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Reduced Order Model Based Non-Linear Preconditioning For Iterative Seismic Data Interpretation Via Full Waveform Inversion

IP.com Disclosure Number: IPCOM000241974D
Publication Date: 2015-Jun-11
Document File: 6 page(s) / 335K

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   REDUCED ORDER MODEL BASED NON-LINEAR PRECONDITIONING FOR ITERATIVE SEISMIC DATA INTERPRETATION VIA FULL WAVEFORM INVERSION

INTRODUCTION

    Seismic data interpretation based on iterative fitting of the measured data with a PDE wave propagation model known as the full waveform inversion (FWI) is the most general yet a very computationally challenging approach. The difficulties arise from the fact that when formulated as non-linear least squares, it yields a highly non-linear optimization problem with a functional that has many local minima. Thus, any local derivative-based method has difficulties obtaining a global minimizer. Even when convergence is achieved, it typically needs many iterations. Since each iteration requires one or more solutions of a large scale wave propagation problem, the overall cost of FWI can become prohibitively expensive. In our novel approach we replace the traditional least squares functional with a non-linearly preconditioned one. The preconditioner maps the data space to the reduced order model (ROM) space, so instead of minimizing the data misfit the optimization seeks a minimizer of a misfit between the ROMs. Since the ROMs are closely related to the full wave propagation models, the optimization functional becomes much better behaved than the standard one. This leads to a robust convergence from poor starting models and to a reduction in the total number of iterations. Our approach is general enough to be applicable to a variety of time domain wave inversion problems. This includes surface and borehole (VSP and cross-well) seismics, acoustic (sonic) logging, etc.

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BACKGROUND

    Time domain seismic data interpretation via FWI is typically formulated as a non- linear least squares optimization problem

minimize ‖𝒅 βˆ’ 𝓕[π’Ž]β€–πŸπŸ + πœΆπ“‘(π’Ž),
π’Ž

where the data measured at receiver locations is 𝑑, the model is π‘š, the forward modeling operator β„± maps the model space to the data space, and the regularization functional β„› is weighted by a penalty parameter 𝛼. The minimization problem is then solved by a derivative-based method such as the steepest descent, non-linear conjugate gradients (NLCG), Gauss-Newton (GN) or quasi-Newton (BFGS, L-BFGS) [1].

    The highly non-linear nature of the minimization problem (1) makes it hard to solve numerically since the derivative-based methods may get stuck in local minima due to cycle skipping, noisy data, etc. A way to alleviate these issues is to modify the optimization functional in (1) by employing a preconditioner. Commonly, the preconditioned functional takes the form

                 π“’(𝐦) = πš«π’…βˆ— 𝑾 πš«π’… + πœΆπ“‘(π’Ž),
where Δ𝑑 = 𝑑 βˆ’ β„±[π‘š] and π‘Š is a weighting matrix [2]. Other approaches rely on a particular optimization method. For example, a conjugate gradient method is often preconditioned with multiplying the gradient by an approximation to the diag...