# Production, Validation and Morphometric Analysis of a Digital Terrain Model for Lake Trichonis Using Geospatial Technologies and Hydroacoustics

^{1}

^{2}

^{3}

^{4}

^{5}

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## Abstract

**:**

^{2}and with a maximum depth of 58 m, Lake Trichonis is the largest and one of the deepest natural lakes in Greece. As such, it constitutes an important ecosystem and freshwater reserve at the regional scale, whose qualitative and quantitative properties ought to be monitored. Depth is a crucial parameter, as it is involved in both qualitative and quantitative monitoring aspects. Thus, the availability of a bathymetric model and a reliable DTM (Digital Terrain Model) of such an inland water body is imperative for almost any systematic observation scenario or ad hoc measurement endeavor. In this context, the purpose of this study is to produce a DTM from the only official cartographic source of relevant information available (dating back approximately 70 years) and evaluate its performance against new, independent, high-accuracy hydroacoustic recordings. The validation procedure involves the use of echosoundings coupled with GPS, and is followed by the production of a bathymetric model for the assessment of the discrepancies between the DTM and the measurements, along with the relevant morphometric analysis. Both the production and validation of the DTM are conducted in a GIS environment. The results indicate substantial discrepancies between the old DTM and contemporary acoustic data. A significant overall deviation of 3.39 ± 5.26 m in absolute bottom elevation differences and 0.00 ± 7.26 m in relative difference residuals (0.00 ± 2.11 m after 2nd polynomial model corrector surface fit) of the 2019 bathymetric dataset with respect to the ~1950 lake DTM and overall morphometry appear to be associated with a combination of tectonics, subsidence and karstic phenomena in the area. These observations could prove useful for the tectonics, geodynamics and seismicity with respect to the broader Corinth Rift region, as well as for environmental management and technical interventions in and around the lake. This dictates the necessity for new, extensive bathymetric measurements in order to produce an updated DTM of Lake Trichonis, reflecting current conditions and tailored to contemporary accuracy standards and state-of-the-art research in various disciplines in and around the lake.

## 1. Introduction

## 2. Study Area

^{2}, approximate volume of 2.6 × 10

^{9}m

^{3}and is the deepest (maximum depth of 58 m) natural lake in Greece. The catchment consists of a 403 km

^{2}semi-mountainous area.

## 3. Materials and Methods

#### 3.1. Topographic and Bathymetric Maps

#### 3.2. DTM Production

#### 3.2.1. DTM Interpolation

#### 3.2.2. DTM Accuracy Analysis

- 1)
- New contours at a 5 m interval (i.e., at half the original 10 m interval) were created from the interpolated DTM (DTM_original);
- 2)
- The 5 m contours, together with the original 135 depth points, were used to create a new DTM using the Topo-to-Raster algorithm;
- 3)
- 4)
- The ratio of interpolated depth to interpolation error was calculated as a proxy for the signal-to-noise ratio and was plotted along the transect (Figure 4).

#### 3.3. Echosoundings and GPS Data

#### 3.3.1. Bathymetric Data Extraction from Hydroacoustic Data

#### 3.3.2. Echosounder Accuracy Analysis

- Horizontal Accuracy (Spatial Resolution)

_{imprint-circle}= h × tan(3.5°)

- Vertical (Depth) Accuracy

- i.
- Bottom Slope

_{m}, the error in depth dz caused by the slope of the bottom (represented as the zenith angle of the bottom surface normal vector ζ) when no correction is applied for slopes smaller than half the beam-width (i.e., 3.5 degrees), such as those of Lake Trichonis, amounts to [27]:

_{max}, was equal to ~0.104 m. To model the error for the total calculation, the 3rd quartile slope value was used (1.76°) to express the error as a function of depth:

- ii.
- Sound Velocity Variations

^{2}(m/s)

^{2}

_{c}= (dc/dT)

^{2}× σ

^{2}

_{T}= (4.624 − 0.0766 × T)

^{2}× σ

^{2}

_{T}

_{c}= ~17.07 m/s. The uncertainty propagation equation given in [27] for the depth error based on sound velocity variations is:

_{zc}

^{2}= (z/c)

^{2}× (σ

_{cm}

^{2}+ σ

_{c}

^{2})

_{cm}accounts for sound velocity measurement variations and is considered to be compensated for through the instrument calibration procedure, i.e., is excluded from the equation. The σ

_{c}parameter represents the spatiotemporal variations in sound velocity and corresponds to the value that was calculated above. Using this value and an average sound velocity based on the average temperature of 18 °C and Equation (4) equal to 1475.85 m/s, Equation (6) becomes:

_{zc}= (17.07/1475.85) × z = 0.011566 × z

- iii.
- Time-Dependent Variations

_{re}

_{s}= c × τ/2

_{z(t)}= ~0.094 m, regardless of the actual depth.

- iv.
- Water-Undulation-Related Variations

_{imprint-circle}) = tan(0.18°)/tan(3.5°) = ~0.05 in the worst case, i.e., ~5% over the total footprint, which is well within the spatial resolution of the produced raster (24 × 30 m), even for the largest depth values (e.g., ~18 cm for a worst-case footprint radius of 3.67 m). Furthermore, the worst-case effect introduced by this divergence on the vertical component of the measurements would be equal to 1/cos(0.18°) = 4.93 ppm, which is too small to produce a noticeable effect on the results.

^{2}for the gravitational acceleration, the maximum east–west seiche period was calculated to be ~31.6 minutes for the east–west waves and ~8.3 min for the north–south waves. These periods constitute fractions of the total time of transect measurements (~4–5 h each), thus introducing a homogeneous distortion over the ensonified area of the lake, with areas of elevation and areas of depression with respect to the mean level being uniformly scattered in spatial terms. Additionally, the amplitude of those waves is expected to be very small on average, to the order of a few cm for moderately-sized lakes [34]. Therefore, it can be expected that those will be cancelled out in the statistical analyses, as the primary focus of this study is on the overall average accuracy of the studied data.

- v.
- Vertical Datum Error

- Overall Uncertainty

#### 3.3.3. Data Processing

#### 3.4. Absolute Elevation Validation

_{DTM}, was retrieved from the DTM model at each point, where the acoustic data depth, d

_{Acoustic}, was also available. To perform a direct comparison, the difference between the vertical reference frames of the DTM and the echosounder-derived measurements, respectively, is assumed to have the form of a simple translation (displacement). Furthermore, this displacement is considered constant over the entire area of the lake. If d

_{DTM}= h

_{Lake_DTM}− h

_{Bottom_DTM}and d

_{Acoustic}= h

_{Lake_Acoustic}− h

_{Bottom_Acoustic}are the depth measurements at a single specific point i, then:

_{i}= d

_{i, Acoustic}− d

_{i, DTM}= (h

_{i, Lake_Acoustic}− h

_{i, Bottom_Acoustic}) − (h

_{i, Lake_DTM}− h

_{i, Bottom_DTM})

_{i}= d

_{i,Acoustic}− d

_{i, DTM}= (h

_{i, Bottom_DTM}− h

_{i, Bottom_Acoustic}) − (h

_{i, Lake_DTM}− h

_{i, Lake_Acoustic})

_{Lake_DTM}− h

_{Lake_Acoustic}) term of this equation represents the differences between the vertical reference frames of the old (DTM) and new (SONAR) dataset. For the sake of simplicity, this difference is assumed constant and independent of the measurement point over the lake area, denoted below as c. Therefore, the depth difference distribution is a translation of the actual distribution of the bottom height differences between the two epochs:

_{i}= d

_{i, Acoustic}− d

_{i, DTM}= (h

_{i, Bottom_DTM}− h

_{i, Bottom_Acoustic}) + c

#### 3.5. Relative Elevation Validation

_{ij}= (d

_{j, Acoustic}− d

_{i, Acoustic}) − (d

_{j, DTM}− d

_{i, DTM}) = (d

_{j, Acoustic}− d

_{j, DTM}) − (d

_{i, Acoustic}− d

_{i, DTM}) = d

_{j}− d

_{i}

_{ij}= (h

_{j, Bottom_DTM}− h

_{i, Bottom_DTM}) − (h

_{j, Bottom_Acoustic}− h

_{i, Bottom_Acoustic}) = dh

_{ij, DTM}− dh

_{ij, Acoustic}

#### 3.6. Bathymetric Model

_{i}represent the unknown model parameters and e

_{i}represents the residual of the observed depth difference between the two datasets. This model corresponds to a common 2nd order corrector surface model from the general polynomial model family. The specific choice was based on a generic attempt to fit polynomials from the 1st up to the 3rd degree, accordingly, with the 2nd degree being found to provide optimal fit statistics.

_{i}| > 3 × σ). The estimation of the unknown parameters and the residuals was carried out using a least-squares fit of (15), the descriptive statistics of the fit were calculated and their histogram and cumulative distribution were appropriately charted. Using the resulting model, the relative residuals were also calculated as:

#### 3.7. Morphometric Analysis

_{average}, y’ = y − y

_{average}):

_{i}= a

_{0}+ (a

_{1}· x′) + (a

_{2}· y′)

_{ij}value. However, one of the primary aims of this study was to look for potential morphological changes with respect to the bathymetry, as it is represented by the DTM itself. Therefore, the depth values of the produced DTM are considered to be the known parametric values in the above analysis, such that ${\sigma}_{d{h}_{ij,DEM}}^{2}=0$, and only the echosounder-derived measurements are examined under the hypothesis:

## 4. Results

#### 4.1. DTM

#### 4.2. Absolute Elevation Error

_{DTM}− h

_{SONAR}) is equal to 3.39 ± 5.26 m, which indicates a noteworthy (with respect to the external effects described in the methodology section) displacement between the two datasets (Figure 11). Approximately 82.5% of the data points fall within the 1-sigma range (|dh

_{SONAR-DTM}− 3.39| < 5.26 m), while ~17.5% of the data points are 1-sigma outliers. In the 1-sigma range subgroup, the average bottom elevation difference is 1.67 ± 2.3 m, whereas in the outlier subgroup, the average bottom elevation difference is 11.49 ± 7.31 m.

_{DTM}− h

_{SONAR}> 0, was equal to ~32 km, with an average elevation difference value equal to 4.77 ± 5.29 m. The total length along the transect, where h

_{DTM}− h

_{SONAR}< 0, was equal to ~12 km, with an average elevation difference value equal to 1.03 ± 0.86 m. These values indicate that the distribution of the morphological changes over the lake transect are not uniform, but ~27% of the current bottom elevation along the SONAR transect (12 km out of a total of 44 km of transect length) actually lies above its older (DTM) level (positive average), and is in fact rather smooth (as indicated by the relatively smaller standard deviation).

^{2}= 0.09, p = 0.2, Figure 12). However, the same analysis performed at the subgroup of elevation difference outliers at the 1-sigma level (|dh

_{SONAR-DTM}− 3.39| < 5.26 m) indicated a relatively stronger correlation with distance from the shore (R

^{2}= 0.31, p = 0.01, Figure 13). The correlation analysis for the latitude–longitude distribution of the elevation difference did not reveal any significant trends at the alpha = 0.05 significance level (R

^{2}< 0.1, p > 0.05). A strong (R

^{2}= 0.74, p = 1.2 ∙ 10

^{−38}, d = 0.013 × h

^{2}

_{DTM}+ 0.563 × h

_{DTM}+ 6.314) 2nd degree polynomial-based correlation and also a relatively strong positive linear correlation (R

^{2}= 0.49, p = 8.2 × 10

^{−15}, d = 0.278 × h

_{DTM}+ 7.145) were detected between lake bottom elevation difference (h

_{DTM}− h

_{SONAR}) and absolute lake bottom elevation based on the DTM model (Figure 14). This result indicates that the magnitude of the bottom differences between the ~1950 and 2019 datasets increases as lake bottom elevations increase for the DTM elevation dataset. Since absolute elevation is also an indicator of depth (considering a constant-altitude lake water level surface), this indicates that higher discrepancies generally occur at smaller depths.

#### 4.3. Relative Elevation Error

#### 4.4. Bathymetric Model

#### 4.5. Morphometry

#### 4.5.1. Absolute Depth and Volume

^{2}(~1.46 km

^{2}) for the SONAR-derived profile (2019), while a value of 1,413,329 m

^{2}(~1.41 km

^{2}) was calculated for the DTM-derived profile (~1950). This indicates a slightly higher water volume along the transect (with the SONAR-derived profile being ~1.0357 times larger) between the two datasets.

_{c}, y

_{c}) to minimize distortion, truncation errors and other numerical artifacts, while a total of 89 outliers (~2.7%) were rejected based on Tukey’s fences with k = 3.0 (thus excluding only extreme outliers farther than 3 interquartile ranges from the lower and upper quartiles, respectively). Table 6 and Figure 21 depict the fit statistics and the distribution of the errors of the model. The wider range and the apparent asymmetry in the distribution are inherited from the original elevation difference distribution and indicate the preservation of the (potentially significant) effects of marginal signals on the result.

_{DTM}− h

_{SONAR}component, whereas in the north–south direction, the elevation difference rate of change for the same component is equal to 0.0004229, or ~42 cm/km. These values indicate a trend of NE-oriented increase in the (h

_{DTM}− h

_{SONAR}) component, and therefore a larger discrepancy between bottom elevations, with DTM values generally being larger than SONAR values. This can be viewed as a gradual NE-directed “subsidence” of the SONAR-measured lake bottom with respect to the produced DTM. The main direction of this apparent subsidence vector based on the determined components is oriented at an azimuth approximately equal to 35.668°. Additionally, the results indicate an average elevation decrease of ~2.913 m since the production of the map sheets that the DTM creation was based on (Table 6).

#### 4.5.2. Relative Depth

_{ij,DTM}) compared to depth differences from the SONAR-derived dataset (dh

_{ij,acoustic}) for the same point pairs i-j (Equation (14) were correlated on a bootstrapping basis of 30 iterations with 5000 pairs. Figure 22 depicts the outcome of one of these instances, however the coefficient of determination and the coefficients of the equation displayed are averages accompanied by their corresponding standard deviations, as acquired from all 30 iterations. Specifically, the final regression coefficients for the equation dh

_{ij,DTM}= a ∙ dh

_{ij,acoustic}+ b were calculated to be a = 1.1996 ± 0.0078, b = 0.0083 ± 0.0093 and the coefficient of determination was calculated to be equal to R

^{2}= 0.8662 ± 0.00339 (Figure 22).

_{ij,DTM}/dh

_{ij,acoustic}= a). As a result, a systematic difference in the intrinsic scale (i.e., not in absolute but in relative terms) is also apparent between the two datasets, with DTM depth differences being, on average, 1.2 times larger in absolute terms than the corresponding SONAR depth differences for the same point pairs.

## 5. Discussion

_{DTM}− h

_{SONAR}component, which expresses absolute bottom subsidence. This demonstrates a generally increasing observed lake bottom subsidence in the NE direction, with a maximum descent direction azimuth of ~35.668°. This direction of maximum descent in relatively reasonable accordance to the NE dip direction of ~46° north (strike angle equal to ~316°) of a NW–SE normal fault was determined to have been the mechanism for the 1975 earthquake event documented by Kiratzi et al. [48], with a dip angle equal to ~71°. In addition, a similar mechanism of a NNW–SSE strike fault zone that also dips in the NE direction was determined to have been ruptured during the April 2007 earthquake swarm (average Mw of 5.2) in the proximity of the lake, while epicenters were found to cluster along the eastern part of the lake shore, following a NNW–ESE distribution trend [48]. The dip direction of this normal fault is also close to the determined azimuth value of 35.67° for the direction of the maximum lake bottom descent of this study, while the focused intense seismic activity at the east part of the lake in 1975 and 2007 might explain the larger observed subsidence in the east part of the lake.

^{2}= 0.51), with signed values decreasing with increasing depth. This decrease in the h

_{SONAR}– h

_{DTM}component indicates a decreasing trend in bottom depth in 2019 compared to ~1950, which is more intense in deeper areas. Because of the karstic network underlying the lake [19], this observation can be attributed in part to the effect of the larger hydrostatic pressure, and hence the larger water column mass, applied to the lake bottom at deeper areas, as pressure increases linearly with depth (p – p

_{0}= ρ·g·h). This result can be contrasted to the result in Figure 14, which effectively indicates that at an elevation level of approximately −21.65 m (global minimum of the 2nd degree polynomial curve), the lake bottom exhibits the lowest overall differences between the two datasets along the overall transect. A possible reason for this could be the steepness of those areas (the contour density around the ~20 m depth contour is higher, Figure 8), meaning that they might not be retaining significant amounts of matter, as the latter could flow towards deeper, flatter areas. In addition, the subsidence at those relatively steeper areas is slightly less affected by the hydrostatic pressure force, due to the deviation of the surface normal from the plumbline (because of the steepness), thus resulting in a slightly smaller overall vertical subsidence component.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Available topographic and bathymetric information of the broader study area from 1:50,000 scaled maps (HMGS).

**Figure 4.**Transect-wise plot of the interpolated depth to interpolation error ratio. Axis Y shown in logarithmic scale.

**Figure 7.**Overall survey accuracy vs. IHO (International Hydrographic Organization) Document S-44 accuracy requirements plot.

**Figure 8.**The final bathymetric model at a resolution of 1″ × 1″ produced from the available digitized HMGS map sheets by interpolation of the relevant topographic and bathymetric information.

**Figure 10.**Absolute lake bottom elevation variation along the measured transect (with spatial reference) as derived from the DTM and the SONAR dataset.

**Figure 11.**Absolute lake bottom elevation differences (h

_{DTM}− h

_{SONAR}) along the lake transect (with spatial reference).

**Figure 12.**Elevation difference (h

_{DTM}− h

_{SONAR}) vs. distance from the shore—correlation analysis plot.

**Figure 13.**Elevation difference (h

_{DTM}− h

_{SONAR}) vs. distance from the shore—correlation analysis plot for the 1-sigma outlier subgroup (data points with absolute elevation differences > 5.2 m).

**Figure 14.**Correlation analysis between elevation differences and lake bottom elevation based on the DTM dataset with 2nd order polynomial fit.

**Figure 16.**The histogram of the residuals of the 2nd order model after the fit to the absolute residuals.

**Figure 21.**Histogram and cumulative distribution of the 1st order (plane) surface fit errors for the (h

_{DTM}− h

_{SONAR}) component.

**Figure 23.**Sorted distribution of t-statistic values produced from point pair differences (dh

_{ij,DTM}− dh

_{ij,acoustic}) along with estimated standard deviations. The line indicates the critical t-value (df → ∞, a = 0.05).

Minimum (m) | −4.6 |

Maximum (m) | 2.3 |

Mean (m) | 0.0 |

Standard Deviation (m) | 0.4 |

Number of pixels | 136,640 |

Statistical Measure | Value |
---|---|

Mean (m) | 3.39 |

Standard Deviation (m) | 5.26 |

RMSE (Root Mean Squared Error) (m) | 5.38 |

RMSE 95% (m) | 3.90 |

RMSE 90% (m) | 2.98 |

Median (m) | 1.51 |

NMAD (Normalized Median Absolute Deviation) (m) | 2.86 |

(Excess) Kurtosis | 2.48 |

Skewness | –1.71 |

Minimum (m) | –3.93 |

Maximum (m) | 24.05 |

Range (m) | 27.98 |

N (number of samples) | 3284 |

Statistical Measure | Value |
---|---|

Mean (m) | 0.00 |

Standard Deviation (m) | 7.49 |

RMSE (Root Mean Square Error) (m) | 7.49 |

RMSE 95% (m) | 6.20 |

RMSE 90% (m) | 5.22 |

Median (m) | 0.72 |

Normalised Median Average Deviation (NMAD) (m) | 2.86 |

(Excess) Kurtosis | 1.17 |

Skewness | 0.04 |

Minimum (m) | –24.80 |

Maximum (m) | 24.67 |

Range (m) | 49.47 |

n (number of samples) | 3284 |

Mean | 0.00 m |

Median | –0.09 m |

Minimum | –6.30 m |

Maximum | 4.60 m |

Standard Deviation | 2.11 m |

**Table 5.**Descriptive statistics for relative residuals of 2nd order bathymetric corrector surface fit.

Mean | 0.29 m |

Median | 0.18 m |

Minimum | −10.42 m |

Maximum | 10.89 m |

Standard Deviation | 2.81 m |

1st Order Surface (Plane) | |

Coefficient of (x^{’}) | 0.0003035 |

Coefficient of (y’) | 0.0004229 |

Constant Term | 2.913 m |

Mean Error | 0.000 m |

Median Error | –0.931 m |

Minimum | –6.645 m |

Maximum | 16.689 m |

Range | 23.334 m |

Mean Absolute Error | 3.148 m |

RMSE | 4.331 m |

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## Share and Cite

**MDPI and ACS Style**

Perivolioti, T.-M.; Mouratidis, A.; Terzopoulos, D.; Kalaitzis, P.; Ampatzidis, D.; Tušer, M.; Frouzova, J.; Bobori, D.
Production, Validation and Morphometric Analysis of a Digital Terrain Model for Lake Trichonis Using Geospatial Technologies and Hydroacoustics. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 91.
https://doi.org/10.3390/ijgi10020091

**AMA Style**

Perivolioti T-M, Mouratidis A, Terzopoulos D, Kalaitzis P, Ampatzidis D, Tušer M, Frouzova J, Bobori D.
Production, Validation and Morphometric Analysis of a Digital Terrain Model for Lake Trichonis Using Geospatial Technologies and Hydroacoustics. *ISPRS International Journal of Geo-Information*. 2021; 10(2):91.
https://doi.org/10.3390/ijgi10020091

**Chicago/Turabian Style**

Perivolioti, Triantafyllia-Maria, Antonios Mouratidis, Dimitrios Terzopoulos, Panagiotis Kalaitzis, Dimitrios Ampatzidis, Michal Tušer, Jaroslava Frouzova, and Dimitra Bobori.
2021. "Production, Validation and Morphometric Analysis of a Digital Terrain Model for Lake Trichonis Using Geospatial Technologies and Hydroacoustics" *ISPRS International Journal of Geo-Information* 10, no. 2: 91.
https://doi.org/10.3390/ijgi10020091