A Target Loss Probability Model-Based Framework for Risk Assessment and Defensive Resource Prioritization of Critical Infrastructure under Unmanned Aircraft System Threats

2.1 Target Loss Probability Model

The Target Loss Probability Model, proposed by Major, is a mathematical model for risk assessment that is mainly used to describe the interaction among RV, RA, and RD [8]. In the context of UAS threats to critical infrastructure and key assets, this study uses the Target Loss Probability Model as a risk assessment function from an attacker–defender perspective. Here, the attacker–defender relationship refers primarily to the resource-based confrontation between hostile UAS actors and the counter-UAS defense system of critical infrastructure and key assets. The core objective is to use the Target Loss Probability Model to generate a computable risk index and further analyze how improvements in defensive resource capability affect risk reduction.

In this study, the model is mainly used to characterize the interaction among attack capability, defensive capability, and RV. For computational tractability, attack behavior is treated from the defender’s perspective as an external pressure under a given threat scenario, and the analysis focuses on the effect of changes in the defensive resource capability index on the Target Loss Probability index. Under this attacker-defender relationship, the attacker’s payoff is regarded as corresponding to the defender’s loss; the expected payoff/loss (EL) is expressed as follows [8]:

$E L=\sum_{i=1}^n V_i \cdot p\left(V_i, A_i, D_i\right)$        (1)

where, $V_i$, $A_i$ and $D_i$ denote the RV, and RD for the target $i$, respectively.

In this study, $V_i$ represents the intrinsic value, strategic importance, and potential consequence severity of critical infrastructure and key assets under UAS threat events. $A_i$ represents the comprehensive attack capability of hostile UAS actors in conducting threat activities, including personnel capability, technical capability, financial support, UAS platform performance, payload capability, intelligence planning capability, and coordinated penetration capability. $D_i$ represents the comprehensive defensive capability of critical infrastructure and key assets against UAS threats, including detection, identification, countermeasures, barriers, and management capabilities. On this basis, the Target Loss Probability Model can be expressed as:

$p=p\left(V_i, A_i, D_i\right)=\exp \left(-\frac{A_i \cdot D_i}{\sqrt{V_i}}\right) \times\left(\frac{A_i{ }^2}{A_i{ }^2+V_i}\right)$         (2)

where, $p\left(V_i, A_i, D_i\right)$ denotes the Target Loss Probability index.

It should be noted that this value is not an empirically observed attack occurrence probability derived from a large number of historical samples. Rather, it is a normalized risk index calculated from the RV, RA, and RD. This index is mainly used to compare relative changes in risk under different defensive states. In the subsequent case study, RV and RA are treated as given scenario conditions, while RD is regarded as the main adjustable variable. Therefore, the model-recommended defensive resource capability value calculated later should be interpreted as a model-based defensive capability target, which is used to guide the improvement of defensive capability.

2.2 Indicator system and quantification

2.2.1 Adaptability analysis of model elements

Factor consistency between the Target Loss Probability Model and traditional risk assessment frameworks forms the theoretical foundation for the proposed model's applicability. Traditional risk assessment theory generally centers on three core factors: Threat likelihood $(T)$, vulnerability $(V)$, and consequence severity $(C)$. By contrast, the Target Loss Probability Model employs RA $(A)$, RD $(D)$, and RV $(V)$ as input parameters [9]. Despite terminology differences, the two models exhibit structural isomorphism in risk transmission mechanisms: threat likelihood is reflected by the attacker's capability and intent; vulnerability mirrors the defender's performance in detection, delay, and response; and consequence severity corresponds to the target's inherent attractiveness and loss magnitude. The factor alignment between the Target Loss Probability Model and traditional risk assessment can therefore be summarized as follows: RA corresponds to threat likelihood, RD corresponds to system vulnerability, and RV corresponds to target attractiveness and consequence severity.

In terms of input and output structures, both models rely on a specific indicator system to derive risk assessment results. The Target Loss Probability Model uses RV, RA, and RD to calculate the Target Loss Probability index and the model-recommended defensive resource capability value. Traditional risk assessment models usually rely on likelihood, vulnerability, and consequence to obtain a quantified risk level. Compared with static indicator scoring, the Target Loss Probability Model better reflects the interaction among attack capability, defensive capability, and target consequences in UAS threat scenarios. Therefore, the model has a certain degree of adaptability to the risk-generation process of UAS threats. The model output includes the Target Loss Probability index and the recommended defensive resource capability value, which can provide quantitative support for defensive resource prioritization.

Based on the above adaptability analysis of model elements, this study defines the three core parameters of the Target Loss Probability Model as follows:

RV: The intrinsic value of critical infrastructure and key assets. This value generates attractiveness to the adversary, while the consequences of a UAS intrusion directly reflect the magnitude of the adversary's expected payoff.

RA: The total resources available to hostile actors conducting UAS threat activities against critical infrastructure and key assets. This parameter aligns closely with the Threat Probability factor in traditional risk assessment models.

RD: The total defensive resources invested to counter UAS intrusions and reduce vulnerabilities in the low-altitude security system.

2.2.2 Construction of the indicator system

To ensure that the indicator system has a sound theoretical basis and is properly adapted to the UAS threat scenario, this study constructs the indicator system through a procedure consisting of theoretical framework determination, literature and case review, standards-based mapping, and expert review and revision. This process is based on the factor mapping discussed above, historical case analysis, and expert consultation. The construction of the indicator system follows a reverse-engineering principle: model parameters are first traced back to assessment elements, and these elements are then decomposed into observable indicators.

Specifically, RA is structured within a logical framework progressing from "intent" to "entity." Given the typical "human-machine coupling" characteristic of UAS attacks, assessing drone capabilities alone is insufficient to reflect the true threat level. Consequently, RA is decomposed into two dimensions: operator control and decision-making capability, and UAS attack capability. RD is based on the “detection–delay–response” functional chain of physical security systems. By integrating the specific requirements of C-UAS defense, this parameter is refined into five aspects to guide indicator construction: detection, identification, countermeasures, physical barriers, and security management. RV reflects both the attractiveness of the target to hostile actors and the potential loss that the defender may suffer under a UAS threat scenario. Accordingly, this indicator is decomposed into two dimensions: inherent attractiveness and the consequences of low-altitude UAS threats.

This mapping strategy ensures that each tertiary indicator is both directly observable and quantifiable while playing a distinct role in the model. It not only provides measurable parameters for the Target Loss Probability Model but also avoids the common issue in traditional assessments where indicators become disconnected from underlying risk generation mechanisms. The final indicator system for UAS threat risk assessment of critical infrastructure and key assets is presented in Tables 1, 2 and 3.

The indicators used in this study were selected according to the structural requirements of the Target Loss Probability Model and the operational characteristics of UAS threats against critical infrastructure. Since the model contains three core inputs, namely RA, RD, and RV, the indicator system was constructed around these three dimensions. The selection followed four principles. First, the indicators should be directly related to the UAS attack–defense process, including attack preparation, flight capability, reconnaissance or strike capability, detection, identification, countermeasure, physical protection, and consequence severity. Second, the indicators should be interpretable and scoreable by experts under data-scarce and security-sensitive conditions. Third, the indicators should avoid excessive overlap so that each level-3 indicator reflects a relatively independent aspect of risk. Fourth, the indicators should apply to critical infrastructure and key assets rather than to only one specific facility type.

Table 1. Indicators for Attacker Resources (RA)

Table 2. Indicators for Defensive Resources (RD)

Table 3. Indicators for target value (RV)

Some potential C-UAS indicators were not included as independent level-3 indicators for reasons of confidentiality, redundancy, or limited generalizability. For example, highly site-specific parameters, such as exact sensor models, exact deployment locations, classified blind spots, real-time response routes, and detailed countermeasure inventories, were excluded because they may involve sensitive security information and cannot be consistently obtained across facilities. Some technical parameters were integrated into broader indicators, such as detection coverage rate, detection latency, detection accuracy, minimum detectable target size, identification accuracy, and countermeasure engagement range. Similarly, environmental and operational factors were reflected through indicators such as low-altitude airspace complexity, resource exposure level, and response and recovery capability, rather than being listed as numerous separate site-specific variables. Therefore, the final indicator system is not intended to exhaustively enumerate all possible C-UAS factors, but to provide a general, scoreable, and model-compatible indicator set for UAS threat risk assessment of critical infrastructure.

2.2.3 Quantification and grading of indicators

To bridge the gap between qualitative description and quantitative calculation, this section establishes a standardized quantification and grading framework. This ensures all tertiary indicators are normalized on a unified scale, providing quantitative inputs for subsequent weighting. Given the mathematical structure of the Target Loss Probability Model—specifically the exponential and fractional terms in Eq. (2)—such standardization is essential to maintain numerical stability. To avoid mathematical singularities associated with zero values, this study deviates from the traditional [0, 1.0] interval for indicator quantification. Drawing upon systems engineering evaluation theory, a non-zero lower bound  is introduced to ensure model stability. A five-level gradient assignment method on [0.2, 1.0] is adopted, with discrete values of 0.2, 0.4, 0.6, 0.8, and 1.0. This design follows a five-level expert scoring logic and introduces a non-zero lower bound to maintain numerical stability. This quantification method can be regarded as a direct mapping of a five-level expert scoring scale, which is consistent with the basic logic of Likert-type ordinal rating scales [10]. It preserves the original ordinal information from expert ratings and prevents information distortion caused by interval compression. Higher values indicate greater attacker threat (for attacker indicators), stronger defensive capability (for defender indicators), or more significant RV and consequences (for RV indicators).

Indicator quantification and grading is grounded in leading international regulatory and technical standards, including: Regulation (EU) 2019/947 on UAS operations issued by the EASA [11]; Directive (EU) 2022/2557 on the resilience of critical entities (CER Directive) [4]; NATO Counter-UAS Technical Guidance [12]; and ISO 31000 Risk Management Guidelines [13]. These cross-domain references ensure the authority and scientific rigor of the proposed framework. The complete five-level scoring rubrics for all 38 level-3 indicators are provided in Supplementary Material A. Additional aviation artificial intelligence guidance and Counter-UAS protection guidance were also considered in refining the indicator descriptions and scoring logic [14, 15].

2.3 Weighting and risk-grading method

2.3.1 Combined weighting via the Maximizing Deviation Method

The Maximizing Deviation Method, also known as the Maximum Dispersion Method, is a combined weighting approach widely used in multi-attribute decision-making and comprehensive evaluation. Its core principle is to determine indicator weights by maximizing the deviation of each indicator across different evaluation objects. Given the hierarchical structure of the evaluation indicators, this study adopts the G1 method, also known as Order Relation Analysis, as the subjective weighting method to avoid the high computational burden and consistency-check difficulties associated with the traditional Analytic Hierarchy Process (AHP). To incorporate data-driven information, the Entropy Weight (EW) method is used as a relatively objective weighting method based on the dispersion of expert scores. The combined weighting strategy takes the maximum and minimum weights obtained from the subjective and relatively objective weighting methods as the upper and lower bounds of the combined weights. The main calculation steps are as follows:

(1) Construct the weight matrix

Assume each of the $m$ weighting methods is employed to assign weights to the $n$ evaluation indicators, yielding the weight matrix E:

$E=\left[\begin{array}{ccc}a_{11} & \cdots & a_{1 m} \\ \vdots & \ddots & \vdots \\ a_{n 1} & \cdots & a_{n m}\end{array}\right]$        (3)

where, $a_{n m}$ denotes the weight of the n-th indicator under the m-th method.

(2) Determination of the combined weight interval

For each indicator i, the range of its combined weight $a_i$ is expressed as:

$a_i \in\left[a_i^{-}, a_i^{+}\right]$        (4)

$a_i^{-}=\min \left(a_{i 1}, a_{i 2}, \ldots, a_{i m}\right)$, $a_i^{+}=\max \left(a_{i 1}, a_{i 2}, \ldots, a_{i m}\right)$.

(3) Construction of the maximum deviation weighting model

The objective function is to maximize the variance of the evaluation results:

$\max s^2=\frac{1}{n-1} \sum_{i=1}^n\left[a_i\left(x_i-\bar{x}\right)\right]^2$        (5)

The constraints are defined as follows:

$\left\{\begin{array}{c}\sum_{i=1}^n a_i=1 \\ a_i^{-} \leq a_i \leq a_i^{+}, i=1,2, \cdots, n\end{array}\right.$        (6)

where, $s^2$ is the variance of the comprehensive evaluation value, $x_i$ is the value of the $i$-th indicator, and $\bar{x}$ is the average value of all indicators.

2.3.2 Determination of indicator weights

Indicator weighting is a critical preprocessing step for the Target Loss Probability Model. It is applied to UAS threat scenarios involving critical infrastructure and key assets. This section addresses the challenge of data scarcity—a fundamental issue in such assessments—to provide reliable weight inputs for subsequent model implementation.

Before calculating the indicator weights, it is necessary to clarify the independence of the factors in the Target Loss Probability Model. The Target Loss Probability Model is not a linear weighted model, but a nonlinear function with multiplicative coupling among three factors, as shown in Eq. (2). Constructing a global weight matrix for all 38 indicators would cause dimensional confusion. For example, Strategic Value and Criticality ($\mathrm{RV}_1$) and Detection Coverage Rate ($\mathrm{RD}_2$) belong to different model dimensions. They lack direct algebraic comparability. Weight deviations would be amplified in cross-dimensional multiplicative operations. This would cause system errors to interact. Consequently, assessment results would exhibit significant deviations. Therefore, this study follows the principle of independent weighting within each system and model-based coupling across systems. We conduct independent weighting calculations within the RA system (12 indicators), RD system (18 indicators), and RV system (8 indicators).

For UAS threat risk assessment of critical infrastructure and key assets, publicly reported successful attacks against nuclear power plants and government core facilities are scarce. This prevents forming large-sample statistical matrices required by the Entropy Weight method. Moreover, data gaps in critical-infrastructure modelling often require expert assumptions and extrapolations, which may introduce uncertainty into model outcomes [7]. To this end, this study adopts an Entropy Weight–Order Relation combined weighting strategy. Expert group scoring is introduced as the input source for information entropy, and ten experts are invited to independently score the three subsystems. The Entropy Weight method is then used to measure the dispersion of expert opinions and to identify indicators with strong discriminative power in the evaluation, thereby supplementing the purely subjective ranking process. For indicators with large score variations, these indicators are considered to have higher uncertainty in the assessment process. Greater weights are therefore assigned to reflect their stronger discriminative power [16]. The G1 method complements the Entropy Weight method by preserving expert priority judgments for key indicators.

To obtain the final combined weights, this study constructs a maximum-deviation weighting model using the G1 and EW weights as boundary conditions. The objective of the model is to maximize the variance of the evaluation results, so that indicators with stronger discriminative power receive relatively higher weights while expert priority judgments are preserved.

To avoid overloading the main text, the complete weight matrix is not presented here. The complete weights of all 38 level-3 indicators calculated by the G1 method, Entropy Weight method, and Maximizing Deviation Method are provided in Supplementary Material B. The representative high-weight indicators mainly include Financial Support (0.203), Intelligence and Planning Capability (0.193), Detection Coverage Rate (0.242), Detection Accuracy (0.151), Strategic Value and Criticality (0.248), and Personnel Casualties (0.179). These results suggest that resource availability, intelligence preparation, front-end detection capability, and consequence severity are the most influential components in the three subsystems.

2.3.3 Target Loss Probability Model Calculation: Construction of the risk grading matrix based on the model output distribution

The risk grading matrix in this study is used to classify the Target Loss Probability index into relative risk levels. It provides a semi-quantitative basis for interpreting model outputs and comparing different defensive states of critical infrastructure and key assets under low-altitude UAS threats. It thereby provides a reference for subsequent model application.

Considering the nonlinear characteristics of the Target Loss Probability Model, simple equal-interval partitioning or extreme-value partitioning cannot adequately reflect the distribution of the model outputs and may lead to overly coarse risk grading. Therefore, this study combines Monte Carlo simulation with the percentile method to construct relative risk grading thresholds for the Target Loss Probability index [17]. It should be noted that these thresholds do not represent universal risk thresholds applicable to all critical infrastructure and key assets. Instead, they are internal grading results derived under the indicator system, quantification interval, and model functional form used in this study. Their main purpose is to compare changes in the Target Loss Probability index under different defensive states and to provide a relative basis for subsequent defensive resource prioritization.

Based on the indicator quantification framework in Section 2.2, the values of RV, RA, and RD are all normalized within the interval [0.2, 1.0]. Since UAS threats against critical infrastructure and key assets are low-frequency but high-consequence events, sufficient historical samples are not available to estimate the empirical probability distributions of (V), (A), and (D). Therefore, this study adopts a uniform distribution as a non-informative prior assumption for the baseline Monte Carlo simulation. Under this assumption, each value within the standardized interval has the same probability of being sampled, which avoids introducing additional subjective preference for any specific risk state at the risk-grading stage.

Specifically, RA, RD, and RV are assumed to follow independent uniform distributions over the interval [0.2, 1.0]. The model undergoes (N = 100,000) random sampling simulations. Each simulation yields a triplet (V, A, D), which is then substituted into the Target Loss Probability Model in Eq. (2) to calculate the corresponding Target Loss Probability index P. The simulation results are shown in Figure 1. The results indicate that the Target Loss Probability index is distributed within [0.0312, 0.5318]. With reference to the quartiles of the simulated data, this study divides the Target Loss Probability index into four relative risk levels: Low, Medium, High, and Extreme. The corresponding thresholds are shown in Table 4.

1.png

Figure 1. Distribution and cumulative distribution of Target Loss Probability index

To examine whether the risk-grading thresholds are sensitive to the assumed input distribution, this study further conducts a distributional robustness check at the model-construction stage. In addition to the baseline uniform distribution, two alternative distributions are considered. The first is a triangular distribution with lower bound 0.2, mode 0.6, and upper bound 1.0. The mode 0.6 is the midpoint of the standardized interval and represents a neutral medium-level assumption. The second is a scaled beta distribution 0.2 + 0.8 × Beta (2, 2), which also concentrates more probability mass around the middle of the standardized interval. These two distributions are used to test whether the quartile-based risk thresholds remain relatively stable when the input parameters are no longer uniformly distributed. The robustness check results under the three input distribution assumptions are summarized in Table 5. The robustness results show that the quartile thresholds vary only within a limited range under different input distribution assumptions. The Q50 threshold remains around 0.21–0.23, and the Q75 threshold remains around 0.27. This indicates that the Monte Carlo-based risk grading matrix is not highly sensitive to the baseline uniform distribution assumption. Therefore, the thresholds obtained in this section can be used as relative internal grading criteria for subsequent case analysis under the proposed model.

Table 4. Relative risk grading thresholds for the Target Loss Probability index

Table 5. Robustness check of Monte Carlo-based risk grading thresholds under different input distributions

This study proposes a risk assessment and defensive prioritization method based on the Target Loss Probability Model for UAS threats against critical infrastructure and key assets. Through the construction of a three-dimensional indicator system incorporating RV, RA, and RD, and through a combined weighting strategy based on the Maximizing Deviation Method, this study addresses the weighting problem under small-sample data conditions. In addition, Monte Carlo simulation is used to establish relative grading thresholds for the nonlinear model output. This study introduces Sobol global sensitivity analysis. It reveals that detection coverage rate and detection accuracy are the first-order dominant factors of system risk. The anonymized nuclear facility case study demonstrates that the proposed method can transform qualitative UAS threat factors into a structured and computable risk assessment process. Increasing the defensive resource capability index by 48.77% reduces the Target Loss Probability index by 19.79%. This results in a one-level reduction in the relative risk level from “High” to “Medium”. The method provides auxiliary decision support for prioritizing counter-UAS defensive resource improvement for critical infrastructure and key assets.

Nevertheless, the case study is based on an anonymized scenario and small-sample expert scoring, which may limit the direct generalization of the numerical results. Future research may further validate the proposed framework using multi-scenario datasets, larger expert panels, and dynamic UAS attack-defense simulations.