High-precision temperature prediction for atmospheric refractivity correction using Kalman spatiotemporal data fusion

Abstract

In absolute distance measurement and positioning applications, atmospheric refraction error is a critical factor limiting measurement accuracy. Temperature plays a dominant role in computing the atmospheric refractive index. However, accurately acquiring the temperature field along the ranging path in complex and dynamic outdoor environments remains challenging due to limited sensor deployment and environmental nonstationarity. We propose a spatiotemporal temperature data fusion method for atmospheric refraction correction, which integrates the strengths of the generalized regression neural network (GRNN) and Kriging interpolation within a Kalman filter. This method achieves dynamic prediction and high-accuracy reconstruction of temperature parameters. The proposed method is systematically validated through simulation analysis as well as indoor and kilometer-scale outdoor experimental measurements. The simulation results demonstrate that Kalman filter expanded fusion (KFEF) outperforms the traditional interpolation method radial basis function (RBF) and the state-of-the-art spatiotemporal interpolation and prediction methods spatiotemporal Kriging (STK) and Gaussian process (GP), in terms of both reconstruction accuracy and stability of the temperature field. Specifically, KFEF achieves a 61.54% reduction in root mean square error (RMSE) compared with RBF and reductions of 34.21% and 32.43% relative to STK and GP, respectively. This indicates its practical value for long-distance high-precision ranging engineering applications. Furthermore, the proposed spatiotemporal data fusion framework is highly general and scalable. It can also be applied to other temperature field prediction and reconstruction problems.

1. Introduction

The absolute distance measurement accuracy and positioning accuracy are largely affected by the atmospheric refraction error (Meiners-Hagen et al., 2017; Ding et al., 2020; Bi et al., 2024). When electromagnetic waves such as lasers and microwaves propagate in the atmosphere, their refractive indices are dynamic and change with temperature, air pressure, humidity, etc. This leads to deviations in distance measurement results. The core of atmospheric refraction error correction is the acquisition of precise atmospheric refractivity along the electromagnetic wave propagation path. Existing studies indicate that within commonly used atmospheric refractivity models, such as the Edlén formula (Rüeger, 2002) and the Ciddor formula (Ciddor, 1996), meteorological parameters involved in refractive index computation exert markedly different levels of influence. Among these parameters, temperature plays a decisive role in determining the refractivity accuracy (Pisani et al., 2018).

In recent years, researchers have proposed various methods for acquiring or eliminating temperature parameters with high precision. The main temperature parameter elimination methods are the dual-wavelength method and the acoustic method. The former is not applicable to the microwave band (Tomberg et al., 2017; Guillory et al., 2024), whereas the latter is susceptible to industrial noise and is characterized by high system complexity and cost (Underwood et al., 2015; Pisani et al., 2018). In terms of high-precision temperature parameter acquisition, multiple sensors are typically deployed along the baseline to reconstruct the temperature meteorological field. The Physikalisch-Technische Bundesanstalt (PTB) in Germany and the National Institute of Metrology in China installed 60 temperature sensors outdoors, effectively correcting the refractive error (Pollinger et al., 2012; Liu XD et al., 2020). However, this approach is difficult to apply in mobile outdoor environments due to the large number of sensors.

In outdoor absolute distance measurements, a limited number of sensors are usually used to collect temperature data. Interpolation methods are then employed to estimate the temperature distribution along the propagation path. Previous studies have suggested various methods, including the equivalent area method (Chen Y et al., 2018) and linear interpolation combined with the radial basis function (RBF) (Gu et al., 2019).