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Bathymetry predicting using the altimetry gravity anomalies in South China Sea
Geodesy and Geodynamics xxx (2017) 1e6
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Geodesy and Geodynamics journal homepages: www.keaipublishing.com/en/journals/geog; http://www.jgg09.com/jweb_ddcl_en/EN/volumn/home.shtml
Bathymetry predicting using the altimetry gravity anomalies in South China Sea Zhongmiao Sun a, b, Mingda Ouyang a, c, *, Bin Guan a, b a
Stake Key Laboratory of Geo-information Engineering, Xi'an 710054, China Xi'an Research Institute of Surveying and Mapping, Xi'an 710054, China c Xi'an Division of Surveying and Mapping, Xi'an 710054, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2016 Received in revised form 7 July 2017 Accepted 9 July 2017 Available online xxx
In South China Sea (112 Ee119 E, 12 Ne20 N), 81159 ship soundings published by NGDC (National Geophysics Data Center) and the altimetry gravity anomalies published by SIO (Scripps Institute of Oceanography) were used to predict bathymetry by GGM (gravity-geologic method) and SAS (Smith and Sandwell) method respectively. The residual 40576 ship soundings were used to estimate precisions of the predicted bathymetry models. Results showed that: the standard deviation of difference between the GGM model and ship soundings was 59.75 m and the relative accuracy was 1.86%. The SAS model is 60.07 m and 1.87%. The power spectral densities of the ETOPO1, SIO, GGM and SAS models were also compared and analyzed. At last, we presented an integrated bathymetry model by weighted averaging method, the weighted factors were determined by precisions of the ETOPO1, SIO, GGM, and SAS model respectively. © 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Gravity-geologic method Smith and Sandwell method Bathymetry Gravity anomaly Power spectral density analysis
1. Introduction Seafloor topography or bathymetry has traditionally been mapped by shipborne echo sounding measurements, which were time-consuming and cost a lot [1,2]. However, the advent of satellite radar altimetry had made it possible to estimate global bathymetry economically and accurately [3e5]. In 1960s, Parker showed how a series of Fourier transforms can be used to calculate the magnetic or gravity anomaly caused by an uneven, nonuniform layer of material. The invented Parker formula changed the discontinuity of models inversed by equivalent source method, and improved the computational efficiency [6]. Watts analyzed the relationship between gravity and bathymetry on 14 profiles of the
* Corresponding author. E-mail addresses: [email protected] (Z. Sun), [email protected] (M. Ouyang). Peer review under responsibility of Institute of Seismology, China Earthquake Administration.
Production and Hosting by Elsevier on behalf of KeAi
Hawaiian Emperor Seamount chain using the Cross-spectral technique, and the transfer functions between gravity and bathymetry had been used to evaluate the state of isostasy along the chain. These functions can also be explained by simple models in which the oceanic lithosphere is treated as a thin elastic plate overlying a weak fluid [7]. Based on the Parker formula and Watts plates theoretical model, Smith and Sandwell proposed a new technology to predict bathymetry by altimetry gravity anomalies and ship soundings, which made use of the Geosat GM data fully, and avoided the structure compensation assumptions. The same method was used by Smith and Sandwell again in 1997, the improvement was the differences between ship soundings and bathymetry model were gridded to be a correction to the original model [8]. Mingzhang Hu [9,10] formed a global bathymetry model using vertical gravity gradient anomalies and ship soundings based on the response function, this method was improved in 2015, he combined ship sounding, gravity anomalies and vertical gravity gradient anomalies, formed a 10 10 bathymetry model over China sea and its adjacent areas. The GGM (Gravity Geological method) was originally developed for predicting the basement depth of low density glacial drift deposits [11], but as the subsurface material density was always changing, the GGM method is not suitable in land areas. The
https://doi.org/10.1016/j.geog.2017.07.003 1674-9847/© 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
2
Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
density difference between the oceanic crust and seawater changed small, this method is quite suitable to predict bathymetry by altimetry gravity anomalies. Presently, this method had been successfully used to inverse bathymetry in East Sea of Japan, Emperor Seamount, South of Greenland and Southern Area of Alaska. The precision of inversed model even reached 20e40 m [12e16]. In this paper, we constructed 10 10 bathymetry models in South China Sea (112 Ee119 E, 12 Ne20 N), by the GGM method and the SAS method, the input datum were altimetry gravity anomalies published by SIO and ship soundings published by NGDC, we compared the results and presented an integrated model at last.
i i i gshort ¼ gobs glong
(4)
By rearranging Eq. (2), the water depth at i can be:
Ei ¼
i gshort þD 2pGDr
(5)
2. Methodology 2.1. GGM (Geology Gravity Method) The relationship between the free air gravity anomaly and bathymetry is nonlinear. In general, a nonlinear problem can be linearized by defining a suitable “reference field” and “residual field” in many geodetic computations [17]. Thus, the altimetry gravity anomalies on the sea surface (gobs) can be divided into a short wavelength residual field (gshort) generated by seafloor undulation and a long wavelength reference field (glong) generated by deeper mass variations:
gobs ¼ gshort þ glong
(1)
Fig. 1 illustrates the principle of GGM, and jn (n ¼ 1,2,3…) are the control points with ship sounding depths. The control points jn in Fig. 1 are used to estimate the residual gravity field that generated from a simple Bouguer slab formula:
jn gshort ¼ 2pGDr Ejn D
(2)
where, G is a gravitational constant (6:672 108 cm3 =gs2 ),Dr is the density contrast between seawater and bedrock,Ejn is the water depth at the jn , D is the reference datum depth, which is usually referenced to the deepest depth at the jn control points. Both Ejn and Dare measured in meters. The reference gravity anomalies at jn can be calculated by: j
j
j
n n n glong ¼ gobs gshort
(3)
jn where glong will be gridded to a 1 min model. The reference gravity i anomaly at any point i(glong ) can be interpolated from this grid, and the residual gravity will be:
Fig. 1. Principle of the GGM
Fig. 2. The gravity anomalies and the shipboard cruises in study areas.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
The geology gravity method is based on an assumption that the density contrast must not change all over the seafloor, the contrast determination is the key to the method. 2.2. SAS (Smith and Sandwell) method
3
A is a contrast, the choice of its value must be considered rigorously and should represent an appropriate compromise between noise reduction and signal detail. The HðkÞ is band pass filtered version of the observed bathymetric grid B0 ðkÞ:
The gravity anomaly is introduced by seafloor undulation in a certain spectrum band. Smith and Sandwell defined the total predicted bathymetry as the sum of 15e160 km band-pass prediction and the long wavelength depth dlp ðxÞ in a region [8]:
HðkÞ ¼ B0 ðkÞWðkÞ
depth ¼ dlp ðxÞ þ SðxÞgðxÞ
where, DLP ðkÞ is Fourier transform of dlp ðxÞ, B0 ðkÞ is bathymetry model gridded from ship soundings in the frequency domain.
(6)
where, gðxÞ is the band-pass-filtered and downward continued gravity,sðxÞ is the scaling factor. In the frequency domain, gðxÞ will be:
GðkÞ ¼ G0 ðkÞWðkÞexpð2pkdÞ
(7)
where GðkÞ is the Fourier transformed g(x), G0 ðkÞ is the observed gravity anomalies in the frequency domain, WðkÞ is a band-pass filter which stabilizes the prediction problem. WðkÞ is defined by a high-pass and a low-pass filter:
WðkÞ ¼ W1 ðkÞW2 ðkÞ
(8)
i h W1 ðkÞ ¼ 1 exp 2ðpksÞ2
(9)
W1 ðkÞ is a high-pass filter, k is the circular frequencyð¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2x þ k2y Þ,
ðkx ; ky Þ ¼ ð1=lx ; 1=ly Þ, ðkx ; ky Þ and lx ; ly is the frequency and wavelength in x and y direction, s is the wiener filter parameters. W2 ðkÞ is a low-pass filter:
h i1 W2 ðkÞ ¼ 1 þ AK 4 expð4pksÞ
(10)
(11)
And then, the B0 ðkÞ are low-pass filtered by
DLP ðkÞ ¼ B0 ðkÞ½1 W1 ðkÞ
(12)
3. Result of bathymetry prediction and comparison In South China Sea (112 Ee119 E, 12 Ne20 N), ship soundings are download from NGDC (National geophysical data center), as showed in Fig. 2(a), some of them are gross errors, and should be deleted manually first. Two-third of the ship soundings (81159 points) are used to determine the bathymetry at long wavelength, Dr and sðxÞ. The other one-third of the ship soundings (40576 points) are used to evaluate the accuracy of results. Altimetry gravity anomalies are download from SIO, as showed in Fig. 2(b), the observed gravity anomalies in control points are obtained by interpolation. The density contrast between seawater and bedrock Dr of GGM method is determined by iterative method. Bathymetry models calculated with different density contrasts, and then comparing with ship soundings. The optimum density contrast was obtained when the STD of differences between predicted model and ship soundings was minimum and their correlation coefficient was maximum [16]. Ouyang Mingda given the best contrast value in South China Sea, which is 1.32 gcm3 [18]. The wavelength of band-pass-filter in SAS method is 10e120 km, the linear regression technique is used to calculate scale factor in every 2 2 windows. Fig. 3 showed the
Fig. 3. The scale factors between residual gravity anomalies and residual depths.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
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Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
distribution of scale factors and the scatter between residual gravity anomalies and residual depths. The right-upper sketch in Fig. 3 illustrates a linear regression of residual gravity anomalies and water depths in row three column three, there are 5860 points, the scale factor is 17.15 m/mGal. The right-lower sketch shows another example in row two column four, there are 8990 points, the scale factor is 18.30 m/mGal. ETOPO1 is a 10 resolution global topographic model published by NGDC (National Geophysical Data Center) in 2008. Bathymetric data sets used in compiling the model were obtained from the JODC (Japan Oceanographic Data Center), NGDC, the CEP (Caspian Environment Programme), and the CIESM (Mediterranean Science Commission). The SIO v24.1model is 10 resolution and issued by Scripps Institution of Oceanography. Fig. 4 showed the SAS, GGM models and the existing ETOPO1, SIO models in South China Sea,
the white parts in these figures were areas above the sea surface. Fig. 5 showed the differences among these models. The difference between ETOPO1 and SIO models was the biggest, areas with large difference were seamounts existed. The GGM and SAS models are similar to ETOPO1 model, their difference was a little. Depths interpolated from these models are compared with measured depths at check points, the statistics of their differences were showed in Table 1. The precisions of the SAS, GGM models are better than ETOPO1 and SIO models. The reason of similar precisions between GGM and SAS models was due to the same control points used in constructing the short wavelength bathymetry models. Fig. 6 showed the power spectral density of bathymetry models. In long wavelength, differences among models were little, in less than 10 km wavelength, the SAS and GGM models are all higher than ETOPO1 and SIO models, actually, there may be ghost signal at
Fig. 4. The bathymetry map.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
5
Fig. 5. The difference among models.
wavelength shorter than 10 km (gridded from residual ship depths), because of the spars distribution of ship soundings, but the ghost signal can't be avoided automatically, so they looked like have a better precision bathymetry model than ETOPO1 and SIO models.
4. The establishment of integrated model The geology gravity method was suitable to inverse bathymetry model in large-areas, but the precision depended on the number
Table 1 The precision statistics of ETOPO1, SIO, SAS, GGM models.
GGM ETOPO1 SAS SIO
Min(m)
Max(m)
Mean(m)
Std(m)
Relative precision
1815.36 1055.41 1819.77 1649.87
1180.58 963.75 1182.62 1654.74
0.23 2.26 0.19 5.38
59.75 154.08 60.07 107.72
1.86% 4.81% 1.87% 3.37%
Fig. 6. The power spectral density analysis.
Fig. 7. The comprehensive bathymetry model.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
6
Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
and distribution of ship soundings more. The SAS method could be used to construct bathymetry model in 2 2 window, the precision may be decreased by boundary effects. We established an integrated model use the GGM/SAS/ETOPO1/SIO models. The weighted averaging method was compatible for different quality data. The poor quality models are given less weight to decline their contribution to the integrated model. According to the precision of different models, the weighting factors of GGM, ETOPO1, SAS and SIO models were 1=59:752 , 1=154:082 , 1=60:072 , 1=107:722 respectively, the integrated model was showed in Fig. 7, and its relative accuracy was 1.9%.
5. Conclusions Bathymetry models were obtained in this article by SAS method and GGM method, an integrated bathymetry model was also presented by weighted averaging method. The results lead us to the following conclusions: (1) Gravity anomaly in 10e120 km wavelength was used in SAS method. This method divided bathymetry model into long (>120 km), median (10e120 km) and short (<10 km) wavelength items. The residual bathymetry model at shorter than 10 km wavelength was gridded from residual ship borne data, this method was calculated in frequency domain, the edge effect should been taken into account. (2) The density difference constant between oceanic crust and seawater in GGM method need to be determined first, which was the key of this method, the precision of GGM model is good, but influenced very much by the distribution and amount of ship borne soundings. . (3) An integrated bathymetry model can be constructed by weighted average method using GGM, SAS and existing models, this method gives small weights to poor quality models, which could be used to reduce their contributions to the integrated model and improve the integrated model's precision and reliability.
Acknowledgement This work is supported by the State Key Laboratory of Geoinformation Engineering (SKLGIE2015-M-1-2, SKLGIE2016-M3-2), the National Natural Science Foundation of China (41674082).
References [1] Luo Jia, Bathymetry Inversion Combining Satellite Altimetry and Marine Gravity Data [Master Thesis], Wuhan University, Wuhan, 2000. [2] G. Ramillien, A. Cazenave, Global bathymetry derived from altimeter data of the ERS-1 geodetic mission, Journal of Geodynamics 23 (2) (1997) 129e149. [3] Li Dawei, Li Jiancheng, Feng Hai, et al., Research on inversion of seafloor topography of China Sea and its adjacent areas from satellite altimetric data, Journal of Geodesy and Geodynamics 29 (4) (2009) 70e73. [4] D.T. Sandwell, W.H.F. Smith, Gille Sarah, Bathymetry from space: rationale and requirements for a new, high-resolution altimetry mission, C.R. Geoscience 338 (2006) 1049e1062. [5] Wang Yong, Xu Houze, Zhan Jingang, High resolution seafloor topography of China sea and its adjacent areas, Chinese Science Bulletin 46 (11) (2001) 956e960. [6] R.L. Parker, The rapid calculation of potential anonmalies, Journal of Geophysical Research 31 (1972) 447e455. [7] A.B. Watts, An analysis of isostasy in the World's Oceans: 1 Hawaiian emperor seament chain, Journal of Geopyhsical Research 83 (1978) 5989e6004. [8] W.H.F. Smith, D.T. Sandwell, Bathymetric prediction from dense satellite altimetry and sparse shipboard bathymetry, Journal of Geopyhsical Research 99 (B11) (1994) 21803e21824. [9] Hu Mingzhang, Li Jiangcheng, Xing Lelin, Global bathymetry model predicted from vertical gravity gradient anomalies, Acta Geodaetica et Cartographica Sinica 43 (6) (2014), 558e565,574. [10] Hu Mingzhang, Li Jiancheng, Xu Xinyu, et al., Recovery of bathymetry over China Sea and its adjacent areas by combination of Multi-source data, Geomatics and Information Science of Wuhan University 9 (40) (2015) 1270e1273. [11] A. Ibrahim, W.J. Hinze, Mapping buried bedrock topography with gravity, Ground Water 10 (3) (1972) 18e23. [12] J.M. Adams, W.J. Hinze, The gravity-geologic technique of mapping buried bedrock topography, in: S.H. Ward (Ed.), Geotechnical and Environmental Geophysics, Investigations in Geophysics Series 5, Society of Exploration Geophysicists, 1990, pp. 99e105. [13] R. Nagarajan, Gravity-geologic Investigation of Buried Bedrock Topography in Northwestern Ohio, Master Thesis, Department of Geological Sciences, The Ohio State University, Columbus, OH, USA, 1994. [14] K.B. Kim, Y.S. Hsiao, J.W. Kim, et al., Bathymetry enhancement by altimetryderived gravity anomalies in the East Sea(Sea of Japan), Marine Geophysical Research 31 (2010) 285e298. [15] Y.S. Hsiao, J.W. Kim, K.B. Kim, et al., Bathymetry estimation by gravitygeologic method: investigation of density contrast predicted by downward continuation, Terrestrial, Atmospheric and Oceanic Sciences 22 (2010) 347e358. [16] Hu Minzhang, Li Jiancheng, Jin Taoyong, Bathymetry inversion with gravity geologic method in emperor seamount, Geomatics and Information Science of Wuhan University 37 (5) (2012), 610e612, 629. [17] C.A. Hwang, Bathymetric model for the South China sea from satellite altimetry and depth data, Marine Geodesy 22 (1999) 37e51. [18] Mingda OuYang, Zhongmiao Sun, Zhenhe Zhai, Predicting bathymetry in South China sea using the gravity-geologic method, Chinese Journal of Geodesy 57 (9) (2014) 1756e1765.
Zhongmiao Sun, is a researcher in Xi'an Research Institute of Surveying and Mapping. He obtained his doctoral degree from Information Engineering University, majored in theory and method of airborne gravimetry.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
Contents lists available at ScienceDirect
Geodesy and Geodynamics journal homepages: www.keaipublishing.com/en/journals/geog; http://www.jgg09.com/jweb_ddcl_en/EN/volumn/home.shtml
Bathymetry predicting using the altimetry gravity anomalies in South China Sea Zhongmiao Sun a, b, Mingda Ouyang a, c, *, Bin Guan a, b a
Stake Key Laboratory of Geo-information Engineering, Xi'an 710054, China Xi'an Research Institute of Surveying and Mapping, Xi'an 710054, China c Xi'an Division of Surveying and Mapping, Xi'an 710054, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2016 Received in revised form 7 July 2017 Accepted 9 July 2017 Available online xxx
In South China Sea (112 Ee119 E, 12 Ne20 N), 81159 ship soundings published by NGDC (National Geophysics Data Center) and the altimetry gravity anomalies published by SIO (Scripps Institute of Oceanography) were used to predict bathymetry by GGM (gravity-geologic method) and SAS (Smith and Sandwell) method respectively. The residual 40576 ship soundings were used to estimate precisions of the predicted bathymetry models. Results showed that: the standard deviation of difference between the GGM model and ship soundings was 59.75 m and the relative accuracy was 1.86%. The SAS model is 60.07 m and 1.87%. The power spectral densities of the ETOPO1, SIO, GGM and SAS models were also compared and analyzed. At last, we presented an integrated bathymetry model by weighted averaging method, the weighted factors were determined by precisions of the ETOPO1, SIO, GGM, and SAS model respectively. © 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Gravity-geologic method Smith and Sandwell method Bathymetry Gravity anomaly Power spectral density analysis
1. Introduction Seafloor topography or bathymetry has traditionally been mapped by shipborne echo sounding measurements, which were time-consuming and cost a lot [1,2]. However, the advent of satellite radar altimetry had made it possible to estimate global bathymetry economically and accurately [3e5]. In 1960s, Parker showed how a series of Fourier transforms can be used to calculate the magnetic or gravity anomaly caused by an uneven, nonuniform layer of material. The invented Parker formula changed the discontinuity of models inversed by equivalent source method, and improved the computational efficiency [6]. Watts analyzed the relationship between gravity and bathymetry on 14 profiles of the
* Corresponding author. E-mail addresses: [email protected] (Z. Sun), [email protected] (M. Ouyang). Peer review under responsibility of Institute of Seismology, China Earthquake Administration.
Production and Hosting by Elsevier on behalf of KeAi
Hawaiian Emperor Seamount chain using the Cross-spectral technique, and the transfer functions between gravity and bathymetry had been used to evaluate the state of isostasy along the chain. These functions can also be explained by simple models in which the oceanic lithosphere is treated as a thin elastic plate overlying a weak fluid [7]. Based on the Parker formula and Watts plates theoretical model, Smith and Sandwell proposed a new technology to predict bathymetry by altimetry gravity anomalies and ship soundings, which made use of the Geosat GM data fully, and avoided the structure compensation assumptions. The same method was used by Smith and Sandwell again in 1997, the improvement was the differences between ship soundings and bathymetry model were gridded to be a correction to the original model [8]. Mingzhang Hu [9,10] formed a global bathymetry model using vertical gravity gradient anomalies and ship soundings based on the response function, this method was improved in 2015, he combined ship sounding, gravity anomalies and vertical gravity gradient anomalies, formed a 10 10 bathymetry model over China sea and its adjacent areas. The GGM (Gravity Geological method) was originally developed for predicting the basement depth of low density glacial drift deposits [11], but as the subsurface material density was always changing, the GGM method is not suitable in land areas. The
https://doi.org/10.1016/j.geog.2017.07.003 1674-9847/© 2017 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
2
Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
density difference between the oceanic crust and seawater changed small, this method is quite suitable to predict bathymetry by altimetry gravity anomalies. Presently, this method had been successfully used to inverse bathymetry in East Sea of Japan, Emperor Seamount, South of Greenland and Southern Area of Alaska. The precision of inversed model even reached 20e40 m [12e16]. In this paper, we constructed 10 10 bathymetry models in South China Sea (112 Ee119 E, 12 Ne20 N), by the GGM method and the SAS method, the input datum were altimetry gravity anomalies published by SIO and ship soundings published by NGDC, we compared the results and presented an integrated model at last.
i i i gshort ¼ gobs glong
(4)
By rearranging Eq. (2), the water depth at i can be:
Ei ¼
i gshort þD 2pGDr
(5)
2. Methodology 2.1. GGM (Geology Gravity Method) The relationship between the free air gravity anomaly and bathymetry is nonlinear. In general, a nonlinear problem can be linearized by defining a suitable “reference field” and “residual field” in many geodetic computations [17]. Thus, the altimetry gravity anomalies on the sea surface (gobs) can be divided into a short wavelength residual field (gshort) generated by seafloor undulation and a long wavelength reference field (glong) generated by deeper mass variations:
gobs ¼ gshort þ glong
(1)
Fig. 1 illustrates the principle of GGM, and jn (n ¼ 1,2,3…) are the control points with ship sounding depths. The control points jn in Fig. 1 are used to estimate the residual gravity field that generated from a simple Bouguer slab formula:
jn gshort ¼ 2pGDr Ejn D
(2)
where, G is a gravitational constant (6:672 108 cm3 =gs2 ),Dr is the density contrast between seawater and bedrock,Ejn is the water depth at the jn , D is the reference datum depth, which is usually referenced to the deepest depth at the jn control points. Both Ejn and Dare measured in meters. The reference gravity anomalies at jn can be calculated by: j
j
j
n n n glong ¼ gobs gshort
(3)
jn where glong will be gridded to a 1 min model. The reference gravity i anomaly at any point i(glong ) can be interpolated from this grid, and the residual gravity will be:
Fig. 1. Principle of the GGM
Fig. 2. The gravity anomalies and the shipboard cruises in study areas.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
Z. Sun et al. / Geodesy and Geodynamics xxx (2017) 1e6
The geology gravity method is based on an assumption that the density contrast must not change all over the seafloor, the contrast determination is the key to the method. 2.2. SAS (Smith and Sandwell) method
3
A is a contrast, the choice of its value must be considered rigorously and should represent an appropriate compromise between noise reduction and signal detail. The HðkÞ is band pass filtered version of the observed bathymetric grid B0 ðkÞ:
The gravity anomaly is introduced by seafloor undulation in a certain spectrum band. Smith and Sandwell defined the total predicted bathymetry as the sum of 15e160 km band-pass prediction and the long wavelength depth dlp ðxÞ in a region [8]:
HðkÞ ¼ B0 ðkÞWðkÞ
depth ¼ dlp ðxÞ þ SðxÞgðxÞ
where, DLP ðkÞ is Fourier transform of dlp ðxÞ, B0 ðkÞ is bathymetry model gridded from ship soundings in the frequency domain.
(6)
where, gðxÞ is the band-pass-filtered and downward continued gravity,sðxÞ is the scaling factor. In the frequency domain, gðxÞ will be:
GðkÞ ¼ G0 ðkÞWðkÞexpð2pkdÞ
(7)
where GðkÞ is the Fourier transformed g(x), G0 ðkÞ is the observed gravity anomalies in the frequency domain, WðkÞ is a band-pass filter which stabilizes the prediction problem. WðkÞ is defined by a high-pass and a low-pass filter:
WðkÞ ¼ W1 ðkÞW2 ðkÞ
(8)
i h W1 ðkÞ ¼ 1 exp 2ðpksÞ2
(9)
W1 ðkÞ is a high-pass filter, k is the circular frequencyð¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2x þ k2y Þ,
ðkx ; ky Þ ¼ ð1=lx ; 1=ly Þ, ðkx ; ky Þ and lx ; ly is the frequency and wavelength in x and y direction, s is the wiener filter parameters. W2 ðkÞ is a low-pass filter:
h i1 W2 ðkÞ ¼ 1 þ AK 4 expð4pksÞ
(10)
(11)
And then, the B0 ðkÞ are low-pass filtered by
DLP ðkÞ ¼ B0 ðkÞ½1 W1 ðkÞ
(12)
3. Result of bathymetry prediction and comparison In South China Sea (112 Ee119 E, 12 Ne20 N), ship soundings are download from NGDC (National geophysical data center), as showed in Fig. 2(a), some of them are gross errors, and should be deleted manually first. Two-third of the ship soundings (81159 points) are used to determine the bathymetry at long wavelength, Dr and sðxÞ. The other one-third of the ship soundings (40576 points) are used to evaluate the accuracy of results. Altimetry gravity anomalies are download from SIO, as showed in Fig. 2(b), the observed gravity anomalies in control points are obtained by interpolation. The density contrast between seawater and bedrock Dr of GGM method is determined by iterative method. Bathymetry models calculated with different density contrasts, and then comparing with ship soundings. The optimum density contrast was obtained when the STD of differences between predicted model and ship soundings was minimum and their correlation coefficient was maximum [16]. Ouyang Mingda given the best contrast value in South China Sea, which is 1.32 gcm3 [18]. The wavelength of band-pass-filter in SAS method is 10e120 km, the linear regression technique is used to calculate scale factor in every 2 2 windows. Fig. 3 showed the
Fig. 3. The scale factors between residual gravity anomalies and residual depths.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003
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distribution of scale factors and the scatter between residual gravity anomalies and residual depths. The right-upper sketch in Fig. 3 illustrates a linear regression of residual gravity anomalies and water depths in row three column three, there are 5860 points, the scale factor is 17.15 m/mGal. The right-lower sketch shows another example in row two column four, there are 8990 points, the scale factor is 18.30 m/mGal. ETOPO1 is a 10 resolution global topographic model published by NGDC (National Geophysical Data Center) in 2008. Bathymetric data sets used in compiling the model were obtained from the JODC (Japan Oceanographic Data Center), NGDC, the CEP (Caspian Environment Programme), and the CIESM (Mediterranean Science Commission). The SIO v24.1model is 10 resolution and issued by Scripps Institution of Oceanography. Fig. 4 showed the SAS, GGM models and the existing ETOPO1, SIO models in South China Sea,
the white parts in these figures were areas above the sea surface. Fig. 5 showed the differences among these models. The difference between ETOPO1 and SIO models was the biggest, areas with large difference were seamounts existed. The GGM and SAS models are similar to ETOPO1 model, their difference was a little. Depths interpolated from these models are compared with measured depths at check points, the statistics of their differences were showed in Table 1. The precisions of the SAS, GGM models are better than ETOPO1 and SIO models. The reason of similar precisions between GGM and SAS models was due to the same control points used in constructing the short wavelength bathymetry models. Fig. 6 showed the power spectral density of bathymetry models. In long wavelength, differences among models were little, in less than 10 km wavelength, the SAS and GGM models are all higher than ETOPO1 and SIO models, actually, there may be ghost signal at
Fig. 4. The bathymetry map.
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Fig. 5. The difference among models.
wavelength shorter than 10 km (gridded from residual ship depths), because of the spars distribution of ship soundings, but the ghost signal can't be avoided automatically, so they looked like have a better precision bathymetry model than ETOPO1 and SIO models.
4. The establishment of integrated model The geology gravity method was suitable to inverse bathymetry model in large-areas, but the precision depended on the number
Table 1 The precision statistics of ETOPO1, SIO, SAS, GGM models.
GGM ETOPO1 SAS SIO
Min(m)
Max(m)
Mean(m)
Std(m)
Relative precision
1815.36 1055.41 1819.77 1649.87
1180.58 963.75 1182.62 1654.74
0.23 2.26 0.19 5.38
59.75 154.08 60.07 107.72
1.86% 4.81% 1.87% 3.37%
Fig. 6. The power spectral density analysis.
Fig. 7. The comprehensive bathymetry model.
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and distribution of ship soundings more. The SAS method could be used to construct bathymetry model in 2 2 window, the precision may be decreased by boundary effects. We established an integrated model use the GGM/SAS/ETOPO1/SIO models. The weighted averaging method was compatible for different quality data. The poor quality models are given less weight to decline their contribution to the integrated model. According to the precision of different models, the weighting factors of GGM, ETOPO1, SAS and SIO models were 1=59:752 , 1=154:082 , 1=60:072 , 1=107:722 respectively, the integrated model was showed in Fig. 7, and its relative accuracy was 1.9%.
5. Conclusions Bathymetry models were obtained in this article by SAS method and GGM method, an integrated bathymetry model was also presented by weighted averaging method. The results lead us to the following conclusions: (1) Gravity anomaly in 10e120 km wavelength was used in SAS method. This method divided bathymetry model into long (>120 km), median (10e120 km) and short (<10 km) wavelength items. The residual bathymetry model at shorter than 10 km wavelength was gridded from residual ship borne data, this method was calculated in frequency domain, the edge effect should been taken into account. (2) The density difference constant between oceanic crust and seawater in GGM method need to be determined first, which was the key of this method, the precision of GGM model is good, but influenced very much by the distribution and amount of ship borne soundings. . (3) An integrated bathymetry model can be constructed by weighted average method using GGM, SAS and existing models, this method gives small weights to poor quality models, which could be used to reduce their contributions to the integrated model and improve the integrated model's precision and reliability.
Acknowledgement This work is supported by the State Key Laboratory of Geoinformation Engineering (SKLGIE2015-M-1-2, SKLGIE2016-M3-2), the National Natural Science Foundation of China (41674082).
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Zhongmiao Sun, is a researcher in Xi'an Research Institute of Surveying and Mapping. He obtained his doctoral degree from Information Engineering University, majored in theory and method of airborne gravimetry.
Please cite this article in press as: Z. Sun, et al., Bathymetry predicting using the altimetry gravity anomalies in South China Sea, Geodesy and Geodynamics (2017), https://doi.org/10.1016/j.geog.2017.07.003