A reduced order model to analytically infer atmospheric CO2 concentration from stomatal and climate data

To address questions related to the acceleration or deceleration of the global hydrological cycle or links between the carbon and water cycles over land, reliable data for past climatic conditions based on proxies are required. In particular, the reconstruction of palaeoatmospheric CO2 content (Ca) is needed to assist the separation of natural from anthropogenic Ca variability and to explore phase relations between Ca and air temperature Ta time series. Both Ta and Ca are needed to fingerprint anthropogenic signatures in vapor pressure deficit, a major driver used to explain acceleration or deceleration phases in the global hydrological cycle. Current approaches to Ca reconstruction rely on a robust inverse correlation between measured stomatal density in leaves (ν) of many plant taxa and Ca. There are two methods that exploit this correlation: The first uses calibration curves obtained from extant species assumed to represent the fossil taxa, thereby restricting the suitable taxa to those existing today. The second is a hybrid eco-hydrological/physiological approach that determines Ca with the aid of systems of equations based on quasi-instantaneous leaf-gas exchange theories and fossil stomatal data collected along with other measured leaf anatomical traits and parameters. In this contribution, a reduced order model (ROM) is proposed that derives Ca from a single equation incorporating the aforementioned stomatal data, basic climate (e.g. temperature), estimated biochemical parameters of assimilation and isotope data. The usage of the ROM is then illustrated by applying it to isotopic and anatomical measurements from three extant species. The ROM derivation is based on a balance between the biochemical demand and atmospheric supply of CO2 that leads to an explicit expression linking stomatal conductance to internal CO2 concentration (Ci) and Ca. The resulting expression of stomatal conductance from the carbon economy of the leaf is then equated to another expression derived from water vapor gas diffusion that includes anatomical traits. When combined with isotopic measurements for long-term Ci/Ca, Ca can be analytically determined and is interpreted as the time-averaged Ca that existed over the life-span of the leaf. Key advantages of the proposed ROM are: 1) the usage of isotopic data provides constraints on the reconstructed atmospheric CO2 concentration from ν, 2) the analytical form of this approach permits direct links between parameter uncertainties and reconstructed Ca, and 3) the time-scale mismatch between the application of instantaneous leaf-gas exchange expressions constrained with longer-term isotopic data is reconciled through averaging rules and sensitivity analysis. The latter point was rarely considered in prior reconstruction studies that combined models of leaf-gas exchange and isotopic data to reconstruct Ca from ν. The proposed ROM is not without its limitations given the need to a priori assume a parameter related to the control on photosynthetic rate. The work here further explores immanent constraints for the aforementioned photosynthetic parameter.