Primary data were obtained from the USDA Forest Service’s Forest Inventory and Analysis database, collected by the Southern FIA unit. The FIA study locations selected represented BLH sites, and plots were chosen based on the criteria to obtain oak-gum-cypress forests in naturally regenerating conditions. Condition, plot, and tree tables for the BLH permanent plots in Alabama, Arkansas, Louisiana, Mississippi, two eastern Texas units, and one unit in western Tennessee ( Figure 1 ) were downloaded from the FIA database and were then merged individually into Microsoft Excel. The data were queried for the most recent seven years of plots surveyed. They were next filtered according to the following criteria: oak-gum-cypress as the primary forest type (which resided on both narrow and wide floodplains and bottomlands), natural origin, live trees of growing stock size (at least 11.7 cm diameter at breast height), land not permanently underwater or in the presence of ridges and gullies (excluding inoperable lands), no record of past silvicultural treatments (natural disturbance included), stand age at least 20 years but no more than 100 years, and growing stock basal area of at least 13.8 m²/ha, which was considered a threshold of full stocking ( Schultz et al., 2010 ). Lastly, we calculated a stand density index ratio, which was the sum of individual trees’ SDI relative to the plot’s SDI at its quadratic mean diameter, and set a minimum threshold of 0.90 to better achieve an even-aged condition ( Shaw & Long, 2007 ). Based on these criteria, a total of 403 sample plots were identified ( Figure 1 ). Site index for the plots was set using an equation for sweetgum (Liquidambar styraciflua) for all resident plot trees and averaged arithmetically at the plot level ( Carmean et al., 1989 ; Walters & Ek, 1993 ). Aboveground oven-dry biomass weight was the whole of three components, the tree stem, branches, and foliage ( Schultz et al., 2013 ). The above ground carbon content was assumed to be 50%. These were calculated for the plot and expanded to a per hectare basis using the trees per hectare expansion factor. Plots were divided into model building and testing sets randomly at a 9:1 ratio.

The carbon stock/yield model’s functional form followed Smith et al. (1975) , when they modeled hardwood yields across a range of sites. Sullivan et al. (1983) also used this model form for modeling oak-gum stands in central Mississippi minor stream bottoms

$\ln C={\text{α}}_0+{\text{α}}_1\ln \left(S\right)A^{-1}+{\text{α}}_2{A}^{-1}+{\text{α}}_3\ln \left(B\right)+\underset{j=1}{\overset{11}{{\displaystyle \sum }}}{\text{γ}}_i{D}_i+\text{ε}$ (1)

where ln was the natural logarithm; C was the dependent variable (carbon stock, tonnes per hectare, as one-half dry biomass weight); S was the plot average sweetgum site index (base age 50 years in meter) calculated using Carmean et al. (1989) ; A was stand age (years); and B was growing stock basal area (m²/ha); α_i were parameters for $\ln \left(S\right)A^{-1}$, $A^{-1}$, and $\ln \left(B\right)$; γ_j were parameters for eleven ecoregion dummy variables for n = 403 study plots ( Figure 1 ) with the East Central Texas Plains being the reference group ( United States Environmental Protection Agency, 2017 ); and ε was the error term. Predicting growth (sequestration) required first constructing a companion basal area equation, which was differentiated with respect to age. Doing so accounted for both growth and mortality as stands progress from many small trees to fewer larger ones ( Buckman, 1962 ).

The basal area model was

$\ln B={\text{β}}_0+{\text{β}}_1\ln \left(S\right)A^{-1}+b{\text{β}}_2{A}^{-1}+{\text{β}}_3\ln \left(N\right)+\text{ε}$ (2),

where N was growing stock trees per hectare (TPH), β_i were parameters, and all other variables being as previously described (Equation (1)). The first derivative of basal area with respect to age defined the annual growth rate

$dB=\hat{B}\left(-b_1\ln \left(S\right)A^{-2}-b_2{A}^{-2}\right)dA$ (3),

with dB being the difference in basal area; $\hat{B}$ was predicted arithmetic mean basal area per hectare adjusted by the model’s correction factor for a particular tract’s site index, age, and trees per hectare from Equation (2); b_i were regression coefficients from Equation (2); and dA was the difference in age. Specifying the derivative with respect to age reduced N to a constant, and the derivative of a constant is zero. Basal area growth therefore occurred at the same rate for all sites’ numbers of trees per hectare. Basal area increment, square meter per hectare per year, can be calculated sequentially from the subsequent age t+1 to some future time at age T. Future basal area, B_T was

$B_T=\hat{B}+\left(\underset{Ag{e}_{t+1}}{\overset{T}{{\displaystyle \sum }}}\hat{B}*dB\right)$ (4).

where predicted arithmetic mean basal area per hectare was added to the sum product of $\hat{B}$ and dB from age t + 1 to the end of the desired growth period, T.

Equations (1) and (2) were estimated simultaneously because the correlation of the two models’ residuals was significant at alpha = 0.05, albeit low in magnitude (r = 0.44). But Equation (3) also contained basal area as an independent variable, whereas it was the dependent variable in Equation (4). Basal area was therefore endogenous to the system. Together, these elements led to using three-stage least squares via SAS 9.4’s SYSLIN procedure at alpha = 0.05 (SAS 2023). Site index and age were exogenous, while trees per hectare served as an instrumental variable that exhibited some correlation with basal area (r = 0.50) but little with carbon stock (r = 0.19). Outliers were identified using standardized residuals with absolute values greater than 2.00 for each equation, removed, and the SYSLIN procedure was rerun. The final equation was evaluated using the validation set to calculate the residuals’ average (bias), standard error (precision), mean absolute deviation (MAD), mean absolute percent error (MAPE). A correction factor of $MSE/2$ accounted for the bias introduced by the transformation ( Baskerville, 1972 ). After model training and testing were completed, the two datasets were merged to discover the model’s final coefficients. Equations (3) and (4) were calculated in MS Excel with the PROC SYSLIN results.

Future carbon sequestration per acre was evaluated for a typical stand of average quality at the study means for stand age (rounded to age 56), site index (rounded to 22), and TPH (rounded to 250). A predicted basal area per acre $\hat{B}$ was found per Equation (2). The average values for age, site index, and $\hat{B}$ were input into Equation (1) to next provide the initial carbon stock. The stand was then plotted onto the southern bottomland hardwood stocking guide from Goelz (1995) . It was assumed the stand would experience 10% mortality to grow in basal area per acre to reach 100% stocking for a timber harvest using our basal area growth equation and the USDA Forest Service’s mortality and growth data for our study region. The needed future basal area ${\hat{B}}_T$ to reach 100% stocking at 225 TPH was 28.6 m²/ha ( Goelz, 1995 ). Future basal areas were determined from age 56 years to the stand age A_T needed to achieve 28.6 m²/ha, which was A_T = 92 years. The future basal areas were exported back to Equation (1) sequentially to calculate each year’s future carbon stock per hectare. The year-on-year changes in the per hectare carbon stock represented the tonnes of carbon sequestered per hectare over the holding period. The carbon captured was considered eligible for payment up to age A_T−1, which was 91 years.

Discounted accumulated present carbon values (APCV) were calculated from two perspectives. The first was as a social value. The formula was

$APCV,\frac{\$}{ha}=P*\underset{t=1}{\overset{35}{{\displaystyle \sum }}}\left(\frac{3.6667\ast \Delta {\hat{C}}_{ha}}{e^{rt}}\right)$, (5),

with P = $190 per tCO₂e ($172.37 per ton for emission year 2020) being the social value of carbon ( US EPA, 2023 ), $\Delta {\hat{C}}_{ha}$ was the carbon sequestered per hectare per year. The carbon sequestered each year was converted to tCO₂e using a multiplier of 3.6667, which equals the atomic weight of carbon dioxide (44) relative to the atomic weight of carbon (12). The denominator was comprised of the base of the natural logarithm e, the social discount r of 2.00%, and t was the sequential number of years (t = 1, 2, 3, …, 35) from age 56 to age A_T−1 = age 91. The social cost of greenhouse gas emissions is, “…a comprehensive metric that includes the value of all future climate change impacts (both negative and positive), including changes in net agricultural productivity, human health effects, property damage from increased flood risk, changes in the frequency and severity of natural disasters, disruption of energy systems, risk of conflict, environmental migration, and the value of ecosystem services” ( US EPA, 2023 ). Conversely then, the APCV provides a social net present benefit metric for removing tCO₂e from the atmosphere and sequestering the carbon in trees.

Discounting the sequestered carbon followed Lundgren’s (1966) calculation of expectation value index, which allowed a range of potential market prices to be studied. The market value of carbon followed Equation (5), with P representing prices of $1.00, $5.00, $10.00, $25.00, and $50.00 per tCO₂e. A discount rate more typical of forest investments was set at r = 5.00%. Both the social and market APCVs were calculated for lengths of t = 5 years to t = 35 years from the present in 5-year intervals. Annual equivalent values (AEV) were calculated at each contract length, so all could be compared on an equal time scale

$AEV,\frac{\frac{\$}{ha}}{yr}=APCV\ast \left(\frac{r{e}^{rt}}{e^{rt}-1}\right)$ (6).

The initial model was run using Equations (1) and (2), and the generated data visualizations found the assumptions regarding the residuals were met. Twenty-eight outliers were identified, removed, and the model was refit. The variables site index, age, and basal area were all significant in the carbon model, but none of the ecoregions significantly differed. The ecoregions were then dropped from Equation (1), the outliers were returned to the dataset and the model was reconstructed. Twenty-eight total outliers were again detected and isolated.

The model development set revealed anticipated positive marginal effects of $\ln \left(S\right)A^{-1}$ and $\ln \left(B\right)$ on carbon yield and a negative marginal effect of $A^{-1}$. The adjusted coefficient of determination was 0.8922. Assessing the model using the validation dataset found mean bias was 0.67 tonnes/ha with a standard error of 14.68; on percentage bases these were −0.08% and 23.0% respectively. The MAD was 9.21 tonnes/ha, and the MAPE was 3.04%. The basal area prediction model’s coefficients were all significant at α = 0.05 with the expected signs. The adjusted R-square was 0.6021. Validation metrics for the basal area model included a mean bias of 0.59 m²/ha (−0.62%) with a standard error of 1.21 (18.8%). The MAD was 4.75 m²/ha, and the MAPE was 24.7%.

The full dataset was then run to find the final coefficients, and those results can be found in Table 2 for the carbon stock and Table 3 for basal area. Thirty outliers were identified and removed. The adjusted R-square increased slightly for both carbon (adj R² = 0.9019) and basal area (adj R² = 0.6364). Other supporting statistics, model mean square error, analysis of variance table F and p values, and correction factor are also provided for each model. All coefficients were significantly below the 0.01 level with expected signs. The derivative of the basal area equation from Table 3 was

$dB=\hat{B}\left(243.1A^{-2}-61.5\ln \left(S\right)A^{-2}\right)dA$ (7).

Adjusted R square = 0.9019; Model Mean Square Error = 0.0260; F = 1141.60; p < 0.0001; Correction Factor = 1.0131.

Adjusted R square = 0.6364; Model Mean Square Error = 0.0770; F = 218.00; p < 0.0001; Correction Factor = 1.0392.

The stand at an average age of 56 years was 77% stocked ( Goelz, 1995 ). It took 36 years to reach 100% stocking for a timber harvest. Carbon thus accumulated for 35 years to age 91 years. Carbon accumulation came to 9.91 tonnes/ha over the first five years and was 54.9 tonnes/ha over 35 years ( Table 4 ). Carbon sequestration provided $6507 per hectare in social APCV in the first five years. By 20 years, social APCV had surpassed $20,000 per hectare and exceeded $25,000 per hectare in 30 years. Over 35 years, $28,146 per acre in social value had accumulated. The social AEV values ranged from $1118 per hectare per year over 35 years to $1368 per hectare per year over 5 years.

The market-based APCVs ranged from $31.40 per hectare for five years when priced at $1.00 per tCO₂e to as much as $5000 per hectare when priced at $50 per tCO₂e for 35 years ( Table 5 ). The market APCV surpassed $100 per hectare over 35 years at the lower $1.00 per tCO₂e price. As much as $500 per hectare could be earned at $5.00 per tCO₂e over 35 years. Total earnings surpassed $1000 per hectare if a landowner received at least $10 per tCO₂e for 35 years. A 20-year contract paying at least $25 per tCO₂e would provide just over $2000 per hectare in present income. At least $50 per tCO₂e was required to accumulate more than $3000 per hectare. The overall range of AEV was from $6.17 per hectare per year over 35 years at $1.00 per tCO₂e to $355 per hectare per year over 5 years at $50 per tCO₂e. The AEV surpassed $30 per hectare per year if paid at least $5 per tCO₂e. It exceeded $60 per hectare per year if paid at least $10 per tCO₂e. The AEV bettered $150 per hectare per year if paid at least $25 per tCO₂e and $300 per hectare per year if paid at least $50 per tCO₂e.