## 1 Introduction  
In the past decade, investigation of the formation of light filaments in air has become a topic of active experimental and theoretical research due to their various potential atmospheric applications, such as remote sensing of chemical/biological agents [1] and meteorological elements [2], high-voltage discharge guiding [3], air waveguiding [4], and weather-related applications [5; 6; 7].  
The widely acceptable physical mechanism that sustains the formation of light filaments is the result of a dynamic balance between the Kerr self-focusing effect and the plasma defocusing effect [8]. Firstly, the optical Kerr effect focuses the pulse and then the beam intensity experienced obviously increases. When the beam intensity becomes sufficiently large, it ionizes the medium and creates an electron plasma [9]. Then, the electron plasma defocuses the trail of the pulse and strongly shortens its leading edge around and beyond the nonlinear focus [10].  
Many filamentation-based applications rely on long distance propagation in atmosphere [11]. It is well known that real atmospheric air is composed of various types of gas molecules, which vary in content. In previous theoretical simulation studies, the ionization of N\({}_{2}\) and O\({}_{2}\) was usually only considered since they are the main components of air gas [12; 13; 14]. The ionization of water vapor molecules is often neglected because their concentration is usually too low. However, influences of water vapor on multiple physical effects during the filamentation process are expected to be very different when a laser pulse wavelength increases [15]. Up to now, only a few studies have considered the influence of relative humidity on the propagation of femtosecond pulses. [PERSON] et al. [16]showed that the spatiotemporal shape of femtosecond pulse at a central wavelength of 800 nm broadens and deforms under high relative humidity conditions. Simulation studies also showed that physical effects like dispersion and linear absorption become more and more important in the formation and propagation dynamics of 10 \(\upmu\)m filaments with the increase of relative humidity [15]. More recently, [PERSON] et al. [17] experimentally identified that the photoionization of water vapor acts as the dominant ionization process in atmospheric air for femtosecond laser pulses at a 248-nm central wavelength and the relevant ionization coefficients were also measured. With increasing air humidity, the plasma channels generated by ultraviolet laser pulses in air become longer and wider [18]. However, the detailed dynamic evolution of UV filamentation and its energy deposition characteristics in an atmospheric environment with different humidity levels has rarely been reported. Allowing for the reported valuable applications of UV laser filaments in meteorology [19; 20; 21], it is also necessary to evaluate how water vapor will affect the transmission of femtosecond lasers in a cloud environment.  
In this paper, we numerically investigate the influences of ionization of water vapor on nonlinear propagation of ultraviolet (UV) filaments. The formation and propagation dynamics of UV filaments under different humidity conditions are studied in detail. The remainder of this paper is organized as follows: the model used to describe the propagation of femtosecond pulse in gas medium is introduced in Section 2; in Section 3, the evolution of filament along propagation distance z in different gas media is investigated; and the conclusions are summarized in Section 4.