MODELING FOR INVERSION IN EXPLORATION GEOPHYSICS
- Project Research
- 1-5 Chapters
- Abstract : Available
- Table of Content: Available
- Reference Style: APA
- Recommended for : Student Researchers
- NGN 5000
Introduction and Background
Understanding the physics of our surrounding has driven science and technology for a long time and while some part of the earth is well know and understood, its subsurface is mostly unknown. The subsurface is one of today’s most important challenges. Firstly, it contains natural resources that are critical to our technologies such as water, minerals, gas and oil. Secondly monitoring of the subsurface such as CO2 sequestration [1, 2], earthquake monitoring and prediction [3] and global seismology [4, 5] are major problems for safety and environments. However, the technologies to monitor or find these resources are scientifically extremely challenging. My thesis addresses some of the challenges that arise from these problems, mainly the computational cost and the access to the right software for research and development. Seismic imaging estimates subsurface parameters such as the velocity of sound waves or the rock’s density from pressure measurements recorded at the surface of the earth or the ocean. This parameter estimation problem can be formulated as a mathematical optimization problem that is usually the minimization of a data misfit between the field recorded data, and numerically generated synthetic data [6]. The optimization problem and its minimization algorithm therefore involves solving the wave-equation, a partial differential equation (PDE), either in the frequency domain via iterative solvers [7, 8, 9, 10] or in the time domain via time-steppers [11, 12, 13]. In practice, the type of seismic sources used in the field to record the data are bandwidth-limited, and, unlike other fields such as Medical Imaging, seismic measurement can only be recorded at the surface and/or at very limited number of well locations. These two physical constraints render the mathematical 1 data fitting problem ill-posed and non-convex in most cases. As a consequence, extensive research has been directed towards finding algorithms to solve seismic inversion and imaging problems with good or less good success depending on the situation and the quality of available datasets, e.g. [6, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. For these reasons, this is a field that is still in active development. While the mathematical difficulty to solve the data-fitting optimization is complex, one of its main requirement is to have access to computationally efficient wave-equation solvers. The problem formulation requires solutions of thousands of wave-equations in order to achieve an acceptable (inversion) results for large scale domains. A standard seismic problem involves an unknown (discrete representation of the subsurface) typically with up to a billion unknowns: minimize x O X (1e4) i=1 fi(x), x ∈ R 1e9 . (1.1) However, this computational complexity and cost cannot become a burden to mathematicians or geophysicist whose domains of expertise and interest are optimization algorithms, subsurface parameters estimation, and imaging and not high-performance computing to carry out the wave simulations. For these reasons, well designed software including an interface to PDE solvers are necessary to provide a workflow that allows separation of concerns and rapid innovation. In addition to the computational demand of having to solve the wave-equation many times for thousands of times steps, gradient calculations of inversion algorithms often require solutions of the adjoint equation. As I will explain below, the derivation of the adjoint wave equation itself including its interaction with the forward wavefield is challenging since it is generally impossible to store this state variable in memory. In practice, storage of this variable may require terabytes of memory because of our model size and number of time steps (O(1e9) grid points and O(1e4) time-steps). This model size calls for advanced methods such as optimal checkpointing [26, 27, 28, 29] or 2 compression methods [30].It is clear that this type of additional requirements add complexity to the wave-equation solver and requires special care in the design of the interface so that these additional requirements can be accommodated and implemented by domain specialists. The concept of separation of concerns in computational physics has led to numerous projects that motivated my core contribution: Devito [13]. Devito is a finite-differences domain-specific language (DSL) that provides a symbolic interface to define partial differential equations (PDE) and implements its own just-int-time compiler. High-level interfaces such as symbolic DSLs and just-in-time compilers are gaining attention and earlier work in computational fluid dynamics (CFD) provided a strong basis and justification for it. The need for a high-level interface for CFD led to the design of symbolic DSLs such as FEniCS [31] or Firedrake [32] that provide a symbolic interface to define the weak form of variational problems. Both of these two frameworks implement the same DSL known as UFL [33]. The success of these DSL laid the ground for Devito’s high-level user interface. Moreover, Devito also relies on just-in-time code generation and compilation to provide state-of-the-art computational performance. A thorough overview of the literature is detailed in each of the Chapters constituting my thesis. The main aim of my thesis is to find an answer to the extreme computational challenges of seismic inversion while providing an interface that enables rapid code development, with a carefully designed DSL, and performance tools such as automatic roofline performance benchmark [34]. My introduction is organized as follows. First, I introduce the seismic inversion problem in more detail including its mathematical formulation. Next, I provide a motivating example that highlights the complexities that arise when dealing with realistic physical models that describe wave motion in the Earth subsurface. After discussing this example, I will define the objectives of my thesis and detail my contributions and conclude with an outline of the three main chapters of my thesis.
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