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.

S U M M A R YSeismic data enable imaging of the Earth, not only of velocity and density but also of attenuation contrasts. Unfortunately, the Born approximation of the constant-density visco-acoustic wave equation, which can serve as a forward modelling operator related to seismic migration, exhibits an ambiguity when attenuation is included. Different scattering models involving velocity and attenuation perturbations may provide nearly identical data. This result was obtained earlier for scatterers that did not contain a correction term for causality. Such a term leads to dispersion when considering a range of frequencies. We demonstrate that with this term, linearized inversion or iterative migration will almost, but not fully, remove the ambiguity. We also investigate if attenuation imaging suffers from the same ambiguity when using non-linear or full waveform inversion. A numerical experiment shows that non-linear inversion with causality convergences to the true model, whereas without causality, a substantial difference with the true model remains even after a very large number of iterations. For both linearized and non-linear inversion, the initial update in a gradient-based optimization scheme that minimizes the difference between modelled and observed data is still affected by the ambiguity and does not provide a good result. This first update corresponds to a classic migration operation. In our numerical experiments, the reconstructed model started to approximate the true model only after a large number of iterations.Seismic imaging provides qualitative, structural information about the subsurface geology. Inversion-a term that we use in the mathematical sense of finding the Earth's parameters that best explain the observed data for a given type of forward modelling-leads to a quantitative description of material properties. Even with a simplified wave propagation model such as constant-density viscoacoustics, the reconstruction of not only the velocity but also the attenuation, as a function of subsurface position, can help in distinguishing between a fluid-or gas-filled rock formation.Inversion for attenuation has been attempted by several authors with different levels of success. Ribodetti & Virieux (1998) considered linearized inversion for density, velocity and attenuation perturbations in a given background model using ray tracing to model the wave propagation. The correction for causality was not included. They claimed successful reconstruction of the model perturbations, although their results had large errors near sharp interfaces. Hicks & Pratt (2001) considered the non-linear inversion problem on real data and obtained convincing results with alternating updates of the velocity and the quality factor. Kamei & Pratt (2008) applied the same approach to cross-well data. Simultaneous inversion caused problems, but inverting for the velocity model first and then for the quality factor, reproduced the correct model with synthetic data. They applied their approach to real data as well. Smithyman et a...

A B S T R A C TMigration maps seismic data to reflectors in the Earth. Reflections are not only caused by small-scale variations of the velocity and density but also of the quality factor that describes attenuation. We investigated scattering due to velocity and attenuation perturbations by computing the resolution function or point-spread function in a homogeneous background model. The resolution function is the migration image of seismic reflection data generated by a point scatterer. We found that the resolution function mixes velocity and attenuation parameter perturbations to the extent that they cannot be reconstructed independently. This is true for a typical seismic setting with sources and receivers at the surface and a buried scatterer. As a result, it will be impossible to simultaneously invert for velocity and attenuation perturbations in the scattering approach, also known as the Born approximation.We proceeded to investigate other acquisition geometries that may resolve the ambiguity between velocity and attenuation perturbations. With sources and receivers on a circle around the scatterer, in 2D, the ambiguity disappears. It still shows up in a cross-well setting, although the mixing of velocity and attenuation parameters is less severe than in the surface-to-surface case. We also consider illumination of the target by diving waves in a background model that has velocity increasing linearly with depth. The improvement in illumination is, however, still insufficient to remove the ambiguity.

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