A method of constructing a dynamic chart depth model for coastal areas

Datum offset correction

In simulating tidal levels, open boundary conditions play a crucial role in determining the accuracy of the results. Unfortunately, the conventional method of setting these conditions is unable to obtain actual tidal information at open boundary nodes, resulting in an inevitable error between the obtained and actual water levels. This error is constant at each open boundary point and is transferred to the entire domain during simulation, leading to a vertical deviation between the simulated tidal level and the actual measured value. Through verification, it has been found that the values of vertical deviations between discrete distribution verification points are consistent with each other, suggesting that the error conforms to the same variation trend in the domain, and can be reduced through interpolation. The harmonic constant datum method (HCDM) is currently used to obtain tidal datum. The datum offset correction can be derived by subtracting the observed tidal water level from the simulated tidal level during the same period (Eq. (2)). The study area’s triangular grids provide the coordinates of each grid node, from which the simulated tidal level is extracted, and the simulated tidal datum value at the grid node is obtained following the tidal datum definition algorithm. Finally, the tidal datum of the study area is obtained by interpolation fitting, and the mathematical equations for calculating the tidal datum are provided below. ${\displaystyle \Delta L_i=L_{P_i}-L_{M_i}}$ ${\displaystyle L={\left(fH\right)}_{K_1}cos{\phi }_{K_1}+{\left(fH\right)}_{K_2}cos\left(2{\phi }_{K_1}+2g_{K_1}-180^{\circ }-g_{K_2}\right)-\sqrt{{\left({\left(fH\right)}_{M_2}\right)}^2+{\left({\left(fH\right)}_{o_1}\right)}^2+2{\left(fH\right)}_{M_2}{\left(fH\right)}_{O_1}cos\left({\phi }_{K_1}+g_{K_1}+g_{O_1}-g_{M_2}\right)}}$ ${\displaystyle -\sqrt{{\left({\left(fH\right)}_{S_2}\right)}^2+{\left({\left(fH\right)}_{P_1}\right)}^2+2{\left(fH\right)}_{S_2}{\left(fH\right)}_{P_1}cos\left({\phi }_{K_1}+g_{K_1}+g_{P_1}-g_{S_2}\right)}-\sqrt{{\left({\left(fH\right)}_{N_2}\right)}^2+{\left({\left(fH\right)}_{Q_1}\right)}^2+2{\left(fH\right)}_{N_2}{\left(fH\right)}_{Q_1}cos\left({\phi }_{K_1}+g_{K_1}+g_{Q_1}-g_{N_2}\right)}+{\left(fH\right)}_{M_4}}$ (2)${\displaystyle cos{\phi }_{M_4}+{\left(fH\right)}_{M_6}cos{\phi }_{M_6}+{\left(fH\right)}_{M{S}_4}cos{\phi }_{M{S}_4}+H_{S_a}cos{\phi }_{S_a}+H_{S_{Sa}}cos{\phi }_{S_{Sa}}}$ ${\displaystyle {\phi }_{M_4}=2{\phi }_{M_2}+2g_{M_2}-g_{M_4},\left(°\right);{\phi }_{M_6}=3{\phi }_{M_2}+3g_{M_2}-g_{M_6},\left(°\right);}$ ${\displaystyle {\phi }_{M{S}_4}={\phi }_{M_2}+{\phi }_{S_2}+g_{M_2}+g_{S_2}-g_{M{S}_4},\left(°\right);}$ ${\displaystyle {\phi }_{M_2}=t{g}^{-1}\left(\frac{{\left(fH\right)}_{o_1}sin\left({\phi }_{K_1}+g_{K_1}+g_{O_1}-g_{M_2}\right)}{{\left(fH\right)}_{M_2}+{\left(fH\right)}_{O_1}cos\left({\phi }_{K_1}+g_{K_1}+g_{O_1}-g_{M_2}\right)}\right)+180^{\circ },\left(°\right);}$ ${\displaystyle {\phi }_{S_2}=t{g}^{-1}\left(\frac{{\left(fH\right)}_{P_1}sin\left({\phi }_{K_1}+g_{K_1}+g_{P_1}-g_{S_2}\right)}{{\left(fH\right)}_{S_2}+{\left(fH\right)}_{P_1}cos\left({\phi }_{K_1}+g_{K_1}+g_{P_1}-g_{S_2}\right)}\right)+180^{\circ },\left(°\right);}$ ${\displaystyle {\phi }_{S_a}={\phi }_{K_1}-\frac{1}{2}{\varepsilon }_2+g_{K_1}-\frac{1}{2}{g}_{S_2}-180^{\circ }-g_{S_a},\left(°\right);}$ ${\displaystyle {\phi }_{S_{Sa}}=2{\phi }_{K_1}-{\varepsilon }_2+2g_{K_1}-g_{S_2}-g_{S_{Sa}},\left(°\right);}$ ${\displaystyle {\varepsilon }_2={\phi }_{S_2}-180^{\circ },\left(°\right);}$

where Q₁, O₁, P₁, K₁, N₂, M₂, S₂, K₂, M₄, MS₄, M₆, S_a and S_Sa are partial tides, expressed by the amplitude and phase (the so-called tidal harmonic constants). The datum offset is obtained by HCDM of the observed water level and simulated water level at the same location in the same time periods, and then calculated by Eq. (2). The parameters of the Eq. (2) are explained in Table 1.

In situations where the values of datum offset correction at discrete verification points exhibit proximity to each other, the inverse distance weighting (IDW) interpolation algorithm is a frequently employed technique to ameliorate the datum offset in the surveyed region. The mathematical formula for this algorithm is as follows: (3)${\displaystyle L_j=\sum _i^n{\lambda }_{ij}\cdot \left[-L_i\right],{\lambda }_{ij}=l_{ji}^{-p}/\sum _{i=1}^n{l}_{ji}^{-p}}$ where L_j is the revised value of the datum offset at depth point j; L_i is the corrected value for the datum offset at gauging station i; λ_ij is the weight of gauging station i at point j; n is the number of tide stations; i is the gauging station number; l_ji is the straight-line distance between station i and point j; and p is the index of the weighting function, and the default value is 2.

Residual water level correction

The residual water level is determined by meteorologically induced changes in sea level, including storm surge heights, local wind set-up, and the inverted barometer effect (Ningsih, Suryo & Anugrah, 2011). Due to the intricate nature of meteorological disturbances, modeling this phenomenon can be challenging.

To calculate the residual water level, the tidal level distribution must be removed from the measured water level. The following mathematical equation illustrates how to compute the residual water level: (4)${\displaystyle R_i\left(t\right)=H_i\left(t\right)-h_0-\sum _{k=1}^m{f}_k{h}_{k}cos\left({\sigma }_{k}t+{\upsilon }_k+u_k-g_k\right)}$ where R_i(t) is the residual water level of gauging station k at time t ; $H_i\left(t\right)$ is the measured water level of gauging station i at time t, and the time interval is usually less than an hour; and h₀ is the mean sea level, which is usually replaced by the calculated mean water level. h_k and g_k are the amplitude and phase, respectively; m is the number of partial tides after analyzing the measured water level; f_k and u_k are a nodal factor and nodal angle; σ_k is the angular rate; and υ_k is the initial phase of the astronomic constituent. The accuracy of the residual water level extraction relies heavily on the precision of the tide harmonic analysis in the measured water level. Due to the ocean water level’s same or similar inducement and inertia, the residual water level demonstrates short-term and local regularity, resulting in strong spatial correlation (Tang, Bao & Jun, 2007; Zhao et al., 2017). The distance weighted interpolation algorithm is widely employed in meteorology (Degaetano & Belcher, 2006) to create a two-dimensional field from finite observational data using a range-dependent weighting function. We improved the interpolation algorithm by considering both distance and correlation factors to reconstruct the residual water level field in the survey region. To obtain the weight value of each site during interpolation, we compared the residual water levels between stations and calculated the mean correlation coefficient between the residual water level and other stations. This mean value is used as the weight value of the site. ${\displaystyle R_j\left(t\right)=\sum _{i=1}^n{\mu }_i\cdot \left[-R_i\left(t\right)\right]\cdot {\lambda }_{ij},{\mu }_i=\frac{n\cdot \sum _{k=1}^{n-1}{\rho }_{ik}}{\sum _{i=1}^n\sum _{k=1}^{n-1}{\rho }_{ik}}}$ (5)${\displaystyle {\rho }_{ik}=\frac{\sum _t^N\left(R_i\left(t\right)-\overline{R_i\left(t\right)}\right)\left(R_{ki}\left(t\right)-\overline{R_k\left(t\right)}\right)}{\sqrt{\sum _t^N{\left(R_i\left(t\right)-\overline{R_i\left(t\right)}\right)}^2\sum _t^N{\left(R_k\left(t\right)-\overline{R_k\left(t\right)}\right)}^2}}}$ where $R_j\left(t\right)$ is the residual water level at point j at time t; μ_i is the correlation coefficient value of tide station i; $R_i\left(t\right)$ is the residual water level of tide station i at time t; λ_ij is the weight of gauging station i at point j; ρ_ik is the simulation deviation correlation coefficient of tide station i with station $k.R_k\left(t\right)$ is the residual water level of tide station k at time t; and N is the number of residual water level records.

Study area and model setting

In this study, we developed a numerical hydrodynamic model using the MIKE21 Flow Model to simulate the tide in the southwest part of the Yellow Sea within the domain of 118°42′∼120°47′ longitude and 34°32′∼36°28′ latitude. The study area includes seven tide stations, six of which are along the coast and one on Cheniushan Island (see Fig. 1 and Table 2). We corrected the chart depth using the seven discrete L values obtained from the tide station distribution and derived the bathymetric data based on the mean sea level. Considering the features of the study area, we established two open boundaries that start from the tide survey stations and run perpendicular to the coastline before converging at a certain point in the sea. To specify the open boundary conditions, we used the tide tables to provide the dynamic tide levels. Using the Time-Difference Method, we interpolated the water levels of the two tide tables to each node of the open boundaries. Both the open boundary water level and bathymetric data were referenced to the low water datum.

The method for setting the domain of this model is based on Tian et al.’s (2023) method. To simulate the dynamic boundary of the intertidal zone during tidal rise and fall, we employed the wet/dry point treatment method. The temperature and salinity of the model were set to constant values of 10 °C and 0.035, respectively. The model’s time step interval was set at 0–10 s, the output data frequency was one hour, and the simulation period was from January 1st, 2007, to March 31st, 2007. To set the initial and boundary conditions, we collected reliable data from ENCs as described in Tian et al. (2023). The shoreline data was referenced to the Mean High Water Springs benchmark, and the bathymetric data was referenced to the low tide datum.

This study utilized tidal tables to provide dynamic tidal levels for open boundaries instead of relying on global tidal models, such as DTU10, due to their low accuracy nearshore. In contrast, the astronomical tide predicted by the tide tables offers high accuracy. We compared the accuracy of the tidal tables and the global tidal model using measured data from several tidal stations. The tidal stations located near the shore (Lianyungang, Binhai, Qingdao, Rizhao, and Yanweigang) were evaluated by comparing the tide table with the astronomical tide predicted by the measured data. For the tidal stations located in the sea (Cheniushan), we compared the DTU10 global tide model with the astronomical tide predicted by the measured data. The specific results are presented in Fig. 2.

As shown in Fig. 2, the tide table provided a more accurate estimate of the actual measured water level than the DTU10 global tide model, making it the preferred option to provide open boundary conditions in this study.

Deviation corrections and verification

To validate the model results, we compared the simulated tidal water level with the observed tidal gauge data from the survey region. As indicated in Table 3 and Fig. 3, the water level deviations from several verification stations in the study area exhibit similar time-series variations, suggesting an inherent temporal characteristic of the study area. Consequently, we applied the deviation correction method to mitigate these deviations. The datum offsets of the four stations were determined using Eq. (2) and presented in Table 4. The residual water levels of the four tide stations calculated using Eq. (4) are displayed in Fig. 4, and the correlation coefficients for each station are provided in Table 5. We first corrected the datum offset for the survey region using Eq. (3), and the results are shown in Fig. 5 and Table 6.

Table 3:

The observed and simulated values of tide at tide station and the water level deviation between them, and the vertical datum is low water datum (unit: cm).

DOI: 10.7717/peerj.15616/table-3

After correcting the datum offsets, we reconstructed the residual water level field using Eq. (5) and compared the corrected and uncorrected results, as illustrated in Fig. 6 and Table 7.

To verify the accuracy of the corrected instantaneous water level in the survey region, we designed a cross-validation method. We selected one of the four tidal stations as the reference point and used the water levels from the other three stations to calculate the instantaneous water level of the selected station. We then compared the results with the actual water level by calculating the root-mean-square deviation.

As indicated in Tables 7 and 8, the results of the two corrections are similar, demonstrating the applicability of the proposed deviation correction method.