An ambiguity in attenuation scattering imaging

Abstract

SUMMARY Migration constructs a subsurface image by mapping band-limited seismic data to reflectors in the Earth, given a background velocity model that describes the kinematics of the seismic waves. Classically, the reflectors correspond to impedance perturbations on length scales of the order of the seismic wavelength. The Born approximation of the visco-acoustic wave equation enables the computation of synthetic data for such a model. Migration then amounts to solving the linear inverse problem for perturbations in density, velocity, and attenuation. Here, the problem is simplified by assuming the density to be constant, leaving only velocity and attenuation perturbations. In the frequency domain, a single complex-valued model parameter that depends on subsurface position describes both. Its real part is related to the classic reflectivity, its imaginary part also involves attenuation variations. Attenuation scattering is usually ignored but, when included in the migration, might provide information about, for instance, the presence of fluids. We found, however, that it is very difficult to solve simultaneously for both velocity and attenuation perturbations. The problem already occurs when computing synthetic data in the Born approximation for a given scattering model: after applying a weighted Hilbert transform in the depth coordinate to a given scattering model, we obtained almost the same synthetic data if the scatterers had small dip and were located at not-too-shallow depths. This implies that it will be nearly impossible to simultaneously determine the real and imaginary part of the scattering parameters by linearized inversion without imposing additional constraints. The oil and gas industry acquires seismic data to obtain an image of the subsurface that may reveal hydrocarbon bearing formations. Because a 3-D full visco-elastic inversion is computationally still out of reach, various approximations of the wave equation are employed, often based on ray tracing or one-way wave equations. These are often sufficiently accurate to obtain a structural image. A precise characterization of the subsurface that allows for volumetric estimates of the amounts of hydrocarbons in place requires a more accurate description. Visco-acoustic or visco-elastic full waveform inversion is computationally tractable in a 2-D approximation. However, the presence of local minima in the least-squares misfit functional makes the solution of the inverse problem difficult. One cause for this problem is the absence of low frequencies, below 8–10 Hz, in the seismic data. The inverse problem becomes considerably simpler when linearized. Classic methods for velocity analysis provide a background velocity model. An operation called migration maps the band-limited seismic data to reflectors in the subsurface. Mathematically, this method amounts to a single iteration of a gradient based minimization of the least-squares misfit functional between observed and modelled data, using the Born approximation of the wave equation. Because this approximation assumes single scattering, the direct arrival and multiple reflections should be removed from the data before migration. Examples (Ostmo et al. 2002; Mulder & Plessix 2004) show that with proper weighting or preconditioning (Plessix & Mulder 2004), one or a few iterations with the conjugate-gradient method suffice to obtain a solution to the inverse problem when using the Born approximation of the constant-density visco-acoustic wave equation. The method works in the frequency-domain and reconstructs scatterers that represent perturbations of the background velocity model. In the frequency-domain, these are represented by complex numbers. The real part is almost entirely related to impedance perturbations, usually caused by abrupt changes in rock properties. If we include the imaginary part, we can formally obtain both velocity and attenuation perturbations. The latter, however, appeared to have no relation to physically realistic values in our numerical experiments. To better understand why it may be difficult to reconstruct the imaginary part of the scattering perturbations, we consider the simple case of a 1-D model consisting of horizontally layered scatterers in a homogeneous background for the constant-density