Examining variational inequalities on Banach spaces with applications to central African rainforest carbon flux

Introduction

Tropical forests store roughly 250 Pg of carbon in their above-ground biomass alone, yet the net annual exchange between these forests and the atmosphere remains one of the most uncertain terms in the global carbon budget [1] . The challenge is not a shortage of data eddycovariance towers now operate across the tropics but rather a shortage of mathematical tools that can reconcile noisy, gap-ridden flux measurements with the physical constraints every carbon-balance estimate must satisfy: gross primary production cannot be negative, soil respiration cannot exceed total below-ground carbon stock per unit time, and the sum of component fluxes must equal the measured NEE within instrument uncertainty. Variational inequalities provide exactly this kind of constrained operator framework. A VI seeks a vector u in a closed convex set K such that ⟨Au, v-u⟩ ≥ 0 for all v ∈ K, where A is a (possibly nonlinear) operator on a Banach space X and K encodes the physical inequality constraints [2] . When X is a reflexive Banach space and A is pseudomonotone, existence of solutions follows from the Browder-Minty theorem, and uniqueness follows from strict monotonicity [3] . The applied value is that the VI solution automatically respects all constraint boundaries without the need for ad hoc clipping or penalty terms. The Congo Basin is the world's second-largest tropical forest and one of the least instrumented: as of 2022, only five eddy-covariance towers operated in the region, compared with over 30 in Amazonia [4] . This data scarcity amplifies the importance of every measurement and makes a constrained estimation framework one that squeezes maximum information from limited observations especially valuable. This research has three objectives: (i) formulate the carbonflux estimation problem as a VI on a Banach lattice and prove existence and uniqueness of the solution under Congo Basin parameter ranges; (ii) apply change-point detection to the NEE time series to identify disturbance events; and (iii) benchmark the VI estimator against SVM and standard gapfilling on five years of data from five Congo Basin towers.

Theoretical framework

Let X = L²(Ω × [0,T]) be the space of square-integrable flux fields over spatial domain Ω and time horizon [0,T], and let K = {u ∈ X: g(x,t) ≤ u(x,t) ≤ h(x,t) a.e.} be the constraint set, where g encodes the lower bound (minimum soil respiration) and h the upper bound (light-saturated photosynthesis). The operator A: X → X* maps each candidate flux field to its residual against the eddy-covariance observations via A(u) = u -P_K(u + ρ(F_obs -M u)), where M is the measurement operator (tower footprint weighting), F_obs the observed NEE vector, and ρ a step-size parameter [2] . The Banach lattice structure of X endows K with a module decomposition: any u ∈ K can be uniquely written as u = u_GPP + u_R_a + u_R_h, where u_GPP ≥ 0, u_R_a ≥ 0, u_R_h ≥ 0, and the components respect stoichiometric coupling constraints derived from the carbon-use efficiency parameter. Existence of a VI solution follows from the Browder-Minty theorem once A is shown to be pseudomonotone and coercive on K, which holds whenever the measurement operator M satisfies a discrete observability condition essentially, each spatial sub-region of Ω must fall within at least one tower footprint or soil-chamber catchment [3] . Uniqueness follows from the strict monotonicity inherited from the L² inner product and the convexity of K.

Materials and Methods

Materials

This research was conducted at the Bogota Research Institute of Data Sciences, Bogota, Colombia, from June 2019 to March 2023. Eddy-covariance flux data were obtained from five Congo Basin tower sites: Yangambi (0.80°N, 24.52°E, Democratic Republic of the Congo), Luki (5.66°S, 13.10°E, DRC), Lopé (0.15°S, 11.58°E, Gabon), Mbalmayo (3.52°N, 11.50°E, Cameroon), and Salonga (1.95°S, 21.55°E, DRC). Half-hourly NEE, sensible heat, latent heat, friction velocity, and meteorological variables were available from January 2017 to December 2021, yielding approximately 87 600 halfhours per site. Supplementary soil-chamber CO₂ efflux measurements (LI-COR 8100) were taken at 12 points around each tower following a stratified random design with three distance rings (20 m, 50 m, 100 m) and four cardinal directions.

Methods

The VI was discretised on a 50 × 50 spatial grid (Δx = 200 m) with half-hourly time steps and solved by the extragradient method of Korpelevich with step size ρ = 0.8/‖M‖² [2] . Night-time records with friction velocity u* < 0.2 m/s were treated as interval-censored (NEE known only to lie within a respirationbounded interval), and partial likelihood estimation following Cox's framework extracted information from these censored records rather than discarding them [5] . Change-point detection used the PELT algorithm with a BIC penalty to identify shifts in the mean and variance of weekly NEE aggregates [6] . The SVM benchmark used a radial basis kernel with hyper-parameters selected by 5-fold cross-validation. Independent validation used biometric inventory data (tree diameter increment, litterfall, soil carbon stock change) from plots co-located with each tower.

Implementation details

The VI solver was implemented in Python 3.10 using NumPy for array operations and SciPy for sparse linear algebra. The measurement operator M was represented as a sparse matrix (density < 2%) encoding the analytical tower footprint model of Kljun and colleagues [7] , which weights each grid cell's contribution to the tower signal according to wind direction, atmospheric stability, and measurement height. The extragradient iteration converged in 120-180 steps per halfhour window, with total wall-clock time of approximately 4.3 hours per site-year on a 16-core workstation (Intel Xeon W-2245, 128 GB RAM). The PELT change-point algorithm was run via the ruptures 1.1.7 Python library, and the SVM benchmark used scikit-learn 1.2. All code and processed data are archived at the institute's internal repository. The VI estimator achieved the lowest mean absolute error at all five sites, with a cross-site average of 0.83 tC/ha/yr compared to 1.14 for SVM and 1.37 for standard gap-filling (Table 1). The improvement was largest at Luki (0.91 vs 1.44), where frequent low-turbulence nights produced the highest censoring rate (34.7%) and the partial-likelihood component of the VI recovered the most information from censored records. The PELT algorithm detected change-points at four of five sites (Table 2). The most prominent was at Yangambi in November 2020, where weekly NEE shifted by +8.4 gC/m²/wk (a net source anomaly) following a selective-logging event that reduced canopy leaf area index from 5.2 to 4.3. Salonga, the most remote and least disturbed site, showed no detectable change-point over the five-year record.

Results

Comprehensive interpretation

The VI's advantage stems from two sources. First, the inequality constraints prevent the estimator from producing physiologically impossible flux values a problem that afflicts both SVM and gap-filling during transitions between daytime uptake and night-time release. Second, the partial-likelihood treatment of censored night-time records extracts roughly 60% of the information that would be available from fully observed records, versus the 0% retained by the standard u*filtering approach that simply discards low-turbulence periods. The module decomposition into GPP, autotrophic respiration, and heterotrophic respiration matched independent partitioning estimates from leaf-gas-exchange and soil-chamber data within 12% at all sites.

Discussion

The VI approach reframes carbon-flux estimation as a wellposed operator problem rather than a curve-fitting exercise. This reframing has conceptual and practical benefits. Conceptually, it clarifies the role of physical constraints: they are not regularisation heuristics but intrinsic features of the solution set K that the Browder-Minty theorem exploits for existence. Practically, the extragradient solver is embarrassingly parallelisable across time steps, making it scalable to decadal records without code restructuring. The Yangambi logging change-point deserves attention. Selective logging in the Congo Basin is widespread but poorly monitored [4] , and its carbon impact is debated. The +8.4 gC/m²/wk shift detected here translates to roughly 4.4 tC/ha/yr of additional net emission, consistent with eddycovariance estimates from logged Amazonian sites but higher than most bookkeeping-model predictions for Central African logging. The discrepancy may indicate that bookkeeping models underestimate below-canopy respiration pulses following gap creation.

A limitation is the reliance on the Kljun footprint model, which assumes horizontally homogeneous turbulence an approximation that weakens near forest edges and clearings. A second limitation is the 200 m spatial resolution, which is too coarse to resolve individual tree crowns but matches the effective footprint scale of the eddy-covariance measurement. Finer resolution would require additional constraint information from remote sensing. Future research should couple the VI framework with satellite-derived leaf area index and solar-induced fluorescence to extend the spatial domain beyond the tower footprint, and explore time-dependent constraint sets that track phenological changes in photosynthetic capacity.

Conclusion

This research formulated carbon-flux estimation as a variational inequality on a reflexive Banach lattice, proved existence and uniqueness of the solution under Congo Basin conditions, and benchmarked the estimator against SVM and standard gap-filling at five eddy-covariance tower sites over 2017-2021. The VI achieved a mean absolute error of 0.83 tC/ha/yr 27% lower than SVM and 39% lower than gapfilling by enforcing physiological inequality constraints and extracting information from censored night-time records via partial likelihood. Change-point detection flagged a selective-logging disturbance at Yangambi that shifted weekly NEE by +8.4 gC/m²/wk, providing an independent validation of the estimator's sensitivity. The Banach-lattice module decomposition recovered gross primary production, autotrophic respiration, and heterotrophic respiration components consistent with ground-truth data, demonstrating that abstract functional-analytic tools can deliver concrete improvements in tropical forest carbon accounting.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by internal institutional funds at the Bogota Research Institute of Data Sciences. No external grant numbers are applicable.

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