceccherini).
0022-4073/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2009.09.001
products in terms of precision and accuracy, spatial and temporal coverage and resolution and overall consistency. Comprehensive reviews of the mathematical methods developed for merging atmospheric measurements from different sensors can be found in the papers by Nirala about data fusion of aerosol optical thickness [1] and total column ozone [2]. Data fusion is a complex problem, indeed the quantities retrieved from the different measurements are in general represented on different grids (defined by the different observation and retrieval methods) and their combination implies some interpolation that produces a loss of information. Furthermore, it is possible that the quantities retrieved from the different measurements contain a priori information that may introduce a bias in the fused data. Interpolations and biases are the two main problems encountered in data fusion. Recently a new method [3], called the MSS method, has been proposed for the optimal use of the information
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provided by indirect measurements of atmospheric vertical profiles. This method provides the retrieved profile as the sum of a component belonging to the measurement space (the space generated by the rows of the Jacobian matrix of the forward model), referred to as measurement space solution (MSS), and a component belonging to the null space (the orthogonal complement space to the measurement space). The observations only provide information on the MSS and leave completely undetermined the component belonging to the null space, an estimation of which can only be obtained by the use of some a priori information. The MSS method has the advantage that, without any a priori information, the retrieved profile can be determined on a vertical grid as fine as desirable. In case that two or more independent measurements of the same portion of the atmosphere are available, we can calculate the relative MSSs on the same vertical grid and from these obtain the MSS in the union space of the measurement spaces of the original measurements. This new MSS includes all the information contained in the observations of the measurements to be fused without any bias due to a priori information. In this paper the application of the MSS method is extended to the case of the column retrieval of an atmospheric constituent. We describe how the MSS method can be used to determine the vertical column of the atmospheric constituent and in particular how this result is obtained in the case of an individual remote sensing measurement and in the case of the data fusion of two or more measurements. The method is applied to the retrieval of partial ozone columns from data fusion of simulated measurements of IASI (infrared atmospheric sounding interferometer) [4] and MIPAS (Michelson interferometer for passive atmospheric sounding) [5] instruments that fly aboard Metop-A (metereological operational) and Envisat (ENVIronmental SATellite) satellites, respectively. MIPAS performs limb observations and mainly provides information on the stratospheric ozone, while IASI performs nadir observations and consequently its measurements contain information also on the tropospheric ozone [6]. IASI and MIPAS measurements have, therefore, complementary altitude coverage and provide a conspicuous example of the data fusion advantages. We describe the products of this data fusion and compare their quality, characterized by errors and averaging kernels (AKs), with that obtained when only one of the two measurements is used. In Section 2 the theory to retrieve the column of an atmospheric constituent from data fusion of remote sensing measurements is described and in Section 3 a test case is presented where this procedure is used for the fusion of two simulated measurements.
2. Theory 2.1. Column of an atmospheric constituent The column of an atmospheric constituent between two altitudes z1 and z2 is defined by Z z2 rðzÞxðzÞ dz; ð1Þ c¼ z1
where r(z) and x(z) are, respectively, the air number density and the volume mixing ratio (VMR) of the constituent as functions of altitude. Expressing the air number density as a function of pressure p(z) and temperature T(z) by means of the ideal gas law we obtain Z 1 z2 pðzÞ c¼ xðzÞ dz; ð2Þ k z1 TðzÞ where k is the Boltzmann constant. The discretization of Eq. (2) can be obtained dividing the altitude range between z1 and z2 in n layers of thickness Dzi (i=1,2,y,n), so that pressure, temperature and VMR can be assumed constant inside a single layer. In this case, replacing the continuous functions p(z), T(z) and x(z) with vectors, Eq. (2) becomes c¼
n 1X pi x Dz : k i¼1 Ti i i
ð3Þ
If we define the discrete air column vector d: di ¼
Dzi pi k Ti
with i ¼ 1; 2; . . . ; n;
ð4Þ
whose i-th element provides the air column in the i-th layer, the column c can be written as the scalar product between d and the VMR profile x: T
c ¼ d x;
ð5Þ
where the superscript T denotes transposed vectors (in the following also transposed matrices). Therefore, the column represents the component of x along d. 2.2. Application of the MSS method to the calculation of the column The unknown VMR profile x is a vector of n elements and, according to the MSS method proposed in [3], can be decomposed as the sum of a vector belonging to the measurement space (the space generated by the rows of the Jacobian matrix of the forward model) and a vector belonging to the null space (the orthogonal complement space to the measurement space): x ¼ xa þ xb ;
ð6Þ
where xa and xb belong to the measurement space and to the null space, respectively. These two components can be expressed as xa ¼ Va;
ð7Þ
xb ¼ Wb;
ð8Þ
where V is a matrix whose columns are an orthonormal basis of the measurement space, W is a matrix whose columns are an orthonormal basis of the null space and a and b are the projections of x on these orthonormal bases: a ¼ VT x;
ð9Þ
b ¼ WT x;
ð10Þ
Substituting Eq. (6) in Eq. (5) we obtain the following expression for the column: T
T
c ¼ d xa þ d xb ¼ ca þ cb
ð11Þ
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where T
ca ¼ d xa
ð12Þ
is the contribution to the column from the component of the profile in the measurement space and T
cb ¼ d xb
ð13Þ
is the contribution to the column from the component of the profile in the null space. By means of Eqs. (7)–(10) the components ca and cb can also be written as follows: T
T
T
ca ¼ d Va ¼ d VVT x ¼ da x; T
T
T
cb ¼ d Wb ¼ d WWT x ¼ db x;
db ¼ WW d
ð21Þ
which is characterized by the error qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eb ¼ dTb Sc db :
ð22Þ
ð15Þ
2.5. Solution and its characterization
ð16Þ
From Eqs. (18) and (21) we obtain the estimation c^ for the column:
ð17Þ
is the component of d in the null space. 2.3. Estimation of the contribution to the column from the component of the profile in the measurement space Following the procedure described in [3] the measurements can be used to calculate the matrix V and the estimation a^ of the vector a. From Eq. (14) we obtain the estimation of the contribution to the column from the component of the profile in the measurement space: T ^ c^ a ¼ d Va;
ð18Þ
which is characterized by the error qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ea ¼ dT VSa VT d;
ð19Þ
^ Sa where Sa is the variance-covariance matrix (VCM) of a. is a diagonal matrix [3] because the elements of a^ are independent of each other. The sensitivity of a^ to the true profile x is given by ([3], Eq. (17)) @a^ ¼ VT : @x
T c^ b ¼ db xc ;
ð14Þ
is the component of d in the measurement space and T
with a climatological profile xc (characterized by a VCM Sc):
The use of the climatological profile for the calculation of c^ b introduces an a priori information in the estimation of the column; however, the a priori information is kept separate from the component retrieved from the observations.
where da ¼ VVT d
509
ð20Þ
2.4. Estimation of the contribution to the column from the component of the profile in the null space The contribution cb depends on the component xb of the profile x in the null space for which the observations do not provide any information. Any available a priori information can be used for the estimation of cb. For instance a regularization method, called null-space regularization, has been used in [3] for the determination of the null-space component. However, in the case of column measurements an a priori information based on the climatological knowledge of the profile is typically used in order to fill the observation gaps that may be present in the measurements. Accordingly, in order to calculate an estimation c^ b of this contribution, we replace x in Eq. (15)
T T c^ ¼ c^ a þ c^ b ¼ d Va^ þ db xc :
ð23Þ
Since the two contributions are determined independently, the error of c^ can be estimated by the square root of the quadratic sum of the errors of the two contributions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ e2a þ e2b ¼ dT VSa VT d þ dTb Sc db : ð24Þ Using Eq. (20) and Eq. (23) we can calculate the sensitivity of the estimated column c^ to the true VMR profile: ^ @c^ T @a T T ð25Þ ¼d V ¼ d VVT ¼ da : @x @x The AK A in the partial column space is the vector whose i-th element is the derivative of the retrieved column c^ with respect to the partial column ci=dixi of the ith layer: Ai ¼
@c^ 1 @c^ d ¼ ¼ ai di @ci di @xi
ð26Þ
Since the MSS method provides the solution as sum of the contribution coming from the observations plus the contribution coming from the a priori information, the AK of this solution is independent from the a priori information. 2.6. Data fusion In the case that two or more instruments sound the same portion of atmosphere the information contained in the observations of all the instruments can be exploited to obtain a more precise estimation of the column of an atmospheric constituent with respect to when a single instrument performs the measurement. In this paper we use the approach of data fusion proposed in [3], where, starting from the MSSs of the VMR profile of the individual measurements, a new MSS (data fusion MSS) is calculated which lies in the union space of the original measurement spaces. Once that this new MSS has been calculated (corre^ the same sponding to a new matrix V and a new vector a) procedure described in the previous subsections can be applied to the data fusion MSS in order to determine a
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new estimation of the column that includes the information coming from all the considered measurements.
3. Retrieval of the ozone column in a simulated case 3.1. Introduction In this section we apply the procedure proposed in the previous section to the retrieval of partial ozone columns from data fusion of simulated atmospheric radiance measurements of IASI and MIPAS instruments. Since in the case of simulated measurements the true ozone VMR profile that determines the observed radiances is available, we can evaluate the performances of the proposed procedure comparing the true columns with the retrieved columns. The simulation is made in the case of a Northern midlatitude climatological atmosphere of April [7] with a modified ozone VMR profile. The calculation of the MSSs of the two individual measurements as described in [3] requires the definition of a predefined vertical grid (common for the two instruments) on which to represent the ozone VMR profile and of the linearization points (in general different for the two instruments) close enough to the true profile in such a way that the linear approximation of the forward models is appropriate. The predefined grid can be chosen as fine as wished and, thanks to this freedom, it can be determined on the basis of the application rather than according to the vertical resolution of the measurements. In this work we have chosen a vertical grid of 1 km step between 0 and 80 km. The linearization points have been obtained interpolating at the predefined grid the ozone VMR profiles obtained by the retrieval codes of the two instruments. With these choices we have calculated the MSSs of both the IASI and MIPAS measurements following the procedure described in [3].
in the frame of a project of the European Spatial Agency (ESA). IASI observations, corresponding to a single IFOV measured at nadir (12 12 km ground pixel from an altitude of approximately 817 km), are simulated using the forward model and adding a random noise based on the nominal values of IASI noise equivalent spectral radiance. The full spectral coverage of IASI measurements (8461 channels) is used for the retrieval. 3.3. Simulation of the MIPAS measurement MIPAS [5] is a Fourier-transform spectrometer operating in the middle infrared that observes the atmospheric emission at the limb for the retrieval of the vertical profiles of several minor atmospheric constituents. The code adopted by ESA for the operational retrieval [11,12] (and used in this work for the calculation of the linearization point) uses a non-linear least-squares fit of the observed spectra with forward model simulations to retrieve the vertical profiles of pressure, temperature, water vapor, ozone, nitric acid, methane, nitrous oxide and nitrogen dioxide between 7 and 72 km of altitude. The simulated spectra correspond to the MIPAS measurement mode adopted after January 2005, for which the unapodized spectral resolution is 0.0625 cm1 and the tangent altitudes are with a 1.5 km step between 7 and 22 km, a 2 km step between 22 and 32 km, a 3 km step between 32 and 47 km, a 4 km step between 47 and 63 km and a 4.5 km step between 63 and 72 km (for a total of 27 tangent altitudes). The simulated observations are obtained adding a realistic random noise to the radiances calculated with the forward model. The microwindow approach, described in [13], is adopted and of the 27 spectra only a subset of 4557 spectral points containing the maximum information on ozone profile is used. 3.4. Results of the simulation
3.2. Simulation of the IASI measurement The IASI instrument [4], launched on-board the sunsynchronous polar orbiting satellite METOP-A on 19 October 2006, is a nadir-viewing Fourier transform spectrometer for passive atmospheric sounding in the thermal infrared region (from 645 to 2760 cm1 with unapodized spectral resolution of 0.25 cm1). IASI observations are mainly devoted to the retrieval of accurate information on meteorological parameters of interest for numerical weather prediction applications. Operational products include, along with vertical profiles of temperature and water vapor and surface temperature and emissivity, total and partial columns of ozone and columnar values of CH4, CO and N2O. Calculations of synthetic IASI radiances and retrieval of ozone VMR profiles, used in this work as inputs for the simulation of the proposed approach to data fusion with MIPAS measurements, are based on a version of the MARC retrieval code [8] recently upgraded for the analysis of the REFIR measurements [9,10] and subsequently optimized
In order to evaluate the gain of information obtained performing the data fusion of IASI and MIPAS measurements with respect to when only one of the two measurements is used we have retrieved three ozone partial columns (corresponding to the vertical ranges: 0–16 km, 16–80 km and 0–80 km, 16 km being the tropopause altitude in the considered atmosphere) with their errors and AKs in the following three cases: using only the IASI measurement, using only the MIPAS measurement and performing the data fusion of the IASI and MIPAS measurements. Once that the MSSs for the ozone VMR profile have been calculated as described in [3] for the IASI measurement, the MIPAS measurement and for the union space of the two measurement spaces (obtained considering the 20 largest singular values for the MSS of IASI and the 50 largest singular values for the MSS of MIPAS), we have calculated the estimations for the three partial columns by means of Eq. (23). The vector d has been calculated with Eq. (4) using the climatological pressure and temperature vertical profiles [7] in the vertical ranges to
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which the partial columns refer to and set to zero outside. The ozone climatological profile xc needed for the estimation of c^ b has been taken from the ozone climatology reported in [14]. Since these climatological ozone profiles are given in the altitude range between 0 and 60 km, they have been extended between 60 and 80 km using the climatology reported in [7]. The ozone climatology provided in [14] is given for latitudinal bands of 101 and for each month, while that provided in [7] is given for wider latitudinal bands (the mid latitude band extends between 201 and 651) and for each season. As a consequence the ozone climatological profiles provided in [14] have standard deviations smaller than those of the profiles reported in [7] and, therefore, their use determines a smaller error on the retrieved column. For this reason we have chosen to use the ozone climatological profiles reported in [14] in the altitude range where they are available. In order to characterize the retrieved columns we have calculated their errors and AKs by means of Eqs. (24) and (26), respectively. The VCM Sc of the climatological profile has been calculated using the climatological variances for the diagonal elements and a correlation that decreases
IASI
511
exponentially with a correlation length of 5 km for the offdiagonal elements. As described in [3] the MSS generally includes a large number of components but only a small number of these is determined with an acceptable error. In the retrieval of the column it is preferable not to use components of the MSS with large errors that introduce large uncertainties, but it is better to reduce the dimension p of the measurement space and to consider components affected by a large error as belonging to the null space (consequently they are determined by means of the climatology). In Fig. 1 the percentage total errors (calculated using Eq. (24)) for the three partial columns retrieved using only the IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)) are reported as a function of the number of singular values considered to determine the measurement space. The corresponding percentage difference between the retrieved values and the true values are shown in panels (d), (e) and (f). The total errors on the retrieved columns depend on the number of considered singular values. Indeed increasing the number of considered singular
MIPAS
FUSION
10 20 30 Number of singular values
10 20 30 Number of singular values
20
Error [%]
15
10
5
Difference [%]
10 5 0 -5 0-16 km 16-80 km 0-80 km
-10 -15 0
2 4 6 Number of singular values
40
Fig. 1. Percentage total errors for the three partial columns retrieved using only the IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)) as a function of the number of singular values considered to determine the measurement space. Panels (d), (e) and (f) show the corresponding percentage difference between the retrieved values and the true values. The number of singular values corresponding to the minimum of the total error is emphasized with a triangle for the column between 0 and 16 km, a diamond for the column between 16 and 80 km and a square for the column between 0 and 80 km. For this condition also the error bars are reported in panels (d), (e) and (f).
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values we have a decrease of eb and an increase of ea and the square root of their quadratic sum has a minimum for a particular number of singular values. This minimum can be taken as a criterion for the choice of the number of singular values. The number of singular values corresponding to the minimum of the total error is emphasized in Fig. 1 with a triangle for the column between 0 and 16 km, a diamond for the column between 16 and 80 km and a square for the column between 0 and 80 km. Panels (d), (e), and (f) of Fig. 1 show that the differences between the retrieved columns and the true values are consistent with the errors.
In Table 1 the errors ea, eb, and e (respectively, of the measurement space component, of the null space component and of the total value), calculated from Eqs. (19), (22) and (24) for the number p of singular values that corresponds to the minimum of the total error e, are reported for the three retrieved partial columns. We can see that the fusion between MIPAS and IASI measurements provides a significant reduction of all the three total errors with respect to the single measurements. In Fig. 2 the AKs calculated by means of Eq. (26) are reported for the three columns retrieved using only the
Table 1 Errors ea, eb, and e (in percentage), calculated from Eqs. (19), (22) and (24) for the number p of singular values that corresponds to the minimum of the total error e, of the three retrieved partial columns. Partial column 0–16 km
IASI MIPAS Fusion
Partial column 16–80 km
Total column 0–80 km
p
ea (%)
eb (%)
e (%)
p
ea (%)
eb (%)
e (%)
p
eb (%)
eb (%)
e (%)
2 12 14
2.52 0.80 1.20
7.53 5.94 2.28
7.94 5.99 2.58
4 14 20
0.77 0.16 0.17
0.85 0.17 0.15
1.15 0.23 0.22
1 23 11
0.09 0.19 0.25
2.34 1.33 0.41
2.34 1.35 0.48
The values are reported for when only the IASI measurement is used, only the MIPAS measurement is used and the data fusion of them is performed.
IASI
FUSION
MIPAS
80 0-16 km 16-80 km 0-80 km
Altitude [km]
60
40
20
0 -1
0
1
Averaging kernel
0
1
Averaging kernel
0
1
2
Averaging kernel
Fig. 2. AKs in the partial column space for the three retrieved ozone columns obtained using only the IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)). For each case the number of considered singular values is the one that corresponds to the minimum of the total error e and is reported in Table 1.
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IASI measurement (panel (a)), only the MIPAS measurement (panel (b)) and the fusion of them (panel (c)). For each case the number of considered singular values is the one that corresponds to the minimum of the total error e and is reported in Table 1. The ideal AK that we would have in case of a complete measurement space would be 1 in the vertical range where the partial column is calculated and 0 outside. We can see that the IASI AKs are significantly different from the ideal ones, while the MIPAS AKs are much closer to the ideal ones with the only exception that those of the columns in the vertical ranges 0–16 and 0–80 km are 0 instead of 1 below 6 km. This is due to the fact that the vertical range below 6 km is not sounded by the MIPAS measurement. In the case of the data fusion the AK of the partial column between 16 and 80 km is quite comparable to the MIPAS AK as far as the capability of reproducing the ideal AK is concerned. The AKs of the columns in the vertical ranges 0–16 and 0–80 km retrieved with the data fusion show instead deviations from the expected value that at high altitude are larger than those observed for the corresponding MIPAS AKs. However, an improvement of the AKs is still obtained because in the data fusion case a better representation of the 0–6 km contribution is provided by
513
IASI and the oscillating values at high altitude are a small contribution to the measured column. The shape of the AKs depends on the number of considered singular values p that we have chosen on the basis of the criterion of the minimum total error. In order to analyze the dependence of the AKs on p we have reported in Fig. 3 the AKs of the column between 0 and 80 km retrieved from the data fusion of the IASI and MIPAS measurements for several values of p. We can see that by increasing the number of p the AK approaches the ideal shape of a constant equal to 1. Looking at Figs. 1(c) and 3 we can see that the choice of p corresponding to the minimum of the total error determines an AK that in some vertical ranges significantly deviates from the ideal shape. The choice of a larger value of p would determine an improved AK but an increase of the total error. The choice of the optimal value for p is, therefore, a compromise between the error value and the AK shape, and it should be driven by our confidence in the estimation of the climatological profile and of its VCM. If we trust in the estimation of these quantities we can safely choose the condition that corresponds to the minimum of the total error because the not measured components are well represented by the climatological values within their
80 p=5
p = 23
p = 17
p = 11
p = 29
Altitude [km]
60
40
20
0 -1
0
1
0
1
0
1
0
1
0
1
2
Averaging kernel Fig. 3. AKs in the partial column space of the ozone column between 0 and 80 km retrieved by the data fusion of the IASI and MIPAS measurements for several values of the number p of considered singular values.
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errors. Otherwise a larger value of p may be more appropriate because it improves the AK.
and application of KLIMA algorithm to OCO and GOSAT validation’’ under ESA-ESRIN contract N. 21612/08/I-OL.
4. Conclusions
References
We have proposed a procedure to determine the vertical column of an atmospheric constituent using the MSS method and extended this procedure to retrieve the column from data fusion of two or more measurements. The use of the MSS method to perform data fusion permits to avoid any loss of information due to interpolation and the propagation of possible biases present in the quantities retrieved from the measurements to be fused into the product of the data fusion. We have applied the proposed procedure to the calculation of the ozone column from simulated measurements of IASI and MIPAS and from their data fusion. We have characterized the quality of the obtained products in terms of retrieval errors and AKs. The retrieval errors of the partial columns obtained from the fusion of the two measurements are significantly smaller than those obtained when only the measurement of one instrument is used. This result demonstrates the ability of the proposed procedure to exploit the complementary information provided by the two instruments. The analysis of the AKs shows that the sensitivity of the retrieved quantities to the true quantities depends on the number of singular values considered to build the MSS. Increasing this number the sensitivity increases but above a certain value also the error increases. Therefore, the choice of the optimal value for this number is a compromise between the error and the sensitivity to the true value.
Acknowledgments The forward and inverse models, that we used for retrieval simulations from IASI spectra, have been optimized and tested in the frame of the project ‘‘Sensitivity analysis
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