#
The Use of an Artificial Neural Network to Process Hydrographic Big Data during Surface Modeling^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Surfaces Used During the Tests

_{i}, y

_{i}): i = 1, 2, …, 100}, S2 = {(x

_{i}, y

_{i}): i = 1, 2, …, 200}, S3 = {(x

_{i}, y

_{i}): i = 1, 2, …, 100}, S4 = {(x

_{i}, y

_{i}): i = 1, 2, …, 200} and S5 = {(x

_{i}, y

_{i}): i = 1, 2, …, 200,000}. The spatial distributions of samples in S1, S2, S3, and S4 data sets are presented in Figure 2. Such distribution was determined on the basis of the real data sets, recorded by hydrographical sensors.

#### 2.2. RBF Network Optimization for Geodata Interpolation

_{i}is the ith radial basis function, K is the number of radial basis functions, W

_{i}is the weight of the ith network, x is the input vector, and t

_{i}is the center of the ith RBF.

_{z}) is calculated for the basis functions based on an independent test set to minimize the RMS error, which can be treated as an approximate solution. For this step, finding a coarse value for the basis function radius was not time-consuming, because only one test set was used. The RMS error is then calculated for different values of the basis function radii, until it reaches its smallest value (these values of the radii are increased iteratively by a small interval, starting from any small value). In the second step, the radius value is computed using the more accurate but time-consuming LOO method; however, the search for the optimal value starts at the value determined from the approximate, coarse solution (σ

_{z}). If we denote the error for a single-element validation set as e

_{i}, then the error for the LOO (E

_{L}) method can be calculated according to the formula:

_{L}, the optimal value was determined. Calculation of the optimal value of the radius of the basis functions was carried out in several stages.

_{i}, for each neuron, which changes across k iterations. The minimal potential, p

_{min}, determines whether a neuron participates in the competition. Changes in neuron potentials can be calculated according to the dependency [19]:

_{i}errors for the polygon generated from the test samples for an algorithmically ordered set of different σ

_{i}parameter values (these values are sorted in ascending order and then incremented by any small interval).

_{z}parameter, which corresponds to the minimum error value from the set of RMS

_{i}values. An important point in the calculation is the construction of a test data polygon. In the case of the coarse σ

_{z}calculation for an ordered test data polygon (built using the self-organizing network), a greater convergence to the optimal solution can be obtained than by using a random spacing of samples. Figure 6 shows the optimal σ and coarse σ

_{z}for ordered and random polygons. These values represent averages for 16 network organizations. In two cases for TF1 (sets S1 and S3), better results were obtained using a random polygon for optimization. In the other cases, calculating σ

_{z}algorithmically using an ordered test polygon proved to be closer to the optimum solution. This approach significantly reduced the calculation time for the next stage.

_{z}.

#### 2.3. Neural Network Optimization for Geodata Reduction

_{i}. After several iterations, a training simple is selected, and the index q of the winning node is defined as

^{s}is the learning rate for the sth training stage [22].

## 3. Results and Discussion

#### 3.1. Interpolation of Bathymetric Data

_{1}and S

_{3}. These results confirm the rule that with less data for the surface creation process and a more irregular surface shape, worse results are obtained. The dependence of RMS error on hidden neuron reduction with MQ RBF for all the cases studied is presented in Figure 9.

#### 3.2. Reduction of Bathymetric Data

^{2}and contained 2,077,651 real measuring points. The minimum depth was 0.55 m and the maximum was 12 m. By using this created method, the set was reduced for three scales: 1:500, 1:1000, and 1:2000. Three sets of bathymetric geodata were obtained after reduction, which contained the following number of measuring points:

- for scale 1:500—4036 points XYZ (minimum depth 0.55 m and maximum depth 10.79 m);
- for scale 1:1000—1306 points XYZ (minimum depth 0.55 m and maximum depth 10.65 m);
- for scale 1:2000—497 points XYZ (minimum depth 0.55 m and maximum depth 10.56 m).

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Surfaces used during the tests: (

**TF 1**) Surface 1 obtained using function (1); (

**TF 2**) Surface 2 obtained using function (2); (

**TF 3**) Surface 3 obtained using function (3).

**Figure 5.**The organization of hidden neurons using the k-means algorithm, shown either (

**a**) without application of the neuron fatigue mechanism, or (

**b**) with the neuronal fatigue mechanism applied. Neurons are marked with the ‘o’ symbol and measuring points with ‘+’.

**Figure 7.**Clusters resulting from the use of: (

**a**) grid topology; (

**b**) hexagonal topology; (

**c**) random topology.

**Figure 8.**Influence of neuron reduction in the hidden layer of radial neural networks on the root mean square error (RMS) of the test surfaces and data sets (4000 epochs).

**Figure 12.**Surfaces modelled without reduction of hidden neurons (

**a**) and with reduction to 30 hidden neurons (

**b**).

Radial Network | |
---|---|

Number of input, hidden, and output layers | 1 |

Number of neurons in input layer/transfer function type | 2/linear |

Number of neurons in output layer/transfer function type | 1/linear |

Number of neurons in hidden layer/transfer function type | various/multiquadric |

Training set pre-processing method | Min-Max normalization |

Training algorithm | k-means with neuron fatigue mechanism |

Min Depth [m] | Max Depth [m] | Mean Depth [m] | Max Error [m] | Mean Error [m] | |
---|---|---|---|---|---|

TF1, scale 1:500 | 5.00 | 29.73 | 14.92 | 0.5932 | 0.0118 |

TF1, scale 1:1000 | 5.00 | 29.55 | 14.69 | 1.9475 | 0.0330 |

TF1, scale 1:2000 | 5.00 | 29.56 | 14.27 | 3.2506 | 0.1370 |

TF2, scale 1:500 | 4.52 | 15.11 | 9.71 | 0.3575 | 0.0058 |

TF2, scale 1:1000 | 4.52 | 15.11 | 9.60 | 1.3723 | 0.0134 |

TF2, scale 1:2000 | 4.52 | 15.11 | 9.37 | 2.5310 | 0.0473 |

TF3, scale 1:500 | 4.88 | 30.27 | 12.13 | 0.7603 | 0.0061 |

TF3, scale 1:1000 | 4.88 | 30.15 | 11.88 | 1.3959 | 0.0135 |

TF3, scale 1:2000 | 4.88 | 29.44 | 11.55 | 1.5159 | 0.0499 |

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**MDPI and ACS Style**

Wlodarczyk-Sielicka, M.; Lubczonek, J.
The Use of an Artificial Neural Network to Process Hydrographic Big Data during Surface Modeling. *Computers* **2019**, *8*, 26.
https://doi.org/10.3390/computers8010026

**AMA Style**

Wlodarczyk-Sielicka M, Lubczonek J.
The Use of an Artificial Neural Network to Process Hydrographic Big Data during Surface Modeling. *Computers*. 2019; 8(1):26.
https://doi.org/10.3390/computers8010026

**Chicago/Turabian Style**

Wlodarczyk-Sielicka, Marta, and Jacek Lubczonek.
2019. "The Use of an Artificial Neural Network to Process Hydrographic Big Data during Surface Modeling" *Computers* 8, no. 1: 26.
https://doi.org/10.3390/computers8010026