II Simpósio Brasileiro de Geofísica II Simpósio Brasileiro da SBGf – Natal 2006. A comparison of methodologies for the selection of regularization parameter in anisotropic traveltime tomography Eduardo Telmo Fonseca Santos, IGEO/UFBA and CEFET/BA, Brazil. Amin Bassrei, IF/UFBA and CPGG/UFBA, Brazil. Jessé C. Costa, UFPA, Brazil. Copyright 2006, SBGf - Sociedade Brasileira de Geofísica Este texto foi preparado para a apresentação no II Simpósio de Geofísica da Sociedade Brasileira de Geofísica, Natal, 21-23 de setembro de 2006. Seu conteúdo foi revisado pela Comissão Técnico-científica do II SimBGf mas não necessariamente representa a opinião da SBGf ou de seus associados. E proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf. ____________________________________________________________________ Abstract Since inverse problems are usually ill-posed it is necessary to use some method to reduce their deficiencies. The method that we choose is the regularization by derivative matrices. When a first derivative matrix is used the order is called the first. Then, second order regularization is when the matrix is formed by second order differences, and order zero means that the regularization matrix is the identity. There is a crucial problem in regularization, which is the selection of the regularization parameter λ. We use the L-curve as a tool for the selection of λ. L-curve was reintroduced in the literature of inverse problems by Hansen (1992a) and we use it in cross hole traveltime tomography. In this approach of tomography the goal is to obtain the 2-D velocity distribution from the measured values of traveltime between sources and receivers. Besides the L- curve, we also propose a new extension of it, which we called θ-curve. We present several simulation results with synthetic data and we validate the feasibility of regularization, as well as both parameter selection approaches. Introduction The main purpose of exploration geophysics for hydrocarbons is to provide trustworthy images of the subsurface, which could indicate potential hydrocarbon reservoirs. Exploration seismology, better known as seismics, is the area of applied geophysics most employed for the subsurface imaging in hydrocarbons reservoirs. And within seismics, tomography was incorporated as a suitable method of data inversion. In this work we use traveltime tomography where the input data is the acoustic traveltime measured at the receivers, and the velocity of the 2-D medium is the inversion output. For the forward modeling we compute the traveltime by acoustic ray tracing from a given 2-D velocity distribution. One common way to calculate inverse matrix is by the generalized inverse through singular value decomposition, but since geophysical tomography is an ill-posed inverse problem, it is necessary to use some tool to reduce this deficiency. The tool that we choose is the regularization of the inverse problem by derivative matrices, known in the literature by several names, specially as Tikhonov regularization. Regularization has an input parameter with crucial role, known as regularization parameter λ, which choice is already a problem. In this work we use the L-curve and an extension of it which we called θ-curve for the selection of regularization parameter in cross hole traveltime tomography. Methodology Consider a modeling process where the input of some system is described by certain parameters contained in m and the output is described as Gm(=d) which is a linear transformation on m. If the vector d describes the observed output of the system, the problem is to "choose" the parameters m in order to minimize in some sense, the difference between the observed d and the prescribed output of the system Gm. If we measure this difference through the norm ||·||, our task is to find the value of m which minimizes , d m − G where the M×N matrix G and the data vector d with M elements are provided to the problem. This is called a least squares problem, which can be formally stated as follows. Considering the basic relationship , m d G = we wish to minimize the error using the following objective function based on the work of Levenberg (1944) and Marquardt (1963): , 2 λL ) Φ( T + = e e m where the error is given by e = d - Gm, λ is a scalar called the damping parameter and L 2 = m T ⋅m. The estimated solution, also called damped least squares (DLS) solution, is . G λI) G (G T - T est d m 1 + = Model estimation can be solved using the method of conjugate gradient (CG), described by Hestenes and Stiefel (1952). This method was developed for the