New generation finite element methods for seismic forward modelling

该项目致力于开发和制作新的、更有效的计算机模拟方法，以模拟地震波在地球地下的传播和反射。这项研究的动力来自Case合作伙伴斯伦贝谢剑桥研究公司，他们希望开发更有效的方法来成像陆地或海洋沉积物下的地下，以定位含油气岩石。特别是，为了在地震反射率较差的更复杂的几何形状中进行准确成像，例如在玄武岩或盐岩构造下成像，最先进的成像方法是使用迭代进行的成像方法，求解描述每个步骤的波传播和反射的完整数学方程。不幸的是，由于复杂的地下几何结构和必须模拟的大的3D区域，用标准的数值方法对地震传播和反射进行全面的模拟在计算资源上是非常昂贵的。与地震波的波长相比，它的直径很大，因此需要非常高的分辨率才能使用标准计算方法准确地可视化波的传播。这些标准方法包括所谓的“有限元方法”，即计算机模拟方法，其中地球次表层被认为是由大量(例如1,000,000-1,000,000)小块(“有限单元”)组成的，其中每个地震波具有非常简单的行为，例如近似恒定。该项目致力于探索一种新的、更复杂的有限元方法在地震模拟中的使用。这类新方法的不同之处在于在每个元素中使用了更复杂的假设行为，即特定的标准波状行为(平面波或平面波的组合)。为了将该项目保持在一个可管理的规模，一个适合概念验证的规模，一个适合博士生在三年半内完成的规模，建模将限于二维模拟(其中假设几何图形在一个水平方向上是恒定的)，以及地震传播的简化声学模型。该项目的主要目标将是：i)扩展这类以前的工作，即所谓的超弱变分公式，以便该方法能够处理空间变化的地震性质(即波速随地下位置逐渐或突然变化的情况)。这种对当前方法的(重大)扩展将需要强大的数学和计算技能，这将是该方法用于通用地震模拟的关键。Ii)应用复杂的数学和数值实验相结合的方法，以了解新算法的行为。Iii)在斯伦贝谢公司提供的具有代表性的二维声学地质模型上测试新方法，比较在计算机软件中实现的新算法与基于标准有限元和所谓的有限差分建模的现有方法的性能。斯伦贝谢已经在现有的计算机软件中实施了这些标准方法。在博士学位的前18个月，学生将在雷丁和斯伦贝谢一流的研究环境中接受数学、计算和地震传播建模标准方法知识的培训，这将是完成该项目所必需的。在这方面，雷丁的学生将受益于我们科学和工业计算数学硕士的课程，我们是数学高级研究生级别培训联盟(魔术组)的成员，以及一个庞大的学术和研究人员社区的成员，以及从事许多令人兴奋的波数学应用的博士生，以及在成像中使用波。

This project is concerned with developing and prototyping new, more effective computer simulation methods for modelling the propagation and reflection of seismic waves in the earth subsurface. The pull for this research comes from the CASE partner, Schlumberger Cambridge Research, who wish to develop more efficient methods for imaging the subsurface below land or marine deposits in order to locate hydrocarbon-bearing rocks. In particular, for accurate imaging in more complex geometries where seismic reflectivity is poor, for instance imaging below basalt or salt structures, the state-of the art is to use imaging methods which proceed iteratively, solving the full mathematical equations describing the wave propagation and reflection at each step. Unfortunately, this full simulation of seismic propagation and reflection by standard numerical methods is hugely expensive in computing resources, because of the complex subsurface geometry and the large 3D region that must be simulated. Large, that is, in diameter in comparison with the wavelengths of the seismic waves, so that a very high resolution is needed to visualise the wave propagation accurately using standard computational methods. These standard methods include so-called 'finite element methods', computer simulation methods in which the earth subsurface is thought of as composed of a large number (e.g. 1,000,000-100,000,000) of small pieces (the 'finite elements') in each of which the seismic wave has a very simple behaviour, e.g. is approximately constant. This project is concerned with exploring the use, for seismic simulations, of a new, more sophisticated class of finite element method. This new class of method differs in using a more sophisticated assumed behaviour in each element, namely a certain standard wave-like behaviour (that of a plane wave or a combination of plane waves). To keep the project to a manageable size, one suitable for proof of concept, and suitable for a PhD student to complete in 3 1/2 years, the modelling will be restricted to two-dimensional simulations (where it is assumed that the geometry is constant in one horizontal direction), and to a simplified acoustic model of the seismic propagation. Main objectives of the project will be: i) To extend previous work of this type, namely the so-called Ultra Weak Variational Formulation, so that the method can deal with spatially varying seismic properties (that is, where the wave speed varies gradually or suddenly with position in the subsurface). This (significant) extension to the current method, which will need both strong mathematical and computing skills, will be essential for the method to be of use for general purpose seismic modelling. ii) To apply a combination of sophisticated mathematics and numerical experiments so as to understand the behaviour of the new algorithm. iii) To test the new method on representative acoustic 2D geological models supplied by Schlumberger, comparing the performance of the new algorithms, as implemented in computer software, with existing methods, based on standard finite elements and so-called finite difference modelling. These standard methods Schlumberger has implemented in existing computer software. In the first 18 months of the PhD the student will receive training, in superb research environments at Reading and Schlumberger, in the mathematics, computing, and knowledge of standard methods for modelling seismic propagation, that will be necessary for completion of the project. In this, the student at Reading will benefit from access to courses forming part of our MSc in Mathematics of Scientific and Industrial Computation, from our membership of an advanced graduate level training consortium in Mathematics (the MAGIC group), and from membership of a large community of academic and research staff and PhD students working on many exciting applications of the mathematics of waves, and the use of waves in imaging.