## 1 Introduction  
Terrestrial gross primary production (GPP) is the largest flux in the global carbon cycle [1]. Anthropogenic disturbances to natural cycles include CO\({}_{2}\) emission to the atmosphere, but terrestrial vegetation provides a sink for a part of the emitted CO\({}_{2}\)--driven primarily by the fact that the CO\({}_{2}\) concentration is nonetheless rising [2]. Currently our best, in situ estimates of GPP are provided by the eddy-covariance method that measures high-frequency net carbon fluxes between the canopy and the atmosphere [3]. Those measurements of net ecosystem exchange (NEE) can be partitioned into GPP and ecosystem respiration allowing empirical evaluation of model simulations of GPP. There are now hundreds of flux-tower sites [4] and collectively they provide the foundation and benchmark for global terrestrial vegetation monitoring. The technique has its difficulties, however--including how best to partition NEE into the component fluxes [5], the need for a reasonably flat and homogenous footprint [6] and under-representation of certain biomass (notably tropical biomes). These problems, together with the desire to provide spatially continuous estimates, have encouraged the development of remote-sensing models where global fields of key drivers are provided by satellite-derived products. The overall goal of such research is to develop a global GPP monitoring system, based on a reliable model, driven by remotely sensed data and evaluated against flux-tower data. The instantaneous controls of leaf-level photosynthesis are described by the FvCB model of C\({}_{3}\) photosynthesis [7] in which instantaneous carbon assimilation is limited either by Rubisco activity (carboxylation) or by light (electron transport). Both rates are influenced by the leaf-internal partial pressure of CO\({}_{2}\) (C\({}_{i}\)). Empirical equations are commonly used to determine C\({}_{i}\), which depends on stomatal behaviour. Current eco-physiological theory, and most biophysical land-surface schemes for climate modelling, make use of the FvCB model. The instantaneous photosynthetic response to light, as observed experimentally and represented in the FvCB model, is a non-linear (saturating) function. A different relationship with light emerges, however, at longer timescales [8; 9]. Based on measurements of crop growth, [PERSON] [10; 11] proposed the light-use efficiency (LUE) model (GPP = LUE \(\times\) IPAR \(\times\) fAPAR) where IPAR is the incident photosynthetically active radiation and fAPAR the fraction absorbed by green vegetation. There are now many LUE models in the literature [12], but no consensus on which is best. There has also been a marked tendency for these models to become more complex and to require more parameters, but without necessarily increasing in accuracy [13]. The P model [14] is a LUE model based on optimality principles that coordinate capacities for CO\({}_{2}\) fixation, water- and electron-transport to simulate GPP consistent with the FvCB framework. The model can be applied universally to C\({}_{3}\) plants with no need for biome-specific parameters. The P model as deployed here is the 'BRC' version presented by [PERSON] et al. [15] with a modification [16] to describe C\({}_{4}\) photosynthesis. This version differs from the original model of [PERSON] et al. principally by representing the temperature-dependence of the intrinsic quantum efficiency of photosynthesis, which was described by [PERSON] et al. [17], but is neglected in many current models [18]. [PERSON] et al. (2020) also introduced a single global calibration parameter to optimize agreement between P model simulations and flux data. In a departure from the approach of [PERSON] et al., we have here assigned the maximum quantum efficiency its theoretical upper bound of 0.125 (eight quanta per CO\({}_{2}\) reduced), eliminating the need for calibration. GPP is expected to be influenced by temperature because of the various temperature-dependent quantities that affect photosynthetic rates [17; 19].