# Improvement of an Operational Forecasting System for Extreme Tidal Events in Santos Estuary (Brazil)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Study Area

^{2}basin area [4]. The estuarine system of Santos consists of three major channels, namely S. Vicente, Santos, and Bertioga, interconnected in the inner area. The Santos and S.Vicente channels cover an approximate area of 44.100 m

^{2}, with an average depth of 15 m in the central dredged channel of Santos and 8 m in S.Vicente channel [1].

^{3}/s average flow. The main forcing agent driving water circulation is the tide, with an average amplitude of 1.43 m (measured at Santos port), being considered a micro-tidal estuary. The estuary is partially mixed to stratified, where the salt transport within the estuary is due to the up-estuary propagation of the salt wedge and eddy diffusion [6], inducing changes in circulation patterns.

_{3}(period of 8.28 h), related to tidal current asymmetries. This constituent amplification was first described by [9], the largest amplitudes being found in the Paranaguá estuarine system [10]. Distortions are found in every two tidal cycles, indicating this component relates to the daily inequalities in the tidal amplitudes [11].

## 2. Materials and Methods

#### 2.1. Hydrodynamic Model

#### 2.1.1. Governing Equations

_{i}is the horizontal velocity vector (i = 1,2 directions) and u

_{j}the 3D velocity vector (j = 1,2,3), $\eta $ the water level, v the turbulent viscosity, p

_{atm}the atmospheric pressure, ρ is the density, ρ

_{0}is the reference density, ρ(η) represents the density at the free surface, g is the gravity acceleration, t is the time, Ω is the earth velocity of rotation and ε is the alternate tensor.

#### 2.1.2. Model Domains

#### 2.2. Water-Level and Wave Data

#### 2.3. Statistical Analysis of Water Level

#### 2.3.1. Astronomical Tide

_{1}, O

_{1}, Q

_{1}, P

_{1}), lunar (M

_{2}, S

_{2}, N

_{2}, K

_{2}) and non-linear (M

_{3}, M

_{4}) constituents. Having both an amplitude (A

_{i}) and phase lag (G

_{i}) of each constituent, the tide height (H) can be calculated at any time (t) as the sum of contributions of each tidal constituent by the following equation:

_{0}represents the mean water level; k the number of tidal constituents; i the index of a constituent; ω

_{i}the angular velocity and (V

_{0}+ υ)

_{i}the astronomical argument. F

_{i}and u

_{i}can be intended as the amplitude and phase corrections.

_{i}) is calculated to compare the amplitude and phase of each harmonic constituent, and also the relative value of the HC (RHC

_{i}) parameter.

#### 2.3.2. Residual Tide

_{0}+ T(t)) are the observed and synthesized level, respectively.

#### 2.3.3. Statistical Parameters

_{obs}and X

_{mod}represent observations and model predictions, respectively.

_{obs}and X

_{mod}, the Correlation Coefficient (R) is expressed as:

#### 2.4. Optimizing Boundary Conditions

_{3}and M

_{4}) would be expected. The CMEMS (meteorological tide), imposed in the model as an oceanic boundary condition, was tested with an hourly resolution, hypothesizing that the water level and barotropic velocity daily solution, currently in use, may not be sufficient to forecast some intense residual tide events during short time periods.

#### 2.5. Study the Relation among Physical Variables

## 3. Results and Discussion

#### 3.1. Data Analysis

#### 3.2. Astronomical Tide Analysis

_{2}, is well reproduced, presenting an amplitude error of 0.21 cm and a phase lag of 6.48° (13 minutes). Highest discrepancies are found for S

_{2}(amplitude error of 2.84 cm and a phase lag of 9.11° (18 minutes)), M

_{3}(3.14 cm and comprehending a phase lag of 22.94° (32 minutes)) and M

_{4}(1.51 cm with a phase lag of 83.82° (1h27 minutes)) constituents. For S

_{2}constituents, discrepancies appear to be related to boundary conditions imposed, since predictions are higher than observations at the S1 station. For the M

_{3}constituent, model underestimation of amplitude may reflect global astronomical solution uncertainties related to Brazilian regional features. It is important to mention that regarding the S3, S4 and S5 stations, M

_{3}is the sixth most important constituent, proving this constituent amplification importance on the Brazilian coast. Regarding M

_{4}, the model overestimates the amplitude from the bay entrance. Further errors may reflect the bathymetric issues or arise from M

_{2}uncertainties. Regarding diurnal constituents, very good agreement is found in terms of amplitude and phase lag, except P

_{1}constituent presenting a phase lag of 19.8° (1hr19 minutes), being this lag amplified throughout the inner estuary. Generally, phase lags and amplitude differences increase towards S5.

_{2}in terms of amplitude is ~44% for all semi-diurnal constituents considered (results not shown). The significance of third-diurnal and quarter diurnal constituents is noticeable, being M

_{3}responsible for 40% of total third-diurnal, whilst M

_{4}contributes with 29% total quarter-diurnal amplitudes (results not shown). The importance of the non-linear terms by the amplification of third and quarter-diurnal constituents throughout the Santos estuary is clearly observed. A similar pattern was reported by [10] for the Paranaguá estuarine system. The harmonic analysis results (Figure 5) show HC intensification as the distance to the estuary mouth increases, regarding predictions and observations. This can be explained by the bathymetric uncertainties in the vast inter-tidal mangrove area of the inner estuary.

_{2}reaches a maximum difference at S5 (0.157 cm). HC values at S1 are less than 0.02 cm for the main constituents, except for S2 (0.33 cm) and for the shallow water constituents M

_{3}and M

_{4}(0.34 cm and 0.36 cm, respectively). Relative to RHC, major errors were found for the main non-linear constituents (M

_{3}and M

_{4}). In general, for higher amplitude constituents, amplitude and phase difference between observations and predictions are lower. Moreover, differences in phase are observed for the shallow water constituents. This analysis concluded the main harmonics are reasonably reproduced since RHC are less than 10%. Conversely, for constituents P

_{1}(RHC > 30%), M

_{3}(RHC > 50%), and M

_{4}(RHC > 130%) the largest discrepancies were found. These results show in a clear way that for the FES2012 the M

_{3}and M

_{4}present a large error in the coastal area (MOHID boundary).

#### 3.3. Residual Tide Analysis

#### 3.4. Influence of Oceanic Boundary Conditions

#### 3.4.1. FES2014 Solution

_{4}constituent regarding amplitude and phase values (RHC < 30%), considering the statistical parameters applied. There is a slight improvement in S

_{2}constituent. The M

_{3}inaccuracies are not solved with FES2014.

#### 3.4.2. Copernicus Marine Environment Monitoring Service (CMEMS) Hourly Solution

#### 3.5. Investigating Wave Influence on Residual Tide Errors

#### Post-Processing Correction

_{s}, residuals observations are higher, unlike for the predictions of residuals. For the extreme H

_{s}events (maximum heights of 4.01 and 4.25 m), the highest errors were found (0.356 and 0.496 cm). A regression technique was applied, considering the H

_{s}and residual error the independent and dependent parameters, respectively (Figure 9).

## 4. Conclusions

_{3}constituent, which is a very particular feature of the Brazilian shelf. Nevertheless, residual errors do not show significant differences along the Santos estuary. The individual study of water level components allowed the improvement of the forecast capacity for each component (astronomic and residual).

_{2}and M

_{4}constituents). However, some inaccuracies persist (e.g., M

_{3}constituent), as errors remain in shallow waters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of the study area with the location of the Port of Santos. Adapted from: [5].

**Figure 2.**Hydrodynamic numerical grids (

**top**) and bathymetry (

**bottom**) with 4 levels (L1, L2, L3, L4); the color scale represents the depth in meters. Adapted from [5].

**Figure 4.**Harmonic constants (amplitude (m) and phase (°)) determined from model predictions and observations, with error bars in the tidal gauges located in the Santos estuary.

**Figure 5.**Mean complex amplitude error (HC,

**top**) in m and Relative mean complex amplitude error (RHC,

**bottom**) in % error for the 5 tide gauges in Santos estuary for 2016–2017 using FES2012.

**Figure 6.**Mean complex amplitude error (HC,

**top**) in m and Relative mean complex amplitude error (RHC,

**bottom**) in % error for the 5 tide gauges in Santos estuary for 2016–2017 using FES2014.

**Figure 7.**Residual tide observed for the S1 station (observations) and Copernicus Marine Environment Monitoring Service (CMEMS, predicted): daily and hourly resolution for 3 distinct time periods (18/04/2016 to 08/05/2016; 10/08/2016 to 28/08/2016; 15/09/2016 to 05/10/2016).

**Figure 8.**Comparison between observed significant wave height and residual tide determined for S1 station.

**Figure 10.**Indices of Hs (

**grey**) higher than 1.5 m, model predictions (

**dark blue**), the model with a correction factor (

**light blue**) and observations (

**orange**) of the residual tide for the year of 2016–2017.

Tidal Gauge | Designation | Latitude | Longitude | Parameter |
---|---|---|---|---|

Ilha das Palmas | S1 | 24°00′81.86″ S | 46°32′50.94″ W | Wave data; Water level |

Praticagem | S2 | 23°99′14.11″ S | 46°30′17.67″ W | Water level |

Capitania | S3 | 23°95′56.79″ S | 46°30′81″ W | Water level |

Ilha Barnabé | S4 | 23°92′26.33″ S | 46°33′35″ W | Water level |

Cosipa | S5 | 23°87′08.84″ S | 46°37’81.53″ W | Water level |

**Table 2.**Statistical parameters (R (Correlation Coefficient), RMSE (Root Mean Square Error), UNBIAS RMSE, BIAS, and S (Skill)) (cm) for water level for the 5 tide gauges in Santos estuary for 2016–2017.

Station | Model Domain | R | RMSE | UNBIAS RMSE | BIAS | S |
---|---|---|---|---|---|---|

S1 | MOHID-Level 4 | 0.95 | 12.9 | 12.5 | 3.5 | 0.97 |

S2 | 0.95 | 17.9 | 12.8 | −13.5 | 0.95 | |

S3 | 0.94 | 15.1 | 14.2 | −5.0 | 0.97 | |

S4 | 0.93 | 17.3 | 16.9 | −3.8 | 0.96 | |

S5 | 0.90 | 19.9 | 19.7 | −2.7 | 0.94 |

**Table 3.**Statistical parameters (R, RMSE, UNBIAS RMSE, and S) (cm) for astronomical tide for the 5 tide gauges in Santos estuary for 2016-2017.

Station | Parameter | ||
---|---|---|---|

R | UNBIAS RMSE | S | |

S1 | 0.98 | 7.3 | 0.99 |

S2 | 0.98 | 6.9 | 0.99 |

S3 | 0.97 | 9.9 | 0.98 |

S4 | 0.95 | 12.9 | 0.97 |

S5 | 0.93 | 17.4 | 0.94 |

**Table 4.**Harmonic constituents’ amplitude (cm) and phase (°) differences for the 5 tide gauges in Santos estuary for 2016–2017 for FES2012 and FES2014 solutions.

Constituents | S1 | S2 | S3 | S4 | S5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

M_{2} | Obs | 36.6 | 161.9 | 35.8 | 167.7 | 38.9 | 170.6 | 42.8 | 172.5 | 33.9 | 175.6 |

Pre | 36.2 | 167.8 | 37.0 | 170.4 | 40.9 | 176.7 | 43.4 | 181.4 | 46.0 | 185.3 | |

S_{2} | Obs | 23.4 | 169.8 | 22.7 | 175.0 | 24.8 | 179.1 | 27.4 | 181.8 | 21.6 | 184.8 |

Pre | 25.3 | 177.3 | 25.8 | 179.4 | 28.3 | 187.1 | 29.9 | 192.5 | 31.6 | 197.3 | |

O_{1} | Obs | 11.1 | 123.2 | 11.2 | 124.8 | 11.4 | 127.5 | 11.5 | 128.0 | 11.4 | 133.1 |

Pre | 11.1 | 123.2 | 11.1 | 124.2 | 11.4 | 126.1 | 11.5 | 128.0 | 11.6 | 129.7 | |

K_{2} | Obs | 7.3 | 160.7 | 7.2 | 166.0 | 7.9 | 168.8 | 8.7 | 171.7 | 6.3 | 178.0 |

Pre | 6.9 | 165.8 | 7.0 | 168.2 | 7.7 | 175.7 | 8.1 | 181.2 | 8.5 | 185.9 | |

K_{1} | Obs | 6.5 | 183.9 | 6.5 | 6.5 | 6.6 | 185.3 | 7.0 | 182.1 | 5.5 | 186.6 |

Pre | 6.5 | 184.8 | 6.6 | 6.6 | 6.7 | 187.5 | 6.8 | 189.2 | 6.8 | 190.5 | |

N_{2} | Obs | 4.9 | 215.4 | 4.9 | 4.9 | 5.4 | 225.1 | 5.9 | 227.4 | 4.6 | 232.8 |

Pre | 4.6 | 226.9 | 4.7 | 4.7 | 5.1 | 235.3 | 5.3 | 239.9 | 5.6 | 244.0 | |

Q_{1} | Obs | 3.2 | 93.8 | 3.2 | 3.2 | 3.1 | 98.5 | 3.6 | 97.7 | 2.6 | 102.0 |

Pre | 3.1 | 99.4 | 3.1 | 3.1 | 3.2 | 101.9 | 3.2 | 103.7 | 3.3 | 105.5 | |

P_{1} | Obs | 2.4 | 173.9 | 2.4 | 2.4 | 2.6 | 177.2 | 2.9 | 191.6 | 2.1 | 184.0 |

Pre | 2.1 | 193.7 | 2.2 | 2.2 | 2.2 | 198.9 | 2.6 | 202.2 | 2.3 | 205.1 | |

M_{3} | Obs | 4.8 | 325.0 | 4.8 | 4.8 | 5.5 | 340.8 | 6.5 | 347.0 | 5.1 | 352.0 |

Pre | 1.7 | 302.1 | 1.7 | 1.7 | 2.0 | 320.1 | 2.2 | 328.5 | 2.4 | 335.4 | |

M_{4} | Obs | 1.9 | 149.1 | 1.6 | 1.6 | 2.3 | 153.7 | 5.6 | 143.9 | 2.4 | 149.5 |

Pre | 3.5 | 232.9 | 2.8 | 2.8 | 2.7 | 277.7 | 2.9 | 302.7 | 3.4 | 317.1 |

**Table 5.**Sum of the amplitudes (m) and relative importance (%) of the main tidal constituents in different frequency bands.

Constituents | S1 | S2 | S3 | S4 | S5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Long Period | 0.16 | 11% | 0.19 | 13% | 0.20 | 12% | 0.23 | 13% | 0.18 | 12% |

Diurnal | 0.29 | 19% | 0.29 | 19% | 0.31 | 19% | 0.36 | 20% | 0.26 | 18% |

Semi-Diurnal | 0.82 | 55% | 0.81 | 54% | 0.88 | 53% | 0.98 | 54% | 0.78 | 54% |

Third-Diurnal | 0.12 | 8% | 0.12 | 8% | 0.14 | 8% | 0.16 | 9% | 0.13 | 9% |

Quarter-Diurnal | 0.07 | 5% | 0.06 | 4% | 0.08 | 5% | 0.11 | 6% | 0.09 | 6% |

Others | 0.03 | 2% | 0.03 | 2% | 0.04 | 2% | 0.06 | 3% | 0.06 | 4% |

Total | 1.49 | 100% | 1.50 | 100% | 1.66 | 100% | 1.80 | 100% | 1.44 | 100% |

**Table 6.**Statistical parameters (R, UNBIAS RMSE, and S) (cm) for residual tide determined for the 5 tide gauges in Santos estuary for 2016–2017.

Station | Parameter | ||
---|---|---|---|

R | UNBIAS RMSE | S | |

S1 | 0.87 | 11.0 | 0.90 |

S2 | 0.89 | 9.68 | 0.92 |

S3 | 0.89 | 12.1 | 0.92 |

S4 | 0.91 | 10.3 | 0.92 |

S5 | 0.85 | 9.48 | 0.91 |

**Table 7.**The correlation coefficient (R) between residual tide (predictions and observations) and Hs (observation).

Station | Parameter | |
---|---|---|

Dataset | R | |

S1 | Obs | 0.64 |

Pre | 0.61 | |

S2 | Obs | 0.65 |

Pre | 0.61 | |

S3 | Obs | 0.64 |

Pre | 0.61 | |

S4 | Obs | 0.65 |

Pre | 0.62 | |

S5 | Obs | 0.62 |

Pre | 0.61 |

**Table 8.**Statistical analysis (UNBIAS RMSE) of the residual tide (cm) for total time series and for Hs higher than 1.5 m.

Hs (m) | Pre (m) | Obs (m) | Error (m) |
---|---|---|---|

1.60 | 0.157 | 0.276 | 0.118 |

1.62 | 0.160 | 0.243 | 0.084 |

1.73 | 0.060 | 0.123 | 0.064 |

1.75 | 0.202 | 0.468 | 0.266 |

1.76 | 0.320 | 0.542 | 0.222 |

1.77 | 0.006 | 0.241 | 0.235 |

1.81 | 0.129 | 0.482 | 0.353 |

1.93 | 0.053 | 0.128 | 0.074 |

2.0 | −0.063 | 0.105 | 0.168 |

2.0 | 0.040 | 0.182 | 0.142 |

2.12 | 0.232 | 0.427 | 0.196 |

2.19 | 0.102 | 0.40 | 0.298 |

2.31 | 0.130 | 0.461 | 0.331 |

2.34 | 0.227 | 0.483 | 0.255 |

2.37 | 0.485 | 0.655 | 0.170 |

2.37 | 0.242 | 0.49 | 0.249 |

2.38 | 0.046 | 0.238 | 0.192 |

2.47 | 0.182 | 0.383 | 0.20 |

2.72 | 0.307 | 0.584 | 0.277 |

2.79 | 0.191 | 0.373 | 0.182 |

2.83 | 0.345 | 0.542 | 0.197 |

3.19 | 0.517 | 0.897 | 0.381 |

2.27 | 0.315 | 0.648 | 0.332 |

4.01 | 0.496 | 0.852 | 0.356 |

4.25 | 0.279 | 0.775 | 0.496 |

**Table 9.**Statistical analysis (UNBIAS RMSE) of the residual tide for Hs higher than 1.5 m with and without correction factor.

Dataset | Model | UNBIAS RMSE | UNBIAS RMSE (Hs >1.5) | Difference |
---|---|---|---|---|

Residual tide | CMEMS daily | 12.10 | 23.77 | 11.67 |

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## Share and Cite

**MDPI and ACS Style**

Mendes, J.; Leitão, P.; Chambel Leitão, J.; Bartolomeu, S.; Rodrigues, J.; Dias, J.M.
Improvement of an Operational Forecasting System for Extreme Tidal Events in Santos Estuary (Brazil). *Geosciences* **2019**, *9*, 511.
https://doi.org/10.3390/geosciences9120511

**AMA Style**

Mendes J, Leitão P, Chambel Leitão J, Bartolomeu S, Rodrigues J, Dias JM.
Improvement of an Operational Forecasting System for Extreme Tidal Events in Santos Estuary (Brazil). *Geosciences*. 2019; 9(12):511.
https://doi.org/10.3390/geosciences9120511

**Chicago/Turabian Style**

Mendes, Joana, Paulo Leitão, José Chambel Leitão, Sofia Bartolomeu, João Rodrigues, and João Miguel Dias.
2019. "Improvement of an Operational Forecasting System for Extreme Tidal Events in Santos Estuary (Brazil)" *Geosciences* 9, no. 12: 511.
https://doi.org/10.3390/geosciences9120511