Full-Wavefield Inversion: An Extreme-Scale PDE-Constrained Optimization Problem

  • Martin-D. LacasseEmail author
  • Laurent White
  • Huseyin Denli
  • Lingyun Qiu
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)


Full-wavefield inversion is a geophysical method aimed at estimating the mechanical properties of the earth subsurface. This parameter estimation problem is solved iteratively using optimization techniques aimed at minimizing some measure of misfit between computer-simulated data and real data measured in a seismic survey. This PDE-constrained optimization problem poses many challenges due to the extreme size of the surveys considered. Practical issues related to the physical fidelity and numerical accuracy of the forward problem are presented. Also, issues related to the inverse problem such as the limitations of the optimization methods employed, and the many heuristic strategies used to obtain a solution are discussed. The goal of this paper is to demonstrate some of the progress achieved over the last decades while highlighting the many areas where further investigation could bring this method to full technical maturity. It is our hope that this paper, together with other contributions in this book, will motivate a new generation of researchers to contribute to this broad and challenging research area.



The authors thank ExxonMobil Research and Engineering Company for permission to publish this work. The authors would also like to thank Fadil Santosa and the Institute of Mathematics and its Applications for hosting the workshop where this work was presented. We would also like to thank Jeremy Brandman, Jerry Krebs, Anatoly Baumstein, and Dimitar Trenev for their insightful suggestions and comments on the original manuscript.


We start the mathematical description of attenuation by considering a rheological model composed of springs and dashpots as shown in Figure 2. The effective modulus c(ω) of this mechanical model can be expressed as a function of auxiliary variables representing relaxation angular frequencies ωl = 2πfl and nondimensional anelastic coefficients al,
$$\displaystyle \begin{aligned} c(\omega) = c_u \left(1 - \sum_{l=1}^{n} \frac{a_l\omega_l}{\omega_l + \mathrm{i}\omega} \right),\end{aligned} $$
where ωl = Δclηl, al = Δclcu where the unrelaxed modulus \(c_u = c(\omega \rightarrow \infty ) = c_r + \sum _{l=1}^{n} \varDelta c_l\), in contrast to the relaxed modulus cr = c(ω → 0). This only says that if one moves the system in Figure 2 very slowly, only spring cr is felt as dashpots are relaxing and not transmitting force, while if one moves it very quickly all springs are fully active. Anything in between depends on the frequency according to Equation (49). This model will have an attenuation quality factor following the ratio of real and imaginary parts of the modulus [54, 71], leading to the following self-consistent relation
$$\displaystyle \begin{aligned} Q^{-1}(\omega) = \frac{\Im\left[c(\omega)\right]} {\Re\left[c(\omega)\right]} = \sum_{l=1}^{n} a_l \frac{\omega_l \omega + \omega_l^2 Q^{-1}(\omega)} {\omega_l^2 + \omega^2}.\end{aligned} $$
The frequency dependence of Q(ω) is set by carefully picking values for ωl and al. This task is usually achieved by sampling frequencies ωl logarithmically in the band of interest and fitting the anelastic coefficients al using a least-squares method [28, 37, 86]. In order to obtain a constant-Q attenuation, i.e., Q(ω) = Qo, we have shown [25] that at least three such relaxation mechanisms are required to obtain a response close to the desired behavior. Figure 9 shows the effect of using a different number of relaxation mechanisms on the frequency response Q(f) of a generalized Maxwell solid. The parameters of the relaxation mechanisms are optimally tuned over a frequency band ranging from 3 Hz to 40 Hz in view of obtaining a constant target quality factor of Qo = 50.
Fig. 9

Frequency response of quality factor Q(f) for generalized Maxwell solids with 1, 2, 3, and 5 relaxation mechanisms over a frequency band ranging from 3 Hz to 40 Hz. The parameters of the relaxation mechanisms are optimized (least-squares) to mimic a constant target quality factor Q = 50 in the frequency band

Each spring and dashpot added to the relaxation model introduce an additional anelastic function ζl(t) (sometimes called memory variable) that has to be solved as part of the governing equations. Each equation in (1) is then replaced by
$$\displaystyle \begin{aligned} \tau(e, t) = c_u e(t) - \sum_{l=1}^{n} a_l\zeta_l(t),\end{aligned} $$
which are coupled to n additional equations for the anelastic functions,
$$\displaystyle \begin{aligned} \frac{d\zeta_l(t)}{dt} + \omega_l\zeta_l(t) = \omega_l e(t). \end{aligned} $$
These equations are obtained after integrating the frequency-dependent moduli while maintaining causality (Boltzmann superposition principle). See [71] for details.

Because of this additional complexity, viscoelastic simulations are more costly by up to an order of magnitude, which can be reduced if special algorithms are used [26].


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Authors and Affiliations

  • Martin-D. Lacasse
    • 1
    Email author
  • Laurent White
    • 1
  • Huseyin Denli
    • 1
  • Lingyun Qiu
    • 2
  1. 1.Corporate Strategic ResearchExxonMobil Research and Engineering CompanyAnnandaleUSA
  2. 2.Petroleum Geo-ServicesHoustonUSA

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