Kermarrec Gaël, Paffenholz Jens-André, Alkhatib Hamza
Geodetic Institute, Leibniz University Hannover, Nienburger Str. 1, 30167 Hannover, Germany.
Sensors (Basel). 2019 Aug 21;19(17):3640. doi: 10.3390/s19173640.
B-spline surfaces possess attractive properties such as a high degree of continuity or the local support of their basis functions. One of the major applications of B-spline surfaces in engineering geodesy is the least-square (LS) fitting of surfaces from, e.g., 3D point clouds obtained from terrestrial laser scanners (TLS). Such mathematical approximations allow one to test rigorously with a given significance level the deformation magnitude between point clouds taken at different epochs. Indeed, statistical tests cannot be applied when point clouds are processed in commonly used software such as CloudCompare, which restrict the analysis of deformation to simple deformation maps based on distance computation. For a trustworthy test decision and a resulting risk management, the stochastic model of the underlying observations needs, however, to be optimally specified. Since B-spline surface approximations necessitate Cartesian coordinates of the TLS observations, the diagonal variance covariance matrix (VCM) of the raw TLS measurements has to be transformed by means of the error propagation law. Unfortunately, this procedure induces mathematical correlations, which can strongly affect the chosen test statistics to analyse deformation, if neglected. This may lead potentially to rejecting wrongly the null hypothesis of no-deformation, with risky and expensive consequences. In this contribution, we propose to investigate the impact of mathematical correlations on test statistics, using real TLS observations from a bridge under load. As besides TLS, a highly precise laser tracker (LT) was used, the significance of the difference of the test statistics when the stochastic model is misspecified can be assessed. However, the underlying test distribution is hardly tractable so that only an adapted bootstrapping allows the computation of trustworthy p-values. Consecutively, the extent to which heteroscedasticity and mathematical correlations can be neglected or simplified without impacting the test decision is shown in a rigorous way, paving the way for a simplification based on the intensity model.
B样条曲面具有诸如高度连续性或其基函数的局部支撑等吸引人的特性。B样条曲面在工程大地测量中的主要应用之一是对来自例如地面激光扫描仪（TLS）获取的三维点云进行曲面的最小二乘（LS）拟合。这种数学近似使得能够在给定的显著性水平下严格测试在不同时期获取的点云之间的变形量。实际上，当在诸如CloudCompare等常用软件中处理点云时，无法应用统计测试，这些软件将变形分析限制在基于距离计算的简单变形图上。然而，为了做出可靠的测试决策并进行有效的风险管理，需要对基础观测的随机模型进行最优指定。由于B样条曲面近似需要TLS观测的笛卡尔坐标，原始TLS测量的对角方差协方差矩阵（VCM）必须通过误差传播定律进行变换。不幸的是，这个过程会产生数学相关性，如果被忽略，可能会强烈影响用于分析变形的所选测试统计量。这可能潜在地导致错误地拒绝无变形的原假设，从而产生风险且成本高昂的后果。在本论文中，我们建议使用来自加载桥梁的真实TLS观测来研究数学相关性对测试统计量的影响。由于除了TLS之外，还使用了高精度激光跟踪仪（LT），因此可以评估随机模型指定错误时测试统计量差异的显著性。然而，基础测试分布很难处理，因此只有经过调整的自助法才能计算出可靠的p值。接着，以严格的方式展示了在不影响测试决策的情况下，可以忽略或简化异方差性和数学相关性的程度，为基于强度模型的简化铺平了道路。
