Crop specific inversion of PROSAIL to retrieve green area index (GAI) from several decametric satellites using a Bayesian framework

Highlights

• Bayesian theory and Hamiltonian Monte Carlo algorithm are applied to RTM inversion.

• Joint distributions of crop-specific input variables in PROSAIL is derived from HMC.

• GAI predictions agree closely with corresponding worldwide ground measurements.

• Use of prior information for some PROSAIL variables minimizes bias in GAI prediction.

Abstract

The objective of this study is to evaluate the performances of a semi-empirical approach based on the Bayesian theory to retrieve Green Area Index (GAI) from multiple decametric satellites. It is designed to overcome some limitations in existing Radiative Transfer Model (RTM) inversion methods, including the high dimensionality of the inverse problem, the convergence problem due to possible equifinality, and the dependence of some RTM variables on the crop-specific architecture. The PROSAIL model is first inverted in a calibration step using the Hamiltonian Monte Carlo (HMC) algorithm over a global dataset of ground GAI measurements (for maize, wheat, and rice) and the corresponding reflectance observations from Landsat-8, Sentinel-2, and Quickbird to derive crop-specific distributions of PROSAIL input variables. These distributions were then used as prior information to predict GAI over an independent set of reflectance observations.

Results show that the full Bayesian approach provides close estimates of GAI to ground truth, with respective Root Mean Square Error (RMSE) of 1.01, 1.33, and 0.97 for maize, wheat, and rice (R²=0.67, 0.76 and 0.63, respectively). The performances are better than those approaches generally reported using radiative transfer models that are non-crop-specific, like the SNAP algorithm for Sentinel-2, but are slightly behind the purely empirical models based on machine learning. However, the proposed approach provides an explicit insight of the joint distribution of PROSAIL variables that are valid for any satellite platform. This constitutes a major advantage against purely empirical models, as it enables to fully exploit large observational datasets from multiple sensors and generalize to other platforms.

Introduction

Accurate monitoring of crop biophysical variables is required for a range of agricultural applications including crop management and yield forecasting. Green Area Index (GAI) -the green vegetation area per unit horizontal ground area (Chen and Black, 1992)- constitutes the plant–atmosphere interface of the active organs. It is therefore an essential variable required by many terrestrial biosphere models to simulate several processes including photosynthesis and transpiration (Liu et al., 1997; Monteith, 1977; Ryu et al., 2011). While the commonly used LAI (Leaf Area Index) is restricted to the leaves and does not make difference between green and senescent parts, GAI includes the green organs such as stems and inflorescence together with the green leaves that may also contribute significantly to both photosynthesis and transpiration (Duveiller et al., 2011), and the remote sensing signal. Therefore, in the following, we will consistently use the term GAI although many authors employ LAI to indicate GAI.

Developing reliable approaches to estimate GAI from remote sensing has attracted increased scientific interests in the recent decades (Boegh et al., 2013; Chen, 1996; Verrelst et al., 2015b). The retrieval of GAI from multispectral satellite observation relies on a model linking GAI with canopy reflectance (Verrelst et al., 2015b). This model can be empirical, calibrated with observations on field experiments (Camacho et al., 2021; Frampton et al., 2013; Qi et al., 2000; Xiao et al., 2011), or physically-based, using models that approximate the actual radiative transfer processes (Koetz et al., 2005; Richter et al., 2011). Empirical models are operationally simple and computationally efficient, while they are prone to perform unreliable when extrapolated over areas, observational configuration, sensors or crop types different from those used in the calibration process (Kang et al., 2016; Verrelst et al., 2012b). By contrast, Radiative Transfer Models (RTMs) provide a mechanistic link between the reflectance and biophysical variables, in which differences in canopy structure, illumination, soil backgrounds, and viewing geometries are taken into account explicitly (Jacquemoud et al., 2009). For that reason, physically-based methods have been widely considered as a more generic and robust approach for retrieving biophysical variables (Koetz et al., 2005; Richter et al., 2011; Shiklomanov et al., 2016; Xu et al., 2019). However, they may also be limited by the assumptions embedded in the description of canopy architecture and leaf optical properties (Baret, 2016), and suffer from the ill-posed problem (Combal et al., 2003) since many RTM input variables combinations can correspond to almost the same set of reflectances.

RTMs simulate canopy reflectance under a given observational configuration from canopy architecture and leaf optical properties. Estimating GAI requires therefore to invert RTMs from the observed spectral reflectance. Over the past decades, various successful strategies have been proposed to invert RTMs, including the iterative optimization (OPT) (Jacquemoud et al., 1995), Look-Up Table (LUT) (Knyazikhin et al., 1998), and non-parametric models (Verrelst et al., 2015a; Yang et al., 2012) such as Artificial Neural Network (ANN) (Li et al., 2015; Weiss and Baret, 1999). OPT is one of the most classical approach to invert RTMs, which consists in iteratively updating the RTM input variables until the model-simulated reflectance fits the observed one. This approach is computer demanding (Bacour et al., 2006) but presents the advantage to be generic and flexible among different observational configurations (Baret and Buis, 2008) while often encountering the problem of trapping in possible local minima during the optimization. The LUT method compensates for some of these problems by locating the global optimal solution that leads to the closest distance to the observed reflectance from a pre-defined set of combinations of RTM-model input variables and their corresponding reflectance (Weiss et al., 2000). Thanks to that, the LUT method greatly speeds up the inversion process (Pan et al., 2019; Dorigo et al., 2007). The ANN-based methods provide another efficient way to invert RTMs, using large datasets simulated with RTMs to train an ANN. This approach combines the physical realism of RTMs and the operational efficiency of empirical algorithms, which proved to be reliable by many authors (Curnel et al., 2011; Delloye et al., 2018; Verger et al., 2008; Weiss and Baret, 2016). Lately, the Gaussian processes regression (GPR) has also been introduced to learn the RTM-simulated database with kernels, which are projection functions evaluating the similarity between the each spectrum and all training spectra (Campos-Taberner et al., 2015; Moreno-Martínez et al., 2018; Verrelst et al., 2012a, Verrelst et al., 2012b). As a probabilistic approach, GPR provides not only the mean GAI estimates but also associated uncertainty intervals as well as an indication of informative bands for each variable (Rasmussen and Williams, 2005; Verrelst et al., 2013).

More recently, Bayesian parameter inference has been adopted to estimate biophysical variables in RTM with promising results (Schraik et al., 2019; Varvia et al., 2017; Varvia et al., 2018; Zhang et al., 2005). In the Bayesian approach, all variables are considered as random, and prior information about them can be introduced to constrain their estimation. This is particularly useful when estimating a large number of RTM input variables (Shiklomanov et al., 2016). A great advantage of the Bayesian approach is that it outputs a posterior distribution rather than a point estimate, i.e., the conditional probability distribution of the unknown variables given the observed reflectance, which statistically provides the uncertainty range of the inversion results. The quantification of uncertainties in the estimated GAI is highly required when assimilating remote-sensing GAI into crop models (Chen and Tao, 2020; Cressie et al., 2009). Moreover, like LUTs, Bayesian methods allow estimating all the model input variables at once, and their joint posterior distribution permits to describe possible compensation effects among them.

As mentioned before, the RTM model inversion constitutes an ill-posed problem (Combal et al., 2003; Zurita-Milla et al., 2015). To mitigate this problem, the use of ancillary information or direct observations of several model input variables is critical to prevent possible robustness and convergence problems. However, this information is often not available, and furthermore, some RTM input variables such as the hot-spot parameter corresponding to the ratio between the average size of the leaves and canopy height in the SAIL model (Verhoef, 1984), or the leaf structure parameter (N) corresponding to the number of equivalent elementary layers in PROSPECT (Jacquemoud and Baret, 1990), cannot be directly measured and are called parameters for this reason. In addition, the RTM assumptions can lead to considerable biases in the inversion process (Duveiller et al., 2011). Particularly for true GAI retrieval, the turbid medium assumption of most RTMs can introduce biases for those crops, like maize, with a canopy architecture that departs from turbid medium (López-Lozano et al., 2007).

This study proposes an original semi-empirical approach based on the Bayesian theory to overcome the above-mentioned problems of classical RTM inversion. It will be applied to the widely known PROSPECT+SAILH model (or PROSAIL), which has been considered as a reference model for the remote sensing retrieval of GAI (Jacquemoud et al., 2009). Our approach assumes a number of PROSAIL variables being crop-specific, and therefore, can be calibrated using a training set from GAI and reflectance observations. The first aim of this study is to retrieve the joint distributions of these crop-specific input variables using a Bayesian Monte-Carlo algorithm. Special attention will be paid to analyse the covariance and compensation effects among the crop-specific input variables. Secondly, those calibrated joint distributions will be applied to predict GAI in two ways: (i) as prior information in a full Bayesian approach to estimate GAI from reflectance, (ii) as inputs to generate the training database for an ANN to predict GAI in an operational context. The corresponding GAI predictions will be compared against a reference ANN-based algorithm: the Sentinel Simplified Level 2 Product Prototype Processor embedded in the ESA Sentinel Application Platform (SNAP, Weiss and Baret, 2016). Finally, we discuss the benefits of the proposed algorithm to identify the distributions of crop-specific PROSAIL variables that help to minimize bias of GAI predictions; and how those distributions compensate, implicitly, the model assumptions of PROSAIL– to predict true GAI. In particular, given that Bayesian inversion algorithm requires a massive volume of simulations to derive the posterior distributions, an emulation strategy of the PROSAIL model will be proposed and evaluated to improve the computational efficiency.