Abstract

Laser systems have been developed to image underwater objects. However, the performance of these systems can be severely degraded in turbid water. We have developed a technique using modulated light to improve underwater detection and imaging. A program, Modulated Vision System (MVS), which is based on a new theoretical approach, has been developed to predict modulated laser imaging performance. Experiments have been conducted in a controlled laboratory environment to test the accuracy of the theory as a function of system and environmental parameters. Results show a strong correlation between experiment and theory and indicate that the MVS program can be used to predict future system performance.

© 2004 Optical Society of America

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References

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  1. M. P. Strand, “Underwater electro-optical system for mine identification,” in Detection Technologies for Mines and Minelike Targets, A. C. Dubey, I. Cindrich, J. M. Ralston, K. A. Rigano, eds., Proc. SPIE2496, 487–497 (1995).
    [CrossRef]
  2. G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
    [CrossRef]
  3. J. W. McLean, “High-resolution 3D underwater imaging,” in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE3761, 10–19 (1999).
    [CrossRef]
  4. M. W. Scott, “Range imaging laser radar,” U.S. patent4,935,616 (19June1990).
  5. J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
    [CrossRef]
  6. L. Mullen, V. M. Contarino, P. R. Herczfeld, “Modulator lidar system,” U.S. patent5,822,047 (13October1998).
  7. L. Mullen, V. M. Contarino, P. R. Herczfeld, “Hybrid lidar-radar ocean experiment,” IEEE Trans. Microwave Theory Tech. 44, 2703–2710 (1996).
  8. L. Mullen, E. Zege, I. Katsev, A. Prikhach, “Modulated lidar system: experiment and theory,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frouin, G. D. Gilbert, eds., Proc. SPIE4488, 25–35 (2001).
    [CrossRef]
  9. I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  10. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803–2812 (1993).
    [CrossRef] [PubMed]
  11. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).
    [CrossRef]
  12. E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
    [CrossRef]
  13. J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.
  14. F. W. Sears, M. W. Zemansky, H. D. Young, University Physics (Addison-Wesley, New York, 1987).

1997 (1)

1996 (1)

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Hybrid lidar-radar ocean experiment,” IEEE Trans. Microwave Theory Tech. 44, 2703–2710 (1996).

1993 (2)

G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803–2812 (1993).
[CrossRef] [PubMed]

Blume, B. T.

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

Bonnier, D.

G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
[CrossRef]

Bruscaglioni, P.

E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
[CrossRef]

Concannon, B.

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

Contarino, V.

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

Contarino, V. M.

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Hybrid lidar-radar ocean experiment,” IEEE Trans. Microwave Theory Tech. 44, 2703–2710 (1996).

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Modulator lidar system,” U.S. patent5,822,047 (13October1998).

Edwards, J. W.

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

Forand, J. L.

G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
[CrossRef]

Foster, J.

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

Fournier, G. R.

G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
[CrossRef]

Herczfeld, P. R.

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Hybrid lidar-radar ocean experiment,” IEEE Trans. Microwave Theory Tech. 44, 2703–2710 (1996).

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Modulator lidar system,” U.S. patent5,822,047 (13October1998).

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).
[CrossRef]

Katsev, I.

L. Mullen, E. Zege, I. Katsev, A. Prikhach, “Modulated lidar system: experiment and theory,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frouin, G. D. Gilbert, eds., Proc. SPIE4488, 25–35 (2001).
[CrossRef]

Katsev, I. L.

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803–2812 (1993).
[CrossRef] [PubMed]

E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
[CrossRef]

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).
[CrossRef]

Laux, A.

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

Lebien, S. M.

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

Ludbrook, G. D.

E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
[CrossRef]

McLean, J. W.

J. W. McLean, “High-resolution 3D underwater imaging,” in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE3761, 10–19 (1999).
[CrossRef]

Mullen, L.

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Hybrid lidar-radar ocean experiment,” IEEE Trans. Microwave Theory Tech. 44, 2703–2710 (1996).

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

L. Mullen, E. Zege, I. Katsev, A. Prikhach, “Modulated lidar system: experiment and theory,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frouin, G. D. Gilbert, eds., Proc. SPIE4488, 25–35 (2001).
[CrossRef]

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Modulator lidar system,” U.S. patent5,822,047 (13October1998).

Nellums, R. O.

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

Pace, P. W.

G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
[CrossRef]

Polonsky, I. N.

Prentice, J.

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

Prikhach, A.

L. Mullen, E. Zege, I. Katsev, A. Prikhach, “Modulated lidar system: experiment and theory,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frouin, G. D. Gilbert, eds., Proc. SPIE4488, 25–35 (2001).
[CrossRef]

Prikhach, A. S.

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
[CrossRef]

Rish, J. W.

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

Scott, M. W.

M. W. Scott, “Range imaging laser radar,” U.S. patent4,935,616 (19June1990).

Sears, F. W.

F. W. Sears, M. W. Zemansky, H. D. Young, University Physics (Addison-Wesley, New York, 1987).

Strand, M. P.

M. P. Strand, “Underwater electro-optical system for mine identification,” in Detection Technologies for Mines and Minelike Targets, A. C. Dubey, I. Cindrich, J. M. Ralston, K. A. Rigano, eds., Proc. SPIE2496, 487–497 (1995).
[CrossRef]

Weidemann, A.

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

Young, H. D.

F. W. Sears, M. W. Zemansky, H. D. Young, University Physics (Addison-Wesley, New York, 1987).

Zege, E.

L. Mullen, E. Zege, I. Katsev, A. Prikhach, “Modulated lidar system: experiment and theory,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frouin, G. D. Gilbert, eds., Proc. SPIE4488, 25–35 (2001).
[CrossRef]

Zege, E. P.

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and oceanic sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803–2812 (1993).
[CrossRef] [PubMed]

E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
[CrossRef]

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).
[CrossRef]

Zemansky, M. W.

F. W. Sears, M. W. Zemansky, H. D. Young, University Physics (Addison-Wesley, New York, 1987).

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (1)

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Hybrid lidar-radar ocean experiment,” IEEE Trans. Microwave Theory Tech. 44, 2703–2710 (1996).

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

G. R. Fournier, D. Bonnier, J. L. Forand, P. W. Pace, “Range-gated underwater laser imaging system,” Opt. Eng. 32, 2185–2190 (1993).
[CrossRef]

Other (10)

J. W. McLean, “High-resolution 3D underwater imaging,” in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE3761, 10–19 (1999).
[CrossRef]

M. W. Scott, “Range imaging laser radar,” U.S. patent4,935,616 (19June1990).

J. W. Rish, S. M. Lebien, R. O. Nellums, J. Foster, J. W. Edwards, B. T. Blume, “Performance of a gated scannerless optical range imager against volume and bottom targets in a controlled underwater environment,” in Information Systems for Divers and Autonomous Underwater Vehicles Operating in Very Shallow Water and Surf Zone Regions II, J. L. Wood-Putnam, ed., Proc. SPIE4039, 114–123 (2000).
[CrossRef]

L. Mullen, V. M. Contarino, P. R. Herczfeld, “Modulator lidar system,” U.S. patent5,822,047 (13October1998).

L. Mullen, E. Zege, I. Katsev, A. Prikhach, “Modulated lidar system: experiment and theory,” in Ocean Optics: Remote Sensing and Underwater Imaging, R. J. Frouin, G. D. Gilbert, eds., Proc. SPIE4488, 25–35 (2001).
[CrossRef]

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).
[CrossRef]

E. P. Zege, I. L. Katsev, A. S. Prikhach, G. D. Ludbrook, P. Bruscaglioni, “Analytical and computer modeling of the ocean lidar performance,” in 12th International Workshop on Lidar Multiple Scattering Experiments, C. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 189–199 (2002).
[CrossRef]

J. Prentice, A. Laux, B. Concannon, L. Mullen, V. Contarino, A. Weidemann, “Comparison of Monte Carlo model predictions with tank beam spread experiments using a Maalox phase function obtained with volume scattering function instruments,” presented at the 2002 Ocean Sciences Meeting, Honolulu, Hawaii, 11–15 February 2002.

F. W. Sears, M. W. Zemansky, H. D. Young, University Physics (Addison-Wesley, New York, 1987).

M. P. Strand, “Underwater electro-optical system for mine identification,” in Detection Technologies for Mines and Minelike Targets, A. C. Dubey, I. Cindrich, J. M. Ralston, K. A. Rigano, eds., Proc. SPIE2496, 487–497 (1995).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

(a) Dependency of φBSN(z, f) and φVS(z, f) on modulation frequency for z = 2.74 m and c = 2.2/m. (b) Dependency of φBSN(z, f) and φVS(z, f) on depth for f = 33 MHz and c = 2.2/m. (c) Dependency of φBSN(z, f) and φVS(z, f) on the beam attenuation coefficient for z = 2.74 m and f = 33 MHz.

Fig. 2
Fig. 2

(a)–(c) P BSN(z, f) and P VS(z, f) corresponding to the phase data in Fig. 1.

Fig. 3
Fig. 3

(a) Total power P(z, f) as a function of modulation frequency for the data in Fig. 1(a) and 2(a). The power was normalized relative to the power at f = 10 MHz. The dashed lines indicate the frequencies at which destructive and constructive interference occurs. (b) φ = φBSN(z, f) - φVS(z, f) (left axis) and η = P BSN(z, f)/P VS(z, f) (right axis) as a function of modulation frequency calculated from the data in Figs. 1(a) and 2(a). The dashed lines indicate the frequencies at which destructive and constructive interference occurs.

Fig. 4
Fig. 4

(a) and (b) Data from Figs. 3(a) and 3(b) plotted along with the data corresponding to an increased object depth of z = 3.25 m.

Fig. 5
Fig. 5

(a) and (b) Data from Figs. 3(a) and 3(b) plotted along with the data corresponding to an increased beam attenuation coefficient of c = 2.6/m.

Fig. 6
Fig. 6

Contrast k(z, f) as a function of modulation frequency for the data in Figs. 3 5.

Fig. 7
Fig. 7

Plot of k destr as a function of η = P BSN(z, f)/P VS(z, f).

Fig. 8
Fig. 8

Images corresponding to the data shown in Figs. 3 and 6. CW, continuous wave. The water-free (WF) image is also shown for comparison. The graphs below the computer-generated images show the normalized energy (black curve, left axis) and η = P BSN(z, f)/P VS(z, f) (gray curve, right axis) plotted as a function of position r.

Fig. 9
Fig. 9

Effect of a change in target albedo on the object image. The albedo of the white portion of the target was varied while all other parameters remained the same as in Fig. 8.

Fig. 10
Fig. 10

Images corresponding to the data shown in Figs. 4 and 6. CW, continuous wave.

Fig. 11
Fig. 11

Images corresponding to the data shown in Figs. 5 and 6. CW, continuous wave.

Fig. 12
Fig. 12

Diagram of the experimental setup used to validate the MVS simulation predictions.

Fig. 13
Fig. 13

Model (black curves) and experimental (gray curves) results for the (a) target contrast, k(z, f) and (b) phase difference between the backscatter and the valid signals φ = φBSN(z, f) - φVS(z, f) for three different water clarities.

Fig. 14
Fig. 14

Data from Fig. 13 plotted along with the results obtained with a decreased target depth of z = 1.83 m and an increased beam attenuation of c = 3.7/m (model data results in black, experimental results in gray).

Fig. 15
Fig. 15

Model (black) and experimental (gray) images for c = 1.2/m and z = 2.74 m [the values for η(r) are plotted on the right axis for the destructive images and are indicated by filled diamonds]. CW, continuous wave.

Fig. 16
Fig. 16

Model (black) and experimental (gray) images for c = 2.1/m and z = 2.74 m [the values for η(r) are plotted on the right axis for the destructive images and are indicated by filled diamonds]. CW, continuous wave.

Fig. 17
Fig. 17

Model (black) and experimental (gray) images for c = 2.5/m and z = 2.74 m [the values for η(r) are plotted on the right axis for the destructive images and are indicated by filled diamonds]. CW, continuous wave.

Fig. 18
Fig. 18

Model (black) and experimental (gray) images for c = 3.7/m and z = 1.83 m [the values for η(r) are plotted on the right axis for the destructive images and are indicated by filled diamonds]. CW, continuous wave.

Tables (2)

Tables Icon

Table 1 System Configurationa and Environmental Parameters Used as Inputs to the MVS Program to Produce the Results Shown in Figs. 1 11

Tables Icon

Table 2 System Configurationa and Environmental Parameters Corresponding to the Experimental Setup Shown in Fig. 12 and Used as Inputs to the MVS Program

Equations (19)

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

Pt=P01+expi2πft,
Pr, t=PVSr, t+Pbr, t,
Pbr, t=Pbrexpi2πft-φbr,
PVSr, t=PVSrexpi2πft-φVSr,
Pbr, t=PBSNr, t+Pbotr, t,
PBSNr, t=PBSNrexpi2πft-φBSNr,
Pbotr, t=Pbotrexpi2πft-φbotr
PVSr, t=Pobr, t-Pshr, t.
Pr=|Pbr, t+PVSr, t|=Pbr1+1η2r+2ηrcos φr1/2,
ηr=Pbr/PVSr,
φr=φbr-φVSr.
kz, f=Pz, f-PBSNz, fPz, f+PBSNz, f,
kconstr=|PVS||PVS|+2|PBSN|=11+2η,
η=|PBSN||PVS|.
kdestr=|PVS|-2|PBSN||PVS|=1-2η.
kdestr=|PVS|2|PBSN|-|PVS|=-12η-1.
kdestr>0 atη<0.5, kdestr<0 atη>0.5.
φr=φBSNr-φVSrconst.
ηr=1-ξr-1, PBSNr>PVSr ηr=1+ξr-1, PVSr>PBSNr,

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