Mathematical and Numerical Problems in Seismic Imaging

一般领域这个项目是在一般领域的数值分析波传播问题。更具体地说，该项目侧重于地震成像。背景如下：作为地震勘探过程的一部分，波从震源发射到地球中，然后由传感器测量这些波从地球地下的反射。对于海洋环境中的地震勘探，震源和测量传感器被拖在船后。地震波可以用时域或频域的偏微分方程（PDE）来描述。给定传感器测量，可以推断地下的地震属性。这个过程被称为全波形反演（FWI）。FWI需要PDE的数值解，以生成预测数据，以及优化步骤，其中预测和测量数据之间的差异最小化，以获得改进的地下模型。早期培训作为CDT桑巴舞的学生，Shaunagh将为一年级博士生进行桑巴舞培训计划，包括参加综合智库，并采取与她的项目领域相关的单元。特别是，她将参加波动方程的阅读课程，并将参加有关大规模优化的4级课程。她将接受的培训的进一步细节在下面的“培训要求”一节中给出。最终目标和方法这个项目将考虑地震成像中的不同数学和数值问题，特别是与FWI过程有关的问题。所使用的数学工具将是数值分析、优化和偏微分方程分析的方法。我们寻求调查的一个初始问题是传感器的最佳定位，以便从地震勘探过程中获得最佳回报，该问题在石油勘探中具有很大的实际意义，并且在FWI的标准方法中没有考虑。这个问题的关键问题是“给定关于地下可能构成的先验信息（以一类通用模型的形式），能否优化传感器的位置以检索关于地下的足够信息？''

GENERAL FIELDThis project is in general area of the numerical analysis ofwave-propagation problems. More specifically, the project focuses on seismic imaging. The context is as follows: Aspart of the seismic exploration process, waves are emitted from a source into the earth and then the reflections ofthese waves from the subsurface of the earth are measured by sensors. For seismic exploration in a marineenvironment, the source and measurement sensors are towed behind boats. The seismic waves can be described bypartial differential equations (PDEs), formulated in the time or frequency domain. Given the sensor measurements,the seismic properties of the subsurface may be inferred. This process is termed Full Waveform Inversion (FWI). FWIrequires the numerical solution of the PDE, to generate predicted data, and an optimisation step, in which thedifference between the predicted and measured data is minimised to obtain an improved modelof the subsurface.EARLY TRAININGAs a student in the CDT SAMBa, Shaunagh will undertake the SAMBa training programme for first-year PhDstudents, including participating in an Integrative Think Tank, and taking units relevant to her project area. Inparticular she will take a reading course on the wave equation and will do a relevant Level 4 course on large-scaleoptimisation. Further details of the training she will undergo are given in the section "Training Requirements" below.FINAL AIMS AND METHODOLOGYThis project will consider different mathematical and numericalproblems in seismic imaging, in particular related to the process of FWI. The mathematical tools used will bemethods from numerical analysis, optimisation, and analysis of PDEs. One initial problem we seek to investigate,which is of great practical interest in petroleum prospecting, and is not considered in the standard approach to FWI,is the optimal positioning of the sensors in order to obtain the best return from the seismic exploration process. Thekey question in this problem would be "given prior information about the likely make-up of the subsurface (in the formof a class of generic models), can one optimise the location of the sensors to retrieve sufficient information about thesubsurface?''