# A Method for Evaluating Systematic Risk in Dams with Random Field Theory
## 1 Introduction  
Dams are widely used in the world for irrigation, shipping, power generation, flood control, water supply, etc. The safe operation of dams has long been a great concern for governments and scholars [1; 2]. According to the statistics of 142 gravity dam accidents between 1802 and 1998, more than 50% were caused by foundation instability and structure damage [3; 4; 5; 6; 7; 8], so making sure the long-term safe operation of gravity dams is particularly important. What is more, the parameters of concrete and rock materials in dam-foundation systems objectively exhibit spatial variability due to construction factors, load effects, etc., significantly affecting the structural response and stability safety risks of the dam. Therefore, it is necessary to reasonably evaluate the safety risks of gravity dams on complex foundations considering the spatial variability of concrete and foundation materials. Dam risk analysis has garnered widespread attention, particularly in areas, such as system risk, stability risk, overtopping risk, etc. [9; 10; 11; 12], one of which is the instability risk of dam-rock foundation systems. The instability of a dam-rock foundation system generally results from pre-existing geological features and uncertainty in the foundation. For a gravity dam foundation, reliability analysis and assessment play a critical role in safety assessment. Thus, much research has been conducted on the reliability analysis of system safety in gravity dams and foundations. [PERSON] and [PERSON] [13] put forward a finite element method-support vector machine-based hybrid methodology to quantifythe reliability of concrete dams. [PERSON] et al. [14] proposed a reliability analysis method for gravity dams based on a probability fuzzy-interval-hybrid model and an improved branch and bound method. [PERSON] et al. [15] developed an advanced first-order second-moment method based on finite elements for the reliability of gravity dam anti-sliding stability. [PERSON] et al. [16; 17] performed a reliability analysis of a gravity roller-compacted concrete (RCC) dam using Monte Carlo simulation (MCS) and the first-order reliability method (FORM). [PERSON] et al. [18] quantified the 12 risk factors for earth dam failure using a Bayesian network, which was helpful for the development of an intelligent BN modeling framework. The above studies show that the random variable model is usually adopted in the risk or reliability analysis of concrete gravity dams and foundations [19; 20], where the spatial variability of material parameters is homogenized by statistical characteristics. However, the rock mass in the foundation is generally natural material, the engineering characteristics of which are significantly affected by deposition, chemical weathering, physical degradation, hydrothermal change, and loading history, performing obvious spatial variability in the material parameters [21]. Moreover, concrete gravity dams strongly show heterogeneity and stochastic characteristics in their material parameters due to the influences of multiple construction zones, complex construction processes, long construction periods, and difficult maintenance, and they are highly vulnerable to environmental impacts. Rock masses and concrete can behave quite differently from homogeneous material due to the spatial variability of the parameters [22], which further affects the system reliability of the dam and rock foundation. Recently, some researchers have paid special attention to the spatial variability of dam and foundation materials. [PERSON] et al. [23] and [PERSON] et al. [24] determined the influence laws of concrete material parameters on the dynamic response characteristics of gravity dams based on the aspects of damage area distribution, crack length statistics, displacement in the crest, energy dissipation, and parameter sensitivity. [PERSON] et al.