Multi-angle polarimetric (MAP) measurements contain rich information for characterization of aerosol microphysical and optical properties that can be used to improve atmospheric correction in ocean color remote sensing. Advanced retrieval algorithms have been developed to obtain multiple geophysical parameters in the atmosphere–ocean system, although uncertainty correlation among measurements is generally ignored due to lack of knowledge on its strength and characterization. In this work, we provide a practical framework to evaluate the impact of the angular uncertainty correlation from retrieval results and a method to estimate correlation strength from retrieval fitting residuals. The Fast Multi-Angular Polarimetric Ocean coLor (FastMAPOL) retrieval algorithm, based on neural-network forward models, is used to conduct the retrievals and uncertainty quantification. In addition, we also discuss a flexible approach to include a correlated uncertainty model in the retrieval algorithm. The impact of angular correlation on retrieval uncertainties is discussed based on synthetic Airborne Hyper-Angular Rainbow Polarimeter (AirHARP) and Hyper-Angular Rainbow Polarimeter 2 (HARP2) measurements using a Monte Carlo uncertainty estimation method. Correlation properties are estimated using autocorrelation functions based on the fitting residuals from both synthetic AirHARP and HARP2 data and real AirHARP measurement, with the resulting angular correlation parameters found to be larger than 0.9 and 0.8 for reflectance and degree of linear polarization (DoLP), respectively, which correspond to correlation angles of 10 and 5

Satellite remote sensing is important for the study of the earth system at a global scale

Uncertainty quantification from MAP retrievals provides information on the quality of the data products and improves our understanding of retrieval sensitivities. These uncertainties depend on the retrieval algorithm as well as the instrument characterization, including the spectral bands, viewing angles, and polarization capability and the measurement accuracy. As shown in Fig.

Current MAPs in terms of the number of spectral bands and total number of viewing angles as summarized in

Acronyms and their full name for the MAP instruments plotted in Fig.

To understand the retrieval uncertainties, an uncertainty model is required to describe the combined uncertainties from the MAP measurements, forward model, and a priori assumptions. These combined uncertainty sources are often assumed to be independent and without correlations; however, measurements with high angular or spectral resolution are likely to have correlated uncertainty, depending on instrument design. For example, a sensor may use the same detector to scan through all measurement view angles (e.g. the Research Scanning Polarimeter, RSP,

Retrieval algorithms that exploit correlation information in retrieval parameters and measurement uncertainties have shown benefits in improving remote sensing capabilities. The Generalized Retrieval of Aerosol and Surface Properties (GRASP) algorithm retrieves multiple pixels simultaneously while considering the spatial correlation of the retrieval parameters

In this study, we provide a practical framework to understand the measurement uncertainty structure, study the impact of correlation in MAP retrievals, and demonstrate the potential for improvement in geophysical retrieval performance when proper correlation information is incorporated into the retrieval algorithm. Angular uncertainty correlations in measurements from the AirHARP and HARP2 instruments are studied as examples. Both instruments measure 60 angles at 670 nm. AirHARP measures 20 angles for the 440, 550, and 870 nm bands, while HARP2 measures at 10 angles for these bands. Angular correlation within each band is considered and modeled separately. Two methods are used to evaluate the retrieval uncertainties under different correlation strengths: (1) the error propagation method is used to evaluate the optimal retrieval uncertainties, by mapping the input uncertainty model describing the total uncertainty of the measurement and forward model to the retrieval parameter domain, and (2) comparative analyses are performed between the retrieval results from synthetic MAP measurements and the “truth data” that were assumed in the generation of that synthetic MAP data. The Monte Carlo error propagation method (MCEP) is adopted to compare the retrieval uncertainties from these two methods. To efficiently conduct retrieval and uncertainty analysis, the FastMAPOL retrieval algorithm is employed in this study, which uses neural-network forward models for coupled atmosphere and ocean systems

Furthermore, we study the angular uncertainty correlation in the measurements and demonstrate that the correlation property can be derived using the autocorrelation function from the retrieval fitting residuals. Studies on both synthetic data with various correlation strengths are conducted with results applied to the real measurement retrievals from AirHARP over multiple ocean scenes. Useful tools are provided to understand and analyze the angular correlated uncertainty structure and models.
Note that autocorrelation analysis based on fitting residuals has been found useful in analyzing performance of machine learning algorithms such as using the Durbin–Watson test

In the following sections, we will discuss how to conveniently include an angular correlated uncertainty model in the retrieval algorithm (Sect.

In this study, the FastMAPOL algorithm is used to retrieve aerosol and ocean optical properties from HARP measurements. The algorithm includes three main components: (1) a set of neural-network-based radiative transfer forward models of the coupled atmosphere and ocean system

The neural-network forward models are trained for both reflectance and degree of linear polarization (DoLP) based on simulations from the successive orders of scattering radiative transfer model (RTSOS) developed by

A total of nine parameters are used to describe the aerosol microphysical properties. There are four parameters for the complex refractive index of fine and coarse mode. Aerosol size distributions are parameterized by five volume densities for five size submodes with fixed effective radius and variance

In this study, we will discuss the retrieval uncertainty and performance in aerosol properties, ocean surface wind speed, and Chl

The maximum likelihood approach is used to retrieve the state parameters in FastMAPOL by minimizing a cost function that represents the difference between the measurements and the forward-model fitting

The error covariance matrix

The ratios between random and calibration uncertainties may be different for reflectance and polarized signals

To better represent stronger correlations when it is close to 1, we define the correlation angles

The pixel-wise retrieval uncertainty can be quantified by mapping the measurement and forward-model uncertainties into retrieval parameter space

The retrieval uncertainties estimated by error propagation (hereafter called theoretical retrieval uncertainty) as shown in Eq. (

Error and uncertainty definitions.

The error covariance matrix with non-diagonal terms is challenging to implement efficiently in optimization algorithms, which typically operate in diagonal space with no correlation between measurements. The error covariance matrix also creates barriers to understand the retrieval uncertainties, as the input uncertainties are not for a single measurement but rather related to multiple measurements. To overcome these issues, we convert the measurements into a new space where the error covariance matrix is diagonalized. Therefore, conventional optimization and error analysis techniques can be readily used.

To achieve this goal, eigenvector decomposition is applied on the error covariance matrix

Equations (

Examples of simulated measurement errors generated for reflectance at the 660 nm band with a correlation angle of 10

Autocorrelation is a useful function to quantify correlation in a discrete data sequence and is defined as

We can estimate the correlation by analyzing the residuals between the measurement and forward model. The autocorrelation function is averaged over multiple pixels to reduce uncertainties for the analysis in both synthetic data and real retrieval residuals. The correlation parameter can then be derived as

Correlation angles,

An example is shown in Fig.

The autocorrelation and partial autocorrelation on the 1000 sets of simulated errors with a correlation angle of 60

The neural-network forward model discussed in Sect.

Example viewing zenith (

To generate realistic measurements, correlated uncertainties with a correlation angle (

Correlated errors for both the AirHARP and HARP2 instruments are generated according to the same 3 % uncertainty for reflectance but 0.01 in DoLP for AirHARP and 0.005 in DoLP for HARP2. These errors are added to the corresponding simulated reflectance and DoLP (

Four scenarios of simulated uncertainties are considered in the synthetic data and retrievals. C1 and C2 indicate simulated errors with a correlation angle of

Using the Monte Carlo error propagation (MCEP) method discussed in

The histogram of the AirHARP retrieval errors for the fine-mode AOD from theoretical and real uncertainty estimations based on the MCEP method.

Both real errors and theoretical uncertainties have occasional outliers with large values possibly due to convergence to local minima instead of global minima, and this has large impacts on the RMSE values. For example, in Fig.

Based on the MCEP method, we analyzed the retrieval uncertainties for synthetic AirHARP measurements from real errors and theoretical errors for various properties in Fig.

Retrieval uncertainties averaged for the AOD within

The real retrieval uncertainties for Scenario C3, in which correlation is considered in the simulated errors but not in the retrieval cost function, are found to be always increasing with the correlation angle. When the correct correlation angle is considered in the retrieval cost function (C4), the real retrieval uncertainty increases until

The results for HARP2 are similar to that of AirHARP as shown in Fig.

The ratio of the real (

To understand more quantitatively how the correlation angle impacts both AirHARP and HARP2 retrievals, the ratios of the real uncertainties between scenarios C3 and C4 are represented as

The ratio of real retrieval uncertainties between scenarios C4 and C3 as denoted in the legend by

To understand retrieval performance when the uncertainty correlation is only in reflectance for scenarios C1 and C2, similar ratios to Fig.

Similar to Fig.

As discussed in

Cost function histogram with a correlation angle of 0, 10, and 60

However, the cost function histogram shifts to a smaller value for Scenario C3 (Fig.

To understand how much overfitting impacts retrievals under a different strength of correlations, we compare the standard deviation of the retrieval residuals with the original simulated uncertainty for all 1000 cases. For reflectance, we consider the ratio between the simulated uncertainty and the reflectance as a convenient way to compare with the 3 % (or 0.03) uncertainty model for reflectance (Fig.

The standard deviation of simulated measurement errors and fitting residual for cases C3 and C4. The model uncertainty of 3 % for reflectance and 0.01 for DoLP are indicated by dashed lines. Results of scenarios C1 and C2 for reflectance are similar, but with estimated DoLP uncertainties closer to the 0.01 line.

Different amount of over fitting also partially removed the correlation in the fitting residuals. As shown in Fig.

The estimated correlation angle

The Aerosol Characterization from Polarimeter and Lidar (ACEPOL) field campaign was conducted from October to November of 2017 with the NASA's ER-2 aircraft at a high altitude of approximately 20 km

The number of total viewing angles (

Correlation properties from real measurements are difficult to quantity. However, we show in Sect.

Autocorrelation

We analyzed the fitting residuals for all the five AirHARP scenes at the four bands with results summarized in Fig.

In this study, we evaluate the impacts of angular correlation on the retrieval uncertainties for various aerosol microphysical and optical properties, ocean surface properties, and water-leaving signals. Theoretical uncertainties are derived based on error propagation, and the real uncertainties are obtained through the comparison of retrieved and true values. The theoretical and real uncertainties are compared and discussed. Only small angular correlation impacts are found on the real retrieval uncertainties unless the correlation strength is large (such as with a correlation angle larger than

Studies on the fitting residuals from both synthetic data and real AirHARP measurements were conducted. Autocorrelation is useful to estimate the angular correlation, though it tends to underestimate when the correlation strength is strong, and thus overfitting of the measurements is likely. Analysis on the real data showed that the angular correlation is stronger in the reflectance data than DoLP, which makes sense because we expect that DoLP is less sensitive to systematic uncertainties that are more likely to be correlated. Partial autocorrelation analysis suggests that the uncertainty model considering a linear Markov process (AR(1)) is sufficient for reflectance but may need to be further studied for DoLP. From AirHARP retrieval residual analysis, the correlation angles for reflectance and DoLP are estimated to be larger than 10 and 5

This work intends to provide basic methodology to analyze the measurement uncertainties with angular correlations, but the methods can also be applied in the spatial and spectral domains that may be more appropriate for other instruments. There are several remaining issues that need discussion in future works.

The retrieval is based on a forward model which also has uncertainties, a portion of which may be correlated. This uncertainty will contribute to the fitting residuals and may impact correlation analysis, but it is difficult to quantify.

The fitting residuals are often not stationary with uniform mean and variance. To reduce this issue, the residuals are normalized, but it would be valuable to analyze how the mean value and variance depend on the angle, as this may provide insight into the modeling uncertainties.

Some residuals are not continuous with angle due to removed cirrus clouds, which may reduce the correlation.

Synthetic data analysis has demonstrated that the retrieval is likely to overfit the data when the correlation is strong.

The angular grids for HARP measurements are slightly non-uniform, which is likely to further reduce the correlation strength from autocorrelation analysis. To evaluate impacts of this feature, an uncertainty model considering the impact of the real angular grids needs to be built. But the variation of the angular grids is less than 1

A total of 15 parameters are used as input of the forward model as discussed in Sect.

Parameters used to train the FastMAPOL forward model. The minimum (min) and maximum (max) values of each parameter are also shown. The a priori uncertainties (

Section

The uncertainties for reflectance and DoLP of the error covariance matrix at the 670 nm band after eigenvector decomposition as discussed in Sect.

The uncertainties in the new measurement space (

The Shannon information content (SIC) and corresponding theoretical retrieval uncertainties for refractive index (

The AirHARP data used in this study are available from the ACEPOL data portal (

MG, KK, BAF, PWZ, and BC formulated the study concept. MG generated the scientific data and wrote the original manuscript. PWZ developed the radiative transfer code used to train the NN models. XX and JVM provided and advised on the HARP data. All authors provided critical feedback and edited the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank the ACEPOL teams for conducting the field campaign. The numerical studies are conducted on the Poseidon supercomputer cluster at the NASA Ocean Biology Processing Group (OBPG). We thank the OBPG system team for supporting the high-performance computing. We thank Yunwei Cui, Zhonghuan Chen, Amir Ibrahim, Can Li, Xu Liu, Andy Sayer, and Jason Xuan for constructive discussions.

Meng Gao, Kirk Knobelspiesse, Bryan A. Franz, and Brian Cairns are supported by the NASA PACE project. Peng-Wang Zhai is supported by NASA (grant no. 80NSSC20M0227). The ACEPOL campaign has been supported by the NASA Radiation Sciences Program, with funding from NASA (ACE and CALIPSO missions) and SRON (NWO/NSO project no. ALWGO/16-09).

This paper was edited by Jian Xu and reviewed by two anonymous referees.