Bathymetry Inversion Using Vertical Deflections: a Case Study in South China Sea


 As an alternative method, an algorithm for bathymetry inversion using vertical deflections is proposed. Firstly, the formulas for the bathymetry inversion from north and east components of vertical deflections are derived and the data processing is introduced. Then a local area in the South China Sea is selected as an example to experiment the method. The bathymetry inversion based on gravity anomaly was also conducted for a comparison. The results show that the bathymetry derived from the north component of the vertical deflections have almost the same accuracy as that derived from gravity anomalies and the results derived from the east component have the poorest accuracy. The experiment’s results also show that accuracy of the derived bathymetry can be improved if the fitting parameters are adjusted according to the water depths. In summary, among the gravity field products used in this study, although the gravity anomaly yielded the best performance in the bathymetry inversion, the vertical defections can still be used as supplements, especially in areas where accurate vertical deflections exist. This is because deriving gravity anomaly from altimetry observations needs additional data and calculation efforts.

anomalies and the results derived from the east component have the poorest accuracy. 23 The experiment's results also show that accuracy of the derived bathymetry can be 24 improved if the fitting parameters are adjusted according to the water depths. In summary, 25 among the gravity field products used in this study, although the gravity anomaly yielded 26 the best performance in the bathymetry inversion, the vertical defections can still be used 27 as supplements, especially in areas where accurate vertical deflections exist. This is 28 because deriving gravity anomaly from altimetry observations needs additional data and 29 calculation efforts. The bathymetry inversion based on altimetry products are mainly based on gravity 41 anomaly. Parks (1972) proposed the spectral method for bathymetry inversion using 42 gravity anomaly. In spectral domain, Smith and Sandwell (1994) proposed an admittance 43 theory and predicted the bathymetry from dense gravity data and sparse shipboard 44 bathymetry for the southern oceans. The spectral method has also been verified in many 45 other areas, such as the South China Sea (Hwang 1999), the western Indian offshore 46 (Bhattachryya and Majumbar 2009), South Atlantic Ocean (Jung and Vogt 1992). In 47 high accuracy, the vertical deflections certainly contain abundant gravity information, 75 including those generated from the bathymetry. Hence, in theory it is feasible to use 76 vertical deflections to inverse bathymetry. 77 This study investigates how to derive bathymetry using vertical deflections as well as 78 its performance. Section 2 proposes the inversion method based on vertical deflections. 79 Data used and study area are introduced in Section 3. Section 4 presents the inversion 80 results, and some discussions are given in Section 5. Conclusions are made in Section 6. 81

The relationship between vertical deflections and bathymetry 83
According to Parks (1972), the disturbing gravitational potential at the ocean surface 84 ( Figure 1) has the following relationship with ocean water depths, 85 where T is the disturbing potential created by the seafloor topography, h is the 87 seafloor topography relative to the mean water depth,

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G is the gravitational constant,   represents the density contrast between the ocean 89 water and seafloor crust, often setting as 1670 kg/m 3 , 0 Z denotes the mean water depths 90 in the observation area and F means fast Fourier transform computation. And then, the gravity anomaly, g  , caused by the seafloor topography can be represented as: This is the formula which has been used widely for bathymetry inversion in the spectral 96 domain (Hu et al. 2014). Indeed, according to Equation (1), we also have 97 According to Heiskanen and Moritz (1967), 99 where  is the normal gravity. Finally, the formulas for bathymetry inversion can be According to Equations (3) and (6),   is an important parameter for the inversion. 104 According to the previous studies, its theoretical value does not always lead to the best 105 inversion results (Annan et al. 2020). Instead, it can be derived by linear regression 106 between the water depths at the control points and gravity anomaly after downward 107 continuation (Smith and Sandwell 1994; Hu et al. 2014). This method is often used in 108 bathymetry inversion with spectral methods. The advantage is that the we do not need to 109 know an accurate density contrast. Similar technique is also adopted for the vertical 7 deflections in this study. The issue is that the ocean depths do not have a linear 111 relationship with the vertical deflections even after downward continued processing. In 112 order to solve this issue, two new values are defined. 113 We call this preprocessing a scale factor correction in this study. Then we have 115 Equation (8) has the same functional relationship as Equation (2). Therefore, the 117 algorithm of bathymetry inversion based on gravity anomaly can all adopted for that 118 based on '  and '  .

Data processing flow 120
According to previous studies, the gravity field products only have high correlation with 121 the bathymetry in a certain wavelength band (Smith and Sandwell 1994). This is because 122 only part of the gravity anomaly or vertical deflections are created by the seafloor 123 topography. Hence, the gravity field products are mainly used to predict the bathymetry 124 in the wavelength in which they have a high correlation with the bathymetry. For the 125 gravity anomaly, it usually resides in the bandwidth of 20~200 km. The sensitive 126 wavelength band varies with geographic areas (Marks and Smith 2012). Although no 127 literature discussed the correlation between the vertical deflections and bathymetry, it is 128 wavelength band and thus can be used to predict bathymetry in the related band only. In 130 this study, the sensitive band will be derived based on the correlation analysis in Section 131 3. Hence, in this study, the bathymetry signals are provided by vertical deflections in the 132 sensitive band of vertical defections and provided by ship depths in other bands. The data 133 processing flow is shown in Figure 2. For comparisons, the data processing for gravity 134 anomaly is given in Figure 3. shown in the southeast corner is unlikely to exist in practice, which may be caused by the 152 gross error in the ship survey data. The topography presented by ETOPO1 are clearer than 153 Figure 5, although not every point of ETOPO1 has higher accuracy than its corresponding 154 ship-depth. According to this comparison, the three-dimensional model of the bathymetry 155 directly from ship-depths in the study area is not accurate due to the lack of ship-depths 156 and thus it would be better to combine other observations to improve the resolution. 157 According to Figures 6 and 9, gravity anomaly distribution has many similar 158 characteristics with ETOPO1, which denotes the high correlation between the two data 159 sets. For example, a sea floor plateau exists near the middle area both in Figures 6 and 9. 160 These similarities indicate the linear relationship between bathymetry and gravity 161

anomaly. 162
Although the similarities between bathymetry and vertical deflections are not 163 equally strong as in Figures 6 and 9, vertical deflections always vary largely in the area 164 where topography varies largely. This denotes that the bathymetry and vertical deflections 165 are also highly correlated, but their relationship is not linear. This is the reason why we 166 define the new quantity in Equation (7). 167

Correlation analysis 169
The correlation analysis is firstly conducted on the data as Equation (9)  The result is shown in Figure 10. It can be seen from the figure that the bands with 184 strong correlation are mainly located in the 20~100 km band. It is also found that the 185 correlation between the north component of vertical deflections and bathymetry is close 186 to that between gravity anomaly and bathymetry. However, the correlation of the east 187 component of vertical deflections is slightly weaker than that of the north component at 188 wavelengths shorter than 100km. In summary, the correlations don't have large 189 differences. The main reason is that both vertical deflections and gravity anomaly reflect 190 the first derivative signals of gravity potential.
According to data processing flow charts shown in Figures 2 and 3 . Therefore, the slopes can be used to derive density contrast information, 214 which are shown in Table 1. 215 According to Table 1 Figure 13. 223

Accuracy assessment 224
Accuracy assessments are conducted by comparing the differences between the 225 derived bathymetry and ship-depths at control and test points. Tables 2 and 3 show the 226

results. 227
In order to remove the impact of the gross errors, the values that deviated from the 228 ship-depths by three times of the initial std were removed. The removal ratios are given 229 in Tables 2 and 3. According to Tables 2 and 3, the inversion results based on gravity 230 anomaly data are better than those based on vertical deflections both at control and test 231 points. This is mainly because that the gravity anomaly products are derived using more 232 data than the vertical deflections and thus, contain more gravity field signals, such as  In order to analyze the error distribution, Figure 14 presents the cumulative ratio of 241 the error magnitudes for the test points and Figure 15 shows the accuracy variation with 242 water depths. Please note, Figure 15 The accuracy of bathymetry derived from vertical deflections is very close to that of 248 gravity anomaly, especially those from the north component. According to Figure 15, all 249 the bathymetry models have large errors in shallow depths. However, the number ratio of 250 the water depths deeper than 3000m exceeds 80%, which means shallow water area only 251 occupy a small part of the study. For the depths 3000~5000m, the gravity field derived 252 results, no matter from gravity anomaly or from the vertical deflections, are all very close 253 to those of ETOPO1. Table 4 gives the related statistics. 254 As to why bathymetries derived from gravity field products have large errors in shallow 255 14 in such area. According to the Figure 15(a), system errors exist in terms of error means. 257 For example, for the shallower area, mean values of the error are all negative, but for the 258 deeper sea, e.g., deeper than 4000m, the error mean value is positive, which is also 259 demonstrated by Table 4. This phenomenon could be attributed to the use of the same 260 fitting parameter for the whole area. Indeed, the density contrast, as well as mean water 261 depths should be changed with variation of the inversion area. However, this is not the 262 case for the above inversion. Hence, although for the whole study area, the mean error is 263 not large, but for some local areas, systematic errors exist. This means it is better to 264 determine different optimal fitting parameters for different areas. One possible method is 265 to divide the study area to some smaller regions and inverse the bathymetry in each 266 subregion. Another alternative method is to determine the optimal fitting parameters in 267 terms of water depths, which is experimented in the next section. 268

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In order to remove the systematic errors, the fitting processing are conducted for different 270 water depths. In other words, for different water depths, different fitting parameters are 271 derived. Figure 16 shows the derived density contrasts for different water depths in the 272 study area, and Figure 17 shows the statistics of corresponding errors like in Figure 15. 273 Since the derived density contrasts varied with water depths, the fitting parameters varies 274 correspondingly. This leads to large accuracy improvements compared to Figure 15 vertical deflections are also important products of the Earth gravity field and certainly 298 and also the topic of this study. Therefore, we suggest that scientists in this field release 300 global vertical deflection products like they do for gravity anomaly products (