# Using Artificial Neural Networks for the Estimation of Subsurface Tidal Currents from High-Frequency Radar Surface Current Measurements

^{*}

## Abstract

**:**

^{2}(0.98), mean absolute error (0.05 ms

^{−1}), and the Nash–Sutcliffe efficiency (0.98). The method demonstrates its high prediction ability using only 2 weeks of training data to predict subsurface currents up to 6 months in the future, whilst a constant surface current input is available. The resulting current predictions can be used to calculate flow power, with only a 0.4% mean error. The method is shown to be as accurate as harmonic analysis whilst requiring comparatively few input data and outperforms harmonics by identifying non-celestial influences; however, the model remains site specific.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Site and Datasets

#### 2.2. Metrics Used in Neural Network

#### 2.3. Neural Network Creation and Analysis

^{2}), root mean square error (RMSE), mean error (ME) (to show prediction bias), mean absolute error (MAE), mean absolute percentage error (MAPE), and the Nash–Sutcliffe hydrological efficiency (used to assess the predictive power of hydrological models where the output ranges from −∞ to 1, and where E = 1 would be a perfect match) [34].

## 3. Results

#### 3.1. Performance of Neural Network Structures

^{2}, and E values, along with its notably lower error values. In addition, the BR model was carried forward due to its reduced complexity whilst having the best capability, along with its smoother velocity profile predictions shown in 3.2. Results for the north component ANN showed slightly inferior performance, but still very high. BR with one hidden neuron was also the best architecture for the north component (Table 2).

#### 3.2. Vertical Velocity Profile

^{−1}, compared to 0.824 ms

^{−1}measured. r and ME could not be calculated above 29 m where the ADCP data had been removed.

#### 3.3. Time Series Prediction

#### 3.4. Total Velocity and Direction

#### 3.5. Tidal Power

^{−2}over the 3.5 months. The inclusion of the cosine response showed that the tides have a very high angular fidelity, showing only 0.58 Wm

^{−2}decrease in power. The power–frequency plot (Figure 8) shows high similarity between network and measured values, showing that using the network predicted or measured values may have little impact on power prediction.

#### 3.6. Application to Other Areas

^{−1}, this translated to a large difference in the resulting mean power; 29.73 Wm

^{−2}measured and 41.25 Wm

^{−2}predicted.

^{−1}and reducing mean power difference to 4.8 Wm

^{−2}(36.83 Wm

^{−2}measured and 41.59 Wm

^{−2}predicted), peaks were also suppressed, although still higher than the ADCP-W predictions, shown by the far higher MAPE (Table 5).

^{−1}(14.60% error). The network similarly overpredicts velocities during ebb, contributing to the overall negative mean error of the network.

## 4. Discussion

#### 4.1. Neural Network Performance and Behavior-ADCP-W

#### 4.1.1. Current Velocity Time Series

^{−1}at peak ebbing tide. As found previously in a wide variety of problems, the LM functions, including BR, which is based on LM, outperform the simple gradient descent and scaled conjugate gradient methods [36,37,38,39]. Based upon r and MAPE values of the BR network (0.99 and 4.63%, respectively), this model can be described as a good predictor for tidal modelling [40]. The high-quality statistics of the BR network in comparison to the measurements show that the HF radar-ANN technique proves a useful tool for analysis of subsurface current time series over any period at the training location.

^{−1}. It appears the network tries to draw from any values in a depth bin, so as the low waters are more often submerged and below the 10% threshold, there is more for the ANN to learn from. At depth bins always in air or the threshold, the network can only predict 0 ms

^{−1}, and heights which are mostly in the upper 10% are inaccurate due to limited data.

#### 4.1.2. Velocity Profiles

^{−1}) than from the seabed to 22 m (ME = 0.0052 ms

^{−1}) (Figure 3a). The likely reason for this is the occurrence of a weather system approaching from the southwest in the first half of the two-week training period [46], the south-westerly winds (with some gusts up to 100 km h

^{−1}) would have accelerated the surface currents during flood tide and reduced the surface velocity during ebb tide, while as the distance from the surface increases, the effects from the wind diminish. The network learns this to be normal. This hypothesis can be complemented by plotting the mean spring ebb current profile predicted by the network (Figure 10). The underprediction of the surface currents nearest to the surface suggests that the abnormally strong south-westerly wind could be slowing the ebb surface current.

#### 4.1.3. Flow Power

#### 4.1.4. Potential of the Radar-Network Technique at a Single Location

^{−1}in the last month in comparison to 0.056 ms

^{−1}in the first two and half months. This higher-than-average MAE could be due to the long time since the training period, where long-period harmonics may have altered the tide, or the approaching winter could bring about different coastal conditions to the summer training period.

#### 4.2. ADCP-E and Beyond

^{2}= 0.84–0.95) unless complex bathymetrical variation occurs. This is precisely where the ANN falters on ADCP-E. The difference in morphology between sites is such that at ADCP-W, the mean spring tide measured by the radar at the surface above the ADCP is 0.877 ms

^{−1}, whereas the highest-depth bin from the ADCP where there is accurate data have a higher mean velocity at spring tide of 0.928 ms

^{−1}. Therefore, the network has learned to predict higher velocities below the surface than the surface data given as the input. In contrast, at ADCP-E, the mean spring surface current measured by the radar is 0.651 ms

^{−1}whilst the ADCP bin highest in the water column has a mean spring velocity of 0.582 ms

^{−1}, lower than the surface velocity. The network has not seen this relationship before and acts as it did at ADCP-W, causing a large overprediction at ADCP-E. Naturally, to test if the error was due to external factors, the network method was reversed, being trained using ADCP-E data and tested on ADCP-W, the network underpredicted ADCP-W currents by a similar bias. Inter-site bathymetric and coastal morphology differences are why almost all tidal range ANN research has concluded the network can only be used for single location estimation [22,23,24,25,26], while research which has attempted multi-point prediction uses astronomical data as network inputs and concludes that prediction error increases with distance and dissimilarity from the training location [52].

^{−1}because it is trained on data where this depth is within the 10% threshold. The error is lowest near the seabed, as water–seabed interactions will be similar at each site but increase with height where differing surface interactions also play an increasing role.

#### 4.3. Comparison to Harmonic Analysis

^{2}of 0.978 and MAE of 0.048 ms

^{−1}in comparison to the harmonic predictions r

^{2}of 0.974 and MAE of 0.058 ms

^{−1}when compared to the ADCP data. Overall, this analysis proves the ADCPs worth in the measurement of small scale non-tidal and large one-off currents in the resource assessment. It also shows that this HF radar and ANN technique is a proficient tool for the long-term prediction of subsurface currents with few data records required, instead of prediction via harmonic analysis.

_{2}and M

_{4}generated tidal currents to combine [56], causing a stronger tide in one direction. The network was able to pick up on this asymmetry as it is an often-recurring feature. However, this resulted in the network predicting a strong asymmetry, sometimes up to 0.2 ms

^{−1}on every ebb tide, whereas the asymmetry was more varied in the measured data. A potential reason for this failing is the recurrent architecture of the network. It uses the happenings at the previous time steps for the prediction of the next time step [27], meaning that at the approach to peak ebb tide, the rate of decrease in velocity may have been such that the network did not expect the peak velocity to occur so soon, hence the overprediction. Despite its failing, it does show the network’s ability to identify asymmetry at a site which is useful for tidal site characterisations as the deployment of tidal developments in asymmetric regions is likely to have a more pronounced impact on sediment dynamics [57,58]

## 5. Conclusions

^{2}= 0.98, STD = 0.46 ms

^{−1}, MAPE = 4.63%).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Map showing the locations of the HF radars at Pendeen and Perranporth and their coverage. The red squares represent ADCP west and east. The rectangle represents the WaveHub test site.

**Figure 2.**Scatterplot of the measured vs. predicted east velocities at 20 m above the seabed by BR, around the exact fit line.

**Figure 3.**(

**a**) Current velocity profile using the BR training function. (

**b**) Current velocity profile using the GDA function. Blue: measured values, orange: network prediction, and black: mean water level.

**Figure 4.**(

**a**) Plot of the predicted current velocity throughout the water column. (

**b**) Error between the predictions and measured data. Red line = water depth. Top 10% is white despite the ANNs prediction as the data were removed in the ADCP comparison due to sidelobe interference.

**Figure 5.**(

**a**) Enhanced comparison of measured velocity variation. (

**b**) Network predicted velocity variation. Red line = water depth. White space below red line is the location of the inaccurate ADCP data.

**Figure 7.**Extract of time series of measured and predicted current power. Blue: measured values, and orange: network prediction.

**Figure 8.**ADCP measured and neural network predicted frequency of tidal power occurrence separated into 50 Wm

^{−2}bins. Blue: measured values, and orange: network prediction.

**Figure 9.**Velocity profile at ADCP-E, trained by SCG method using ADCP-W. Blue: measured values, and orange: network prediction.

**Figure 10.**Velocity profile at ADCP-W averaged over 16 spring ebb tides using the BR network. Blue: measured values, and orange: network prediction.

**Figure 11.**(

**a**) Network predicted current, alongside harmonic prediction and residual current between the two. (

**b**) Measured tides, alongside harmonic prediction, and residual tide between the two. Blue: harmonic prediction, orange: network prediction, black: measured values, and red: residual value.

**Table 1.**Best-performing east network for each training function with their respective statistics, rounded to three decimal places.

Network Function | Hidden Layers | r | r^{2} | STD (ms^{−1}) | RMSE (ms^{−1}) | ME (ms ^{−1}) | MAE (ms^{−1}) | MAPE (%) | E |
---|---|---|---|---|---|---|---|---|---|

GDA | 22 | 0.986 | 0.972 | 0.514 | 0.082 | 0.0114 | 0.060 | 9.915 | 0.971 |

SCG | 27 | 0.987 | 0.974 | 0.511 | 0.081 | 0.0036 | 0.054 | 1.372 | 0.974 |

LM | 1 | 0.988 | 0.976 | 0.511 | 0.081 | 0.0037 | 0.056 | 2.757 | 0.975 |

BR | 1 | 0.989 | 0.978 | 0.464 | 0.067 | 0.002 | 0.048 | 4.630 | 0.979 |

**Table 2.**Best-performing network for north component with statistics. MAPE = N/A due to zeros in data, rounded to three decimal places.

Network Function | Hidden Layers | r | r^{2} | STD (ms^{−1}) | RMSE (ms^{−1}) | ME (ms ^{−1}) | MAE (ms^{−1}) | MAPE (%) | E |
---|---|---|---|---|---|---|---|---|---|

GDA | 22 | 0.961 | 0.924 | 0.247 | 0.127 | −0.018 | 0.059 | N/A | 0.917 |

SCG | 27 | 0.975 | 0.951 | 0.251 | 0.050 | −0.008 | 0.046 | N/A | 0.949 |

LM | 1 | 0.982 | 0.964 | 0.263 | 0.043 | −0.008 | 0.039 | N/A | 0.963 |

BR | 1 | 0.979 | 0.958 | 0.239 | 0.050 | −0.005 | 0.0037 | N/A | 0.958 |

**Table 3.**r and ME of the different trained functions on the predictions of vertical velocity profiles.

Function | GDA | SCG | LM | BR |
---|---|---|---|---|

r | 0.925 | 0.989 | 0.996 | 0.997 |

ME | −0.0519 | −0.0170 | 0.0155 | 0.0087 |

**Table 4.**Measured mean power before accounting for the angle of current (raw power), considering incident angle (measured), and each training method’s best-performing networks prediction of mean power and error.

Mean Power (Wm^{−2}) | Network Percentage Error from Mean Measured (%) | Max Power (Wm^{−2}) | |
---|---|---|---|

Raw power | 127.72 | - | 904.95 |

Measured | 127.14 | - | 904.82 |

GDA | 124.03 | −2.48 | 928.03 |

SCG | 126.17 | −0.77 | 777.68 |

LM | 128.10 | 0.75 | 931.42 |

BR | 126.63 | −0.40 | 875.88 |

Network Function | Hidden Layers | r | r^{2} | STD (ms^{−1}) | RMSE (ms^{−1}) | ME (ms^{−1}) | MAE (ms^{−1}) | MAPE (%) | E |
---|---|---|---|---|---|---|---|---|---|

SCG | 27 | 0.976 | 0.953 | 0.377 | 0.079 | −0.013 | 0.054 | 10.68 | 0.941 |

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Bradbury, M.C.; Conley, D.C.
Using Artificial Neural Networks for the Estimation of Subsurface Tidal Currents from High-Frequency Radar Surface Current Measurements. *Remote Sens.* **2021**, *13*, 3896.
https://doi.org/10.3390/rs13193896

**AMA Style**

Bradbury MC, Conley DC.
Using Artificial Neural Networks for the Estimation of Subsurface Tidal Currents from High-Frequency Radar Surface Current Measurements. *Remote Sensing*. 2021; 13(19):3896.
https://doi.org/10.3390/rs13193896

**Chicago/Turabian Style**

Bradbury, Max C., and Daniel C. Conley.
2021. "Using Artificial Neural Networks for the Estimation of Subsurface Tidal Currents from High-Frequency Radar Surface Current Measurements" *Remote Sensing* 13, no. 19: 3896.
https://doi.org/10.3390/rs13193896