## 1 Introduction  
Attitude measurement is the foundation for controlling carrier motion to a specified coordinate point, which occupies an essential position in various fields, such as aircraft, land carriers, ships and underground energy extraction [1; 2]. The attitude measurement of the wellbore is crucial for locating the reservoir position. A highly accurate wellbore attitude can improve oil and gas recovery and reduce operating costs in an internationally tight energy inventory situation [3]. Generally, after completing the drilling operation, the driller drops the attitude measurement instrument at the specified survey site to obtain 3D attitude information from static sensor data. Then, the communication system is employed for data transmission to the drilling platform. Such a single-point measurement mode is widely applied to vertical drilling or sidetracking wells. Wellbore attitude measurement technology has been developed for decades. In the 1970s, patents reported electronic devices consisting of three-axis magnetometers and three-axis accelerometers to measure well inclination and azimuth angles [4; 5]. With the development of technology and the iteration of research findings, the current measuring instruments have been diversified to include gyroscopes, accelerometers, magnetometers and combinations between sensors [6]. To enhance borehole planning and reservoir positioning, numerous researchers have concentrated on improving wellbore attitude accuracy through optimal algorithms and system construction. [PERSON] et al. proposed a robust Mag/INS-based orientation estimation algorithm that utilizes the failure detection and integration techniques to establish a federal Kalman filter framework fusing gravity and the magnetic field [7]. The robust combined algorithm effectively eliminates the anomalous magnetic azimuth by comparing it to the true azimuth from INS. [PERSON] et al. developed a motor-driven IMU-based Measurement While Drilling (MWD) system to enhance the gyroscope bias estimation effect and achieve high accuracy in attitude measurements [8]. The MWD systems with motors may not be suitable for strong downhole vibrations. However, the rotary modulation scheme is effective for continuous trajectory measurements in cased wells. [PERSON] et al. established a neural network model that combines the K-proximity method to enhance the accuracy of drilling tool attitude [9]. The proposed Support Vector Classification Machine method was utilized to train the measured data for calibrating the errors of magnetometer-based MWD. [PERSON] et al. introduced a Bi-orthonormal Optimal Signal Search-based de-noising for improving wellbore continuous MWD surveying utilizing a MEMS-reduced inertial sensor system algorithm together with an extended Kalman filtering module [10]. Only four-axis low-cost sensors are used to calculate the wellbore attitude. However, accurate attitude information is hard to obtain for some specific boreholes such as horizontal wells. [PERSON] et al. reported a novel state-space model based on the rotational angular velocity of a rotary-steerable drilling system in each axial direction. The dynamic measuring algorithm with two Kalman Filters was established to solve the spatial attitude of the bottom rotating drilling string and minimize the inclination azimuth errors [11]. The various optimization techniques discussed in the literature have proven to be successful in improving wellbore attitude accuracy. However, the attitude measurement in near-vertical boreholes has been addressed with less attention. The near-vertical attitudes affected by radial accelerometers have low-precision horizontal attitudes causing large azimuth deviation. The present near-vertical measurements are performed with the equivalent tool face angle measured by radial magnetometers or gyroscopes together with the marked points of the instrument to judge the approximate direction. This particular angle is defined as the magnetic tool face or gyroscopic tool face. [PERSON] et al.