# Application of Seismic Interferometry by Multidimensional Deconvolution to Earthquake Data Recorded in Malargüe, Argentina

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## Abstract

**:**

## 1. Introduction

## 2. Method

**V**,

**U**, and

**Δ**

_{r}can be obtained by applying SVD to the matrix ${V}_{B}^{\text{}}$. In this work, we have implemented the SVD procedure in the frequency domain and have selected singular values for each frequency separately. The matrix

**V**is a right singular matrix, the matrix

**U**a left singular matrix, and matrix

**Δ**

_{r}is a diagonal matrix whose (diagonal) components are the nonzero singular values ordered from large to small, where r indicates the rank of the matrix ${V}_{B}^{\text{}}$.

## 3. MalARRgüe

## 4. Application to Data and Results

#### 4.1. Application to Synthetic Surface Waves

**U**is an 11 × 11 matrix,

**V**is a 19 × 19 matrix, and we have 11 singular values, which are sorted in order of decreasing value. The threshold S introduced in Section 2 determines the number of singular values (the rank) that will be used to construct the Moore-Penrose pseudoinverse ${V}_{B}^{+}$. This may vary between frequencies. We set S to 85, 90, 95, 97, and 99 and evaluate the stability of the retrieved MDD responses for each of these thresholds. For instance, a threshold of 97 indicates that ${g}_{\mathrm{d}}^{\left(\mathrm{est}\right)}$ can be reconstructed by using 97% of the energy.

_{rec}(see Equation (5)). The comparison demonstrates that the phase of the responses retrieved through SI by CC deviates more from the phase of the true (directly modelled) responses than the phase of the responses retrieved through SI by MDD. This is the case for all frequency ranges and all thresholds, which is in line with the larger travel-time deviations of the CC responses (compared to the travel-time deviations of the MDD responses) observed in Figure 5 and Figure 7. We do not observe large variations in the phase difference between the directly modelled response and the MDD responses associated with different thresholds.

#### 4.2. The Effect of Site Amplification

#### 4.3. Application to Field Data from Earthquake-Generated Surface Waves

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Source configuration associated with (

**a**) SI by crosscorrelation (CC) and (

**b**) SI by multidmensional deconvolution (MDD)

**.**Stars indicate sources. Triangles indicate receivers. For successful application of SI by CC, the sources (black and grey stars in (

**a**)) need to be characterised by the same source-time function and be regularly distributed along a boundary in the far field of the receivers; successful application of SI by MDD does not require this, and hence, the sources can be distributed randomly and have different source-time functions (illustrated by varying sizes of the stars in (

**b**). The red triangle depicts the position of a (possible) virtual source along the north-south oriented line of receivers, whereas the blue triangle indicates the position of a receiver along the east-west oriented line of receivers.

**Figure 2.**(

**a**) Location of MalARRgüe and location and magnitude of the earthquakes generating surface waves used in this study. (

**b**) Configuration of the T-array of MalARRgüe, with the triangles presenting the station locations.

**Figure 4.**(

**a**) Surface-wave phase velocity of the modelled surface waves as a function of frequency. (

**b**) Amplitude spectrum of the sources used in our modelling.

**Figure 5.**(

**a**) Virtual-source responses at stations TE03–TE09 obtained from SI by CC (blue colour) for a virtual source at station TN08 in the frequency range of 0.2–0.3 Hz. (

**b**) Responses retrieved by means of SI by MDD (black colour), filtered between 0.2 and 0.3 Hz, for a threshold of 85. (

**c**–

**f**) Same as (

**b**) but for a threshold of 90, 95, 97, and 99, respectively. The green colour indicates directly modelled (DM) responses.

**Figure 6.**(

**a**) The singular values for individual frequencies. (

**b**) Rank associated with the five different thresholds for the frequencies in (

**a**).

**Figure 7.**Comparison between the directly modelled response (green) and the retrieved responses using SI by CC (blue colour) and SI by MDD (black colour) at station TE07 from a virtual source at station TN08 in the frequency range of (

**a**) 0.1–0.2 Hz, (

**b**) 0.2–0.3 Hz, (

**c**) 0.3–0.4 Hz, and (

**d**) 0.4–0.5 Hz.

**Figure 8.**The average absolute phase difference between the directly modelled responses and the SI responses for virtual sources at stations TN06–TN16 (after having unwrapped the phases). The responses to these virtual sources are retrieved at stations TE03–TE09. We have differentiated between SI by CC (white) and SI by MDD (grey to black) for the different thresholds (85, 90, 95, 97, 99). Averages over the various virtual sources, receivers, and discrete frequencies are computed independently for the different frequency ranges.

**Figure 9.**Comparison of the responses retrieved using different methods (SI by CC and SI by MDD for different thresholds) for all realizations from bootstrapping. The responses are retrieved at station TE07 from a virtual source at station TN11 and filtered between 0.1 and 0.2 Hz.

**Figure 10.**The absolute values of the phase difference between the directly modelled responses and the interferometric response at TE07 for a virtual source at TN11 for all frequency ranges and all realizations of the bootstrapping method: (

**a**) SI by MDD with a threshold of 97 (different colours are associated with different ranks); (

**b**) SI by CC.

**Figure 11.**Distributions of the (

**a**) phase and (

**b**) amplitude of the retrieved responses with respect to their mean values for SI by CC (blue histogram) and SI by MDD with different thresholds (grey histograms) for all realizations. Distributions are computed from all virtual-source responses at station TE07 between 0.1 Hz and 0.5 Hz. The amplitudes of the responses are normalized with respect to the mean amplitude for each frequency, virtual source-receiver pair, and method individually.

**Figure 12.**Relative amplitude of the responses retrieved at TE07 from virtual sources at TN06-TN16 in the frequency range of 0.2–0.3 Hz and for all realizations of the bootstrapping method using SI by CC (blue lines) and SI by MDD with threshold 97 (black lines) (

**a**) in the absence of site amplification; and (

**b**) in the presence of site amplification. All amplitudes are normalized with respect to the mean amplitude of the responses from virtual sources TN05–TN07. Red circles show site amplification prescribed for the different TN stations.

**Figure 13.**As in Figure 11 but in the presence of site amplification. (

**a**) Phase (deviation from average; radians). (

**b**) Relative amplitude (deviation from average).

**Figure 14.**Surface-wave arrivals at station TE07 for the chosen 11 earthquakes (compare with Figure 3). Traces are sorted according to the distance between an earthquake’s epicentre and TE07. The recordings are bandpass filtered between 0.1 Hz and 0.5 Hz.

**Figure 15.**As in Figure 11 but for the field data. (

**a**) Phase (deviation from average; radians). (

**b**) Relative amplitude (deviation from average).

**Figure 16.**(

**a**) Virtual-source responses at stations TE03–TE09 for a virtual source at station TN06 obtained from SI by CC (blue colour) and SI by MDD for a threshold of 97 (black colour) for application of field data in the frequency range of 0.1–0.2 Hz and (

**b**) same as (

**a**) but for a virtual source at TN08.

**Table 1.**Epicentres and magnitudes of the earthquakes from Figure 2a, the direct surface waves of which are used for interferometric surface-wave retrieval.

Num. | Longitude (DD) | Latitude (DD) | Magnitude Scale | Magnitude |
---|---|---|---|---|

1 | −73.981 | −38.148 | MW | 4.5 |

2 | −74.237 | −37.455 | ML | 4.1 |

3 | −73.723 | −37.658 | MB | 4.5 |

4 | −73.547 | −37.654 | ML | 4.2 |

5 | −73.397 | −37.199 | MB | 5.0 |

6 | −72.985 | −37.512 | MB | 4.8 |

7 | −73.462 | −35.776 | MB | 4.8 |

8 | −72.766 | −35.541 | MB | 4.3 |

9 | −72.012 | −35.127 | MB | 4.7 |

10 | −71.075 | −36.036 | MB | 5.0 |

11 | −70.570 | −36.074 | MW | 6.0 |

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**MDPI and ACS Style**

Shirmohammadi, F.; Draganov, D.; Hatami, M.R.; Weemstra, C.
Application of Seismic Interferometry by Multidimensional Deconvolution to Earthquake Data Recorded in Malargüe, Argentina. *Remote Sens.* **2021**, *13*, 4818.
https://doi.org/10.3390/rs13234818

**AMA Style**

Shirmohammadi F, Draganov D, Hatami MR, Weemstra C.
Application of Seismic Interferometry by Multidimensional Deconvolution to Earthquake Data Recorded in Malargüe, Argentina. *Remote Sensing*. 2021; 13(23):4818.
https://doi.org/10.3390/rs13234818

**Chicago/Turabian Style**

Shirmohammadi, Faezeh, Deyan Draganov, Mohammad Reza Hatami, and Cornelis Weemstra.
2021. "Application of Seismic Interferometry by Multidimensional Deconvolution to Earthquake Data Recorded in Malargüe, Argentina" *Remote Sensing* 13, no. 23: 4818.
https://doi.org/10.3390/rs13234818