Abstract

Geostationary ocean colour sensors have not yet been launched into space, but are under consideration by a number of space agencies. This study provides a proof of concept for mapping of Total Suspended Matter (TSM) in turbid coastal waters from geostationary platforms with the existing SEVIRI (Spinning Enhanced Visible and InfraRed Imager) meteorological sensor on the METEOSAT Second Generation platform. Data are available in near real time every 15 minutes. SEVIRI lacks sufficient bands for chlorophyll remote sensing but its spectral resolution is sufficient for quantification of Total Suspended Matter (TSM) in turbid waters, using a single broad red band, combined with a suitable near infrared band. A test data set for mapping of TSM in the Southern North Sea was obtained covering 35 consecutive days from June 28 until July 31 2006. Atmospheric correction of SEVIRI images includes corrections for Rayleigh and aerosol scattering, absorption by atmospheric gases and atmospheric transmittances. The aerosol correction uses assumptions on the ratio of marine reflectances and aerosol reflectances in the red and near-infrared bands. A single band TSM retrieval algorithm, calibrated by nonlinear regression of seaborne measurements of TSM and marine reflectance was applied. The effect of the above assumptions on the uncertainty of the marine reflectance and TSM products was analysed. Results show that (1) mapping of TSM in the Southern North Sea is feasible with SEVIRI for turbid waters, though with considerable uncertainties in clearer waters, (2) TSM maps are well correlated with TSM maps obtained from MODIS AQUA and (3) during cloud-free days, high frequency dynamics of TSM are detected.

© 2009 Optical Society of America

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2007

G. Lacroix, K. Ruddick, Y. Park, N. Gypens, and C. Lancelot, "Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll a images," J. Mar. Syst. 64(1-4), 66-88 (2007).
[CrossRef]

M. Fettweis, B. Nechad,  et al, "An estimate of the suspended particulate matter (SPM) transport in the southern North Sea using SeaWiFS images, in situ measurements and numerical model results," Continental Shelf Research 27,1568-1583 (2007).
[CrossRef]

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107,479-507 (2007).
[CrossRef]

2006

K. Ruddick, V. De Cauwer, Y. Park, and G. Moore, "Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters," Limnol. Oceanogr. 51, 1167-1179, (2006).
[CrossRef]

M. Fettweis, F. Francken, V. Pison, and D. Van Den Eynde, "Suspended particulate matter dynamics and aggregate sizes in a high turbidity area," Marine Geology 235,63-74 (2006).
[CrossRef]

2003

E. Wolanski and S. Spagnol, "Dynamics of the turbidity maximum in King Sound, tropical Western Australia," Estuar Coast Mar Sci 56,877-890 (2003).

M. Babin, D. Stramski,  et al, "Variations in the light absorption coefficients of phytoplankton, nonalgal particles and dissolved organic matter in coastal waters around Europe," J. Geophys. Res. 108(C7), 3211 (2003), doi:10.1029/2001JC000882.
[CrossRef]

2001

R. P. Stumpf, "Applications of satellite ocean color sensors for monitoring and predicting harmful algal blooms," Human and Ecological Risk Assessment 7,1363-1368 (2001).
[CrossRef]

2000

1998

W. Van Raaphorst, C. J. M. Philippart,  et al, "Distribution of suspended particulate matter in the North Sea as inferred from NOAA/AVHRR reflectance images and in situ observations," J. Sea Res. 39,197-215 (1998).
[CrossRef]

1997

W. Ebenhoeh, J. G. B. Bekker, and J. W. Baretta, "The primary production module in the marine ecosystem model ERSEM II, with emphasis on the light forcing," J. Sea Res. 38,173-193 (1997).
[CrossRef]

1996

R. Frouin, M. Schwindling, and P. Y. Deschamps, "Spectral dependence of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications," J. Geophys. Res. 101(C6), 14361-14371 (1996).
[CrossRef]

1994

R. Doerffer and J. Fischer, "Concentrations of chlorophyll, suspended matter, gelbstoff in case II waters derived from satellite coastal zone color scanner data with inverse modeling methods," J. Geophys. Res. 99(C4), 7457-7466 (1994).
[CrossRef]

H. R. Gordon and M. Wang, "Influence of oceanic whitecaps on atmospheric correction of ocean-color sensors," Appl. Opt. 33,7754-7763 (1994).
[CrossRef] [PubMed]

1990

L. Otto, J. T. F. Zimmerman,  et al., "Physical Oceanography of the North Sea," Netherlands J. Sea Res. 26,161-238 (1990).
[CrossRef]

1989

R. P. Stumpf and J. R. Pennock, "Calibration of a general optical equation for remote sensing of suspended sediments in a moderately turbid estuary," J. Geophysical Res. 94,14363-14371 (1989).
[CrossRef]

P. M. Holligan, T. Aarup, and S. B. Groom, "The North Sea satellite colour atlas," Continental Shelf Research 9,665-765 (1989).
[CrossRef]

1987

D. Eisma, "The North Sea: an overview," Phil. Trans. R. Soc. Lond. B 316,461-485 (1987).
[CrossRef]

1981

D. Eisma, "Supply and deposition of suspended matter in the North Sea," Spec. Publs int. Ass. Sediment 5,415-428 (1981).

1980

M. Viollier, D. Tanré, and P.Y. Deschamps, "An algorithm for remote sensing of water color from space," Boundary Layer Meteorology 18,247-267 (1980).
[CrossRef]

Aarup, T.

P. M. Holligan, T. Aarup, and S. B. Groom, "The North Sea satellite colour atlas," Continental Shelf Research 9,665-765 (1989).
[CrossRef]

Babin, M.

M. Babin, D. Stramski,  et al, "Variations in the light absorption coefficients of phytoplankton, nonalgal particles and dissolved organic matter in coastal waters around Europe," J. Geophys. Res. 108(C7), 3211 (2003), doi:10.1029/2001JC000882.
[CrossRef]

Baretta, J. W.

W. Ebenhoeh, J. G. B. Bekker, and J. W. Baretta, "The primary production module in the marine ecosystem model ERSEM II, with emphasis on the light forcing," J. Sea Res. 38,173-193 (1997).
[CrossRef]

Bekker, J. G. B.

W. Ebenhoeh, J. G. B. Bekker, and J. W. Baretta, "The primary production module in the marine ecosystem model ERSEM II, with emphasis on the light forcing," J. Sea Res. 38,173-193 (1997).
[CrossRef]

De Cauwer, V.

K. Ruddick, V. De Cauwer, Y. Park, and G. Moore, "Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters," Limnol. Oceanogr. 51, 1167-1179, (2006).
[CrossRef]

Deschamps, P. Y.

R. Frouin, M. Schwindling, and P. Y. Deschamps, "Spectral dependence of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications," J. Geophys. Res. 101(C6), 14361-14371 (1996).
[CrossRef]

Deschamps, P.Y.

M. Viollier, D. Tanré, and P.Y. Deschamps, "An algorithm for remote sensing of water color from space," Boundary Layer Meteorology 18,247-267 (1980).
[CrossRef]

Deuze, J. L.

E. F. Vermote, D. Tanre, J. L. Deuze, M.  Herman, and J. J. Morcette, "Second Simulation of the Satellite Signal in the Solar Spectrum,6S: an overview," IEEE Tran. Geoscie. Remote Sensing  35(3),675-686 (1997).
[CrossRef]

Deuzé, J. L.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107,479-507 (2007).
[CrossRef]

Doerffer, R.

R. Doerffer and J. Fischer, "Concentrations of chlorophyll, suspended matter, gelbstoff in case II waters derived from satellite coastal zone color scanner data with inverse modeling methods," J. Geophys. Res. 99(C4), 7457-7466 (1994).
[CrossRef]

Ebenhoeh, W.

W. Ebenhoeh, J. G. B. Bekker, and J. W. Baretta, "The primary production module in the marine ecosystem model ERSEM II, with emphasis on the light forcing," J. Sea Res. 38,173-193 (1997).
[CrossRef]

Eisma, D.

D. Eisma, "The North Sea: an overview," Phil. Trans. R. Soc. Lond. B 316,461-485 (1987).
[CrossRef]

D. Eisma, "Supply and deposition of suspended matter in the North Sea," Spec. Publs int. Ass. Sediment 5,415-428 (1981).

Fettweis, M.

M. Fettweis, B. Nechad,  et al, "An estimate of the suspended particulate matter (SPM) transport in the southern North Sea using SeaWiFS images, in situ measurements and numerical model results," Continental Shelf Research 27,1568-1583 (2007).
[CrossRef]

M. Fettweis, F. Francken, V. Pison, and D. Van Den Eynde, "Suspended particulate matter dynamics and aggregate sizes in a high turbidity area," Marine Geology 235,63-74 (2006).
[CrossRef]

Fischer, J.

R. Doerffer and J. Fischer, "Concentrations of chlorophyll, suspended matter, gelbstoff in case II waters derived from satellite coastal zone color scanner data with inverse modeling methods," J. Geophys. Res. 99(C4), 7457-7466 (1994).
[CrossRef]

Francken, F.

M. Fettweis, F. Francken, V. Pison, and D. Van Den Eynde, "Suspended particulate matter dynamics and aggregate sizes in a high turbidity area," Marine Geology 235,63-74 (2006).
[CrossRef]

Frouin, R.

R. Frouin, M. Schwindling, and P. Y. Deschamps, "Spectral dependence of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications," J. Geophys. Res. 101(C6), 14361-14371 (1996).
[CrossRef]

Gordon, H. R.

Groom, S. B.

P. M. Holligan, T. Aarup, and S. B. Groom, "The North Sea satellite colour atlas," Continental Shelf Research 9,665-765 (1989).
[CrossRef]

Gypens, N.

G. Lacroix, K. Ruddick, Y. Park, N. Gypens, and C. Lancelot, "Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll a images," J. Mar. Syst. 64(1-4), 66-88 (2007).
[CrossRef]

Herman, M.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107,479-507 (2007).
[CrossRef]

E. F. Vermote, D. Tanre, J. L. Deuze, M.  Herman, and J. J. Morcette, "Second Simulation of the Satellite Signal in the Solar Spectrum,6S: an overview," IEEE Tran. Geoscie. Remote Sensing  35(3),675-686 (1997).
[CrossRef]

Holligan, P. M.

P. M. Holligan, T. Aarup, and S. B. Groom, "The North Sea satellite colour atlas," Continental Shelf Research 9,665-765 (1989).
[CrossRef]

Lacroix, G.

G. Lacroix, K. Ruddick, Y. Park, N. Gypens, and C. Lancelot, "Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll a images," J. Mar. Syst. 64(1-4), 66-88 (2007).
[CrossRef]

Lafrance, B.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107,479-507 (2007).
[CrossRef]

Lancelot, C.

G. Lacroix, K. Ruddick, Y. Park, N. Gypens, and C. Lancelot, "Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll a images," J. Mar. Syst. 64(1-4), 66-88 (2007).
[CrossRef]

Lenoble, J.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107,479-507 (2007).
[CrossRef]

Moore, G.

K. Ruddick, V. De Cauwer, Y. Park, and G. Moore, "Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters," Limnol. Oceanogr. 51, 1167-1179, (2006).
[CrossRef]

Morcette, J. J.

E. F. Vermote, D. Tanre, J. L. Deuze, M.  Herman, and J. J. Morcette, "Second Simulation of the Satellite Signal in the Solar Spectrum,6S: an overview," IEEE Tran. Geoscie. Remote Sensing  35(3),675-686 (1997).
[CrossRef]

Nechad, B.

M. Fettweis, B. Nechad,  et al, "An estimate of the suspended particulate matter (SPM) transport in the southern North Sea using SeaWiFS images, in situ measurements and numerical model results," Continental Shelf Research 27,1568-1583 (2007).
[CrossRef]

B. Nechad, K. G. Ruddick, and Y. Park, "Calibration and validation of a generic multisensor algorithm for mapping of Total Suspended Matter in turbid waters," Subm. to Rem. Sens. Env.

Otto, L.

L. Otto, J. T. F. Zimmerman,  et al., "Physical Oceanography of the North Sea," Netherlands J. Sea Res. 26,161-238 (1990).
[CrossRef]

Ovidio, F.

Park, Y.

G. Lacroix, K. Ruddick, Y. Park, N. Gypens, and C. Lancelot, "Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll a images," J. Mar. Syst. 64(1-4), 66-88 (2007).
[CrossRef]

K. Ruddick, V. De Cauwer, Y. Park, and G. Moore, "Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters," Limnol. Oceanogr. 51, 1167-1179, (2006).
[CrossRef]

B. Nechad, K. G. Ruddick, and Y. Park, "Calibration and validation of a generic multisensor algorithm for mapping of Total Suspended Matter in turbid waters," Subm. to Rem. Sens. Env.

Pennock, J. R.

R. P. Stumpf and J. R. Pennock, "Calibration of a general optical equation for remote sensing of suspended sediments in a moderately turbid estuary," J. Geophysical Res. 94,14363-14371 (1989).
[CrossRef]

Philippart, C. J. M.

W. Van Raaphorst, C. J. M. Philippart,  et al, "Distribution of suspended particulate matter in the North Sea as inferred from NOAA/AVHRR reflectance images and in situ observations," J. Sea Res. 39,197-215 (1998).
[CrossRef]

Pison, V.

M. Fettweis, F. Francken, V. Pison, and D. Van Den Eynde, "Suspended particulate matter dynamics and aggregate sizes in a high turbidity area," Marine Geology 235,63-74 (2006).
[CrossRef]

Rijkeboer, M.

Ruddick, K.

G. Lacroix, K. Ruddick, Y. Park, N. Gypens, and C. Lancelot, "Validation of the 3D biogeochemical model MIRO&CO with field nutrient and phytoplankton data and MERIS-derived surface chlorophyll a images," J. Mar. Syst. 64(1-4), 66-88 (2007).
[CrossRef]

K. Ruddick, V. De Cauwer, Y. Park, and G. Moore, "Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters," Limnol. Oceanogr. 51, 1167-1179, (2006).
[CrossRef]

Ruddick, K. G.

K. G. Ruddick, F. Ovidio, and M. Rijkeboer, "Atmospheric correction of SeaWiFS imagery for turbid coastal and inland waters," Appl. Opt. 39(6), 897-912 (2000).
[CrossRef]

B. Nechad, K. G. Ruddick, and Y. Park, "Calibration and validation of a generic multisensor algorithm for mapping of Total Suspended Matter in turbid waters," Subm. to Rem. Sens. Env.

Santer, R.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107,479-507 (2007).
[CrossRef]

Schwindling, M.

R. Frouin, M. Schwindling, and P. Y. Deschamps, "Spectral dependence of sea foam in the visible and near-infrared: In situ measurements and remote sensing implications," J. Geophys. Res. 101(C6), 14361-14371 (1996).
[CrossRef]

Spagnol, S.

E. Wolanski and S. Spagnol, "Dynamics of the turbidity maximum in King Sound, tropical Western Australia," Estuar Coast Mar Sci 56,877-890 (2003).

Stramski, D.

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Supplementary Material (1)

» Media 1: AVI (1283 KB)     

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Figures (13)

Fig. 1.
Fig. 1.

Spatial extent of SEVIRI full disk imagery and viewing angle in degrees of SEVIRI on the MSG1-Meteosat8 platform located at 3.5°W.

Fig. 2.
Fig. 2.

Normalized spectral response, ω(λ), of the SEVIRI solar channels (source: [14]) and two-way atmospheric transmittances for water vapor, ozone and molecular scattering for a vertical atmospheric path and the US standard atmosphere model simulated with LOWTRAN.

Fig. 3.
Fig. 3.

Viewing angle of SEVIRI (on MSGl-Meteosat8 platform located at 3.5°W) over Western-Europe. The purple box delimits the study area, for which the northern limit corresponds to a 64° satellite viewing angle. The white dots are the locations for which daily variability of airmass and Rayleigh scattering are presented in Fig. 4. The red polygons bound the clear water pixels from which the ratio of aerosol reflectance ε (6,8) is obtained (see further in the text).

Fig. 4.
Fig. 4.

Variability of Rayleigh reflectance for VIS0.6 (red lines) and total airmass (blue lines) on 29th June 2006 for two locations in the middle (dashed lines) and at the top (solid lines) of the SEVIRI subscene in Fig. 3.

Fig. 5.
Fig. 5.

Marine reflectances in the VIS0.6 and VIS0.8 bands obtained from optimal in situ above-water marine reflectance measurements collected between 2001 and 2006 in Southern North Sea waters. The parameter σ is calibrated through linear regression (black line) of 33 reflectance measurements for which ρw (0.8)<0.011.

Fig. 6.
Fig. 6.

Estimation of VIS0.6:VIS0.8 ratio of aerosol reflectances. (a) Rayleigh corrected reflectances for a set of clear water pixels in VIS0.6 and VIS0.8 bands on June 29th 2006 at 11:30UTC. (b) The corresponding histogram of the VIS0.6:VIS0.8 Rayleigh corrected reflectance ratios (N is the number of pixels).

Fig. 7.
Fig. 7.

Regression between 63 in situ reflectance measurements of ρw (0.6) and TSM concentration obtained from ship borne measurements in the Southern North Sea between 2001 and 2006.

Fig. 8.
Fig. 8.

Schematical depiction of the processing steps in the atmospheric correction of the SEVIRI VIS0.6 and VIS0.8 channels (SSS= sun-sea-satellite). The second pass in the two-pass algorithm is represented by the blue lines (in the first pass, shown by the orange lines, γ = 1).

Fig. 9.
Fig. 9.

(a) (Media 1) TSM (mg/l) concentration in the Southern North Sea from SEVIRI on June 29th 2006 at 13:00 UTC. Five pixels P1–P5 were selected in clear and turbid waters for which high frequency TSM dynamics are presented in Fig. 12). (b) Estimated aerosol correction uncertainty on TSM concentration (from Eq. (24)). White areas are clouds or have aerosol optical thicknesses higher than 0.5. Grey areas are land.

Fig. 10.
Fig. 10.

TSM (mg/l) maps obtained from SEVIRI (a) and MODIS (b) on July 18th 2006 at 13:15 UTC.

Fig. 11.
Fig. 11.

Regression of TSM (mg/l) obtained from MODIS and from SEVIRI on July 18th 2006 at 13:15 UTC. The error bars for MODIS TSM show the standard deviation from the spatial mean obtained from downsampling the MODIS pixels to the SEVIRI grid. The error bars for SEVIRI TSM show the estimated uncertainty on the TSM value assicated with the aerosol correction and computed from Eq. (24). Zero TSM values were omitted.

Fig. 12.
Fig. 12.

High frequency variability of TSM concentration at P1–P3 (a) and P4–P5 (b) on a cloudfree day (June 29th 2006). The error bars denote the estimated uncertainty on the TSM concentration arising from the aerosol correction and as expressed by Eq. (24).

Fig. 13.
Fig. 13.

Estimation of VIS0.6:NIR1.6 ratio of aerosol reflectances. (a) Rayleigh corrected reflectances for a set of clear water pixels in VIS0.6 and NIR1.6 bands on June 29th 2006 at 11:30UTC. (b) The corresponding histogram of the VIS0.6:NIR1.6 Rayleigh corrected reflectance ratios (N is the number of pixels).

Tables (3)

Tables Icon

Table 1. Calibration parameters ([14]) and correction factors (A0, [24]) for SEVIRI channels VIS0.6, VIS0.8 and NIR1.6

Tables Icon

Table 2. Statistics of the regression analysis between TSM (mg/l) obtained from MODIS and from SEVIRI.

Tables Icon

Table 3. Estimation of uncertainties on ρw (0.6) associated with assumptions (Eq. (9) and Eq. (10) in the main text) for the (VIS0.6,VIS0.8) band pair and assumption (Eq. (A5) and Eq. (A6)) for the (VIS0.6,NIR1.6) band pair, obtained from Eq. (A4) and Eq. (A7) respectively. Values typical of the Southern North Sea were used: ε (6,8) = 1.1 ± 0.3, ε (6,16) = 3.1 ± 2.1, σ = 6.1 ± 0.3, τa (0.8) between 0.05 and 0.5 and ρw (0.8) between 0.001 and 0.010. Some simplifications: t 0,v a,r,(0.6) = 1, γ = 1. Estimated uncertainties on TSM concentration obtained from Eq. (A11) are also shown.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

ρw=πLw0+Ed0+
LtotTOA=10(cfK+r0)λ02
ρtotTOA=πd2LtotTOAA0E0cosθ0
ρtotTOA=ρrTOA+ρaTOA+ρwcTOA+ρgTOA+T0Tvρw
Tv=tvatvrtvgandT0=t0at0rt0g
ρtotTOA=ρrTOA+ρaTOA+t0,vaρw
ρc=ρtotρr=ρa+t0,vaρw
ρc(0.6)=ρa(0.6)+t0,va(0.6)ρw(0.6)
ρc(0.8)=ρa(0.8)+t0,va(0.8)ρw(0.8)
σ=ρw(0.6)ρw(0.8)
ε(6,8)=ρa(0.6)ρa(0.8)
ρw(0.6)=πλVIS0.6Lw0+(λ)ω(λ)λVIS0.6Ed0+(λ)ω(λ)
α=ln(ε(6,8))ln(0.6350.810)
t0,va(0.6)ρw(0.6)=t0,va(0.6)t0,va(0.8)t0,va(0.8)σρw(0.8)=γσt0,va(0.8)ρw(0.8)
γ=t0,va(0.6)t0,va(0.8)
ε(6,8)ρa(0.8)+γσt0,va(0.8)ρw(0.8)=ρc(0.6)
ρa(0.8)+t0,va(0.8)ρw(0.8)=ρc(0.8)
ρa(0.8)=γσρc(0.8)ρc(0.6)γσε(6,8)
ρw(0.8)=ρc(0.6)ε(6,8)ρc(0.8)t0,va(0.8)(γσε(6,8))
ρa(0.6)=ε(6,8)γσρc(0.8)ρc(0.6)γσε(6,8)
ρw(0.6)=σρc(0.6)ε(6,8)ρc(0.8)t0,va(0.8)(γσε(6,8))
TSM=Aρw(0.6)Cρw(0.6)+B
Δρw(0.6)=1t0,va(0.8) [(ρa(0.8)σ(γσε(6,8))Δε(6,8))2+(ρw(0.8)ε(6,8)t0,va(0.8)γσε(6,8)Δσ)2]1/2
Δρw(0.6)[(ρa(0.8)σσε(6,8)Δε(6,8))2+(ρw(0.8)ε(6,8)Δσσε(6,8))2]1/2
ΔTSM=ACΔρw(0.6)(Cρw(0.6))2
â=xiyixi2
Sa2=1(n1)xi2(yi2(xiyi)2xi2)
σ=6.1±0.3
ε(6,8)=ε̂(6,8)±2Sε(6,8)2
Δρw(0.6)=[(ρw(0.6)σΔσ)2+(ρw(0.6)ε(6,8)Δε(6,8))2]1/2
Δρw(0.6)=1t0,va(0.8)[(ρa(0.8)σ(γσε(6,8))Δε(6,8))2+(ρw(0.8)ε(6,8)t0,va(0.8)γσε(6,8)Δσ)2]1/2
ρa(1.6)=ρc(1.6)
ε(6,16)=ρa(0.6)ρa(1.6)
Δρw(0.6)=ρw(0.6)ε(6,16)Δε(6,16)=1t0,va(0.6)ρa(0.6)ε(6,16)Δε(6,16)=ρa(1.6)t0,va(0.6)Δε(6,16)
ρa(1.6)=ρa(0.8)ρa(0.6)ρa(0.8)ρa(1.6)ρa(0.6)=ρa(0.8)ε(6,8)ε(6,16)
ΔTSM=1Cρw(0.6)[(ρw(0.6)ΔA)2+(ACΔρw(0.6)Cρw(0.6))2]1/2
ΔTSMTSM=[(CΔρw(0.6)Cρw(0.6))2+(ΔAA)2]1/2
ΔTSM=ACΔρw(0.6)(Cρw(0.6))2

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