Physics-based preconditioned multidimensional deconvolution in the time domain

Physics-based preconditioned multidimensional deconvolution in the time domain

Item Type Conference Paper

Authors Vargas, David;Vasconcelos, Ivan;Ravasi, Matteo;Luiken, Nick Citation Vargas, D., Vasconcelos, I., Ravasi, M., & Luiken, N. (2022).

Physics-based preconditioned multidimensional deconvolution in the time domain. Second International Meeting for

Applied Geoscience & Energy. https://doi.org/10.1190/

image2022-3745244.1 Eprint version Post-print

DOI 10.1190/image2022-3745244.1

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removing complex overburden effects, and retrieving ampli- tude consistent image gathers for impedance inversion. Despite its potential, the deconvolution process relies on the solution of an ill-conditioned linear inverse problem sensitive to noise artifacts due to incomplete acquisition, limited sources, and band-limited data. Typically, this inversion is performed in the Fourier domain where the estimation of optimal regularization parameters hinders accurate waveform reconstruction. We re- formulate the problem in the time domain - long believed to be computationally intractable - and introduce several physical constraints that naturally drive the inversion towards a reduced set of reliable, stable solutions. This allows to successfully reconstruct the overburden-free reflection response beneath a complex salt body from noise-contaminated data.

INTRODUCTION

To formally describe wavefield propagation between two ob- servation points in terms of data measured at such locations, we rely on integral representations based on Green’s functions.

Typically, cross-correlation performed at each station inside a closed energy radiating boundary provides access to the desired wavefields (Schuster et al., 2004; Bakulin and Calvert, 2006).

The extension of this method to dissipative media with open boundaries results in an implicit representation and the prob- lem must be solved using Multi-Dimensional Deconvolution (MDD). This problem is particularly challenging to solve, as both the data and modeling operator are comprised of noisy measurements. Traditionally, MDD is approached in the fre- quency domain where multiple inverse problems are solved independently. Inverting each frequency one by one is challeng- ing since little to no theory is available to consistently define a series of frequency-dependent damping parameters. Boiero and Bagaini (2020) enforce similarity of the solution between neighbor receivers as a regularizer, however, they are still con- fronted with a hand-tuned damping selection to stabilize the inverse process. Alternatively, MDD can be solved in the time domain (e.g., van der Neut and Herrmann, 2012; Luiken and van Leeuwen, 2020; Vargas et al., 2021a), with the advantage that using the entire spectrum at once acts as a natural regular- izer. Moreover, we can exploit the nature of time-domain fields by including preconditioners that honor the expected physics of waveform data. We apply this approach to full-waveform trans-

Figure 1: (a) Sub-salt model along with the MDD interact- ing quantities. Noise-contaminated Scattering Marchenko (b) Down- and (c) up-going gathers at receiver 75. (d) Noise-free common-receiver gather of the up-going field.

mission wavefields from a challenging subsalt synthetic dataset where the high impedance contrast of the salt body introduces a strong internal-multiple regime promoting cross-talk artifacts leaking into the solution.

DATA-DRIVEN WAVEFIELD REDATUMING

In the space-frequency domain, the band-limited field observed at receiverxrdue to an impulsive monopole sourcexscan be defined by the following relation (Amundsen, 2001; Wapenaar et al., 2011; Ravasi et al., 2015)

Pˆ⁻(xr,xs;ω) = Z

∂D^r

Pˆ⁺(x⁰_r,xs;ω)Gˆp(x⁰_r,xr;ω)dx⁰_r. (1) This representation is valid in the far-field approximation for dis- sipative and lossless media with lateral variations and describes the interaction of the up-going broadband pressure wavefield Gˆpat receiverx⁰_rdue to an impulsive monopole sourcexr, with the up/down-going pressure recordings ˆP^±that arise after ignit- ing the sourcexs(Figure 1a). Solving such an equation for ˆGp

produces the overburden-free reflection response at∂Dr. Frequency domain MDD

A discrete version of the integral representation in equation (1) reads asPˆ⁻=Pˆ⁺Gˆp, where all terms are matrices and Gˆprepresents the wavefield we wish to recover. MDD in the frequency domain can be solved as a regularized least-squares problem:

minGˆ_p

Pˆ⁺Gˆp−Pˆ⁻

2 2+λ

Gˆp

2

2, (2)

Time domain MDD

Figure 2: Subsurface-induced blurring effects in interferometric redatuming. (a) time-domain MDD reconstruction using wavefields in the simplified sediments-only medium. (b) Corresponding solution for up/downgoing wavefields propagating in the complex salt medium. The red and dashed-green line in the trace overlay corresponds to that in the gathers, whereas the solid-black line is extracted from the benchmark at the same location.

where thel2-norm is used to evaluate the distance from the ob- served data to the numerical estimation, and the regularization parameter,λ, weights the information in the data against that in the prior. The solution to equation (2) is then written in terms of the normal equations,

Gˆp=

(Pˆ⁺)^HPˆ⁺+λI−1

(Pˆ⁺)^HPˆ⁻. (3) This expression can be physically interpreted as follows: the point spread function (PSF),ΓΓΓ= (Pˆ⁺)^HPˆ⁺, acts as a blurring operator on the sought after solution, resulting in a band-limited cross-correlation function (CCF),C= (Pˆ⁺)^HPˆ⁻. Inversion of the normal equations,C=ΓΓΓGˆp, is performed for every frequency independently; this can be accomplished by means of either direct or block Krylov methods (Luiken et al., 2019).

The main difficulty in this approach lies in the selection of the λparameters as no formal theory exists to determine optimal frequency-dependent regularizations that are consistent with each other.

Time domain MDD

An alternative implementation of the MDD problem is derived by transforming equation (1) from the frequency to the time domain:

P⁻(xr,xs;t) = Z

∂Dr

Z ∞

−∞

P⁺(x⁰_r,xs;t−τ)Gp(x⁰_r,xr;τ)dτdx⁰_r. (4) Its discretized version can be written in compact matrix-vector notationp⁻=P⁺gp, where bothp⁻andgpare now repre- sented by vectors containing stacked traces. Contrary to the frequency domain formulation, here the convolutional kernel in (4) cannot be decoupled on a frequency-by-frequency basis. In the time-domain, the operatorP⁺is too large to be explicitly formed and the use of direct solvers is prohibited, therefore, the solutiongpis usually approximated using an iterative solver such as LSQR (Paige and Saunders, 1982). The main advantage of this approach is that no frequency-dependent damping pa- rameters need to be identified, leading to an inversion scheme

naturally constrained by all frequencies contributing as pre- scribed by the operator’s power spectrum. In general, the prob- lem (4) remains ill-posed. To alleviate this issue, MDD in the time-domain is posed as a constrained least-squares problem that enforces the reconstruction ofgpto belong to a restricted subset of physically meaningful solutionsG (Luiken and van Leeuwen, 2020),

ming_p

p⁻−P⁺gp

2 2+λ

gp

2

2 s.t. gp∈G. (5) Introducing a projection operatorPthat enforces the solution to remain within the set of admissible solutionsG, the constrained problem can be equivalently written as a preconditioned uncon- strained least-squares problem:

minz

P⁻−P⁺Pz

2 2+λ

z

2

2, (6)

wheregp=Pz. In this case, introducing the preconditionerP is a necessary condition to retrieve a physically reliable and stable solution. Our prior belief on the solution is enforced in terms of physical constraints imposed at each iteration as the optimization scheme progresses, thereby, we can adequately capture the physics of wave propagation while driving the in- version towards a meaningful set of solutions. First, we seek causal solutions for which any event arriving before the direct wave is rejected (Luiken and van Leeuwen, 2020). Such precon- ditioner is typically built using eikonal travel times computed with approximated velocities at the redatuming level∂Dr,

Θ ΘΘ

Gp(x⁰_r,xr;t) =

(Gp(x⁰_r,x_r;t), t≥τd(x⁰_r,x_r)

0, otherwise, (8)

whereτdis the direct wave’s travel time. Second, we leverage on the symmetry properties of the sought wavefield by intro- ducing a reciprocity preconditioner that enforces the reflection response to remain unchanged when sources and receivers are interchanged. Formally, the prescribed prior is written as

ϒϒϒ

Gp(x⁰_r,xr;t) = 1

2 Gp(xr,x⁰_r;t) +G^T_p(x⁰_r,xr;t)

, (9)

Figure 3: Noise-free (a)ω- and (b)t-MDD estimation ofGpfollow by trace comparison (c) extracted from receiver 115. Noisy recordsP_mko^± result in quality degradation for both (d)ω- and (e)t-MDD. (c), (f) Trace overlay indicates overestimated amplitudes for theω-domain implementation (dashed-blue/red lines) in comparison to those predicted byt-MDD (dashed-green/magenta lines).

and makes sure that any solution is forced to satisfy reciprocity.

A third prior is designed in the frequency-wavenumber domain to constrain the spectrum of our solution in a way that it remains within the expected signal cone and rejects solutions outside a specific frequency range (Vargas et al., 2021b). We introduce the 3D f-kfilter

Wˆ

Gp(k_x⁰_r,kxr;ω) = (Gp(k_x0

r,k_xr;ω), k_x²0

r+k²_xr≤ _c^ω

min

2

0, otherwise,

(10) wherecminis a design parameter that corresponds to the mini- mum velocity to be retained. This preconditioner is motivated by the fact that as iterations progress, many coherent short- scale length features and fringes leak into the solution. This is likely due to small singular values amplifying any noise present in the data. In the frequency-wavenumber domain, most of this unwanted information corresponds to low-velocity events mapping in the rejected region ofW, which our preconditionerˆ removes. In the space-time domain, this preconditioner takes the form ofW=F⁻¹WF, whereˆ FandF⁻¹are forward and inverse Fourier transforms. To better constrain the inversion, preconditioners can be chained together with the idea that the combined effect of multiple priors in the setP={ΘΘΘ,ϒϒϒ,W}via composition generally leads to improved model reconstruction.

NUMERICAL EXAMPLES

To validate the effectiveness of the proposed constrained time- domain MDD, the scattering Marchenko redatumed wavefields P^±_mkopresented in Vargas et al. (2021b) are used in our nu- merical example. These noise-polluted wavefields propagate through a large overburden that contains an inhomogeneous (dirty) salt body (Figure 1a). They correspond to the transmis- sion response of 201 sources at the surface of the Earth to an array of 151 virtual receivers regularly distributed every 20 m beneath the complex overburden. Such overburden exhibits sharp impedance contrasts and supposes an additional degree of complexity for MDD. The situation is exemplified by assuming that we also have access to redatumed fields in a simplified version of the medium, where the salt body is removed and only the laterally variant sediment structure of the background

remains. In such a case, the downgoing response is mostly dominated by its ballistic wave, opposite to Figure 1b, where a complex coda emerges due to the scattering induced by the overburden. Note that the PSF in the sediments (Figure 2a) is closer to a band-limited delta function around zero offsets, whereas its counterpart, for the salt medium, is corrupted with long unbalanced tails distorting the expected delta pulse (Figure 2b). In the simplified subsurface, the CCF is better resolved with limited cross-talk artifacts, and even an unpreconditioned MDD results in a plausible solution. In the second case, the

Figure 4: f-kamplitude spectra for (a) the CCF, (b) the uncon- strained, and (c) the preconditioned reflectivity reconstructed by t-MDD. The dashed-red line delineates the extent ofW. (d)-(f)ˆ Space-time reflectivities associated with panels (a)-(c).

Time domain MDD

Figure 5:t-MDD reconstructed impulse responsesGpusing physical priors. (a) Numerically-modeled reflectivity. (b) The causality- and (c) reciprocity-constrained solutions. Inverting with hybrid priors, (e)ΘΘΘWand (e)ΘΘΘϒϒϒW, conveys the physical properties of an actual local response into the sought solution. (f) Matching traces extracted from receiver 75 (the colored lines in the gathers) uncovering phase and amplitude differences in thet-MDD andω-MDD (dashed-blue line) against the benchmark (solid-black line).

cross-talk between uncorrelated events leak into the solution, which is severely affected by the combined effect of noise in both data and operator, plus subsurface-induced blurring. A noise-free scenariois considered next. Here,P⁺_mkoacts as an aerial source convolved with the numerically-modelledgpto buildp⁻_mod(Figure 1d). DeconvolvingP⁺_mkofromp⁻_modreveals well-defined arrivals with a wide range of spectral contribu- tions (Figures 3a-c). When compared with thenoisy scenario usingP^±_mko, the solution is further degraded and unphysical events systematically occur in the reconstructed shot gathers (Figures 3d-f). In both cases, theω-domain solution is of lower quality in comparison to that of the unpreconditionedt-domain MDD. Here, the suboptimal choice ofω-dependent regulariza- tion parameters leads to results of lower quality than those of t-MDD where all frequencies are implicitly balanced. In the first case, we observe substantial phase misalignment along with uncompensated amplitudes, whereas in the latter, the syn- thesized virtual response exhibits significantly fewer spurious events plus a radiating pattern with improved causality proper- ties. Trace evaluation (Figure 3c) reveals that the unconstrained time-domain MDD (i.e.,P=I) takes care of crucial waveform information otherwise underestimated or neglected, in contrast to itsω-domain counterpart.

Iter I ΘΘΘ W ϒϒϒ ΘΘΘW ΘΘΘϒϒϒ ΘΘΘϒϒϒW 5 23.52 23.61 23.60 23.56 22.65 23.65 23.70 10 23.60 23.74 23.69 23.67 22.55 23.79 23.87 20 17.81 18.02 18.81 22.85 22.86 23.06 22.99 Table 1: PSNR across iterations for multiple preconditioners Preconditioning the MDD is not only an operation aiming at controlling noise amplification but also a way to mitigate subsurface-induced blurring. For instance, artificial waves whose dips are rather high in the CCF (Figure 4d) are am- plified as iterations progress in an unconstrained inversion, therefore,t-MDD quickly diverge from a satisfactory solu- tion. We observe strong short-wavelength events arriving at high dipping angles (Figure 4b,e), which can not be physi- cally explained. UsingP=F⁻¹WFˆ as a preconditioner, we remove unphysical energy mapping in the periphery of the filter Wˆ (Figure 4c), but still retain crucial information otherwise overlapping with linear moveout events and high-frequency

uncorrelated noise (Figure 4f). Similarly, we effectively re- ject any acausal wave by imposingP=ΘΘΘ, which prevents energy outside the time window from leaking into the causal part of the solution (Figure 5b). Finally, one can expect the wavefields to obey spatial reciprocity (Wapenaar, 1998). In Figure 5c, the reconstructed response usingP=ϒϒϒmore closely approximates the benchmark. Figure 5d-e shows the result of using chained preconditioners. Acausal arrivals are con- trolled throughout iterations, being removed from the solution and allowing the solver to better estimate amplitudes at causal times. Likewise, the frequency-wavenumber constraint rejects all low-velocity artifacts and prevents sharp fringes to leak into the solution. When the reciprocity prior is enforced, the signals in the common shot and common receiver gathers are identical, indicating that only reciprocally-consistent events are maintained in the solution. Finally, we assess the perfor- mance of the different reconstructions using the peak signal to noise ratioPSNR=10×log₁₀[max(Gp)/MSE(Gp,G^inv_p )]

as a quality measure. In general, the solution improves when chained preconditioners are enforced across iterations (Table 1).

Nevertheless, in all scenarios we observe a semi-convergence behavior (i.e., the solution starts to degrade after a certain num- ber of iterations), highlighting the reliance of our method on a good stopping criterion (Ravasi et al., 2021).

CONCLUSION

We introduce physics-based priors to significantly improve the conditioning of the MDD problem. By implementing the inver- sion in the time domain and enforcing the solution vector to lie in a subset of physically meaningful solutions, we show that MDD can be successfully applied to highly complex subsur- face wavefields even in the presence of strong coherent noise both in the data and the kernel of the modeling operator. Our regularization-free approach consistently outperforms the stan- dard Tikhonov-stabilized inversion in theω-domain and tackles the misconception that MDD is unreliable due to its dependence on hand-tuned regularization parameters. The use of chained preconditioners results in responses with noticeably broader temporal bandwidth, lower spurious events, and measurably superior amplitude fidelity than for the unconstrained approach.

without wavefield separation: Geophysical Journal International,202, 1–16.

Ravasi, M., T. Selvan, and N. Luiken, 2021, Stochastic multi-dimensional deconvolution.

Schuster, G. T., J. Yu, J. Sheng, and J. Rickett, 2004, Interferometric/daylight seismic imaging: Geophysical Journal International, 157, 838–852.

van der Neut, J., and F. J. Herrmann, 2012, Interferometric redatuming by sparse inversion: Geophysical Journal International,192, 666–670.

Vargas, D., I. Vasconcelos, M. Ravasi, and N. Luiken, 2021a, Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation: Remote Sensing,13.

Vargas, D., I. Vasconcelos, Y. Sripanich, and M. Ravasi, 2021b, Scattering-based focusing for imaging in highly complex media from band-limited, multi-component data: Geophysics,0, 1–64.

Wapenaar, K., 1998, Reciprocity properties of one-way propagators: Geophysics,63, 1795–1798.

Wapenaar, K., J. van der Neut, E. Ruigrok, D. Draganov, J. Hunziker, E. Slob, J. Thorbecke, and R. Snieder, 2011, Seismic interferometry by crosscorrelation and by multidimensional deconvolution: A systematic comparison: Geophysical Journal International,185, 1335–1364.