# Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data

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## Abstract

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## 1. Introduction

## 2. Constrained Least Squares Inversion

- $\delta m$ is the perturbation in the vicinity of the current model ${m}_{n}$,
- $\delta {d}_{n}$ are the data residuals for the model ${m}_{n}$,
- $\Delta {m}_{n}$ is the difference between the current model ${m}_{n}$ and the a priori model ${m}_{prior}$
- ${G}_{n}$ is the linear function tangent to $g$ at the model ${m}_{n}$,
- $g$ is the function mapping the model space $m\in \mathit{M}$ into the data space $d=g\left(m\right)\in \mathit{D}$
- ${C}_{D}$ is the covariance matrix on the data space.
- ${C}_{M}$ is the covariance matrix on the model space (defining the Gaussian a priori probability density).

## 3. Constraints on the Data Space

#### 3.1. Cross-Correlations and Polarization

#### 3.2. An Offset VSP Synthetic Example

## 4. Constraints on the Model Space

#### 4.1. Quantification of Number of Degrees of Freedom

#### 4.1.1. General Case

#### 4.1.2. Spatial Correlation on the Model Space

#### 4.2. Seismic Crosswell Numerical Experiment Example

#### 4.2.1. Description of the Crosswell Experiment and Observed Data

#### 4.2.2. Multiscale Constrained Inversion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Generalities

#### Appendix A.2. Sequential Simulation

#### Appendix A.3. The $\mathit{L}\mathit{U}$ Simulation Technique

#### Appendix A.4. Interpretation of the $\mathit{L}\mathit{U}$ Decomposition of the Covariance Matrix

#### Appendix A.5. The $\mathit{L}\mathit{U}$ Decomposition of the Correlation Matrix $\mathit{R}$

## Appendix B

**Figure A1.**The joint probability density $f\left(x,y\right)$ of the couple of random variables $\left(X,Y\right)$ is plotted using contour levels (at level $0.9\xb7{f}_{max}$, $0.7\xb7{f}_{max}$, $0.5\xb7{f}_{max}$, $0.3\xb7{f}_{max}$ and $0.1\xb7{f}_{max}$), the marginal probability densities for the variables themselves (${f}_{x}\left(x\right)$ for $X$ and ${f}_{y}\left(y\right)$ for $Y$) are plotted on the $x$ and the $y$ axes. The value ${x}_{0}=1.5$ has been fixed and the conditional probability density ${f}_{c}\left(y\right)={f}_{y|x}\left(y|{x}_{0}\right)$ is plotted, the mean ${m}_{x}={m}_{y}=0$ and ${m}_{c}$ has been reported as the standard deviation ${\sigma}_{x}={\sigma}_{y}=1$ and ${\sigma}_{c}$, the correlation $\rho =0.8$ appears only in the ellipticity of the ellipses for the joint p.d.f. $f$. As Equation (A17) shows, the number of free parameters of the couple $\left(X,Y\right)$ is $1+\frac{{\sigma}_{c}}{{\sigma}_{y}}=1+\sqrt{1-{\rho}^{2}}=1.6$.

## References

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**Figure 1.**Offset Vertical Seismic Profile (OVSP) numerical experiment: (

**a**) Synthetic experiment configuration where the asterisk near the surface denotes the source position for the unique shot point, whereas the triangles in the well denote the location of the 40 two-component geophones. The “true” model is composed of nine homogeneous layers, the first one being the water layer. This true model is used both to model the observed data and as a reference to check inversion results; (

**b**) the horizontal (top) and vertical (bottom) components of the ground velocity field recorded at geophones—the fields are obtained by modeling the wave equation in the true model (finite difference approximation). The color represents the local polarization estimated using a 2-period time window.

**Figure 2.**Two inversion experiments with and without polarization constraints. (

**a**) Without polarization constraints for P-wave velocity estimated model. (

**b**) With polarization constraint for P-wave velocity estimated model. (

**c**) Without polarization constraints for S-wave velocity estimated model. (

**d**) With polarization constraints for S-wave velocity estimated model. In all plots: red thick solid line denotes the true model we want to retrieve by inverting the observed data; gray solid line denotes the starting model in the inversion process; blue, green, and black solid lines denote estimated models from inversion, respectively, at iterations 10th, 30th, and 80th. The vertical thick black solid line indicates the level of the receiver antenna. The constrained inversion contributes to the recovery of the sharp interfaces, especially at the level of the receiver antenna.

**Figure 3.**Degree of freedom per point (for a regular grid) as a function of the range (expressed in multiples of the grid step) of the correlation for a Laplace correlation function.

**Figure 4.**The reference velocity models: (

**a**) Reference P-wave velocity field; (

**b**) reference S-wave velocity field. The source borehole is located at 0 m distance and the nine asterisks (*) denote the source locations for the nine shots. The receiver borehole is located at 280 m offset and the small triangles denote the locations of the receivers. On can notice that the structures of these reference velocity fields are the same for P- and S-wave. It is composed of 1D regions (invariance along the horizontal axis) in the upper part of the model and at the bottom with homogeneous layers and other layers with a velocity increasing linearly with depth (vertical constant gradient). The middle part of the model exhibits complex 2D structures.

**Figure 5.**The 5th shot, at 1200 m deep, of the observed data obtained by full-wave modeling from the reference models: (

**a**) The horizontal component; (

**b**) the vertical component. The source function is a first derivative of a Gaussian with a central frequency of 100 Hz and it acts as a stress source with the following radiation coefficients ${\sigma}_{xx}=1,$ ${\sigma}_{zz}=0.5,$ ${\sigma}_{xz}=0.$ These data are used as observed data in the multiscale inversion process.

**Figure 6.**The results of the multiscale inversion process for the P-wave velocity field. Stars denote the source location while the small triangles denote the receiver location: (

**a**) The P-wave velocity starting model; (

**b**) the estimated P-velocity field obtained from the first inversion scale using correlation ranges of ${r}_{x}=320\mathrm{m}$ and ${r}_{z}=24\mathrm{m}$ in the 2D region; (

**c**) the estimated P-velocity field obtained from the second inversion scale using correlation ranges of ${r}_{x}=80\mathrm{m}$ and ${r}_{z}=10\mathrm{m}$ in the 2D region; (

**d**) the estimated P-velocity field obtained from the third inversion scale using correlation ranges of ${r}_{x}=20\mathrm{m}$ and ${r}_{z}=6\mathrm{m}$ in the 2D region; (

**e**) the estimated P-velocity field obtained from the fourth and last inversion scale using correlation ranges of ${r}_{x}=5\mathrm{m}$ and ${r}_{z}=2\mathrm{m}$ in the 2D region; (

**f**) the reference P-wave velocity model. The results for the first scale of the inversion do not allow recovery of the 2D structures, as the correlation ranges are greater than the size of the objects. The 2D structures start to appear from the second scale, improving from the third scale, and are well recovered for the last scale.

**Figure 7.**The results of the multiscale inversion process for the S-wave velocity field. Stars denote the source location while the small triangles denote the receiver location: (

**a**) The S-wave velocity starting model; (

**b**) the estimated S-velocity field obtained from the first inversion scale using correlation ranges of ${r}_{x}=320\mathrm{m}$ and ${r}_{z}=24\mathrm{m}$ in the 2D region; (

**c**) the estimated S-velocity field obtained from the second inversion scale using correlation ranges of ${r}_{x}=80\mathrm{m}$ and ${r}_{z}=10\mathrm{m}$ in the 2D region; (

**d**) the estimated S-velocity field obtained from the third inversion scale using correlation ranges of ${r}_{x}=20\mathrm{m}$ and ${r}_{z}=6\mathrm{m}$ in the 2D region; (

**e**) the estimated S-velocity field obtained from the fourth and last inversion scale using correlation ranges of ${r}_{x}=5\mathrm{m}$ and ${r}_{z}=2\mathrm{m}$ in the 2D region; (

**f**) the reference S-wave velocity model. The S-wave estimated field exhibits a better spatial resolution than the P-wave (see Figure 6) due to the smaller wavelength content associated with this mode. Small artifacts can be observed around the 2D triangle structure, pointing out the fact that the problem is less constrained in this part of the model.

**Figure 8.**Comparison between results from the constrained inversion using the multiscale inversion process results of the multiscale inversion process and results from the unconstrained inversion: (

**a**) The estimated P-wave velocity field for the multiscale inversion for a misfit of 0.3%; (

**b**) the reference P-wave velocity field; (

**c**) the estimated P-wave velocity field for the unconstrained inversion for a misfit of 0.3%; (

**d**) the estimated S-wave velocity field for the multiscale inversion for a misfit of 0.3%; (

**e**) the reference S-wave velocity field; (

**f**) the estimated S-wave velocity field for the unconstrained inversion for a misfit of 0.3%. Both P- and S-wave estimated fields show that the multiscale inversion results are smoother than the unconstrainted inversion results, with well artifact and better reliability. The main artifacts are near the sources, below the triangle structure, above the source or receiver zones and below (less visible due to the color range), and in the constant gradient 1D layers. As expected, there is no additional information outside the square delimited by the wells for the case of unconstrained inversion.

Depth (m) | Region Type | Number of Vertical Points |
---|---|---|

800–1120 | Quasi 1D | 161 |

1120–1180 | Transition | 29 |

1180–1340 | 2D | 81 |

1340–1400 | Transition | 29 |

1400–1600 | Quasi 1D | 101 |

**Table 2.**Horizontal and vertical correlation ranges for the different a priori region types and inversion scales.

Region Type | Correlation Ranges (m) | Inversion Scale (b) | Inversion Scale (c) | Inversion Scale (d) | Inversion Scale (e) |
---|---|---|---|---|---|

Quasi 1D | $({r}_{x},{r}_{z})$ | (1000, 24) | (1000, 10) | (1000, 6) | (1000, 2) |

2D | $({r}_{x},{r}_{z})$ | (320, 24) | (80, 10) | (20, 6) | (5, 2) |

Inversion Parameters | Inversion Scale (b) | Inversion Scale (c) | Inversion Scale (d) | Inversion Scale (e) |
---|---|---|---|---|

Initial Misfit | 45.5% | 27.9% | 6.55% | 0.81% |

Final Misfit | 27.9% | 6.55% | 0.81% | 0.14% |

Iteration # | 30 | 30 | 50 | 50 |

Region Type | Inversion Scale (b) | Inversion Scale (c) | Inversion Scale (d) | Inversion Scale (e) |
---|---|---|---|---|

Quasi 1D | 1919 | 2812 | 3416 | 4554 |

Transition | 267 | 468 | 622 | 857 |

2D | 1022 | 2922 | 6792 | 15,725 |

All | 3208 | 6202 | 10,830 | 21,136 |

DoF/point | 2.8% | 5.5% | 9.6% | 18.8% |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Charara, M.; Barnes, C.
Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data. *Geosciences* **2019**, *9*, 45.
https://doi.org/10.3390/geosciences9010045

**AMA Style**

Charara M, Barnes C.
Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data. *Geosciences*. 2019; 9(1):45.
https://doi.org/10.3390/geosciences9010045

**Chicago/Turabian Style**

Charara, Marwan, and Christophe Barnes.
2019. "Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data" *Geosciences* 9, no. 1: 45.
https://doi.org/10.3390/geosciences9010045