## 1 Introduction  
A millimeter-wave (MMW) fuze is a device that uses the MMW frequency band for target detection and detonation, and is widely employed in missile, artillery, and other weapon systems owing to its high precision, reliability, and strong resistance to electromagnetic interference [1, 2]. Given that MMW fuzes operate in complex and variable climatic conditions, they must demonstrate superior environmental resilience to ensure reliable performance. The wavelength of MMW signals is comparable to the size of aerosol particles such as raindrops, snow, and smoke. Consequently, MMW signals are susceptible to attenuation, scattering, and depolarization effects caused by these particles [3, 4]. Among these particles, raindrops with wavelengths closest to MMW signals have the most significant impact on MMW fuze detection [5, 6]. Therefore, studying the characteristics of MMW fuze echo signals under rainfall conditions is essential. To explore the impact of rainfall on the echo signal of MMW detectors, scholars have delved into the impact of factors such as beam width, rainfall intensity, and operating frequency on the performance of MMW detectors in rainy environments, across various application fields including MMW Doppler fuzes, pulse radars, and automotive MMW radars. [PERSON] et al. analyzed the influence of rain clutter on MMW Doppler fuzes and highlighted the need to consider the impact of rain clutter on fuze detection when the radar cross-section area of the target is small or the antenna beam is wide [7, 8]. [PERSON] and [PERSON] proposed a method for simulating the effects of rainfall environments on automotive radars by stratifying the rain field [9]. [PERSON] et al. examined rain attenuation and backscatter, noting that rainfall reduces the detection range of MMW radars by 45% [10]. [PERSON] et al. compared wide-beam and narrow-beam radars, emphasizing the superior detectability of narrow-beam radars in rainy conditions [11]. [PERSON] et al. analyzed the influence of the rainfall rate and radar operating frequency on the rain backscatter noise using MMW pulse radar echo data under rainy conditions [12]. [PERSON] et al. developed a model for MMW ground fuze echo signals under rainfall conditions using the single scattering method, and preliminarily analyzed the impact of rainfall, fuze frequency, and detection distance on the characteristics of the echo signal [13]. None of above analyses considered the effects of multiple scattering processes of MMW signals under rainy conditions. To address the issue of multiple scattering of MMW signals in rainy environments, scholars gradually established high-order scattering theory. [PERSON] et al. developed a second-order multiple scattering theory for MMW signals in such conditions [14]. [PERSON] and [PERSON] used an exhaustive method to sample third-order and higher-order scattering paths and investigated the characteristics of backward scattering enhancement effects of MMW signals under rainy conditions [15]. [PERSON] et al. simulated the attenuation and reflection effects of MMW plane-wave signals in rainy environments using the Monte Carlo method. The method theoretically possesses a formidable capability to simulate scattering paths with unlimited numbers of orders. Results suggested that the simulation outcomes from multiple scattering calculations were more consistent with measured data than those from single scattering [16]. However, these studies focused on evaluating attenuation and backward scattering of MMW signals due to rainfall, without considering detection targets, and thus cannot be directly applied to MMW fuze target detection under rainy conditions. To integrate multiple scattering methods with target detection, this study establishes an MMW fuze echo signal model for rainy environments. Given the complexity of multiple signal scattering paths in the echo signal model, the adoption of Monte Carlo methods as a research tool is particularly suitable.