A MAP Algorithm for AVO Seismic Inversion Based on the Mixed (L 2, non-l 2 ) Norms to Separate Primary and Multiple Signals in Slowness Space

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1 A MAP Algorithm for AVO Seismic Inversion Based on the Mixed (L 2, non-l 2 ) Norms to Separate Primary and Multiple Signals in Slowness Space Harold Ivan Angulo Bustos Rio Grande do Norte State University - UERN BR 110, Km 46 - Mossoro, RN, Brazil haroldivan@hotmail.com Marcelino Pereira dos Santos Silva Claubio Landney Lima Bandeira Rio Grande do Norte State University marcelinopereira@uern.br claubiobandeira@yahoo.com.br Abstract AVO (Amplitude Vs Offset) seismic inversion is a technique of tomographic seismic imaging for creating a model in stack-velocity space that can correctly reconstruct the measured AVO seismic dataset. This is usually implemented by minimizing a least squares inversion algorithm. This algorithm has limitations because it reconstructs seismic images with artifacts yield by impulsive noise contained in the input raw seismic dataset. Recently, superior seismic images were reconstructed using a MAP (Maximum A Posterior) approach, based on the Norm L p. In this paper, we demonstrate similar results and even superior ones via minimizing a MAP approach built through L 2 norm of dataset misfit and a non-l 2 Lorentzian error norm of the model energy. Keywords: travel time sub-surface seismic imaging, seismic inversion, MAP algorithm. 1. Introduction The velocity stack domain is an alternative domain for processing seismic datasets. The input raw dataset of seismic inversion algorithms has axes of time, offset. The seismic stack-velocity space has axes of time, velocity or slowness [1]. Many seismic data processing applications are much simpler in the stack-velocity space. E.g., multiple attenuation can be performed as an operation of a noise filter in the stack-velocity space. In this paper, the forward transform or forward modeling of input data is defined and the inverse transform can be implemented using an iterative solver. The usual process is to compute the inverse as the minimization of a least squares problem. The least squares solution has some attributes that may be undesirable. If the model space is overdetermined, the least squares solution will usually be spread out over all the possible solutions. Other methods may be more useful if we desire a parsimonious representation. In this work, we demonstrate some limitations of the L 2 norm used at the Stanford exploration project [1] and [2] to generate images obtained by seismic inversion from a raw input seismic dataset corrupted by impulsive noise. We obtained this dataset from Madagascar [3] data repository, a specialized free software on seismic imaging and signal processing. Its components were used to develop our seismic inversion algorithm using Scons technology (software construction), a well-know opensource program. Madagascar started in 2003, and was publicly released in It is an open-source package developed at the University of Texas in Austin, USA. It is distributed under the standard GPL (General Public License) license. In [4] further results of the Stanford exploration project based on a MAP approach are presented. It is implemented an objective function build on the L p norm of data misfit, with (p=1.1), and a L 1..1 error norm that represent the prior energy of the model. This error norm has the purpose of image regularization in stack-velocity space. It was designed to reduce the effect of impulsive noise yield by multiple signals on the stack-velocity model. The multiple seismic signals are compressional waves that suffer multiple reflections in the subsurface layers and increase in intensity with respect to the primary waves of lower energy that do not suffer from any multiple reflection. In [5] it is presented a method for robust seismic inversion based on the norm Huber of data misfit. As measures of data misfit, they show considerably less sensitivity to large measurement errors than least squares L 2 measures. In [6] it is presented a travel time seismic inversion algorithm based on the weighted L 2 norm such that some ray tracing has more weight than others on the full L 2 norm. This behavior of the algorithm makes it less sensible to measurement errors on data misfit. In [7] it is presented a travel time inversion algorithm that can give the best least-squares fit to the measured travel time data. This algorithm was applied using a technique based on a neural network.

2 In our work, the objective function is build on two mixed (L 2, non-l 2 ) norms. The L 2 norm is the energy of the measured and modeled misfit data. The non-l 2 error norm represents the prior energy on the stack-velocity model and is given by Lorentzian robust norm [8, [9]. It is used to image regularization in the stack-velocity space. We developed a regularized inversion algorithm to generate images in the stack-velocity space from raw input seismic dataset in which the multiple and primary energy of seismic signals must be well separated. This goal is achieved using Lorentzian robust norm, what is presented in this paper. Although the L 2 norm is sensitive to large errors of measurement at the data misfit, produced by multiple signals, the effect of those errors, in the stack-velocity space, is minimized by image regularization executed by robust Lorentzian norm. 2. MAP Formulation of the inverse problem 2.1 Least squares inversion In velocity stack inversion, it is assumed that the forward operator H, that maps from stack-velocity space to offset space, can be implemented. In the equation (1), d represents the measured travel-time dataset in offset space and belongs to Hilbert space, m represents the slowness object model in stack-velocity space and H m is the forward modeling operator, which represents the modeled travel-time dataset in offset space. d = H m (1) The objective function or MAP estimator to find least squares optimal solution is given below. m = d H m 2 k f m (2) We seek a solution to the problem of finding the model in stack-velocity space m, given the data d. This is usually posed as a least squares optimization problem that minimizes the energy of the measured and modeled data misfit. In the Stanford exploration project [1] it is implemented a conjugate direction optimization algorithm to find optimal solution for equation (2) with (k=0). We recreated this problem in this work using the methodology based on reproducible experiment based on Madagascar package [3]. 2.2 Regularized seismic inversion based on mixed (L 2, non-l 2 ) norms The L 2 inversion is the optimal choice in the presence of Gaussian noise in the seismic input dataset. However, in our case, we are working with spurious seismic input dataset that contain non-gaussian impulsive noise. In this situation, least squares inversion yield reconstructed images with undesirable artifacts that degrades the images [1]. [4] presents more results of Stanford exploration project that show a novel MAP approach built on the L p non-gaussin norm of data misfit and on the L p non-gaussian error norm of the model energy. This algorithm was design to remove artifacts on reconstructed images in stack-velocity space, generated by impulsive noise associated with multiple seismic signals. In our work, we designed an objective function built on the L 2 Gaussian norm that represents the energy of the measured and modeled data misfit and a non-l 2 Robust Lorentzian error norm that represents the energy of the model [8], [9]. When it is applied the descent gradient algorithm on Robust Lorentzian norm, it is obtained the Perona-Malik anisotropic diffusion MAP filter. This filter is part of the Madagascar library seismic processing package. However, it is fundamentally applied as a postprocessing filter from any formed generic input image. We are applying it, in this work, for image regularization (in the stack-velocity space) on reconstructed images from raw input seismic dataset containing impulsive noise. This filter allows the small image-edges regularization while it preserves stronger edges. Our objective function is defined by equation (3). The first term represents the energy of the measured and modeled data misfit. The second term represents the prior energy on the stackvelocity model given by Lorentzian robust norm [8], [9]. This norm is a Non-Gaussian Markov Random Field and it is a function of the image intensity differences, (m p -m s ) between pixel s and its neighboring pixels p. The scale parameter of the norm has the function of output image regularization. In this case, if the image intensity differs, (m p -m s ) is below this threshold, the output image-edges are diffused but image-edges are preserved above it. The parameter n s represents the spatial neighborhood of the pixel s, and n s is the number of neighbors (usually four, except at image boundaries), such that p n s. This Robust Lorentzian norm is appropriate for image regularization obtained from raw seismic input datasets contaminated by impulsive noise. This norm trends to forbid the formation of artifacts associated with image-edges yield by L 2 norm. Adjusting the scale parameter, it is possible to smooth the whole output image while preserving stronger image-edges. 1 m = d H m 2 m p m s, s m p n s (3)

3 This Lorentzian norm is given by equation (4). m p m s, =log[1 1 m p m s 2 ] 2 2 (4) The optimal solution MAP of the objective function is represented by the following expression. argmin m d H m 2 m p m s, s m p n s The numerical representation of the posterior solution m is given by equation (5) through an iterative-recursive scheme. The two first terms of this equation represent the classic suboptimal solution based on the L 2 norm used in Madagascar [1]. It is obtained from conjugated direction optimization algorithm that can converge to local minimum [1], where m i 1 is the preceding estimative of m, m i is the new estimative of m, p i 1 denotes the step direction to be specified in the space model, and i is a optimization parameter (or direction weight factor). The equation (6) represents the residual or difference between input dataset and forward model in the iteration (i). The equation (7) gives the relation between the residuals in the iteration (i) and (i-1). In our case, we developed a strategy for output image regularization generated from classic suboptimal solution of Madagascar. This strategy was carried out using the Perona-Malik MAP filter [6], which acts on Madagascar output image from a input raw seismic dataset corrupted by impulsive noise. The discretization of Perona-Malik for their anisotropic diffusion equation is given by the sum of the third and first terms in (5) [8], [9]. The g(.) function in equation (5) has the goal of output image regularization generated by the L 2 norm used at Madagascar. This is the image reconstruction algorithm of our proposal. i m s =m i 1 s i p i 1 g m i 1 s, p (5) Y m i = d H,m i (6) Y m i =Y m i 1 i H p i 1 (7) The function g(.) is given by equation (8) and named the influence function [8], [9]. In this equation, the constant λ represents a positive scalar that determines the rate of image intensity diffusion of function w(.), which is given by equation (10). The image intensity differences (m p -m s ), between pixel s and its neighboring pixels p, is represented by the gradient operator. The influence function is quasi-zero for very small image-edges and will not update those pixels. Pixels with stronger image-edges will be updated adjusting the scale parameter according to the behavior of Perona-Malik function given by equation (10). g m s, p = λ n s p n s w m s, p m s, p (8) m s, p =m p m s i The Perona-Malik function is presented at [8]. It was designed to execute image intensity diffusion on small edges, but stopping the diffusion when finding big edges. Adjusting the scale parameter, it is possible to diffuse the output image-artifacts associates even with multiple signals, while reducing its width in stack-velocity space. Thus, this seismic imaging technique allows to separate appropriately, in the stack-velocity space, the primary and multiple signals. w m s, p = 3. Experimental results 1 (9) [1 m s, p 2 ] (10) 2 2 The following figures show the effects of using the various inversion methods on the raw seismic input datasets which are part of the Madagascar dataset repository. These raw datasets are obtained from an offshore oil reservoir in the north sea (Scotland). Such datasets were publicly released by Chevron Mobil petroleum company [10] for basic research and pre-stack AVO (Amplitude Vs Offset) analysis. Figures (1a, 1b, 1c, 1d) display four stages in the processing of the raw input seismic dataset. These images were generated in Stanford exploration project [1] and reproducible in this work. Figure 1a is the raw seismic input dataset, figure 1b is its L 2 stack-velocity inverse, figure 1c is the modeled dataset from the inverse, and figure 1d is the residual (difference between the input seismic dataset and the modeled seismic dataset). The raw dataset is heavily contaminated by multiples and there are high amplitude bursts at the far offsets. The impulsive noise produces long curved artifacts in the stack-velocity space of figure 1b. They are the dominant features in the inversion and it is difficult to

4 identify the primary velocity trend. However, despite the unpleasant appearance in the stack-velocity space, the reconstructed dataset is a good match to the input (see figure 1c) and the residual is small over the whole domain (see figure 1d). This is the characteristic behavior of L 2 inversion, which attempts to minimize the global energy in the residual. Figures (2a, 2b, 2c, 2d) show the same four stages of processing when our algorithm in (5) is implemented. These figures display the effect of regularized inversion in the stack-velocity space using Robust Lorentzian error norm. In what follows, it was applied a 2D median filter in the stack-velocity space using a 6X6 mask. It was applied on the output image from the inversion algorithm (5) at each iterative-recursive step of it. The events in velocity-stack space are much more compact and the noise streaks do not cross the trend of the primary energy. This is a very desirable result, if we intend to forbid the noisy data in velocity-stack space. It is much easier to separate the primary trend (on the left of figure 2b) from the multiple ones and the effects of the impulsive noise contained in the raw input seismic dataset. The modeled data in figure 2c is still a good match to most of the raw input dataset in figure 2a, but the algorithm has diffused the lower amplitudes and enhanced the higher amplitudes such as expected by Perona-Malik diffusion. intensity is still relatively low. The reason for this behavior is that the residual of figure 2d is equivalent to the residual of figure 1d plus the g(.) function, therefore increasing its intensity value. Figure (1b): L 2 inversion Figure (1a) :raw input seismic dataset Figure (1c): L 2 inversion + modeling The residual in figure 2d is stronger than in the L 2 case (see figure 1d) on the whole domain, although its

5 Figure (1d): Residual after L 2 inversion Figure (2b): regularized L 2 inversion Figure (2a): raw input seismic dataset Figure (2c): L 2 inversion + modeling

6 sedimentary rocks containing hydrocarbons. Other relevant aspect of this work lies in the fact that we developed this novel application of Madagascar free software package. Therefore, it enables our team to use our Madagascar-application without any kind of restriction, including the possibility of technology transfer. Acknowledgments This research has been supported by CAPES, a Brazilian Government Agency for human resource development, and also by CNPq, the Brazilian National Council for Scientific and Technological Development. References 4. Conclusion Figure (2d): Residual after L 2 inversion The choice of a suitable norm for data misfit is dependent on expectations of the noise character. If the noise in raw input seismic dataset is known to be Gaussian and uncorrelated, then a good stack-velocity model may be reconstructed from seismic dataset via minimizing the L 2 norm of the data misfit. If the noise in seismic dataset is known to be impulsive, it is shown in this work that the separation of primary signals from multiple ones, in stack-velocity space, will be obtained via minimizing the objective function made on the mixed (L 2, non-l 2 ) norms. The Lorentzian norm trends to forbid the artifacts formation with small intensity image-edge that are yield by the L 2 norm of dataset misfit. On the other hand, the stack-velocity image regularization based on this Lorentzian error norm allows that only stronger artifacts associated with impulsive noise rise in stack-velocity space. Thus, the effect of the regularization based on Lorentzian norm is to separate appropriately, in the stackvelocity space, the stronger multiple signals of high energy (associated with impulsive noise) from the primary signals of low energy. This is one of the most relevant tasks in multidimensional seismic signal processing, because it leads to super-resolution imaging. Features of the high-resolution seismic images in stackvelocity space will give important information about porosity formation and pressure in sub-surface [1] S. Fomel, Least-square inversion with inexact adjoints, method of conjugate direction: a tutorial, Stanford Exploration Project, vol. SEP-92, pp , [2] J. G. Berryman, Analysis of Approximate Inverses in Tomography II. Iterative Inverses, Optimization and Engineering vol. 1, pp , 2000 [3] S. Fomel and G. Hennenfent, Reproducible computational experiments using scons, In Proc. IEEE International Conference on Acoustic, Speech, and Signal Processing, no. paper 4099, April 18, pp , 2007 [4] D. Nichols, Velocity-stack inversion using norms, Stanford Exploration Project, vol. SEP-82, pp. 1-16, [5] A. Guitton and W. W. Symesz, Robust inversion of seismic data using the Huber norm, Geophysics, vol. 68, No. 4, pp , July-August [6] J. G Berryman, Weighted least-squares criteria for seismic traveltime tomography, IEEE Transactions on Geoscience and Remote Sensing, vol 27, issue 3, May 1989, pages [7] M. Ning; W. Yanping; H. Zhengyi; B. Zongdi., An iterative algorithm using a neural network for nonlinear traveltime tomography, In Proc. IEEE 3rd International Conference on signal processing, vol.1, issue 14-18, pp , Oct., [8] M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, Robust Anisotropic Diffusion, IEEE Trans. on Image Processing, vol. 7, No. 3, pp , March [9] P. Perona and J. Malik, IEEE Trans. on Pattern Analysis and Machine Intelligence. Vol. 12. n. 7. pp , July [10] D. Lumley, D. Nichols and T. Rekdal, Amplitudepreserved multiple suppression, Stanford Exploration Project, vol. SEP 82, pp , 1994.

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