A Connection between Geometrical Spreading and the Adjoint Field in Travel Time Tomography

• 作者: William Menke
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: ADJOINT State METHOD EIKONAL EQUATION GEOMETRICAL SPREADING Gradient-Descent METHOD Transport EQUATION Travel Time Tomography
• 年，卷（期）: 2020,（2）
• 页码范围: 84-96
• 页数: 13页
• 分类号: O17

摘要

The goal of tomography is to reconstruct a spatially-varying image function s(x,m), where x is position and m is a finite-length vector of parameters. Many reconstruction methods minimize the total L2 error E ≡ eTe, where individual errors ei quantify misfit between predictions and observations, to quantify goodness of fit. So-called adjoint state methods allow the gradient ∂E/∂mi to be computed extremely efficiently from an adjoint field, facilitating image reconstruction by gradient-descent methods. We examine the structure of the differential equation for the adjoint field under the ray approximation and find that it has the same form as the transport equation, whose solution involves the well-known geometrical spreading function R Consequently, as R is routinely tabulated as part of a ray calculation, no extra work is needed to compute the adjoint field, permitting a rapid calculation of the gradient?∂E/∂mi.