Abstract
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 âE/âmi 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 âE/âmi.
Highlights
Acoustic, electromagnetic and seismic waves are routinely used to probe the media through which they propagate, and especially to image the spatially-varying velocity field
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
Our analysis is divided into four sections: first, we review how the adjoint state method is used to streamline the computation of a critical quantity need to perform tomography; second, we review the concept of the geometrical spreading of rays and its connection to the transport equation; third, we use the adjunct state method to derive and solve the differential equation for the adjoint field; and lastly, we show that the adjoint equation is very closely related to the transport equation and that its solution can be trivially constructed when the solution to the transport equation is known
Summary
Electromagnetic and seismic waves are routinely used to probe the media through which they propagate, and especially to image the spatially-varying velocity field. Over the last several decades, the development of the so-called adjoint state method [9] has allowed tomographic imaging to be applied in cases where it was hitherto fore infeasible, because of vastly reduced computational effort To date, this efficiency mainly has used to enable computationally-intensive forms of tomography, and especially to full wavefield tomography [7] [8]. We study the mathematical structure of the differential equation that arises out of the adjoint state method (the equation for the so-called adjoint field) and show that it is very closely related to and in important cases identical to the transport equation of ray theory This relationship provides an intuitive understanding of the adjoint field and suggests further ways of obtaining further computational efficiency. Our analysis is divided into four sections: first, we review how the adjoint state method is used to streamline the computation of a critical quantity need to perform tomography; second, we review the concept of the geometrical spreading of rays and its connection to the transport equation; third, we use the adjunct state method to derive and solve the differential equation for the adjoint field; and lastly, we show that the adjoint equation is very closely related to the transport equation and that its solution can be trivially constructed when the solution to the transport equation (the geometrical spreading function) is known
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