## 1 Introduction
Because the empirical formula is fitted by a large number of field monitoring data, the parameters of the fitting function are closely related to the actual construction conditions, and lack of a clear theoretical basis, the accuracy of the results predicted by this method is often difficult to meet the requirements, and its  
Figure 1: Sources of stratum movements of shield tunnel. defects are evident. Comparatively, the solving process of the analytic method and modified analytic method is relatively complex, but through more rigorous analysis, as long as the parameters are selected properly, it has better universality. [PERSON] [35] deduced the expression of three-dimensional displacement of stratum by using the mirror source-sink method. This formula can not only calculate the curve of surface settlement trough, but also calculate the displacement and stress of deep soil, so it is widely used [36; 37; 38; 39; 40; 41; 42; 43]. However, the movement pattern of actual volume loss is not consistent with the assumed equivalent radial movement pattern in this method and has some deviation. Based on field monitoring data, [PERSON], [PERSON], and other researchers [44; 45; 46; 47; 48; 49; 50; 51] modified the stratum displacement pattern by exponential function and deduced the prediction formula of surface subsidence under non-equivalent radial movement pattern. However, most of these methods are only suitable for three-dimensional orthogonal plane or two-dimensional cases, which have great defects and need to be improved urgently. Many other researchers concerned the field description, soil classification for engineering purposes [52; 53; 54], and the analysis methods of data such as neural networks [55; 56; 57; 58]. [PERSON] et al. [59] used measured settlements from the tunneling excavation for the extension of Line 3 of the Athens Metro for the verification and calibration of the empirical formulae. [PERSON] et al. [60] evaluated the potential and the limitations of artificial neural networks (ANN) for predicting surface settlements caused by Earth pressure balance (EPB) shield tunneling and to develop optimal neural network models for this objective. [PERSON] et al. [61] created a database from previous research [62; 63; 64; 65] and studied the capability of adaptive neuro-fuzzy inference system (ANFIS) and gene expression programming (GEP) methods for settlement prediction. However, those methods belong to the fuzzy solutions without mechanism analysis, training neural networks requires a lot of data from databases or numerical simulation, and there are problems such as difficult convergence or slow convergence speed. This paper studies the Greenfield stratum movements caused by shield tunnel construction combining an elastic half-space model of mirror source-sink method with the use of modified analytical method. Through theoretical derivation and empirical function modification, the prediction formula of three-dimensional Greenfield stratum movements is improved, and it shows good consistency with the centrifugal test results. The new solution can calculate the three-dimensional stratum displacement accurately, which is unavailable for the previous methods. It can be used in the process of tunnel excavation rather than only for the ultimate settlement. Based on this improved solution, it is convenient to evaluate the effect on the other skewed underground structures through the elastic foundation beam and other similar methods; therefore, this paper can provide a wide guidance and service for the design and construction of underground engineering in the future.