Linearization of the eikonal equation

Geophysics ◽  
1994 ◽  
Vol 59 (10) ◽  
pp. 1631-1632 ◽  
Author(s):  
David F. Aldridge
Keyword(s):  
Inverse Problem ◽  
Perturbation Theory ◽  
Path Integral ◽  
Seismic Waves ◽  
Eikonal Equation ◽  
Nonlinear Inverse Problem ◽  
Nonlinear Functional ◽  
Arrival Times ◽  
Traveltime Tomography ◽  
Fermat's Principle

Seismic traveltime tomography is a nonlinear inverse problem wherein an unknown slowness model is inferred from the observed arrival times of seismic waves. Nonlinearity arises because the raypath connecting a given source and receiver depends on the slowness. Specifically, if L(s) designates a raypath through the slowness model s between two fixed endpoints, then the path integral for traveltime [Formula: see text] is a nonlinear functional of s because it does not, in general, satisfy the superposition condition (i.e., [Formula: see text] where [Formula: see text] and [Formula: see text] are two different slowness models). The tomographic inverse problem can be solved after linearizing the traveltime expression about a known slowness model [Formula: see text]. This linearized expression is usually obtained by appealing to Fermat’s principle (e.g., Nolet, 1987). Alternately, the required relation can be rigorously derived via ray‐perturbation theory (Snieder and Sambridge, 1992). The purpose of this note is to present a straightforward derivation of the same result by linearizing the eikonal equation for traveltimes. Wenzel (1988) adopts this approach, but his method of proof cannot be generalized to heterogeneous 3-D media. A full 3-D treatment is given here. The proof is remarkably simple, and thus it is quite possible that others have discovered it previously.

Seismic Rays

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Inverse Problem ◽  
Seismic Tomography ◽  
Seismic Waves ◽  
Seismic Station ◽  
Three Dimensional ◽  
Arrival Times ◽  
The Earth ◽  
Straight Lines ◽  
Ray Path ◽  
Uniform Composition

This chapter focuses on the underlying mathematics of seismic rays. Seismic waves caused by earthquakes and explosions are used in seismic tomography to create computer-generated, three-dimensional images of Earth's interior. If the Earth had a uniform composition and density, seismic rays would travel in straight lines. However, it is broadly layered, causing seismic rays to be refracted and reflected across boundaries. In order to calculate the speed along the wave's ray path, the time it takes for a seismic wave to arrive at a seismic station from an earthquake needs to be determined. Arrival times of different seismic waves allow scientists to define slower or faster regions deep in the Earth. The chapter first presents the relevant equations for seismic rays before discussing how rays are propagated in a spherical Earth. The Wiechert-Herglotz inverse problem is considered, along with the properties of X in a horizontally stratified Earth.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

Inversion of induced polarization data

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1327-1341 ◽  
Author(s):  
Douglas W. Oldenburg ◽  
Yaoguo Li
Keyword(s):  
Inverse Problem ◽  
Objective Function ◽  
Effective Conductivity ◽  
Optimization Theory ◽  
Induced Polarization ◽  
Dc Resistivity ◽  
Earth Structure ◽  
Nonlinear Inverse Problem ◽  
Polarization Data ◽  
The Earth

We develop three methods to invert induced polarization (IP) data. The foundation for our algorithms is an assumption that the ultimate effect of chargeability is to alter the effective conductivity when current is applied. This assumption, which was first put forth by Siegel and has been routinely adopted in the literature, permits the IP responses to be numerically modeled by carrying out two forward modelings using a DC resistivity algorithm. The intimate connection between DC and IP data means that inversion of IP data is a two‐step process. First, the DC potentials are inverted to recover a background conductivity. The distribution of chargeability can then be found by using any one of the three following techniques: (1) linearizing the IP data equation and solving a linear inverse problem, (2) manipulating the conductivities obtained after performing two DC resistivity inversions, and (3) solving a nonlinear inverse problem. Our procedure for performing the inversion is to divide the earth into rectangular prisms and to assume that the conductivity σ and chargeability η are constant in each cell. To emulate complicated earth structure we allow many cells, usually far more than there are data. The inverse problem, which has many solutions, is then solved as a problem in optimization theory. A model objective function is designed, and a “model” (either the distribution of σ or η)is sought that minimizes the objective function subject to adequately fitting the data. Generalized subspace methodologies are used to solve both inverse problems, and positivity constraints are included. The IP inversion procedures we design are generic and can be applied to 1-D, 2-D, or 3-D earth models and with any configuration of current and potential electrodes. We illustrate our methods by inverting synthetic DC/IP data taken over a 2-D earth structure and by inverting dipole‐dipole data taken in Quebec.

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