Best basis construction and comparison of trial functions for ill-posed inverse problems in Earth sciences - studied at the examples of global-scale seismic tomography and gravitational field modelling

地球科学中不适定反问题的最佳基础构建和试验函数比较——以全球尺度地震层析成像和重力场建模为例进行研究

• 批准号: 437390524
• 负责人: Professor Dr. Volker Michel
• 金额: --
• 依托单位: Arbeitsgruppe Geomathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/437390524?language=en
• 关键词: Best; basis; construction; comparison; trial

The choice of basis functions can essentially influence the result of an inverse problem. In view of today's demands on the accuracy of models, we are, consequently, confronted with the question how the obtained results can be confirmed or improved, respectively, by verifying or correcting (if necessary) the used basis functions. The arising questions are: can there be artefacts due to the used numerical method in individual structures in the mapping of e.g. a seismic velocity field or of the gravitational field? To what extent are data sensitive to single regional changes in the solution? Vice versa, how can local variations in the Earth or at the Earth's surface or, alternatively, known local errors in a current model be considered in a model, without deteriorating the model elsewhere? The latter appears to be better to achieve with local basis functions (such as radial basis functions) than with global basis functions (such as spherical harmonics).In the recent years, the principal investigator and his research group have developed several algorithms (the Regularized Functional Matching Pursuit, RFMP, and its enhancements) which are able to iteratively construct a kind of a best basis for an inverse problem. These methods were particularly elaborated for scenarios on the sphere or the ball. The applicability to several problems has already been demonstrated. However, the methods still have some limitations. E.g. large data sets, as they are common for the gravitational field, cannot be handled up to now, and the traveltime tomography does not allow efficient formulae for the forward calculations.Within this project, these algorithms shall be further enhanced, in order to make them better applicable to realistic problems in Earth sciences. We particularly consider global-scale seismic tomography and high-dimensional modelling of the gravitational potential. We expect especially new insights into these two practical problems. In the former case, the above mentioned question arises concerning possible artefacts in velocity models. RFMP and its variants yield the possibility to automatize the multi-scale adaptation of grid structures, which has previously been done manually. We anticipate an improvement of the accuracy of the calculated models. Moreover, the set of trial functions (which is called a dictionary) may be varied, in order to test how stable some aspects of a model are. This way, artefacts due to the choice of the basis functions can be better identified. In the case of gravitational field modelling, efficient ways shall be found which enable us to approximate local anomalies as locally concentrated and as accurately as possible, also in high-resolution models, by the choice of optimal additional basis functions (splines, wavelets, Slepians). For this purpose, new innovative ways of improving existing algorithms need to be found.

基函数的选择可以本质上影响反问题的结果。鉴于当今对模型准确性的要求，我们因此面临着如何通过验证或纠正（如果必要）所使用的基函数来分别确认或改进所获得的结果的问题。出现的问题是：由于在映射中的各个结构中使用了数值方法，例如，是否会出现伪影。地震速度场还是重力场？数据在多大程度上对解决方案中的单个区域变化敏感？反之亦然，如何在模型中考虑地球或地球表面的局部变化，或者当前模型中已知的局部误差，而不会使其他地方的模型恶化？后者似乎用局部基函数（如径向基函数）比全局基函数（如球谐函数）更好地实现。近年来，主要研究者和他的研究小组开发了几种算法（正则化功能匹配追踪、RFMP 及其增强）能够迭代地构建逆问题的最佳基础。这些方法是专门针对球体或球体上的场景而精心设计的。已经证明了它对几个问题的适用性。然而，这些方法仍然存在一些局限性。例如。迄今为止，引力场中常见的大型数据集还无法处理，并且走时层析成像不允许有效的正演计算公式。在这个项目中，这些算法将进一步增强，以便使它们成为可能。更好地适用于地球科学中的现实问题。我们特别考虑全球尺度地震层析成像和重力势的高维建模。我们期待对这两个实际问题有特别新的见解。在前一种情况下，出现了上述关于速度模型中可能的人为因素的问题。 RFMP 及其变体使网格结构的多尺度适应自动化成为可能，而这以前是手动完成的。我们预计计算模型的准确性将会提高。此外，试验函数集（称为字典）可能会有所不同，以便测试模型某些方面的稳定性。这样，可以更好地识别由于基函数的选择而产生的伪影。在重力场建模的情况下，应找到有效的方法，使我们能够通过选择最佳的附加基函数（样条、小波、Slepians）尽可能准确地近似局部异常，在高分辨率模型中也是如此。 ）。为此，需要找到改进现有算法的新创新方法。