Seismic attenuation imaging with causality

Abstract

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...

High-resolution seismic attenuation imaging from wide-aperture onshore data by visco-acoustic frequency-domain full-waveform inversion

“…For the complex‐valued wave speed given by , the partial derivative of the mass term with respect to c and Q gives, respectively, for d and The approximate expression of d shows that the virtual source associated with c and Q are approximately related by a Hilbert transform through the imaginary term i . This prompt Mulder & Hak (2009), Hak & Mulder (2010) and Hak & Mulder (2011) to conclude that many combinations of c and Q can produce nearly identical data, and, therefore, cannot be retrieved by linear waveform inversion, in particular from short‐aperture seismic reflection data. However, when Q decreases, the relationships between the phase of the radiation pattern of c and Q might become more linearly independent (through the complex coefficients (1 + i / Q ) and (1 + i 3/2 Q ) in and ), that can help to unambiguously reconstruct c and Q in particular when wide‐aperture data are considered and non‐linear inversion is performed.…”

“…If this condition is not satisfied, the asymptotic Hessian is singular. Hak & Mulder (2011) showed however that using an attenuation model with a causality correction term allows to successfully reconstruct the attenuation by non-linear FWI, provided a sufficiently high number of iterations is performed.…”

New geophysical models for subsurface velocity structure in the Nechako–Chilcotin plateau from 2.5-D waveform tomography

“…Numerous authors have argued for the benefits of phase-only inversion or phase-first inversion (e.g., Operto et al 2004;Malinowski and Operto 2008;Malinowski et al 2011;Bleibinhaus and Hilberg 2012), in part because inversion of data amplitudes is not necessarily well posed when modelling data using the 2-D viscoacoustic wave equation. Mulder and Hak (2009) and Hak and Mulder (2011) also point out ambiguities between the resolution of velocity and attenuation when updating the models using full-waveform inversion (discussed by Smithyman and Clowes 2013). Whereas 2.5-D forward modelling and inversion yield some benefits in this regard (as 3-D geometric spreading is modelled, precluding the need for geometric spreading correction), the simultaneous inversion of logarithmic amplitude and phase also requires handling of the energy loss in the model, lest this be attributed solely to scattering effects.…”

“…These techniques may affect the recovered attenuation structures. Some ambiguities also exist in the simultaneous recovery of P-wave velocity and numerical attenuation (e.g., Hak and Mulder 2011). However, models of numerical attenuation may be useful in distinguishing lithological boundaries based on the scattering potential of the rock units (Malinowski et al 2011).…”

Imaging velocity and attenuation scatterers with wave equation migration

“…The chapter includes an example that has noise and a more realistic synthetic marine example. It has been submitted as a journal article (Hak and Mulder, 2010b). The seventh chapter summarises the main findings.…”