The research team, led by Fareeha Siddique of Nanjing University of Information Science and Technology, together with colleagues in Belgium, Singapore and China, focused on a specific gap in the LKIF literature. The framework quantifies causality as a directional, signed rate of information transfer derived from the entropy evolution of an underlying dynamical system, rather than from statistical association or improved prediction. A key theoretical property, the principle of nil causality, guarantees that if the evolution of one variable is independent of another, the estimated information flow between them is zero. But uncertainties have persisted about how the estimated causal graph changes when so-called conditioners — variables that either mediate a causal pathway or confound it — are introduced alongside the driver and response.

Conditioners matter for a concrete reason. A mediator lies on the causal route between a driver and a response, so conditioning on it helps separate direct information flow from indirect contributions transmitted through the middle variable. A confounder influences both the driver and the response, and omitting it can bias the estimated coupling by inducing spurious information flow. Previous work had shown that omitting a key state variable can make bivariate and multivariate estimates diverge substantially, but no conclusive criteria existed for when the simpler bivariate form of LKIF might become unreliable, or what physical conditions drive the differences between the two formulations.

To build those criteria, the authors first ran a controlled synthetic experiment using a five-dimensional linear autoregressive process in which two hidden confounders influence the observable variables. The results were reassuring on one front: LKIF preserved correct causal directions even as the strength of unobserved confounding increased, with true connections consistently recovered and nonexistent connections remaining near zero. Yet differences between bivariate and multivariate estimates still appeared, reflecting the influence of the unobserved conditioners. Notably, the response to conditioning differed between pathways in the toy system — conditioning reduced the estimated flow for one link while leaving another essentially unchanged — demonstrating that the effect of adding variables is pathway-specific rather than universal.

The real-world application then tested these ideas on two humid forest biomes: Southeast China and the Amazon Basin. Using satellite-derived leaf area index (LAI) and gross primary productivity (GPP) from the GLASS dataset, together with soil moisture from ERA5-Land, temperature and surface solar radiation from ERA5, and vapor pressure deficit (VPD) estimated from reanalysis temperatures, the team computed time-varying information flows from soil moisture and VPD to vegetation over 1981–2018. Because of uncertainties in early vegetation products and the need for adequate preceding observations in the Kalman filter estimation, only results from 2004 to 2018 were presented and interpreted.

The central finding is a striking structural–functional divide. The influence of soil moisture on LAI, a measure of vegetation structure, remained consistently stable and showed little sensitivity to the selected conditioners, with the bivariate and multivariate estimates agreeing almost always — a convergence rate near 100 percent in the Southeast China analysis. In contrast, the influence of soil moisture on GPP, a measure of vegetation function, and the influence of VPD on vegetation exhibited substantial divergence between bivariate and multivariate estimates. In Southeast China, the bivariate soil-moisture-to-GPP flow was statistically significant in only 13 of 180 months, or 7.2 percent, whereas conditioning on VPD raised that figure to 116 months, or 64.4 percent. The Amazon Basin showed the opposite response, with significant months falling from 73 (40.6 percent) in the bivariate form to 41 (22.8 percent) when fully conditioned — evidence that the effect of conditioning is region-dependent and can operate in either direction.

To make this divergence diagnostically interpretable, the team introduced four indices built on a sequential decomposition of the differential information flow: the Conditioner Dominance Index, which quantifies each conditioner’s share of the total restructuring; Moderation Gain, capturing how strongly the flow decreases after a conditioner is added; Conditioning Pressure, measuring how strongly it increases; and the Convergence Rate, summarizing how often the bivariate and fully conditioned estimates agree. In the Southeast China test case, VPD emerged as the dominant conditioner for both soil-moisture pathways, accounting for roughly two-thirds of the total restructuring for LAI and nearly all of it for GPP. Temperature produced the largest moderation gain for the GPP pathway, and the accumulated conditioning pressure was much larger for GPP than for LAI, indicating that the bivariate representation can markedly understate the inferred flow for the productivity coupling.

The environmental state of the system turned out to be central to when conditioners matter. Binning the data into joint soil-moisture and VPD quartile regimes, the researchers found that soil-moisture-driven couplings showed uniformly high correlation and low divergence across nearly all regime combinations, whereas VPD-driven couplings varied widely in both metrics. An analysis of variance confirmed the pattern: for VPD-driven pathways, the soil-moisture-by-VPD interaction was the only dominant component, and when temperature or surface solar radiation was added as a third factor, the three-way interaction became the dominant contributor to variability in divergence across all four pathways. The divergence between bivariate and multivariate estimates peaked under compound dry–hot conditions, with VPD and temperature emerging as dominant conditioners precisely when drought and heat jointly stressed the vegetation.

The authors are careful about what these results do and do not show. A small difference between bivariate and multivariate estimates does not establish causal sufficiency, because opposing pathway contributions may compensate and yield a small overall difference, and important variables may remain unobserved. The time-varying formulation relies on a linear approximation over short intervals, which may not fully represent strongly nonlinear responses under extreme conditions; the analysis is limited to two humid forest biomes and cannot be generalized to drylands, where soil-moisture deficits constrain vegetation more directly, or to high-latitude systems governed by temperature and radiation. The indices themselves depend on the conditioning design and should be read as diagnostic summaries of pathway sensitivity rather than universal causal measures, and the monthly temporal framework means lag-specific relationships remain unexplored.

Even so, the practical message is considerable. The same conditioners can affect the temporal evolution of the same causal relationships very differently at different locations, which means that spatial and temporal averaging can mask the relevance of conditioning variables and that stationary assumptions may oversimplify causal structure. Because the framework identifies when inferred pathways are stable and when they are strongly conditioning-dependent, the authors suggest it could serve as a process-based benchmark for evaluating how Earth system models, including those in CMIP6, represent vegetation–climate coupling — testing whether models reproduce not just the magnitude of the coupling but its sensitivity to co-varying drivers and its reorganization under compound environmental stress.

Subject of Research: Benchmarking conditioning variables in Liang–Kleeman Information Flow analysis of soil moisture, vapor pressure deficit and vegetation coupling in humid forests

Article Title: Benchmarking conditioners in Liang–Kleeman Information Flow: application to land–atmosphere interactions in humid forests

Article References: Siddique, F., Hagan, D. F. T., Wang, G., Docquier, D., & Vannitsem, S. (2026). Benchmarking conditioners in Liang–Kleeman Information Flow: application to land–atmosphere interactions in humid forests. Earth System Dynamics, 17(5), 1551-1572. https://doi.org/10.5194/esd-17-1551-2026

Image Credits: AI Generated

DOI: 10.5194/esd-17-1551-2026

Keywords: causality, Liangle information flow, soil moisture, vapor pressure deficit, vegetation, GPP, LAI, humid forests, Amazon Basin, Southeast China, confounders, drought