An ambiguity in attenuation scattering imaging

Abstract

S U M M A R YMigration 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.

Imaging velocity and attenuation scatterers with wave equation migration

“…Here, we will demonstrate that a similar statement holds for a 1D layered scattering model m(z) in a 3D homogeneous background. For that case, we showed earlier (Mulder and Hak, 2009) that we obtain approximately the same data for a model and its scaled, depth-weighted Hilbert transform. These data obey 6) with µ = m(z)/z.…”

“…For the constant-density acoustic case with the Born approximation, we found that the resolution function cannot unambiguously distinguish between perturbations in velocity or in the quality factor that describes attenuation for an acquisition geometry with sources and receivers at the surface and a constant background model. This ambiguity was described in detail for horizontally layered scatterers in an earlier paper (Mulder and Hak, 2009). Here, we observed a similar behaviour for a 2D scattering model: data generated in a constant background model for the resolution function of a point scatterer were nearly identical when either the real part of this resolution function was used as a scattering model or i times its imaginary part.…”

“…The real and imaginary part of the resolution function are approximately related by a z-weighted Hilbert transform for 1D horizontally layered scattering models and also on the symmetry line through a localised scatterer. We discussed this in detail in Mulder and Hak (2009). We showed that two scattering models, related by a depth-weighted Hilbert transform, will produce approximately the same data.…”

“…A method for the estimation of multiple parameters was presented by Kennett et al (1988) and Kennett and Sambridge (1998), but this approach will not be considered here. Mulder and Hak (2009) show that simultaneous imaging of velocity and attenuation perturbation with the Born approximation of the constant-density visco-acoustic wave equation leads to an ambiguity: the data can be equally well explained by a velocity perturbation or by an attenuation perturbation. There exists a simple transformation between these two models for 1D, horizontally layered scattering models in a constant background.…”

“…The explanation is accompanied by a synthetic data example. This chapter is also available as journal article (Mulder and Hak, 2009). In the fifth chapter we will investigate different acquisition geometries and their influence on the occurrence of the ambiguity.…”