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

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Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

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.

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Notes

  1. 1.

    In reality, cables will not be straight but will follow ocean currents. The hardware and controls required to keep the cables apart and monitor their locations are a real engineering accomplishment.

  2. 2.

    This relation can be derived as follows. For a cubic survey of dimensions L3, the number of spatial points Nx to compute will scale as L3∕h3, where h is the discretization length set by h ∼ λo = v∕fo. If using explicit time integration, the number of time integration steps Nt = T∕Δt, where T is the listening time, and Δt the time step. As \(\varDelta t \sim h/v \sim f_o^{-1}\), then the number of operations \(N_{op}\sim N_x N_t \sim f_o^4\), and therefore \({\mathcal {O}}(f_o^4)\). For larger fo, the distance between the different shots at the earth surface also need to be smaller to maintain resolution, and therefore more shots are used (and need to be computed), resulting in \({\mathcal {O}}(f_o^6)\) when the inversion is performed separately for each source.

  3. 3.

    We purposely picked the word best to emphasize that the problem has nonunique solutions and that the chosen solution might be the result of applying some additional measures of merit, sometimes even including some subjective domain expertise.

  4. 4.

    A third element called model qualification determines the level of adequacy of the model for the intended application. This aspect will not be discussed here.

  5. 5.

    We will be using the Einstein convention where repeated indices imply a sum over these indices.

  6. 6.

    It is interesting to note that Green, Cauchy, and Poisson were part of a lengthy controversy in which the last two argued that the number of coefficients could not exceed 15. See [93] and references therein.

  7. 7.

    In material science, these two end-value members for binary composite materials are termed the Reuss and Voigt averages, respectively. For more details, see, for example, [66, 67].

  8. 8.

    Nepers are not part of the SI units. They have dimensionless units and refer to the natural logarithm of ratios of measurements.

  9. 9.

    Note that ultrasonic lab measurements are typically performed in the MHz range while the frequency bandwidth used in reflection seismology covers about 2 orders of magnitude ranging from 1 Hz to 100 Hz.

  10. 10.

    To be more precise, in some cases the dimensionality of u may be higher than the number of spatial points when the numerical method introduces additional degrees of freedom, such as in the case of the discontinuous-Galerkin method.

  11. 11.

    Stability criterion named after Courant, Friedrichs, and Lewy stating that vΔt ≤ h.

  12. 12.

    Graphics Processing Units and Field-Programmable Gate Arrays.

  13. 13.

    This simple approach is very useful, however, for providing test cases for verifying gradient computations.

  14. 14.

    By the notation \(L^2(\mathbb {R}^a \times \left [0,\mathrm{T}\right ];\mathbb {R}^b)\), we express a b-dimensional vector defined over an a-dimensional space over a time interval [0, T], with a norm that is square-integrable over \(\mathbb {R}^a\times \left [0,\mathrm{T}\right ]\).

  15. 15.

    Current generation of geophones uses accelerometers that can measure the three orthogonal components of acceleration. Such receivers are termed multicomponents or 3-C.

  16. 16.

    This relation is also referred to as the Lagrange identity.

  17. 17.

    See, e.g., Equation (32). A similar result is obtained for the acoustic wave equation as derived in joint tutorial [13].

  18. 18.

    Some authors refer to this operation as a zero-lag correlation.

  19. 19.

    In the derivation for a simple 1-D acoustic wave equation, one obtains (see, e.g., [13]) \(\mathbf {{R}(\kappa )} = \frac {1}{\kappa ^2}\frac {\partial ^2}{\partial t^2}\). In practical computations, using the linearity of the wave equation, it is more efficient to apply this operator to the source time functions.

  20. 20.

    Also less affectionately known as trivially or embarrassingly parallel problems.

  21. 21.

    Also called Taylor’s formula with remainder or the mean-value theorem when truncated to first order. See, e.g., [46]. At second order, it states that \(f(\mathbf {x+h}) = f(\mathbf {x}) + \boldsymbol {\nabla } f(\mathbf {x})^T \mathbf {h} + \frac {1}{2}{\mathbf {h}}^T \boldsymbol {\nabla }^2 f(\mathbf {x} + t\mathbf {h})\mathbf {h} \), where t ∈ (0,  1).

  22. 22.

    Sometimes also called true model, or target model.

  23. 23.

    Named from its inventors Broyden, Fletcher, Goldfarb, and Shanno.

  24. 24.

    Including the ability to use Gauss-Newton methods for the inversion.

  25. 25.

    Also called Wasserstein metric, or more descriptively earth mover’s distance.

  26. 26.

    Note that \(\mathcal {R}^{(1)}\) is not differentiable with respect to m. However, some differentiable approximations can be used. See [106] for more detail.

  27. 27.

    For example, nth-order Tikhonov, Tikhonov-Miller, Phillips-Twomey, Total Variation, etc.

  28. 28.

    These curves are log-log plots generated by plotting the misfit value \(\left | \mathbf {F}({\mathbf {m}}_{\mathbf {n}}) - {\mathbf {d}}^\dag \right |\) as a function of the value of the residual \(\mathcal {R}({\mathbf {m}}_{\mathbf {n}}; \beta )/\beta \) at the “final” nth iteration obtained with different values of β.

  29. 29.

    By multiparameter, we refer to systems being described by multiple, distinct, spatially varying physical parameters such as density, bulk, and shear moduli for a three-parameter inversion.

  30. 30.

    Some receivers (termed multicomponent) can provide vectorial information on the displacement that can be exploited through a continuation strategy.

  31. 31.

    Term borrowed from telecommunications describing signals transferring from one channel to another due to unintentional coupling (e.g., poor electromagnetic insulation).

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Acknowledgements

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.

Appendix

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,

$$\displaystyle \begin{aligned} c(\omega) = c_u \left(1 - \sum_{l=1}^{n} \frac{a_l\omega_l}{\omega_l + \mathrm{i}\omega} \right),\end{aligned} $$
(49)

where ωl = Δcl∕ηl, al = Δcl∕cu 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

$$\displaystyle \begin{aligned} Q^{-1}(\omega) = \frac{\Im\left[c(\omega)\right]} {\Re\left[c(\omega)\right]} = \sum_{l=1}^{n} a_l \frac{\omega_l \omega + \omega_l^2 Q^{-1}(\omega)} {\omega_l^2 + \omega^2}.\end{aligned} $$
(50)

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
figure 9figure 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

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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

$$\displaystyle \begin{aligned} \tau(e, t) = c_u e(t) - \sum_{l=1}^{n} a_l\zeta_l(t),\end{aligned} $$
(51)

which are coupled to n additional equations for the anelastic functions,

$$\displaystyle \begin{aligned} \frac{d\zeta_l(t)}{dt} + \omega_l\zeta_l(t) = \omega_l e(t). \end{aligned} $$
(52)

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

  1. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, 08801, USA

    Martin-D. Lacasse, Laurent White & Huseyin Denli

  2. Petroleum Geo-Services, Houston, TX, USA

    Lingyun Qiu

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  1. Martin-D. Lacasse
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Correspondence to Martin-D. Lacasse .

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Editors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Harbir Antil

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Drew P. Kouri

  3. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Martin-D. Lacasse

  4. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Denis Ridzal

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Lacasse, MD., White, L., Denli, H., Qiu, L. (2018). Full-Wavefield Inversion: An Extreme-Scale PDE-Constrained Optimization Problem. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_6

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