# Seismic Coda-Waves Imaging Based on Sensitivity Kernels Calculated Using an Heuristic Approach

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## Abstract

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## 1. Introduction

## 2. The Theoretical (Energy Transport) Model

#### 2.1. The Energy Transport Model and Its Approximations

#### 2.2. 2D Formulations of ETM

#### 2.3. Scattering Regimes

- (1)
- ${l}_{a}$ is the “absorption” or “intrinsic attenuation” mean path. ${l}_{a}=\frac{v{Q}_{i}}{2\pi f}$.
- (2)
- ${l}_{g}$ is the “scattering” mean free path. ${l}_{g}=\frac{v{Q}_{s}}{2\pi f}$.

#### 2.4. Application to Real Data to Estimate the Seismic Attributes

#### 2.5. The Time Evolution of Coda Waves Studies

## 3. The Heuristic Method to Calculate Sensitivity Kernels

#### 3.1. Early Q-Coda Images

#### 3.2. The Introduction of Peak Delay Time and the First Attempts to Separate Intrinsic from Scattering Q

#### 3.3. Sensitivity Kernels for Scattering Radiation

#### 3.4. Numerical Simulation to Estimate the Sensitivity Kernels for Coda Waves

## 4. Analytical Approximation of the Space Weighting Functions

## 5. Sensitivity Kernel for Q-Coda

## 6. Imaging Methods Based on SWF

#### 6.1. Imaging Methods Based on SWF’s in Diffusive Media

#### 6.2. The Projection Method

#### 6.3. The Inversion Method

#### 6.4. Equivalency of the Two Approaches

#### 6.5. Sensitivity Tests

#### 6.6. The “Resolution” Function for the Projection Method

- The weighting functions (each normalized for their maximum) are represented by the quantities ${w}_{ij}$ where i is the event-source index and j represents the j-$th$ pixel of generic coordinates $\{x,y\}$. i spans from 1 to N where N is the number of source-receiver couples in the data set. j spans from 1 to M where M is the number of pixels (square regions in which the input image is divided).
- $\overline{{q}_{ij}}$ is the j-$th$ Q-value (or its inverse) measured for the i-$th$ source-receiver couple
- ${q}_{j}$ is the output of the method for the j-$th$ pixel.

## 7. Application to Real Data

#### 7.1. Projection Method in the Diffusion Assumption

#### 7.2. Projection Method Applied to Stromboli Volcano: A Revision of the Results

#### 7.3. Inversion Method in the Single Scattering Approximation

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Explanation |

N | Number of wave-particles in the simulation |

${\eta}_{s}$ | Scattering coefficient. $g={\eta}_{s}=\frac{2\pi f}{v{Q}_{s}}$ where f is the frequency |

v | Wave speed |

${t}_{lapse}$ | Lapse time (measured from origin) |

${\eta}_{i}$ | Intrinsic attenuation coefficients. ${\eta}_{i}=\frac{2\pi f}{v{Q}_{i}}$ where f is the frequency |

${B}_{0}$ | Seismic Albedo. ${B}_{{}^{0}}=\frac{{\eta}_{s}}{{\eta}_{i}+{\eta}_{s}}$ |

$Le$ | Extinction Length. $L{e}^{-1}=$${\eta}_{i}+{\eta}_{s}$ |

${n}_{sc},{n}_{i}$ | Space density of scatterers and paths, respectively, or Space Weighting Functions |

$\{{x}_{s},{y}_{s}\}$ | Source coordinates |

$\{{x}_{r},{y}_{r}\}$ | Receiver coordinates |

D | $\sqrt{{({x}_{s}-{x}_{r})}^{2}+{({y}_{s}-{y}_{r})}^{2}}$ source-receiver distance |

$\delta t$ | time step used in simulations |

${Q}_{i},{Q}_{s}$ | Intrinsic and Scattering Quality Factor |

${K}_{num}{}^{3D},{K}_{num}{}^{2D}$ | Analytical expression of ${n}_{sc},{n}_{i}$ for high heterogeneity (diffusion) |

E | Energy envelope |

${E}_{d}$ | Energy envelope in the Diffusion approximation |

${E}_{ss}$ | Energy envelope in case of Single scattering |

$\overline{q}$ | Measured Q-value |

${w}_{ij}$ | The same of $Knum$ (to simplify notations) |

${R}_{j}$ | Resolution function at j-$th$ pixel |

Acronym | Explanation |

ETM | Energy Transport Model |

DM | Diffusion Model |

SSM | Single Scattering Model |

SEE | Seismogram Energy Envelope |

SWF | Space Weighting Functions |

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**Figure 1.**The units in the axes are normalized for $\lambda $, the wavelength. The blue transparent zone marks the domain in which Q definition fails. The central strip (multiple scattering) separates the zone where the “Diffusion” and “Single Scattering” regimes properly hold.

**Figure 2.**Scattering ellipsoid in case of surface source and receiver, at a distance of 230 km for max lapse time of 100 s.

**Figure 3.**Energy particles emitted by the source travel (dashed paths) inside the scattering ellipse, following the Fermat’s rules. Collisions with heterogeneities are marked with dark grey circles. When arrived at the Receiver, the particles are counted and a time histogram is calculated.

**Figure 4.**Redrawn from [16]. Contour plots (upper panels) of the numerically generated SWF (2D) for ${\eta}_{i}$ = 0.001 km${}^{-1}$, ${\eta}_{s}$ = 0.628 km${}^{-1}$${t}_{lapse}$ = 15 s v = 2 km/s. The time step in the simulation was set at 0.05 s. Source and receiver are positioned at {0 km, 0 km}, and $\{5$ km, 0 km}, respectively. Both SWF are normalized at the middle point between source and receiver. In the central panels, the values of normalized ${n}_{i}$ and ${n}_{sc}$ at y = 0.0 are plotted as a function of x (

**left panel**) and the corresponding pattern at x = 0.0 as a function of y (

**right panel**). In the lowermost panels the values of normalized ${n}_{i}$ and ${n}_{sc}$ at x = 2.5 are plotted as a function of y (

**left panel**) and the corresponding pattern at x = 5.0 as a function of y (

**right panel**). In this example, scattering attenuation predominates and ${n}_{i}$ and ${n}_{sc}$ are practically coincident.

**Figure 5.**Three examples of 2D Kernels. Sources are indicated with a small white circle. Receivers with a small white triangle. Isoline values in the color scale. Kernel values are normalized at the midpoint between the source and receiver.

**Figure 6.**Isolines of the 3D Kernel are depicted in a plane (parallel to x-axis) and in a vertical plane, both cutting source and receiver. Kernel values are normalized at the midpoint between source and receiver.

**Figure 7.**Contours of ${Q}_{c}^{-1}$ as a function of ${Q}_{i}^{-1}$ and ${Q}_{s}^{-1}$ calculated fitting ETM with SSM in the time interval between origin time and 30s lapse. (

**Left**) ${Q}_{c}^{-1}$ does not depend on ${Q}_{s}^{-1}$ for media with low ${Q}_{s}^{-1}$. (

**Right**) in the case of high heterogeneity (${Q}_{s}^{-1}>0.1)$ the trade off appears.

**Figure 9.**Qualitative scheme to show how SEE should be fitted. SEE measured from single seismograms recorded in media with increasing heterogeneity and shallow source depths are candidate for being fitted to DM.

**Figure 11.**2D space distribution of intrinsic attenuation parameters. Instead of the absolute values of inverse ${Q}_{i}^{-1}$, we plot their percent fluctuations respect to the space average of ${Q}_{i}^{-1}$. In this way, the reader is focused to the attenuation space changes, more useful for the geological interpretation. The six panels represent the intrinsic attenuation fluctuations in six frequency bands, indicated at the top of each panel.

**Figure 13.**To the input checkerboard (rightmost panel, 1.5 × 1.5 km cells) is superimposed a black cell when ${R}_{j}$ of Equation (25) is smaller than 0.4 (upper left panel), 0.6 (upper right panel), 0.8 (down left panel), and 0.9 (down right panel). Once assumed, an acceptable level for ${R}_{j}$, for example, ${R}_{j}>0.6$, the cells with insufficient resolution are colored in black. We remark here that the quantity ${R}_{j}$ is just proportional to the Resolution. In other words these plots are useful for a relative comparison among different images, once that a given value of ${R}_{j}$ is assumed for all the images.

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**MDPI and ACS Style**

Del Pezzo, E.; Ibáñez, J.M. Seismic Coda-Waves Imaging Based on Sensitivity Kernels Calculated Using an Heuristic Approach. *Geosciences* **2020**, *10*, 304.
https://doi.org/10.3390/geosciences10080304

**AMA Style**

Del Pezzo E, Ibáñez JM. Seismic Coda-Waves Imaging Based on Sensitivity Kernels Calculated Using an Heuristic Approach. *Geosciences*. 2020; 10(8):304.
https://doi.org/10.3390/geosciences10080304

**Chicago/Turabian Style**

Del Pezzo, Edoardo, and Jesús M. Ibáñez. 2020. "Seismic Coda-Waves Imaging Based on Sensitivity Kernels Calculated Using an Heuristic Approach" *Geosciences* 10, no. 8: 304.
https://doi.org/10.3390/geosciences10080304