Deep learning and process understanding for data-driven Earth system science.

Machine learning approaches are increasingly used to extract patterns and insights from the ever-increasing stream of geospatial data, but current approaches may not be optimal when system behaviour is dominated by spatial or temporal context. Here, rather than amending classical machine learning, we argue that these contextual cues should be used as part of deep learning (an approach that is able to extract spatio-temporal features automatically) to gain further process understanding of Earth system science problems, improving the predictive ability of seasonal forecasting and modelling of long-range spatial connections across multiple timescales, for example. The next step will be a hybrid modelling approach, coupling physical process models with the versatility of data-driven machine learning.

Compensatory water effects link yearly global land CO2 sink changes to temperature.

Large interannual variations in the measured growth rate of atmospheric carbon dioxide (CO2) originate primarily from fluctuations in carbon uptake by land ecosystems. It remains uncertain, however, to what extent temperature and water availability control the carbon balance of land ecosystems across spatial and temporal scales. Here we use empirical models based on eddy covariance data and process-based models to investigate the effect of changes in temperature and water availability on gross primary productivity (GPP), terrestrial ecosystem respiration (TER) and net ecosystem exchange (NEE) at local and global scales. We find that water availability is the dominant driver of the local interannual variability in GPP and TER. To a lesser extent this is true also for NEE at the local scale, but when integrated globally, temporal NEE variability is mostly driven by temperature fluctuations. We suggest that this apparent paradox can be explained by two compensatory water effects. Temporal water-driven GPP and TER variations compensate locally, dampening water-driven NEE variability. Spatial water availability anomalies also compensate, leaving a dominant temperature signal in the year-to-year fluctuations of the land carbon sink. These findings help to reconcile seemingly contradictory reports regarding the importance of temperature and water in controlling the interannual variability of the terrestrial carbon balance. Our study indicates that spatial climate covariation drives the global carbon cycle response.

A hybrid generative adversarial network for weakly-supervised cloud detection in multispectral images.

Cloud detection is a crucial step in the optical satellite image processing pipeline for Earth observation. Clouds in optical remote sensing images seriously affect the visibility of the background and greatly reduce the usability of images for land applications. Traditional methods based on thresholding, multi-temporal or multi-spectral information are often specific to a particular satellite sensor. Convolutional Neural Networks for cloud detection often require labeled cloud masks for training that are very time-consuming and expensive to obtain. To overcome these challenges, this paper presents a hybrid cloud detection method based on the synergistic combination of generative adversarial networks (GAN) and a physics-based cloud distortion model (CDM). The proposed weakly-supervised GAN-CDM method (available online https://github.com/Neooolee/GANCDM) only requires patch-level labels for training, and can produce cloud masks at pixel-level in both training and testing stages. GAN-CDM is trained on a new globally distributed Landsat 8 dataset (WHUL8-CDb, available online doi:https://doi.org/10.5281/zenodo.6420027) including image blocks and corresponding block-level labels. Experimental results show that the proposed GAN-CDM method trained on Landsat 8 image blocks achieves much higher cloud detection accuracy than baseline deep learning-based methods, not only in Landsat 8 images (L8 Biome dataset, 90.20% versus 72.09%) but also in Sentinel-2 images ("S2 Cloud Mask Catalogue" dataset, 92.54% versus 77.00%). This suggests that the proposed method provides accurate cloud detection in Landsat images, has good transferability to Sentinel-2 images, and can quickly be adapted for different optical satellite sensors.

Systematic Assessment of MODTRAN Emulators for Atmospheric Correction.

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.

Partitioning net carbon dioxide fluxes into photosynthesis and respiration using neural networks.

The eddy covariance (EC) technique is used to measure the net ecosystem exchange (NEE) of CO2 between ecosystems and the atmosphere, offering a unique opportunity to study ecosystem responses to climate change. NEE is the difference between the total CO2 release due to all respiration processes (RECO), and the gross carbon uptake by photosynthesis (GPP). These two gross CO2 fluxes are derived from EC measurements by applying partitioning methods that rely on physiologically based functional relationships with a limited number of environmental drivers. However, the partitioning methods applied in the global FLUXNET network of EC observations do not account for the multiple co-acting factors that modulate GPP and RECO flux dynamics. To overcome this limitation, we developed a hybrid data-driven approach based on combined neural networks (NNC-part ). NNC-part incorporates process knowledge by introducing a photosynthetic response based on the light-use efficiency (LUE) concept, and uses a comprehensive dataset of soil and micrometeorological variables as fluxes drivers. We applied the method to 36 sites from the FLUXNET2015 dataset and found a high consistency in the results with those derived from other standard partitioning methods for both GPP (R2  > .94) and RECO (R2  > .8). High consistency was also found for (a) the diurnal and seasonal patterns of fluxes and (b) the ecosystem functional responses. NNC-part performed more realistic than the traditional methods for predicting additional patterns of gross CO2 fluxes, such as: (a) the GPP response to VPD, (b) direct effects of air temperature on GPP dynamics, (c) hysteresis in the diel cycle of gross CO2 fluxes, (d) the sensitivity of LUE to the diffuse to direct radiation ratio, and (e) the post rain respiration pulse after a long dry period. In conclusion, NNC-part is a valid data-driven approach to provide GPP and RECO estimates and complementary to the existing partitioning methods.

Multispectral high resolution sensor fusion for smoothing and gap-filling in the cloud.

Remote sensing optical sensors onboard operational satellites cannot have high spectral, spatial and temporal resolutions simultaneously. In addition, clouds and aerosols can adversely affect the signal contaminating the land surface observations. We present a HIghly Scalable Temporal Adaptive Reflectance Fusion Model (HISTARFM) algorithm to combine multispectral images of different sensors to reduce noise and produce monthly gap free high resolution (30 m) observations over land. Our approach uses images from the Landsat (30 m spatial resolution and 16 day revisit cycle) and the MODIS missions, both from Terra and Aqua platforms (500 m spatial resolution and daily revisit cycle). We implement a bias-aware Kalman filter method in the Google Earth Engine (GEE) platform to obtain fused images at the Landsat spatial-resolution. The added bias correction in the Kalman filter estimates accounts for the fact that both model and observation errors are temporally auto-correlated and may have a non-zero mean. This approach also enables reliable estimation of the uncertainty associated with the final reflectance estimates, allowing for error propagation analyses in higher level remote sensing products. Quantitative and qualitative evaluations of the generated products through comparison with other state-of-the-art methods confirm the validity of the approach, and open the door to operational applications at enhanced spatio-temporal resolutions at broad continental scales.

Deep Gaussian processes for biogeophysical parameter retrieval and model inversion.

Parameter retrieval and model inversion are key problems in remote sensing and Earth observation. Currently, different approximations exist: a direct, yet costly, inversion of radiative transfer models (RTMs); the statistical inversion with in situ data that often results in problems with extrapolation outside the study area; and the most widely adopted hybrid modeling by which statistical models, mostly nonlinear and non-parametric machine learning algorithms, are applied to invert RTM simulations. We will focus on the latter. Among the different existing algorithms, in the last decade kernel based methods, and Gaussian Processes (GPs) in particular, have provided useful and informative solutions to such RTM inversion problems. This is in large part due to the confidence intervals they provide, and their predictive accuracy. However, RTMs are very complex, highly nonlinear, and typically hierarchical models, so that very often a single (shallow) GP model cannot capture complex feature relations for inversion. This motivates the use of deeper hierarchical architectures, while still preserving the desirable properties of GPs. This paper introduces the use of deep Gaussian Processes (DGPs) for bio-geo-physical model inversion. Unlike shallow GP models, DGPs account for complicated (modular, hierarchical) processes, provide an efficient solution that scales well to big datasets, and improve prediction accuracy over their single layer counterpart. In the experimental section, we provide empirical evidence of performance for the estimation of surface temperature and dew point temperature from infrared sounding data, as well as for the prediction of chlorophyll content, inorganic suspended matter, and coloured dissolved matter from multispectral data acquired by the Sentinel-3 OLCI sensor. The presented methodology allows for more expressive forms of GPs in big remote sensing model inversion problems.

Gaussian processes retrieval of LAI from Sentinel-2 top-of-atmosphere radiance data.

Retrieval of vegetation properties from satellite and airborne optical data usually takes place after atmospheric correction, yet it is also possible to develop retrieval algorithms directly from top-of-atmosphere (TOA) radiance data. One of the key vegetation variables that can be retrieved from at-sensor TOA radiance data is leaf area index (LAI) if algorithms account for variability in atmosphere. We demonstrate the feasibility of LAI retrieval from Sentinel-2 (S2) TOA radiance data (L1C product) in a hybrid machine learning framework. To achieve this, the coupled leaf-canopy-atmosphere radiative transfer models PROSAIL-6SV were used to simulate a look-up table (LUT) of TOA radiance data and associated input variables. This LUT was then used to train the Bayesian machine learning algorithms Gaussian processes regression (GPR) and variational heteroscedastic GPR (VHGPR). PROSAIL simulations were also used to train GPR and VHGPR models for LAI retrieval from S2 images at bottom-of-atmosphere (BOA) level (L2A product) for comparison purposes. The BOA and TOA LAI products were consistently validated against a field dataset with GPR (R2 of 0.78) and with VHGPR (R 2 of 0.80) and for both cases a slightly lower RMSE for the TOA LAI product (about 10% reduction). Because of delivering superior accuracies and lower uncertainties, the VHGPR models were further applied for LAI mapping using S2 acquisitions over the agricultural sites Marchfeld (Austria) and Barrax (Spain). The models led to consistent LAI maps at BOA and TOA scale. The LAI maps were also compared against LAI maps as generated by the SNAP toolbox, which is based on a neural network (NN). Maps were again consistent, however the SNAP NN model tends to overestimate over dense vegetation cover. Overall, this study demonstrated that hybrid LAI retrieval algorithms can be developed from TOA radiance data given a cloud-free sky, thus without the need of atmospheric correction. To the benefit of the community, the development of such hybrid models for the retrieval vegetation properties from BOA or TOA images has been streamlined in the freely downloadable ALG-ARTMO software framework.

Retrieval of aboveground crop nitrogen content with a hybrid machine learning method.

Hyperspectral acquisitions have proven to be the most informative Earth observation data source for the estimation of nitrogen (N) content, which is the main limiting nutrient for plant growth and thus agricultural production. In the past, empirical algorithms have been widely employed to retrieve information on this biochemical plant component from canopy reflectance. However, these approaches do not seek for a cause-effect relationship based on physical laws. Moreover, most studies solely relied on the correlation of chlorophyll content with nitrogen, and thus neglected the fact that most N is bound in proteins. Our study presents a hybrid retrieval method using a physically-based approach combined with machine learning regression to estimate crop N content. Within the workflow, the leaf optical properties model PROSPECT-PRO including the newly calibrated specific absorption coefficients (SAC) of proteins, was coupled with the canopy reflectance model 4SAIL to PROSAIL-PRO. The latter was then employed to generate a training database to be used for advanced probabilistic machine learning methods: a standard homoscedastic Gaussian process (GP) and a heteroscedastic GP regression that accounts for signal-to-noise relations. Both GP models have the property of providing confidence intervals for the estimates, which sets them apart from other machine learners. Moreover, a GP-based sequential backward band removal algorithm was employed to analyze the band-specific information content of PROSAIL-PRO simulated spectra for the estimation of aboveground N. Data from multiple hyperspectral field campaigns, carried out in the framework of the future satellite mission Environmental Mapping and Analysis Program (EnMAP), were exploited for validation. In these campaigns, corn and winter wheat spectra were acquired to simulate spectral EnMAP data. Moreover, destructive N measurements from leaves, stalks and fruits were collected separately to enable plant-organ-specific validation. The results showed that both GP models can provide accurate aboveground N simulations, with slightly better results of the heteroscedastic GP in terms of model testing and against in situ N measurements from leaves plus stalks, with root mean square error (RMSE) of 2.1 g/m2. However, the inclusion of fruit N content for validation deteriorated the results, which can be explained by the inability of the radiation to penetrate the thick tissues of stalks, corn cobs and wheat ears. GP-based band analysis identified optimal spectral settings with ten bands mainly situated in the shortwave infrared (SWIR) spectral region. Use of well-known protein absorption bands from the literature showed comparative results. Finally, the heteroscedastic GP model was successfully applied on airborne hyperspectral data for N mapping. We conclude that GP algorithms, and in particular the heteroscedastic GP, should be implemented for global agricultural monitoring of aboveground N from future imaging spectroscopy data.

Fusing optical and SAR time series for LAI gap fillingwith multioutput Gaussian processes.

The availability of satellite optical information is often hampered by the natural presence of clouds, which can be problematic for many applications. Persistent clouds over agricultural fields can mask key stages of crop growth, leading to unreliable yield predictions. Synthetic Aperture Radar (SAR) provides all-weather imagery which can potentially overcome this limitation, but given its high and distinct sensitivity to different surface properties, the fusion of SAR and optical data still remains an open challenge. In this work, we propose the use of Multi-Output Gaussian Process (MOGP) regression, a machine learning technique that learns automatically the statistical relationships among multisensor time series, to detect vegetated areas over which the synergy between SAR-optical imageries is profitable. For this purpose, we use the Sentinel-1 Radar Vegetation Index (RVI) and Sentinel-2 Leaf Area Index (LAI) time series over a study area in north west of the Iberian peninsula. Through a physical interpretation of MOGP trained models, we show its ability to provide estimations of LAI even over cloudy periods using the information shared with RVI, which guarantees the solution keeps always tied to real measurements. Results demonstrate the advantage of MOGP especially for long data gaps, where optical-based methods notoriously fail. The leave-one-image-out assessment technique applied to the whole vegetation cover shows MOGP predictions improve standard GP estimations over short-time gaps (R2 of 74% vs 68%, RMSE of 0.4 vs 0.44 [m 2 m -2]) and especially over long-time gaps (R2 of 33% vs 12%, RMSE of 0.5 vs 1.09 [m 2 m -2]). A second assessment is focused on crop-specific regions, clustering pixels fulfilling specific model conditions where the synergy is profitable. Results reveal the MOGP performance is crop type and crop stage dependent. For long time gaps, best R2 are obtained over maize, ranging from 0.1 (tillering) to 0.36 (development) up to 0.81 (maturity); for moderate time gap, R2 = 0.93 (maturity) is obtained. Crops such as wheat, oats, rye and barley, can profit from the LAI-RVI synergy, with R2 varying between 0.4 and 0.6. For beet or potatoes, MOGP provides poorer results, but alternative descriptors to RVI should be tested for these specific crops in the future before discarding synergy real benefits. In conclusion, active-passive sensor fusion with MOGP represents a novel and promising approach to cope with crop monitoring over cloud-dominated areas.

Synergistic integration of optical and microwave satellite data for crop yield estimation.

Developing accurate models of crop stress, phenology and productivity is of paramount importance, given the increasing need of food. Earth observation (EO) remote sensing data provides a unique source of information to monitor crops in a temporally resolved and spatially explicit way. In this study, we propose the combination of multisensor (optical and microwave) remote sensing data for crop yield estimation and forecasting using two novel approaches. We first propose the lag between Enhanced Vegetation Index (EVI) derived from MODIS and Vegetation Optical Depth (VOD) derived from SMAP as a new joint metric combining the information from the two satellite sensors in a unique feature or descriptor. Our second approach avoids summarizing statistics and uses machine learning to combine full time series of EVI and VOD. This study considers two statistical methods, a regularized linear regression and its nonlinear extension called kernel ridge regression to directly estimate the county-level surveyed total production, as well as individual yields of the major crops grown in the region: corn, soybean and wheat. The study area includes the US Corn Belt, and we use agricultural survey data from the National Agricultural Statistics Service (USDA-NASS) for year 2015 for quantitative assessment. Results show that (1) the proposed EVI-VOD lag metric correlates well with crop yield and outperforms common single-sensor metrics for crop yield estimation; (2) the statistical (machine learning) models working directly with the time series largely improve results compared to previously reported estimations; (3) the combined exploitation of information from the optical and microwave data leads to improved predictions over the use of single sensor approaches with coefficient of determination R ≥ 2 0.76 ; (4) when models are used for within-season forecasting with limited time information, crop yield prediction is feasible up to four months before harvest (models reach a plateau in accuracy); and (5) the robustness of the approach is confirmed in a multi-year setting, reaching similar performances than when using single-year data. In conclusion, results confirm the value of using both EVI and VOD at the same time, and the advantage of using automatic machine learning models for crop yield/production estimation.

Gradient-Based Automatic Lookup Table Generator for Radiative Transfer Models.

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.

Soil and vegetation water content identify the main terrestrial ecosystem changes.

Environmental change is a consequence of many interrelated factors. How vegetation responds to natural and human activity still needs to be well established, quantified and understood. Recent satellite missions providing hydrologic and ecological indicators enable better monitoring of Earth system changes, yet there is no automatic way to address this issue directly from observations. Here, we develop an observation-based methodology to capture evidence of changes in global terrestrial ecosystems and attribute these changes to natural or anthropogenic activity. We use the longest time record of global microwave L-band soil moisture and vegetation optical depth as satellite data and build spatially explicit maps of change in soil and vegetation water content and biomass reflecting large ecosystem changes during the last decade, 2010-20. Regions of prominent trends (from [Formula: see text] to 9% per year) are observed, especially in humid and semi-arid climates. We further combine such trends with land cover change maps, vegetation greenness and precipitation variability to assess their relationship with major documented ecosystem changes. Several regions emerge from our results. They cluster changes according to human activity drivers, including deforestation (Amazon, Central Africa) and wildfires (East Australia), artificial reforestation (South-East China), abandonment of farm fields (Central Russia) and climate shifts related to changes in precipitation variability (East Africa, North America and Central Argentina). Using the high sensitivity of soil and vegetation water content to ecosystem changes, microwave satellite observations enable us to quantify and attribute global vegetation responses to climate or anthropogenic activities as a direct measure of environmental changes and the mechanisms driving them.

Exploring interactions between socioeconomic context and natural hazards on human population displacement.

Climate change is leading to more extreme weather hazards, forcing human populations to be displaced. We employ explainable machine learning techniques to model and understand internal displacement flows and patterns from observational data alone. For this purpose, a large, harmonized, global database of disaster-induced movements in the presence of floods, storms, and landslides during 2016-2021 is presented. We account for environmental, societal, and economic factors to predict the number of displaced persons per event in the affected regions. Here we show that displacements can be primarily attributed to the combination of poor household conditions and intense precipitation, as revealed through the interpretation of the trained models using both Shapley values and causality-based methods. We hence provide empirical evidence that differential or uneven vulnerability exists and provide a means for its quantification, which could help advance evidence-based mitigation and adaptation planning efforts.

Inferring causal relations from observational long-term carbon and water fluxes records.

Land, atmosphere and climate interact constantly and at different spatial and temporal scales. In this paper we rely on causal discovery methods to infer spatial patterns of causal relations between several key variables of the carbon and water cycles: gross primary productivity, latent heat energy flux for evaporation, surface air temperature, precipitation, soil moisture and radiation. We introduce a methodology based on the convergent cross-mapping (CCM) technique. Despite its good performance in general, CCM is sensitive to (even moderate) noise levels and hyper-parameter selection. We present a robust CCM (RCCM) that relies on temporal bootstrapping decision scores and the derivation of more stringent cross-map skill scores. The RCCM method is combined with the information-geometric causal inference (IGCI) method to address the problem of strong and instantaneous variable coupling, another important and long-standing issue of CCM. The proposed methodology allows to derive spatially explicit global maps of causal relations between the involved variables and retrieve the underlying complexity of the interactions. Results are generally consistent with reported patterns and process understanding, and constitute a new way to quantify and understand carbon and water fluxes interactions.

Predicting regional coastal sea level changes with machine learning.

All ocean basins have been experiencing significant warming and rising sea levels in recent decades. There are, however, important regional differences, resulting from distinct processes at different timescales (temperature-driven changes being a major contributor on multi-year timescales). In view of this complexity, it deems essential to move towards more sophisticated data-driven techniques as well as diagnostic and prognostic prediction models to interpret observations of ocean warming and sea level variations at local or regional sea basins. In this context, we present a machine learning approach that exploits key ocean temperature estimates (as proxies for the regional thermosteric sea level component) to model coastal sea level variability and associated uncertainty across a range of timescales (from months to several years). Our findings also demonstrate the utility of machine learning to estimate the possible tendency of near-future regional sea levels. When compared to actual sea-level records, our models perform particularly well in the coastal areas most influenced by internal climate variability. Yet, the models are widely applicable to evaluate the patterns of rising and falling sea levels across many places around the globe. Thus, our approach is a promising tool to model and anticipate sea level changes in the coming (1-3) years, which is crucial for near-term decision making and strategic planning about coastal protection measures.

Correction: Kernel methods and their derivatives: Concept and perspectives for the earth system sciences.

[This corrects the article DOI: 10.1371/journal.pone.0235885.].

Emergent vulnerability to climate-driven disturbances in European forests.

Forest disturbance regimes are expected to intensify as Earth's climate changes. Quantifying forest vulnerability to disturbances and understanding the underlying mechanisms is crucial to develop mitigation and adaptation strategies. However, observational evidence is largely missing at regional to continental scales. Here, we quantify the vulnerability of European forests to fires, windthrows and insect outbreaks during the period 1979-2018 by integrating machine learning with disturbance data and satellite products. We show that about 33.4 billion tonnes of forest biomass could be seriously affected by these disturbances, with higher relative losses when exposed to windthrows (40%) and fires (34%) compared to insect outbreaks (26%). The spatial pattern in vulnerability is strongly controlled by the interplay between forest characteristics and background climate. Hotspot regions for vulnerability are located at the borders of the climate envelope, in both southern and northern Europe. There is a clear trend in overall forest vulnerability that is driven by a warming-induced reduction in plant defence mechanisms to insect outbreaks, especially at high latitudes.

A unified vegetation index for quantifying the terrestrial biosphere.

Empirical vegetation indices derived from spectral reflectance data are widely used in remote sensing of the biosphere, as they represent robust proxies for canopy structure, leaf pigment content, and, subsequently, plant photosynthetic potential. Here, we generalize the broad family of commonly used vegetation indices by exploiting all higher-order relations between the spectral channels involved. This results in a higher sensitivity to vegetation biophysical and physiological parameters. The presented nonlinear generalization of the celebrated normalized difference vegetation index (NDVI) consistently improves accuracy in monitoring key parameters, such as leaf area index, gross primary productivity, and sun-induced chlorophyll fluorescence. Results suggest that the statistical approach maximally exploits the spectral information and addresses long-standing problems in satellite Earth Observation of the terrestrial biosphere. The nonlinear NDVI will allow more accurate measures of terrestrial carbon source/sink dynamics and potentials for stabilizing atmospheric CO2 and mitigating global climate change.

Explicit Granger causality in kernel Hilbert spaces.

Granger causality (GC) is undoubtedly the most widely used method to infer cause-effect relations from observational time series. Several nonlinear alternatives to GC have been proposed based on kernel methods. We generalize kernel Granger causality by considering the variables' cross-relations explicitly in Hilbert spaces. The framework is shown to generalize the linear and kernel GC methods and comes with tighter bounds of performance based on Rademacher complexity. We successfully evaluate its performance in standard dynamical systems, as well as to identify the arrow of time in coupled Rössler systems, and it is exploited to disclose the El Niño-Southern Oscillation phenomenon footprints on soil moisture globally.