Tomographic full waveform inversion (TFWI) by combining full waveform inversion with wave-equation migration velocity analysis

Introduction

Conventional seismic imaging relies on a separation of scales between migration velocity model (long-wavelength components) and reflectivity (short-wavelength components). The migration velocity model is estimated first, and then it is used as input to migration for imaging reflectivity. Even when wave-equation operators are employed to estimate the velocity, such as in wave-equation migration velocity analysis (WEMVA) methods, reflectivity is used only indirectly to measure the focusing power of the velocity model (Shen and Symes, 2008; Biondi and Sava, 1999). The only important exceptions in current practice occur when migrated volumes are used: 1) to interpret boundaries of geobodies (e.g. salt bodies), whose interior are assigned predefined velocities, and 2) to estimate predominant dips in the geologic layering that are then used to constraint a tomographic velocity updating (Clapp et al., 2004).

As the industry strives to widen the data frequency band at both the low and high end, the advantages of overcoming the limitations of conventional imaging, and of exploiting reflectivity information for velocity estimation, are becoming more relevant to important imaging problems. One of the main attractions of full waveform inversion (FWI) (Pratt, 1999; Tarantola, 1987) is to overcome the limitations imposed by the conventional approach that may limit the quality of the imaging results by finding a suboptimal solution. However, FWI suffers from well-known convergence problems when the starting model is far from the correct one and low frequencies are missing from the data.

We discuss an inversion framework that overcomes FWI difficulties by supplementing an FWI-like data-fitting objective function with a WEMVA-like term that measures the reflection-focusing power of the velocity model. The method fits the recorded data with data modeled using a generalized version of the acoustic wave equation; the domain of velocity model is extended to include subsurface offsets. The extension of the reflectivity along the subsurface offset axes (or reflection angles) is a well-established technique for migration, linearized waveform inversion, and WEMVA (Biondi, 2006). In a data-fitting inversion, extending reflectivity to the prestack domain has the critical advantage that the kinematics of the modeled data will not be too distant from the ones of the recorded data, no matter the magnitude of the background velocity error.

Symes (2008) introduced the idea of using a wave equation with an extended velocity. By extending velocity the convergence difficulties of conventional FWI are overcome and all scales can be solved simultaneously. In the same paper, he also introduced the waveform-inversion formulation used in this paper, and described its application to the solution of 1D inversion problem in presence of multiple reflections.

The main goal of our research is not to tackle the problem of multiples, but to perform simultaneous inversion for all scales of the velocity model. Therefore, we apply the theory and numerically solve the extended wave equation in 2D. We also derive an effective scheme to linearize the extended wave equation and to compute the gradient of the objective-function by an adjoint-state method. In 3D the proposed method would be extremely expensive. In a companion report, (Almomin and Biondi, 2012) we present an approximation to the method presented here that drastically reduces the computational cost, but still retains the capability of simultaneously solving for all the wavelengths of the velocity model.

Another potential problem with strict coupling of velocity with reflectivity arises when the assumption of constant density cannot be made, as is the case in most of field data problems. In this case density variations may create reflections that do not correspond to velocity contrasts. However, we still would like to avoid the addition of density to the problem parameters for computational and convergence reasons. The approximate method proposed in Almomin and Biondi (2012) has the potential of being more flexible in accommodating these discrepancies.

Tomographic full waveform inversion (TFWI) by combining full waveform inversion with wave-equation migration velocity analysis