The research, led by Guopeng Wu, Ruiying Zhang, Zhijie Zhao, Guanghao Yu, Zhuohang Tian, Jiacheng Sui, and Kai Cui of the Silk Road Cultural Heritage Protection Research Institute and the Western Center of Disaster Mitigation in Civil Engineering at Lanzhou University of Technology, tackles a problem that has long frustrated heritage scientists. Spalling is not a uniform phenomenon. It manifests differently across a rock face depending on the local mineralogy, the density of microcracks, the strength of the surface, and the interplay of moisture, temperature swings, and salt weathering. Traditional condition surveys tend to rely on qualitative visual grading, which is inherently subjective and difficult to repeat or compare over time. The Damaidi team set out to replace that subjectivity with a reproducible numerical framework grounded in field measurement and laboratory testing.

The scale of the fieldwork alone is striking. The researchers conducted detailed surveys and laboratory analyses covering 144 individual instances of spalling distributed across eight densely carved zones of the conservation area. For each instance, they recorded a battery of physical and mechanical properties of the stone substrate, ultimately distilling the data into twelve indicators, labeled C1 through C12, that together characterize the vulnerability of the rock surface. These indicators span the mechanical integrity of the stone, its microstructural condition, and the geometry of the fracture networks that thread through it. By standardizing the description of each spalling instance through this common set of twelve variables, the team created a shared diagnostic language that can be applied consistently across the entire site and, in principle, at other rock art locations facing similar threats.

The analytical heart of the study is a hybrid method that combines cloud modeling with a game theory-based weighting scheme. Cloud modeling, a technique developed within the field of uncertainty artificial intelligence, is designed to handle the fuzzy boundary between qualitative descriptions and quantitative data. Rather than forcing each spalling instance into a rigid category, the cloud model represents concepts as clouds of numerical values defined by three parameters: expectation, entropy, and hyper-entropy. Expectation captures the central tendency of a grade, entropy describes its spread or fuzziness, and hyper-entropy quantifies the randomness of that spread. When a measured rock surface is evaluated, the model computes a membership degree describing how strongly it belongs to each severity grade, allowing borderline cases to be represented honestly rather than artificially forced into a single box.

The second ingredient, game theory-based weighting, addresses an equally stubborn problem in multi-indicator assessment: how much should each of the twelve indicators count toward the final severity score? Objective weighting methods derive weights from the statistical structure of the data itself, while subjective methods encode expert judgment about which factors matter most physically. Each approach has blind spots. The researchers treated the two families of weights as players in a cooperative game and solved for a combined weighting vector that minimizes the divergence between the combined result and each individual weighting scheme. The result is a set of composite weights that neither ignores the mathematics of the data nor the domain knowledge of conservators, producing a more defensible synthesis than either method alone could deliver.

With the hybrid model in place, the team classified spalling severity at Damaidi into four risk grades, each tied to a corresponding conservation strategy. This grading transforms the assessment from a static description into an actionable management tool. Zones and individual spalling instances falling into the highest risk grades can be prioritized for urgent intervention, such as consolidation or protective sheltering, while lower grades can be scheduled for routine monitoring. The researchers validated the framework, confirming both its accuracy and its rationality, which is a critical step for any quantitative method intended to inform real-world conservation budgets and interventions. Validation matters because a model that cannot be trusted to reproduce known conditions cannot be safely used to direct scarce preservation resources.

One of the most revealing components of the study is its correlation analysis. Using Pearson correlation, the team examined how strongly each of the twelve indicators related to the assessed severity of spalling. Four factors emerged as the most strongly correlated: rebound strength, designated C1, which reflects the hardness and integrity of the rock surface as measured by rebound testing; particle density, C6, which characterizes the compactness of the mineral grains forming the substrate; nearest fracture distance, C8, which measures how close the nearest crack lies to a given point on the surface; and fracture aperture, C10, which quantifies how wide those cracks are open. Together, these four indicators sketch a coherent physical picture of spalling as a process governed by surface mechanical weakness and the proximity of discontinuities through which moisture and stress can act.

That physical interpretation carries direct practical consequences. Rebound strength and particle density describe the intrinsic quality of the stone: a weak, loosely packed surface has little resistance to the cyclic stresses of daily heating and cooling, freeze-thaw action, and salt crystallization that drive flaking. Nearest fracture distance and fracture aperture, meanwhile, describe the pathways of attack: cracks act as conduits for water infiltration and as stress concentrators where failure initiates. A surface that is both mechanically weak and threaded with wide, closely spaced fractures is therefore the most likely candidate for severe spalling. For conservators, this suggests that monitoring programs should prioritize mapping fracture networks and tracking surface hardness over time, and that consolidation treatments aimed at strengthening the near-surface matrix and sealing microfractures may offer the greatest protective return.

Beyond its findings at a single site, the study’s authors frame the work as a preliminary scientific basis for the conservation of Damaidi and, importantly, as a potentially transferable model for analogous heritage sites. Rock art survives on every inhabited continent, from the painted caves of Europe to the engravings of the Australian outback and the deserts of the Americas, and spalling-type deterioration is a near-universal enemy of carved stone exposed to the elements. A framework built from twelve measurable indicators, processed through an uncertainty-aware model with transparent weighting, does not depend on the specific geology of Ningxia. Other research teams can adapt the indicator set to their local stone types, recalibrate the grade boundaries, and apply the same cloud-based evaluation pipeline, creating the possibility of genuinely comparable condition data across sites and continents.

The work also signals a broader shift in cultural heritage science, a field historically dominated by art historical judgment and artisanal intervention, toward the quantitative, data-driven methodologies familiar in engineering and the environmental sciences. The study was supported by the National Natural Science Foundation of China, the China Postdoctoral Science Foundation, the National Natural Science Foundation of Gansu, and programs at Lanzhou University of Technology, reflecting sustained institutional investment in heritage protection along the Silk Road corridor. For the petroglyphs of Damaidi, whose engraved panels have endured for thousands of years, the immediate benefit is a rigorous, repeatable diagnosis of where the stone is failing and why. For the discipline at large, the message is that the tools needed to save humanity’s oldest art may lie not only in the conservator’s kit but in the mathematics of uncertainty, provided researchers are willing to walk the rock faces and measure, instance by instance, exactly how the stone is letting go.

Subject of Research: Quantitative assessment of spalling deterioration in Damaidi petroglyphs using cloud modeling and game theory-based weighting

Article Title: A comprehensive evaluation of spalling disease in Damaidi petroglyphs using cloud modeling

Article References: Wu, G., Zhang, R., zhao, Z., Yu, G., Tian, Z., Sui, J., & Cui, K. (2026). A comprehensive evaluation of spalling disease in Damaidi petroglyphs using cloud modeling. npj Heritage Science. https://doi.org/10.1038/s40494-026-02972-z

Image Credits: AI Generated

DOI: 10.1038/s40494-026-02972-z

Keywords: Damaidi petroglyphs, rock art conservation, spalling disease, cloud model, game theory weighting, stone deterioration, cultural heritage science, fracture analysis, rebound strength, Pearson correlation, Ningxia, risk grading

The core problem the researchers set out to solve is deceptively simple to state. Tunnel large deformation is not triggered by any single factor. It emerges from the coupled effects of rock mass strength and integrity, groundwater action, discontinuity geometry, and construction disturbance. Yet conventional grading methods draw hard lines between deformation grades, treating a sample just above or below a threshold as fundamentally different even when the underlying conditions are nearly identical. When grade boundaries are blurred and indicator definitions vary between projects, these threshold-based approaches lose both accuracy and interpretability, particularly when a model trained on one tunnel is applied to another.

The team’s answer is built on cloud model theory, a mathematical framework designed to express the mapping between qualitative concepts and quantitative values while simultaneously capturing randomness and fuzziness. A basic cloud model describes any grade concept with three numerical characteristics: expectation, which marks the center of the concept; entropy, which reflects its dispersion and boundary uncertainty; and hyper-entropy, which describes how much that entropy itself fluctuates. A forward cloud generator then scatters thousands of cloud drops across the domain, each drop carrying a quantitative position and a membership degree between zero and one, producing a visual distribution that shows exactly where a grade concept is certain and where it dissolves into ambiguity.

What makes the new framework distinctive is its asymmetry. In real tunnels, grade intervals are rarely equal in length, and transitions toward the risk side, where deformation grows more severe, tend to be more sensitive than transitions in the safe direction. A single symmetric entropy cannot represent that physics. The researchers therefore introduced separate left and right entropies for each indicator and grade, derived from the spacing between adjacent grade centers, so the membership function decays at different rates on either side of the grade center. At the extreme outer grades, a saturation rule fixes membership at one once an indicator enters the endpoint zone, keeping the boundary response continuous and consistent with engineering semantics.

From these asymmetric clouds, the method extracts membership intervals rather than single values. By proportionally scaling the entropy to 0.9 and 1.1 times its nominal value, sensitivity analysis showed a stable intermediate range, the team constructs upper and lower envelopes of membership degree for every indicator at every grade. The bandwidth between those envelopes quantifies how dispersed and variable a sample is near a boundary, effectively softening the sensitivity that plagues hard-threshold grading. When an indicator value sits close to a grade interface, its support for neighboring grades rises simultaneously, which is precisely what field engineers observe in progressively evolving deformation rather than abrupt jumps.

Weighting the thirteen indicators in the system posed its own dilemma. Subjective weights, derived here from an expert judgment matrix built on the Saaty 1-to-9 scale and checked for logical consistency, capture hard-won engineering experience but carry randomness. Objective weights, computed statistically, have a firmer mathematical basis yet can drift from engineering intent. Rather than choosing one, the team computed four competing objective schemes, the entropy weight method, CRITIC, PCA, and TOPSIS, each emphasizing a different notion of importance, from information dispersion to inter-indicator correlation to variance structure to distance from ideal solutions. The objective scheme achieving the highest grading accuracy was selected, and then fused with the subjective weights in equal measure to form comprehensive weights.

The framework was validated on three tunnels with markedly different geology: the Telmo Tunnel in Sichuan, a 7.827-kilometer alignment contributing forty data groups; the Huangjiagou Tunnel in Hubei, 6.514 kilometers long with twenty data groups; and the Muzhailing twin-tube tunnel in Gansu, with tubes of 15.226 and 15.168 kilometers and thirteen data groups dominated by deep-burial squeezing conditions. Using indicators such as uniaxial compressive strength, rock quality designation, groundwater inflow, strike-tunnel axis angle, and discontinuity condition, the method achieved grading accuracies of 88 percent at Telmo, 90 percent at Huangjiagou, and 85 percent at Muzhailing, with a pooled accuracy of 88 percent across all seventy-three samples.

Perhaps the most striking result is how comprehensively the fused weighting outperformed every single scheme. At Telmo, accuracies for the entropy, CRITIC, PCA, TOPSIS, and purely subjective schemes ranged from 55 to 83 percent, all below the 88 percent of the comprehensive weights. At Muzhailing, where samples were few and boundaries most sensitive, the gap widened dramatically: objective schemes managed only 39 to 62 percent, while the fused weights reached 85 percent. Across the three tunnels, accuracy fluctuation shrank to just five percentage points under the comprehensive scheme, compared with up to 46 points for individual objective methods, evidence that the fusion strategy delivers both accuracy and cross-project stability.

Equally important is what the method does with its own mistakes. Misclassified samples clustered at adjacent-grade transitions, notably near the Grade IV to V and Grade III to IV boundaries, and the team showed these are not random failures but meaningful signals. In one representative Telmo case, the discontinuity orientation indicator supported Grade IV while rock strength leaned toward Grade III, placing the sample squarely in a transition zone where risk escalation was driven by discontinuity geometry rather than weak rock. The practical implication is direct: pre-reinforcing segments with unfavorable discontinuity orientations and adjusting support closure timing offers more improvement potential than trying to further strengthen the rock mass.

Aggregating the three projects produced a global ranking of all thirteen indicators, with uniaxial compressive strength at 0.2019 and rock quality designation at 0.1623 standing far above the rest, confirming that surrounding-rock strength and integrity are the fundamental controls on deformation grades. Strike-tunnel axis angle at 0.1241 and groundwater inflow at 0.0928 followed, with support closure time at 0.0900 highlighting the role of construction sequencing. The authors caution that the entropy perturbation factors remain empirical, the validation dataset is modest, and the global ranking reflects the cases included so far. Even so, the framework offers something tunnel engineering has lacked: a unified, interpretable, and transferable tool that turns fuzzy boundaries from a liability into a source of actionable early-warning guidance.

Subject of Research: A cloud-model-based fuzzy evaluation framework for grading nonlinear large deformation in tunnels

Article Title: A multi-dimensional cloud-model-based fuzzy evaluation method for grading nonlinear tunnel large deformation

Article References: Yao, S., Yang, P., & Zhou, J. (2026). A multi-dimensional cloud-model-based fuzzy evaluation method for grading nonlinear tunnel large deformation. Environmental Earth Sciences, 85(16), Article 413. https://doi.org/10.1007/s12665-026-13108-8

Image Credits: AI Generated

DOI: 10.1007/s12665-026-13108-8

Keywords: tunnel large deformation, cloud model, fuzzy evaluation, asymmetric membership, subjective-objective weighting, geotechnical engineering, weak rock, groundwater, discontinuities, risk grading, early warning, Muzhailing Tunnel