Improving structural-information recovery in joint migration inversion using an image-based regularization

Joint Migration Inversion (JMI) can be formulated to simultaneously estimate parameters of scattering operators associated to the seismic image and of the slowness or velocity model. The inversion strategy can be implemented by alternating updates of each parameter class or in a simultaneous manner. Since JMI is not a well-posed inverse problem, regularization can help to stabilize the problem and/or to select an acceptable subset of the possible solutions according to their predefined desirable characteristics, which are then embedded in the choice of the regularizing function. We propose to take advantage of the structural information available in the inverted image by means of a simple Tikhonov type regularization applied to the tomography part of the JMI method. Tests with two synthetic data sets indicate the effectiveness of the proposed regularization. The methodology developed here is readily applicable to any seismic tomography at practically no additional cost if a migrated image is available. INTRODUCTION Joint Migration Inversion (JMI) is an approach to the seismic inverse problem based on scattering and propagation operators (Berkhout, 2014b). In its most simple form, the scattering operator is given by the stacked seismic image. Primarily, it carries information related to the subsurface impedance contrasts at the boundaries between geologic formations. In contrast, the estimated velocities carry information about each layer interval. Therefore, given the nature of the information present in the image and velocity models, it is quite not unusual to think of these quantities as being uncorrelated. However, layer boundaries actually correlate with strong velocity variations. Thus, with additional analytical relations between these quantities, it is possible to make them approximately correlate in the inversion process, for example, by taking the derivative of the velocity profile to transform it into a reflectivity series or by integrating the image to transform it into impedance (see, e.g., Assis et al., 2019). Establishing relationships between different classes of parameters or different data types is very common in geophysical inverse problems. For example, Gallardo and Meju (2003) propose the cross-gradients constrain which relates the cross-product of the model-parameters gradient to enforce structural similarity in the inversion. Also, imposing structural information from the image to the velocity model has already been proposed using different methodologies. Costa et al. (2008) applied a structurally motivated smoothing constraint in the direction of a potential reflector in slope tomography and showed that this constraint led to geologically more consistent models. Williamson et al. (2011) solved a differential equation to smooth the updated velocity model consistently with structural information. In the context of JMI, Maciel et al. (2015) applied morphological operators directly to the updated velocity model. Masaya and Verschuur (2018) proposed a regularization for the tomography part of the JMI that relates the image and an approximate image obtained from the velocity model. In this manner, 44 Annual WIT report 2019 Masaya and Verschuur (2018) improved the structural information in the velocity model update and not in the velocity directly like Williamson et al. (2011) and Maciel et al. (2015). Regularization of ill-posed inverse problems is fundamental. It has basically two roles, one being to reduce instability and the other to select solution candidates according to desired characteristics of the solution. These are carried over to the problem by means of the regularization function (Zhdanov, 2015). For example, defining the regularization term as the first derivative of the model parameter penalizes roughness and favors smooth estimates or, in the words of Constable et al. (1987), the simplest model that explains the data. Inspired by cross-gradients regularization, we investigate a cross-correlation minimization between the image and velocity models in order to make use of the high spatial-frequency information from the image in the velocity updates. We also analyze the inclusion of each parameter’s gradient in the regularization function. We demonstrate the effectiveness of the methodologies on two synthetic models. The regularization developed here is very similar to the one of Masaya and Verschuur (2018), but there is no need of an approximate or exact analytical relationship between the parameters. The only assumption needed here is that the events in the image are, to some extent, correlated with velocity contrasts. REGULARIZING FUNCTIONS We want to make use of the intrinsic connection between the velocity distribution v(x, z) and the image R(x, z), in which x indicates the horizontal coordinate and z is depth. ImageR represents the reflection coefficient for incidence from above on a boundary between adjacent layers. It is assumed to be equivalent to the stacked image estimated in most least-squares migration methods. In the following subsections we define two regularizing functions that use this connection, with focus on using the image information in the velocity or slowness update. Function I: Correlation of the image and velocity parameters We define the first regularization function as the zero-lag cross-correlation between velocity and image, i.e., Jreg1(v) = ∫ Ω v(x′, z′)R+(x′, z′)dx′dz′ , (1) where Ω is the domain of the model parameters. With this regularization function, we try to incorporate the fact that areas of constant velocities have low reflectivity. The kernel of its first derivative with respect to velocity is the image, i.e., ∂Jreg1 ∂v = R(x, z) . (2) Function II: Velocity and image gradients We define our second regularization function as the zero-lag cross-correlation between gradients of velocity and image, i.e., Jreg2(v) = 1 2 ∫ Ω ∇v(x′, z′) · ∇R+(x′, z′)dx′dz′