# The Influence of Interpolated Point Location and Density on 3D Bathymetric Models Generated by Kriging Methods: An Application on the Giglio Island Seabed (Italy)

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, the RMSE of the differences between measured and interpolated values falls below 1 m, while a further increment of soundings is required in the presence of a high level of variation of seabed morphology.

## 1. Introduction

^{9}points/km

^{2}.

## 2. Study Area and Datasets

^{2}(one point per square meter). The original dataset counts a total of 825,602 points and is provided by I.I.M.M., who also carried out the bathymetric survey in 2012.

_{1}= 658,640 m, E

_{2}= 659,240 m, N

_{1}= 4,690,639 m, and N

_{2}= 4,691,039 m.

## 3. Methods

#### 3.1. Kriging Interpolation

_{i}and x

_{i}+ h indicate the positions of each couple of points.

#### 3.1.1. Ordinary Kriging

- (1)
- The global, constant mean $\mu \in \mathrm{R}$ of the random function Z(x) is unknown; and
- (2)
- The data derive from an intrinsically stationary random function Z(x) with known variogram function γ(h).

- n is the number of the input point;
- i = 1, 2, …. n;
- j = 1, 2, … n;
- ${h}_{ij}\left({h}_{11},{h}_{12},{h}_{13},\dots ,{h}_{21},{h}_{22},{h}_{23},\dots \right)$ is the distance between input point i and input point k; and
- $\gamma ({h}_{ij})\left(\left(\gamma ({h}_{11}\right),\left(\gamma ({h}_{12}\right),\left(\gamma ({h}_{13}\right),\dots ,\left(\gamma ({h}_{21}\right),\left(\gamma ({h}_{22}\right),\left(\gamma ({h}_{23}\right),\dots \right)$ is the value of the semi-variogram for the distance between input point i and input point k.

- ${h}_{pi}\left({h}_{p1},{h}_{p2},{h}_{p3},\dots ,{h}_{pn}\right)$ is the distance between the output pixel p and input point i and
- $\gamma ({h}_{pn})\left(\left(\gamma ({h}_{p1}\right),\left(\gamma ({h}_{p2}\right),\left(\gamma ({h}_{p3}\right),\dots ,\left(\gamma ({h}_{pn}\right)\right)$ is the value of the semi-variogram for distance ${h}_{pi}$ between the output pixel p and input point i.

#### 3.1.2. Universal Kriging

- ${a}_{l}$ is the coefficient to be estimated from the data (generally unknown) and characterizes the local trend (drift coefficient);
- ${f}_{l}$ is the lth basic function (generally polynomials) of spatial coordinates that describes the drift; and
- k is the number of functions used in modelling the drift.

- ${x}_{i},{y}_{i}$ are the x, y coordinates of the input points and
- ${a}_{0},{a}_{1},\dots ,{a}_{5}$ are the unknown drift coefficients.

- n is the number of the input point;
- i = 1, 2, … n;
- j = 1, 2, … n;
- ${h}_{ij}\left({h}_{11},{h}_{12},{h}_{13},\dots ,{h}_{21},{h}_{22},{h}_{23},\dots \right)$ is the distance between input point i and input point k;
- $\left({h}_{p1},{h}_{p2},{h}_{p3},\dots ,{h}_{pn}\right)$ is the distance between the output pixel p and input point i, and is the value of the semi-variogram for the distance between input point i and input point k;
- $\gamma ({h}_{pn})\left(\left(\gamma ({h}_{p1}\right),\left(\gamma ({h}_{p2}\right),\left(\gamma ({h}_{p3}\right),\dots ,\left(\gamma ({h}_{pn}\right)\right)$ is the value of the semi-variogram for distance h
_{pi}between the output pixel p and input point i; - ${x}_{i},{y}_{i}$ are the x, y coordinates of the input points;
- ${a}_{0},{a}_{1},\dots ,{a}_{5}$ are the unknown drift coefficients; and
- ${x}_{p},{y}_{p}$ are the x, y coordinates of the output pixel.

#### 3.2. Adopted Methodological Approach

#### 3.2.1. Model Construction and Global Accuracy Evaluation

^{2}is limited, it is necessary to fix a very large cell size. Nevertheless, in our study, the attention was focused on the influence of the morphology of the seabed on the determination of the number of points needed, therefore the grid cell was set at 1 m in every case and the number of points used for modeling varied from time to time. The choice of the pixel dimension equal to 1 m allows for the direct comparison of the constructed models with the starting grid obtained from the multibeam survey.

- -
- Cross validation (CV); and
- -
- Direct comparison with the MBD.

- N is the number of the depth points (that, in this case, coincides with the number of the grid cells);
- Z
_{i}(x,y) is the measured depth at the location i(x, y); and - ${Z}_{i}^{\prime}\left(x,y\right)$ is the interpolated depth at the same location i(x, y).

#### 3.2.2. Morphological Variability and Local Accuracy Evaluation

_{xy}) is related to the variation of the direction of the slope (ascending or descending) while the second component (S

_{xy}) takes into account the variation of the value of the slope. Both these components are considered in relation to x and y directions.

_{xy}, the difference between the consecutive pixel values is considered along the x direction (i.e., along each row) and along the y direction (i.e., along each column); if this difference always has the same sign, the direction does not change, i.e., the seabed continues to rise or fall. By assigning the value 0 to the conservation of the direction and 1 to its variation, the sum of all the values divided by the total number of the pixels of the grid gives a percentage value that expresses the variability of the seabed in terms of the frequency of the ascend–descent inversion. Since two values are calculated (one for the x direction and the other for the y direction) for each pixel, I

_{xy}is the mean of them, referring to all the grid cells.

_{xy}, the difference between the consecutive pixel values (depth values) related to the pixel dimension is considered along the x direction as well as the y direction to calculate slope. Then, the variation of slope is determined and normalized by dividing each value by the maximum (theoretical) value, which is 180°. Additionally, in this case, two values were calculated (one for the x direction and the other for the y direction) for each pixel, thus S

_{xy}is the mean of them, referring to all the grid cells.

_{xy}, six values of S

_{xy}, and, consequently, six values of MVI are calculated, namely one for each sector.

## 4. Results and Discussion

_{xy}.

_{xy}was introduced in the MVI to take into account the level of inclination of the seabed and its variation. The study area has both low slope and steep slopes, often alternating with each other even at short distances, thus high values of abrupt variations were found.

_{xy}.

^{2}is sufficient to make the RMSE values fall below 1 m for both Ordinary Kriging and Universal Kriging. On the contrary, for the sectors where the MVI indicator has the highest values, the RMSE below 1 m is much higher. In sector B, characterized by the highest value of MVI, 1 point per 100 m

^{2}is just enough to obtain an RMSE of approximately 1 m but only in the case of the Ordinary Kriging. In fact, in the case of Universal Kriging, the same concentration determines an RMSE value that remains higher than 1 m (RMSE = 1.214 m).

## 5. Conclusions

_{xy}) is related to the frequency of the variation of the slope direction (ascending or descending) and the second (S

_{xy}) is associated to the variation of the value of the slope, both considered for the x direction as well as y direction.

^{2}is sufficient to produce an accurate model in areas characterized by a low level of variation of seabed morphology (not only RMSE but also minimum and maximum values fall below 1 m). On the contrary, a density 10 times greater than that is necessary to produce an accurate model in areas characterized by a high level of variation of seabed morphology; in this case, even if the RMSE drops below 1 m, there may be strong differences between the predicted and observed depths in some places, thus a further increment of the measured points is required.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three-dimensional representation of the seafloor obtained from the entire multi-beam dataset (MBD): the upper image represents the point cloud of the initial MBD; the lower image represents the continuous 3D model of the seafloor.

**Figure 2.**The study area: on the left, localization of Giglio Island into Tyrrhenian Sea, in equirectangular projection and WGS84 geographic coordinates (EPSG: 4326); on the right, Giglio Island visualization in RGB composition of Sentinel-2B images in UTM/WGS 84 plane coordinates expressed in meters (EPSG: 32632), with the study area highlighted in the red rectangle.

**Figure 3.**The study area with 2400 chosen points (in red) in UTM/WSG 84 plane coordinates expressed in meters (EPSG: 32632).

**Figure 4.**Slopes chart of the initial MBD in UTM/WSG 84 plane coordinates expressed in meters (EPSG: 32632).

**Figure 6.**Direct comparison of Ordinary Kriging models with MBD: the variation of RMSE related to the number of points analyzed in each sector.

**Figure 7.**Direct comparison of Universal Kriging models with MBD: the variation of RMSE related to the number of points analyzed in each sector.

**Figure 8.**Three-dimensional representation of the seafloor in sector B. The detail permits us to appreciate the high variability of the seafloor.

**Figure 10.**The image of the slope inversions along y direction in the study area (the pixels in which slope inversions occur are colored white).

**Figure 12.**RMSE values in the case of 2400 points using Ordinary Kriging and Universal Kriging for each sector (

**upper**) and MVI values calculated in the six sectors (

**lower**).

Count | Min (m) | Max (m) | Mean (m) | St. Dev. (m) | RMSE (m) |
---|---|---|---|---|---|

24 | −13.495 | 11.618 | −0.303 | 4.479 | 4.489 |

48 | −12.587 | 8.487 | 0.139 | 3.014 | 3.017 |

120 | −9.895 | 5.095 | −0.134 | 1.892 | 1.897 |

240 | −9.062 | 11.159 | −0.042 | 1.331 | 1.332 |

480 | −8.810 | 8.211 | 0.021 | 1.130 | 1.130 |

1200 | −7.366 | 5.305 | 0.002 | 0.714 | 0.714 |

2400 | −5.945 | 4.760 | −0.002 | 0.503 | 0.503 |

**Table 2.**Statistical terms of the residuals between each Ordinary Kriging model and initial dataset values.

Count | Min (m) | Max (m) | Mean (m) | St. Dev. (m) | RMSE (m) |
---|---|---|---|---|---|

24 | −30.799 | 6.526 | −0.213 | 4.047 | 4.053 |

48 | −17.098 | 11.578 | −0.180 | 2.052 | 2.060 |

120 | −12.337 | 21.097 | 0.056 | 2.020 | 2.020 |

240 | −10.694 | 10.399 | −0.110 | 1.344 | 1.349 |

480 | −11.457 | 7.816 | −0.059 | 1.087 | 1.089 |

1200 | −8.067 | 7.703 | 0.007 | 0.663 | 0.663 |

2400 | −14.037 | 7.720 | −0.008 | 0.537 | 0.537 |

Count | Min (m) | Max (m) | Mean (m) | St. Dev. (m) | RMSE (m) |
---|---|---|---|---|---|

24 | −14.745 | 12.489 | −0.233 | 5.100 | 5.105 |

48 | −16.002 | 8.574 | −0.297 | 3.297 | 3.310 |

120 | −15.212 | 10.861 | −0.017 | 2.168 | 2.168 |

240 | −11.032 | 11.916 | −0.013 | 1.581 | 1.581 |

480 | −11.188 | 7.590 | −0.041 | 1.211 | 1.212 |

1200 | −6.799 | 4.830 | 0.006 | 0.752 | 0.752 |

2400 | −8.307 | 5.835 | −0.003 | 0.628 | 0.628 |

**Table 4.**Statistical terms of the residuals between each Universal Kriging model and initial dataset values.

Count | Min (m) | Max (m) | Mean (m) | St. Dev. (m) | RMSE (m) |
---|---|---|---|---|---|

24 | −32.563 | 10.076 | −0.340 | 4.851 | 4.863 |

48 | −24.201 | 21.304 | −0.321 | 3.459 | 3.474 |

120 | −15.360 | 31.617 | 0.098 | 2.319 | 2.321 |

240 | −12.154 | 17.509 | −0.108 | 1.566 | 1.570 |

480 | −12.320 | 17.700 | −0.006 | 1.180 | 1.180 |

1200 | −15.795 | 8.532 | 0.013 | 0.795 | 0.795 |

2400 | −17.151 | 8.575 | −0.016 | 0.646 | 0.647 |

**Table 5.**Statistical terms of the residuals between the Ordinary Kriging model with 2400 total points and initial dataset values for B and D sectors.

Sector | Min (m) | Max (m) | Mean (m) | St. Dev. (m) | RMSE (m) |
---|---|---|---|---|---|

B | −14.037 | 7.720 | −0.015 | 1.061 | 1.061 |

D | −0.405 | 0.352 | −0.003 | 0.067 | 0.067 |

**Table 6.**Statistical terms of the residuals between the Universal Kriging model with 2400 total points and initial dataset values for B and D sectors.

Sector | Min (m) | Max (m) | Mean (m) | St. Dev. (m) | RMSE (m) |
---|---|---|---|---|---|

B | −14.469 | 7.499 | −0.046 | 1.214 | 1.214 |

D | −0.486 | 0.353 | −0.001 | 0.064 | 0.064 |

Sector | Min (m) | Max (m) | Mean (m) | St. Dev. (m) |
---|---|---|---|---|

A | −5.250 | 6.540 | −0.041 | 0.260 |

B | −8.030 | 9.050 | −0.091 | 0.503 |

C | −5.280 | 3.890 | −0.200 | 0.242 |

D | −0.440 | 0.270 | −0.091 | 0.040 |

E | −2.810 | 4.240 | −0.069 | 0.090 |

F | −2.670 | 2.310 | −0.068 | 0.063 |

Sector | Min (m) | Max (m) | Mean (m) | St. Dev. (m) |
---|---|---|---|---|

A | −4.570 | 7.830 | 0.258 | 0.287 |

B | −8.690 | 9.320 | 0.287 | 0.552 |

C | −3.020 | 4.280 | 0.105 | 0.254 |

D | −0.280 | 0.420 | 0.033 | 0.055 |

E | −1.830 | 4.490 | 0.050 | 0.138 |

F | −1.510 | 3.330 | 0.017 | 0.089 |

Sector | MVI |
---|---|

A | 0.0041 |

B | 0.0075 |

C | 0.0026 |

D | 0.0014 |

E | 0.0017 |

F | 0.0011 |

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## Share and Cite

**MDPI and ACS Style**

Alcaras, E.; Amoroso, P.P.; Parente, C.
The Influence of Interpolated Point Location and Density on 3D Bathymetric Models Generated by Kriging Methods: An Application on the Giglio Island Seabed (Italy). *Geosciences* **2022**, *12*, 62.
https://doi.org/10.3390/geosciences12020062

**AMA Style**

Alcaras E, Amoroso PP, Parente C.
The Influence of Interpolated Point Location and Density on 3D Bathymetric Models Generated by Kriging Methods: An Application on the Giglio Island Seabed (Italy). *Geosciences*. 2022; 12(2):62.
https://doi.org/10.3390/geosciences12020062

**Chicago/Turabian Style**

Alcaras, Emanuele, Pier Paolo Amoroso, and Claudio Parente.
2022. "The Influence of Interpolated Point Location and Density on 3D Bathymetric Models Generated by Kriging Methods: An Application on the Giglio Island Seabed (Italy)" *Geosciences* 12, no. 2: 62.
https://doi.org/10.3390/geosciences12020062