# Modeling Tidal Datums and Spatially Varying Uncertainty in the Texas and Western Louisiana Coastal Waters

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## Abstract

**:**

## 1. Introduction

## 2. Model, Data, and Methods

#### 2.1. Hydrodynamic Model and Its Configuration

#### 2.1.1. Model Configuration

- (1)
- nonlinear quadratic bottom friction with a spatially constant bottom friction coefficient of 0.002;
- (2)
- a spatially constant horizontal eddy viscosity of 5.0 m
^{2}/s for the momentum equations; - (3)
- wetting and drying process enabled with a minimum water depth of 0.05 m as a wet node/element criterion;
- (4)
- a spatially uniform Generalized Wave-Continuity Equation (GWCE) weighting factor of 0.02;
- (5)
- advective terms were included;
- (6)
- no atmospheric forcing and river flow were imposed;
- (7)
- tidal potential body force of eight principal tidal constituents (K1, O1, P1, Q1, M2, S2, N2, and K2) was included;
- (8)
- water elevations from the same eight principal tidal constituents: K1, O1, P1, Q1, M2, S2, N2, and K2 were used at the open ocean boundary. That is, open ocean boundary forcing equals the sum of the elevations of the eight tidal harmonic constituents, which were extracted from the EC2015 tidal database [20,21].

- (9)
- a total of 67 days of the ADCIRC model run. A hyperbolic tangent ramp function was specified, and the beginning six days were used to ramp up ADCIRC forcings from zero. The time step for the ADCIRC model run is 3 s. The output from the ADCIRC model run is the 6-min water level time series at each model grid point from the final 60-day run, which were used for computing tidal datums at each model grid point.

#### 2.1.2. Model Domain

#### 2.2. Data

#### 2.2.1. NOAA’s Continually Updated Shoreline Product (CUSP)

#### 2.2.2. Bathymetry Data

#### 2.2.3. Observed Tidal Datums and Associated Root-Mean-Square (RMS) Errors

#### 2.3. Methods

#### 2.3.1. Calculation of Observed Tidal Datums

#### 2.3.2. Calculation of ADCIRC Tidal Datums

#### 2.3.3. Statistical Interpolation of Tidal Datums and Their Associated Spatially Varying Uncertainties

^{2}). ${\sigma}_{{n}_{1}}$ and ${\sigma}_{{n}_{2}}$ are the standard deviations of the model errors at nodes ${n}_{1}$ and ${n}_{2}$, respectively, and are assumed to be constant at all the model nodes which were equal to the standard deviation of the modeled errors at all the tide stations. The correlation between two points is calculated using a three-day moving average tidal datum time series. Here, the covariance is adjusted and decreases exponentially over the distance between nodes ${n}_{1}$ and ${n}_{2}$. Also, the weight matrix $W$ determines the weight of $R$ in the computation of the analysis field $f$. The diagonal element ${w}_{ii}$ (0 ≤ ${w}_{ii}$≤ 1, 1 ≤$i$ ≤$m$) is the weight of the observation error variance ${r}_{ii}$ at station $i$ in the determination of analysis field $f$. The weight matrix $W$ was determined through iteration following the predetermined constraint; that is, the discrepancy between the analysis field and the observations at all tide stations is equal to or less than 1 cm or the CO-OPS’s uncertainty value (observed rms error), whichever is less.

#### 2.3.4. Estimates of Non-Tidal Zones and VDatum Marine Grid Population

## 3. Results and Discussion

#### 3.1. Observed Tidal Datums

#### 3.2. The Assessment and Improvement of ADCIRC Modeled Tidal Datums

#### 3.3. Statistical Interpolation of Modeled Tidal Datums and Associated Uncertainties

#### 3.4. Non-Tidal Polygon Upgrade and VDatum Marine Grid Population

#### 3.5. Comparisons of the Updated Tidal Model with the Previous Tidal Model

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Model grid extension into smaller rivers and the Intracoastal Waterway (ICW) with updated bathymetry: before (

**left**) and after (

**right**) the model update. The two pink arrows in the right panel show the two areas with a minor extension of model grids in the middle of the TX coast. The extended model grid points in the two areas are shown as pink dots in the enlarged plot in the right lower corner of the right panel.

**Figure 3.**The locations, types, and years of the new bathymetry data used for the model update. The areas with a pink color represent data other than NOS bathymetry. The region with the extended water paths in the western LA’s ICW area is enlarged in the box in the lower right corner to show details. “A. O.” represents data provided by ACE’s Andrew Oakman.

**Figure 4.**A schematic diagram of the workflow of the model update with the SVU statistical interpolation step enclosed within the box.

**Figure 6.**The mean tidal ranges from the 75 tide stations. The blue polygon includes 43 tide stations from Houston to the east (with a mean tidal ranges of 0.33 m), and the red polygon includes 32 tide stations from Houston to the west (with a mean tidal ranges of 0.23 m).

**Figure 7.**Modeled tidal datums vs observed tidal datums before (

**a**) and after (

**b**) model adjustments. The dashed lines represent 0.10 m error limits.

**Figure 8.**Modeled and observed tidal datums (in units of meters) at Station #8766072 before (

**left**) and after (

**right**) correcting the station’s coordinates. “Obs” and “O” refer to “Observation”. “M” refers to “Model”. “ND ID” refers to “model node (grid point) identification”.

**Figure 9.**Modeled MHHW (

**upper row**) and MHW (

**lower row**) tidal datums. The first column shows ADCIRC modeled tidal datums; the second column shows the tidal datums after the SVU statistical interpolation; the third column shows the associated SVU spatially varying uncertainties. Model grid points in pink represent the modeled non-tidal grid points (modeled MHW-MLW < 0.09 m).

**Figure 10.**Modeled MLW (

**upper row**) and MLLW (

**lower row**) tidal datums. The first column shows ADCIRC modeled tidal datums; the second column shows the tidal datums after the SVU statistical interpolation; the third column shows the associated SVU spatially varying uncertainties. Model grid points in pink represent the modeled non-tidal grid points (modeled MHW-MLW < 0.09 m).

**Figure 11.**Modeled DTL (

**upper row**) and MTL (

**lower row**) tidal datums. The first column shows ADCIRC modeled tidal datums; the second column shows the tidal datums after the SVU statistical interpolation; the third column shows the associated SVU spatially varying uncertainties. Model grid points in pink represent the modeled non-tidal grid points (modeled MHW-MLW < 0.09 m).

**Figure 12.**The concave geographic shape and shallow bathymetry characteristics of the northern Texas-Louisiana shelf within the black box.

**Figure 13.**Non-tidal polygons (closed black lines) before (

**left**) and after (

**right**) the upgrade. Modeled non-tidal grid points are marked as pink dots. Area 1 and Area 2 (within the closed red lines) are the two areas which had major adjustments in the non-tidal upgrade.

**Figure 14.**An example of the marine grid population for the modeled MHHW after the SVU interpolation in the western LA coastal region by using the non-tidal polygons before (

**a**) and after (

**b**) the upgrade.

**Figure 15.**An example of the detailed marine grid field surrounding a narrow water path (light gray dots). The marine grids at land and in water are marked in brown and blue, respectively. The artificially added water layers 1 to 5 are marked in cyan, pink, green, purple, and red, respectively.

**Figure 16.**The locations of the three tide stations (“1”, “2”, and “3”) in the western LA region, which were included in both the previous (

**a**) and current (

**b**) tidal model domains. The coordinates of the tide stations 1, 2, and 3 are [91.3381° W, 29.4496° N], [−91.8800° W, 29.7134° N], and [−91.8800° W, 29.7134° N], respectively. The red dots are the tide stations in the CO-OPS tidal datum data used for this work.

**Table 1.**Statistical values of observed and modeled tidal datums and the associated SVU spatially varying uncertainties (in meters). Model errors from observations are given in parentheses when applicable; note that errors do not necessarily correspond to the categorical value reported next to them (e.g., min, max, mean), but are instead the categorical error over the entire model domain. For example, the maximum model error (0.063) next to the maximum value of ADCIRC modeled MHHW (0.28) refers to the maximum model error of ADCIRC modeled MHHW in comparison with the observations at the 75 tide stations. Note also: Model errors refer to modeled tidal datums minus observed tidal datums; “Mean Value” of ADCIRC model errors refers to Mean (Abs(ADCIRC model error)); “Mean Value” of ADCIRC-SVU model errors refers to Mean (Abs(ADCIRC-SVU model error)); “STD” stands for Standard Deviation.

Data Type | MHHW | MHW | MLW | MLLW |
---|---|---|---|---|

Maximum Value | ||||

Observation | 0.32 | 0.26 | −0.25 | −0.38 |

ADCIRC | 0.28 (0.063) | 0.24 (0.067) | −0.31 (0.072) | −0.43 (0.132) |

ADCIRC-SVU | 0.31 (0.010) | 0.25 (0.010) | −0.27 (0.010) | −0.45 (0.010) |

SVU Uncertainty | 0.036 | 0.033 | 0.034 | 0.046 |

Minimum Value | ||||

Observation | 0.05 | 0.05 | −0.05 | −0.05 |

ADCIRC | 0.03 (−0.010) | 0.03 (−0.088) | −0.03 (−0.093) | −0.03 (−0.089) |

ADCIRC-SVU | 0.03 (−0.010) | 0.03 (−0.010) | 0.00 (−0.010) | −0.03 (−0.010) |

SVU Uncertainty | 0 | 0 | 0 | 0 |

Mean Value | ||||

Observation | 0.16 | 0.14 | −0.15 | −0.20 |

ADCIRC | 0.16 (0.028) | 0.15 (0.021) | −0.18 (0.035) | −0.21 (0.032) |

ADCIRC-SVU | 0.17 (0.005) | 0.15 (0.005) | −0.16 (0.005) | −0.22 (0.005) |

SVU Uncertainty | 0.015 | 0.013 | 0.015 | 0.018 |

STD | ||||

Observation | 0.066 | 0.052 | 0.053 | 0.090 |

ADCIRC | 0.066 (0.035) | 0.056 (0.028) | 0.074 (0.033) | 0.102 (0.042) |

ADCIRC-SVU | 0.069 (0.006) | 0.055 (0.006) | 0.057 (0.007) | 0.095 (0.006) |

SVU Uncertainty | 0.004 | 0.004 | 0.005 | 0.006 |

**Table 2.**The statistical values of the model errors from the current and previous tidal models (in meters).

Tide Station | MHHW (M-O) | MHW (M-O) | MLW (M-O) | MLLW (M-O) |
---|---|---|---|---|

1 | −0.011 (0.009) | 0.009 (0.032) | −0.093 (−0.022) | −0.078 (−0.090) |

2 | −0.057 (−0.063) | −0.029 (−0.046) | −0.001 (0.013) | 0.050 (0.049) |

3 | −0.025 (−0.030) | −0.016 (−0.030) | −0.012 (0.005) | 0.037 (0.038) |

Mean |Error| | 0.031 (0.034) | 0.018 (0.036) | 0.035 (0.013) | 0.055 (0.059) |

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## Share and Cite

**MDPI and ACS Style**

Wu, W.; Myers, E.; Shi, L.; Hess, K.; Michalski, M.; White, S.
Modeling Tidal Datums and Spatially Varying Uncertainty in the Texas and Western Louisiana Coastal Waters. *J. Mar. Sci. Eng.* **2019**, *7*, 44.
https://doi.org/10.3390/jmse7020044

**AMA Style**

Wu W, Myers E, Shi L, Hess K, Michalski M, White S.
Modeling Tidal Datums and Spatially Varying Uncertainty in the Texas and Western Louisiana Coastal Waters. *Journal of Marine Science and Engineering*. 2019; 7(2):44.
https://doi.org/10.3390/jmse7020044

**Chicago/Turabian Style**

Wu, Wei, Edward Myers, Lei Shi, Kurt Hess, Michael Michalski, and Stephen White.
2019. "Modeling Tidal Datums and Spatially Varying Uncertainty in the Texas and Western Louisiana Coastal Waters" *Journal of Marine Science and Engineering* 7, no. 2: 44.
https://doi.org/10.3390/jmse7020044