Deformation estimation of a circular tunnel from a point cloud using elliptic Fourier analysis

Abstract

In recent years, light detection and ranging technology has been applied to tunnel engineering. Although more accurate than traditional measurement techniques, estimating the tunnel deformation from the light detection and ranging point cloud remains challenging. This paper proposes a deformation estimation method using elliptic Fourier analysis, whereby the tunnel deformation is estimated from the difference in the Fourier series representations of the tunnel outline before and after deformation. The applicability of the proposed method for a circular tunnel cross-section is theoretically considered. Numerical results show that there is almost no error in the estimation of radial displacement, regardless of the measurement conditions. In contrast, there is an error in the estimation of circumferential displacement because this does not modify the circular tunnel outline. Therefore, tunnel deformation cannot be accurately estimated using point cloud data obtained only from distance measurements. As a special case, when the full-slip condition is imposed on the ground–lining interface and the estimated circumferential displacement of the inner surface is regarded as that of the middle surface, circumferential displacement can also be accurately estimated. Moreover, the difference in the deformation patterns under the full-slip and no-slip conditions decreases as the relative stiffness of the lining with respect to the ground increases. As a result, the deformation estimation accuracy is expected to be high for tunnels with a high possibility of deformation. When the distance measurement error is considered, the estimation accuracy of the displacement decreases, especially that of the circumferential displacement. Smoothed data significantly improve the accuracy, and high-precision estimation can be achieved when the measurement error is small compared with the tunnel deformation. However, small displacement errors produce significant stress errors in the higher-order terms of the Fourier series. Therefore, only the lower-order terms can be used for stress estimations.