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How to account for temporal correlations with a diagonal correlation model in a nonlinear functional model: a plane fitting with simulated and real TLS measurements
Journal of Geodesy ( IF 3.9 ) Pub Date : 2020-12-23 , DOI: 10.1007/s00190-020-01456-7
Gaël Kermarrec , Michael Lösler

To avoid computational burden, diagonal variance covariance matrices (VCM) are preferred to describe the stochasticity of terrestrial laser scanner (TLS) measurements. This simplification neglects correlations and affects least-squares (LS) estimates that are trustworthy with minimal variance, if the correct stochastic model is used. When a linearization of the LS functional model is performed, a bias of the parameters to be estimated and their dispersions occur, which can be investigated using a second-order Taylor expansion. Both the computation of the second-order solution and the account for correlations are linked to computational burden. In this contribution, we study the impact of an enhanced stochastic model on that bias to weight the corresponding benefits against the improvements. To that aim, we model the temporal correlations of TLS measurements using the Matérn covariance function, combined with an intensity model for the variance. We study further how the scanning configuration influences the solution. Because neglecting correlations may be tempting to avoid VCM inversions and multiplications, we quantify the impact of such a reduction and propose an innovative yet simple way to account for correlations with a “diagonal VCM.” Originally developed for GPS measurements and linear LS, this model is extended and validated for TLS range and called the diagonal correlation model (DCM).

中文翻译:

如何在非线性函数模型中考虑对角线相关模型的时间相关性:具有模拟和真实 TLS 测量的平面拟合

为了避免计算负担,首选对角方差协方差矩阵 (VCM) 来描述地面激光扫描仪 (TLS) 测量的随机性。如果使用正确的随机模型,这种简化会忽略相关性并影响具有最小方差的可信的最小二乘 (LS) 估计。当执行 LS 函数模型的线性化时,要估计的参数及其离散会出现偏差,这可以使用二阶泰勒展开来研究。二阶解的计算和相关性的解释都与计算负担有关。在此贡献中,我们研究了增强型随机模型对该偏差的影响,以根据改进来衡量相应的收益。为了这个目标,我们使用 Matérn 协方差函数对 TLS 测量的时间相关性进行建模,并结合方差的强度模型。我们进一步研究扫描配置如何影响解决方案。因为忽略相关性可能很容易避免 VCM 反转和乘法,我们量化了这种减少的影响,并提出了一种创新但简单的方法来解释与“对角 VCM”的相关性。该模型最初是为 GPS 测量和线性 LS 开发的,后来针对 TLS 范围进行了扩展和验证,称为对角线相关模型 (DCM)。因为忽略相关性可能很容易避免 VCM 反转和乘法,我们量化了这种减少的影响,并提出了一种创新但简单的方法来解释与“对角 VCM”的相关性。该模型最初是为 GPS 测量和线性 LS 开发的,后来针对 TLS 范围进行了扩展和验证,称为对角线相关模型 (DCM)。因为忽略相关性可能很容易避免 VCM 反转和乘法,我们量化了这种减少的影响,并提出了一种创新但简单的方法来解释与“对角 VCM”的相关性。该模型最初是为 GPS 测量和线性 LS 开发的,后来针对 TLS 范围进行了扩展和验证,称为对角线相关模型 (DCM)。
更新日期:2020-12-23
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