## 1 Introduction  
Methane (CH\({}_{4}\)), as an important greenhouse gas, has attracted extensive global attention due to its potential impact on climate change [1, 2, 3]. The greenhouse effect in the atmosphere has been increasing due to the CH\({}_{4}\) concentration steadily rising. CH\({}_{4}\) emissions are gradually becoming an urgent environmental problem in different areas, including natural gas extraction, landfills, and agriculture [4, 5, 6]. Therefore, the development of an accurate, efficient, and sensitive method for CH\({}_{4}\) concentration measurement is of critical importance. Such methods not only enable precise environmental monitoring, but also provide data support for the management and mitigation of greenhouse gas emissions [7, 8, 9]. Absorption spectroscopy has been widely applied in chemical analysis [10; 11], environmental monitoring [12; 13], and life sciences [14; 15] due to its high selectivity and sensitivity. Among them, the off-axis integrated cavity output spectroscopy (OA-ICOS) has demonstrated outstanding measurement accuracy and sensitivity in gas measurement, and has become an advanced technology in the gas concentration measurement [16; 17; 18]. By extending the effective optical interaction path between light and matter, which significantly improves the measurement of low concentration gases. [PERSON] et al. [19] proposed a wavelength modulation spectroscopy (WMS) combined with the OA-ICOS for the measurement of CH\({}_{4}\) concentration. The 60 cm cavity with an effective absorption path length of 2150 m was fabricated, and the minimum detection limit (MDL) of 13 ppb was obtained. [PERSON] et al. [20] developed an open-based OA-ICOS sensor system for the measurement of oxygen concentration. The light from a distributed feedback (DFB) laser around 761 nm was coupled into a Fabry-Perot cavity, and the MDL of 0.0038% was obtained at 20 s. [PERSON] et al. [21] demonstrated a portable carbon dioxide sensor system based on the OA-ICOS. Fluid dynamics were used to establish a stable optical resonance cavity with an optical path length of 891 m and a physical length of 30 cm. The sensor exhibited a wide dynamic measurement range of 0.011-800 ppm and an MDL of 11 ppm. However, cavity mode noise as the main noise source in the OA-ICOS systems seriously affects the measurement accuracy and sensitivity [22; 23; 24]. To meet this challenge, various methods are proposed to suppress the cavity mode noise. [PERSON] et al. [25] proposed a method combining Variational-Mode-Decomposition (VMD) and Savitzky-Golay (SG) algorithms to effectively filter out optical noise and improve the signal-to-noise ratio (SNR). The received photoelectric signals were decomposed into multiple frequency components by the VMD algorithm and subsequently reconstructed by the SG algorithm. The Extreme Learning Machine algorithm optimized by the Dragonfly Algorithm was applied to CH\({}_{4}\) concentration inversion for further improving the measurement accuracy. [PERSON] et al. [26] presented a multi-input multi-output laser-to-cavity coupling scheme for suppressing cavity mode noise. The validation of the measurement scheme was confirmed by simulations and experiments. [PERSON] et al. [27] developed a comprehensive 3D ray tracing model to simulate the optical re-injection process. The re-injection effect was improved by the larger diameter re-injection mirrors. The transmission intensity and SNR were also improved. [PERSON] et al. [28] injected RF white noise into the current driving of the laser to broaden the laser emission linewidth. The cavity mode noise was effectively suppressed. In this work, a CH\({}_{4}\) sensor based on direct absorption off-axis integrated cavity output spectroscopy (DA-OA-ICOS) and wavelength modulation off-axis integrated cavity output spectroscopy (WM-OA-ICOS) was presented. The sensor was designed with a compact optical resonance cavity to enable rapid and efficient CH\({}_{4}\) concentration measurements with minimal gas samples.