New non-linear adjustment methods for application in geodesy and related fields

工程或其他技术科学中的一个常见问题是，通常无法直接确定期望的目标值。因此，需要通过应用功能模型将未知参数与相应的测量联系起来。一个例子是根据观察到的距离和方向确定物体的三维坐标。为了确保精度和可靠性，总是进行比需要更多的测量，以唯一确定未知参数。在这种情况下，在考虑不可避免的随机测量误差的情况下，会出现调整问题，导致可能的最佳结果。根据1980年以来在数理统计领域进行的科学努力，不仅考虑了随机测量误差，而且考虑了函数模型参数中的误差，这种情况被称为“变量误差（EIV）模型”。为了解决由此产生的非线性平差问题，提出了“总最小二乘”方法。在一定条件下，这种方法将期望参数的确定转化为一个特征值问题，该问题具有有效的求解方法。然而，这种方法在关于功能和/或随机模型的特殊结构的情况下失效。在拟议的项目中，应发展新的非线性调整方法，使eiv模型能够得到普遍解决。一般的方法是建立适当的非线性正态方程，以获得关于数值效率和收敛性的最佳可能解。新方法还可用于病态问题的正则化、存在异常值的鲁棒参数估计以及考虑任意结构的色散矩阵。具有先验信息的eiv模型是一个以前没有处理过的问题，它应该服从于TLS的配置。这些新的平差方法可以应用于大地测量学（如坐标变换）、地质统计学（如克里格）和计算机视觉（如三维表面匹配）。由于新的非线性调整方法的普遍化，适应其他主题领域的挑战应该是可能的。

A common problem in engineering or other technical sciences is that oftentimes desired target val-ues cannot be determined directly. Hence the unknown parameters need to be connected to corre-sponding measurements by applying a functional model. An example is the determination of 3D coordinates of an object based on observed distances and directions.In order to ensure accuracy and reliability always more measurements are conducted than needed for the unique determination of the unknown parameters. In such cases an adjustment problem arises which leads to the best possible result under consideration of inevitable random measurement errors.Based upon scientific efforts that have been carried out from 1980 onwards within the field of mathematical statistics not only random measurement errors but also errors in the parameters of the functional model have been considered, a case which is referred to as "Errors-In-Variables (EIV) Model".In order to solve the resulting non-linear adjustment problem the "Total Least-Squares (TLS)" approach has been developed. This approach transfers, under certain conditions, the determination of desired parameters into an eigenvalue problem for which efficient solution methods are available.This approach, however, fails in case of special structures regarding the functional and/or stochastic model. Within the proposed project new non-linear adjustment methods should be developed that enable a generalised solution of the EIV-model.The general methodology is based on setting up appropriate non-linear normal equations for gen-eralised problems in order to obtain the best possible solution regarding numerical efficiency and convergence behaviour. The new methods should also be applied for regularisation of ill-posed problems, robust parameter estimation in presence of outliers and consideration of dispersion matrices of arbitrary structure. A problem which previously has not been treated is the EIV-model with prior information which should yield to TLS collocation.These novel adjustment methods should be applied in Geodesy (e.g. for coordinate transformation), geostatistics (e.g. kriging) and computer vision (e.g. 3D surface matching). The adaptation to challenges in other subject areas should be possible due the generalised formulation of the new non-linear adjustment methods.