A new method for constructing the Hessian in elastic full waveform inversion Vegard Stenhjem Hagen * and Børge Arntsen, Norwegian University of Science and Technology; Espen Birger Raknes, Aker BP and Norwegian University of Science and Technology SUMMARY We propose a method for constructing the Hessian in Elastic Full Waveform Inversion (EFWI) as a series of Hessian-vector products with model perturbations. The assembled Hessian of both simple and complex synthetic models are studied to draw conclusions regarding spatial resolution and parameter retrieval. The results indicate a strong cross-talk between the different model parameters of a common density-velocity parametrisation in EFWI. Furthermore we are able to comment on the influence of offset by comparing zero- and intermediate- offset results. INTRODUCTION The ill-posedness of EFWI introduces local minima which must be avoided to perform successful recovery of subsurface param- eters (Operto et al., 2013). Commonly, the gradient update in EFWI is estimated by linearising the problem and approximat- ing the Hessian with a scalar as a way of eliminating some computational complexity, but at the cost of information loss. We propose calculating the action of the Hessian matrix on a subset of the model (Hessian-vector products) as a way of estimating the resolution and parameter cross-talk in a defined region, helping us give an estimate of the accuracy of the in- version. By utilising second-order events we are able to construct the Hessian kernel (Marquering et al., 1999; Fichtner and Tram- pert, 2011b). We can then analyse the Hessian kernel to help us better understand the inversion results obtained from EFWI. For instance the Hessian contain information about parameter cross-talk and sensitivity to model changes (Sager et al., 2017), indicating how well different areas of the model is resolved in terms of uncertainty and resolution. We have calculated the Hessian-vector product for a vertical slice of the model for different model perturbations along the slice. The products have then been assembled to construct the Hessian of the model slice which we use for studying the resolution and uncertainty of the model slice. The Hessian can also be directly used in a Newton inversion scheme (Pratt et al., 1998; Epanomeritakis et al., 2008), but these methods will not be discussed here. THEORY FWI is a technique for iteratively recovering model parameters using the entire recorded waveform (Tarantola, 1984; Mora, 1987). For a comprehensive overview of modern FWI we refer to Virieux and Operto (2009), but we will state a brief overview of the basic in elastic media using the adjoint formulation in the time-domain (Fichtner et al., 2006). The elastic displacement field u(x, t ) in a model m(x, t ) with space-coordinates x ∈ G ⊂ R 3 and time t ∈ [0, T ] ⊂ R can be described by the wave-operator L(u, m) defined as L(u, m) = ρ(x) ¨ u(x, t ) -∇σ (x, t ) = f (x, t ), (1) where f (x, t ) is the driving force, density is denoted by ρ, and the stress tensor σ (x, t ) given by the 4th order stiffness tensor C as σ ij = C ijkl ∂ k u l , (2) using the Einstein summation convention. Next we define a misfit function Ψ = Ψ(u(m, x r ), d 0 ) (3) as a measure of how good a fit our modelled recording u(x r , t ) is to the true recorded data d 0 (x r , t ) at recording locations x r . By calculating the gradient of the misfit with respect to the model parameters we can then find a model update that will decrease the misfit. This is done by introducing the Jacobian as J(m + δm) = ∇ m Ψ(m + δm), (4) and linearising around m resulting in J(m + δm) ’ J(m) + ∇ m J(m) δm = 0. (5) Thus introducing the Hessian H(m) = ∇ m J(m) = ∇ m ∇ m Ψ(m) . (6) The model update δm can then be obtained from equation (5) by solving H(m) δm = -J(m) (7) for δm. To compute the full Hessian one would need to calculate the second partial derivative of every model point which is compu- tationally prohibitive for reasonable large models. A common adequate approximation is substituting the Hessian with a scalar and performing a line search for the optimal α ∈ R to instead solve (Raknes, 2014) δm ’ αJ, (8) thus losing 2nd order information in the process. In this work we will be calculating the action of the Hessian on model per- turbations (Hessian-vector products) using an adjoint approach proposed by Fichtner and Trampert (2011a). We can calculate the Jacobian in equation (4) by introducing the adjoint wavefield u † . This wavefield can be obtained by time-reversing the kernel χ of the misfit function (3) as Ψ = Z T Z G χ  u(m; x r , t ), d 0  dt dG, (9) 10.1190/segam2018-2998233.1 Page 1359 © 2018 SEG SEG International Exposition and 88th annual Meeting Downloaded 11/24/18 to 111.89.211.186. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/