Infrared attitude phase compensation for mountain interference using beta distribution and hyperbolic tangent function Open Access

In recent years, with the rapid development of infrared technology, its application prospects in fields such as military, aerospace, and meteorology have garnered increasing attention. Particularly, in the field of projectile attitude measurement, infrared sensors—especially long-wave infrared (LWIR) sensors have become a research hotspot due to their advantages of low cost, all-weather capability, and high performance. However, in practical applications, the output signals of infrared sensors are prone to environmental noise and other interferences, resulting in amplitude and phase errors. One prominent source of such interference is mountain interference, which refers to signal disturbances in infrared attitude measurement caused by abrupt changes in surface elevation. These terrain variations can lead to partial or complete occlusion of the natural horizon. In addition, differences in slope orientation, surface materials, and thermal properties can produce irregularities in surface infrared radiation. Such anomalies disrupt the expected periodicity and intensity of the infrared signals, thereby further complicating accurate attitude estimation. Therefore, accurately identifying and correcting these errors has become key to improving measurement accuracy.

Researchers have conducted extensive discussions and studies on the simplified infrared attitude measurement model. The low-cost roll control system developed by Frank et al. greatly reduces costs, but may exhibit significant control errors in high-precision applications, and the accuracy of aerodynamic modeling may be limited in complex environments, affecting system stability.¹ The directional state estimator proposed by Du et al. enhances attitude estimation accuracy using infrared sensors but it may experience inaccuracies in environments with significant thermal radiation interference, leading to less reliable results.² The real-time horizon estimation method proposed by Zhao improves estimation accuracy, but online calibration may be influenced by sensor noise and environmental changes, thus affecting the accuracy of attitude estimation.³ Yu et al. combined the extended Kalman filter to improve estimation accuracy, but in systems with strong nonlinearity, there may be slow convergence or local optima issues.⁴ The infrared focal plane array attitude measurement model proposed by Cao has advantages in fault classification, but it relies on model assumptions, which may not adapt to complex real-world scenarios, leading to reduced classification accuracy.⁵ The attitude determination method based on ellipse observations proposed by Dario may be limited by sensor resolution and environmental factors in practical applications, leading to larger attitude estimation errors.⁶ 

In addition, since the simplified infrared attitude measurement model is derived from actual data, there is inevitably some error between it and real-world measurements. Researchers have conducted various studies to compensate for and correct these errors. Gu proposed a method for classifying and identifying space targets based on their infrared radiation characteristics, which has proven effective. However, because it relies on models tailored to different attitude targets, its accuracy is affected by environmental variations, leading to reduced classification and recognition performance.⁷ John introduced a multi-sensor infrared channel calibration method that improves the measurement of infrared and water vapor channels in geostationary imagers, but its applicability may be constrained by specific sensors and application scenarios.⁸ Xiong developed a Q-learning-based target selection algorithm that enhances the performance of the extended Kalman filter, yet the slow convergence rate of Q-learning and the heavy computational burden in high-dimensional observation spaces pose challenges.⁹ Cao established a modified infrared attitude measurement model based on odd-power functions, which improves attitude calculation accuracy. However, its dependence on specific model assumptions limits adaptability to complex real-world environments, potentially reducing accuracy.¹⁰ Xu proposed a four-axis attitude compensation algorithm to mitigate the impact of solar infrared radiation interference. Nevertheless, the algorithm may require high-performance hardware, and its compensation effectiveness may fall short in rapidly changing environments.¹¹ Barraquand et al. proposed a bio-inspired geomagnetic navigation model that effectively utilizes geomagnetic information for autonomous navigation without prior maps. However, in environments with geomagnetic anomalies, the model’s performance may degrade due to magnetic field distortions.¹² In such complex environments, mathematical statistics methods, such as large deviation theory, can be employed to analyze and address phase error issues, thereby enhancing navigation system performance.

This paper proposes a method for identifying and correcting infrared sensor output errors based on statistical distributions, focusing on the impact of ground infrared radiation on mountain inclination angle. Mountain inclination angle significantly influences phase errors in the infrared sensor output curve, so this study uses the Beta distribution to identify phase errors and the hyperbolic tangent function to correct the error curve, mitigating these errors. Since LWIR is sensitive to ground radiation interference, the proposed method enhances LWIR sensor accuracy in complex terrain. A comparative study with traditional correction methods validates its effectiveness, showing that the hyperbolic tangent function offers a better fit and greater robustness under mountain interference conditions.

The main contributions of this article are as follows:

1. A phase error extraction method for infrared sensors based on the Beta distribution is proposed to address the impact of mountain ranges on output errors.

2. The error curve is corrected using the hyperbolic tangent correction function (HTC), enhancing the accuracy of the measurement model.

3. The correction method’s effectiveness is validated through semi-physical experiments, demonstrating its reliability under various mountain interference conditions.

The remainder of this paper is organized as follows: Sec. II introduces the mathematical model for infrared sensor output errors. Section III describes the mountain extraction method based on the Beta distribution. Section IV details the design and implementation of the compensation algorithm. Section V presents the semi-physical experiments, and Sec. VI summarizes the research findings and potential applications.

The infrared radiation field within Earth’s atmosphere consists of a complex mixture of components, primarily solar radiation, atmospheric emission, and surface emission, with heat exchange occurring between them through processes such as radiation, convection, and conduction, as depicted in Fig. 1. Specifically, within the 8– $14μ m$ wavelength range, which corresponds to the long-wave atmospheric window, the radiation field is predominantly composed of emissions from the surface and the atmosphere. At an observation point at altitude h, when the observation angle $ϕ$ is between 0 and $π$⁠, the sensor primarily detects downward infrared radiation from the atmosphere. Conversely, when the observation angle $ϕ$ is between $π$ and 2 $π$⁠, the sensor captures a combination of upward atmospheric radiation, surface emission, and the surface-reflected downward radiation. However, due to the negligible nature of the surface-reflected radiation, it is typically disregarded. By analyzing the infrared radiation field within the 8– $14μ m$ spectral range, critical information about the Earth’s infrared radiation environment and mountain infrared radiation characteristics can be extracted. This paper primarily researches the impact of ground infrared radiation on long waves in the $8∼14μ m$ wavelength range, with a particular focus on the infrared radiation effects of mountain ranges within this band.⁴ 

When an infrared sensor is used to detect terrain, mountains typically create a distinct boundary within the observation field.²⁰ By obstructing radiation from the horizon along the sensor’s detection path, they form a unique occlusion boundary that affects the phase information recorded by the infrared sensor. The mountain inclination angle, denoted as $α m$⁠, as shown in Fig. 2.

The complex variations in terrain can significantly interfere with infrared sensor detection, especially in mountainous and hilly regions.²¹ When projectile-mounted infrared sensors fly over such terrain, their line of sight may be obstructed by mountain ridges. Additionally, since the infrared radiation intensity of mountains is higher than that of the sky,²² this difference can lead the sensor to mistakenly identify ridge lines as the horizon, causing abnormal changes in the infrared radiation signal. Therefore, it is essential to conduct an in-depth analysis of the infrared radiation characteristics of projectiles flying in mountainous environments to enhance attitude measurement accuracy.

Based on the relative position between the projectile-mounted infrared sensor and the mountains, this paper conducts a theoretical analysis of the infrared radiation interference caused by the mountains, as shown in Fig. 3, $h p$ represents the flight altitude of the projectile, $h m$ indicates the height of the mountains, $d p m$ is the horizontal distance between the projectile and the mountains, and $β m$ denotes the angle of the ridge line in the field of view of the projectile.

Interference from mountain infrared radiation is approximated as an equivalent effect of surface infrared radiation. Therefore, under the mountain infrared radiation interference, the measurement model for the projectile-mounted infrared sensor is represented by Eq. (4), where $ψ$ sets the angular integration range, $ϕ 0$ indicates the main radiation direction, $λ$ affects propagation characteristics, $τ si$ reflects atmospheric attenuation, and $τ m$ describes terrain blocking effects.

The term $Δδ( α m)$ represents the phase shift correction term influenced by the mountain inclination angle $α m$⁠, used to describe the effect of the mountain in a specific direction on the signal phase, while the parameter $k 1$ is a proportional coefficient that quantifies the influence of the mountain angular span on the phase shift. Additionally, n is an empirical exponent used to capture the nonlinear effects induced by the geometric characteristics of the mountain inclination angle.

Compared with existing models that assume ideal terrain and uniform surface emissivity, the proposed model enhances accuracy by incorporating the mountain inclination angle. These improvements enable more reliable attitude estimation in realistic environments where traditional models often fail. Although the model introduces additional computational overhead and depends on auxiliary geographic data, such requirements are increasingly feasible with advances in onboard processing and the accessibility of remote sensing datasets. Therefore, while some trade-offs exist, the overall gains in precision and adaptability make the model highly promising for practical applications.

The simulation results shown in Fig. 4 indicate that when the ridge line enters the infrared sensor’s field of view, the infrared radiation intensity from the mountain is higher than that of the sky, resulting in an increase in the sensor’s output signal strength. Mountain interference is manifested as a local phase shift on the infrared sensor’s output curve, and the size of this shifted region is related to the mountain inclination angle $α m$⁠. As the angle $α m$ increases, the impact on the output of the airborne infrared sensor also becomes significantly greater.

In the output signal of the infrared sensor, the interference caused by the mountain inclination angle introduces complex nonlinear characteristics in the phase error. The variation in phase error is influenced by the shape of the mountain range and the position of the sensor. Therefore, an effective feature extraction method is needed to quantify the interference intensity and enable subsequent correction.

As shown in Fig. 5, as the mountain inclination angle $α m$ increases, the phase shift in the infrared sensor output also increases. To verify the effectiveness of this method, the complete phase interval affected by mountain interference (⁠ $270 °$– $630 °$⁠) is extracted, and the sensor output data are fitted using the least squares method. This interval is selected because it fully encompasses the phase shift caused by mountain interference, ensuring that the fitting process comprehensively captures the characteristics of the interference. The fitted Beta distribution curve undergoes a corresponding shift, leading to an increase in skewness. This approach accurately captures the trend of phase shifts and effectively differentiates the phase shift patterns caused by different mountain inclination angles.

To more accurately quantify the phase shift phenomenon in the signal, this study introduces the calculation method of skewness as a statistical feature to measure the asymmetry of the data distribution. Skewness effectively reveals the asymmetry of the signal and is used to detect potential phase shift phenomena in the infrared sensor output curve. If the skewness of the signal is nonzero, it indicates an asymmetric distribution and a possible phase shift, suggesting that the signal may be affected by mountain interference.

The research results demonstrate that by fitting the probability distribution of the sensor signal using the Beta distribution and combining it with the calculation of skewness, the interference features of different mountain zenith angles can be effectively extracted, providing theoretical support for subsequent error correction.

Because other common correction methods, such as traditional polynomial fitting and linear calibration methods, often fail to effectively handle the nonlinear characteristics and interference of the signal, resulting in significant error accumulation, a function correction method is applied to adjust the infrared radiation brightness signal to approximate a single-frequency sine wave. Compared to these methods, the function correction method demonstrates better adaptability and lower errors in subsequent attitude calculations. Within the framework of a simplified infrared attitude measurement mathematical model, the selected correction function should remain continuous and monotonically increasing across its entire domain, while maintaining symmetry about the coordinate origin, assuming that any removable signal bias has been calibrated.

As the parameter $a$ increases, the function becomes steeper, causing the curve to change more rapidly near the origin. This allows the function to quickly approach its asymptotic values of $y=1$ and $y=−1$⁠, improving the correction of nonlinear features and interference. Thus, a larger $a$ enhances the correction effect, making the process more precise and reducing error accumulation.

This study employs HTC with varying coefficients to approximate the signal as a single-frequency sinusoidal wave. Using the least squares method to optimize the parameters, three sets of HTC are selected to fit the sinusoidal signal. The corrected curve of infrared radiation intensity variation with roll angle and the corresponding fitting results are presented in Fig. 7.

The analysis results clearly show that the correction of infrared radiance data using HTC effectively improves the curve fitting accuracy, significantly outperforming the uncorrected data. It is noteworthy that as the correction function coefficient increases, the fitting effect further improves, but this also leads to a decrease in computational efficiency, which may negatively affect subsequent attitude estimation. Therefore, in practical applications, it is necessary to balance accuracy and efficiency by selecting an appropriate optimization coefficient based on the specific requirements of the scenario. Building upon this, by combining HTC with the existing simplified infrared attitude measurement model, a corrected attitude measurement theoretical framework can be further developed to support the design of practical systems.

The correction of the infrared attitude measurement theoretical model is based on Eq. (20), where $A 2$ and $B 2$ denote the amplitude and the baseline offset of the infrared signal during calibration, respectively. During the calibration phase, the original signal offset is removed, and the infrared measurement signal is adjusted using HTC. The amplitude and phase of the corrected signal are then fitted and calibrated using the least squares method. Once these calibration parameters are determined under the same background conditions, they can be applied to subsequent measurements. In the measurement process, the same hyperbolic tangent function with the calibrated parameters is used to correct the signal, eliminating offset effects. The resulting fitted sinusoidal signal with a single fixed frequency is then utilized for further attitude estimation.

To validate the effectiveness of the modified infrared attitude measurement model based on the hyperbolic tangent function, an experimental system was constructed, as illustrated in Fig. 8. The experiment utilized the HMS P11 sensor from Heimann as the core testing device. This sensor is a miniature thermopile infrared sensor, designed for non-contact temperature measurements. It features a spectral response range of 8– $14μ m$⁠, a field of view (FOV) ranging from $− 43.5 °$ to $43.5 °$ and supports an operating temperature range from $−20 ° C$ to $120 ° C$⁠. An integrated thermistor provides an ambient temperature reference, enhancing measurement accuracy. Data acquisition, storage, and USB communication were managed by an STM32F767 microcontroller (MCU).

As shown in Fig. 9, the infrared testing equipment was mounted on a triaxial turntable, which simulated the flight trajectory of a projectile. During the experiment, the pitch angle of the equipment was maintained at 60 $°$⁠, while the roll angle continuously varied. Simultaneously, the MCU collected the infrared signals in real-time and transmitted the data to the host computer for analysis.

As shown in Fig. 10, the feature extraction process consists of three key steps: selecting curve segments containing interference, fitting these segments to a beta distribution curve, and calculating the skewness of the fitted curve. The correction process involves selecting appropriate correction parameters based on the curve’s skewness, applying a hyperbolic tangent function for correction, and ultimately obtaining accurate infrared data. Through this integrated approach, attitude information is extracted from the corrected curve, ensuring higher accuracy and reliability in dual-band infrared signal analysis.

The experiment shows that the infrared radiation curves collected by the sensor closely align with the theoretical model, and larger mountain angles result in more significant interference. To simulate mountain conditions, artificial vegetation was attached to a square plate, and the mountain angle was modeled by adjusting the tilt angle of the plate. During the experiment, the triaxial turntable was controlled to rotate steadily at a speed of 100 $°/s$ in the Z-axis direction for two full rotations, while the infrared signals were observed. Figure 11 illustrates the state of the square plate at different inclination angles (30 $°$⁠, 60 $°$⁠, and 90 $°$⁠) to simulate various mountain conditions. As the inclination angle of the square plate increases, the degree of interference or the phase shift in the curve also increases. This phenomenon indicates that changes in the inclination angle affect the phase shift of the curve, thereby influencing the accuracy of measurement results.

Additionally, under flat terrain conditions, the measured data exhibited a slight deviation, primarily due to differences in the surface materials on both sides of the test area, which led to uneven infrared radiation reflection characteristics. Specifically, different surface materials have varying absorption and scattering properties in the infrared spectrum, causing slight phase variations in the infrared signals received by the sensor. This uneven surface influence is largely overshadowed by mountain-induced interference at other inclination angles but becomes more noticeable in flat terrain conditions. Therefore, when analyzing the experimental data, it is necessary to account for the impact of surface materials on infrared signals to enhance the reliability of the measurement results.

By calculating the skewness of the infrared radiation curves at different inclination angles, the impact of mountain angles on infrared signals can be quantified. The specific results are shown in Table I.

The table presents $S k$ at different inclination angles (30 $°$⁠, 60 $°$⁠, and 90 $°$⁠). Through a quantitative analysis of these data, it can be observed that as the inclination angle increases, $S k$ gradually becomes more negative, with values of $−$0.036, $−$0.074, and $−$0.11, respectively. This indicates that the increase in inclination angle leads to a more pronounced asymmetry in the signal distribution, further validating the impact of the mountain inclination angle on infrared measurement results.

As shown in Eq. (19), the correction coefficient $a$ is determined based on the signal’s skewness $S k$⁠. According to this formula, the correction coefficient $a$ increases as the absolute value of skewness $| S k|$ increases, meaning that as the intensity of mountain interference becomes stronger, a stronger correction is needed to eliminate the signal offset. For instance, when the absolute value of skewness $| S k|$ is small (e.g., $| S k|$=0.01), the correction coefficient $a$ will be close to 1; as the skewness increases, the correction coefficient also increases, providing a stronger signal correction. This approach effectively handles mountain interference and corrects the infrared radiation signal to approximate the ideal single-frequency sine wave.

As shown in Fig. 12, the correction results for mountain terrain with 30 $°$⁠, 60 $°$⁠, and 90 $°$ slopes are presented. The corrected curves show a reduction in both amplitude and phase deviations, particularly at larger mountain angles, further validating the effectiveness and applicability of the correction method.

After obtaining the statistical characteristics of the correction parameters, extensive experiments were conducted to compare the proposed method with other commonly used algorithms. Specifically, five correction methods were compared: Direct Fitting Method (DFM), Odd-Power Function Correction (OPFC), Tangent Function Correction (TFC), Hyperbolic Sine Correction (HSC), and Hyperbolic Tangent Correction (HTC).

By analyzing the errors between the corrected infrared radiation curves and the interference-free curve, the results shown in Fig. 13 and Table II demonstrate that the HTC method achieved the best performance, exhibiting the lowest root mean square error (RMSE), mean absolute error (MAE), and the highest coefficient of determination (R $2$⁠). Quantitative analysis of the data in the table reveals that the DFM method has an average value of 0.349, an RMSE of 4.8871, an MAE of 3.91, and an R $2$ of 0.41, indicating poor correction performance. The OPFC and TFC methods have similar performances, with average values of 0.127 and 0.129, RMSEs of 1.8672 and 1.9306, MAEs of 1.49 and 1.54, and R $2$ values of 0.75 and 0.73, respectively, showing moderate effectiveness. The HSC method shows better performance, with an average value of 0.033, an RMSE of 1.3382, an MAE of 1.07, and an R $2$ of 0.84. Finally, the proposed HTC method outperforms all others, with an average value of 0.05, an RMSE of 0.6859, an MAE of 0.55, and the highest R $2$ of 0.92, demonstrating significant advantages in reducing both absolute and squared errors. Overall, the HTC method yields the most accurate and reliable correction among all compared approaches.

A joint compensation algorithm integrating Beta distribution and hyperbolic tangent function (Beta-HTC) was proposed to address infrared attitude measurement accuracy degradation caused by mountainous terrain interference. A theoretical model characterizing infrared radiation under mountainous interference was developed, leveraging Beta distribution to statistically analyze interference features at 30 $°$⁠, 60 $°$⁠, and 90 $°$ inclination angles, enabling precise identification of phase shift distortion and signal asymmetry intervals. Nonlinear compensation for interference signals was achieved by dynamically correcting infrared radiation curve phases using the hyperbolic tangent function (HTC). Semi-physical experiments validated the effectiveness of the Beta-HTC method, demonstrating a minimum root mean square error (RMSE) of 0.6859 across three typical inclination scenarios and outperforming the best traditional model by a reduction of 0.6523. Future work will focus on enhancing the model’s dynamic adaptability through multi-sensor fusion frameworks and advancing high-precision attitude estimation methods based on Beta-HTC, thereby providing robust theoretical foundations and practical solutions for infrared measurements in complex mountainous environments.

The work was supported, in part, by the National Natural Science Foundation of China under Grant Nos. 62101287 and 62203226; in part, by the Key Research and Development Program of Nantong under Grant No. GZ2024002; and, in part, by the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant No. KYCX25_3721.

The authors have no conflicts to disclose.

Miaomiao Xu: Data curation (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Ling Shuai: Data curation (equal); Validation (equal); Visualization (equal); Writing – original draft (lead). Enwu Du: Resources (equal). Xiaoyu Han: Data curation (equal); Visualization (equal). Qiang Sun: Supervision (equal); Visualization (equal). Xiongzhu Bu: Resources (equal). Yihan Cao: Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.