4  Next Steps

This tutorial provides the basic understanding and code to run the numerical P-Model coupled to an energy balance model. As shown in Chapter 3, there are a few short-comings to the model that can be addressed as listed below.

4.1 Refining Numerical P-Model

The optimization criteria in Equation 2.10 seems to fail to optimize \(J_{max}\) and has other peculiarities to it. To resolve this issue one could work on the following tasks:

  • The entire formulation for the cost criteria could be revised. Here are a few starting points:

    • Instead of optimizing gross assimilation, the traits could be optimized with respect to the assimilation rate they constitute (\(A_c\) for \(V_{cmax}\), \(A_j\) for \(J_{max}\))

    • Instead of taking the relative carbon costs, one could take the absolute costs.

    • Instead of using a cost-minimization approach, one could use a profit-maximization approach (Joshi et al., 2022).

    • Higher leaf temperatures directly affect respiration, which is not considered yet. But according to Michaletz et al. (2016) plants may purposefully ramp up their transpiration to avoid the carbon loss due to lethal temperature stress. Thus, one could consider net assimilation (gross assimilation - respiration) instead.

  • Earlier tests - not included in this tutorial - showed that the initially assumed value for optimization (the starting values for vcmax, jmax, gs in optimize_traits_and_costs()) can lead to different optimal values. An optimization problem should be independent of initialization conditions, so this chaotic behavior could be investigated.

  • The current model only applies to long-term acclimation conditions. One could couple an instantaneous photosynthesis model (see {rpmodel} package) to predict on shorter timescales. However, to couple an energy balance model to the instantaneous photosynthesis model, one would first need a numerical formulation of the latter.

  • The current implementation uses the {optimr} package, which may or may not be suitable. To test whether another optimization algorithm finds a better solution, one could implement the {gensa} package instead.

4.2 Refining Energy Balance Model

  • Not all parameters used in the current implementation may stay constant. For example, a leaf’s emissivity may depart from the chosen 0.95. One could add such variables to the function’s input and run a sensitivity analysis.

  • The current energy balance model is based on the Penman-Monteith equation but one could use the simpler Priestly-Taylor equation as well (e.g. as compared in Dong et al. (2017)).

  • The current energy balance model is based on the {plantecophys} package (Duursma, 2015). One could also implement and test the more exhaustive model implemented in the {tealeaves} package (Muir, 2019).

4.3 Test Cases for Model Evaluation

  • \(V_{cmax}\) and \(J_{max}\): One could use global leaf-measurements from Peng et al. (2021) to evaluate the prediction of \(V_{cmax}\) and \(J_{max}\) by the numerical P-Model. Also, using this data one could revise the cost factor \(c\) following the methods by Wang et al. (2017).

    • To predict \(V_{cmax}\) at the canopy-level, one could also implement the methodology by Jiang et al. (2020) and additionally predict land surface temperatures as explained next.

    • Note: The cost factor \(c\) is hard-coded for now. The P-Model functions would need to be adjusted to create a working calibration routine.

  • Land Surface Temperatures: To evaluate the prediction of leaf temperatures, one could use MODIS Land Surface Temperature data over dense ecosystems (high leaf area index) at FLUXNET sites without water stress (see Stocker et al. (2018)).

  • Long-Term Leaf Temperature Measurements: To evaluate whether the predicted leaf temperature resembles an optimal long-term leaf temperature, one could use long time-series of leaf temperature measurements from e.g. thermal imaging (Still et al., 2022). Another data source could be the photosynthesis-weighted leaf temperature that can be inferred from \(^{18}\)O isotopes (Helliker & Richter, 2008).

  • \(^{13}\)C Isotopes: To evaluate the prediction of the long-term ratio of internal to ambient CO\(_2\) concentrations (\(\chi\)), one could repeat the analyses done in Stocker et al. (2020). This could also be used to revise the cost factor \(\beta\).