Franks et al. (2014) tested the model on four species of field-grown trees (three gymnosperms and one angiosperm) and one conifer grown in chambers at 480 and 1270 ppm CO2. The average error rate (absolute value of estimated CO2 minus measured CO2, divided by measured CO2) was 5 %. Follow-up work with three field-grown tree species (Maxbauer et al., 2014; Kowalczyk et al., 2018), CO2 experiments on seven tropical trees species (Londoño et al., 2018), and experiments on two fern and one conifer species (Milligan et al., 2019) indicate somewhat higher error rates (Fig. 1). Combined, the average error rate is 20 % (median 13 %).

In these studies, two of the key physiological inputs were measured directly with an infrared gas analyzer: the assimilation rate at a known CO2 concentration (A0) and/or the ratio of operational to maximum stomatal conductance to CO2 (gc(op)/gc(max), or ζ), the latter of which is important for calculating the total leaf conductance (gc(tot)). These two inputs cannot be directly measured on fossils; thus, the error rates associated with Fig. 1 may not be representative for fossil studies. Franks et al. (2014) argue that within plant functional types growing in their natural environment, mean A0 is fairly conservative, leading to the recommended mean A0 values in Franks et al. (2014) (12 µmol m-2 s-1 for angiosperms, 10 for conifers, and 6 for ferns and ginkgos). Along similar lines, the mean ratio gc(op)/gc(max) tends to be conserved across plant functional types; Franks et al. (2014) recommend a value of 0.2, which may correspond to the most efficient set point for stomata to control conductance (Franks et al., 2012). This conservation of physiological function is one of the underlying principles in the Franks model.

Here we test this assumption by estimating CO2 from 40 phylogenetically diverse species of field-grown trees. In making these estimates, we use the recommended mean values of A0 and gc(op)/gc(max) from Franks et al. (2014) instead of measuring them directly (see also Montañez et al., 2016, for other ways to infer assimilation rate from fossils). Thus, this dataset should be a more faithful gauge for model accuracy as applied to fossils. Of the 40 species, 21 were previously published in Londoño et al. (2018), who collected sun-adapted canopy leaves of angiosperms using a crane in Parque Nacional San Lorenzo, Panama. To test the method in temperate forests, we collected leaves from 11 angiosperm and 7 conifer species from Dinosaur State Park (Rocky Hill, Connecticut), Wesleyan University (Middletown, Connecticut), and Connecticut College (New London, Connecticut) during the summer of 2015. Here, all trees grew in open, park-like settings; one to three sun leaves were sampled from the lower outside crown of each tree. In January of 2015, we also sampled sun-exposed leaves from the tree fern Cyathea arborea in El Yunque National Forest, Puerto Rico (near the Yokahú Tower).

Stomatal size and density were measured either on untreated leaves using epifluorescence microscopy with a 420–490 nm filter or on cleared leaves (using 50 % household bleach or 5 % NaOH) using transmitted-light microscopy. For most species, whole-leaf δ13C comes from Royer and Hren (2017); the same leaves were measured for δ13C and stomatal morphology. The UC Davis Stable Isotope Facility measured some additional leaf samples. Atmospheric CO2 concentration (400 ppm) and δ13Cair (-8.5 ‰) come from Mauna Loa, Hawaii (NOAA/ESRL, 2019), which we assume are representative of the local conditions under which we sampled (e.g., Munger and Hadley, 2017). Table S1 summarizes for these 40 species all of the inputs needed to run the Franks model, along with the estimated CO2 concentrations. Uncertainties in the estimates are based on error propagation using Monte Carlo simulations (Franks et al., 2014).

The Franks model can be configured for any temperature. Franks et al. (2014) recommend that the photosynthesis parameters A0 and Γ*, and the air physical properties affecting the diffusion of CO2 into the leaf (the ratio of CO2 diffusivity in air to the molar volume of air, or d/v), correspond to the mean daytime growing-season leaf temperature (more precisely, assimilation-weighted leaf temperature). The reasoning behind this is that (i) the assimilation-weighted leaf temperature corresponds to the mean ci/ca derived from fossil leaf δ13C, and (ii) both theory (Michaletz et al., 2015, 2016) and observations (Helliker and Richter, 2008; Song et al., 2011) indicate that the control of leaf gas exchange leads to relatively stable assimilation-weighted leaf temperatures (∼19–25 ∘C from temperate to tropical regions) despite large differences in air temperature. This is mostly due to the effects of transpiration on leaf energy balance. Franks et al. (2014) chose a fixed temperature of 25 ∘C because much of the Mesozoic and Cenozoic correspond to climates warmer than the present day. When applying the Franks model to known cooler paleoenvironments, improved accuracy may be achieved with leaf-temperature-appropriate values for A0, Γ*, and d/v.

Bernacchi et al. (2003) proposed the following temperature sensitivity for Γ* based on experiments: 6Γ*=e19.02-37.83RT, where R is the molar gas constant (8.31446×10-3 kJ K-1 mol-1) and T is leaf temperature (K). Marrero and Mason (1972) describe the sensitivity of water vapor diffusivity to temperature as 7d=1.87×10-10T2.072P, where P is atmospheric pressure, which we fix at 1 atmosphere. Lastly, the temperature sensitivity of the molar volume of air follows ideal gas principles: 8v=vSTPTTSTPPPSTP, where TSTP is 273.15 K, PSTP is 1 atmosphere, and vSTP is the air volume at TSTP and PSTP (0.022414 m3 mol-1).

Using Eqs. (6)–(8), we can describe how, conceptually, the sensitivities of Γ* and d/v to leaf temperature affect estimates of CO2 from the Franks model. We apply these relationships to a suite of 409 fossil and extant leaves from 62 species of angiosperms, gymnosperms, and ferns. These data come from the current study (see Sect. 2.1 and 2.4) and Londoño et al. (2018), Kowalczyk et al. (2018), and Milligan et al. (2019).

To experimentally test more generally how the Franks model is influenced by temperature, we grew six species of plants inside two growth chambers with contrasting temperatures (Conviron E7/2; Winnipeg, Canada). Air temperature was 28±0.5 ∘C (1σ) and 20±0.3 ∘C during the day and 19±0.7 ∘C and 11±1.1 ∘C during the night. We note that the difference in leaf temperature was probably smaller than that in air temperature during the day (8 ∘C; see earlier discussion). We held fixed the day length (17 h with a 30 min simulated dawn and dusk) and CO2 concentration (500±10 ppm). Light intensity at the heights at which we sampled leaves ranged from 100 to 400 µmol m-2 s-1. Humidity differed moderately between chambers (76.5±1.8 % and 90.0±3.6 %). To minimize any chamber effects, we alternated plants between chambers every 2 weeks.

Four of the species started as saplings purchased from commercial nurseries: bare-root, 30 cm tall saplings of Acer negundo and Carpinus caroliniana, 30 cm tall saplings of Ostrya virginiana with a soil ball, and bare-root, 10 cm tall saplings of Ilex opaca. We grew the other two species from seed: Betula lenta from a commercial source and Quercus rubra from a single tree on Wesleyan University's campus. All seeds were soaked in water for 24 h and then cold-stratified in a refrigerator for 30 and 60 days, respectively.

All seeds and saplings grew in the same potting soil (Promix Bx with Mycorise; Premier Horticulture; Quakertown, Pennsylvania, USA) and fertilizer (Scotts all-purpose flower and vegetable fertilizer; Maryville, Ohio, USA). They were watered to field capacity every other day, and we discarded any excess water passing through the pots. After 3 months of growth in the chambers, for each species–chamber pair we harvested the three newest fully expanded leaves whose buds developed during the experiment. In most cases, we harvested five plants per species–chamber pair; the one exception was I. opaca, for which we were limited to three plants in the warm treatment and two in the cool treatment.

We measured stomatal size and density on cleared leaves (using 50 % household bleach) with transmitted-light microscopy. Whole-leaf δ13C comes from the UC Davis Stable Isotope Facility and the Light Stable Isotope Mass Spec Lab at the University of Florida; the same leaves were measured for δ13C and stomatal morphology. We used either a hole punch or razor to remove two adjacent sections of leaf tissue near the leaf centers, avoiding major veins. Because we used the same CO2 gas cylinder (δ13C =-11.8 ‰) and laboratory space (δ13C =-10.4 ‰) as Milligan et al. (2019), we used their two-end-member mixing model (1/CO2 vs. δ13C; Keeling, 1958) to calculate the δ13C of the chamber CO2 at 500 ppm (-10.6 ‰). We used the recommended values from Franks et al. (2014) for the physiological inputs A0 and gc(op)/gc(max). Table S1 summarizes all of the inputs from this experiment needed to run the Franks model, along with the estimated CO2 concentrations. The standard errors for the inputs are based on plant means.

To test if leaf δ13C and stomatal morphology (stomatal density, stomatal pore length, and single guard cell width) differed between temperature treatments across species, we implemented a mixed model in R (R Core Team, 2016) using the lme4 (Bates et al., 2015) and lmerTest (Kuznetsova et al., 2017) packages, with temperature and species as the two fixed factors. To test if there was a significant difference between CO2 estimates from the two temperature treatments, we ran a Kolmogorov–Smirnov (KS) test in R. For each species, we first estimated CO2 for each plant in the warm and cool treatments based on simulated inputs constrained by their means and variances. In the typical case with five plants per chamber, this produced five CO2 estimates for the warm chamber and the same for the cool chamber. A KS test was then used to test for a significant temperature effect. We repeated this procedure 10 000 times, with 10 000 associated KS tests. The fraction of tests with a p value < 0.05 was taken as the overall p value. An advantage of this approach is that it incorporates both within- and across-plant variation.

ci/ca is estimated in the Franks model following Farquhar et al. (1982): 9Δleaf=a+(b-a)×cica, where a is the carbon isotope fractionation due to the diffusion of CO2 in air (4.4 ‰; Farquhar et al., 1982), b is the fractionation associated with RuBP carboxylase (30 ‰; Roeske and O'Leary, 1984), and Δleaf is the net fractionation between air and assimilated carbon ([δ13Cair-δ13Cleaf]/[1+δ13Cleaf/1000]).

Equation (8) can be expanded to include other effects, including photorespiration (Farquhar et al., 1982): 10Δleaf=a+(b-a)×cica-fΓ∗ca, where f is the carbon isotope fractionation due to photorespiration. Photorespiration occurs when the enzyme rubisco fixes O2, not CO2 (i.e., RuBP oxygenase). One product of photorespiration is CO2 (Jones, 1992), whose δ13C is lower than the source substrate glycine. If this respired CO2 escapes to the atmosphere, the δ13C of the leaf carbon becomes more positive. Thus, if ci/ca is calculated using Eq. (8), as is common practice, the calculation may be falsely low, leading to an underprediction of atmospheric CO2.

Measured values for f vary from ∼9 to 15 ‰ (see compilation in Schubert and Jahren, 2018), which is in line with theoretical predictions (Tcherkez, 2006). At a 400 ppm atmospheric CO2 and Γ* of 40 ppm, Eq. (9) implies that ∼1 ‰ of Δleaf is due to photorespiration, meaning that ci/ca should be ∼0.04 higher relative to Eq. (8). Here, using the suite of fossil and extant leaves described in Sect. 2.2, we explore how the carbon isotopic fractionation associated with photorespiration affects CO2 estimates with the Franks model. Because ci/ca is present in both of the fundamental equations (Eqs. 2 and 3), we solve them iteratively until ci/ca converges.

The composition of air close to the forest floor can differ considerably from the well-mixed atmosphere. Of relevance to the Franks model, soil respiration can lead to a locally higher CO2 concentration and lower δ13Cair (Table 2). This effect is strongest at night, when the forest boundary layer is thickest (e.g., Munger and Hadley, 2017), but we focus here on daylight hours because that is when most plants take up CO2. In wet tropical forests, which can have very high soil respiration rates, CO2 during the day near the forest floor can be elevated by tens of parts per million, and the δ13Cair can be 2–3 ‰ lower; in temperate forests, the deviations are smaller (Table 2). Above ∼2 m, CO2 concentrations and air δ13C during the daytime largely match the well-mixed atmosphere.

As a result, leaves that grow close to the forest floor may cause the Franks model to produce CO2 estimates higher than that of the mixed atmosphere for at least two reasons. First, the concentration of CO2 near the forest floor is elevated; that is, the model may correctly estimate a CO2 concentration that the user is not interested in. Second, because the δ13Cair that a forest-floor plant experiences is lower than the global well-mixed value, if the user chooses the well-mixed value for model input (inferred, for example, from the δ13C of marine carbonate; Tipple et al., 2010), then ci/ca and thus atmospheric CO2 will be overestimated (see Eq. 2).

We sought to test how the Franks model is affected by the forest-floor microenvironment for five tropical angiosperm species and 15 temperate angiosperm and fern species. The tropical leaves were sampled at ∼1–2 m of height from Parque Nacional San Lorenzo, Panama. In contrast to the canopy dataset from San Lorenzo (Sect. 2.1), these CO2 estimates have not been previously reported. In the summer of 2015, seven fern species were sampled at ∼0.5 m of height from Connecticut College and Wesleyan University. Also, we used leaf vouchers from Royer et al. (2010), who sampled eight herbaceous angiosperm species at ∼0.1–0.2 m of height from Reed Gap, Connecticut. For all 20 species, stomatal and carbon isotopic measurements follow the methods described in Sect. 2.1. Table S1 contains all of the inputs needed to run the Franks model, along with the estimated CO2 concentrations.

We also investigated if we could include the forest-floor δ13Cair effect in our estimates of atmospheric CO2. We did not measure the CO2 concentration and δ13Cair around our plants, so we could not directly compare our values. But, if the only CO2 inputs close to the forest floor are from the soil and well-mixed atmosphere, then the system can be modeled as a two-end-member mixing model in which δ13Cair has a positive, linear relationship with 1/CO2 (Keeling, 1958). If the CO2 concentration and δ13C of both end-members are known, the forest-floor microenvironment should fall somewhere on the modeled line. Importantly, the Franks model provides a second constraint on the system. Here, δ13Cair has a negative, nonlinear relationship with 1/CO2 because δ13Cair is positively related to ci/ca and CO2. The Franks model thus provides a second calculation for the relationship between δ13Cair and estimated CO2 concentration. The intersection between the two curves should be the correct δ13Cair and CO2 concentration for the forest-floor microenvironment.

To estimate the soil CO2 end-member, we measured the δ13C of soil organic matter collected from the A horizons of 13 soil sites at San Lorenzo and of five each at Reed Gap and Connecticut College. For all soils, we assume a 5000 ppm CO2 concentration for a depth that is below the zone of CO2 diffusion from the atmosphere (∼0.3 m; Cerling, 1999; Breecker et al., 2009). The true value for wet temperate and tropical forest soils may be somewhat less or substantially more than 5000 ppm (Medina et al., 1986; Cerling, 1999; Hirano et al., 2003; Hashimoto et al., 2004; Sotta et al., 2004). Because the mixing model uses 1/CO2, a much higher CO2 concentration (e.g., 10 000 ppm) has little impact on our results.

Estimates of CO2 across the 40 tree species sampled in the field range from 275 to 850 ppm, with a mean of 478 ppm and median of 472 ppm (Fig. 2); two-thirds of the estimates (a close equivalent to ±1 standard deviation) range between 353 and 585 ppm. In 28 % of the tested species, the estimated CO2 concentrations overlap the target concentration (400 ppm) at 95 % confidence; for these species, the CO2 estimates do not differ significantly from the target. There are no strong differences across taxonomic orders or between leaves from tropical and temperate forests. The mean error rate across the estimates is 28 % (median 24 %), which is higher than estimates that include direct measurements of the physiological inputs A0 and gc(op)/gc(max) (mean 20 %; median 13 %; Fig. 1). Along similar lines, if the estimates presented in Fig. 1 are reestimated using the values for A0 and gc(op)/gc(max) recommended by Franks et al. (2014), the mean error rate increases to 37 % (median 33 %). If only the default values of A0 are used, the median error rate is 27 %, and for only default values of gc(op)/gc(max) it is 22 %.

These results indicate that CO2 accuracy is generally improved when A0 and/or gc(op)/gc(max) are measured. These measurements require expensive gas-exchange equipment and are not always easy or practical to make. Moreover, A0 and gc(op)/gc(max) cannot be measured on fossils. Some gains in accuracy are possible by measuring A0 and gc(op)/gc(max) on extant relatives of the fossil species (e.g., the same genus). Absent of this, our analysis using the recommended mean values of Franks et al. (2014) indicates an error rate, on average, of approximately 28 %. This is comparable to or better than other leading paleo-CO2 proxies (Franks et al., 2014).

One reliable way to improve accuracy is to estimate CO2 with multiple species because the falsely high and falsely low estimates partly cancel each other out. The grand mean of estimates presented in Fig. 2 (478 ppm) is 20 % from the 400 ppm target, which is less than the 28 % mean error rate of individual estimates.

Dow et al. (2014) observed that gc(op)/gc(max) inversely varies with CO2 in Arabidopsis thaliana, but primarily at subambient concentrations (yellow triangles in Fig. 3). At elevated CO2, gc(op)/gc(max) is close to 0.2, which is the value recommended by Franks et al. (2014). Data from 11 species of angiosperms, conifers, and ferns at present-day (or near present-day) and elevated CO2 concentrations support the view of a limited effect at high CO2 (Fig. 3; Franks et al., 2014; Londoño et al., 2018; Milligan et al., 2019). More data at subambient CO2 are needed, but for CO2 concentrations similar to or higher than the present day, we see no strong reason to include a CO2 sensitivity in gc(op)/gc(max). The rather low values for Cedrus deodara and many of the tropical angiosperms (<0.1) are likely due to stress imposed by their growth-chamber environment; these gc(op)/gc(max) values are probably not representative of field-grown trees, which tend to be closer to 0.2 (Franks et al., 2014).

The temperature sensitivities of the ratio of diffusivity of CO2 in air to the molar volume of air (d/v) and the CO2 compensation point in the absence of dark respiration (Γ*) have little effect on estimated CO2 in the Franks model (Fig. 4). Given that assimilation-weighted leaf temperature only varies about 7 ∘C across plants today, the differences shown in Fig. 4, which are based on leaf temperatures of 25 and 15 ∘C, are likely a maximum effect. As such, we consider the use of a fixed leaf temperature (e.g., 25 ∘C) in the model to be a defensible simplification.

Other inputs in the model may respond to temperature, though. In our growth-chamber experiments for which daytime air temperatures were 28 and 20 ∘C, the effect on estimated CO2 was mixed (Fig. 5). In five out of six species, estimated CO2 was higher in the warm treatment, but for all species these differences were not statistically significant (p>0.05 based on a KS test; see Methods). Incorporating the temperature sensitivities in d/v and Γ* had little effect (“adj” estimates in Fig. 5), as expected from Fig. 4.

None of the measured inputs – stomatal density, stomatal pore length, single guard cell width, and leaf δ13C – were significantly affected by temperature across all species (p>0.05 for each of the four inputs based on a mixed model; see Sect. 2.2). These small differences probably cannot account for the differences in estimated CO2 between temperatures. It is more likely that some of the inputs that we did not directly measure, such as assimilation rate (A0), the gc(op)/gc(max) ratio, or mesophyll conductance (gm), differ from the true mean value. In the cases for the five species for which estimated CO2 is higher in the warm treatment, our mean A0 for the warm plants must be falsely high, or gc(op)/gc(max) or gm is falsely low.

In summary, we see no strong reason to expand the parameterization of temperature in the model, though more growth-chamber experiments may be warranted. We note that in three out of the six species from the warm treatment, the estimated CO2 cannot be distinguished from the 500 ppm target at 95 % confidence; for the cool treatment, this is true for four of the species. Also, the across-species means of estimated CO2 for the warm and cool treatments are reasonably close to the target (590 and 502 ppm, respectively) and overall have a mean error rate of 25 %. This level of uncertainty is similar to our field estimates for which we did not measure A0 or gc(op)/gc(max) (28 %; see Fig. 2). This too provides support for our recommendation that it is not critical to include a broader treatment of temperature in the model.

The theoretical effects of photorespiration do not strongly impact estimates of CO2 in the Franks model. The average effect for our 409 extant and fossil leaves is to increase estimated CO2 by 2.2 % plus 38 ppm (Fig. 6). At 1000 ppm, for example, estimates would increase by 60 ppm. This calculation assumes a photorespiration fractionation (f) of 9.1 ‰, which is the value estimated for Arabidopsis thaliana (Schubert and Jahren, 2018). If a fractionation towards the upper bound of published estimates is used instead (15 ‰), estimated CO2 increases on average by 3.8 % plus 61 ppm. Across this range in f, the associated uncertainty in estimated CO2 is well within the method's overall precision (∼+35/-25 % at 95 % confidence; Franks et al., 2014). As such, CO2 estimates made without these photorespiration effects (i.e., using Eq. 9 instead of Eq. 10) should be reliable, although some improvement is possible using Eq. 10 in cases in which f is accurately known.

We note that both f and Γ* are also affected by atmospheric O2 concentration. Because O2 is directly responsible for photorespiration, f should scale with O2 (or, more precisely, the O2 : CO2 molar ratio). Unfortunately, this effect is poorly constrained (Beerling et al., 2002; Berner et al., 2003; Porter et al., 2017). In contrast, the theoretical effect of O2 on Γ* is known: it is linear with an approximate slope of 2 (Farquhar et al., 1982; see their Eq. B13). During the Phanerozoic, O2 likely ranged from 10 % to 30 %, with lows during the early Paleozoic and early Triassic and highs during the Carboniferous to early Permian and Cretaceous (Berner, 2009; Glasspool and Scott, 2010; Arvidson et al., 2013; Mills et al., 2016; Lenton et al., 2018). Assuming a present-day Γ* of 40 ppm (at 21 % O2), Γ* would be 60 ppm at 30 % O2 and 20 ppm at 10 % O2. Running the Franks model on our library of 409 extant and fossil leaves, we find little effect on estimated CO2: estimates are 7.4 % higher on average at 30 % O2 than at 10 % O2 (see also McElwain et al., 2016).

CO2 estimates for tropical understory leaves from five species at San Lorenzo, Panama, are very high, ranging from 1903 to 18863 ppm (species mean 6837 ppm). For two of the species, Londoño et al. (2018) also analyzed canopy leaves from trees nearby, and these within-species comparisons highlight the vast discrepancy (Ocotea sp.: 541 vs. 5737 ppm; Tontelea sp.: 622 vs. 18 863 ppm). The primary difference in the inputs between the canopy and understory leaves is the δ13Cleaf: Londoño et al. (2018) report a species mean δ13Cleaf of -30.0 ‰ for the 21 canopy species versus -35.6 ‰ for the five understory species. This difference leads to very different mean estimates of ci/ca: 0.69 for canopy leaves versus a highly unrealistic (Diefendorf et al., 2010) 0.93 for understory leaves.

It is likely that the high CO2 estimates from understory leaves are mostly driven by falsely high δ13Cair inputs. Following the mixing model strategy outlined in Sect. 2.4 (and based on a soil organic matter δ13C of -28.2 ‰ measured at San Lorenzo), we calculate a species mean δ13Cair of -16.7 ‰ (mean of intersection points in Fig. 7). When this δ13Cair is used to estimate CO2 with the Franks model (instead of -8.5 ‰), the species mean drops to 699 ppm. This is somewhat higher than the species mean from canopy leaves in the same forest (563 ppm; red triangles in Fig. 2; Londoño et al., 2018).

Understory leaves from Connecticut temperate forests show similar but less dramatic patterns, which we attribute to a more open canopy with stronger atmospheric mixing. CO2 estimates for the 15 species range from 447 to 1567 ppm (mean 794 ppm). Our intersection method identifies a mean δ13Cair of -11.2 ‰ for the Wesleyan and Connecticut College campuses (based on a soil δ13C of -27.6 ‰ measured at Connecticut College) and -10.3 ‰ for Reed Gap (soil δ13C =-26.4 ‰). Using these adjusted δ13Cair, the species mean of estimated CO2 drops to 566 ppm, which is somewhat higher than the species mean from canopy leaves in the same areas (449 ppm; blue circles in Fig. 2).

We acknowledge that this analysis is too simple. First, we did not measure the understory CO2 concentration and δ13Cair (this would require repeated measurements during different daytime hours over a growing season to calculate a time-integrated value); instead, we assumed a simple two-end-member mixing model between the soil and well-mixed atmosphere. Second, other factors probably contribute to the differences in estimated CO2 between canopy and understory leaves. Our predicted δ13Cair values are too low (∼8 ‰ and 2 ‰ lower than the well-mixed atmosphere for the tropical and temperate forests) and our estimated CO2 too high (∼100 ppm higher than that from canopy leaves). In the lowermost 1–2 m of the canopy, previous work suggests up to a -3 ‰ and +70 ppm deviation in tropical forests and -1 ‰ / +20 ppm in temperate forests (Table 1). One input that could help to resolve this discrepancy is the assimilation rate (A0). We assumed a fixed A0 of 12 µmol m-2 s-1 for all leaves, regardless of canopy position. Shade leaves often have lower assimilation rates than sun leaves (Givnish, 1988). Substituting lower A0 values for understory leaves would lower estimated CO2 roughly in proportion (Eqs. 2–3). Using lower A0 values for shade leaves in the model is appropriate, but determining the best value is difficult. Typical A0 values for leaves growing at the top of the canopy in full sun are far more consistent because photosynthesis in these leaves is usually at its maximum capacity (saturated at full sunlight) for the prevailing atmospheric CO2 concentration. Because the degree of shadiness near the forest floor is highly variable, photosynthesis (A0) in these leaves will be acclimated to some fraction of the full-sun maximum in a sun-exposed leaf, but careful thought must go into determining what this fraction is.

We note that our mixing model strategy cannot be applied to fossils because the global atmospheric CO2 concentration is needed (one end point for dashed line in Fig. 7). Instead, our motivation for the analysis is to demonstrate that (1) leaves growing in the lowermost 2 m of the canopy should be considered with caution in the context of the Franks model, and (2) the failure of the model is due to faulty inputs (mostly δ13Cair), not the model itself.

In most fossil leaf deposits, shade morphotypes are comparatively rare (e.g., Kürschner, 1997; Wang et al., 2018) because – relative to sun leaves – they are less durable, do not travel as far by wind, and are produced at a slower rate (Dilcher, 1973; Roth and Dilcher, 1978; Spicer, 1980; Ferguson, 1985; Burnham et al., 1992). Our recommendation is to exclude such leaves. There are several ways to differentiate sun vs. shade morphotypes: overall shape (Talbert and Holch, 1957; Givnish, 1978; Kürschner, 1997; Sack et al., 2006), shape of epidermal cells (larger and with a more undulated outline in shade leaves; Kürschner, 1997; Dunn et al., 2015), vein density (lower in shade leaves; Uhl and Mosbrugger, 1999; Sack and Scoffoni, 2013; Crifò et al., 2014; Londoño et al., 2018), and range in δ13Cleaf (high when both sun and shade leaves are present, for example in our study; Graham et al., 2014). Not all shade leaves grow within 2 m of the forest floor, but excluding all such leaves would eliminate the forest-floor bias.