## 1 Introduction  
Beamforming technology with spherical microphone arrays (SMAs) is widely used in localizing and identifying noise sources in the mid-high-frequency range because of its small size, full space directivity, and good resolution in medium and high frequencies [1, 2, 3]. In recent years, low-frequency noise sources have attracted extensive research attention. Therefore, it is essential to develop a low-frequency-noise-source localization and identification method using a small-aperture SMA. An SMA, used in combination with the spherical harmonic beamforming (SHB) algorithm, is the most commonly used noise-source localization and identification method in the mid-high-frequency range. This method decomposes the sound field into a superposition of spherical harmonic functions and achieves the localization and identification of noise sources through the orthogonality of the spherical harmonic functions [1, 4]. However, it is necessary for the SHB algorithm to reasonably select the truncation order of the spherical harmonic function according to the number of SMA elements. Furthermore, the SHB algorithm is vulnerable to noise and sidelobe interference in the application process, which leads to a poor localization effect [3, 4]. In addition, the small aperture of the SMA limits the frequency range of the SHB algorithm's application, especially due to its poor localization effect of low-frequency noise sources [5; 6]. In order to more accurately localize noise sources, scholars have provided solutions in two aspects, based on extensive research. On the one hand, scholars have improved the beamforming effect using algorithms. [PERSON] proposed a filter and sum (FAS) beamforming algorithm to effectively suppress the side lobes of the beam [7]. Scholars have improved the spatial resolution of sound source localization by using clear algorithms for sound source identification, such as DAMAS, CLEAN, etc. [8]. On the other hand, researchers have improved the applicable frequency range by redesigning the physical structure of a spherical array. [PERSON] discussed the design method of an SMA and provided an optimal layout scheme of microphones [9]. [PERSON] and [PERSON] et al. designed a double-layer open SMA to overcome the instability of the single-layer open SMA at the nadal of the spherical Bessel functions, as well as broadening the applicable frequency range of a single-layer open SMA [10; 11]. [PERSON] put forward another design and optimization method of double-layer SMA that arranged an open sphere at a certain distance outside the rigid SMA. Furthermore, the author discussed the advantages of this SMA in improving the frequency range and the optimal design of the SMA [12]. Using different traditional methods, in this paper, we consider both algorithms and the extrapolated aperture technique to achieve a better localization performance for the localization of low-frequency noise sources, which does not affect the aperture and the element number of the actual SMA. In this paper, a virtual SMA with a large aperture is first obtained by extrapolating the SMA, and its ability to localize low frequencies is verified by generalized inverse beamforming algorithm (GIB) [5]. Then, the virtual SMA and actual SMA are combined to form a virtual double-layer open SMA, and a virtual SMA with velocity elements is obtained by the finite difference for all the same azimuth elements. In addition, because vector signal processing technology can improve the effect of low-frequency applications by simultaneously using sound pressure and velocity, this paper presents the transfer function of'sound pressure-velocity' and proposes the joint processing method of sound pressure and velocity of the obtained virtual vector SMA. The proposed method can effectively improve the low-frequency localization performance of the actual SMA. The remainder of this paper is arranged as follows: Section 2 gives the theory of the virtual SMA extrapolation method. In Section 3, an SMA noise source localization method based on GIB is briefly described.