*3.1. Overall Flatness*

The overall flatness of the initial support surface of the tunnel is mainly determined by the dispersion degree of the normal vector distance from the original point cloud to the flatness calculation datum. If the dispersion is large, it means that the original point cloud and the flatness calculation datum are quite different. The rougher the surface. If the degree of dispersion is small, it means that the difference between the original point cloud and the flatness calculation reference plane is small, and the surface is flatter.

To intuitively express the overall flatness of the surface of the initial support of the tunnel, the concept of standard deviation is introduced. Simply put, the standard deviation is a measure of the degree to which a set of values are scattered from the average. A larger standard deviation means that the difference between most of the values and its average is larger; a smaller standard deviation means that these values are closer to the mean.

To sum up, the formula for calculating the overall flatness of the initial support surface of the tunnel is as follows:

$$m\_0 = \pm \sqrt{\frac{\sum d\_i^2}{n-1}} (i = 1, 2, \dots, n) \tag{4}$$

In the formula, *m*0 is the overall flatness of the initial support surface of the tunnel, di is the normal vector distance, and n is the no. of point clouds collected during flatness detection.
