Understanding the uncertainties in the retrieval of aerosol and surface properties is very important for an adequate characterization of the processes that occur in the atmosphere. However, the reliable characterization of the error budget of the retrieval products is a very challenging aspect that currently remains not fully resolved in most remote sensing approaches. The level of uncertainties for the majority of the remote sensing products relies mostly on post-processing validations and intercomparisons with other data, while the dynamic errors are rarely provided. Therefore, implementations of fundamental approaches for generating dynamic retrieval errors and the evaluation of their practical efficiency remains of high importance.
This study describes and analyses the dynamic estimates of uncertainties in aerosol-retrieved properties by the GRASP (Generalized Retrieval of Atmosphere and Surface Properties) algorithm. The GRASP inversion algorithm, described by
In this study, we analyse the efficiency of the GRASP error estimation approach for applications to ground-based observations by a sun/sky photometer and lidar. Specifically, diverse aspects of the error generations and their evaluations are discussed and illustrated. The studies rely on a series of comprehensive sensitivity tests when simulated sun/sky photometer measurements and lidar data are perturbed by random and systematic errors and inverted. Then, the results of the retrievals and their error estimations are analysed and evaluated. The tests are conducted for different observations of diverse aerosol types, including biomass burning, urban, dust and their mixtures. The study considers observations of AErosol RObotic NETwork (AERONET) sun/sky photometer measurements at
The analysis shows overall successful retrievals and error estimations for different aerosol characteristics, including aerosol size distribution, complex refractive index, single scattering albedo, lidar ratios, aerosol vertical profiles, etc. Also, the main observed tendencies in the error dynamic agree with known retrieval experience. For example, the main accuracy limitations for retrievals of all aerosol types relate to the situations with low optical depth. Also, in situations with multicomponent aerosol mixtures, the reliable characterization of each component is possible only in limited situations, for example, from radiometric data obtained for low solar zenith angle observations or from a combination of radiometric and lidar data. At the same time, the total optical properties of aerosol mixtures are always retrieved satisfactorily.
In addition, the study includes an analysis of the detailed structure of the correlation matrices for the retrieval errors in mono- and multicomponent aerosols. The conducted analysis of error correlation appears to be a useful approach for optimizing observation schemes and retrieval set-ups. The application of the approach to real data is provided.
Remote sensing is one major tool for monitoring atmosphere and surface properties at large scales. These observations have a nondestructive character and allow for dynamic local, regional or global monitoring of the ambient atmosphere. Correspondingly, diverse remote sensing observations are employed for routine observations and characterization of the Earth atmosphere. One of the key challenges in implementing remote sensing is the development of the retrieval algorithms. While remote sensing retrievals have substantially evolved during the last few decades, a significant need for further advancing various aspects of the retrieval algorithms remains. One of the most challenging and important, while underdeveloped, aspects is the evaluation of the errors in the retrieval products. For example, the review by
Here we discuss and analyse the approach implemented in the GRASP (Generalized Retrieval of Atmosphere and Surface Properties) retrieval algorithm. The GRASP concept is based on the statistical optimization fitting designed for the retrieval of detailed aerosol properties from diverse observations
One of the most visible data sets of ground-based radiometric observations is provided by AERONET (AErosol RObotic NETwork;
The GRASP is a state-of-the-art, highly versatile inversion algorithm of a new generation that can be applied to a variety of remote sensing and laboratory observations. For example, GRASP has been applied for several satellite instruments
The evaluation of the retrieval accuracy in all those studies was realized by comparing and validating the retrieval results with independent reliable data. It should be noted, however, that practically none of these studies discuss the retrieval error estimates, while the formalism of the error estimate has been realized in GRASP for a while. As a result, the validity of the retrieval error estimates provided by the GRASP approach remains unattended and unverified. Therefore, in order to address this issue, the current work proposes a discussion of the main aspects of the GRASP error generation and attempts to provide an evaluation of the retrieval error estimates provided by GRASP. The study is focused on the considerations of aerosol retrieval from sun/sky photometers and lidar ground-based observations.
As mentioned in the introduction, in this work we make use of the GRASP (Generalized Retrieval of Atmosphere and Surface Properties) algorithm. It is a rigorous, versatile and open-source algorithm capable of providing information of aerosol properties from the measurements of different instruments and dynamic error estimates
The retrieval error estimates in GRASP are calculated by modelling the propagation of measurement errors based on a statistical estimation approach. In addition, the formulation used for estimating errors accounts for some contribution of the systematic errors that could originate from biases in the measurement or some modifications implemented in the algorithm for improving the retrieval convergence of nonlinear solutions. Below, the descriptions of the overall concept and specific key implementations of the error estimation in GRASP are provided.
The multiterm LSM employed in GRASP searches for the solution using statistically optimized fitting under multiple a priori constraints
It can be noted that Eq. (
It can be noted that, if any of correlation coefficients are
For the general case of nonlinear functions
The asymptotic limit of the minimized quadratic form, for most applications, can be written as follows:
It should be noted that the LSM solution of Eq.(
In the frame of a multiterm approach, the use of weighting matrices additionally allows for making the contribution of different data sources more explicit. Indeed, using the weighting matrices
In this formulation, the relative contributions of the data from different data sources are scaled by the corresponding Lagrange parameters,
Therefore,
As discussed in detail by
At the same time, in practice, there are always two different types of data sets, i.e. measurements and the a priori constraint on the unknowns
Therefore, Eq. (
As discussed by
While the multiterm LSM concept allows flexible utilizations of nearly arbitrary a priori constraints, the GRASP algorithm is fully adapted for using the most popular and physically transparent a priori constraints, such as direct a priori estimates of unknowns
The a priori constraints defined by the second line
For the vector of unknowns
Thus, for a case where only direct a priori estimates and smoothness constraints are used, Eq. (
The realization of the inversion in GRASP, in principle, is based on the general Eq. (
The direct a priori estimates
The smoothness a priori constraints can be applied for each group of parameters describing a continuous function (e.g.
It should be noted that GRASP considers two types of a priori constraints, namely the single-pixel a priori constraints for the retrieved parameters that correspond to simultaneous and co-located observations, and the multipixel constraints that limit the variability for unknowns in different groups of similar parameters when several such groups of unknowns are retrieved simultaneously from coordinated but not fully coincident or not fully co-located observations (see details in
Since most of atmospheric remote sensing applications are strongly nonlinear, the Levenberg–Marquardt optimization
Correspondingly, using this additional requirement, an additional term will be introduced in Eq. (
The variance
Also, following the common Levenberg–Marquardt procedure, the impact of the correction
Thus, in the case of nonlinear
Estimations of the retrieval errors in GRASP are based on LSM equations expressed for the case of multiterm solutions written via weighting matrixes
The estimation of not only the random retrieval error but also the error retrieval bias
A rather obvious tendency can be seen from the analysis of this equation because the higher the contributions of the second and the third terms, the smaller the random errors are, i.e. the stronger a priori constraints used, the lower the random errors will be in the retrieval. However, in practice, a priori constraints can be unintentionally inadequate and therefore introduce some systematic uncertainties, i.e. biases. In principle, there is no guaranteed approach for detecting those biases unless a comprehensive analysis and validation of the retrievals have been done. Nonetheless, some biases can manifest themselves via misfit of measurements
In this equation, the contribution of a priori estimates to bias is probably the most significant in many applications, since it is never possible to have fully accurate a priori values (widely used in optimum estimation approaches) for constraining. In a similar way, the a priori biases are estimated in the case when multipixel a priori constraints are used.
The Levenberg–Marquardt optimization of the convergence, discussed in Sect.
Note that, from fundamental viewpoint, the a priori information is used in order to make solution unique, and if it is fully and adequately added, no dependence on the initial guess should be observed. At the same time, in practice, such a dependence often appears to some extent, especially in cases when the state vector includes a large number of unknowns. Moreover, if the retrieval is not optimally set, such a dependence can be rather significant, while unnoticed, because the retrieval continues to converge to local minima once the Levenberg–Marquardt optimization of the retrieval convergence is used. Therefore, in order to account for such an effect, the Levenberg–Marquardt contribution was added into the formalism for accounting possible biases. According to our evaluation, this term is nearly negligible if there is no dependence on the initial guess, while it increases if such a dependence appears.
Equations (
Also, in practice, the users may not directly need the retrieved parameters
Finally, the effect of biases in the measurements on the solution bias
In addition, in this work we also study the structure of the covariance matrix for different aerosols and configurations. Apparently, such a matrix provides interesting information about the error estimates (focusing on the diagonal elements) and the relation between the retrieval parameters (from the covariance values, i.e. non-diagonal elements). The representation of the covariance matrix for the parameters has the following structure:
In order to study the error correlation structure of the error, the following correlation matrix will be considered in this work that can be obtained from the covariance matrix (Eq.
The calculation of retrieval estimates in GRASP is based on rigorous formulations of statistical estimations described above. At the same time, a practical evaluation of the developed error formalism and possible tuning is desirable for a comprehensive evaluation of the approach and gaining full confidence in the practical efficiency of the approach. In this regard, one can probably state that the error estimate always tends to be less accurate than the retrievals themselves. Indeed, in remote sensing, the retrieval relies on the formalism of the electromagnetic light interaction theory that is fundamentally very accurate and well established, while the factors contributing to the uncertainties can be very diverse, not fully formalized and often not even fully understood. For example, the forward model is nonlinear, while the error propagations are usually (and in this work specifically) estimated in linear approximations, as commented previously, and the retrieval can be affected by not fully predicted biases in the measurements or by impercipient of aerosols or surface models (in our applications). Therefore, an important part of establishing error estimates is their evaluation and validation.
As mentioned above, in this study, we attempt to evaluate the GRASP estimations based on an extensive series of the numerical tests with added random noise that covers a wide range of practical situations. Moreover, we complement this study, assuming different biases, in order to see how the error estimates are represented in the cases with both random noise and bias. This section describes the design of the numerical experiment, including the following:
the instruments and retrieval scenarios used, the description of the overall experiment, the assumed atmospheric properties and covered distinct specific situations of interest, and the assumptions made for generated random errors, the considered retrieved parameter, the considered error characteristics and so on.
As mentioned earlier, this study evaluates the GRASP error estimates produced for the aerosol properties retrieved from ground-based observations. The details of the used observations and considered aerosol retrieval scenarios are provided in the next sections.
The analysis is focused on two widely known, and probably the most popular, retrieval scenarios used for deriving detailed aerosol optical properties.
Retrieval of columnar properties of aerosol from the measurements by ground-based sun/sky-scanning radiometers alone. Simultaneous retrieval of both columnar aerosol properties and their vertical distribution from the combined observations by sun/sky-scanning radiometers and multiwavelength lidar.
All the tests and analyses in this study include the spectral observations by the ground-based sun/sky-scanning radiometers. These radiometers were used for more than 30 years by the worldwide AERONET project (
Summary of general input data and the set of parameters retrieved by the GRASP algorithm used in this work for two configurations, i.e. sun/sky radiometer only and sun/sky radiometer plus lidar.
The detailed aerosol properties in the total atmospheric column provided by the AERONET inversion of sun/sky-scanning radiometers has been widely recognized as being rather unique, reliable data. For example, AERONET retrievals provided the first reliable data about aerosol spectral absorption and other detailed aerosol optical characteristics
As mentioned in earlier sections, the evaluations of the accuracy of retrieved aerosol parameters mainly relied on extensive sensitivity studies by
The inversion of co-located observations by sun/sky radiometers and lidar is another popular retrieval approach in the aerosol community. Indeed, radiometer direct sun and multiangular polarimetric observations of diffuse Sun radiation transmitted through the atmosphere have a significant sensitivity to the atmospheric aerosol amount, its particles size, shape and morphology; however, they have practically no sensitivity to the vertical variability in aerosols. The lidar observations, on the other hand, provide the information about the vertical distribution of aerosol, while their sensitivity to other aerosol properties is more limited compared to radiometer observations. Therefore, the information from co-located photometric measurements and lidar systems is complementary and always desirable for the enhanced characterization of aerosol properties. This complementarity is well recognized by the research community, and a large number of joint observational sites with both radiometer and lidar observations have been established in last decade. In these regards, the European ACTRIS (Aerosols, Clouds and Trace gases Research Infrastructure Network) infrastructure (
GRASP retrieval has been successfully adapted by
In these studies, we consider the aerosol retrieval from the base GARRLiC/GRASP input data set that includes AERONET sun/sky-scanning observations in the solar almucantar at four wavelengths and the lidar backscattering attenuation profile at three wavelengths at
As was already mentioned, GRASP has a capability to provide the full covariance matrix of the retrieval errors, and this study aimed to evaluate and illustrate the efficiency of these estimated covariance matrices. At the same time, retrieval error evaluations in most of the practical applications rely on the consideration of mainly diagonal elements of the covariance matrices, while non-diagonal elements of covariance matrices are much less common. Indeed, in spite of the fact that non-diagonal elements of covariance matrices provide valuable and interesting information about retrieval error correlations, these non-diagonal elements are not often available in practice, and the analysis of error correlations requires more sophisticated considerations compared to straightforward analysis diagonal elements only, and therefore, it is less popular. Considering these aspects, in the present study, as a first step, we make a more detailed and extensive analysis of the error variances, and then, as a second step, we illustrate the usefulness of obtained non-diagonal elements.
The performance of the GRASP error variances estimated, provided by Eqs. (
General scheme for the validation of the error estimates.
First, as showed in Fig.
where
The GRASP-generated variances of the retrieval errors are evaluated in the presence of random errors and analysed using a series on the numerical tests conducted for a statistically representative set of random error realizations. These results are then summarized for the whole series of the tests by figures and tables. The tests with added systematic errors are discussed for most of the separate systematic error types, while some overall summaries are also provided.
As was mentioned before, in addition to the standard deviation, the non-diagonal elements of covariance matrices provide additional important insight about the retrieval quality. This additional information mainly relates to non-zero correlation coefficients. Therefore, in order to illustrate the correlation structure, in this work we also analysed the correlation matrix that contains the covariance matrix elements normalized by the respective variances, as shown by Eq. (
The synthetic tests were performed for several preselected realizations of aerosol in the atmosphere. These realizations were selected based on extensive experience with aerosol retrieval from sun/sky radiometer data and their combination with co-located lidar data. It is expected that the selected aerosol realization scenarios are representative of the majority of distinct actual observations of atmospheric aerosols.
Two main observational scenarios are considered for different total aerosol optical depth at Single aerosol, such as biomass burning (BB), urban and dust for different aerosol loads The mixture of dust with BB and with urban (BB–Dust and Urban–Dust) is given. For each mixture, we have selected nine different scenarios that correspond to three different aerosol loads,
The single-aerosol and aerosol mixture observational scenarios are used in the generation of synthetic tests with sun/sky-photometer-only observations. By considering both a single-aerosol and two aerosol types, in this work we evaluated how the accuracy of the retrieved data evolves once a larger number of parameters are derived from the same information content. In contrast, the retrieval based on the synergy between lidar and sun/sky photometers aimed for the retrieval of the properties of two fine- and coarse-mode aerosol components; therefore, the numerical tests for this type of the retrieval rely on the mixed aerosol observation scenario. At the same time, the error estimation is also checked in the case when the joint radiometer and lidar observations of single aerosol are analysed. The aerosol properties description used for the synthetic cases can be founded in Table 1 of
The retrieval settings were used similar to those that conventionally used in the retrieval of aerosol from AERONET sun/sky radiometer observations by
As mentioned above, in the case of the joint processing of AERONET radiometer and lidar data, we considered the approach developed earlier by
Several tests were realized to evaluate the error estimates reliability and usefulness in the presence of both random and systematic uncertainties for aerosol retrievals from the observations of sun/sky radiometers alone and in combination with lidar. In this section, the results for the following two scenarios are presented: (i) a simpler case when only one type of aerosol is present and (ii) a more complex case in which two distinct types of aerosol are present at the same time. Moreover, we estimates the correlation matrices for both scenarios and illustrate their usefulness for understanding retrieval error tendencies, thus optimizing the retrieval approach.
In a series of these tests of all inverted the synthetic measurements, we added random noise with a standard deviation of
This section describes the evaluation of the error estimates, assuming the presence of only one type of aerosol, i.e. BB, urban or dust. As mentioned before, the retrieval aerosol properties under the assumption of the presence of a single-aerosol type composed of homogeneous particles is a well-established approach for deriving detailed aerosol properties from ground-based observations by a sun/sky radiometer that is adapted by the operational AERONET retrievals by
Figures
Aerosol properties retrieved from simulated sun photometer data, with random noise added for BB aerosols for
Aerosol properties retrieved from simulated sun photometer data, with random noise added for urban aerosol for
Aerosol properties retrieved from simulated sun photometer data, with random noise added for dust aerosol for
The retrievals improve and the errors decrease when the aerosol load increases, specially for IRI and SSA (absorption information). For BB and urban, the SSA error increases with the wavelength. On the other hand, SSA error decreases with the wavelength for dust. This is an expected behaviour, since the scattering efficiency is more pronounced at short wavelengths for small particles, while it is somewhat increasing with wavelength for large particles. Furthermore, as shown in Fig.
To evaluate the error estimates in the presence of random noise, a set of the simulations for 300 different realizations of noise modelled using a random number generator has been analysed in this work. The results of such numerical tests conducted with a statistically representative set of random errors are summarized and illustrated, using box plots of the errors, as demonstrated in Fig.
Comparison of the variance SSA(675) values estimated by the GRASP algorithm with actual errors obtained for extensive tests with randomly added modelled errors. In the upper panel, the box represents
Figure
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from sun/sky-photometer-simulated measurements (a case with
On the other hand, the RRI errors seem to be similar at the different wavelengths. This is likely related to the fact that, spectrally, RRI retrievals rely on rather strong smoothness constraints on spectral variability in RRI
Table
Errors provided by GRASP for the RRI, IRI and SSA are represented by the mean values of each box plot for their respective wavelength. Absolute errors are given for RRI and SSA and relative errors for IRI. Mean values of actual errors are provided in parenthesis.
As already mentioned, most conventional aerosol retrievals from ground-based radiometer measurements
At the same time, since the retrieval of multicomponent aerosol from radiometers only is not often used and not employed for operational retrievals, the tests in this section are limited to several illustrations only, and no statistical evaluation is performed. The illustrations are produced for the observations of a mixture of Urban–Dust and BB–Dust (see Sect.
Figures
Aerosol properties retrieved from simulated sun photometer data with random noise added for a mixture of Urban–Dust aerosols. The solid lines indicate the simulated properties (SD, RRI, IRI and SSA), and the dashed lines are the retrieved parameters. The shaded areas indicate the error estimated by the GRASP algorithm. The magnified plots represent the effective refractive index and total SSA.
Aerosol properties retrieved from simulated sun photometer data with random noise added for a mixture of BB–Dust aerosols. The solid lines indicate the simulated properties (SD, RRI, IRI and SSA), and the dashed lines are the retrieved parameters. The shaded areas indicate the error estimated by the GRASP algorithm. The magnified plots represent the effective refractive index and total SSA.
The most obvious difficulties in the separation of modes are evident when the properties of each mode are not very different. For example, such a situation can be seen for IRI of the Urban–Dust mixture (Fig.
It should be noticed that the retrieval of the multicomponent aerosol from radiometric observations was added here to illustrate error transformation tendencies that are not evident. Indeed, the retrieval of multicomponent aerosol from the AERONET observation is very uncertain, as was shown, for example, by
The GRASP aerosol retrieval from combined sun/sky radiometers and lidar observations were always designed for the retrieval of bicomponent aerosol
Figures
Aerosol properties retrieved from simulated sun photometer and lidar data with random noise added for a mixture of Urban–Dust aerosols. The solid lines indicate the simulated properties (SD, RRI, IRI and SSA), and the dashed lines are the retrieved parameters. The shaded areas indicate the error estimated by the GRASP algorithm. The magnified plots represent the effective refractive index and total SSA.
Aerosol properties retrieved from simulated sun photometer and lidar data with random noise added for a mixture of BB–Dust aerosols. The solid lines indicate the simulated properties (SD, RRI, IRI and SSA), and the dashed lines are the retrieved parameters. The shaded areas indicate the error estimated by the GRASP algorithm. The magnified plots represent the effective refractive index and total SSA.
Figure
The aerosol lidar ratio (LR) retrieved from simulated sun photometer and lidar data with random noise added for a mixture Urban–Dust aerosols (above) and BB–Dust (below). The solid lines indicate the simulated properties (SD, RRI, IRI and SSA), and the dashed lines are the retrieved parameters. The shaded areas indicate the error estimated by the GRASP algorithm. The magnified plots represent the results for total LR retrievals.
Figure
The aerosol vertical profiles (AVPs) retrieved from simulated sun photometer and lidar data with random noise added for a mixture of Urban–Dust aerosols
In order to evaluate the error estimates in the presence of random errors,
a set of simulations, adding 300 realization of random noise values, is analysed. Figures
Figures
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of Urban–Dust aerosol. The distributions were obtained using 300 realizations of added random errors. The median values of the errors are shown by a line in the box plot, along with the 25th–75th percentiles indicated by a box, and 5th–95th percentiles are indicated using whiskers. The mean values are represented by the black dot. The red colour shows the error estimates provided by GRASP, and the blue colour shows the calculated actual errors (Eq.
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of BB–Dust aerosol. The distributions were obtained using 300 realizations of added random errors. The median values of the errors are shown by a line in the box plot, along with the 25th–75th percentiles indicated by a box, and 5th–95th percentiles are indicated using whiskers. The mean values are represented by the black dot. The red colour shows the error estimates provided by GRASP, and the blue colour shows the calculated actual errors (Eq.
Figures
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of Urban–Dust aerosol. The distributions were obtained using 300 realizations of added random errors. The median values of the errors are shown by a line in the box plot, along with the 25th–75th percentiles indicated by a box, and 5th–95th percentiles are indicated using whiskers. The mean values are represented by the black dot. The red colour shows the error estimates provided by GRASP, and the blue colour shows the calculated actual errors (Eq.
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of BB–Dust aerosols. The distributions were obtained using 300 realizations of added random errors. The median values of the errors are shown by a line in the box plot, along with the 25th–75th percentiles indicated by a box, and 5th–95th percentiles are indicated using whiskers. The mean values are represented by the black dot. The red colour shows the error estimates provided by GRASP, and the blue colour shows the calculated actual errors (Eq.
The error evaluation for LR is represented in Figs.
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of Urban–Dust aerosols. The distributions were obtained using 300 realizations of added random errors. The median values of the errors are shown by a line in the box plot, along with the 25th–75th percentiles indicated by a box, and 5th–95th percentiles are indicated using whiskers. The mean values are represented by the black dot. The red colour shows the error estimates provided by GRASP, and the blue colour shows the calculated actual errors (Eq.
Comparison of estimated and actual error distributions for spectrally dependent aerosol parameters retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of BB–Dust aerosols. The distributions were obtained using 300 realizations of added random errors. The median values of the errors are shown by a line in the box plot, along with the 25th–75th percentiles indicated by a box, and 5th–95th percentiles are indicated using whiskers. The mean values are represented by the black dot. The red colour shows the error estimates provided by GRASP, and the blue colour shows the calculated actual errors (Eq.
The results illustrated by the figures are summarized in Tables
The mean values of RRI, IRI, SSA and LR retrieval errors estimated by GRASP for the synthetic test for a mixture of the Urban–Dust aerosol mixture. The mean values represent the distributions obtained using 300 realizations of the added random errors for the situation with total
The mean values of RRI, IRI, SSA and LR retrieval errors estimated by GRASP for the synthetic test for a mixture of the BB–Dust aerosol mixture. The mean values represent the distributions obtained using 300 realizations of the added random errors for the situation with total
The mean values of aerosol vertical profile (AVP) retrieval errors estimated by GRASP for the synthetic test for a mixture of the Urban–Dust aerosol mixture. The mean values represent the distributions obtained using 300 realizations of the added random errors for the situation with total
For retrieval errors in fine mode in the case of urban aerosol parameters,
the mean values for RRI are around
The mean values of the error estimates provided by GRASP for dust present a good agreement in the case of both mixtures. In general, the mean values for RRI error estimates vary between
Once again, it is important to note that the errors in the parameters characterizing the total aerosol are generally accurately estimated. For both cases of Urban–Dust and BB–Dust mixtures, the mean values of the total SSA error estimates vary between
Figure
Comparison of estimated and actual error distributions for AVP retrieved from measurements by simulated sun/sky photometer values and lidar for a mixture of Urban–Dust aerosols
Finally, a lower sensitivity to the retrieval of fine-mode properties can be observed as a clear tendency in the evaluation of the retrieval errors for the cases when mixed aerosols are analysed. In particular, quite high errors were obtained for the complex refractive index. Then, these errors consequently propagate to the errors in other optical properties, such as the SSA of fine mode, as was found in the earlier study by
In Sect.
As mentioned above, in Eq. (
The potential effect of the systematic errors is analysed in series of numerical tests with possible assumed systematic errors. Following previous studies by In AOD, there is a nominal bias of In radiances, there is a nominal bias of
To evaluate the effects of biases, the above values were added to the synthetic direct measurements of AOD and sky radiance by an AERONET-like ground-based radiometer. These data were inverted by the GRASP code, and the retrieved values of aerosol parameters were compared to the values assumed in the synthetic simulation as a truth. In addition, the deviations in the retrieved values from the true ones are compared to the errors estimates generated by GRASP based on Eq. (
In this section, the study is focused on the analysis of the effects of the biases and on estimating contribution of systematic errors in the retrievals of aerosol from ground-based observations by a radiometer. In a similar manner to the analysis of random errors, we first considered the observations dominated by two types of aerosols, i.e. BB and dust. The effect of measurement biases is expected to be manifested in the situations with low and moderate aerosol loading; therefore, the analysis is focused on the scenarios with AOD
The following two situations were considered:
when a single bias in AOD or radiances is present, and when the biases can be present in both AOD and radiances simultaneously. In this case, the different combinations of positive and negative biases in AOD and radiances are considered.
The estimations of the errors introduced by the biases were calculated as follows:
Figures
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for BB aerosol for
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for BB aerosol for
The figures show different and clear tendencies, which are in agreement with general expectations, and with the tendencies already observed in previous studies by
Overall, the estimated systematic error agrees well with the actual manifestations of the bias in the retrieval. The quantitative estimations are also quite convincing and shown in Figs.
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for dust aerosol for
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD simulated data for dust aerosol for
It can be seen that, among all considered aerosol parameters, the main differences between the bias effects and the obtained error estimates are observed for the real part of the refractive index (RRI). In these cases, the bias is not fully covered by the systematic component of the retrieved error results. Similarly, an apparent underestimation of the RRI errors was also seen by
Figures
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for BB aerosol for
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for BB aerosol for
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for dust aerosol for
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD-simulated data for dust aerosol for
It should be also noted that we observe an anticorrelation between the radiance bias and the retrieval of the imaginary part of the refractive index. This effect is opposite to the one observed in the case of AOD and with significantly smaller differences. Thus, when the biases are positive (
Figure
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD- and radiance-simulated data for BB (left) and dust (right) aerosol for
As can be seen from Fig.
Figure
Aerosol properties retrieved from simulated sun/sky photometer data with assumed bias in AOD- and radiance-simulated data for BB (left) and dust (right) aerosol for
It can be seen that the total error estimates capture the deviations for all parameters in the presence of random and systematic noise. These results confirm that the estimations using Eq. (
The present section tries to understand how the bias is affected when inhomogeneous aerosol are observed. The example is not commonly considered in practical application, e.g. in AERONET operational processing. At the same time, since GRASP can consider this type of bicomponent inversion that is fundamentally of high interest, we are analysing this situation in the presence of biases. In Sect.
Different tests were performed for this study. In particular, we focus on the case of BB–Dust, since Sect.
Aerosol properties retrieved from simulated sun/sky photometer data with assumed random noise and bias in AOD- and radiance-simulated data for BB–Dust for
On the other hand, in this section, some illustrations for the lidar ratio are also provided in order to demonstrate how the retrievals and the error estimates are affected by the bias in the measurements. Figure
This section considers the same example as in Sect.
Figure
Aerosol properties retrieved from simulated sun/sky photometer and lidar data with assumed random noise and bias in AOD-, radiance- and lidar-simulated data for BB–Dust for
With regard to the accuracy of the error estimation, in Sect.
Thus, using the synergy of both instruments can provide more accurate retrievals of LR, and the error can be estimated rather accurately using the developed methodology for both aerosol components, even for an aerosol mode with a lower presence. Figure
Aerosol properties retrieved from simulated sun/sky photometer and lidar data with assumed random noise and bias in AOD-, radiance- and lidar-simulated data for BB–Dust for
The values of non-diagonal elements of covariance provide important and interesting information about the retrieved parameters. For example, if the values
From this equation, the importance of the correlation coefficient
When
From these equations, it can be seen that if the correlation coefficient
In practical cases, when the derived parameter
Figure
Correlation matrices of the estimated errors for aerosol retrieval from sun/sky radiometer observations
The correlation for biomass burning case is shown in Fig.
Figure
Thus, the analysis of the correlation matrices itself provide very useful insight that helps to understand and interpret retrieval results. For example, such artefacts as the appearance of tails (unrealistically high concentrations) at extremes of the size distribution have been noticed and widely discussed
In this section, the correlation matrix for bicomponent aerosol retrieved from the synthetic observations of sun/sky radiometers of two aerosol mixtures is illustrated in Fig.
Correlation matrices of the estimated errors for mixed aerosol retrieval from sun/sky radiometer observations
The area of the correlation matrix that contains SD, RRI, IRI and the sphericity fraction is quite similar to the correlation matrix obtained for aerosol AERONET-like retrieval. The main difference is the separation into two modes, since strong negative correlations can be observed between the corresponding parameters of the fine and coarse mode. For example, strong negative correlations can be observed between IRI fine and coarse mode. These correlations mean that overestimating the amount or absorption of one aerosol mode is likely compensated by underestimation of the amount or absorption of another aerosol mode. Another interesting anticorrelation can be observed for the last three bins of SD fine mode and the first three bins of SD coarse mode. Actually, both volume distributions have these three bins in common. This overlapping zone is never easy to properly separate for the code, but at the same time, it coincides with a local minimum value of most size distributions found in the real retrievals.
Figure
Correlation matrices of the estimated errors for aerosol retrieval from joint sun/sky radiometer and lidar observations for a mixture of urban and desert dust using the GRASP algorithm. The values close to
The area of the correlation matrix that contains SD, RRI, IRI and the sphericity fraction is quite similar to the correlation matrix described in the previous section, considering that the aerosol mixture retrievals are from AERONET-like observations only.
As expected, the block of the correlations of AVP retrieval shows strong negative correlations between the errors in the retrieved parameters of fine and coarse mode. Thus, an overestimation of one mode is highly correlated with an underestimation of another AVP mode. Furthermore, strong positive correlations can be observed between AVP values corresponding to the same fine or coarse mode, i.e. the AVP values of each mode have tendency to be simultaneously overestimated or underestimated. This is related to the limited sensitivity of the used lidar data for distinguishing the contributions of different modes and also to the use of smoothness constraints on vertical variations in AVP of each fraction. On the other hand, a near-zero correlation can be seen for AVP parameters at altitudes with a significant presence of one or both aerosol components. Correspondingly, there is a high sensitivity of both lidar and radiometer observations to the aerosol parameters at those altitudes. It should be noted that all the above-discussed retrievals suggested from the analysis of the correlation matrices were actually observed in the retrievals from real data, as discussed by
This section illustrates the GRASP error estimates performance for the retrieval from real data. For that purpose, the lidar and sun/sky photometer measurements collected at the Aeroparque (34
Observations from two different biomass burning events in Argentina were selected for the illustrations. Specifically, 3 d were chosen with different aerosol loads and SZA. The lidar range-corrected signals (RCSs) corresponding to each event are shown in Fig.
RCSs at 1064 nm in arbitrary units from Villa Martelli, Argentina, on 19 August 2014
The two first cases selected correspond to an important event of biomass burning that occurred in the bordering countries to the north of Argentina in August 2014, particularly in the south of Brazil and Paraguay. It was detected in Buenos Aires between 19 and 23 August. For illustration purposes, the measurements corresponding to 19 August are used, which present a low aerosol load at
Figure
Comparison of columnar properties retrieved by GRASP from a combination of sun/sky photometer and lidar data and data retrieved conventionally by AERONET. SD, RRI, IRI and SSA retrieved by GRASP are shown in solid lines of blue (fine mode) and green (coarse mode). The shaded area in colours of blue and green represents the total error provided by GRASP, and black shaded areas are the uncertainties provided by AERONET. Magnified panels show the RI effective and total SSA provided by GRASP (black solid line) and AERONET (black dashed line). Their associated errors are represented in the grey shaded area for GRASP and with error bars for AERONET.
The error tendencies for SD that can be seen from Fig.
The errors in RRI, IRI and SSA were retrieved for each mode separately by GRASP, and they are significantly higher than the error for RRI, IRI and SSA of the total components. The effective RRI and IRI and the total SSA obtained by GRASP are in the middle of the retrieved values for fine and coarse mode separately. The total values shown in the magnified plots agree well with the RRI, IRI and SSA provided by AERONET.
On the other hand, the case corresponding to 25 September has an AOD
Figure
Retrieved aerosol vertical profiles (AVPs) by GRASP from a combination of sun/sky photometer and lidar data. The blue solid line represents the AVP fine mode and green the AVP coarse mode. The shaded areas correspond to the total error provided by GRASP.
Moreover, Fig.
Retrieved LR by GRASP from a combination of sun/sky photometer and lidar data. The blue solid line represents the LR fine mode and green the LR coarse mode. Magnified plots show, with black solid lines, the LR provided by GRASP and, with dashed lines, the LR provided by AERONET. Their GRASP-associated errors are represented in shaded areas.
Thus, the retrieved parameters and error estimates from the GRASP application to the real data and their comparisons to the AERONET retrieval results showed an encouraging agreement between the columnar properties of aerosol. At the same time, GRASP provide the error estimates for the retrieved properties in both the fine and coarse mode and also for the total components. Moreover, GRASP has also the advantage of providing the dynamic error estimates in all configurations. As seen above, AERONET error estimates are only provided in some particular situations when the AOD at 440 nm is
In this work, we reviewed the approach realized in the GRASP algorithm for estimating errors in the parameters retrieved from remote sensing observations. The employed approach relies on the rigorously realized concept of statistical estimations and tends to account for the propagation of both random and systematic errors. Then we evaluated the performance of the GRASP error estimates for aerosol parameters retrieved from ground-based observations. We considered AERONET-like retrievals from observations by sun/sky-scanning radiometers and GRASP synergy aerosol retrievals from joint observations by radiometers and lidar. GRASP generates the full covariance matrices that are expected to be used for generating error bars for retrieved parameters and provides an interesting insight for understanding retrieval tendencies. Therefore, we studied the quantitative reliability of the obtained covariance diagonal elements and analysed the structure of correlation coefficient of covariance matrices.
The performance of the GRASP estimates of error variances in the presence of random errors was evaluated in a series of numerical tests and illustrated the capabilities of the GRASP algorithm to provide rigorous estimates of dynamic retrieval errors. In the frame of these tests, the synthetic proxy observations perturbed by 300 random-noise-generated realizations were inverted using the GRASP algorithm. Then, the retrieved parameters were compared to those used for the generation of the synthetic data, and the obtained error estimates were compared with actual deviations in the retrieved parameters from assumed values. This analysis was realized for the synthetic observations for three different types of aerosols and for the mixture of them. Observations of dust were modelled using AERONET retrieval climatology at the Solar Village (Riyadh, Saudi Arabia) site. The AERONET retrieval climatologies from the African savanna (Zambia) and the GSFC (Maryland, USA) were used to simulate urban and BB aerosol observations, respectively. The Urban–Dust aerosols and BB–Dust mixture were considered for modelling the properties of mixed aerosols. For each observed aerosol type or mixture thereof, different aerosol loads were tested. First, we modelled observations of aerosols of only one type aerosol for
The tests evaluated the situations when only radiometer data were inverted, and then radiometer data were inverted jointly with coincident lidar data. Two GRASP retrieval set-ups were tested, where (i) the retrieval assumes that aerosol is composed of homogeneous particles, and parameters of only one aerosol component are retrieved, and (ii) then the aerosol is assumed to be an external mixture of two aerosol components, and the parameters of each component are retrieved separately. In the case when the lidar data were used, the vertical profiles of the concentration were also retrieved for each aerosol component. The illustrations for aerosol retrieval from all simulated data sets were provided, using both approaches. However, the full statistical analysis provided only the two conventional retrieval scenarios, i.e. AERONET-like, single-component retrieval from radiometric observations and GRASP bicomponent aerosol retrieval from combinations of radiometer and lidar data.
The results of the tests showed that the complete set of aerosol parameters for each aerosol component can be robustly derived with acceptable accuracy in almost all considered situations. The retrieval of bicomponent aerosol was evaluated using radiometer-only simulated measurements and then adding lidar observations. These tests allowed us to observe that, by using the synergy of two instruments, there are some improvements in the retrieval of the aerosol properties of each component of the observed aerosol mixture and in the estimations of the retrieval errors. The test for selected cases with different presences of different aerosol components (
The results of the statistical tests with randomly generated noise showed that the GRASP error estimates, in most cases, are comparable to or exceed the actual errors by 20 % to 30 % and therefore can be safely used for assuring uncertainties in the actual retrieval products. In addition, the observation of typical error values was summarized for different situations and retrieval scenarios. Namely, the study confirmed that the detailed properties of aerosol mixtures can be rather reliably retrieved from a combination of radiometer and lidar data, provided that there is a sufficient amount of both aerosol components. For example, for the case when the total
The effects of the systematic errors in the retrievals were also analysed in a series of limited dedicated numerical tests. The results of the tests were used for adjusting the GRASP retrieval estimates to the potential effects of the systematic errors. The results show enhancements of total error estimates from the assumption of bias in the equation of systematic components.
In addition to the evaluation of the error bar estimates and the effects of systematic errors, in this paper we illustrated and discussed the correlation structures of the error covariance matrices for all main considered retrieval scenarios. The results showed that analysis of the correlation structure can be very useful for understanding the observed retrieval tendencies and optimizing retrieval. For example, for conventional AERONET-like aerosol retrievals from radiometer data only, the strong negative correlation between errors is in the real part of the refractive index and size distribution values for small sizes. This agrees well with the tendency commonly observed in actual retrieval when the underestimations of the real part are coincident with the overestimation of the fine-mode size distribution. Also, the presence of a high positive correlation between the errors in the size distribution for extreme sizes and between the errors in the refractive index at different wavelengths agrees well with the known possibilities of possible overestimations of aerosol concentrations for very small or very large particles and joint overestimations/underestimations of the refractive value at different wavelengths. For bicomponent retrievals, strong negative correlations can be observed between nearly all corresponding parameters of the fine and coarse mode. This means, for example, that the overestimation of the amount or absorption of one aerosol mode is likely compensated by underestimation of the amount or absorption of another aerosol mode. The decrease in some of these correlations was observed when inverted radiometer data were inverted simultaneously with the lidar data. The high positive correlations were seen for the errors in the vertical profile of the fine and coarse concentrations, with the exception of the values for the altitudes where one or both of the aerosol modes had substantial loads. These and other less obvious, but quite interesting, correlation structures and tendencies can be identified using the analysis of the correlation matrix structure. Thus, the availability and analysis of not only the error variances but also the correlation patterns appear to be a useful and promising approach for optimizing observation schemes and retrieval set-ups.
Finally, the utilization of GRASP for deriving detailed aerosol properties and estimations of their errors was demonstrated for the coincident lidar and sun photometer observations from Buenos Aires, Argentina. The GRASP retrievals and the error estimates of the columnar aerosol properties were shown to be fully adequate in a comparative analysis with the aerosol products available from AERONET operational retrievals. The retrieval of the vertical profiles of fine and coarse aerosol modes showed results consistent with the expectation and the predictions of back-trajectory analysis.
Thus, the results presented in this work show promising potential for the utilization of GRASP-retrieved dynamic error estimates for the detailed retrieved aerosol parameters from measurements of ground-based radiometers and lidars, considering different geometries and the presence of diverse aerosol loads. These studies are expected to be completed in future by a more extensive analysis of the error estimates for such detailed parameters as vertical profiles of SSA and LR.
GRASP is an open-source software that is available, upon registration, from
The data are available upon request to the corresponding author. AERONET data used for this study can be downloaded from NASA website
OD contributed to the development of the overall algorithm methodology, research planning and article writing. BT contributed to the result discussions and the article writing. BT, TL, DF, AL, PL, CC and MEH contributed to the algorithm designs, development and software support. PL, CC, JABO and JLB contributed to the result discussions and polishing the article text. PR contributed to the data preparation and analysis. MEH prepared the paper, including the co-author contributions, and performed the main part of the presented research.
At least one of the (co-)authors is a member of the editorial board of
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The authors acknowledge the support from the H2020 Marie Skłodowska-Curie RISE Actions (grant no. 778349), and the authors acknowledge the support from the European Metrology Program for Innovation and Research (EMPIR) within the joint research project of EMPIR 19ENV04 MAPP “Metrology for aerosol optical properties”. The EMPIR is jointly funded by the EMPIR participating countries within EURAMET and the European Union. Tatyana Lapyonok has been supported by the Chemical and Physical Properties of the Atmosphere Project funded by the French National Research Agency through the Programme d'Investissement d'Avenir (contract no. ANR-11-LABX-0 0 05-01), the Regional Council, Hauts-de-France, the European Funds for Regional Economic Development (grant no. URF/1/2180-01-01), and the Combined Radiative and Air Quality Effects of Anthropogenic project. The authors also acknowledge AERONET and LALINET, for the scientific and technical support. We acknowledge the use of imagery from the NASA Worldview application (
This research has been supported by the European Metrology Programme for Innovation and Research (grant no. 19ENV04 MAPP) and the French National Research Agency (grant no. ANR-ll-LABX-0005-01).
This paper was edited by Daniel Perez-Ramirez and reviewed by Feng Xu and one anonymous referee.