Abstract

A simple algorithm that solves the inverse problem is numerically tested for estimating the ratio of the absorption to the transport coefficients from measurements of the irradiance and scalar irradiance at depths sufficiently beneath the surface of seawater. Results of radiative transfer calculations with a phase function for San Diego harbor seawater were used to test the algorithm and show that the ratio is quite insensitive to the depth of the measurements and to the direction of incident solar radiation unless the absorption is very weak. A second algorithm, based on the assumption that the radiance in the diffusion regime is proportional to a (one-parameter) Henyey-Greenstein shape, is examined for estimating the ratio of the backward scattering to absorption coefficients from only irradiance measurements. The algorithm has the advantage that no scalar irradiance measurements are required, but it provides significantly poorer estimates because the radiance is not sufficiently close to the Henyey-Greenstein shape.

© 1989 Optical Society of America

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References

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  1. N. J. McCormick, “Transport Scattering Coefficients from Reflection and Transmission Measurements,” J. Math. Phys. N.Y. 20, 1504–1507 (1979).
    [CrossRef]
  2. R. Sanchez, N. J. McCormick, “Numerical Evaluation of Optical Single-Scattering Properties Using Multiple-Scattering Inverse Transport Methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1983).
    [CrossRef]
  3. J. C. Oelund, N. J. McCormick, “Sensitivity of Multiple-Scattering Inverse Transport Methods to Measurement Errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
    [CrossRef]
  4. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  5. H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications, Vols. 1 and 2 (Academic, New York, 1980).
  6. N. Højerslev, “A Spectral Light Absorption Meter for Measurements in the Sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
    [CrossRef]
  7. N. G. Jerlov, Marine Optics (Elsevier, New York, 1976).
  8. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York, 1983).
  9. I. N. Melnikova, “The Field of Scattered Solar Radiation in a Cloud Layer,” Izv. Acad. Sci. U.S.S.R. Atmos. Oceanic Phys. 14, 928–931 (1978).
  10. N. J. McCormick, “Methods for Estimating the Similarity Parameter of Clouds from Internal Measurements of the Scattered Radiation Field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
    [CrossRef]
  11. T. Duracz, N. J. McCormick, “Estimating the Similarity Parameter from Radiation Measurements Within Weakly Absorbing Optically Thick Clouds, J. Atmos. Sci. 43, 486–492 (1986).
    [CrossRef]
  12. J. T. O. Kirk, “Estimation of the Scattering Coefficient of Natural Waters Using Underwater Irradiance Measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
    [CrossRef]
  13. W. N. Vant, R. J. Davies-Colley, “Factors Affecting Clarity of New Zealand Lakes,” N. Z. J. Mar. Freshwater Res. 18, 367–377 (1984).
    [CrossRef]
  14. A. D. Weidemann, T. T. Bannister, “Absorption and Scattering Coefficients in Irondequoit Bay,” Limnol. Oceanogr. 31, 567–583 (1986).
    [CrossRef]
  15. R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903–929 (1984).
    [CrossRef]
  16. E. Aas, “Two-Stream Irradiance Model for Deep Waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  17. V. I. Haltrin, “Exact Solution of the Characteristic Equation for Transfer in the Anisotropically Scattering and Absorbing Medium,” Appl. Opt. 27, 599–602 (1988).
    [CrossRef] [PubMed]
  18. L. C. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  19. D. M. Di Toro, “Optics of Turbid Estuarine Waters: Approximations and Applications,” Water Res. 12, 1059–1068 (1978).
    [CrossRef]
  20. T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Institution of Oceanography Publication72–78 (1972).
  21. I. Kuščer, “Milne’s Problem for Anistropic Scattering,” J. Math. Phys. Cambridge, Mass. 34, 256–266 (1956).
  22. M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds Through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031–2044 (1981).
    [CrossRef]
  23. IMSL Reference Manual, International Mathematical and Statistical Library, Inc., Houston, TX, Edition 10.0 (1987).
  24. G. W. Kattawar, T. J. Humphreys, G. N. Plass, “Radiative Transfer in an Atmosphere–Ocean System: a Matrix Operator Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 160, 123–131 (1978).
  25. G. E. Rinaldi, “An Inverse Diffusion Method for Estimating Optical Properties of a Scattering Medium,” M.S. Thesis, U. Washington (1988).
  26. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  27. D. Bauer, A. Morel, “Étude aux petits angles de l’indicatrice de diffusion de la limière par les eaux de mer,” Ann. Geophys. 23, 109–123 (1967).
  28. R. D. O’Dell, F. W. Brinkley, D. R. Marr, “User’s Manual for ONEDANT: a Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report, LA-6941-MS (Feb.1982); available as Oak Ridge National Laboratory Radiation Shielding Information Center Computer Code Collection 428.

1988

1987

1986

A. D. Weidemann, T. T. Bannister, “Absorption and Scattering Coefficients in Irondequoit Bay,” Limnol. Oceanogr. 31, 567–583 (1986).
[CrossRef]

T. Duracz, N. J. McCormick, “Estimating the Similarity Parameter from Radiation Measurements Within Weakly Absorbing Optically Thick Clouds, J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

1985

N. J. McCormick, “Methods for Estimating the Similarity Parameter of Clouds from Internal Measurements of the Scattered Radiation Field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
[CrossRef]

J. C. Oelund, N. J. McCormick, “Sensitivity of Multiple-Scattering Inverse Transport Methods to Measurement Errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
[CrossRef]

1984

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

W. N. Vant, R. J. Davies-Colley, “Factors Affecting Clarity of New Zealand Lakes,” N. Z. J. Mar. Freshwater Res. 18, 367–377 (1984).
[CrossRef]

1983

R. Sanchez, N. J. McCormick, “Numerical Evaluation of Optical Single-Scattering Properties Using Multiple-Scattering Inverse Transport Methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1983).
[CrossRef]

1981

J. T. O. Kirk, “Estimation of the Scattering Coefficient of Natural Waters Using Underwater Irradiance Measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
[CrossRef]

M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds Through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

1979

N. J. McCormick, “Transport Scattering Coefficients from Reflection and Transmission Measurements,” J. Math. Phys. N.Y. 20, 1504–1507 (1979).
[CrossRef]

1978

I. N. Melnikova, “The Field of Scattered Solar Radiation in a Cloud Layer,” Izv. Acad. Sci. U.S.S.R. Atmos. Oceanic Phys. 14, 928–931 (1978).

D. M. Di Toro, “Optics of Turbid Estuarine Waters: Approximations and Applications,” Water Res. 12, 1059–1068 (1978).
[CrossRef]

G. W. Kattawar, T. J. Humphreys, G. N. Plass, “Radiative Transfer in an Atmosphere–Ocean System: a Matrix Operator Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 160, 123–131 (1978).

1975

N. Højerslev, “A Spectral Light Absorption Meter for Measurements in the Sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[CrossRef]

1972

T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Institution of Oceanography Publication72–78 (1972).

1967

D. Bauer, A. Morel, “Étude aux petits angles de l’indicatrice de diffusion de la limière par les eaux de mer,” Ann. Geophys. 23, 109–123 (1967).

1956

I. Kuščer, “Milne’s Problem for Anistropic Scattering,” J. Math. Phys. Cambridge, Mass. 34, 256–266 (1956).

1941

L. C. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aas, E.

Bannister, T. T.

A. D. Weidemann, T. T. Bannister, “Absorption and Scattering Coefficients in Irondequoit Bay,” Limnol. Oceanogr. 31, 567–583 (1986).
[CrossRef]

Bauer, D.

D. Bauer, A. Morel, “Étude aux petits angles de l’indicatrice de diffusion de la limière par les eaux de mer,” Ann. Geophys. 23, 109–123 (1967).

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Brinkley, F. W.

R. D. O’Dell, F. W. Brinkley, D. R. Marr, “User’s Manual for ONEDANT: a Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report, LA-6941-MS (Feb.1982); available as Oak Ridge National Laboratory Radiation Shielding Information Center Computer Code Collection 428.

Davies-Colley, R. J.

W. N. Vant, R. J. Davies-Colley, “Factors Affecting Clarity of New Zealand Lakes,” N. Z. J. Mar. Freshwater Res. 18, 367–377 (1984).
[CrossRef]

Di Toro, D. M.

D. M. Di Toro, “Optics of Turbid Estuarine Waters: Approximations and Applications,” Water Res. 12, 1059–1068 (1978).
[CrossRef]

Duracz, T.

T. Duracz, N. J. McCormick, “Estimating the Similarity Parameter from Radiation Measurements Within Weakly Absorbing Optically Thick Clouds, J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

Gordon, H. R.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York, 1983).

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Haltrin, V. I.

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Højerslev, N.

N. Højerslev, “A Spectral Light Absorption Meter for Measurements in the Sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[CrossRef]

Humphreys, T. J.

G. W. Kattawar, T. J. Humphreys, G. N. Plass, “Radiative Transfer in an Atmosphere–Ocean System: a Matrix Operator Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 160, 123–131 (1978).

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, New York, 1976).

Kattawar, G. W.

G. W. Kattawar, T. J. Humphreys, G. N. Plass, “Radiative Transfer in an Atmosphere–Ocean System: a Matrix Operator Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 160, 123–131 (1978).

King, M. D.

M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds Through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

Kirk, J. T. O.

J. T. O. Kirk, “Estimation of the Scattering Coefficient of Natural Waters Using Underwater Irradiance Measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
[CrossRef]

Kušcer, I.

I. Kuščer, “Milne’s Problem for Anistropic Scattering,” J. Math. Phys. Cambridge, Mass. 34, 256–266 (1956).

Marr, D. R.

R. D. O’Dell, F. W. Brinkley, D. R. Marr, “User’s Manual for ONEDANT: a Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report, LA-6941-MS (Feb.1982); available as Oak Ridge National Laboratory Radiation Shielding Information Center Computer Code Collection 428.

McCormick, N. J.

T. Duracz, N. J. McCormick, “Estimating the Similarity Parameter from Radiation Measurements Within Weakly Absorbing Optically Thick Clouds, J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

N. J. McCormick, “Methods for Estimating the Similarity Parameter of Clouds from Internal Measurements of the Scattered Radiation Field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
[CrossRef]

J. C. Oelund, N. J. McCormick, “Sensitivity of Multiple-Scattering Inverse Transport Methods to Measurement Errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
[CrossRef]

R. Sanchez, N. J. McCormick, “Numerical Evaluation of Optical Single-Scattering Properties Using Multiple-Scattering Inverse Transport Methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1983).
[CrossRef]

N. J. McCormick, “Transport Scattering Coefficients from Reflection and Transmission Measurements,” J. Math. Phys. N.Y. 20, 1504–1507 (1979).
[CrossRef]

Melnikova, I. N.

I. N. Melnikova, “The Field of Scattered Solar Radiation in a Cloud Layer,” Izv. Acad. Sci. U.S.S.R. Atmos. Oceanic Phys. 14, 928–931 (1978).

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Morel, A.

D. Bauer, A. Morel, “Étude aux petits angles de l’indicatrice de diffusion de la limière par les eaux de mer,” Ann. Geophys. 23, 109–123 (1967).

Morel, A. Y.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York, 1983).

O’Dell, R. D.

R. D. O’Dell, F. W. Brinkley, D. R. Marr, “User’s Manual for ONEDANT: a Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report, LA-6941-MS (Feb.1982); available as Oak Ridge National Laboratory Radiation Shielding Information Center Computer Code Collection 428.

Oelund, J. C.

Petzold, T. J.

T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Institution of Oceanography Publication72–78 (1972).

Plass, G. N.

G. W. Kattawar, T. J. Humphreys, G. N. Plass, “Radiative Transfer in an Atmosphere–Ocean System: a Matrix Operator Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 160, 123–131 (1978).

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Rinaldi, G. E.

G. E. Rinaldi, “An Inverse Diffusion Method for Estimating Optical Properties of a Scattering Medium,” M.S. Thesis, U. Washington (1988).

Sanchez, R.

R. Sanchez, N. J. McCormick, “Numerical Evaluation of Optical Single-Scattering Properties Using Multiple-Scattering Inverse Transport Methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1983).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications, Vols. 1 and 2 (Academic, New York, 1980).

Vant, W. N.

W. N. Vant, R. J. Davies-Colley, “Factors Affecting Clarity of New Zealand Lakes,” N. Z. J. Mar. Freshwater Res. 18, 367–377 (1984).
[CrossRef]

Weidemann, A. D.

A. D. Weidemann, T. T. Bannister, “Absorption and Scattering Coefficients in Irondequoit Bay,” Limnol. Oceanogr. 31, 567–583 (1986).
[CrossRef]

Ann. Geophys.

D. Bauer, A. Morel, “Étude aux petits angles de l’indicatrice de diffusion de la limière par les eaux de mer,” Ann. Geophys. 23, 109–123 (1967).

Appl. Opt.

Astrophys. J.

L. C. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aust. J. Mar. Freshwater Res.

J. T. O. Kirk, “Estimation of the Scattering Coefficient of Natural Waters Using Underwater Irradiance Measurements,” Aust. J. Mar. Freshwater Res. 32, 533–539 (1981).
[CrossRef]

Izv. Acad. Sci. U.S.S.R. Atmos. Oceanic Phys.

I. N. Melnikova, “The Field of Scattered Solar Radiation in a Cloud Layer,” Izv. Acad. Sci. U.S.S.R. Atmos. Oceanic Phys. 14, 928–931 (1978).

J. Atmos. Sci.

T. Duracz, N. J. McCormick, “Estimating the Similarity Parameter from Radiation Measurements Within Weakly Absorbing Optically Thick Clouds, J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

M. D. King, “A Method for Determining the Single Scattering Albedo of Clouds Through Observation of the Internal Scattered Radiation Field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

J. Math. Phys. Cambridge, Mass.

I. Kuščer, “Milne’s Problem for Anistropic Scattering,” J. Math. Phys. Cambridge, Mass. 34, 256–266 (1956).

J. Math. Phys. N.Y.

N. J. McCormick, “Transport Scattering Coefficients from Reflection and Transmission Measurements,” J. Math. Phys. N.Y. 20, 1504–1507 (1979).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

R. Sanchez, N. J. McCormick, “Numerical Evaluation of Optical Single-Scattering Properties Using Multiple-Scattering Inverse Transport Methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1983).
[CrossRef]

N. J. McCormick, “Methods for Estimating the Similarity Parameter of Clouds from Internal Measurements of the Scattered Radiation Field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
[CrossRef]

Limnol. Oceanogr.

N. Højerslev, “A Spectral Light Absorption Meter for Measurements in the Sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[CrossRef]

A. D. Weidemann, T. T. Bannister, “Absorption and Scattering Coefficients in Irondequoit Bay,” Limnol. Oceanogr. 31, 567–583 (1986).
[CrossRef]

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

N. Z. J. Mar. Freshwater Res.

W. N. Vant, R. J. Davies-Colley, “Factors Affecting Clarity of New Zealand Lakes,” N. Z. J. Mar. Freshwater Res. 18, 367–377 (1984).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

G. W. Kattawar, T. J. Humphreys, G. N. Plass, “Radiative Transfer in an Atmosphere–Ocean System: a Matrix Operator Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 160, 123–131 (1978).

Scripps Institution of Oceanography Publication

T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Institution of Oceanography Publication72–78 (1972).

Water Res.

D. M. Di Toro, “Optics of Turbid Estuarine Waters: Approximations and Applications,” Water Res. 12, 1059–1068 (1978).
[CrossRef]

Other

N. G. Jerlov, Marine Optics (Elsevier, New York, 1976).

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York, 1983).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications, Vols. 1 and 2 (Academic, New York, 1980).

IMSL Reference Manual, International Mathematical and Statistical Library, Inc., Houston, TX, Edition 10.0 (1987).

G. E. Rinaldi, “An Inverse Diffusion Method for Estimating Optical Properties of a Scattering Medium,” M.S. Thesis, U. Washington (1988).

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

R. D. O’Dell, F. W. Brinkley, D. R. Marr, “User’s Manual for ONEDANT: a Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report, LA-6941-MS (Feb.1982); available as Oak Ridge National Laboratory Radiation Shielding Information Center Computer Code Collection 428.

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Figures (5)

Fig. 1
Fig. 1

Inverse diffusion coefficient C2 vs similarity parameter s.

Fig. 2
Fig. 2

Percent error in s vs depth z in m for ω = 0.99.

Fig. 3
Fig. 3

Percent error in s vs depth z in m for ω = 0.96.

Fig. 4
Fig. 4

Percent error in s vs depth z in m for ω = 0.84.

Fig. 5
Fig. 5

Percent error in s vs depth z in m for ω = 0.66.

Tables (6)

Tables Icon

Table I Phase Function Expansion Coefficients for San Diego Harbor

Tables Icon

Table II Depth for Which the Error in Estimated s is <1 %.

Tables Icon

Table III Effect of Measurement Errors on the Estimated Similarity Parameter at Deep Depths for σu = 1 % and σd = 1 %

Tables Icon

Table IV Effect of Measurement Errors on the Estimated Similarity Parameter at Deep Depths for σu = 5% and σd = 5%

Tables Icon

Table V Effect of Measurement Errors on the Estimated Similarity Parameter at Deep Depths for σu = 5 % and σd = 1 %

Tables Icon

Table VI Estimated Value of bb/a at Deep Depths

Equations (53)

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

s = [ 1 + ( b / a ) ( 1 β ¯ ) ] 1 / 2 .
E d ( z ) = 0 2 π d ϕ 0 1 d μ μ L ( z , μ , ϕ ) ,
E u ( z ) = 0 2 π d ϕ 0 1 d μ μ L ( z , μ , ϕ ) .
E 0 d ( z ) = 0 2 π d ϕ 0 1 d μ L ( z , μ , ϕ ) ,
E 0 u ( z ) = 0 2 π d ϕ 0 1 d μ L ( z , μ , ϕ ) .
E ( z ) = E d ( z ) E u ( z ) ,
E 0 ( z ) = E 0 d ( z ) + E 0 u ( z ) ,
R ( z ) = E u ( z ) / E d ( z ) ,
R 0 ( z ) = E 0 u ( z ) / E 0 d ( z ) .
L ( z , μ ) = 0 2 π L ( z , μ , ϕ ) d ϕ
μ z L ( z , μ ) + ( a + b ) L ( z , μ ) = b 1 1 β ( μ , μ ) L ( z , μ ) d μ .
β ( μ , μ ) = 1 2 n = 0 N ( 2 n + 1 ) f n P n ( μ ) P n ( μ ) ,
f n = 1 1 P n ( μ ) β ( μ , 1 ) d μ , f 0 = 1 , f 1 = β ¯ .
L ( z , μ ) = A 1 exp ( K E z ) P ( μ ) + A 2 exp ( K E z ) P ( μ ) .
E ( z ) = ( a / K E ) [ A 1 exp ( K E z ) A 2 exp ( K E z ) ] ,
E 0 ( z ) = A 1 exp ( K E z ) + A 2 exp ( K E z ) .
1 1 μ P ( μ ) d μ = a / K E
1 1 P ( μ ) d μ = 1 .
a = μ ¯ K E ,
μ ¯ = E ( z ) / E 0 ( z ) , K E = [ E ( z ) ] 1 d E ( z ) / d z , [ ln E ( z 1 ) ln E ( z 2 ) ] / ( z 1 z 2 ) .
E 2 ( z ) = ( a / K E ) 2 [ A 1 2 exp ( 2 K E z ) 2 A 1 A 2 + A 2 2 exp ( 2 K E z ) ] , E 0 2 ( z ) = A 1 2 exp ( 2 K E z ) + 2 A 1 A 2 + A 2 2 exp ( 2 K E z ) ,
Δ E 2 ( z 1 , z 2 ) = E 2 ( z 1 ) E 2 ( z 2 ) ,
Δ E 0 2 ( z 1 , z 2 ) = E 0 2 ( z 1 ) E 0 2 ( z 2 )
s 2 = C 2 Δ E 2 ( z 1 , z 2 ) / Δ E 0 2 ( z 1 , z 2 ) .
C 2 = K E 2 / { a [ a + b ( 1 β ¯ ) ] } .
s = C E ( z ) / E 0 ( z ) = μ ¯ C .
K E = 3 a [ a + b ( 1 β ¯ ) ] 1 / 2 { 1 2 5 a / [ a + b ( 1 f 2 ) ] + } ,
C 2 = 3 { 1 s [ ( a 1 a 2 s ) / ( a 3 a 4 s ) ] } 2 .
s 2 3 { 1 s [ ( a 1 a 2 s ) / ( a 3 a 4 s ) ] } 2 Δ E 2 ( z 1 , z 2 ) / Δ E 0 2 ( z 1 , z 2 ) = 0 ,
s 3 { 1 s [ ( a 1 a 2 s ) / ( a 3 a 4 s ) ] } E u ( z ) / E d ( z ) = 0 .
P ( μ ) = 1 2 ( 1 μ ¯ 2 ) / ( 1 + μ 2 2 μ ¯ μ ) 3 / 2 , = 1 2 n = 0 ( 2 n + 1 ) μ ¯ n P n ( μ ) .
β ( μ , μ ) = α δ ( μ μ ) + [ ( 1 α ) / 2 ] × n = 0 P n ( μ ) P n ( μ ) , 0 α 1 ,
f n = ( 2 n α + 1 ) / ( 2 n + 1 ) , 0 α 1 .
L n ( z ) = 1 1 P n ( μ ) L ( z , μ ) d μ ,
L n ( z ) = μ ¯ n .
L ( z , μ ) = A 1 / 2 + A 2 [ ( 1 β ¯ ) z μ ] / 2 ,
R = [ ( 1 μ ¯ ) / ( 1 + μ ¯ ) ] [ ( 1 + μ ¯ 2 ) 1 / 2 μ ¯ ] 2 .
μ ¯ = [ 1 + ( 4 + 2 2 ) b b / a ] 1 / 2 .
s 2 = [ 1 + 2 3 ( 2 + 2 ) b b / a ] 1 .
b / a = 1 2 ( μ ¯ 2 1 ) / ( 1 α ) .
[ E d ( z ) E u ( z ) ] 2 = [ F d F u ] 2 [ A 1 2 exp ( 2 K E z ) 2 A 1 A 2 + A 2 2 exp ( 2 K E z ) ] 2 , [ E d ( z ) + E u ( z ) ] 2 = [ F d + F u ] 2 [ A 1 2 exp ( 2 K E z ) + 2 A 1 A 2 + A 2 2 exp ( 2 K E z ) ] 2 ,
F d = 0 1 μ P ( μ ) d μ , F u = 0 1 μ P ( μ ) d μ ,
Δ ( z 1 , z 2 ) = Δ E 2 ( z 1 , z 2 ) = [ E d ( z 1 ) E u ( z 1 ) ] 2 [ E d ( z 2 ) E u ( z 2 ) ] 2 ,
Δ + ( z 1 , z 2 ) = [ E d ( z 1 ) + E u ( z 1 ) ] 2 [ E d ( z 2 ) + E u ( z 2 ) ] 2
Δ ( z 1 , z 2 ) / Δ + ( z 1 , z 2 ) = [ ( F d F u ) / ( F d + F u ) ] 2 , = [ ( 1 R ) / ( 1 + R ) ] 2 .
β ( μ , μ ) = α δ ( μ μ ) + [ ( 1 α ) / 2 ] × n = 0 ( 2 n + 1 ) f ˜ n P n ( μ ) P n ( μ ) , 0 α 1 .
a 1 = 0 . 30633 , a 3 = 6 . 2114 , a 2 = 0 . 32405 , a 4 = 4 . 5767 .
β ( θ ) / b = 0 . 00328 ( 1 . 0006 cos θ ) 1 . 4 , 0 θ 10 ° , = 0 . 00224 ( 1 . 017 cos θ ) 1 . 8 , 10 ° θ 90 ° , = 0 . 00629 + 0 . 00272 cos θ 0 . 00411 sin θ , 90 ° θ 180 ° .
a 1 = 0 . 31253 , a 3 = 6 . 2423 , a 2 = 0 . 31189 , a 4 = 4 . 6071 ,
τ * = z [ a + b ( 1 α ) ] ,
f n * = [ b ( 1 α ) ] f ˜ n / [ a + b ( 1 α ) ] .
μ r * L ( τ * , μ ) + L ( τ * , μ ) = 1 1 β * ( μ , μ ) L ( τ * , μ ) d μ
β * ( μ , μ ) = 1 2 n = 0 N * ( 2 n + 1 ) f n * P n ( μ ) P n ( μ ) .

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