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A Methodology to Validate the InSAR Derived Displacement Field of the September 7^{th}, 1999 Athens Earthquake Using Terrestrial Surveying. Improvement of the Assessed Deformation Field by Interferometric Stacking

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## Abstract

**:**

## 1. Introduction

^{th}century was the September 7, 1999, 11

^{h}56

^{m}50

^{s}UTC, Mw (moment magnitude) = 5.9 Athens earthquake. It claimed the lives of 143 people, and caused the collapse of several buildings, mainly in the northwest suburbs of the Greek capital. The approximate location of the earthquake epicenter was 38.10°N, 23.56°E, roughly 20 km northwest from the center of Athens [1].

## 2. Input Data

#### 2.1. ERS-1/2 InSAR Data

#### 2.2. Leveling Data Along the Mornos Aqueduct

## 3. Rendering InSAR Data Comparable to Leveling Data

- InSAR processing provided wrapped interferograms, consequently only the fractional part mod
_{2π}Φ of the full phase difference Φ was known. - InSAR results correspond to the projection of the true vertical deformation along the LOS vector.
- The reference systems of the leveling data and the InSAR data were different. InSAR data were referring to ED 50 UTM zone 34 while the leveling data were referring to the mean sea level and the height reference positions to the Hellenic Geodetic Reference System 87 (HGRS 87).
- The interferograms were “noisy” mainly due to temporal decorrelation, orbital and tropospheric disturbances.

#### 3.1. Wrapped Interferogram Filtering

_{i,j}) and imaginary sin(ψ

_{i,j}) parts of a virtual unitary magnitude signal e

^{jψi,j}= cos(ψ

_{i,j}) + j sin(ψ

_{i,j}) . The phase of this signal is the unfiltered interferometric phase ψ

_{i,j}In other words, the 2D space filter was applied on a unitary signal to which the phase of the input interferogram was projected. The phase ψ

_{flt/i,j}, comprising the filtered interferogram, was extracted through an arctan operation from the filtered real and imaginary parts of the virtual signal. The filtering procedure is best defined by the following formula:

#### 3.2. Phase Unwrapping

#### 3.3. Incidence Angle Correction

**Φ**

_{LOS}(E,N) , and not the vertical differential displacements

**Φ**

_{dU}(E,N) themselves, as is the case of leveling (Figure 6). These two quantities are related through the incidence angle In(E, N):

**r**(t) derived by ERS 1/2 operational orbits provided in the header file of every SAR image. These expressions simply provide the satellite position vectors in the orbit's terrestrial geocentric reference frame as a function of time. Three polynomials were derived, one for every coordinate X, Y and Z. Exactly the same procedure was applied for the satellite velocity vector

**ṙ**(t) and three additional equations were also obtained. Therefore, for every single target (i, j) the following procedure was followed:

- The map projection coordinates of the target were converted to geocentric Cartesian coordinates in the geodetic terrestrial reference frame in which the satellite orbits were provided (in this particular case from HGRS 87 map coordinates to ITRF 96 geocentric Cartesian coordinates).
- The mean Doppler frequency shift was computed by the CNES DIAPASON software and was assumed to be the same for every single pixel target. The Doppler frequency shift f(i,j) was expressed as a function of the satellite position, the satellite velocity vectors and the target position
**r**(i, j), by the following equation (λ denotes the SAR sensor wavelength):$$\text{f}(\text{i},\text{j})=\frac{2(\mathbf{\text{r}}(\text{i},\text{j})-\mathbf{\text{r}}({\text{t}}_{\text{i}}))\cdot \dot{\mathbf{\text{r}}}(\text{i},\text{j})}{\lambda |\mathbf{\text{r}}(\text{i},\text{j})-\mathbf{\text{r}}({\text{t}}_{\text{i}})|}$$ - A total of seven equations were accumulated, and an equal number of unknowns was introduced, three for the satellite position vector, three for the satellite velocity vector and one for the time t
_{i}. Hence, a non linear seven-equation system was created for the estimation of the seven unknowns. The system was linearised with Taylor series expansion and solved iteratively. - Knowing the satellite and target position vectors, the unitary LOS vector could be calculated simply from the following vector equation:$$\mathbf{\text{LOS}}(\text{i},\text{j})=\frac{\mathbf{\text{r}}(\text{i},\text{j})-\mathbf{\text{r}}{(\text{t}}_{\text{i}})}{|\mathbf{\text{r}}(\text{i},\text{j})-\mathbf{\text{r}}{(\text{t}}_{\text{i}})|}$$
- The target position ellipsoidal coordinates φ
_{i,j}, λ_{i,j}were then calculated on the same geodetic terrestrial frame, which was used to express the orbits and the target coordinates in the previous step. - Knowing the target's latitude and longitude φ
_{i,j}, λ_{i,j}the LOS vector components were transformed to the local geodetic reference system (delta north - DN, delta east - DE, delta up -DU) by means of a rotation matrix:$$[\begin{array}{l}\begin{array}{c}\text{DX}(\text{t})\\ \text{DY}(\text{t})\\ \text{DZ}(\text{t})\end{array}]\end{array}\text{R}({\phi}_{\text{i},\text{j}}{,\lambda}_{\text{i},\text{j}})=\left[\begin{array}{l}\text{DN}(\text{t})\\ \text{DE}(\text{t})\\ \text{DU}(\text{t})\end{array}\right]$$ - The third component of the
**LOS**vector as expressed in the local geodetic reference system is actually the direction cosine for the “up” axis of the system, and consequently the cosine of the incidence angle In. Thus the incidence angle can be derived as:$$\text{In}=\text{arctan}(\text{DU})$$

#### 3.4. Stacking

_{1}(i, j), I

_{2}(i, j),… I

_{n}(i, j)), where n represents the number of the available interferograms and I

_{m}(i, j) the unwrapped interferometric phase of the m

^{th}interferogram at pixel location (i,j). Consequently the produced interferogram depicting the mean deformation field, was released from high and intermediate frequencies [21], which corresponded to non-earthquake related interferometric disturbances (Figure 7).

#### 3.5. Geodetic Reference System Conversion

- The ED 50 UTM map coordinates (Eastings and Northings - E, N) were converted to ED 50 ellipsoidal coordinates (latitude and longitude -φ, λ), assigning to each pixel the corresponding orthometric height (H) derived from the input DEM.
- The orthometric heights were converted to geometric ones (h), by implementing a constant additive geoid undulation value (N) for the entire area of interest, since the geoid in this area is relatively “flat” exhibiting a very low gradient. This value was obtained by the Ohio State University OSU 91 Geoid Model, and was recomputed for ED 50.
- The ED 50 ellipsoidal coordinates were converted to ED 50 Cartesian coordinates (X, Y, Z).
- Subsequently, the ED 50 geocentric Cartesian coordinates were converted to HGRS 87 geocentric ones, assuming only a parallel shift between the two systems. The latter assumption was expected to successfully provide the conversion due to the small size of the area of interest.
- Then, the HGRS 87 geocentric Cartesian coordinates were translated to HGRS 87 ellipsoidal (φ, λ) coordinates.
- Ultimately, the HGRS 87 ellipsoidal (φ, λ) coordinates were converted to HGRS 87 Transverse Mercator projection coordinates (E, N).

#### 3.6. Differential Vertical Displacement Modeling

**Φ**

_{dU}(E, N). After a series of adjustments, a successful fit according to the chi-square (χ

^{2}) test was achieved, using the value of 6 mm as a-priori standard deviation for the observations. By the application of the error propagation law (given the estimated model parameters and their a-posteriori standard deviation values), it was concluded that the 3D-mathematical surface would provide the vertical deformation estimate for each target pixel (E, N), with an estimated a-priori deviation not higher than 0.2 mm. In order to ensure that the mathematical model represents the best fit to the displacement pattern observed, the most general form of m

^{th}degree surface was tested:

## 4. Results

## 5. Conclusions – Discussion

## Acknowledgments

## References

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**Figure 1.**(a) Plots of the Mornos aqueduct (blue) and height network (red) projected on 1:50000-scale map and (b) onto an ERS-2 SAR image interferogram.

**Figure 2.**Set of interferometric pairs used in the study. The vertical dashed line indicates the date of the earthquake occurrence.

**Figure 3.**Leveling path legs plot (red) and aqueduct plot (blue) projected onto a wrapped interferogram. For clarity purposes, only the segments connecting the height references are displayed. The actual leveling path follows the channel.

**Figure 5.**(a) & (b) Wrapped and unwrapped versions of the same interferogram. Note the unrealistic fringe pattern due to inaccuracies in the orbital data used. (c) & (d) The effect of the “tilting” and “shifting” operation on the same interferogram; the orbital fringes are removed.

**Figure 9.**Differential vertical deformation profiles derived by the, (a) conventional terrestrial surveying, (b) single “highest quality” interferogram, (c) mean stacked interferogram, (d) windowed maximum coherence interferogram. HR65 indicates the starting point of leveling.

**Table 1.**Average and standard deviation of the vertical displacement differences between the leveling data and the InSAR methods.

Highest quality interferogram | Mean stacked interferogram | Weighted mean stacked interferogram | Maximum coherence stacked interferogram | Windowed maximum coherence stacked interferogram | |
---|---|---|---|---|---|

Average difference (m) | −0,0096 | −0,0016 | −0,0030 | 0,0047 | 0,0020 |

Standard deviation (m) | 0,0056 | 0,0048 | 0,0055 | 0,0150 | 0,0056 |

© 2008 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Kotsis, I.; Kontoes, C.; Paradissis, D.; Karamitsos, S.; Elias, P.; Papoutsis, I. A Methodology to Validate the InSAR Derived Displacement Field of the September 7^{th}, 1999 Athens Earthquake Using Terrestrial Surveying. Improvement of the Assessed Deformation Field by Interferometric Stacking. *Sensors* **2008**, *8*, 4119-4134.
https://doi.org/10.3390/s8074119

**AMA Style**

Kotsis I, Kontoes C, Paradissis D, Karamitsos S, Elias P, Papoutsis I. A Methodology to Validate the InSAR Derived Displacement Field of the September 7^{th}, 1999 Athens Earthquake Using Terrestrial Surveying. Improvement of the Assessed Deformation Field by Interferometric Stacking. *Sensors*. 2008; 8(7):4119-4134.
https://doi.org/10.3390/s8074119

**Chicago/Turabian Style**

Kotsis, Ioannis, Charalabos Kontoes, Dimitrios Paradissis, Spyros Karamitsos, Panagiotis Elias, and Ioannis Papoutsis. 2008. "A Methodology to Validate the InSAR Derived Displacement Field of the September 7^{th}, 1999 Athens Earthquake Using Terrestrial Surveying. Improvement of the Assessed Deformation Field by Interferometric Stacking" *Sensors* 8, no. 7: 4119-4134.
https://doi.org/10.3390/s8074119