Abstract

We present a model to simulate receiver waveforms from an airborne sea-depth-sounding lidar to compare the influence that is due to different shapes of objects placed on the sea bottom. The objects are of size 1 m3, and the bottom depths are 5–12 m. We use an existing analytical beam-propagation model and divide the bottom into squares. For each element on the bottom grid we create a transmitted and a reflected waveform. The waveforms are summed, yielding a total contribution from all bottom elements. We compare two object types, cylinder and cube, and find that the difference in the receiver waveform is small between these objects. Simulated waveforms are compared with experimental data from the Swedish Hawk Eye system and show good agreement.

© 1999 Optical Society of America

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References

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  1. G. C. Guenther, R. W. L. Thomas, “Prediction and correction of propagation-induced depth measurement biases plus signal attenuation and beam spreading for airborne laser hydrography,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1984).
  2. K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Experimental evaluation of an airborne depth-sounding lidar,” Opt. Eng. 32, 1307–1321 (1993).
    [CrossRef]
  3. T. Kaijser, “A Monte Carlo simulation model of airborne hydrographic laser systems,” (Defence Research Establishment, Linköping, Sweden, 1990).
  4. K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Simulation of laser bathymetry for irregular bottoms,” (Defence Research Establishment, Linköping, Sweden, 1994).
  5. W. H. Wells, “Diffusion of light in the sea,” Opt. Eng. 16, 112–127 (1977).
    [CrossRef]
  6. O. V. Kopelevich, I. M. Levin, “The main problems of the optics of the sea,” J. Opt. Technol. 64, 230–239 (1997).
  7. D. Arnush, “Underwater light-beam propagation in the small-angle-scattering approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [CrossRef]
  8. L. R. Thebaud, S. J. Gayer, “Calculation of lidar beam spread in stratified media,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 236–246 (1984).
    [CrossRef]
  9. R. F. Lutomirski, “An analytic model for optical beam propagation through the marine boundary layer,” in Ocean Optics V, M. B. White, ed., Proc. SPIE160, 110–122 (1978).
    [CrossRef]
  10. V. J. Feigels, “Lidars for oceanological research: criteria for comparison, main limitations, perspectives,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. SPIE1750, 473–484 (1992).
    [CrossRef]
  11. H. R. Gordon, “Interpretation of airborne oceanic lidar: effects of multiple scattering,” Appl. Opt. 21, 2996–3001 (1982).
    [CrossRef] [PubMed]
  12. K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
    [CrossRef]
  13. G. C. Guenther, “Airborne laser hydrography, system design and performance factors,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1985).
  14. L. S. Dolin, I. M. Levin, “Optics, underwater,” in Vol. 12 of Encyclopedia of Applied Physics, G. L. Trigg, ed. (VCH, New York, 1995), pp. 571–601.
  15. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).
  16. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [CrossRef] [PubMed]
  17. J. W. McLean, J. D. Freeman, R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
    [CrossRef]
  18. H. C. van de Hulst, G. W. Kattawar, “Exact spread function for a pulsed collimated beam in a medium with small-angle scattering,” Appl. Opt. 33, 5820–5829 (1994).
    [CrossRef] [PubMed]

1998

1997

O. V. Kopelevich, I. M. Levin, “The main problems of the optics of the sea,” J. Opt. Technol. 64, 230–239 (1997).

1994

1993

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Experimental evaluation of an airborne depth-sounding lidar,” Opt. Eng. 32, 1307–1321 (1993).
[CrossRef]

1982

1977

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

W. H. Wells, “Diffusion of light in the sea,” Opt. Eng. 16, 112–127 (1977).
[CrossRef]

1973

1972

Arnush, D.

Bucher, E. A.

Dolin, L. S.

L. S. Dolin, I. M. Levin, “Optics, underwater,” in Vol. 12 of Encyclopedia of Applied Physics, G. L. Trigg, ed. (VCH, New York, 1995), pp. 571–601.

Feigels, V. J.

V. J. Feigels, “Lidars for oceanological research: criteria for comparison, main limitations, perspectives,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. SPIE1750, 473–484 (1992).
[CrossRef]

Freeman, J. D.

Gayer, S. J.

L. R. Thebaud, S. J. Gayer, “Calculation of lidar beam spread in stratified media,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 236–246 (1984).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

Gordon, H. R.

Guenther, G. C.

G. C. Guenther, “Airborne laser hydrography, system design and performance factors,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1985).

G. C. Guenther, R. W. L. Thomas, “Prediction and correction of propagation-induced depth measurement biases plus signal attenuation and beam spreading for airborne laser hydrography,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1984).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

Kaijser, T.

T. Kaijser, “A Monte Carlo simulation model of airborne hydrographic laser systems,” (Defence Research Establishment, Linköping, Sweden, 1990).

Karlsson, U. C. M.

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Experimental evaluation of an airborne depth-sounding lidar,” Opt. Eng. 32, 1307–1321 (1993).
[CrossRef]

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Simulation of laser bathymetry for irregular bottoms,” (Defence Research Establishment, Linköping, Sweden, 1994).

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

Kattawar, G. W.

Kopelevich, O. V.

O. V. Kopelevich, I. M. Levin, “The main problems of the optics of the sea,” J. Opt. Technol. 64, 230–239 (1997).

Koppari, K. R.

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Experimental evaluation of an airborne depth-sounding lidar,” Opt. Eng. 32, 1307–1321 (1993).
[CrossRef]

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Simulation of laser bathymetry for irregular bottoms,” (Defence Research Establishment, Linköping, Sweden, 1994).

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

Levin, I. M.

O. V. Kopelevich, I. M. Levin, “The main problems of the optics of the sea,” J. Opt. Technol. 64, 230–239 (1997).

L. S. Dolin, I. M. Levin, “Optics, underwater,” in Vol. 12 of Encyclopedia of Applied Physics, G. L. Trigg, ed. (VCH, New York, 1995), pp. 571–601.

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

Lutomirski, R. F.

R. F. Lutomirski, “An analytic model for optical beam propagation through the marine boundary layer,” in Ocean Optics V, M. B. White, ed., Proc. SPIE160, 110–122 (1978).
[CrossRef]

McLean, J. W.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

Steinvall, K. O.

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Experimental evaluation of an airborne depth-sounding lidar,” Opt. Eng. 32, 1307–1321 (1993).
[CrossRef]

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Simulation of laser bathymetry for irregular bottoms,” (Defence Research Establishment, Linköping, Sweden, 1994).

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

Thebaud, L. R.

L. R. Thebaud, S. J. Gayer, “Calculation of lidar beam spread in stratified media,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 236–246 (1984).
[CrossRef]

Thomas, R. W. L.

G. C. Guenther, R. W. L. Thomas, “Prediction and correction of propagation-induced depth measurement biases plus signal attenuation and beam spreading for airborne laser hydrography,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1984).

van de Hulst, H. C.

Walker, R. E.

Wells, W. H.

W. H. Wells, “Diffusion of light in the sea,” Opt. Eng. 16, 112–127 (1977).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Technol.

O. V. Kopelevich, I. M. Levin, “The main problems of the optics of the sea,” J. Opt. Technol. 64, 230–239 (1997).

NBS Monog.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monog. 160, D.C. (1977).

Opt. Eng.

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Experimental evaluation of an airborne depth-sounding lidar,” Opt. Eng. 32, 1307–1321 (1993).
[CrossRef]

W. H. Wells, “Diffusion of light in the sea,” Opt. Eng. 16, 112–127 (1977).
[CrossRef]

Other

G. C. Guenther, R. W. L. Thomas, “Prediction and correction of propagation-induced depth measurement biases plus signal attenuation and beam spreading for airborne laser hydrography,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1984).

T. Kaijser, “A Monte Carlo simulation model of airborne hydrographic laser systems,” (Defence Research Establishment, Linköping, Sweden, 1990).

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Simulation of laser bathymetry for irregular bottoms,” (Defence Research Establishment, Linköping, Sweden, 1994).

L. R. Thebaud, S. J. Gayer, “Calculation of lidar beam spread in stratified media,” in Ocean Optics VII, M. A. Blizard, ed., Proc. SPIE489, 236–246 (1984).
[CrossRef]

R. F. Lutomirski, “An analytic model for optical beam propagation through the marine boundary layer,” in Ocean Optics V, M. B. White, ed., Proc. SPIE160, 110–122 (1978).
[CrossRef]

V. J. Feigels, “Lidars for oceanological research: criteria for comparison, main limitations, perspectives,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. SPIE1750, 473–484 (1992).
[CrossRef]

K. O. Steinvall, K. R. Koppari, U. C. M. Karlsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

G. C. Guenther, “Airborne laser hydrography, system design and performance factors,” (National Oceanic and Atmospheric Administration, Rockville, Md., 1985).

L. S. Dolin, I. M. Levin, “Optics, underwater,” in Vol. 12 of Encyclopedia of Applied Physics, G. L. Trigg, ed. (VCH, New York, 1995), pp. 571–601.

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Figures (29)

Fig. 1
Fig. 1

Schematic description of the laser bathymetry system and our coordinate system. In fact the platform altitude H is larger in relation to the spot size on the water surface and the receiver FOV diameter. Thus the beam divergence and the FOV angle are much smaller than shown. The coordinate system is left oriented, with its origin in the hit on surface of the optical axis of collocated, collinear transmitter and receiver. The laser inclination angle θ0 is always in the xz plane.

Fig. 2
Fig. 2

Behavior of the irradiance distribution according to the Lutomirski model.

Fig. 3
Fig. 3

Standard deviation in the x direction versus depth for irradiance distribution according to the Lutomirski model; water type, clear; θ0 = 20°.

Fig. 4
Fig. 4

Standard deviation in the x direction versus depth for the ray vector distribution according to the Lutomirski model; water type, clear; θ0 = 20°. The B x axis shows the dimensionless standard deviation of the projection of the unit ray vector along the x direction.

Fig. 5
Fig. 5

Maximum radiance ray direction along the x direction versus depth for ray vector distribution according to the Lutomirski model; water type, clear; θ0 = 20°. The sample is taken at x = 0, y = 0, and the nadir angles are given in degrees for clarity.

Fig. 6
Fig. 6

Irradiance distribution of the inverse transmitter (the receiver).

Fig. 7
Fig. 7

(a), (b), (c) Local system centered horizontally on top of each object or element A, B, and C, respectively.

Fig. 8
Fig. 8

Cartesian and spherical coordinates of the local coordinate system, where η̂, is the beam maximum direction or the unit ray vector.

Fig. 9
Fig. 9

Reflection geometry for the impulse response calculation from one object or element.

Fig. 10
Fig. 10

Zoom of the small area component dA k in Fig. 9.

Fig. 11
Fig. 11

Shadowing along the x axis of the cylinder. The shadow can fall on either side for both the illumination rays and the viewing rays.

Fig. 12
Fig. 12

Shadowing along the y axis of the cylinder. The shadow can fall on either side, for both the illumination rays and the viewing rays.

Fig. 13
Fig. 13

Geometry of the system. The lidar nadir angle θ0 is always in the xz plane in the left-oriented coordinate system.

Fig. 14
Fig. 14

From Dolin and Levin14: Dimensionless effective pulse duration tr′ = 2c w at r as a function of the effective distance ζ from a source for an initially δ-pulsed light beam with different initial beam cross sections S 0. The value of dimensionless beam cross section S 0′ = 0.25a 2σ D 2 S 0 is indicated for each curve. A receiver with a wide field angle is placed on the beam axis.

Fig. 15
Fig. 15

Impulse response function from bottom depth 12 m with our model. Parameter values: σ s = 0.1 m; θ0 = 20°; water type, average; a FOV diameter on the water surface equal to the bottom depth.

Fig. 16
Fig. 16

Comparison of 25% and 50% widths of IRF’s at bottom depths of 4, 8, 12, and 16 m. Typical IRF’s from Monte Carlo simulations by Guenther and Thomas1 with the Navy phase function. Parameter values: σ s = 0.1 m; θ0 = 20°; water type, average; and FOV diameter equal to the bottom depth. Our model results are plotted with the standard deviation that occurs when the bottom sampling grid is moved horizontally. The standard deviation is near zero, except for the bottom depth 4 m, where the variation is explained by the fact that our assumption (Subsection 2.A) of a bottom sampling grid (1 m × 1 m) that is small in relation to the beam width is less realistic. See also Fig. 17.

Fig. 17
Fig. 17

Comparison of 50% widths of IRF’s for two different bottom grid side lengths, 0.5 and 1.0 m. Parameter values: σ s = 0.6 m; θ0 = 20°; water type, clear; and FOV of 20 mrad. Our model results are plotted with the standard deviation that occurs when the bottom sampling grid is moved horizontally. For a grid side of 1 m the variation at 4 and 5 m (and shallower) bottom depth is explained by the fact that our assumption (Subsection 2.A) of a bottom sampling grid (1 m × 1 m) that is small in relation to the beam width is less realistic. The remaining standard deviation for bottom depths of >5 m is the numerical noise that is due to the temporal sampling (we used 0.5 ns).

Fig. 18
Fig. 18

Definition of the height and width characteristics of the pulses in the received waveforms.

Fig. 19
Fig. 19

Comparison of measured waveforms and model waveforms from four different bottom depths z. The depth-axis scaling shows the elapsed time from the surface hit multiplied by c w /2. The y axis shows the logarithm of the received power in relative units. Noticeable errors occur for 16-m depth caused by violation of the validity range for the element parameterization; see conditions C1–C4 in Appendix B.

Fig. 20
Fig. 20

Comparison of bottom pulse widths (width is defined in Fig. 18) from measured and model waveforms from a flat bottom at three depths. Measured data were taken from five waveforms at each depth; water type, clear.

Fig. 21
Fig. 21

Comparison of measured waveforms and model waveforms. Hawk Eye data from a 1 m × 1 m × 1 m cube at two horizontally adjacent positions along the y axis with two model checks. The cube was mounted upon a flat 10 m × 10 m frame. The top of the cube was at 6.4-m depth, and the frame was at 8.7-m depth. Sounding density, 2 m; θ0 = 20°; σ s = 0.6 m; FOV, 30 mrad; water type, clear. The exact xy positions of the cube are not known from measurements but are given as intervals. In the simulations the reflectances were set to ρ m = 10% (cube) and ρ m = 6% (frame).

Fig. 22
Fig. 22

Simulation with the model. Zoom of waveforms with the top of object at 7-m depth and a flat bottom at 8-m depth. Two different horizontal positions. Water type, clear; θ0 = 20°; σ s = 0.6 m; FOV, 30 mrad.

Fig. 23
Fig. 23

Comparison of bottom and object pulse height normalized to the maximum bottom pulse height (height is defined in Fig. 18) as a function of the object position on the x axis when y = 0. Model waveforms from the object on a flat bottom at two depths. Water type, clear; θ0 = 20°; σ s = 0.6 m; FOV, 30 mrad. The object pulse height is plotted only when the bottom peak and the object peak are separated, as shown in Fig. 22(a). The center of the downwelling distribution (x 1) at the sea bottom is marked with an ! on the x axis.

Fig. 24
Fig. 24

Typical impulse response for a cylinder together with its fit to the γ-distribution function. A maximum sliding temporal mean between the fit and the numerically obtained impulse response of 0.5 ns corresponds to an approximately 5-cm maximum depth error at the receiver.

Fig. 25
Fig. 25

Simple illustration of the impulse response from two cases of illumination and viewing ray directions on a diffuse reflecting surface.

Fig. 26
Fig. 26

Geometrical reflectance ρ̂ g as a function of Θ i for fixed Φ i = Φ v = Θ v = 0. Center value and maximum error values for the three objects, B xi = B yi = 0.15, B xv = B yv = 0.30. Note the nonlinear scale on the ρ g axis.

Fig. 27
Fig. 27

Geometrical reflectance ρ̂ g as a function of B xv = B yv for fixed Φ i = Φ v = Θ i = Θ v = 0. Center value and maximum error values for the three elements, B xi = B yi = 0.15.

Fig. 28
Fig. 28

Standard deviation σ̂ of the impulse response as a function of Θ i = Θ v for fixed Φ i = Φ v = 0. Center value and maximum error values for the three elements, B xi = B yi = 0.15, B xv = B yv = 0.30.

Fig. 29
Fig. 29

Standard deviation σ̂ of the impulse response as a function of Θ i = Θ v for fixed Φ i = Φ v = 90°. Center value and maximum error values for the three elements, B xi = B yi = 0.15, B xv = B yv = 0.30.

Tables (4)

Tables Icon

Table 1 Three Element Types for Modeling the Seabed

Tables Icon

Table 2 Parameters for Simulation Unless Otherwise Quantified in Text

Tables Icon

Table 3 Theoretical Comparison of Necessary Shot-to-Shot Distance on the x Axis If the Shots Lie on the line y = 0 and a Separate Object Pulse Peak is Required

Tables Icon

Table 4 Element Model Parameters with Output Error Estimates

Equations (41)

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P=Pb+Pbsc+Ps+N,
Pbsc=0.5×t0c0βπnw×1-ρs2P0ARηopt exp-2K zcos θwFTatm2 cos2 θ0nwH+z2,
Ex, y, z=Pz 12πAxAy exp-x-x122Ax2-y22Ay2,
x1=z sin θ0nwRx,
Pz,1W=1-ρsexp-Kz1-sin2 θ0nw2Rx1/2RxRy,
Pz=1-ρsP0 exp-Kz1-sin2 θ0nw2Rx1/2TatmRxRy.
Rx, y, z, kˆ=12πBxBy exp-kx-ηxx22Bx2-ky-ηyy22By2,
PH|Gj  PGj|H,
RΩ=AR cos2 θ0πnwH2,
Pbj=ARηopt cos2 θ0πnwH2×1-ρsPvjexp-Kz1-sin2 θ0nw2Rx1/2TatmRxRy,
σsFOV=DiamFOV12,
Pvj=AGRIDEvjexp-x-x1v22Axv2-y22Ayv2,
Rvx, y, z, kˆ=exp-kx-ηxvx22Bxv2-ky-ηyvy22Byv2.
Evjx, y, z=Eijx, y, zρmρg,
tk=r·kˆi+r·kˆvcw,
ht=k cosΘi,dAkcosΘv,dAkdAkδt-tk
hi,vt=ωiωv RiRvhtdωidωv,
ρg=1π  hi,vtdt,
hi,vt=ρghnt,
Fsh,x=1-l tan|Θx|,
Fsh,y=1-l tan|Θy|,
Fsh,x=1-l-stan|Θx|+ssin|Θx|tan|Θx|-1+cos|Θx|,
Fsh,y=1-l-s+πs4tan|Θy|.
tdelay,d=x2+y2+z21/2cw,
tdelay,u=zcw cos θw+x-z tan θwsin θ0c0,
ζ=0.5 σD21-ωsacr,
σD2=2ωsaΘ4521-ωsa1/2
Ax=σscos θ02+z12wx2+θs2/6/Rx1/2,
Ay=σs2+z12wy2+θs2/6/Ry1/2,
wx2=σxμ02+σβcos θ0/nw2,
wy2=σyμ02+σβ/nw2
Rx=1+Kz1wx2+θs2/6,
Ry=1+Kz1wy2+θs2/6
Bx2=Qx-1wx2+θs2/2+1+zKσs/cos θ02×Kbθs2/6wx2+θs2/8,
Qx=Rx+z12wx2+θs2/6σs/cos θ02.
ηx=xzwx2+θs2/4Qxσs/cos θ02+sin θwQx×1-Kzθs2121+zKσs/cos θ02,
ηy=yzwy2+θs2/4Qyσs2.
hˆnt=λλtα-1 exp-λtΓα,  t>0,
σˆ=L10+L11BxiByiBxvByv1/2+L12cosΦi-Φv2sin Θi sin Θv+εσ,
ρˆg=expL20+L21BxiByi+L22BxvByv+L23+L24+ερ,
tˆstart=L30+L31BxiByiBxvByv1/2+L32cosΦi-Φv2sin Θi sin Θv+L33+εt,

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