## 1 Introduction  
The resilience of critical infrastructure in formerly used conflict zones is increasingly challenged by the presence of unexploded ordnance (UXO), which poses both environmental and structural risks, as well as risk of bodily injury. These UXOs are often encountered during the construction or repair of infrastructure lifelines such as water, gas, communication, and electric lines, as well as roads, tunnels, and foundations. Thus, comprehensive risk management of infrastructures at these locations requires that these sites be remediated [1]. Effective remediation requires accurate prediction of the depth of burial (DoB) of munitions, as detection and extraction technologies depend on depth-specific capabilities. However, DoB prediction is complicated by significant uncertainties in soil properties and impact conditions, necessitating a probabilistic approach. Thus, uncertainty in munition detection remains a pressing concern [2]. Therefore, selecting the most appropriate technology for UXO detection depends on an initial estimation of the DoB for the ordnance at a specific site considering the spatial uncertainty of geotechnical conditions [3; 4] and uncertainty of ballistic conditions [5]. This has led to a growing interest in applying statistical methods for studying UXO sites [6; 7; 8; 9]. However, these methods have not been applied specifically to the DoB problem. Engineers and scientists have studied DoB in soil for over two centuries dating back at least to Robins [10]. In recent years, problems of interest ranged from the penetration of torpedo anchors and dynamic penetrometers into the sea floor [11; 12; 13; 14], to planetary impact [15]. Many studies have been performed under either controlled laboratory conditions [16], or using verified numerical simulations [17; 18; 19]. These have led to several deterministic models; however, a stochastic treatment of the problem is required to consider the effect of uncertainties in soil properties. Unlike materials in many other engineering disciplines, soil deposits exhibit large variations in stratigraphy not only between sites but even within seemingly homogeneous deposits at a single site. In particular, geotechnical data is more appropriately represented as a histogram of values rather than a single point [20; 21]. In addition, failure to account for uncertainties in geotechnical models can introduce bias and influence the predictions. For example, [PERSON] et al. [22] examined the role of randomness on the predicted failure of a slope using probabilistic slope stability analysis employing the finite element method. Their findings demonstrate that in slopes comprising soil with spatial variability, the critical failure mechanism tends to follow the path of least strength within the soil, which may not be the same as that observed in deterministic slope stability analysis. In addition, it has been observed that failure probabilities calculated without considering model uncertainty do not represent the actual failure probabilities of geotechnical systems [23]. Reliability analysis within geotechnical engineering has experienced substantial advancements in recent decades, with the application of various probabilistic methods to a wide range of geotechnical problems. For instance, Monte Carlo simulations have been utilized to characterize uncertainties in geotechnical models, including the at-rest earth pressure coefficient and shallow foundation bearing capacity [24], as well as for probabilistic site characterization using multiple cone penetration test data [25]. On the other hand, the penetration of projectiles has been a vexing problem that has eluded engineers and scientists for two centuries. The problem does not lend itself to deterministic analysis because of the great variability associated with the various soil and impact parameters.