1 Introduction

Models are abstract and simplified representations of reality; we use models when they are easier to work on than a real system (Smith and Smith 2009). Models are widely used in studying plant stable isotopes by including processes important to tree physiology and environmental biophysics. Model complexity spans from the conceptual to detailed mathematical representations of system fluxes. Conceptual or diagrammatic models in this book use diagrams or word descriptions instead of equations, such as conceptual models that describe the links between δ13C and δ18O in plants (Grams et al. 2007; Scheidegger et al. 2000; Chap. 16). Classic mathematic models for plant stable isotopes represent the processes in stable isotope fractionations, such as the widely used model for δ13C developed by Farquhar and colleagues (Farquhar et al. 1982, 1989; Chap. 9) and the Craig-Gordon model for δ18O of liquid water pools undergoing evaporation (Craig and Gordon 1965; Farquhar et al. 2007; Chap. 10). Moreover, many other equations in this book explicitly describe processes in stable isotopic fractionation driven by physiology, and/or environmental biophysics. These equations are mechanistic mathematical models; each equation describes a unique aspect underlying the variations of tree-ring stable isotopes.

Because many isotopic models for specific processes are explicitly described in previous chapters, we will not repeat that work in this chapter. Instead, we focus on larger and more complex models that integrate these equations and our knowledge to simulate tree-ring stable isotopes based on processes of water uptake, gas exchange, and tree-ring formation. These models simulate the interactions between environment and plant, and integrate major processes in tree physiology, including photosynthesis, respiration, transpiration, carbon allocation, tree-ring formation, and the corresponding fractionations of stable isotopes in different tissues. One might think that an ideal process-based model would include all the processes to fully represent the variation in tree-ring stable isotopes; however, such a model would be impossible to parameterize (i.e. to determine appropriate values for an accurate quantitative description of a process). Therefore, all models must simplify process representation; the simplification strategies should match the objectives and constraints of the modeling exercise. We cover a range of tree-ring stable isotope models optimized for different objectives from those optimized for gas-exchange fluxes to those for carbon allocation addressing issues of carbon storage and timing of ring formation.

2 How Tree-Ring Stable Isotopes Help Modeling of Ecosystem Processes

Process-based forest modeling requires significant amounts of data including meteorological and environmental information to drive models, and physiological and mensurational data to parameterize and validate the models. Data shortage—the lack of and need for detailed and long-term data—is a common problem in many forest modeling applications. Tree-ring stable isotopes can alleviate this need because they can serve as a proxy for some key data requirements, especially long-term physiological information (van der Sleen et al. 2017). Tree rings store decadal- to millennial-scale information about how plants responded physiologically to climatic and environmental changes as well as ontogenetic and size changes. Additionally, the time step of tree-ring records is limited only by our extraction capabilities as detailed intra- and inter-annual information is stored within tree rings (Gessler et al. 2014; McCarroll and Loader 2004; Chaps. 7 and 15). In contrast, observational meteorological records began only ~200 years ago, and studies of plant physiology began only in the last several decades. Moreover, tree-ring stable isotopes are relatively easy to obtain, and sample preparation and measurement procedures have been well developed (Chaps. 4–7). Published stable isotope chronologies are also readily available for modelers (e.g. compiled lists in Keller et al. 2017; McCarroll and Loader 2004).

Tree-ring stable isotopic data are ideal for testing model structure and parameterizing complex models. Models of ecosystem processes (e.g., photosynthesis and transpiration) simulate key physical and biochemical processes within the system and often lead to conclusions that are not intuitive. However, as models have become more comprehensive, they are not only more difficult to parameterize, but also more difficult to test. When models become too complex, we begin to rely on empirical calibrations of simplified models to parameterize the complex model. However, with limited available data used to calibrate and test the model, parameters are often tuned without reference to mechanistic constraints. This practice is dangerous when many parameters need to be tuned, but few observations are available. Tuning can “successfully” fit simulations to observations in a way that does not faithfully represent nature; thus, we get the “right” answer for the wrong reasons. As addressed by Aber (1997): “Perhaps one of the worst characteristics of a calibrated model is that it cannot fail…When models cannot fail, we cannot learn from them”. Even worse, an incorrectly parameterized model would likely yield erroneous answers. Therefore, modelers must (1) test the mechanistic basis of models and (2) calibrate the models within mechanistic constraints. Tree-ring stable isotopes may help to solve this dilemma as they can be related to underlying mechanisms (Fig. 26.1). A prerequisite for successful modeling remains the sound knowledge of the physiological processes and their responses to environmental drivers.

Fig. 26.1
figure 1figure 1

Simplified schemes of simulating tree-ring stable C (δ13C) a and O isotopes (δ18O) b, and using them as tests for modeling. Major components in process-based physiological models are included in gray boxes. The blue box indicates where stable isotopes can be used for model testing, either providing information for modeling calibration or validating the model. Post-photosynthetic fractionation (PPD) and storage are modeled less comprehensively than other physiological processes. Explicit PPD (I) may include fractionations in respiration, transportation, allocation, and organic matter synthesis. Mixing of stored and new carbohydrates further complicates the simulations of stable isotopes. Therefore, some models apply a simplified PPD scheme (II), such as a fixed offset between stable isotopes in new photosynthate and tree rings. Blue solid lines indicate comparison and testing, and blue dotted lines indicated post-photosynthetic processes. Chapter (CH) numbers are marked in boxes to indicate locations of corresponding contents in this book

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Accurate simulations of δ13C and δ18O require appropriate parameterization of key equations that describe both the fluxes and the isotopic values (Fig. 26.1). Small changes in parameters can result in similar flux values but may create large discrepancies between measured and modeled stable isotopes (Wei et al. 2014a), thus the stable isotopes provide a test for the calibration process. For example, Fig. 26.2 illustrates how a stable isotope component could test alternative model formulations. The simulations were based on a ~80 year-old natural stand of grand fir (Abies grandis) in the Mediterranean climate of northern Idaho, where low summer precipitation limits gas exchange (Wei et al. 2014a). Three scenarios in the example represent alternative means to parameterize the model: (1) using default gas exchange parameters; (2) calibrating the model with all available sap flow observations from a nearby site with shallower soil; and (3) calibrating the model with part of the sap flow observations until the soil dried out. Scenario 1 did not reasonably simulate transpiration and the response of canopy conductance to vapor pressure deficit (VPD), but Scenarios 2 and 3 did (Fig. 26.2a, b). Although the differences were small between Scenarios 2 and 3 in both the calibrated parameter values and simulated transpiration, simulated tree-ring δ13C were clearly separated (~2‰) between Scenarios 2 and 3, and only Scenario 3 matched the δ13C observations (Fig. 26.2c). With δ13C as a constraint, small parameterization errors in gas exchange can be detected, increasing our confidence in the simulations under Scenario 3. Note that the isotopic data were used here not as parameterizations themselves, but as a test of other means of parameterization. As this example illustrates, stable isotopes can serve as a powerful means to test alternative parameterization strategies (Aranibar et al. 2006; Duarte et al. 2017; Ulrich et al. 2019; Walcroft et al. 1997; Wei et al. 2014a).

Fig. 26.2
figure 2

Example of using tree-ring δ13C to validate a process-based forest model (3-PG; Wei et al. (2014a) for more details). Three scenarios demonstrate possible means to calibrate a model, where three parameters differed among scenarios, including maximum canopy conductance (gcmax, m s−1) the slope of canopy conductance to vapor pressure deficit (VPD; kg, mbar−1) and quantum yield (αcx, mol C mol−1 photon). Scenario 1 used the default parameter set for simulations (gcmax = 0.020, kg = 0.05, αcx = 0.055). Scenario 2 calibrated these parameters with all available sap flow observations at a nearby site with similar forest type but where the soil dried earlier than at the modeling site (gcmax = 0.011, kg = 0.08, αcx = 0.0615). Scenario 3 calibrated the model using only part of the sap flow observations at this nearby site before the soils dried, which should be the most reasonable approach (gcmax = 0.014, kg = 0.08, αcx = 0.0615). Although Scenario 2 and 3 had very similar simulations of transpiration a and similar VPD response curve (b “observed” shows the 99% percentile of all sap flux-based measurements), only Scenario 3 reasonably simulated tree-ring δ13C c. Because reliable δ13C simulations require accurate simulations of photosynthesis and stomatal conductance (Eq. (26.2)), simulations of gas exchanges of Scenario 3 are hence more reliable. We can then accept the simulated GPP of Scenario 3 as a better simulation than the other two scenarios d. This figure is redrawn from Wei et al. (2014a), with permission

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3 Components for Modeling Tree-Ring δ13C and δ18O

Our knowledge about causes of variation in tree-ring δ13C and δ18O has continuously improved (Chaps. 9, 10, 16 and 17). The key processes which have been commonly included in process-based modeling of tree-ring δ13C and δ18O can be broken into various components such as fractionation during carbon fixation within leaves, during carbon allocation, or during cellulose formation within the stem (Fig. 26.1). Models tend to focus on one or two of these boxes depending on the model objectives, but the level of process detail varies greatly between models. As mentioned earlier, more complex modeling requires more detailed data for parameterizing and testing the models. Model predictions are generally validated with data either after simulating leaf-level fractionation or after cellulose formation within the stem. At the leaf level, simulated δ13C can be tested with phloem δ13C, and simulated water δ18O in tissues can be assessed with leaf or stem water observations (Fig. 26.1). However, these observations are normally short-term and non-continuous. Tree-ring stable isotopes provide long-term and continuous data that can be used to validate models over years of simulation.

Leaf-level photosynthetic fractionation for δ13C–CO2 is relatively well understood. Models for photosynthetic fractionations of δ13C–CO2 vary in complexity from “simplified” to “comprehensive” and modelers should pick the model that serves their purpose (Ubierna and Farquhar 2014). The comprehensive model includes fractionations related to Rubisco, stomatal conductance, mesophyll conductance, respiration, photorespiration, and ternary effects (Farquhar et al. 1982; Farquhar and Cernusak 2012; Ubierna and Farquhar 2014). The comprehensive model requires many parameters, some of which (e.g. mesophyll conductance) are difficult to estimate. Therefore, few process-based models for tree-ring δ13C have applied the comprehensive model (e.g. Eglin et al. 2010; Rogers et al. 2017). A simplified model is more widely used in studies that simulate tree-ring δ13C; it highlights the control of intercellular CO2 concentration (ci) on fractionation and lumps the impact of other factors into a single parameter that is mostly determined by Rubisco carboxylation (b, 27‰) (Farquhar et al. 1982). δ13C of new photosynthate can then be approximated as:

$${\delta }^{13}C\approx {\delta }^{13}{C}_{a}-a-(b-a)({c}_{i}/{c}_{a})$$
(26.1)

where δ13Ca is the δ13C of atmospheric CO2, a is the fractionation from diffusion through air (4.4‰), and ca is the atmospheric CO2 concentration. Most components in Eq. (26.1) can either be treated as constants (a, b) or are easily obtained from observations (Ca and δ13Ca). In process-based models, ci is driven by the photosynthetic rate (A) and the stomatal conductance to CO2 (gs) (Farquhar and Sharkey 1982) as follows:

$$A/{g}_{s}=({c}_{a}-{c}_{i})$$
(26.2)

Therefore, process-based models can be conveniently enabled to simulate δ13C of photosynthate.

Another key source of variation for simulating δ13C is atmospheric CO2; both ca and δ13Ca in Eq. (26.1) are changing due to human activities. ca has increased and δ13Ca decreased due primarily to the burning of fossil fuels (Francey et al. 1999, see also Chap. 9). Additionally, ca and δ13Ca vary seasonally, mainly due to seasonal changes in global photosynthesis (Keeling et al. 2017). Tree-ring stable isotopes records have been used extensively to understand both the secular change in CO2 and plant physiological responses to this change (Chap. 24). However, the concurrent ontogenetic changes in tree age and size are difficult to separate from CO2 changes over time (Monserud and Marshall 2001). Models have the potential to separate these temporal concurrent influences on tree-ring isotopic records (see Sect. 26.5).

Less well understood are fractionations in other tissues (i.e., post-photosynthetic fractionations) (Badeck et al. 2005; Gessler et al. 2014; Offermann et al. 2011; Chap. 13). These include fractionations upon phloem loading (Busch et al. 2020) and unloading, composition changes in phloem contents (Bögelein et al. 2019), leakage and refilling during transport (Gessler et al. 2014; Offermann et al. 2011), and biosynthetic processes during wood formation (Bowling et al. 2008). We do not assess these in detail here as they were already presented in Chap. 13. However, models can assess likely consequences of these fractionations in the tree-ring data (Danis et al. 2012; Eglin et al. 2010).

Stored carbohydrates from previous years are sometimes used to build wood of the current year (Barbour and Song 2014; Chap. 14); this phenomenon is called the carry-over effect, legacy effect, or time-lag effect (Fritts 1966; Zweifel and Sterck 2018). Modeling the carry-over effect requires simulations of the dynamics of carbon storage and their impacts on tree-ring stable isotopes over time. Such components have been included in some process-based models (e.g. Danis et al. 2012; Eglin et al. 2010; Hemming et al. 2001) (see Sect. 26.4). The results of simulating the carry-over effect have varied among studies. For example, storage did not play a significant role in simulating tree-ring δ13C for Pinus halepensis (Klein et al. 2005), while it was important in some other studies that simulated tree-ring δ13C with Pinus arizonica and Fagus sylvatica (Hemming et al. 2001; Skomarkova et al. 2006).

Simulations of tree-ring δ18O are mostly extensions of the Craig-Gordon model, which describes evaporative enrichment at the evaporation site in the leaf and exchange with atmospheric water vapor (Fig. 26.1b, Chap. 10). These predictions are often corrected for the Péclet effect, which describes the diffusion of enriched water into the unenriched xylem water (Farquhar and Lloyd 1993; Barbour 2007; Barbour et al. 2004; Cernusak et al. 2016). This back-diffusion of water from the evaporation site isotopically enriches bulk leaf water where carbohydrates are formed. The Péclet effect is directly influenced by transpiration, whereas the Craig-Gordon model is mainly influenced by relative humidity. These models may also include isotopic variation due to phloem transport (Gessler et al. 2014; Chap. 10) and the impact of δ18O in water vapor on δ18O in plant water and organic compounds (Lehmann et al. 2018, 2020). Models of δ18O vary in the detail of the δ18O evaporative enrichment process in leaf water, and one should choose a model that is most suitable to specific objectives (Cernusak et al. 2016). The influence of the Péclet effect should be damped in simulating flows downstream from leaf water pools, so it may not be necessary to include it in simulating tree-ring δ18O (Cernusak et al. 2016; Ogée et al. 2009). However, including the Péclet effect did improve simulations of tree-ring δ18O in at least one study (Ulrich et al. 2019). All models of δ18O require information on the δ18O of water taken up by the trees (see Chap. 18).

Accurate modeling of tree-ring stable isotopes may require allocating the correct amount of new biomass in the tree stem at the right time to integrate seasonal influences into isotopic values measured within the rings (Chap. 14). Therefore, these models should reasonably simulate tree radial growth across the growing season (tree-ring width, the increment of basal area, or tree diameter). Simulating radial growth requires not only reasonable simulations of the carbon supply, but also reasonable descriptions of allocating new and stored carbohydrates to tree-rings, and the phenology of ring production. Process-based models with reasonable simulations for δ13C and δ18O have been used to validate simulations of productivity (see Sect. 26.2), which benefited the simulations of radial growth. Several examples (Danis et al. 2012; Ogée et al. 2009; Ulrich et al. 2019) have reasonably simulated both radial growth and stable isotope composition of the ring (see in Sect. 26.4.3).

Scenario and sensitivity analysis of tree-ring isotope models can also provide important insights into potential causes of variation, both during research planning and in post hoc analyses of tree-ring data. For example, Wei et al. (2018) used alternative modeling scenarios to conclude that elevation differences in temperature had little direct effect on productivity in a montane watershed. Lin et al. (2019) analyzed drought effects on δ18O in tree rings, using a sensitivity analysis to conclude that a change in source water was the most probable explanation for the observed δ18O response. Given the many sources of variation in δ18O, this kind of formal analysis may be especially valuable.

Perhaps even more valuable is the notion of using models in research planning. For example, had the Lin et al. (2019) analysis been done before the measurements were conducted, perhaps measurements of xylem water δ18O would have been prioritized. This idea of close collaboration between modelers and experimentalists, from research planning to publication, is common in other disciplines and has begun to take hold in ecology (Medlyn et al. 2015, 2016). Perhaps the tree-ring isotope community has room to grow in this direction.

4 Models that Simulate Tree-Ring Stable Isotopes

Several process-based models have been used to predict stable isotopic composition at varying levels of complexity. At a small spatial scale, simple leaf- or stand-level models can simulate forest physiology and growth with simple stand structures; at the other end, more complex models can simulate forests with different species, tree sizes, and canopy structures at watershed, regional, and global scales. Here we describe several representative models that have advanced our abilities in modeling tree-ring δ13C and δ18O. All these models have been used to simulate tree-ring stable isotopes for more than one year and have been tested with observed isotopic data. Most models here could simulate time series of tree-ring stable isotopes, but we also include models that simulate δ13C in new photosynthate and/or δ18O in foliage water as they have the potential to also predict tree-ring isotope time series after simple upgrades or modifications.

4.1 Models for Predicting δ13C of Foliage and Tree-Rings

Panek and Waring 1997

Panek and Waring (1997) simulated foliage and tree-ring δ13C in a semi-mechanistic way. A process-based model (FOREST-BGC, Running and Coughlan 1988) was used to capture climate constraints on gs. δ13C was modeled empirically based on multiple linear regressions between δ13C and climatic constraints on stomata. The study tested how much variance in tree-ring δ13C can be explained by climate controls on gs, neglecting all other sources of variation discussed above. The model explained 34% (r = 0.58) of the annual variation in tree-ring δ13C across 22 years for Douglas-fir in Oregon, USA.

Walcroft 1997

Walcroft et al. (1997) combined process-based modeling with tree-ring δ13C to expand our knowledge in the connections between long-term climate and tree-ring stable isotopes. This study also indicated the possibility of using tree-ring δ13C to validate models. Driven by daily meteorology data, a leaf-level model simulated stomatal conductance, leaf photosynthesis, and stand water balance for Pinus radiata over two growing seasons at two sites in New Zealand. The authors used the gas-exchange data to simulate seasonal variations in leaf-level ci. They compared this to seasonal canopy-level ci estimated from δ13C of intra-annual tree-ring slices using the simplified equation from Farquhar et al. (1982) (Chap. 9). The two sets of ci had similar seasonal variations except that simulated ci varied over a larger amplitude. This indicated the importance of post-photosynthetic processes, including mixing of stored carbohydrates s, in damping the signals of gas exchange in tree-rings.

SICA

Trees may adjust their stomatal regulation of ci in response to rising atmospheric CO2 (Voelker et al. 2016), which would change water-use efficiency and tree-ring δ13C. In one of the early attempts to describe these phenomena, Berninger et al. (2000) modeled annual variations of tree-ring carbon isotope values for a long period (1877–1995) using a modified version of the SImple CAnopy model (SICA) model (Berninger 1997). Two kinds of stomatal responses to elevated CO2 were used: a sensitive scheme that reduced gs with increasing CO2 and an insensitive scheme that did not. It turned out that the insensitive scheme better predicted tree-ring carbon isotope values with a correlation coefficient (r) between simulations and observations as high as 0.74. In contrast, r reached only 0.58 with the sensitive scheme. This indicated that gs had not decreased with increasing CO2 and Δ13C increased since the 1950s. The SICA model did not consider ontogenetic effects or post-photosynthetic fractionations; therefore, there was a 4.6‰ offset between simulated photosynthetic Δ13C and observed Δ13C in tree-ring cellulose.

TREERING

The TREERING model (Fritts et al. 1999; Hemming et al. 2001) was among the first to incorporate post-photosynthetic discriminations in simulations of tree-ring δ13C. It included the impact of daily cambium activity and carbon storage on the simulations of tree-ring δ13C. The dynamics of two carbohydrate pools were simulated: sucrose as new photosynthate and starch as storage. Sucrose is first spent each day on respiration and growth. If there were remaining sucrose, a fixed percentage of remaining new photosynthate was allocated to starch storage in each day. The results indicated that the stored carbon had a greater impact on tree-ring δ13C than new photosynthate in 16 years of Pinus arizonica in Arizona USA (Hemming et al. 2001).

ISOCASTANEA

ISOCASTANEA (Eglin et al. 2010) is one of the most comprehensive models to simulate tree-ring δ13C. It captures how short-term δ13C signals at the leaf-level are integrated into tree rings. It incorporates two procedures that are currently rare in simulating plant stable isotopes and results in better estimates of post-photosynthetic fractionation with explicit carbon dynamics. First, the model simulates post-photosynthetic translocation and allocation of carbohydrates based on the physiological processes in the phloem. These processes included the phloem loading in canopy and unloading in stem and roots, and mass flow in sieve tubes which include volume and conductance of phloem elements. This model also simulates the dynamics of storage, respiration, and structural biomass formation, which make its post-photosynthetic carbon dynamics more explicit than many other similar models. Second, tree rings are not formed instantaneously in the ISOCASTANEA model. Tree-ring formation is separated into three main stages in nature: cambial cell division, cell expansion, and cell-wall thickening (Eglin et al. 2010; Samuels et al. 2006). The model simulates tree-ring growth by dividing tree rings into growth layers formed continuously in the cambial division stage. After formation, carbohydrate allocation to the growth layer decreases exponentially over time. The later phases of growth represented lignification and cell wall thickening.

4.2 Models for Predicting δ18O of Leaf Water and Tree Rings

Roden–Lin–Ehleringer 2000

The Roden–Lin–Ehleringer model (Roden et al. 2000) simulates δ18O and hydrogen isotope ratios (δ2H) in tree-ring cellulose. Two major components of the model predict the evaporative enrichment and vapor exchange in leaf water using the Craig-Gordon model, and isotopic exchange during cellulose synthesis. Stable isotope ratios simulated by the Roden–Lin–Ehleringer model were compared to observed values from either hydroponically grown trees in the lab (Roden and Ehleringer 1999a) or trees grown along stream banks at field sites (Roden and Ehleringer 1999b). The lab trees were grown in altered water vapor δ18O and hydroponic solution δ18O. Simulated δ18O had a good 1:1 relationship with observed δ18O from tree-ring cellulose across experiments. This model is so simple that it might be included as an algorithm inside many existing models of more complexity.

Péclet Effect

The introduction of the Péclet effect into modeling leaf and tree-ring δ18O directly included transpiration effects into leaf water enrichment models (Farquhar and Lloyd 1993). However, the Péclet effect also introduced an effective path length variable (L), which describes the distance water moves inside the leaf lamina and is not directly measurable. Using the same data used to develop the Roden-Lin-Ehleringer model above, Barbour et al. (2004) simulated δ18O in leaf water and tree-rings respectively with and without a Péclet effect; simulations with the Péclet effect had better predictive power than without it. Modeled differences in leaf water across 17 Eucalyptus species was largely explained by varied L across species; however, L was highly unconstrained (Kahmen et al. 2008). Process modeling that predicts transpiration can facilitate estimating L values (e.g. Loucos et al. 2015; Song et al. 2013; Ulrich et al. 2019). However, Kahmen et al. (2011) found that most variance in leaf and tree-ring δ18O was related to vapor pressure deficit, regardless of the model complexity used to predict it. The debate about including the Péclet effect in tree-ring δ18O models continues, and largely depends on modeling objectives. Like the Roden-Lin-Ehleringer model above, the Péclet effect is simple enough to be included in a model with a few lines of code. More details can be found in Chap. 10.

4.3 Models for Predicting Both δ13C and δ18O of Tree Rings

Single- Substrate Model

Ogée et al. (2009) simulated tree-ring formation from a single well-mixed water-soluble sugar pool that varied in size and δ13C and δ18O signatures. This pool was assumed to be large enough to fulfill the metabolic demand in the growing season, and was filled with new photosynthate and drained by respiration and biomass formation. The dynamics of the carbon pool accompanied the changes in δ13C and δ18O in this pool, which was controlled by the isotopic fractionation of photosynthesis, respiration, and biomass synthesis (Ogée et al. 2009). In a 2 year study with two Pinus pinaster trees in France, the single substrate model relied upon model simulations of photosynthesis and respiration from the well-calibrated MuSICA model (multi-layer simulator of the interactions between a coniferous stand and the atmosphere) to predict stable isotopes in tree-ring cellulose (Ogée et al. 2003a). MuSICA is a process-based multilayer gas exchange model, which simulates the fluxes of energy, water, and CO2 through soil, vegetation, and atmosphere (Ogée et al. 2003a, b). The single substrate model achieved reasonable simulations for seasonal variation in δ13C and δ18O when compared with measured values from 100-μm thick samples within tree rings.

3-PG with Isotopic Modules

A widely used forest model, 3-PG (Physiological Principles Predicting Growth) (Landsberg and Waring 1997), has been modified to simulate tree-ring δ13C and δ18O. 3-PG is a big-leaf model describing physiological principles applied to the whole canopy. The model estimates gross primary production (GPP) and a VPD-dependent canopy conductance. The modification by Wei et al. (2014a) used these parameters to calculate δ13C via the simplified (Farquhar et al. 1989) model. The difference between δ13C in new photosynthate and the tree-ring was assumed as a constant offset to bypass the simulations of post-photosynthetic fractionation and keep the model simple (Wei et al. 2014a). The model has also been modified to estimate tree-ring δ18O (Ulrich et al. 2019). Here the modification included δ18O enrichment at the site of evaporation, the Péclet effect, and exchange and fractionation during cellulose formation. The fidelity of modeled GPP and canopy conductance can then be tested by comparing simulated tree-ring δ13C and δ18O with observations; this approach greatly constrained the parameter space and avoided artifacts in simulations of gas exchange, which improved the reliability of this model (Ulrich et al. 2019; Wei et al. 2014a). The modified 3-PG model has also been used to estimate variations of tree-ring δ13C across an elevational gradient in a watershed (Wei et al. 2018) and to estimate responses to forest management practices on tree-ring δ13C (Wei et al. 2014b) (also see Chap. 23).

MAIDENiso

MAIDENiso (Danis et al. 2012) was designed specifically for modeling tree-ring width and stable isotopes. This upgraded version of the MAIDEN model (Misson 2004) captures ecophysiological processes to simulate carbon allocation to the stem as well as δ13C and δ18O of tree-ring cellulose. Storage of carbon was explicitly modeled with carbon reservoirs in leaf, bole, and root (Boucher et al. 2013; Misson 2004); this enabled carbon storage impacts to be included on stem growth, δ13C, and δ18O simulations. MAIDENiso was able to simulate tree-ring cellulose δ13C and δ18O with r above 0.5 between simulations and observations for both δ13C and δ18O in an oak species in France (Danis et al. 2012), and 0.52 and 0.62 for δ18O in Picea mariana in northeastern Canada and Nothofagus pumilio in western Argentina (Lavergne et al. 2017).

A notable use of MAIDENiso applied an inverse modeling approach to change the meteorological data and see if the changes would provide a better fit of simulated tree-ring width, δ13C, and δ18O to their respective observations (Boucher et al. 2013). This approach was designed to find the best combination of the climatic inputs as a robust paleoclimatic reconstruction. This model also enabled the separation of climatic impact and CO2 influences on plant growth, which is useful for paleoclimatic reconstructions (Boucher et al. 2013; Guiot et al. 2014).

5 Future Directions

Better modeling of tree-ring stable isotopes relies on the improvement and incorporation of physiological mechanisms that influence plant isotopic values. A particular challenge that modeling could address relates to including ontogenetic changes in CO2 responses. Ontogenetic change is often attributed to age in the tree-ring community, perhaps because age is easiest to measure from tree-ring chronologies. However, no evidence of an age effect exists per se in trees (e.g. Mencuccini et al. 2005), except as caused by increasing tree height. Tall trees generally face a challenge in lifting water to the canopy, which causes an increase in δ13C with height (Brienen et al. 2017; Marshall and Monserud 1996; McDowell et al. 2011; Voelker et al. 2016). These issues are particularly important when tree-ring δ13C is used to infer responses to CO2 because the CO2 effect and the height effect are often confounded. Future work will need to remove the height signal from tree-ring data to draw solid inferences regarding the CO2 effect (Brienen et al. 2017; Marshall and Monserud 1996). Similarly, the proportion of oxygen atoms exchanged with xylem water upon cellulose formation may be variable (Cheesman and Cernusak 2016) and may need to be accounted for in the traditional model of δ18O in tree rings (Roden and Ehleringer 1999b). As mesophyll conductance becomes easier to determine, there may be value in modeling how it differs among species and growing conditions; this critical parameter influences the relationship between intrinsic water-use efficiency (iWUE) and δ13C. It describes the drawdown in concentration between the intercellular spaces, where iWUE is controlled, to the chloroplasts, where δ13C is set by photosynthesis (Stangl et al. 2019). Finally, the continuing debate about the many forms of post-photosynthetic fractionation will certainly continue to modify our tree-ring isotope models (Gessler et al. 2014; Chap. 13).

Another way to promote tree-ring stable isotope modeling is creating comprehensive databases for tree-ring stable isotopes chronologies. Although sources exist for such chronologies (e.g. compiled lists in Keller et al. 2017; McCarroll and Loader 2004), these data have not been well archived into a single database. The International Tree-Ring Data Bank (ITRDB) (Grissino-Mayer and Fritts 1997) had stable isotope measurements from only 24 sites in 2017 (Babst et al. 2017). The global TRY database had only 59 observations of wood δ13C and none of wood δ18O data in 2020 (Kattge et al. 2020). In comparison, the TRY database includes ~15,000 observations from 468 species for leaf δ13C and 533 observations from 96 species for leaf δ18O in 2020. Moreover, data from 4200 sites are available for tree-ring width (Babst et al. 2017), which made tree-ring width chronologies more readily available to facilitate testing models than stable isotopes. Therefore, we suggest archiving of stable tree-ring stable isotope chronologies whenever possible.

Tree-ring stable isotopes can also provide benchmarks for dynamic global vegetation models (DGVMs) in the future. DGVMs attempt to represent the dynamics of the global biosphere (Schulze et al. 2019). They integrate our understanding of physical, chemical, and biological processes of terrestrial ecosystems at varied scales from sites (represented as grids) to the globe. Several of these models have been upgraded to simulate plant stable isotopes, including SiB2 (Suits et al. 2005), ISOLSM (Aranibar et al. 2006; Riley et al. 2002, 2003; Still et al. 2009), LPJ-DGVM (Scholze et al. 2003), JULES (Bodin et al. 2013), ORCHIDEE (Churakova et al. 2016; Risi et al. 2016), LPX-Bern (Keel et al. 2016; Keller et al. 2017; Saurer et al. 2014), and CLM (Duarte et al. 2017; Keller et al. 2017; Raczka et al. 2016). Although these DGVMs simulate δ13C in new photosynthate or leaves, and/or δ18O in leaf water, most of them (except LPX-Bern) do not simulate tree-ring stable isotopes at this point and therefore tree-ring stable isotopes have not been used for testing DGVMs. However, because of the modular structure of DGVMs, they can be conveniently upgraded to simulate tree-rings stable isotopes by adding new modules or modifying current ones.

Inverse modeling for paleoclimate is a new frontier of using tree-ring stable isotopes (Boucher et al. 2013; Guiot et al. 2014). This approach rejuvenated the use of tree-ring information for paleoclimate reconstruction. The reconstruction traditionally relies on empirical relationships between tree-ring chronologies (width and stable isotopes), which means only the most limiting climatic factors would have left clear signals in the tree-ring chronologies. Additionally, these traditional reconstructions have obtained the clearest climate signals from trees growing in an extreme environment. Inverse modeling revolutionized the reconstruction by applying process-based approaches. Many climatic factors can be tested at the same time by mathematically estimating their collective impact on forests and statistically testing and improving the climatic variables that are provided as a priori.

Modeling is an important tool to study ecosystem processes because it can integrate physiological and growth processes. Tree-ring isotopic records likewise integrate these processes so the two approaches are certain to continue to develop in tandem, each tested against the other. Moreover, as demonstrated above, stable isotopes are convenient testing for models, and including a stable isotope component in models is relatively simple. Models can add value to completed isotopic datasets by post hoc analyses of the likely controls over isotopic composition. We urge that more process-based forest models include a stable isotope component to help parametrize and validate the models and improve model credibility.