1 Introduction

A portion of the radiated energy to the canopy is stored by the process of photosynthesis in organic substances; this is known as Gross Primary Productivity (GPP). The other portion is either re-radiated or lost as latent heat [1]. GPP and its derivatives have been the focus of various terrestrial sciences, and it is a common term in interdisciplinary research. In hydrology, there have been attempts at quantifying the available root zone soil moisture sink according to the connection between evapotranspiration (ET) and Leaf Area Index (LAI). Since LAI is a proxy for the anatomized portion of the GPP in ecology, it connects GPP to soil moisture in ecohydrology [2,3,4,5,6,7]. Not all GPP incarnates into the Net Primary Productivity (NPP) since a part of it is consumed for the plant autotrophic respiration (\({\mathrm{R}}_{\mathrm{a}}\)) process through which the plant respires carbon for growth (\({\mathrm{R}}_{\mathrm{G}}\)) and maintenance (\({\mathrm{R}}_{\mathrm{M}}\)) [8]. Each major organ of a plant (leaf, stem, and root) respires carbon for its growth and maintenance. This can be studied in the combined field of ecological and plant phenological sciences to connect the concepts of GPP and NPP [9,10,11]. Subtracting \({\mathrm{R}}_{\mathrm{a}}\) from GPP, the first derivative of GPP will be calculated as NPP which is measured for estimating the rate of biomass accumulation [12]. NPP represents the net C uptake from the atmosphere into vegetation [12] and can be calculated as the sum of biomass increment, mortality, and turnover of foliage and fine roots [13]. The total biomass can be transformed into crop yield in farmlands using the Harvest Index (\({\mathrm{HI}}\)) if the index is multiplied by the accumulated amount of biomass during the crop growing period [14,15,16]. NPP, biomass, and yield are estimated in agricultural studies while they are still connected to ecohydrology and biology through GPP and \({\mathrm{R}}_{\mathrm{a}}\) [17]. Heterotrophic respiration (\({\mathrm{R}}_{\mathrm{h}}\)) is the total flux of carbon to the biosphere caused by the decomposition process (\({\mathrm{R}}_{\mathrm{D}}\)) and by the respiration of non-plant organisms (\({\mathrm{R}}_{\mathrm{O}}\)) [18]. Both \({\mathrm{R}}_{\mathrm{a}}\) and \({\mathrm{R}}_{\mathrm{h}}\) estimations are within the fields of ecology, biogeochemistry, and dynamic biogeography sciences; these can be subtracted from GPP to produce the second derivative of GPP known as net ecosystem exchange (NEE). NEE can be directly monitored by site level measurements through the global eddy covariance (EC) tower network [19,20,21]. GPP can be calculated indirectly by in situ measured NEE values that can be extrapolated to the gridded global NEE values using earth observation data [22]. GPP is calculated indirectly because it is impossible to directly observe daytime GPP or ecosystem respiration, and that is why it is typically inferred from measurements of NEE [23]. Recently, solar-induced chlorophyll fluorescence (SIF) has been shown to be well correlated to GPP, thus offering a path to improve the NEE partitioning by constraining GPP [24].

GPP can also be estimated at global scale using the models that connect soil, vegetation, and atmosphere parameters. These connections are reflected in conceptualizing soil, vegetation, and atmosphere dynamic models for the biosphere that capture the effect of water loss, water stress, nutrition, and age on the stomatal operation and the subsequent amount of LAI. Examples of these models are soil‐vegetation‐atmosphere transfer (SVAT), dynamic global vegetation model (DGVM) and ORganizing Carbon and Hydrology In Dynamic EcosystEm (ORCHIDEE) [25,26,27,28,29,30]. In these models, LAI is one of the determiners for ET, in addition to other predictors, such as precipitation, runoff, infiltration, soil texture, and topography because LAI can be connected to the available moisture in the root zone by ET. Instead of estimating terrestrial GPP using multiparameter models, the estimation process can be simplified using an ecological concept called the efficiency model (PEM) or the light use efficiency (LUE) model [31], Despite the apparent simplicity of the LUE model that estimates GPP by multiplying three terms, the photosynthetically active radiation (PAR), the fraction of absorbed PAR (fPAR), and the LUE factor, however, fPAR and LUE factors are controversial. LUE is often calculated as a function of maximum daily LUE (\({\upvarepsilon }_{\mathrm{max}}\)) regulated by environmental controls (temperature, soil water, vapor pressure deficit, etc.) [32,33,34]. Traditionally, the whole canopy proxies were used to define fPAR and LUE, while recently, the controversy behind the fPAR and LUE definitions refers to the assumption that the absorption of PAR by the canopy’s chlorophyll part is effective in producing GPP rather than the absorption by the whole canopy [35]. Solar-induced fluorescence (SIF) is the proxy of chlorophyll content which can be sensed remotely and has great potential for estimating GPP due to a strong observed relationship between them from both ground-based and satellite observations [36]. From the environmental perspective, the importance of the in situ NEE measurements and subsequent GPP calculation using complicated dynamic or LUE models is to understand the role of carbon in the global climate system. Hence, GPP and its derivatives have been studied to investigate the process of carbon sequestration and release, feedback between climate and vegetation change, heat, momentum, and moisture fluxes as well as global greenhouse gas concentrations [37,38,39,40].

GPP and NPP estimations also exist in regional-scaled studies on ecosystem preservation, reclamation, and production in wetlands, grasslands, and farmlands [41,42,43,44]. From the agricultural perspective, data on the accumulated amount of biomass during the early growing stage until harvest time is important to meet the information demand for food security planners and policymakers [45,46,47]. Spatiotemporal distribution of NPP and biomass on farmlands can help scientists to determine the potential and scope of improvements on agricultural lands for predicting yield values before harvest and for providing early warnings to traders, farmers, and insurance companies [48, 49]. Different approaches for regional estimation of GPP and NPP can be classified into three main categories: mechanistic, statistical, and functional [48]. Mechanistic models are usually known as crop models based on the dominant physical and biophysical principles in a system consisting of crop, soil, and atmosphere. If there is enough detailed information from plant peripherals, litter circumstances, and agricultural managing inputs and in integration with assimilated remote sensing (soil moisture and LAI) data, crop models can provide an acceptable estimation of NPP [50,51,52]. When there is a lack of detailed information, statistic and functional models are used to make assumptions on the relationship among the existing field-measured predictors and NPP or to identify the best combination of predictors that contribute the most variance to NPP. Functional models are indeed simplified versions of the complex mechanistic models in combination with statistical schemes [53,54,55]. After modeling and identification of effective predictors, the monitored values of these predictors are imported into forecasting methods to estimate future crop yield and produce early warnings [48, 56]. In classical forecasting approaches, an attempt is made to differentiate seasonality and trend components of the predictors’ time series, and then by assuming that these discovered trends are constant, the forecast is made for the target parameter, while in newly developed approaches, models are updated each time for new observation [57,58,59]. However, required information about weather, soil properties, and precise land cover data to build a ground-data-dependent functional model are typically not available in developing countries, where reliable yield predictions are most needed [54]. In this case, machine learning ideas like deep learning architectures by which the model automatically learns useful features only from multispectral remote sensing images seem to be the only solution for making crop yield predictions. In an attempt, remotely sensed multispectral images have been used in the deep long short-term memory (LSTM architecture to estimate and forecast crop yield [60].

Inferred from the previous explanations, Fig. 1 shows the contribution of different fields of knowledge in studying GPP and its derivatives and indicates their estimation approaches in environmental and agricultural perspectives. As shown in Fig. 1, ecohydrological investigations with environmental perspective require the consideration of several crucial parameters (such as GPP, ET, LAI, and soil moisture) to the understanding of the complex interactions between ecological and hydrological processes within a biosphere. While in ecology and biochemistry the attempt is to understand respiration processes, they are used to generate NPP and NEE derivatives from GPP. The figure also depicts that the environmental perspective aims to monitor carbon cycle in the biosphere including forest and wetland land use. On the other hand, the agricultural perspective aims to study food security in grassland and farmland land use. While GPP and its first-hand derivatives are the target parameters in environmental studies, the latest derivatives of it like biomass and yield are the specific targets of interest for agricultural studies. In environmental studies, in situ NEE measurements are used for conducting direct estimations or for validating the dynamic models, while the LUE model is used for reducing the complicacy of the estimation process instead of dynamic models. Conversely, within the realm of agricultural studies, the utilization of crop models, statistical models, and newly developed functional models that leverage remote sensing data has facilitated the research process pertaining to agricultural lands. These models prove particularly valuable in cases where the measured biomass or yield serves as the validation indicator. In this paper, after a review on the use of global models from the environmental perspective in the "Introduction" section, the rest of the paper focus on the application of the LUE and functional models from the agricultural perspective. Figure 2 shows the relationship between the main terms reviewed and their proximity to environmental and agricultural perspectives, and Table 1 describes the contents clarified in each paragraph or subsection and their interconnections. Here, with a special emphasis on the most recent improvements in terrestrial GPP estimations using chlorophyll-based fPAR and LUE concepts, the importance of future studies on the application of chlorophyll-based predictors in functional models became apparent. From the agricultural perspective, an argue on the regional application of chlorophyll-based fPAR on the LUE model was raised and recommended to be tested in future studies using downscaled version of remote sensing chlorophyll products. At the end, a new chlorophyll-based remote sensing predictor was introduced for being tested in functional models in estimating NPP.

Fig. 1
figure 1

Contribution of different fields of knowledge in studying GPP and its derivatives outlined into the two environmental and agricultural perspectives

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Fig. 2
figure 2

The relationship between the main reviewed terms and their proximity to environmental and agricultural perspectives

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Table 1 Contents clarified in each paragraph or subsection and their interconnections
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2 Crop Models

Crop models are categorized into mechanistic model classes that can be used to model the crop growth duration and NPP for agricultural purposes [61, 62]. A crop model simulates crop growth based on information about the crop litter and surrounding atmosphere,furthermore, it can be used to build a comprehensive dynamic biosphere model if it is integrated with a climate change model. A crop model also can be embedded in a hydrological model to explain the interactions between the crop and the hydrological circumstances around it [63,64,65]. Many crop models have been developed for each major agricultural crop such as wheat, maize, and rice in different parts of the world [66,67,68,69]. Equations that include explanatory variables and parameters in relation to \({CO}_{2}\), solar radiation, temperature, humidity, wind speed, precipitation, available water, and nutrients provide crop models to determine the required state variables [70]. In some of these models, a linear or a nonlinear temperature response function with an upper limit for the developmental rate is considered to simulate crop growth, while other models consider a function with an optimum temperature above which the growth rate decreases with the temperature increase [71, 72]. Among the crop models, those assume that NPP is driven by the amount of solar radiation and \({CO}_{2}\) and is moderated by temperature use different concepts such as canopy light use efficiency model, single‐leaf photosynthesis light‐response model, and Farquhar‐von Caemmerer‐Berry biochemical model [31, 73, 74]. Because not all model parameters can be determined directly by measuring explanatory variables, a crop model needs to be carefully calibrated, including from the lowest calibration level only containing end-season agronomic measurements to the highest level containing both in-season and end-season measurements based on the calibration goals. By fitting the overall model to a portion of the observed data and then critically validating the model by using the remainder of the observed data, the robustness of the models across environments can be assessed [75]. After the calibration and validation processes, it is possible to analyze the response of a crop in different scenarios by replacing the measured values with scenario-based values for different variables [70, 76].

Crop models vary in terms of the way they involve variables in growth simulation. They simulate the effect of temperature on the rate of crop development [77] and on spikelet fertility [78, 79] in addition to their different \({CO}_{2}\)‐fixation algorithms that lead to various responses to climate change factors [80]. Studies have investigated the single-leaf photosynthesis response to the amount of light and the biochemical responses of the plant depending on the amount of \({CO}_{2}\) concentration to determine how the effect of \({CO}_{2}\) should be incorporated into the crop models [81, 82]. Crop models cannot solely be used to generate spatial distribution of NPP. Integration of crop models into the Geographic Information System (GIS)-based Environmental Policy Integrated Climate (GEPIC) (including six types of input data: DEM, soil map, land use map, climate data, plant data, and management data such as irrigation and fertilizer) simulates the effect of elevation on crop dynamics and provides gridded estimations of NPP [83, 84]. Taking advantage of DEM, a semi-distributed hydrological model like Soil and Water Assessment Tool (SWAT) configures sub-basins to connect embedded crop modelers and hydrologic analyzer systems together for estimating the distribution of NPP and biomass on farmlands [85, 86]. Embedded crop modelers of SWAT can also be used for environmental purposes like for estimating NPP in the forest ecosystems by adjusting SWAT default parameters for this type of ecosystems [87]. Process-based mechanistic crop models are effective means to estimate NPP and crop yield,however, these models create large uncertainty in simulations [88]. Application of the LUE model in estimating crop NPP instead of a mechanistic crop model can simplify the complex physiological, biophysical, and biogeochemical processes of plant photosynthesis and respiration [89].

3 The LUE Model

In addition to its application as a crop model, the simplicity of the LUE model has made it a suitable alternative to complex biosphere dynamical models. The LUE model reflects the effects of the physical and biophysical factors that determine growth rate in the concept of efficiency. The efficiency component in this model is defined as the net amount of solar energy stored by photosynthesis divided by the solar constant, and it is calculated by multiplying the geometrical, atmospheric, spectral, photochemical, diffusion, interception, and respiration factors that reduce the efficiency component [31]. The final production of the LUE model is the net amount of solar energy stored by photosynthesis, while the essence of the final product will be changed by eliminating each factor. As it is difficult to quantify all reducing factors, a disregard for any factor causes an overestimation in the calculation of net production by the LUE model. Thus, proper model structures and rigorous model parameterization and calibration should be adopted in the modeling process. The general form of the LUE model is written as follows:

$$\mathrm{GPP}=\left(\mathrm{fPAR}\right)\times \left(\mathrm{PAR}\right)\times \left(\mathrm{LUE}\right),$$
(1)

in which LUE is often calculated as a function of the maximum daily light use efficiency (\({\mathrm{LUE}}_{\mathrm{max}}\)) multiplied by environmental regulators [35]. LUE values cannot be used or compared when they are derived from different bases. Different approaches in terms of describing how LUE is controlled by ecosystem limiting factors have shaped various structures for LUE models [90]. Even having the same structure, \({\mathrm{LUE}}_{\mathrm{max}}\) values may not match in different studies. The reason for the incompatibility of reported \({\mathrm{LUE}}_{\mathrm{max}}\) values from different sources can be due to the data used in calibrating the model or the concepts through which fPAR was defined in each research project. In addition to different fPAR definitions, consideration of the effect of diffusive radiation on \({\mathrm{LUE}}_{\mathrm{max}}\) has also resulted in a discrepancy in \({\mathrm{LUE}}_{\mathrm{max}}\) reported values. In some research, an attempt was made to improve the LUE model by differentiating \({\mathrm{LUE}}_{\mathrm{max}}\) for sunlit and shaded leaves based on the fact that diffuse radiation results in an increase of carbon uptake [91, 92] .

3.1 LUEmax Based on Training Dataset

The LUE model has been used often for estimating GPP in the biosphere and forest ecosystems for environmental research purposes [34, 35, 93,94,95]. The general form of the LUE model in Eq. 1 has also been used to calculate GPP for agricultural farmlands [89]. Given that the NPP is more informative than GPP for agricultural purposes, some studies have estimated NPP using the same concept as in the LUE model as follows [96,97,98]:

$$\mathrm{NPP}=\left(\mathrm{fPAR}\right)\times \left(\mathrm{PAR}\right)\times \left(\mathrm{LUE}\right).$$
(2)

Despite that Eqs. 1 and 2 have no difference in appearance, application of the same calibrated \({\mathrm{LUE}}_{\mathrm{max}}\) that was used for calculating GPP in this equation can cause NPP overestimation. Estimation of NPP requires the consideration of the respiration term and recalibration of the LUE parameter. It is also possible to put the dry material (DM) on the left-hand side of Eq. 1 [99], while any change in the left-hand side requires a reinterpretation of \({\mathrm{LUE}}_{\mathrm{max}}\). Dry material is the dried body of a plant or a crop including stems, leaves, roots, flowers, and fruits. Neglecting the amount of NPP that is consumed by other creatures such as insects and organisms during the growth period, the DM is equal to the accumulated amount of NPP minus the weight of the liquid component of the body. Based on which parts are weighed and if the consumed portion is counted, the calibrated \({\mathrm{LUE}}_{\mathrm{max}}\) can be case-specific. It is essential to differentiate the reported \({\mathrm{LUE}}_{\mathrm{max}}\) values in the literature since the left-hand side of the equation may differ for different research objectives. The training dataset and the process of calibration used for the study should also be considered before applying the reported potential LUE for another study. A biome-independent \({\mathrm{LUE}}_{\mathrm{max}}\) for NPP estimation was calibrated in the CASA model [100], while in the MODIS-GPP algorithm, \({\mathrm{LUE}}_{\mathrm{max}}\) changes across the biome type [1, 101]. In the case of using a calibrated \({\mathrm{LUE}}_{\mathrm{max}}\), which has been obtained from a GPP training dataset, it is necessary to calculate and import the autotrophic respiration term in Eq. 1 and rewrite it as follows to estimate NPP:

$$\mathrm{NPP}=\left(\mathrm{fPAR}\right)\times \left(\mathrm{PAR}\right)\times \left(\mathrm{LUE}\right)-{\mathrm{R}}_{\mathrm{a}}.$$
(3)

In the MODIS daily GPP and annual NPP (MOD17A2/A3) algorithm, the annual \({\mathrm{R}}_{a}\) is calculated by summing the annual \({\mathrm{R}}_{\mathrm{G}}\) and the annual \({\mathrm{R}}_{\mathrm{M}}\). The MOD17A2/A3 assumes that the \({\mathrm{R}}_{\mathrm{G}}\) is considered to be equal with 25% of GPP and the live wood \({\mathrm{R}}_{\mathrm{M}}\) is calculated using the equations that relate live wood mass to leaf mass [102].

3.2 LUEmax Based on the fPAR Definitions

Despite the Leaf Area Index (LAI) that ignores the complexities of canopy geometry (such as leaf angle distribution, canopy height, or shape), it is only an abstract image of the basic size of the canopy. FPAR is a radiation term that directly relates to those remotely sensed variables which are affected by the reflectance properties of different canopy structures [103]. If it is considered that the intercepted fPAR by the whole canopy contributes to the production of GPP, then fPAR is defined as \({\mathrm{fPAR}}_{\mathrm{Canopy}}\), and this concept leads \({\mathrm{LUE}}_{\mathrm{max}}\) to be defined as \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Canopy}}\). Through the \({\mathrm{fPAR}}_{\mathrm{Canopy}}\) concept, a sort of constant \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Canopy}}\) values have been calculated for each of 11 biome classes and gathered into a table known as Biome-Property-Look-Up-Table (BPLUT) for the MODIS GPP/NPP algorithm [102]. The intercepted part of the incoming radiation by the non-photosynthetic part of the canopy turns to heat loss instead of GPP. Considering the photosynthetic part of the canopy as the part that can produce GPP, only the absorbed portion of \({\mathrm{fPAR}}_{\mathrm{Canopy}}\) by the chlorophyll \({\mathrm{(fPAR}}_{\mathrm{Chl}})\) is used in the LUE model and leads to a constant \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Chl}}\) value across all biome types. \({\mathrm{fPAR}}_{\mathrm{Chl}}\) can be calculated using the Solar induced fluorescence (SIF) as a remotely sensed proxy for the amount of chlorophyll content of the canopy as follows [35]:

$$({{\mathrm{fPAR}}\mathrm{)}}_{\mathrm{Chl}}=\frac{\left(\mathrm{SIF}\right)}{\left(\mathrm{PAR}\right)\times \left(\mathrm{FE}\right)},$$
(4)

where FE is the fluorescence efficiency observed at the top of the canopy.

3.3 Application of Chlorophyll Proxies in the LUE Model

Satellite chlorophyll proxies that are available for download can be categorized into two SIF and chlorophyll index products. SIF is almost equal to 1–2% of the energy absorbed by the photosynthetic part which is remitted between 734 and 758 nm by the chlorophyll content of the canopy. Interest in SIF data has grown exponentially, and the retrieval of SIF and the provision of SIF data products have become an important and formal component of spaceborne Earth observation missions [104]. Different space projects have tried to capture this fluorescence emission with various spatial and temporal resolutions (see Table 2) validated in many research studies by in situ measurements over various environments [105,106,107,108]. The SIF products are applied in the LUE model to replace \({\mathrm{fPAR}}_{\mathrm{Canopy}}\) and \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Canopy}}\) terms with \({\mathrm{fPAR}}_{\mathrm{Chl}}\) and \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Chl}}\) terms improving GPP estimations for global-scale environmental purposes [35]. To the best of our knowledge, the application of satellite chlorophyll proxies in the LUE model for agricultural purposes for tracing NPP, biomass, and yields in farmlands has not been assessed yet. Negligence on the application of SIF products by researchers for agricultural purposes can be due to the coarse spatial resolution of these products, especially before 2016, when 1.3 km\(\times 2\).25 km (each pixel with an approximate area of 300 ha that can include variety of crops, soils and management methods in agricultural regions) was the highest available spatial resolution as show in Table 1. Possibly, it also can be due to insignificant contribution of non-photosynthetic woody limbs in crop anatomy, supposing that remote sensing of the crop canopy provides a good proxy for the crop chlorophyll content. Any probable improvement in NPP, biomass, and yield estimations on croplands using the new SIF products (with finer resolution) or the downscaled old SIF products can be investigated in future research to make decisive comments. The LUE equation for estimating GPP using the \({\mathrm{fPAR}}_{\mathrm{Chl}}\) concept is written as follows:

Table 2 Satellite products of NASA and European space agency for chlorophyll proxies and their spatiotemporal resolution. Continuity to the MTCI is provided by OLCI on-board the Sentinel-3 missions from 2016 with a much higher spatiotemporal resolution
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$${\mathrm{GPP}}= \left\{{(\mathrm{fPAR})}_{\mathrm{Chl}}\times (\mathrm{PAR})\times {\left(\mathrm{LUE}\right)}_{{\mathrm{max}}_{\mathrm{Chl}}}\times (\mathrm{regulators})\right\}-{\mathrm{R}}_{{\varvec{a}}}.$$
(5)

Regulators in Eq. 5 are the environmental factors that affect the calculation of LUE such as temperature, soil, and water [109]. As NPP has always been the case for agricultural studies, Eq. 5 can be written as:

$$\mathrm{NPP }\mathrm{=}{(\mathrm{fPAR})}_{\mathrm{Chl}}\times \left(\mathrm{PAR}\right)\times {\left(\mathrm{LUE}\right)}_{{\mathrm{max}}_{\mathrm{Chl}}}\times \left(\mathrm{regulators}\right).$$
(6)

In Eq. 6, the \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Chl}}\) parameter can be calibrated using NPP training dataset. It should be investigated if Eq. 6 (\({\mathrm{fPAR}}_{\mathrm{Chl}}\) and \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Chl}}\) concepts) could contribute to improve NPP estimation on farmlands in agricultural research. Given that traditional vegetation indices (VIs) tend to exhibit saturation effects in dense canopies and considering the high correlation between satellite SIF datasets and crop productivity [110], it is recommended to test the use of SIF products in estimating crop production to explore their potential for improving accuracy in such estimations. The test may require the use of a downscaled version of the SIF products as the spatial resolution of the available SIF products (especially before 2016) does not allow researchers to calibrate \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Chl}}\) for specific crops (one pixel may cover different types of land use).

3.4 Downscaled Version of the SIF Products

3.4.1 Regression Approach

Apparent canopy SIF yield of corn, soybean, forest, and grass/pasture from two satellite instruments, OCO-2 and TROPOMI, showed clear seasonal and spatial patterns [111]. This suggests that the ability of SIF to observe spatial variances is worth considering despite the coarse resolution of some SIF products. Despite the development of new SIF products with fine resolutions, the endeavor to use coarse products is still ongoing due to a lack of continuous satellite SIF data for long-term evaluations [112]. One of the approaches to deal with the spatial coarseness of the remote sensing products is downscaling [113]. In the context of remote sensing, downscaling refers to a decrease in the pixel size of remotely sensed images; this is a scaling process, converting from a low to a high spatial resolution preserving the original image radiometry. Simplistically, the average of the simulated high-resolution subpixel values is equal to the pixel value of the original low-resolution image. The effective emissivity is derived from the high-resolution NDVI-composite image which has been used as the scaling factor to downscale the LST image [114]. Along with the same concept, the following equation downscales a SIF pixel with \({0.05}^{^\circ }\) spatial resolution (from GOME-2) using the number of NDVI pixels with 250 m spatial resolution (obtained from bands 1 and 2 of MODIS) covered by the SIF pixel:

$${(\mathrm{SIF})}_{250\mathrm{m}}={(\mathrm{SIF})}_{{0.05}^{^\circ }}\times \frac{{(\mathrm{NDVI})}_{250\mathrm{m}}}{{(\mathrm{NDVI})}_{{0.05}^{^\circ }}},$$
(7)

where \({\mathrm{SIF}}_{250\mathrm{m}}\) is the downscaled SIF pixel with 250 m spatial resolution, \({\mathrm{SIF}}_{{0.05}^{^\circ }}\) is the original SIF pixel with \({0.05}^{^\circ }\) spatial resolution, \({\mathrm{NDVI}}_{250\mathrm{m}}\) is the NDVI pixel calculated by red and near infrared bands of the MODIS with 250 m spatial resolution, and \({\mathrm{NDVI}}_{{0.05}^{^\circ }}\) is the NDVI pixel with \({0.05}^{^\circ }\) spatial resolution obtained from averaging \({\mathrm{NDVI}}_{250\mathrm{m}}\) pixels underlaid the original SIF pixel. Both \({\mathrm{SIF}}_{{0.05}^{^\circ }}\) and \({\mathrm{NDVI}}_{250\mathrm{m}}\) pixels should correspond to similar weather conditions, so the downscaling process requires images of the same dates. In another method, a linear regression model is established in the downscaling procedure [115]. After plotting \({\mathrm{SIF}}_{{0.05}^{^\circ }}\) pixels against \({\mathrm{NDVI}}_{{0.05}^{^\circ }}\) pixels, the following equation is fitted:

$${(\mathrm{SIF})}_{{0.05}^{^\circ }}=a+b\times {(\mathrm{NDVI})}_{{0.05}^{^\circ }}.$$
(8)

Assuming that inside a \({\mathrm{SIF}}_{{0.05}^{^\circ }}\) pixel the regression equation for \({SIF}_{250m}\) pixels versus \({NDVI}_{250m}\) pixels has the same coefficients with Eq. 8, then it can be written as:

$${(\mathrm{SIF})}_{250\mathrm{m}}=a+b\times {(\mathrm{NDVI})}_{250\mathrm{m}}.$$
(9)

The following revision is required on \({\mathrm{SIF}}_{250\mathrm{m}}\) pixels to preserve Eq. 8 for the downscaled SIF image:

$${{(\mathrm{SIF})}_{250\mathrm{m}}^{\mathrm{r}}=(\mathrm{SIF})}_{250\mathrm{m}}\pm \sigma ,$$
(10)

where \({\mathrm{SIF}}_{250\mathrm{m}}^{\mathrm{r}}\) is the revised value of \({SIF}_{250m}\) and σ is calculated using the following equation:

$$\sigma ={{(\mathrm{SIF})}_{{0.05}^{^\circ }}-(\mathrm{SIF})}_{{0.05}^{^\circ }}^{\mathrm{avg}},$$
(11)

where \({\mathrm{SIF}}_{{0.05}^{^\circ }}^{\mathrm{avg}}\) is the average of the \({\mathrm{SIF}}_{250\mathrm{m}}\) in pixels underlaid a \({\mathrm{SIF}}_{{0.05}^{^\circ }}\) pixel:

$${(\mathrm{SIF})}_{{0.05}^{^\circ }}^{\mathrm{avg}}=\frac{{\sum }_{1}^{\mathrm{q}}{(\mathrm{SIF})}_{250\mathrm{m}}}{q},$$
(12)

where \(q\) is the number of pixels underlaid a \({SIF}_{{0.05}^{^\circ }}\) pixel.

3.4.2 Area-to-Point Regression Kriging

Downscaling changes the size associated with each data value (support) which means that in geostatistics, it can be considered as a change of support problem (COSP). The GSOP is concerned with inference about the values of a variable at points (point-referenced or simply point data) or blocks (block data) different from those at which it was observed [116]. Kriging processes have been applied to take the kriged values for the blocks (larger support) from the point-level observations (point-to-area prediction) [117] and reversely to predict a support that is smaller than that of the original data (area-to-point prediction) applicable for the downscaling procedure [118]. For example, one might be interested in the significance of the correlation between point observations (point support) and data derived from a regional model (areal support). In such cases, the latter areal support predictions must also be transformed to the point support level coherently in a way that these point support predictions can reproduce exactly the corresponding areal data when they are convolved with the discrete sampling kernel (convolution kernel). The uncertainty in downscaled data derived from coarser resolution using area‐to‐point approach can be assessed in a Monte Carlo framework [119]. In remote sensing, since values lie in similar supports (pixels of the same size), the problem is simpler than for the cases with irregular supports (like zips in a census or hydrologic response units in a semi distributed model) [120].

The prediction of area-to-point regression kriging (ATPRK) for fine pixels \({Z}_{\mathrm{v}}^{\mathrm{l}}\left(x\right)\) of band k underlaid the coarse pixels \({Z}_{\mathrm{V}}^{\mathrm{l}}\left(x\right)\) of band l is [113]:

$${Z}_{v}^{l}\left(x\right)={Z}_{v}^{l}(x)+{Z}_{v2}^{l}\left(x\right)$$
(13)

where \({Z}_{\mathrm{v}1}^{\mathrm{l}}\left(x\right)\) and \({Z}_{\mathrm{v}2}^{\mathrm{l}}\left(x\right)\) are the predictions of the regression and ATPK parts. The regression part takes advantage of fine ancillary data. For example, in case of downscaling a coarse band of MODIS, one of the fine bands with the greater correlation coefficient can be selected as a band that provides ancillary data to be fused for the coarse band. \({Z}_{\mathrm{v}1}^{\mathrm{l}}(x)\) is the prediction of the linear regression between the coarse band \({Z}_{\mathrm{V}}^{\mathrm{l}}\) and its corresponding ancillary fine band \({Z}_{\mathrm{v}}^{\mathrm{lk}}\) (used as covariate) that can be calculated as follows;

$${Z}_{\mathrm{v}1}^{l}(x)= {{a}_{\mathrm{l}}\times Z}_{\mathrm{v}}^{\mathrm{lk}}\left(x\right)+{b}_{\mathrm{l}}.$$
(14)

To estimate \({a}_{\mathrm{l}}\) and \({b}_{\mathrm{l}}\), Eq.15 assumed to be exist at coarse spatial resolution:

$${Z}_{\mathrm{V}1}^{\mathrm{l}}(x)= {{a}_{l}\times Z}_{\mathrm{V}}^{\mathrm{lk}}\left(x\right)+{b}_{\mathrm{l}},$$
(15)

where \({Z}_{\mathrm{V}}^{\mathrm{lk}}(x)\) is the coarse image produced by upscaling \({Z}_{\mathrm{v}}^{\mathrm{lk}}\) using the point spread function (PSF) and the convolution operator. The residuals of the regression should be calculated to reproduce the spectral properties of the observed coarse data. ATPK part downscales the residuals \({Z}_{V2}^{l}\left(x\right)\) to fine spatial resolution residuals \({Z}_{v2}^{l}\left(x\right)\). Based on ATPK, the fine residual \({Z}_{v2}^{l}\left(x\right)\) is a linear combination of \(N\) coarse residuals of band l:

$${Z}_{\mathrm{v}2}^{\mathrm{l}}\left(x\right)=\sum_{i=1}^{N}{(\lambda }_{\mathrm{i}}{\times Z}_{\mathrm{V}2}^{l}\left({x}_{\mathrm{i}}\right)), s.t. \sum_{i=1}^{N}{\lambda }_{\mathrm{i}}=1,$$
(16)

where \({\lambda }_{\mathrm{i}}\) is the weight vector {\({\lambda }_{1},\dots , {\lambda }_{N}\)} that yields the minimum prediction error variance among all weighed linear combinations of \(N\) coarse residuals. \({\lambda }_{\mathrm{i}}\) is obtained from the corresponding kriging system consists of coarse-to-coarse residual covariance between coarse pixels, fine-to-coarse residual covariance between fine and coarse pixels, and the Lagrange multipliers. The \(N\) coarse residuals are from the \(N\) coarse pixels surrounding the pixel in a \(N\times N\) window. In case the area-to-point regression kriging method is used to downscale a coarse \(SIF\) image, a finer \(NDVI\) image obtained from MIDIS or Landsat bands can provide the required ancillary data.

3.4.3 Super-Resolution Mapping

Super-resolution mapping (SRM) uses auxiliary data to increase the spatial resolution of each pixel based on the classification techniques [121]. The objective is to increase the spatial resolution of an image with a coarse spatial resolution using other image(s) with the desired fine spatial resolution but of a different spectral band(s) [122]. Unlike hard classification approaches that assign one class for each pixel, soft classification approaches predict the proportional cover of each land cover class within each pixel. However, the location of the land cover classes in the mixed pixels (the spatial resolution of the thematic map) is not increased relative to that of hard classification. SRM predicts the location of land cover classes within a pixel based on the proportion images produced by soft classification using different approaches like image fusion [123]. Since soft classification methods are limited in terms of detail and accuracy of the resulting thematic map, it is suggested the information available at a finer spatial resolution should be used. Several sources of information at a finer resolution are geostatistical data, fused images, and panchromatic imagery [124]. A proliferation of SRM methods includes artificial neural networks, subpixel-swapping methods, spatial attraction models, Markov random fields, geostatistical solutions, interpolation-based approaches, and other advanced methods [125].

Generally, an SRM analysis has two main steps: first downscaling the coarse resolution fraction image to a fine spatial resolution indicator image which represents the possibility of each fine resolution pixel belonging to a specified land cover class and, second, combining fine fraction images to produce a fine resolution land cover map [126]. SRM models use different techniques for the downscaling step. SRM models based on spatial dependence that define the spatial pattern explicitly may be inadequate for the representation of complex land cover mosaics for which a learning-based model may work better in SRM [127, 128]. The aim of a learning-based model like back-propagation neural networks and support vector regression is to learn the spatial pattern of land cover from existing fine-resolution land cover maps assuming that the pattern is constant while their performance is limited in the case of existing complex nonlinear relationships between the coarse and fine resolution data [126]. Deep learning methods have been shown to have considerable potential in SRM to produce more accurate maps than traditional machine learning approaches [129]. Figure 3 shows a schematic of a SRM process through which a downscaled pixel is produced from the proportion image and the auxiliary pixels. In the case of downscaling a coarse SIF image, the SIF image is used to produce proportion images, and the next one or multiple auxiliary images such as a panchromatic image or a vegetation index image obtained from other satellites with higher resolution can be tested in RSM. By evaluating the downscaled SIF images obtained from the application of different sets of auxiliary images, the best set of auxiliary images for downscaling SIF images can be determined.

Fig. 3
figure 3

A schematic of a downscaled pixel using RSM, assuming that the goal is to increase the resolution of a coarse pixel by 4 times according to the resolution of the existing auxiliary pixels. It is also assumed that the proportional value of the coarse pixel is estimated to 50% in the proportional image. Based on the above assumptions, 6 modes can be imagined for the downscaled pixel, and the downscaled pixel is one of these modes. An auxiliary pixel or auxiliary pixels are used to determine one of the modes with the highest probability as the downscaled pixel

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3.4.4 Long-Term Temporally Corrected and Spatially Downscaled SIF Product

Due to the temporal inconsistency induced by sensor degradation and to correct the impacts of the degradation, generation of long-term temporally corrected and spatially downscaled SIF products is one of the latest attempts in making the SIF products practical for studies [112]. The process includes three main steps: spatial downscaling, temporal correction, and multi-sensor fusion. The process leaded to generate a long-term temporally corrected SIF product from July 1995 to December 2018 with a high spatial resolution (0.05°). Downscaling phase can be conducted using the LUE-based method and machine learning methods [130, 131].

To conduct temporal correction phase, a long-term time series of satellite SIF in a region as the benchmark is required to correct the trend. A stable desert with a very sparse vegetation is recommended as a benchmark since the SIF signals is expected to be very small and constant across the recent years in such a land cover [112]. Average monthly time series of satellite SIF are extracted for the benchmark to get the normalized SIF time series which is used for correcting the temporal trend of satellite SIF. Finally, cumulative distribution frequency (CDF) matching approach is used to fuse these three independent SIF datasets into a long-term consistent data record [131].

4 Functional Models

Functional models are simplified versions of the complex mechanistic models or a combination with statistical schemes and are more suitable for operational crop yield forecasting due to their minimal data input requirements, and the key processes can be parametrized using approximate equations [48, 132]. In recent years, functional models have tended to integrate both climate and remote sensing indices into unified yield prediction models [133,134,135,136]. Functional models can be trained through a single predictor regression, a multiple linear regression, or a nonlinear relation analysis approach [54, 55, 60],). Along with the agricultural perspective, functional models in combination with forecast statistical algorithms are used for yield forecasting and produce early warnings [48]. Figure 4 is a flow inferred from the reviewed research to show the contribution of statistical and mechanistic models and remote sensing data in a functional model that can estimate NPP and yield values and produce early warning for yield reduction before harvest.

Fig. 4
figure 4

Development of the concept of functional model through the incorporation of statistical and mechanistic models, as well as the utilization of remote sensing data to estimate spatiotemporal distribution of NPP and yield values and its combination with forecasting algorithms to produce early warnings for probable yield reduction before harvest (inferred from the reviewed research)

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4.1 Different Methods in Functional Models

Functional models include different approaches through which process-based methods are combined with remotely sensed data. Figure 5 describes one of these approaches based on an empirical equation between \(\mathrm{fPAR}\) and \({\mathrm{LAI}}\) [55]. According to the proposed methodology, \(\mathrm{LUE}\) is calibrated using the data obtained from running a crop model. The model is run for different scenarios for each DOY to estimate \({\mathrm{NPP}}\) and \({\mathrm{LAI}}\) values for that DOY. The \({\mathrm{NPP}}\) and \({\mathrm{LAI}}\) values for each DOY obtained from the crop model are shown with CM subscript (\({\mathrm{NPP}}_{\mathrm{CM}}\) and \({\mathrm{LAI}}_{\mathrm{CM}}\), respectively). Several values for \({\mathrm{NPP}}_{\mathrm{CM}}\) and \({\mathrm{LAI}}_{\mathrm{CM}}\) are calculated based on different scenarios on each DOY for which the \({{(\mathrm{R}}^{2})}_{\mathrm{Year }1,\mathrm{ DOY }1}\) value is obtained by plotting them versus each other. The process is applied for different years for which the required key parameters for running the crop model are available. The highest amount of \({\mathrm{R}}^{2}\) determines the DOY on which the \({\mathrm{LAI}}\) (shown with \({\mathrm{LAI}}_{\mathrm{Best}}\)) returns the best estimation of \({\mathrm{NPP}}\). The \({\mathrm{LAI}}_{\mathrm{Best}}\) value returns the \(\mathrm{fPAR}\) value using a linear equation which is shown with CM subscript (\({\mathrm{fPAR}}_{\mathrm{CM}}\)) since it is obtained from running a crop model. Using the \({\mathrm{fPAR}}_{\mathrm{CM}}\) value, \({\mathrm{LUE}}_{\mathrm{CM}}\) is calculated by the LUE equation. Finally, the NPP value for each pixel (\({\mathrm{NPP}}_{\mathrm{CM}+\mathrm{RS}}\)) is calculated from the calibrated \({\mathrm{LUE}}\) value (\({\mathrm{LUE}}_{\mathrm{CM}}\)) and the remotely sensed \({\mathrm{fPAR}}\) value (\({\mathrm{fPAR}}_{\mathrm{RS}}\)) using the LUE equation.

Fig. 5
figure 5

A proposed process for estimating the spatiotemporal distribution of NPP values in which \({\mathrm{LUE}}\) is calibrated by the data obtained from running a crop model for several scenarios. The calibrated \({\mathrm{LUE}}\) is used in integration with the \({\mathrm{fPAR}}\) product of satellites to calculate NPP for each pixel

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Figure 6 shows another approach for building a functional model based on converting \({\mathrm{LAI}}_{\mathrm{CM}}\) to pseudo-LAI observations (\({\mathrm{LAI}}_{\mathrm{PO}}\)) and a multiple linear regression equation [54]. The proposed method provides a way to estimate the spatiotemporal distribution of the NPP values using remotely sensed vegetation indexes integrated with weather data. In the first step, the LAI values obtained from a crop model are converted to pseudo-vegetation index observations using empirical equations. The pseudo-values are used to calibrate the coefficients of a multiple linear regression model after which the real remotely sensed values of a vegetation index are imported into the model to calculate the \({\mathrm{NPP}}_{\mathrm{CM}+\mathrm{RS}}\) values. Instead of training a simple linear or multiple linear regression equation, grid data on crop yield, remotely sensed data, and climate data can be used to train a machine learning method such as the deep learning algorithm to estimate crop yield [137].

Fig. 6
figure 6

A proposed process for estimating the spatiotemporal distribution of NPP values in which a multiple linear regression equation is calibrated using pseudo-vegetation index observations

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4.2 Application of Chlorophyll Proxies in Functional Models

Based on the promising results obtained from the use of chlorophyll proxies in estimating GPP using LUE models, the present paper proposes the evaluation of these proxies in functional models in estimating NPP and yield for agricultural purposes. Application of chlorophyll proxies in functional models can be followed by introducing new NPP and yield chlorophyll-based predictors for these models. The proposed predictors that can be evaluated in future studies in estimating NPP and yield can be a chlorophyll index or a multiplied sentence including \({\mathrm{fPAR}}_{\mathrm{Chl}}\). The proposed predictors can be used to build a single linear regression model shown in Eqs. 17 and 18.

$$\mathrm{NPP}= a\times (\mathrm{Chl)}+b,$$
(17)
$$\mathrm{NPP}=a{\;\times\;(\mathrm{fPAR)}}_{\mathrm{Chl}}\times (\mathrm{P}\mathrm{AR)}\times {{\mathrm{(LUE)}}_{\mathrm{max}}}_{\mathrm{Chl}}+b,$$
(18)

where \({\mathrm{Chl}}\) is the chlorophyll index such as MTCI and OTCI, \({\mathrm{fPAR}}_{\mathrm{Chl}}\) is the \({\mathrm{fPAR}}\) relating to chlorophyll content of the canopy and \({{\mathrm{LUE}}_{\mathrm{max}}}_{\mathrm{Chl}}\) is the maximum daily LUE based on PAR absorption by canopy chlorophyll. The predictors can be used in a multiple linear regression model shown in Eqs. 19 and 20. In a multiple linear regression model, a chlorophyll-based predictor can be integrated with other predictors (P) as below:

$$\mathrm{NPP}={a}_{1} \, \times ({\mathrm{Chl})} \, +{a}_{2}\times P+{a}_{3},$$
(19)
$$\mathrm{NPP}={a_1\times\left(\mathrm{fPAR}\right)}_{\mathrm{Chl}}\times\left(\mathrm{PAR}\right)\times{\left(\mathrm{LUE}\right)}_{\max\nolimits_{\mathrm{Chl}}}+a_2\times P+a_3.$$
(20)

Since the amount of actual evapotranspiration in the anthesis stage (\({\mathrm{ET}}_{\mathrm{Anthesis}}\)) has been recognized as a differentiating factor to categorize wheat fields into high-productive and low-productive classes [138], a model in which P is replaced by \({\mathrm{ET}}_{\mathrm{Anthesis}}\) can be evaluated for estimating yield before harvest. Remotely sensed chlorophyll-based predictors in addition to other predictors relating to weather, soil type, soil moisture, topography, fertilizers, pesticides, etc. can also be used to train a complex machine learning model for further investigation of NPP and yield estimation. Despite an observed strong seasonal correlation between SIF and GPP, changes in plant growth stages can affect this relationship. Recently, the relationship is investigated for maze from C4 plant category, and it has found that canopy structure impacts seasonal variations of SIF and its relation to GPP [36]. It has been found that exponential regression is the best method to capture the nonlinearity at the site level, while the degree of the nonlinearity varies among different biomes [139]. Indeed, SIF-GPP relationships tend to vary not only between vegetation types but also between crop species specifically [140]. However, additional measurements for different vegetation types are necessary for fully understanding the effects of factors on SIF signals for on croplands. The fact that the relationship between SIF and photosynthesis changes both seasonally and by events may lead to test the nonlinear functions or non-constant coefficients in building functional models for future studies.

5 Conclusion

A comprehensive overview and categorization of the literature that include GPP and NPP studies were presented to clarify the different approaches applied to estimate GPP and its derivatives with environmental and agricultural purposes. From an environmental perspective, GPP as the output from dynamic global vegetation models was reviewed to identify its role in quantification of vegetation production and in simulation of ecosystem processes and the hydro-biochemical cycle [141]. The terrestrial carbon cycle also can be traced using the simple LUE model instead of multiparameter complex models [89]. Application of the LUE model for quantification of GPP in the biosphere has improved by introducing \({\mathrm{fPAR}}_{\mathrm{Chl}}\) and \({\mathrm{LUE}}_{\mathrm{Chl}}\) concepts based on remotely sensed SIF products [35]. The review indicated that just as the GPP estimations are important for environmental purposes, so are its derivatives, NPP, biomass, and yield for agricultural purposes where attempts have focused on crop yield predictions made before harvest [48, 138]. The review showed that the LUE model, crop models, statistical models, and their incorporation into so-called functional models are the common approaches for estimating the NPP as the most important derivative of GPP in agricultural perspective. Due to a promising application of chlorophyll proxies in the utilization of the LUE model at global scale environmental studies for estimating GPP [35], this study proposes the test of these proxies in reginal scale agricultural studies for estimating NPP, biomass, and yield. In the process of estimating NPP using chlorophyll proxies, coarseness of the spatial resolution of historical remotely sensed SIF products can limit their application for agricultural purposes. Hence, different downscaling methodologies that can be used in the downscaling process of the SIF products were reviewed. Generation of long-term temporally corrected and spatially downscaled SIF product with cumulative distribution frequency (CDF) matching approach was of the latest attempts to increase the resolution of SIF products up to 0.05° [112, 131].

With the increase in access to high resolution products of SIF and chlorophyll proxies, the present review paper proposed their application for being tested in both the LUE and functional models for estimating NPP, biomass and yield in agricultural studies. Since different functional models can be built by different combinations of predictors including ground measured or remote sensing predictors, introducing any new remote sensing predictor that can reduce dependency on the field measured data is useful to fulfill the objective of the functional model. Non-linearity generally exists between total emitted SIF and GPP at the canopy level that may require the change of the proposed linear equations into nonlinear forms or employment of different coefficients for crop growth stages. Hence, future studies on testing remote sensing chlorophyll-based predictors will make it clear if they are able to increase the accuracy of NPP estimations with the LUE model or reduce the dependency of functional models on field measured data.