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

基函数的选择可以从根本上影响反问题的结果。鉴于当今对模型精度的要求，因此，我们面临的问题是如何通过验证或校正（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。由此产生的问题是：在绘制例如地震速度场或重力场的地图时，是否会由于在各个结构中使用的数值方法而产生伪影？数据对解决方案中的单个区域更改的敏感程度如何？反之亦然，如何才能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，而不使其他地方的模型恶化？后者似乎是更好地实现与局部基函数（如径向基函数）比全局基函数（如球谐函数）。在最近几年，主要研究者和他的研究小组已经开发了几种算法（正则函数匹配追踪，RFMP，及其增强），能够迭代构造一种最佳基的反问题。这些方法特别针对球体或球上的场景进行了详细说明。对几个问题的适用性已经得到证明。然而，这些方法仍然有一些局限性。例如，大数据集，因为它们是常见的重力场，到目前为止还不能处理，走时层析成像不允许用于正演计算的有效公式，在本项目中，这些算法将进一步增强，以便使它们更好地适用于地球科学中的现实问题。我们特别考虑全球规模的地震层析成像和高维建模的重力位。我们特别期待对这两个实际问题的新见解。在前一种情况下，上述问题涉及速度模型中可能的伪影。RFMP及其变体产生了自动化网格结构的多尺度适应的可能性，这在以前是手动完成的。我们预计计算模型的准确性的提高。此外，为了测试模型的某些方面有多稳定，试验函数集（称为字典）可以变化。这样，可以更好地识别由于基函数的选择而导致的伪影。在重力场建模的情况下，将找到有效的方法，使我们能够通过选择最佳的附加基函数（样条、小波、Slepians），在高分辨率模型中尽可能精确地近似局部异常。为此，需要找到改进现有算法的新的创新方法。

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.