A Method to Construct Depth Datum Geodesic Height Model for GNSS Bathymetric Survey

Abstract

Water depth measurement requires the establishment of one or more tidal stations in the survey area for synchronous water level observation, and finally the water depth is estimated to the depth datum. The non-tidal observation measuring has high efficiency and avoids the water level correction error caused by tidal observation in traditional sounding. Therefore, non-tidal observation measuring has become an effective water depth measurement method in offshore and inland water. However, datum conversion in non-tide operation is mostly based on the polynomial fitting method. The accuracy of this method is influenced by the distribution of datum control points, topographic relief and operation ranges. In this paper, we present a method to construct a depth datum geodesic height model, which can directly obtain a bathymetric database of depth data in a GNSS bathymetric survey. The model incorporates the continuous depth datum and the mean sea level of geodetic height in the same area. Through the numerical simulation of tidal wave motion in regional water, the tidal model is obtained. Based on the grid model, the tidal level is extracted from the tidal model for harmonic analysis, and a continuous depth datum model is established. Mean sea level geodetic height is from the CNES-CLS2015 Average Sea Surface Model. In this paper, the model is confirmed in the South Yellow Sea area. The results show that the accuracy of the depth datum model, and the depth datum geodetic height model meets the accuracy requirements of the datum.

1. Introduction

At present, researchers have carried out extensive research in the field of non-tidal observation measuring technology, and most studies have verified that GNSS technology combined with multi-beam sounding technology can well improve sounding accuracy [1,2,3]. Non-tidal observation measuring effectively eliminates the influence of factors such as draft and surge. At the same time, the water level correction error caused by tidal observation is avoided [4]. Therefore, non-tidal observation measuring has become an effective water depth measurement method in offshore and inland water. Geodetic height measured by non-tidal observation measuring technology is based on the reference ellipsoid of WGS-84, and the water depth based on the depth datum plane needs to be converted. Datum conversion accuracy is an important consideration affecting non-tidal observation measuring, and it is the difficulty and focus of non-tidal observation measuring. However, datum conversion in non-tidal observation measuring is mostly built on a polynomial fitting method. The accuracy of this method is influenced by the distribution of datum control points, topographic relief and operation range [5]. Therefore, converting the measured high-precision GNSS bathymetry data to the depth datum is an important issue for non-tidal observation measuring.

Some researchers have done a lot of research on vertical datum transformation. Based on the global gravity field model EGM2008 [6], Wang et al. (2016) established the geodetic height model of depth datum in the survey area by an interpolation algorithm. The problem of constructing a vertical reference model in multi-beam tide depth measurement data processing without a tide gauge is solved [7]. Jiang et al. (2019) adopted the EGM2008 model, combined with the measured levelling data along the coast. The refined quasi-geoid model in the survey area is established by an interpolation algorithm. The high-precision GNSS elevation and water depth data measured by multi-beam system are converted to the depth datum plane [8]. However, the research areas of the above methods are all near the coast where tidal stations are widely distributed, which has certain limitations. Moreover, the precision control index of the datum plane is not clear. Ke et al. (2020) used the Earth Gravitational Model 2008 geoid model, the Indian Ocean 2010 tide model and a sea surface topography model of a satellite altimeter to establish a seamless chart datum and the conversion model between the WGS84 ellipsoidal reference surface and the chart datum by the fusion method. The accuracy of the model was verified in Prydz Bay, Antarctica [9]. This method uses an open tidal model, which has high accuracy in deep waters, but the accuracy is not ideal in coastal areas. Feng et al. (2018) proposed the non-tidal sounding technology based on the EGM2008 model, using the EGM2008 model to calculate the height anomaly, so as to solve the non-tidal sounding measured by GNSS instantaneous WGS84 reference ellipsoid height to normal height conversion problem [5]. However, this method is still affected by the distribution of control points.

In this paper, we proposed a method to construct a depth datum geodesic height model for a GNSS bathymetric survey, which can directly obtain a bathymetric database of the depth datum in the GNSS bathymetric survey. A continuous depth datum plane is set up based on numerical simulation of tidal wave motion. Combined with the average sea surface geodetic height model, the deep datum geodetic height model is established. The conversion of GNSS sounding data to depth datum is realized. The tolerance of 1/3 times of the water depth measurement accuracy is used as the datum accuracy control index. Experimental verification was conducted in the South Yellow Sea region. The results demonstrate that the error of the depth datum geodetic height model is 15.77 cm, which meets the requirement of datum accuracy.

2. Method

In this paper, the tidal model of the study area is established based on the numerical simulation of tidal wave motion. Based on a grid model, tidal level is extracted from a tidal model for harmonic analysis. The continuous seamless depth datum model is obtained by interpolation fitting. On this basis, using the hierarchical modelling method, combined with the average sea surface geodetic height model, the depth datum geodetic height model is constructed. To ensure the accuracy of bathymetry, 1/3 times the tolerance of bathymetry accuracy is used as the datum level accuracy requirement.

2.1. Construction of Continuous Depth Reference Surface Model

Since 1956, the theoretical lowest tide level (theoretical depth datum) has been used as the chart datum in China’s sea area. The chart datum is determined by the vertical difference relative to the average sea surface, which is usually called the L-value. Therefore, the determination of the chart datum often refers to the calculation of the L-value. Constructing a regional depth datum model based on a tidal model is a common current approach. The idea is to obtain the L-value data set on the regional grid points based on a high-precision tidal model. At the same time, the tide gauge data distributed in the region is used to calculate an accurate L-value of the station. Finally, the whole L-value data set is corrected using the L-value of the tide gauge stations to obtain the regional L-value model.

The tide model is obtained by numerical simulation of the tide level. In the numerical simulation of the tidal level, the most important factors affecting the simulation results are water depth data and open boundary conditions. The conventional method is unable to obtain real tidal information on open boundary nodes. At present, there are two methods to set open boundary conditions. One is to extract the open boundary water level from large-scale tidal models such as global or regional tidal models. The other is to interpolate the open boundary points based on the measured water level of coastal tidal stations [10,11]. Large-scale tidal models generally have low accuracy and are suitable for large-scale research in the open sea. Generally, the water level interpolation method is used to configure the water level in the hydrodynamic research of offshore shallow water areas [12]. Water level observation includes tide level and residual water level, and the residual water level cannot be simulated by the tidal hydrodynamic model to simulate its propagation law. In this paper, to avoid the effect of the residual water level, the tide gauge forecast tide level is used to provide dynamic water level forcing data for the model’s open boundary.

In this study, bathymetric data and the shoreline were sourced from the ENCs produced by the Department of Navigation Guarantee, Chinese People’s Liberation Army Navy Command, where the spatial resolution of the bathymetric data is approximately 100 m. The open boundary water level and the bathymetric data are both referenced to the Theoretical Bathymetrical Datum, and the shoreline is referenced to the Mean High Water Springs. According to the characteristics of the simulated area, two open boundaries are set perpendicular to the shoreline and intersect at a point in the sea. The starting points of the two open boundaries are located at two stations, Qingdao and Binhai. Tidal tables at Qingdao and Binhai stations provide dynamic tidal levels for the open boundary of the model. The tide levels of the two tide tables are interpolated to each node of the open boundary separately using the time difference method [13]. The minimum simulation time is one year.

Tidal level data are extracted from each grid point according to the divided grid. The L value on each grid point is calculated according to the depth datum algorithm. Then the depth datum model is formed by interpolation simulation.

According to the tidal data of long-term tidal stations along the coast, the L value of the model is corrected by inverse distance weighted interpolation, and the correction formula is shown in Equation (1) [14]. Finally, a continuous depth datum is obtained with high accuracy and consistently with the characteristics of tidal variation.

$$\Delta L=\frac{{{\displaystyle \sum }}_{i=1}^n\raisebox{1ex}{$\Delta L_i$}\!\left/ \!\raisebox{-1ex}{$S_i$}\right.}{{{\displaystyle \sum }}_{i=1}^{n}1/S_i}$$

(1)

where ΔL is the depth datum revision value, $\Delta L_i$ is the tide gauge station depth datum difference value, $S_i$ is the distance from the tide gauge station to the point, n is the number of tide gauge stations, and $i$ is the tide gauge station number.

2.2. Chart Datum Geodetic Height Model Construction

To construct a conversion model between two different vertical benchmarks, it is necessary to clarify the spatial relationship between the two benchmarks. Figure 1 shows the spatial geometric relationship between several different vertical datum planes. According to the spatial relationship between vertical benchmarks in Figure 2, Equation (3) can be obtained. H_MSL is the average sea surface geodetic height, L is the constructed continuous chart datum, and $H_{cd}$ is the distance from the chart datum to the reference ellipsoid, namely the geodetic height of the chart datum to be determined.

$$H_{cd}=H_{MSL}-L$$

(2)

At present, many institutions have released satellite altimetry average sea surface product data after various corrections. The CNES_CLS2015 mean sea level model [15] is calculated based on 20 years of satellite altimetry data, which includes T/P, Jason-1, Jason-2, ERS-1, ENVISAT, GFO, GryoSat-2, etc. The latitude of the model altimeter satellite is in the range of 80º S–84º N, and the grid resolution is 1^′ × 1^′. The satellite altimetry data used in the model corrects the time-varying error of the ocean, thus reducing the influence of ocean variability in a large area and making the ocean changes tend to average at any time. Specific data processing is used near the coast to keep the accuracy of the model [16].

In this paper, we extract the mean sea surface height points in the study area based on the CNES_CLS2015 mean sea surface model, interpolate them into the grid model, and then construct the mean sea surface geodetic height model of the study area. According to Equation (3), the chart datum geodetic height can be obtained by combining the mean sea surface geodetic height and the depth datum L value, to establish the chart datum geodetic height model.

2.3. Datum Accuracy Requirements

This paper starts from ensuring the accuracy of the bathymetry. For the L value accuracy of long-term tide gauge stations, the 1/3 times tolerance of bathymetry accuracy is used as the L value accuracy requirement. The specific calculation is as follows:

$$H=h+L$$

(3)

where H represents the true water depth, h represents the measurement result, and L represents the depth reference level. According to the error propagation formula, we can get:

$$\Delta H^2=\Delta h^2+\Delta L^2$$

(4)

$$\Delta H=\sqrt{\Delta h^2+\Delta L^2}$$

(5)

where ∆H represents the true water depth error, ∆h represents the measurement error, which is mainly derived from the accuracy of the equipment and the process of data output, and ∆L represents the reference error.

Assuming $\Delta L=\frac{1}{3}\Delta h$, then

$$\Delta H=\sqrt{\frac{10}{9}\Delta h^2}\approx \sqrt{1.11}\Delta h=1.05\Delta h$$

(6)

When the L value is 1/3 times the tolerance of the measurement accuracy, a measurement accuracy with a confidence level of 95% can be obtained. Therefore, this assumption is feasible.

According to Equation (2), the error of the depth datum geodetic height model should mainly come from the error of the average sea surface geodetic height model and the error of the depth datum model. Combined with the error propagation rate, the error in the geodetic height model of depth datum is:

$$\Delta \zeta =\pm \sqrt{\Delta MS{L}^2+\Delta L^2}$$

(7)

In actual bathymetry, the spatial interpolation of the datum is implicit in the spatial interpolation of the water level. Therefore, this paper deduces the accuracy requirements of the depth datum in bathymetry based on the water level correction accuracy. There are detailed requirements for bathymetry accuracy in the International Hydrographic Organization (IHO) specifications [17]. Considering all error factors, such as tidal observation error, vertical datum determination, and vertical datum conversion, within a 95% confidence interval, the allowable total vertical uncertainty is:

$$TU{V}_{max}=\pm \sqrt{a^2+b^2{d}^2}$$

(8)

where $TU{V}_{max}$ represents the maximum allowable vertical uncertainty, $a$ represents a fixed constant that does not vary with depth, $b$ is a coefficient, and $d$ represents water depth. In water depths less than 100 m, the above parameters $a$ and $b$ are 0.5 m and 0.013, respectively. The datum limit error ranges for different water depths are shown in

Table 1. Datum accuracy requirements (confidence 95 %) (unit: m).

Depth Range Z	Sounding Limit Error δ	Datum Limit Error ζ
0 < Z ≤ 20	0.50 < δ ≤ 0.56	0.167 < ζ ≤ 0.187
20 < Z ≤ 30	0.56 < δ ≤ 0.63	0.187 < ζ ≤ 0.210
30 < Z ≤ 50	0.63 < δ ≤ 0.82	0.210 < ζ ≤ 0.273
50 < Z ≤ 100	0.82 < δ ≤ 1.39	0.273 < ζ ≤ 0.463

.

3. Case Study

3.1. Overview of the Study Area

We construct a depth datum geodetic height model in the South Yellow Sea region. The South Yellow Sea region (118°42′–120°47′, 34°12′–36°28′) is selected as the study area, as shown in Figure 2, mainly from the following aspects. Firstly, the range of water depth in the study area is about 5~37 m, which belongs to the shallow water area, and the global tidal model has low accuracy. Secondly, the South Yellow Sea is rich in submarine oil, and it is also an important shallow sea fishing ground in China. It is of great significance for human social and economic activities to establish a depth datum geodesic height model for the GNSS bathymetric survey. Finally, the tidal observation data of six long-term tidal stations along the South Yellow Sea have been collected in this paper. According to these data, the accuracy and reliability of tidal simulation results can be fully tested. In summary, this paper selects the South Yellow Sea waters as the study area.

3.2. Chart Datum Geodetic Height Model Construction of Depth Datum in South Yellow Sea Region

3.2.1. Irregular Triangular Mesh Generation

The Surface Water Model System (SMS) software is the most commonly used grid generation tool and has a strong terrain grid generation technology [18,19]. This paper will use SMS software to divide triangular meshes. In the grid division, the grid is denser in the offshore area with many islands, a complex coastline and shallow water depth, which is conducive to a more realistic calculation of the complex tidal system in the region. In the outer sea close to the open boundary and the central area of the calculation area, there are fewer islands, deeper water depths, a relatively flat sea floor and a relatively stable tidal wave system, and the grid in this area is sparse. Sparse grids are easy to simulate and save computing resources.

The resulting grid is shown in Figure 3. The open boundary resolution is about 0.01º–0.05º, and the average resolution near the coastline and islands is about 0.01º. There are 13,468 grid nodes and 26,115 horizontal grid units. The grid fits the seabed topography and coastline well, which can reflect the real topography.

3.2.2. Construction of Continuous Seamless Depth Datum

Mike21 software is a two-dimensional mathematical simulation software, developed by the Danish Institute of Hydraulics, which is widely used in domestic and international hydrodynamic simulations, has achieved good results, and is one of the more advanced numerical simulation software packages in the international arena [20]. It has been applied to tidal simulations in various marine, coastal and estuarine environments [21,22,23]. In this paper, MIKE21 will be used for tidal model construction. The simulation range is from 118°42′ E to 120°47′ E and 34°12′ N to 36°28′ N. To simulate the dynamic boundary of intertidal tidal rise and fall, the dry and wet point treatment is used in the calculation. Other initial condition parameters are shown in

Table 2. MIKE21 Flow Model parameters.

Parameter	Value
Time	Time step interval 0–10 s, frequency of output data 1 h, and simulation period from 2007/01/01–2007/03/31.
Flood and dry	Include, model recommended default value
Density	Barotropic
Coriolis force	Varying in domain
Boundary conditions	Water level interpolated by Tide Table data
Bathymetry	Electronic navigational charts
Bed resistance	Quadratic drag coefficient $C_f=\frac{g}{M^2\Delta h^{-3}}$, where $h$ is the total water depth, $M$ is the manning coefficient, and g is gravitational acceleration.
Salinity and temperature	Salinity and temperature average value of simulation period in domain; temperature is 10 ℃, salinity is 0.035

.

Finally, according to the divided grid, 13 main tidal components contained in each grid point are extracted for harmonic analysis, and the L value is calculated according to the definition algorithm of 13 tidal components of the depth datum. According to the tide table data collected from four long-term tide stations distributed along the coast of the study area, namely Lianyungang, Yanweigang, Rizhaogang and Lanshangang, the L value of the model is revised.

The overall revision scheme for the L-value model is to pass the difference between the tide gauge L-value at the tide gauge station and the model L-value to each grid point by an inverse distance weighted interpolation [24]. Taking a grid point as an example, the steps are as follows:

1.

The number of tide gauge stations in the study area is n = 4, the difference in L values of tide gauge stations is $\Delta L_i$, and the distance from the grid point to each tide gauge station is $S_i$.

2.

Using the inverse of the distance to set the weights (since there is no information on short-term tide gauge stations, the weights of long- and short-term tide gauge stations are not given here), the revised positive L-value ΔL for this grid point can be obtained by Equation (1).

The revised L value is interpolated into the grid to obtain the regional continuous depth datum model, as shown in Figure 4. It can be seen that the contour distribution of the depth datum model shows obvious regularity. When the external tidal wave system is introduced into the offshore, due to the uplift of the offshore topography, the potential energy of the sea water is superimposed, the shallow water tide gradually increases, and the tidal range increases closer to the shoreline. Therefore, in the direction perpendicular to the shoreline, the depth datum L value gradually decreases from the shoreline direction.

The model L values (called model values) at each tide gauge station can be obtained from the depth datum model and compared with the calculated tide table values at each tide gauge station. The results are listed in

Table 3. Comparison between model values of tide gauge stations and calculated values of tide tables (unit: cm).

Station Name	Model Value	Calculated Values	Difference
Lianyungang	290.1	298.6	−8.5
Yanweigang	261.6	251.6	10.0
Rizhaogang	270.42	261.8	8.62
Lanshangang	286.7	298.7	−10.0

.

Table 3 shows that the modelled values of the tide gauge station do not differ much from the calculated values of the tide table and generally achieve high accuracy. The depth datum contour distribution obtained from the simulation also conforms well to the tidal variation characteristics of the study area.

3.2.3. Chart Datum Geodetic Height Model Construction

Based on the CNES_CLS2015 average sea surface model to extract the average sea surface geodetic height within the scope of the study, and on inverse distance weighted interpolation to SMS grid points, the average sea surface geodetic height model of the South Yellow Sea region is obtained, as shown in Figure 5.

Finally, the geodetic height of the mean sea surface in the South Yellow Sea region was fused with the continuous depth datum to establish a chart datum geodetic height model, as shown in Figure 6.

3.3. Model Accuracy Evaluation and Analysis

Model accuracy assessment is an important indicator of the model, and its accuracy level will directly determine its practical significance. From Equation (3), the error of the chart datum geodetic height model should mainly come from two aspects: the error of the mean sea surface geodetic height model and the error of the depth datum model.

According to the CNES_CLS2015 model, the maximum median error of the mean sea surface height in the study area is 12.2 cm, and we choose 12.2 cm as the mean square error of the mean sea surface model in the South Yellow Sea. The continuous depth datum is established based on the results of 3D numerical simulation of tidal wave motion. By comparing the simulation results with the calculation results of several long-term tide gauge stations in the South Yellow Sea, the maximum absolute value of error is 10 cm, which is taken as the mean square error of the depth datum model of the whole South Yellow Sea region. According to the error propagation rate and combined with the model expressions in Equation (3), the medium error of the chart datum geodetic height model is 15.8 cm.

The range of water depth in the study area is about 5~37 m, and according to Equation (4), the $TU{V}_{max}$ is 50.4~69.4 cm. Select 50.4 cm as the accuracy limit of water depth measurement in the study area, and the accuracy indicators of the remaining different benchmarks are shown in

Table 4. Comparison of precision indexes of different datums.

Datum Precision Index	Precision
1-time water depth measurement accuracy tolerance	50.4 cm
1/2-time water depth measurement accuracy tolerance	25.2 cm
1/3-time water depth measurement accuracy tolerance	16.8 cm
error in the mean sea level	12.2 cm
error in the depth datum model	10 cm
error in the depth datum geodetic model	15.8 cm

.

In this paper, 1/3 of the tolerance of water depth measurement accuracy is selected as the datum accuracy requirement, so the datum accuracy is required to be within the range of 0~16.8 cm. In this study, the error in the depth datum model is 10 cm, and the error in the depth datum geodetic model is 15.8 cm, both of which meet the accuracy requirements of the datum.

4. Discussion

Existing codes all specify the relationship between the accuracy of bathymetry and the changing conditions of bathymetry. The “IHO Standards for Hydrographic Surveys” promulgated by the International Hydrographic Organization (IHO) [18] stipulates that the accuracy of water level correction not be greater than the corresponding sounding accuracy. However, the L-value accuracy evaluation problem has not been considered in the actual measurement operation. At present, researchers have put forward different precision indexes of the depth datum, including 1-time water depth measurement accuracy tolerance, 1/2-time water depth measurement accuracy tolerance and 1/3-time water depth measurement accuracy tolerance [25,26]. In this study, to ensure the accuracy of bathymetry, for the L value accuracy of long-term tide gauge stations, 1/3 times the tolerance of bathymetry accuracy is used as the datum level accuracy requirement. The depth datum, mean sea level, and geodetic height of the depth datum must meet the accuracy requirements of the datum.

The accuracy of the depth datum geodetic model depends on the accuracy of the depth datum and the accuracy of the mean sea level. The mean sea surface geodetic height data in this paper comes from the CNES_CLS2015 mean sea surface model, and the accuracy of the model is a given definite value. The accuracy of the depth datum will directly affect the geodetic accuracy of the depth datum. Therefore, this paper adopts the numerical simulation method to obtain the tidal model and obtains the high-precision continuous depth datum through interpolation fitting. The accuracy of the depth datum model is determined by comparing the calculated value of the depth datum with the model value of the coastal tide gauge station. From the results, this method achieves better accuracy. The accuracy of the final deep datum geodetic model meets the datum accuracy requirements.

5. Conclusions

In this paper, the construction principle and method of vertical datum transformation model in continuous depth datum and non-tidal observation measuring are studied in detail. Modelling experiments and accuracy evaluation analysis were carried out in the South Yellow Sea region. The experimental results show that the error in the continuous depth datum model is 10 cm and the error in the depth datum geodetic model is 15.8 cm, both of which meet the datum accuracy requirements.

Compared with previous methods, the improved method in this paper has the advantage that the sounding data based on the depth datum in GNSS sounding can be directly obtained in a large range, and the datum precision index is defined. However, this method still has the following problems: The global mean sea surface model CNES_CLS2015 has low accuracy in shallow waters. If the accuracy of the mean sea surface model can be improved, the accuracy of the depth datum geodetic height model will be further improved.