Nonstationary stochastic processes in least squares collocation --- NonStopLSC

通过反向建模和平差技术，大地测量仪试图从他们的测量中推导出数学模型，以更好地了解地球系统中的过程。复杂的确定性和随机性模型被开发出来，以实现对现实和剩余不确定性的最佳反映。虽然确定性建模已经得到了很大的改进，但应用的随机模型和表示法仍然存在严重的缺陷。特别是在搭配方法中，经常使用移除-恢复技术来一方面保证平稳性，另一方面更好地访问不同频率的内容，这些内容通常隐藏在经验协方差序列中。在这个项目中，提出了随机信号的自回归过程表示，它能够描述完整的频谱。应当强调的是，这不仅适用于平稳过程，也适用于时变信号。但在理论上，这种过程表示仅限于等间距的无穷大测量序列。因此，这两种表示法--协方差表示法和过程表示法--各有优缺点。提出了一种PRO融合的框架。这一建议的主要焦点是进一步发展随机模型表示，它能够反映完整的信号内容，并具有从通常的时间平稳假设切换到时变随机模型的能力。我们希望建立和扩展一个方法论框架，将过滤器和以自回归过程和最小二乘配置为代表的协方差方法联系起来。我们使用‘魔方’机制以严格形式化的方式做到这一点，这打开了在这两种方法之间切换的可能性。将时间离散过程推广到随机常微分方程组，开辟了从离散协方差序列导出连续协方差函数的途径。因此，将建立一族协方差函数，它将能够描述整个信号内容以及随机过程的时间变异性。为了研究这一方法框架的实际应用，我们将改进来自大地测量数据集(来自专用大地测量卫星GOCE、GRACE、GRACE/GRACE-FO和GRAV-D航空重力数据)的真实测量序列的分析，特别是关于它们的时变随机信号特征。

Through inverse modeling and adjustment techniques, the geodesists try to derive mathematical models from their measurements to get a better understanding of the processes in the system Earth. Sophisticated deterministic and stochastic models are developed to achieve the best possible reflection of reality and the remaining uncertainty. While deterministic modeling has been improved by much effort, there are still serious weaknesses in the applied stochastic models and representations. Especially in the collocation approach a remove-restore technique is often used to, on the one hand side, guarantee stationarity and, on the other side, get better access to the different frequency contents, which is often hidden in the empirical covariance sequence. Within this project, the representation of stochastic signals with autoregressive processes is proposed which are able to describe the complete frequency spectrum. It shall be highlighted that this is not only the case for stationary processes, but also for time-variable signals. But in theory, this process representation is restricted to equispaced infinite measurement series. Therefore, both representations - the covariance and the process representation - have their pros and cons. A framework for the fusion of the pros is proposed here. The main focus of this proposal is a further development of stochastic model representations, which can reflect the full signal content and have the capability to switch from the usual assumption of time-stationary to time-variable stochastic models. We want to build up and extent a methodical framework to connect the filter and the covariance approach represented by autoregressive processes and least squares collocation. We do this in a strictly formalized way using the 'Magic Square' mechanism which opens the possibility to switch between these two approaches. The proposed extension from the time-discrete processes to stochastic ordinary differential equations opens the way to derive continuous covariance functions from discrete covariance sequences. As a result a family of covariance functions will be established, which will be able to describe the entire signal content as well as the time-variability of stochastic processes.To study this methodical framework with real applications we will refine the analysis of real measurement series from geodetic data sets (from dedicated geodetic satellite missions GOCE, GRACE, GRACE/GRACE-FO and GRAV-D airborne gravity data) especially with respect to their time-variable stochastic signal characteristics.