Bathymetry inversion using the deflection of the vertical:A case study in South

Xioyun Wn , Bo Liu , Xiohong Sui ,*, Richrd F. Annn , Ruijie Ho , Yijun Min

a School of Land Science and Technology, China University of Geosciences (Beijing), Beijing 100083, China

b Qian Xuesen Laboratory of Space Technology, Beijing 100094, China

Keywords:Bathymetry Deflection of the vertical Gravity anomaly Satellite altimetry

ABSTRACT The deflection of the vertical is one of the essential products of altimetry. However, unlike gravity and vertical gravity gradients, it is seldom used in bathymetry inversion. In this study, an algorithm for bathymetry inversion using the deflection of the vertical is proposed. First, we separately derive the formulas for the bathymetry inversion from the north and east components of the vertical deflection and introduce the data processing. Then a local area in the South China Sea is selected as an example to test the method. The bathymetry inversion based on gravity anomaly is also conducted for comparison.Assuming the ship-borne depths are the true values, the error standard deviations (STDs) of the bathymetry derived by north and east components of the vertical deflection are 156.64 m and 165.57 m,respectively. It indicates that the north component has a better performance in bathymetry inversion than the east component.The inversion results from the combination of both components show a higher accuracy of bathymetry than that from a single component.The difference between the error STD of the combination results and that of the gravity anomaly is less than 0.2 m. The experiment's results also show that the precision of the derived bathymetry can be improved if the parameters of linear regression are adjusted according to water depths.In summary,among the gravity field products used in this study,the gravity anomaly yielded the best performance in the bathymetry inversion. However, since additional data and computation time are required to derive gravity anomalies from altimetric observations,the vertical defections can still be used as supplements, especially in areas where accurate vertical deflections exist.

1. Introduction

Bathymetry information is important for human activities and Earth science. Different methods have been developed for obtaining bathymetry information,such as multibeam echo sounding,the derivation based on gravity data [1-5], detection with laser technology [6], and inversion based on imagery [7,8]. Among these methods,altimetry-derived gravity products play a significant role in global bathymetry inversion.

The bathymetry inversion derived from altimetry products is mainly based on gravity anomaly. Parks [9] proposed the spectral method for bathymetry inversion using gravity anomaly.Smith and Sandwell [10] proposed an admittance theory and predicted the bathymetry from dense gravity data and sparse shipboard bathymetry for the southern oceans. The spectral method has been verified in many areas,such as the South China Sea [2], the western Indian offshore [11], and the South Atlantic Ocean [12]. In the spatial domain, the so-called gravity geological method (GGM) [13] is widely used for bathymetry inversion.For example, GGM was used to enhance the bathymetry of the East Sea by Kim et al. [14]. Hisao et al. [15] investigated the density contrast for bathymetry inversion using GGM and concluded that the predicted density contrast could enhance the precision of 3-4 m for GGM. Kim et al. [16] predicted the bathymetry on the eastern end of the Shackleton Fracture in the Drake Passage. Bathymetry of the South China Sea was also predicted using GGM with a precision of 76.95 m [17]. Xiang et al. [18] proposed an adaptive mesh method for modeling long-wavelength gravity in GGM.

Recently, gravity gradients have been used to predict bathymetry in some areas.Wang[19]proposed a least-squares method for bathymetry inversion using vertical gravity gradients. However, it has not been tested using actual gravity gradient data.Hu et al.[20]used vertical gravity gradients to predict the bathymetry over a part of the North Pacific,and the precision was improved by combining gravity gradients and gravity anomalies. Fan et al. [21] proposed a nonlinear iterative least-squares method for seafloor topography by combining gravity anomalies and gravity gradients. The feasibility of seafloor topography estimation from airborne gravity gradients was discussed using the real data [22]. These studies indicate that the gravity gradients can contribute to bathymetry inversion, especially with the precision improvement of marine gravity gradients.

As mentioned above, gravity anomaly and vertical gravity gradients are the most commonly used gravity field products for bathymetry inversion. However, the deflection of the vertical, as another vital set of gravity field products,is seldom used.Indeed,it can be derived with high precision using altimetry observations and is often further used to derive gravity anomalies[23-25].Since the gravity anomalies derived from the deflection of the vertical have high precision, the deflection of the vertical contains abundant gravity information, including those caused by bathymetry.Hence,it is theoretically feasible to use the deflection of the vertical to inverse bathymetry.

This study proposes a method for deriving bathymetry using the deflection of the vertical and analyzes its performance.The effect of different weights for the combination of the bathymetry derived by each component of the vertical deflection is also investigated.

2. Method

2.1. The relationship between the deflection of the vertical and bathymetry

To present the relationship between bathymetry and gravity field products,a coordinate system is constructed(Fig.1).The XOY plane represents the plane at the datum depth, z= z0represents the mean sea surface, and z=h(r) denotes the heights of seafloor topography. According to Parker [9], the disturbing gravitational potential at the ocean surface has the following relationship with the water depth:

where γ is the mean normal gravity. Finally, the formulas for bathymetry inversion can be constructed as follows:

According to Equations(3)and(6),Δρ is an essential parameter for the inversion. According to the previous studies, its theoretical value does not always lead to the best inversion results [29].Instead, it can be derived by linear regression between the water depths at the control points and gravity anomaly after downward continuation from the sea surface to the datum depth, i.e., by multiplying e|k|z0[10,27].This method is often used in bathymetry inversion with spectral methods. The advantage is that the knowledge of an accurate density contrast is not required.A similar technique is also adopted for the deflection of the vertical in this study.The issue is that even after downward continuation,there is no linear relationship between the ocean depth and vertical deflection due to the presence of factor (|k|/ikx). To resolve this issue, two new values are defined:

Equation (8) has same form as Equation (3). Therefore, the algorithm of bathymetry using gravity anomaly can also be used as inputs for the inversion.

2.2. Data processing flow

According to previous studies, the gravity field products have high coherence with the bathymetry in certain wavelength bands[10]. This is because only part of the gravity anomaly or the deflection of the vertical is created by the seafloor topography.Hence, the gravity field products are mainly used to predict the bathymetry in wavelength bands where they have a high coherence with the bathymetry. For gravity anomaly, it usually resides in the bandwidth of 20～200 km. The sensitive wavelength band varies with geographic areas [32]. Although little or no literature has discussed the correlation between the deflection of the vertical and bathymetry, it is reasonable to infer that the vertical deflection is only sensitive to bathymetry in a limited wavelength band. Thus, it can be used to predict bathymetry in the related band. In this study, the sensitive band is derived based on the coherence analysis (see Sections 2.2 and 4.1). The data processing flow is shown in Fig. 2.

Please note that ξ′and η′used in Fig.2 are not the initial values of the deflection of the vertical (see Equation (7)). Downward continuation means the following processing:

2.3. Coherence analysis

To determine the sensitive bands of the deflection of the vertical and gravity anomaly for bathymetry, a coherence analysis is conducted as [32]:

where a and b represent the two signals; ω denotes frequency;Sab(ω)is the cross power spectral density between signals a and b;Sa(ω) and Sb(ω) are self-power spectral densities of a and b,respectively.More details on the coherence analysis method can be found in Ref. [32].

3. Data and study area

The study area is located in the South China Sea, i.e,112°E - 119°E, 12°N - 20°N. The ship-borne depth data were downloaded from National Oceanic and Atmospheric Administration (NOAA) (https://maps.ngdc.noaa.gov/viewers/geophysics/).The total number of ship-borne depth points is 125,228, 90% of which are randomly selected as control points, and the remaining 10% points are used as test points. Fig. 3 shows the location of the study area, the distribution of the ship sounding tracks, and the bathymetry from ship-borne depths. For comparison, the bathymetry from Earth Topographical Database 1(ETOPO1)with the resolution of 1′× 1′is also presented in Fig. 3.

Fig. 3(c) and (d) shows that the bathymetry from the grids of ship-borne depths and ETOPO1 has significant differences in some areas. For example, the burr in the southeast corner is unlikely to exist in practice and may be caused by the gross error in the ship survey data.The topography presented by ETOPO1(Fig.3(d))shows more details than Fig.3(c).To eliminate the effect of gross errors in ship depths,the values of ship depths that deviate from ETOPO1 by 1000 m are removed.

Fig. 2. Flowchart of bathymetry inversion based on the deflection of the vertical or gravity anomaly.

The vertical deflection with a grid of 1′× 1′was provided by Zhu et al. [33]. This data is derived from multi-satellite altimeter observations, including ERS-1, Jason-1, HY-2A, CryoSat-2, and SRL/DP [33]. The gravity anomaly derived from the deflection of the vertical has a precision of 2.78 mGal[34],which indicates the accuracy is high. For comparison, the gravity anomaly from Technical University of Denmark (DTU) with a resolution of 1′× 1′is also used to derive the bathymetry. According to Ref.[35], the DTU gravity anomaly data sets have very high accuracy.Fig. 4 shows the distribution of gravity anomaly and the deflection of the vertical.

According to Figs. 4(a) and 3(d), gravity anomaly distribution has similar characteristics as ETOPO1, such as the central“plateau” (16°N - 17°N, 113°E - 115°E). These similarities indicate the high coherence between the two data sets.Although the similarities between bathymetry and the deflection of the vertical are not strong as in Figs. 4(a) and 3(d), there is always a large variation in vertical deflection in areas with large topographic variations. This denotes that the bathymetry and the deflection of the vertical are also highly correlated, but their relationship is not linear. This is why we define the new quantity in Equation (7).

4. Results and analysis

4.1. Coherence analysis results

Before the bathymetry inversion, we firstly conducted the coherence analysis. To avoid the impact of insufficient ship survey data on the calculation,GEBCO(Grid General Bathymetric Chart of the Oceans)2020 grid data[34]are used for the analysis.Moreover,the northern region of the study area is shallow, especially in the northwest corner. The coherence between altimetry results and depths is weak in this area. Thus, only the region within latitudes 12°～16°is selected for the coherence analysis. It also needs to be noted that ξ′and η′used in this analysis are not the initial vertical deflection.Thus,the final bathymetry inversion is based on Equation(8) using linear regression. Fig. 5 shows the coherence analysis result.

Fig. 3. (a) Location of the study area; (b) the ship tracks (blue dots: control points; red dots: test points); (c) bathymetry from ship-borne depths; and (d) ETOPO1.

Fig. 4. Distribution of gravity anomaly and the deflection of the vertical: (a) Δg, (b) ξ and (c) η.

It can be seen from Fig. 5 that strong coherence exists in the 20-100 km band.The coherence between the north component of vertical deflections and bathymetry is close to that between gravity anomaly and bathymetry. However, the coherence of the east component of vertical deflections is slightly weaker than that of the north component at wavelengths shorter than 100 km. In summary, there are no significant differences between the three coherences. The main reason is that both components of vertical deflections and gravity anomaly reflect the first derivative signals of gravity potential.

Fig. 5. Coherence (Ca2b) between GEBCO_2020 Grid and the gravity signals (ξ′, η′ and Δg).

4.2. Inversion results

According to the data processing flowchart shown in Fig. 2, a band-pass filtering should be conducted on gravity anomaly,ξ′and η′.Based on the results of Section 4.1,20-100 km is selected as the pass band, as shown in Fig. 6. The signals outside this band are provided by ship-borne depths data, which can be obtained from the filtering process. Fig. 7 presents the results from the filtering.Gravity anomaly,ξ′and η′show abundant short wavelength signals.

Fig. 6. Band-pass filter used in this study.

With signals in the pass band, three linear regressions are conducted using ship-borne depths as the dependent variable and each of the three gravity signals as separate independent variables.To remove the effect of gross errors, the linear regression is complemented twice for each pair of variables,i.e.,Δg and depth,ξ′and depth,η′and depth. The mean and standard deviations of the differences between the predicted values and ship-borne depths are obtained from the first fitting results.And then points at which the ship-borne depths deviate from the predicted values by three times of standard deviations are removed.The ratios of the deleted data for gravity anomaly,ξ′and η′are about 2.09%,2.10%and 1.96%,respectively.The final fitting results are shown in Fig.8,in which ξ′and η′are divided by γ to transform their units to arcsecond for visualization purposes. Gravity anomaly, ξ′and η′are all linearly correlated with ship-borne depths in the pass wave band.Based on the fitting parameters, the bathymetric models are obtained(Fig. 9).

4.3. Precision assessment

(1) Evaluation using ship-borne depths

Precision assessments are conducted by comparing the differences between the derived bathymetry and ship-borne depths at the control and test points.In this study,the numbers of control and test points are 112,705 and 12,523, respectively. Tables 1 and 2 show the results.

To remove the impact of the gross errors, the values that deviated from the ship-borne depths by three times the std were removed. The removal ratios are given in Tables 1 and 2. Then the statistics were conducted to the rest of the data. The inversion results based on gravity anomaly data are better than those based on the vertical deflections. This is mainly because gravity anomaly is derived from both components of vertical deflections.Thus,gravity anomaly products contain more signals than a single component of the vertical deflection. However, it should be noted that the inversion results based on ξ′are very close to those based on gravity anomaly data,and their differences are less than 3 m in terms of the mean and std values.According toTables 1 and 2,the results from η′lead to larger errors compared to Δg and ξ′.We attribute this to the east component of the vertical deflection having lower precision than the north component.

Fig. 7. Depth and gravity field data after filtering processing: (a) Ship-borne depths outside of the pass band; Band-pass filtered (b) Δg, (c) ξ′, (d) η′.

Fig. 8. Fitting results between gravity anomaly, the deflection of the vertical and water depths: (a) Δg; (b) ξ′; (c) η′.

Fig. 9. Bathymetry: (a) ETOPO1; (b) derived from gravity anomaly; (c) derived from ξ; (d) derived from η.

Table 1 Statistics of differences between the predicted and ship-borne depths at the control points.

Table 2 Statistics of differences between the predicted and ship-borne depths at the test points.

Fig.10. Point ratio according to error magnitudes.

In this study,the ship-borne depths are used as the true values,and the differences between the derived results and the corresponding ship-borne depths are defined as the inversion errors. To analyze the error distribution, Fig.10 presents the cumulative ratio of the error magnitudes for the test points, and Fig.11 shows the precision variation with water depths. Fig. 11(a)-(c)present the error mean, std values and the ratio of the number of points with depths in the ranges of 0-1000 m, 1000-2000 m,2000-3000 m, 3000-4000 m, and 4000-5000 m, respectively.The precision of bathymetry derived from the vertical deflection is very close to that of gravity anomaly, especially those from the north component. According to Fig.11, almost all the bathymetry models have large errors in shallow depths. However, the ratio of the water depths deeper than 3000 m exceeds 80%, which means the area of shallow water only occupies a small part. For 3000-5000 m depths, results derived from gravity field products are very close to those of ETOPO1. Table 3 shows the related statistics.

The depths derived from gravity field products have large errors in shallow waters. One major reason is that the gravity field products have low signal-to-noise ratio in this area. According to Fig. 11(a), system errors exist in terms of error means. For the shallower area, the mean values of the error are all negative;however, for the deeper sea, e.g., deeper than 4000 m, the error mean value is positive (Table 3). This phenomenon could be attributed to using the same fitting parameter for the whole area,i.e., the intercept and slope of the fitted lines in Fig. 7. Indeed, the density contrast and mean water depths both change with variations in the inversion area. Because of this, the parameters of the linear regression should also be changed correspondingly. However, this is not the case for the above inversion. For gravity anomaly and the deflection of the vertical,linear regression is used for the whole region,and the same regression parameters are used in the bathymetry inversion for the entire region irrespective of terrain variations. Therefore, for the whole study area, the mean error is not large;but for some local areas,systematic errors exist.This means it is better to determine different optimal fitting parameters for different areas. One possible method is to divide the study area into smaller regions and derive the bathymetry in each subregion. Another alternative way is to determine the optimal fitting parameters in terms of water depths,which is experimented in the next section.

(2) Evaluation using other more recent bathymetry models

To further demonstrate the precisions of the derived bathymetry, three bathymetric models recently published are used as the standard values for comparisons,i.e.,DTU18BAT,GEBCO_2020 Grid,and SRTM15+V2.0.DTU18BAT is developed by DTU Space with the resolution of 1′× 1′. GEBCO_2020 Grid is the latest global bathymetric product released by GEBCO. SRTM15 + V2.0 was published in 2019 by the Scripps Institution of Oceanography [36].GEBCO_2020 Grid and SRTM15+V2.0 have the same resolution of 15′′× 15′′. These models are derived by combining multiple datasets (ship-borne depths and gravity anomalies) and have high accuracy. The differences between DTU18BAT, GEBCO_2020 Grid,SRTM15 + V2.0, and the derived bathymetry are presented in Table 4.For each comparison,the first row gives the initial statistics and the second row gives the corresponding results after removing the gross errors(they are defined as values that deviated from the mean value by three times of the initial standard deviations). In terms of the standard deviation of the differences, the gravity anomaly-derived bathymetry performs better than the vertical deflection-derived bathymetry.

5. Discussion

5.1. Effect of water depths on the fitting results

To remove the systematic errors, the fitting processing is conducted for different water depths. In other words, for different water depths,different fitting parameters are derived.Fig.12 shows the statistics of corresponding errors at different water depths. By adopting different fitting parameters for different water depths, a larger improvement in accuracy is obtained compared to Fig.11. It proves that the results derived from the north component of the vertical deflection are closer to gravity anomaly than the east component. It also shows that the inversion error is larger in the shallow water regions(usually in coastal regions)than in the deep waters. For example, the error STDs derived by gravity field products are larger than 250 m when the water depth is shallower than 1500 m,but they are smaller than 200 m when the depth is deeper than 4000 m.One reason for the low accuracy in the shallow water is the low accuracy of altimetry gravity field products in shallow and coast regions [35,37]. Therefore, to improve the bathymetry inversion accuracy in these regions, one ought to improve the accuracy of the gravity field products.

Fig.11. Error variation with water depths: (a) error mean; (b) error std; (c) point number ratio.

5.2. Effect of weights on the combination results

To fuse the two bathymetry grids derived by both components of vertical deflection,the weighted values are obtained as follows:

where hξ, hηand hnewdenote bathymetry from ξ, η and the combined results, respectively; f is the weight of bathymetry result from ξ. If f = 1, the result is hξ, and if f = 0, the result is hη. Supposing gridded data of ship-borne depths are true values, the error standard deviations of hneware presented in Fig. 13.

According to Fig. 13, the optimal result is obtained when the weight of hξis 0.73; and the corresponding error standard deviation is 159.43 m,which is virtually equal to that of gravity anomaly(159.28 m). This result verifies the reliability of the vertical deflection due to the gravity anomalies provided by DTU with very high accuracy.To obtain the optimal result,the weights assigned to hξ and hη should be different because of the different accuracies of ξ and η.

Table 3 Statistics of differences between the predicted results and ship-borne depths at the test points within 3000～5000 m depths.

Table 4Differences between DNSC08BAT,GEBCO_2020 Grid, SRTM+ V2.0 and the derived bathymetry.

Fig.12. (a) Means and (b) standard deviations of the differences between depths from models and measurements.

Fig.13. Error standard deviations of the combined results.

The control point distribution also influences the inversion results. According to previous studies, more control points are needed in regions with large topographic variations[29],and fewer control points are required for gentle topographies.This is also true for the inversion based on the deflection of the vertical. Since the main objective of this study is bathymetry inversion based on vertical defections, the influence of the control points would be studied in future research.

Although the deflection of the vertical is derived from altimetry satellite observations,it can also be derived by combining satellite altimetry observations, shipborne gravimetry [38], and even gravity gradiometer observations. In the future, with the precision enhancement of the deflection of the vertical, the precision of the derived bathymetry would also be improved further.

6. Conclusion

This study proposed an alternative method for bathymetry inversion using the deflection of the vertical. Numerical tests verified the effectiveness of the method.The results show that the north component of the vertical deflection derived more accurate bathymetry than the east component. Although the precision derived from the deflection of the vertical is slightly lower than that from the gravity anomaly, it still can be a supplement since the derivation of gravity anomaly needs additional data and calculation efforts. If the vertical deflections from altimetry observations are used to derive bathymetry,it is better to assign a greater weight to the north component since it usually has higher precision than the east component.

Authors'contributions

Conceptualization and investigation: Xiaoyun Wan, Bo Liu and Xiaohong Sui; data curation and methodology: Xiaoyun Wan and Xiaohong Sui;funding:Xiaoyun Wan and Bo Liu;writing-original draft:Xiaoyun Wan and Xiaohong Sui;writing-review and editing:all the authors.All authors read and approved the final manuscript.

Availability of data and materials

The ship depth data and gravity anomaly data used in this study are available from NOAA and DTU respectively.The deflection of the vertical is from the corresponding author of Zhu et al. [33].

Conflicts of interest

The authors declare that there is no conflicts of interest.

Acknowledgments

We are very grateful to Prof.Guo Jinyun for kindly providing the deflection of the vertical in the South China Sea. We would like to thank DTU for providing gravity anomaly, bathymetry model; and also thank NOAA and GEBCO Compilation Group for providing shipborne depth datasets and bathymetry grids. This research was funded by the National Natural Science Foundation of China (No.42074017,41674026);Fundamental Research Funds for the Central Universities(No.2652018027);Open Research Fund of Qian Xuesen Laboratory of Space Technology, CAST (No. GZZKFJJ2020006); National Defense Science and Technology Innovation Special Zone Project and Qian Xuesen Lab DFH Sat. Co. Joint Research and Development Fund under grants (M-2017-006); China Geological Survey (No. 20191006).