Satellite remote sensing is the primary source of global aerosol observations, providing essential data for understanding aerosol-climate interactions and constraining global climate models. To solve the inverse problem at the heart of the retrieval process, traditional algorithms must make simplifications and often cannot quantify uncertainty. In this study, we explore the use of invertible neural networks (INNs) for retrieving aerosol optical depth (AOD) from spectral top-of-atmosphere (TOA) reflectance. INNs can handle the inherent uncertainty of underdetermined inverse problems. They model the forward and inverse processes simultaneously while learning additional random latent variables used to recover full nonparametric posterior distributions for the inverse predictions. We develop location-specific INNs for MODIS sensor data, training on synthetic datasets generated by combining atmospheric reflectance from MODIS dark target (DT) lookup tables (LUTs) and surface reflectance from a MODIS bidirectional reflectance product. The INNs successfully emulate the forward problem and achieve accurate AOD inversion results on synthetic test sets (RMSE$\approx ~0.05$). The posterior distributions obtained are reliable (mean absolute calibration error (MACE)$\approx ~2.5$%), efficiently providing informative predictive uncertainty estimates. In addition, the INNs’ invertible architecture is found to promote physically consistent predictions and uncertainties. To further validate them in a real-world context, the INNs are applied to MODIS L1B reflectance observations to produce full-resolution AOD estimates with pixel-level uncertainties. The retrievals are compared to collocated ground measurements from the Aeronet network. The INNs obtain good accuracy in all tested locations in line with the operational DT AOD product (RMSE$\approx ~0.1$, 74% within DT expected error (EE) bounds). The INNs are also able to retrieve AOD over bright surfaces where DT cannot be applied. Despite uncovered limitations out-of-distribution, the INNs show consistent skill in target domains across diverse land surfaces. The INNs’ unique modeling and uncertainty quantification features have the potential to enhance aerosol and climate studies in various real-world contexts.

Statistical regression methods are widely used in remote sensing applications but tend to lack physical interpretability. In this paper, we introduce a methodological framework to improve model emulation and its understanding with machine learning feature selection. Our wrapper-forward feature selection method seamlessly integrates physics knowledge into model emulation, improving the trade-off between accuracy and interpretability. We illustrate our methodology by applying it to atmospheric radiative transfer models in the context of global sensitivity analysis (GSA) and emulation. Our approach consistently aligns with variance-based GSA, pinpointing the critical features of aerosol properties, solar zenith angle, and water vapor. While our physically-based emulators yield only a modest accuracy improvement of 0.2% over conventional Gaussian Processes emulators, its introduction signifies a step forward to physics-aware machine learning-based emulation. The emulator performance remains steadfast, unaffected by substantial changes, further underscoring the reliability of our approach.

Atmospheric radiative transfer models (RTMs) are widely used in satellite data processing to correct for the scattering and absorption effects caused by aerosols and gas molecules in the Earth’s atmosphere. As the complexity of RTMs grows and the requirements for future Earth Observation missions become more demanding, the conventional lookup-table (LUT) interpolation approach faces important challenges. Emulators have been suggested as an alternative to LUT interpolation, but they are still too slow for operational satellite data processing. Our research introduces a solution that harnesses the power of multifidelity methods to improve the accuracy and runtime of Gaussian process (GP) emulators. We investigate the impact of the number of fidelity layers, dimensionality reduction, and training dataset size on the performance of multifidelity GP emulators. We find that an optimal multifidelity emulator can achieve relative errors in surface reflectance below 0.5% and performs atmospheric correction of hyperspectral PRISMA satellite data (one million pixels) in a few minutes. Additionally, we provide a suite of functions and tools for automating the creation and generation of atmospheric RTM emulators.

Atmospheric radiative transfer models (RTMs) simulate the light propagation in the Earth's atmosphere. With the evolution of RTMs, their increase in complexity makes them impractical in routine processing such as atmospheric correction. To overcome their computational burden, standard practice is to interpolate a multidimensional lookup table (LUT) of prestored simulations. However, accurate interpolation relies on large LUTs, which still implies large computation times for their generation and interpolation. In recent years, emulation has been proposed as an alternative to LUT interpolation. Emulation approximates the RTM outputs by a statistical regression model trained with a low number of RTM runs. However, a concern is whether the emulator reaches sufficient accuracy for atmospheric correction. Therefore, we have performed a systematic assessment of key aspects that impact the precision of emulating MODTRAN: 1) regression algorithm; 2) training database size; 3) dimensionality reduction (DR) method and a number of components; and 4) spectral resolution. The Gaussian processes regression (GPR) was found the most accurate emulator. The principal component analysis remains a robust DR method and nearly 20 components reach sufficient precision. Based on a database of 1000 samples covering a broad range of atmospheric conditions, GPR emulators can reconstruct the simulated spectral data with relative errors below 1% for the 95th percentile. These emulators reduce the processing time from days to minutes, preserving sufficient accuracy for atmospheric correction and providing model uncertainties and derivatives. We provide a set of guidelines and tools to design and generate accurate emulators for satellite data processing applications.

Many fields of science and engineering require the use of complex and computationally expensive models to understand the involved processes in the system of interest. Nevertheless, due to the high cost involved, the required study becomes a cumbersome process. This paper introduces an interpolation procedure which belongs to the family of active learning algorithms, in order to construct cheap surrogate models of such costly complex systems. The proposed technique is sequential and adaptive, and is based on the optimization of a suitable acquisition function. We illustrate its efficiency in a toy example and for the construction of an emulator of an atmosphere modeling system.

Physically based radiative transfer models (RTMs) are widely used in Earth observation to understand the radiation processes occurring on the Earth's surface and their interactions with water, vegetation, and atmosphere. Through continuous improvements, RTMs have increased in accuracy and representativity of complex scenes at expenses of an increase in complexity and computation time, making them impractical in various remote sensing applications. To overcome this limitation, the common practice is to precompute large lookup tables (LUTs) for their later interpolation. To further reduce the RTM computation burden and the error in LUT interpolation, we have developed a method to automatically select the minimum and optimal set of input-output points (nodes) to be included in an LUT. We present the gradient-based automatic LUT generator algorithm (GALGA), which relies on the notion of an acquisition function that incorporates: 1) the Jacobian evaluation of an RTM and 2) the information about the multivariate distribution of the current nodes. We illustrate the capabilities of GALGA in the automatic construction and optimization of MODTRAN-based LUTs of different dimensions of the input variables space. Our results indicate that when compared with a pseudorandom homogeneous distribution of the LUT nodes, GALGA reduces:1) the LUT size by >24%; 2) the computation time by 27%; and 3) the maximum interpolation relative errors by at least 10%. It is concluded that an automatic LUT design might benefit from the methodology proposed in GALGA to reduce interpolation errors and computation time in computationally expensive RTMs.

Accurately disentangling the tiny Solar–Induced Chlorophyll Fluorescence (SIF) from canopy reflected solar irradiance by using passive remote sensing techniques is always challenging. Regardless the scale at which SIF is measured, i.e., proximal sensing, airborne or satellite level; instrumental and atmospheric effects must be accounted for and compensated as part of the SIF retrieval strategy. Regarding the instrumental effects, the use of very high spectral resolution spectrometers makes mandatory an accurate characterization of the Instrument Spectral Response Function (ISRF); and – in the case of imager spectrometers – an accurate characterization of the full instrument response in the spectral and in the spatial domain. Otherwise, spectral distortions derived by an inaccurate instrument's response characterization would rapidly derive into errors on SIF estimations. With regards to the atmospheric effects, when exploiting the oxygen absorption features to estimate SIF, aerosol characterization generally becomes the bottleneck of the atmospheric correction strategy due to its strong impact on these spectral regions. In this work, instrumental and atmospheric effects are analyzed twofold: (1) focusing on their detection by inspecting the at–sensor radiance and surface apparent reflectance, and (2) proposing an atmospheric and instrument compensation strategy to mitigate these effects.

This paper introduces a methodology to construct emulators of costly radiative transfer models (RTMs). The proposed methodology is sequential and adaptive, and it is based on the notion of acquisition functions in Bayesian optimization. Here, instead of optimizing the unknown underlying RTM function, one aims to achieve accurate approximations. The Automatic Multi-Output Gaussian Process Emulator (AMO-GAPE) methodology combines the interpolation capabilities of Gaussian processes (GPs) with the accurate design of an acquisition function that favors sampling in low density regions and flatness of the interpolation function. We illustrate the promising capabilities of the method for the construction of an emulator for a standard leaf-canopy RTM.

An End-to-end simulator (E2ES) is a tool to evaluate the performance of a satellite mission. Once a mission is approved for operation, E2ES evolves during Phase C/D to become a supporting tool for the development and validation of the ground data processor, as well as for simulating data sets to test the Prototype and Operational Processors. FLEX-E is the E2ES of the FLEX/Sentinel-3 tandem mission, which was selected in 2015 as ESA's eighth Earth Explorer. The FLEX-E evolution implies the consolidation of all the retrieval algorithms (e.g. fluorescence, reflectance, biophysical variables), the implementation of new scientific developments, as well the improvement of the co-registration process, the atmospheric correction, and the retrieval of the Level-2 products. The high-level modular design of the improved Level-2 retrieval module will permit a detailed analysis of the errors propagating through the processing chain.

The collection of field data (biophysical variables with associated spectral data) are an essential part of the development and validation of imaging spectroscopy vegetation products. Yet, their quality can only be assessed in the subsequent analysis, and often it appears that there is a wish for extra data to fill up gaps. In an attempt to generate such additional data, we propose to exploit emulation, i.e. variables-based reconstruction of spectral data through statistical learning. We evaluated emulation against classical interpolation techniques using an experimental field dataset with associated airborne hyperspectral HyMap reflectance spectra to produce HyMap-like spectra for any combination of input variables. Results indicate that: (1) emulation produces reflectance spectra more accurately than interpolation when validating against a split part of the field dataset (8% vs 12% errors), (2) emulation produces spectral data multiple times (tens to hundreds) faster than interpolation, and (3) emulation enables meaningful extrapolation outputs. Consequently, this technique opens various new analysis opportunities, e.g., emulators not only allow to produce large experimental-like datasets in a fraction of a second, but they also can be implemented into computationally intensive processing routines to speed up processing, such as global sensitivity analysis or inversion schemes.

ESA's FLEX/Sentinel-3 tandem mission aims at mapping Sun-induced fluorescence (SIF) as a proxy to quantify photosynthetic activity of terrestrial vegetation. Due to the complexity of the mission concept and stringent requirements for the data processing algorithms, ESA developed a Phase A/B1 End-to-End Mission Performance Simulator (E2ES) tool to reproduce the expected mission performance and check the mission and instrument concepts. In the current Phase C/D, the E2ES concept must evolve to consolidate the whole data processing chain, providing an accurate figures of the whole mission error budget and serving as a roadmap for the future development of FLEX Ground Segment. This paper gives an overview of the activities involving the FLEX E2ES, from the conceptual development phase (Phase A/B1) to the implementation of the Ground Segment and operations (Phase C/D). Particularly, we will focus on the high-level description of the Level-2 data processing chain from an architecture and software design point of view.

End-to-end mission performance simulators (E2ES) are suitable tools to accelerate satellite mission development from concet to deployment. One core element of these E2ES is the generation of synthetic scenes that are observed by the various instruments of an Earth Observation mission. The generation of these scenes rely on Radiative Transfer Models (RTM) for the simulation of light interaction with the Earth surface and atmosphere. However, the execution of advanced RTMs is impractical due to their large computation burden. Classical interpolation and statistical emulation methods of pre-computed Look-Up Tables (LUT) are therefore common practice to generate synthetic scenes in a reasonable time. This work evaluates the accuracy and computation cost of interpolation and emulation methods to sample the input LUT variable space. The results on MONDTRAN-based top-of-atmosphere radiance data show that Gaussian Process emulators produced more accurate output spectra than linear interpolation at a fraction of its time. It is concluded that emulation can function as a fast and more accurate alternative to interpolation for LUT parameter space sampling.

During the design phase of a satellite mission, end-to-end mission performance simulator (E2ES) tools allow scientists and engineers evaluating the mission concept, consolidating system technical requirements and analyzing the suitability of the implemented technical solutions and data processing algorithms. The generation of synthetic scenes is one of the core parts of an E2ES, providing scenes (ground truth) as would be observed by satellite instruments and used as reference against simulated retrieved mission products. An appropriate generation of the scene also allows assessing the performance of the ground data processing chain replacing real instrument data before the mission is in orbit, for which the fidelity of the scene generation is critical. This paper describes the design of a generic scene generator (GSG) with capabilities to generate complex 3-D synthetic scenes that combine the effects of surface, heterogeneity, topography and atmosphere, and viewing/illumination geometry. The proposed design allows generating consistent high spatial and spectral resolution top-of-atmosphere radiance scenes for multiple instruments based on the use of thematic maps, radiative transfer models, and reflectance databases. The described GSG was implemented within the FLuorescence EXplorer (FLEX) E2ES software tool and showed its capabilities to generate compatible scenes for the fluorescence imaging spectrometer, ocean and land color instrument, and sea and land surface temperature radiometer instruments of ESA’s FLEX/Sentinel-3 tandem mission and to validate the fulfillment of the FLEX mission requirements.

This paper introduces an automatic methodology to construct emulators for costly radiative transfer models (RTMs). The proposed method is sequential and adaptive, and it is based on the notion of the acquisition function by which instead of optimizing the unknown RTM underlying function we propose to achieve accurate approximations. The Automatic Gaussian Process Emulator (AGAPE) methodology combines the interpolation capabilities of Gaussian processes (GPs) with the accurate design of an acquisition function that favors sampling in low density regions and flatness of the interpolation function. We illustrate the good capabilities of the method in toy examples and for the construction of an optimal look-up-table for atmospheric correction based on MODTRAN5.

Since oxygen (O2) absorption of light becomes more pronounced at higher pressure levels, even a few meters distance between the target and the sensor can strongly affect canopy-leaving Solar-Induced chlorophyll Fluorescence (SIF) retrievals. This study was conducted to quantify the consequent error propagation and the impact of ignoring oxygen absorption effects on proximal sensing SIF measurements based on the O2-A absorption band with field-acquired and simulated data. It was demonstrated that the uncorrected oxygen transmittance between target and sensor distance of 10 m can lead to SIF relative errors ranging from 66% to higher than 100% when using a Spectral Fitting (SF) technique or the 3FLD retrieval method, respectively. A proposed strategy to include oxygen transmittance effects on the well-known 3FLD and SF techniques is presented here and applied to the NASA-GSFC multi-angular spectral system known as FUSION over a field of corn plants (Zea mays L.) during the second half of the 2014 growing season. Daily averages of oxygen-corrected SIF measurements from FUSION were related to daily averages of heat and energy fluxes obtained from a nearby Eddy-Covariance (EC) flux tower, and showed a consistent behaviour with similar experiments performed at leaf level.

The FLuorescence EXplorer (FLEX) mission, selected as the European Space Agency's eighth Earth Explorer, aims to globally measure the sun-induced-chlorophyll-fluorescence spectral emission from terrestrial vegetation. In the frame of the FLEX mission, several industrial and scientific studies have analyzed the instrument design, image processing algorithms, or modeling aspects. At the same time, a common tool is needed to address the overall FLEX mission performance by combining all these features. For this reason, an end-to-end mission performance simulator has been developed for the FLEX mission (FLEX-E). This paper describes the FLEX-E software design, which combines the generation of complex synthetic scenes with an advanced modeling of the instrument behavior and the full processing scheme up to the final fluorescence product. The results derived from FLEX-E simulations indicate that the instrument and developed image processing algorithms are able to retrieve the sun-induced fluorescence with an accuracy below the 0.2$\text{mW}\cdot\text{m}^{-2}\cdot\text{sr}^{-1}\cdot \text{nm}^{-1}$mission requirement. It is expected that FLEX-E will not only optimize the FLEX retrieval algorithms and technical requirements, but also serve as the baseline for the ground processing implementation and testing of calibration/validation procedures.

A new fluorescence retrieval method is proposed to support ESA's 8th Earth Explorer FLuorescence EXplorer/Sentinel-3 (FLEX-S3) candidate tandem mission. FLEX is the first mission specially dedicated to measure the Sun-Induced vegetation chlorophyll fluorescence (SIF) strongly related with the vegetation photosynthetic activity. Most hyperspectral fluorescence retrieval algorithms available in the literature are very sensitive to true reflectance modelization and/or they assume the atmospheric status as known. The proposed algorithm delivers the retrieval of full fluorescence spectrum at canopy level by using only Top Of Atmosphere (TOA) radiances from S3 and FLEX as input. Once the spatial co-registration and cross-calibration of S3 and FLEX images have been performed, the proposed method starts with (1) the atmospheric correction of TOA radiances, characterizing the state of the atmosphere, (2) performing a first estimation of fluorescence values in main oxygen absorption bands without any approximation of true reflectance spectrum, and using this fluorescence estimation to initialize a Spectral Fitting Method (SFM) to finally retrieving a full fluorescence spectrum. This proposed fluorescence retrieval method is currently being implemented at the Level-2 Retrieval Module (L2RM) of the FLEX/End-To-End Simulator (E2ES).

The Hyperspectral Imager for the Coastal Ocean (HICO) is an imaging spectrometer designed with a very high signal-to-noise ratio to monitor coastal ocean and inland waters. The processing of Top-Of-Atmosphere radiance data down to surface reflectance is fundamental for the retrieval of water quality products. However, the current HICO processing chain does not provide atmospheric corrected data nor higher-level water quality products. This paper describes the algorithms implemented within an HICO data processing chain that includes image pre-processing, atmospheric correction and the retrieval of water quality parameters. The implemented algorithms have been validated over a set of HICO images showing a good match with in-situ surface reflectance data and correlation (R2= 0.95) between in-situ measured and retrieved Chl-a over a water body. It is expected that the presented algorithms will ease the processing of HICO data down to surface reflectance allowing to derive water quality parameters.