## 1 Introduction  
Spatio-temporal dynamic data is a series of data with time and space dimensions, obtained by sensors, GPS, and other devices [1, 2]. It is widely present in many domains, including ecology, meteorology, economics, traffic, and so on. Often, we need to use spatio-temporal data to describe or predict the rules of occurrence and development, further guiding our decision making. However, due to the inherent characteristics of spatio-temporal data in the three aspects of time, space, and attribute, it features the complexity of multi-dimensional and spatio-temporal dynamic correlation. Therefore, we require a flexible, interpretable, and accurate model to fit the spatio-temporal data and obtain a latent dynamic model [3]. Process spatial-temporal dynamic data consists of two stages: A sensor data acquisition stage and a data processing stage, as demonstrated in Figure 1. In the data acquisition stage, due to the limitations of current technology and cost, missing data [4, 5] may occur in each round of data collection, and its distribution may differ. As shown in the second sub-figure, black indicates effective data collection and white indicates missing data. Among them, the effective collected data are mainly composed of real data and random systematic error, which are represented by blue and red, respectively. At the data processing stage, with these incomplete sensor data, we expect to achieve interpolation estimation for generic positions, estimation for future moments, and filtering approximation for real data. The characteristics of the above observed data have been extensively studied and various methods have been explored. Researchers have used existing sensor observation data to fill in the missing data by interpolation, including linear, spline [6], and Lagrangeinterpolations [7]. However, this strategy may lead to some deviations, especially the lack of consideration of spatio-temporal correlation uncertainties in describing local variations of dynamic fields. Similarly, data errors can be eliminated by filtering or statistical analysis, but this approach is mostly biased towards temporal or spatial dimensions. Various commonly used methods have been applied to model spatio-temporal field, including the Bayesian model [8], State Space model [9], and Kalman filter [10]. In practical application, these spatio-temporal dynamic models often require sufficient or consistent available sensor observations to achieve a balance between estimation accuracy and computational burden. Therefore, developing an effective and flexible spatio-temporal dynamic field modeling method from sensor observation data is essential. In this paper, we propose the FUKSS model, approximating the stochastic equation of state representation. This model retains the characteristics of universal Kriging, and uses best linear unbiased estimation (BLUE) [11] to reconstruct the estimation function. By the recursion and calculation of the state equation, we obtain the recursive variables and the calculation process, similar to the Kalman filter error correction. In this way, we make use of universal Kriging analysis of spatial correlation, while maintaining the Kriging calculation burden for each function in the sequence, in order to achieve the error recursive correction of inconsistent data and the storage of consistent intermediate variables. In essence, the algorithm is more biased to the Kalman-Kriging filtering model estimation in the spatial domain; that is, the weight of the current data in the current spatial domain estimation is increased, to some extent, in the calculation process. The result is similar to the extension of the unbiased vector Kalman filter proposed by [PERSON] [12] to the Kriging space domain. In FUKSS, the problems to be solved and main contributions can be summarized as follows:  
* The estimation of general position is realized by the statistical characteristics of Kriging [13; 14].