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

基函数的选择本质上会影响反问题的结果。鉴于当今对模型精度的要求，我们面临这样一个问题：如何通过验证或修正(如有必要)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引力场的图时，个别结构中使用的数值方法会不会产生伪影？数据对解决方案中的单个区域更改的敏感度有多高？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，而不会在其他地方恶化模型？后者似乎是用局部基函数(如径向基函数)比用全局基函数(如球谐函数)更好地实现的。近年来，主要研究者和他的研究小组开发了几种算法(正则化函数匹配追踪及其增强)，能够迭代地构造一种反问题的最优基。这些方法针对球体或球上的场景进行了特别详细的阐述。对几个问题的适用性已经得到了证明。然而，这些方法仍有一定的局限性。例如，大数据集，因为它们是引力场中常见的，到目前为止还不能处理，旅行时间层析成像不能提供有效的正演计算公式。在这个项目中，这些算法应该进一步增强，以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引力势的高维建模。我们期待着对这两个实际问题有特别新的见解。在前一种情况下，关于速度模型中可能出现的伪影，出现了上述问题。RFMP及其变种使网格结构的多尺度适应自动化成为可能，这在以前是手动完成的。我们预计计算模型的精确度会有所提高。此外，为了测试模型的某些方面的稳定性，试函数集(称为词典)可以是不同的。这样，可以更好地识别由于基函数的选择而产生的伪影。在重力场模拟的情况下，应该找到有效的方法，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准确地逼近局部集中的局部异常。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.