Digital superresolution in seismic AVO inversion

Abstract

The sparseness promoted by the total variation norm is utilized to achieve superresolution amplitude-variation-with-offset (AVO) inversion. The total variation norm promotes solutions that have constant values within unspecified regions and thus are well suited for an earth model consisting of layers bounded by faults and erosion surfaces. Algorithmic developments from digital image and video restoration are utilized to solve the geophysical problem. A spatial point spread function is used to model the resulting effect of wave propagation, migration, and processing. The methodology is compared to current alternatives and discussed in the context of AVO inversion. Good results are obtained in a Barents Sea test case. Introduction The concept of superresolution has its origin in seismic inversion. In recent decades, it has found a range of applications in digital image and video restoration. Increased attention to the topic has resulted in theorems defining the limits and in algorithms ready for large-scale problems. Here we show how this approach can be applied in seismic amplitude-variation-with-offset (AVO) inversion and discuss the method in relation to other inversion methods. The methodology we outline gives a type of sparse-spike inversion that is phrased in a spatial context, accounting for the spatial extent of the point spread function as well as the spatial dependencies in a total variation regularization. Traditionally, AVO methods are categorized as either reflection-based or impedance-based methods. Reflection-based methods tend to work on amplitude maps extracted from peak amplitude, whereas impedance methods work on the full waveform. In the last decade or so, a third line of inversion methods has emerged that can be seen as an augmentation of the impedance methods by including rock-physics models. The superresolution method discussed here is an impedance approach, but it aspires to be a simple inversion that competes with the ease of use of the reflection-based methods. Comparing the proposed approach to that of using a rockphysics model, it is seen that both methods sharpen the edges by including step changes in the transition. In the facies-based approach, the steps changes are between discrete levels of predefined lithology and fluid classes. The main limitation of the applicability is then the need for these prior classes. The rockphysics approach is thus less suited in regions where rock-physics knowledge is limited. The superresolution approach makes a minimum set of assumptions for the inversion and still manages to coordinate the contrasts in the inversion. Odd Kolbjørnsen1,2, Andreas Kjelsrud Evensen1, Espen Harris Nilsen1, and Jan-Erik Lie1 Geophysical roots of superresolution The origin of superresolution is in the geophysical tradition. When working with deconvolution, use of the L1 norm rather than the traditional L2 norm as a mean of regularization (Claerbout and Muir, 1973) created a deconvolution consisting of spike trains (Taylor et al., 1979). This approach was thus a sparse-spike inversion. In the noiseless case, the method allowed for perfect reconstruction of the generating spike train (Levy and Fullagar, 1981; Oldenburg et al., 1983). This property was later proved mathematically (Santosa and Symes, 1983, 1986), thus creating the first proof of superresolution. The 1D case discussed in these papers uses the L1 norm on the reflection coefficients. The impedance relates to the integrated reflection coefficients, so the impedance will have a blocky structure, i.e., step changes at the spike locations. The total variation norm on the impedance corresponds to an L1 norm on the reflection coefficients in the 1D case. When generalizing the approach to 2D and 3D, the total variation norm presented in the following provides a natural extension to this 1D methodology. Geophysical model The geophysical model validity is a primary concern for AVO inversion. In the reflection-based methods, there is an issue due to secondary lobes of the wavelet and tuning effects that are not accounted for in the model. In most impedance and rock-physics approaches, the geophysical model is based on a 1D convolution of a wavelet with the reflection coefficients. This 1D model does not fully account for the spatial nature of the wave phenomena under study. We formulate the model using a 3D point spread function. In the context of depth-migrated data, an angle stack is related to the elastic impedance, gθ, through a convolution with a point spread function, Ψθ , d(x,y,z,θ) = Ψθ * gθ(x,y,z) + ε(x,y,z,θ). (1) This geophysical relation is illustrated in Figure 1; see Lecomte et al. (2016) for details. The 3D point spread function is an improvement over the 1D wavelet, but it does not account for a variable distortion of the overburden as it is assumed to be stationary. The error term ε is included to highlight that the relation is an approximation. In a regular grid, we get dθ = Ψθ gθ + εθ , (2) where the operator Ψθ is diagonalized by the discrete 3D Fourier transform; thus, it is efficiently stored and computed. The parameter 1Lundin-Norway, Lundin Geolab, Lysaker, Norway. E-mail: [email protected]; [email protected]; [email protected]; [email protected]. 2University of Oslo, Department of Mathematics, Oslo, Norway. https://doi.org/10.1190/tle38100791.1