Assessment of theoretical approaches to derivation of internal solitary wave parameters from multi-satellite images near the Dongsha Atoll of the South China Sea
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Abstract: This study assesses the accuracy and the applicability of the Korteweg-de Vries (KdV) and the nonlinear Schrödinger (NLS) equation solutions to derivation of dynamic parameters of internal solitary waves (ISWs) from satellite images. Visible band images taken by five satellite sensors with spatial resolutions from 5 m to 250 m near the Dongsha Atoll of the northern South China Sea (NSCS) are used as a baseline. From the baseline, the amplitudes of ISWs occurring from July 10 to 13, 2017 are estimated by the two approaches and compared with concurrent mooring observations for assessments. Using the ratio of the dimensionless dispersive parameter to the square of dimensionless nonlinear parameter as a criterion, the best appliable ranges of the two approaches are clearly separated. The statistics of total 18 cases indicate that in each 50% of cases, the KdV and the NLS approaches give more accurate estimates of ISW amplitudes. It is found that the relative errors of ISW amplitudes derived from two theoretical approaches are closely associated with the logarithmic bottom slopes. This may be attributed to the nonlinear growth of ISW amplitudes as propagating along a shoaling thermocline or topography. The test results using three consecutive satellite images to retrieve the ISW propagation speeds indicate that the use of multiple satellite images (>2) may improve the accuracy of retrieved phase speeds. Meanwhile, repeated multi-satellite images of ISWs can help to determine the types of ISWs if mooring data are available nearby.
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Key words:
- internal solitary waves /
- KdV equation /
- NLS equation /
- South China Sea /
- satellite images
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Figure 1. Satellite images of the internal solitary waves in the northern South China Sea. MODIS images were acquired on July 10 (a), July 11 (b), July 12 (e) and July 13, 2017 (g); CBERS-4 image (c) and GF-1 image (d) were acquired on July 12, 2017; Landsat-7 image (f) was acquired on July 13, 2017.
Figure 2. Location of the mooring station (yellow pentagram) and crest lines (color lines) of the leading ISWs extracted from satellite images in the northern South China Sea. Numbers 1 to 7 denote the ISWs on Images 1 to 7 in Table 1, respectively.
Figure 3. Vertical profiles of water temperature measured by the mooring CTD chain deployed near Dongsha Atoll in the northern South China Sea from July 10−13, 2017.
Figure 4. Comparisons of daily multi-scale ultra-high resolution sea surface temperature (MURSST), mooring observed temperature and Copernicus Marine Environment Monitoring Service (CMEMS) temperature at the mooring location during July 10−13, 2017.
Figure 5. A scatter plot of lg
$\chi $ and characteristic half width of ISWs. Open and solid circles represent the data derived from this study and cited from previous studies, respectively.
Figure 6. Relation of the relative error (dA) of satellite estimated ISW amplitude to the bottom slope (dh). Data points represent the results of ISWs observed on the seven satellite images during July 10−13, 2017.
Figure 7. Locations of the mooring station (yellow pentagram), ISWs observed by CBERS-4/PAN (C0712), GF-1/WFV3 (G0712) and Aqua/MODIS (M0712) (red pentagrams) on July 12, 2017 and Landsat-7/ETM+ (L0713) and Terra/MODIS (M0713) (purple pentagrams) on July 13, 2017. The locations of the mooring station and ISWs observed by Landsat-7/ETM+ on July 13 and GF-1/WFV3 on July 12 are marked as S0, S1 and S2.
Table 1. Information of satellite images
Image No. Image ID Time Satellite (senor) Spatial resolution/m 1 M0710 July 10, 2017, 05:30 UTC Aqua (MODIS) 250 2 M0711 July 11, 2017, 03:10 UTC Terra (MODIS) 250 3 C0712 July 12, 2017, 02:59 UTC CBERS-4 (PAN) 5 4 G0712 July 12, 2017, 03:37 UTC GF-1 (WFV3) 16 5 M0712 July 12, 2017, 05:20 UTC Aqua (MODIS) 250 6 L0713 July 13, 2017, 02:43 UTC Landsat-7 (ETM+) 15 7 M0713 July 13, 2017, 02:55 UTC Terra (MODIS) 250 
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Table 2. The characteristic parameters of the ISWs observed by satellites and the mooring CTD chain
Image ID Water depth/m l/m Amplitude/m S-KdV S-NLS Mooring M0710 370 500 4.9 55.1 68.5 M0711 390 750 2.3 44.0 48.1 C0712 390 175 42.0 189.6 45.7 G0712 370 176 39.4 158.4 45.7 M0712 327 500 4.3 34.3 45.7 L0713 408 120 67.9 419.5 66.1 M0713 398 750 1.7 61.9 66.1 Note: l is the peak-to-peak distance; S-KdV and S-NLS are amplitudes estimated using the KdV and the NLS equation approaches, respectively; the number in bold represents that the estimated amplitude is closer to the mooring observation.
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Table 3. ISW amplitudes derived from satellite observations in previous studies
Literature Satellite (sensor) Spatial resolution/m Region Water depth/m l/m Amplitude/m KdV NLS Observation Small et al. (1999) ERS-1 (SAR) 30 Malin Shelf 500 350 24.2 − 49.0 600 400 26.0 − 49.0 700 400 31.7 − 49.0 500 350 49.6 − 30.0 Zheng et al. (2001) ERS-1 (SAR) 30 Portugues Continental Shelf 140 110 26.0 − 23.0 140 220 7.0 − 10.0 140 240 6.0 − 5.0 Radarsat-1 (SAR) 100 SCS 350 900 48.0 − − 350 1100 37.0 − − Li et al. (2008) Envisat (ASAR) 150 SCS 70 99 8.0 − 6.0−10.0 Li et al. (2013) ERS-1 (SAR) 30 Malin Shelf 500 350 − 42.2 49.0 600 400 − 37.8 49.0 700 400 − 40.8 49.0 Huang and Zhao (2014) Aqua (MODIS) 250 SCS 3179 4301 124.0 − 126.0 Zhang et al. (2016) Envisat (ASAR) 150 SCS 92 317 0.7 18.1 18.0 Aqua (MODIS) 250 117 882 − 14.2 15.0 Note: l is the peak-to-peak distance; − represents no data.
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Table 4. The relative error of satellite estimated ISW amplitude and bottom slope
Image ID Relative error dA/% Bottom slope dh/% M0710 19.5 9.3 M0711 8.7 4.4 C0712 8.1 4.4 G0712 13.7 9.3 M0712 24.8 19.9 L0713 2.7 0 M0713 6.3 2.5
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Table 5. Propagation speeds of the ISWs at CBERS-4 and GF-1 locations
Data sources Satellite derived
propagation
speed/(m·s−1)KdV-derived
propagation
speed/(m·s−1)CBERS-4 CBERS-4 and GF-1 2.04 1.62 CBERS-4 and MODIS 1.34 CBERS-4, GF-1 and mooring 1.73 mean 1.70±0.35 GF-1 GF-1 and CBERS-4 2.04 1.58 GF-1 and MODIS 1.08 GF-1, CBERS-4 and MODIS 1.65 mean 1.59±0.48
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Table 6. The time that the ISWs passed the locations of the mooring (S0), Landsat-7 (S1) and GF-1 (S2)
Time at S0 Time at S1 Time at S2 July 10, 2017 03:01 04:10 05:27 July 11, 2017 02:08 02:42 04:13 July 12, 2017 01:48 02:17 03:37 July 13, 2017 02:31 02:43 04:19 Note: Time at S0 is mooring measured. The number in italic represents the ISW arrival time at S1 or S2 calculated by the phase speed derived from the KdV equation. The number in bold represents satellite-imaging time.
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