# Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation

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## Abstract

**:**

## 1. Introduction

## 2. Seismic Wavefield Redatuming

#### 2.1. Two-Way Multidimensional Deconvolution

#### 2.2. One-Way Multidimensional Deconvolution

#### 2.3. Frequency Domain MDD—Regularized Least Squares

#### 2.4. Time Domain MDD—Constrained Least Squares

#### 2.5. Physics-Inspired Preconditioners

## 3. Numerical Examples

#### 3.1. Noise-Free Modelled Wavefields

#### 3.2. Noise-Contaminated Marchenko Wavefields

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OBS | Ocean-bottom seismic |

MDC | Multidimensional convolution |

MDD | Multidimensional deconvolution |

CCF | Cross correlation function |

PSF | Point spread function |

CSG | Common shot gathers |

CRG | Common receiver gathers |

SVD | Singular-value decomposition |

SRM | Surface related multiples |

SRME | Surface related multiples elimination |

FWM | Full-Wavefield Migration |

RMSE | Root means square error |

SSIM | Structural similarity index measure |

FISTA | Fast Iterative Shrinkage-Thresholding Algorithm |

SPGL1 | Spectral projected gradient 1-norm Algorithm |

HPC | High-performance computing |

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**Figure 1.**Unbounded configuration defining the interacting quantities involved in the MDD problem. According to Equation (1), the Green’s function $\overline{G}({\mathbf{x}}_{\mathbf{r}}^{\prime},{\mathbf{x}}_{\mathbf{r}})$ in the reference configuration (

**a**), can be obtained by deconvolving the wavefield observed at ${\mathbf{x}}_{\mathbf{r}}^{\prime}$ from that in ${\mathbf{x}}_{\mathbf{r}}$ (

**b**). Seismic waves due to sources ${\mathbf{x}}_{\mathbf{s}}$ (grey stars) distributed along the surface $\partial {\mathbb{D}}_{s}$ are measured by receivers (triangles) at $\partial {\mathbb{D}}_{r}$. After MDD, the receiver at location ${\mathbf{x}}_{\mathbf{r}}$ (blue triangle) is turned into a virtual source and its response is observed at the other receivers (red triangle). Note that $G({\mathbf{x}}_{\mathbf{r}},{\mathbf{x}}_{\mathbf{s}})$ and $G({\mathbf{x}}_{\mathbf{r}}^{\prime},{\mathbf{x}}_{\mathbf{s}})$ are full transmission responses carrying all orders of internal as well as surface related multiples. (

**c**) Schematic representation of the reference response $\overline{G}$ for a fixed time. Rows correspond to common shot gathers associated with an specific virtual source, whereas Columns are common receiver gathers; numbers indicate the rows and columns used to display the estimated reflection response in the subsequent figures.

**Figure 2.**Schematic representation of the effect of the acausal contribution ${g}_{i}({\mathbf{x}}_{r}^{\prime},{\mathbf{x}}_{r};\overline{t})$ in the model estimate at i-th iteration on the the gradient $\nabla J$ used to update the model. (

**a**) Forward modelling step, (

**b**) Adjoint modelling step. The gray zones in the $Q{g}_{i}$ and ${r}_{i}$ panels indicate the portion of the modelled data that is only populated by noise contributions arising from the single time-space sample ${g}_{i}({\mathbf{x}}_{r}^{\prime},{\mathbf{x}}_{r};\overline{t})$. Note that whilst only the positive time axis is shown here, non-physical contributions map all the way to the negative axis.

**Figure 3.**(

**a**) Sub-salt velocity model supporting decomposed wavefields at the receiver array, the magenta line extending horizontally at 4.4 km below the overburden. The source array, solid red line, is deployed at the surface. The dashed black line indicates the target area for which reflectivity is recovered. (

**b**) Down- and (

**c**) up-going common-receiver gathers at receiver 75 obtained by means of scattering Marchenko redatuming. (

**d**) Common-receiver gather of the up-going field at the same receiver computed by means of the representation theorem in Equation (5) using the down-going wavefield in panel (

**b**) and the true directly modelled reflection response in the target area shown in Figure 4.

**Figure 4.**(

**a**) Virtual survey configuration depicting the target area for which MDD retrieves a reflection response. The solid red line represents a set of virtual receivers whereas the white dots are associated with different sources locations. (

**b**–

**d**) Multiple common source gathers extracted from the numerically modeled pressure response at positions coinciding with the white dots in (

**a**) are used to benchmark MDD reconstructed reflectivities.

**Figure 5.**Key elements of seismic interferometry by MDD. (

**a**) The One way PSF in the time domain is a noise-contaminated band-limited delta function blurring the sought broadband impulse response. (

**b**) Acausal events appear in the pressure reflection response obtained by cross-correlation of up- and down-going wavefields, while spurious cross-talk events interfere with reflections from the target area. (

**c**) Close-up view of normalized traces corresponding to receiver 75 (red-dashed line in (

**b**)) extracted from the true response (Figure 4c) and the CCF. (

**d**,

**e**) The space-frequency domain versions of the PSF and CCF, respectively.

**Figure 6.**Frequency versus time-domain MDD reconstruction of the local reflection response for noise-free (

**top**) and noisy (

**bottom**) scenarios. Panels in (

**a**,

**c**) are common source gathers (CSG) associated with the second red-tagged slice in (Figure 1c), whereas (

**b**,

**d**) match with the common receiver gather (CRG) indicated by the second blue slice in the referenced figure. The first panel in figures (

**a**–

**d**) correspond to reconstructed fields in the frequency domain implementation, while the second to those in the time domain. An overlay of traces indicates overestimated amplitudes for the frequency-domain implementation in both source (red-dashed line) and receiver (blue-dashed line) gathers when compared with the same trace section in the true response (solid black line). In contrast, a better fit is instead observed for the time domain reconstructions when the magenta-dashed line (CSG) and the green-dashed line (CRG) are evaluated against the benchmark.

**Figure 7.**Comparison of Time-domain reconstructed impulse responses using physical priors for MDD. Panels tagged with the solid red and blue line follow the CSG and CRG convention in Figure 1. (

**a**,

**e**) Slices of the numerically modeled pressure field in the truncated section of the model, Figure 4a, below the focusing datum at $z=4.4$ km. (

**b**,

**f**) The unconstrained reflectivity solution is densely contaminated with non-physical events amplified by low singular values as iterations go on. (

**c**,

**g**) Introducing the time window $\mathbf{P}=\mathsf{\Theta}$ as preconditioner derives in a solution depleted of acausal events but does not prevent noise from leaking into R. In (

**d**,

**h**) an overlay of traces indicates a fairly good amplitude prediction in the receiver side (red-dashed line), but a significant phase and amplitude mismatch is appreciated in the source side (blue-dashed line).

**Figure 8.**Wavenumber-frequency amplitude spectrum for (

**a**) the cross-correlation function, (

**b**) the unconstrained, and (

**c**) the preconditioned reflectivity reconstruction. (

**d**–

**f**) Top view of the spectra in (

**a**–

**c**). The red-dashed line delineates the f-k band-pass filter used as a preconditioner to constrain the time-domain MDD. Note that spurious energy outside the filter, at high wavenumbers in the correlation spectrum, is amplified in the unconstrained inversion with $\mathbf{P}=\mathbf{I}$. Luckily, the undesired low-velocity artifacts map outside the f-k window, whereas the actual physical energy lies at a low-frequency band. Enforcing a restricted spectrum, with $\mathbf{P}={\mathbf{F}}^{-\mathbf{1}}\mathbf{WF}$, as iterations progress in the inversion, removes most of the dipping events observed in Figure 6c,d.

**Figure 9.**Same as in Figure 8, but now the 3D f-k filter constrains the iterative reconstruction. (

**a**,

**e**) slices of the benchmark reflection response at ${\mathbf{x}}_{\mathbf{r}}=8.6$ km. (

**b**,

**f**) As expected, despite the conservative character of $\mathbf{W}$, many coherent low-wavelength structures and dipping events are removed from the spectrum, producing a reasonable solution within the prescribed bandwidth. (

**c**,

**g**) When source-receiver reciprocity, $\mathbf{Y}$, controls the set of accepted solutions, a significantly more accurate reconstruction is achieved. (

**d**,

**h**) Trace inspection at receiver 35 reveals a substantially better fit, opposite to the situation in Figure 6 and Figure 8.

**Figure 10.**Same as in Figure 8 and Figure 9, except that different chained preconditioners are used as leverage to enhance Green’s function reconstruction. (

**a**,

**e**) slices of the benchmark reflection response in the middle of the virtual at ${\mathbf{x}}_{\mathbf{r}}=7.5$ km. (

**b**,

**f**) Applying causality in combination with a flexible smoothing constrain eliminates excessively large coherent noise and controls the maximum resolvable wavelength in the gathers. Likewise, considering a hybrid operator, $\mathsf{\Theta}\mathbf{Y}{\mathbf{F}}^{-\mathbf{1}}\mathbf{WF}$, conveys the physical properties of an actual local response into the sought solution (

**c**,

**g**). Ultimately the benefits of exploiting joint preconditioners are revealed in traces (

**d**,

**h**), not only this estimation is kinematically closer to the benchmark but the amplitudes are better resolved.

**Figure 11.**Error propagation a function of iterations for multiple single and hybrid preconditioners. (

**a**) Structure similarity index measure, (

**b**) root mean square error, and (

**c**) LSQR residuals. (

**a**,

**b**) expose the semi-convergence behavior of the iterative method. Initially, an accurate solution is retrieved at low iterations, but as the algorithm progresses, the noise gets amplified and the reconstruction is degraded.

**Table 1.**Summary of acoustic MDD representations in compact matrix-vector notation for one- and two-way wavefields according to the general representation (7). The fifth and sixth column, $\mathbf{C}$, $\mathsf{\Gamma}$, comprise cross-correlation and point spread functions, respectively.

MDD | U | Q | G | C | $\mathsf{\Gamma}$ | Equation |
---|---|---|---|---|---|---|

Two-way | $\mathbf{P}$ | $\mathbf{D}$ | $\overline{\mathbf{G}}$ | ${\left(\mathbf{D}\right)}^{H}\mathbf{P}$ | ${\left(\mathbf{D}\right)}^{H}\mathbf{D}$ | (3) |

One-way | ${\mathbf{P}}^{-}$ | ${\mathbf{V}}_{\mathbf{n}}^{+}$ | ${\overline{\mathbf{G}}}_{\mathbf{p}}$ | ${\left({\mathbf{V}}_{\mathbf{n}}^{+}\right)}^{H}{\mathbf{P}}^{-}$ | ${\left({\mathbf{V}}_{\mathbf{n}}^{+}\right)}^{H}{\mathbf{V}}_{\mathbf{n}}^{+}$ | (4) |

${\mathbf{P}}^{-}$ | ${\mathbf{P}}^{+}$ | ${\overline{\mathbf{G}}}_{{\mathbf{v}}_{\mathbf{n}}}$ | ${\left({\mathbf{P}}^{+}\right)}^{H}{\mathbf{P}}^{-}$ | ${\left({\mathbf{P}}^{+}\right)}^{H}{\mathbf{P}}^{+}$ | (5) |

**Table 2.**Different preconditioners options explored in this study. The attributes of the transformation rule, $\mathbf{P}$, in the optimization problem (12) are given via composition of preconditioners (14), (15), and (16). Here, ∗ indicates the physical constraints enforced by the chained operator.

Opt | Causality | Reciprocity | f-k Filtering | Preconditioner |
---|---|---|---|---|

1 | $\mathbf{I}$ | |||

2 | ∗ | $\mathsf{\Theta}$ | ||

3 | ∗ | $\mathbf{Y}$ | ||

4 | ∗ | ${\mathbf{F}}^{-\mathbf{1}}\mathbf{WF}$ | ||

5 | ∗ | ∗ | $\mathsf{\Theta}\mathbf{Y}$ | |

6 | ∗ | ∗ | $\mathsf{\Theta}{\mathbf{F}}^{-\mathbf{1}}\mathbf{WF}$ | |

7 | ∗ | ∗ | ∗ | $\mathsf{\Theta}\mathbf{Y}{\mathbf{F}}^{-\mathbf{1}}\mathbf{WF}$ |

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**MDPI and ACS Style**

Vargas, D.; Vasconcelos, I.; Ravasi, M.; Luiken, N.
Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation. *Remote Sens.* **2021**, *13*, 3683.
https://doi.org/10.3390/rs13183683

**AMA Style**

Vargas D, Vasconcelos I, Ravasi M, Luiken N.
Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation. *Remote Sensing*. 2021; 13(18):3683.
https://doi.org/10.3390/rs13183683

**Chicago/Turabian Style**

Vargas, David, Ivan Vasconcelos, Matteo Ravasi, and Nick Luiken.
2021. "Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation" *Remote Sensing* 13, no. 18: 3683.
https://doi.org/10.3390/rs13183683