Abstract

Two iterative methods of inverting lidar backscatter signals to determine altitude profiles of aerosol extinction and altitude-resolved aerosol size distribution (ASD) are presented. The first method is for inverting two-wavelength lidar signals in which the shape of the ASD is assumed to be of power-law type, and the second method is for inverting multiwavelength lidar signals without assuming any a priori analytical form of ASD. An arbitrary value of the aerosol extinction-to-backscatter ratio (S 1) is assumed initially to invert the lidar signals, and the ASD determined by use of the spectral dependence of the retrieved aerosol extinction coefficients is used to improve the value of S 1 iteratively. The methods are tested for different forms of altitude-dependent ASD’s by use of simulated lidar-backscatter-signal profiles. The effect of random noise on the lidar backscatter signals is also studied.

© 1998 Optical Society of America

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    [CrossRef]
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1996

1994

1993

V. A. Kovalev, “Lidar measurement of the vertical aerosol extinction profiles with range-dependent backscatter-to-extinction ratios,” Appl. Opt. 32, 6053–6065 (1993).
[CrossRef] [PubMed]

K. Parameswaran, K. O. Rose, “Spectral dependence of the relationship between backscattering and extinction coefficients in turbid atmospheres with application to lidar signal inversion,” Indian J. Radio Space Phys. 22, 165–179 (1993).

1992

M. J. Post, C. J. Grund, A. O. Langford, M. H. Proffitt, “Observations of Pinatubo ejecta over Boulder, Colorado by lidars of three different wavelengths,” Geophys. Res. Lett. 19, 195–198 (1992).
[CrossRef]

1991

K. K. Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

W. R. Leaitch, G. A. Isaac, “Tropospheric aerosol size distribution from 1982 to 1988 over eastern North America,” Atmos. Environ. 25A, 601–609 (1991).

K. Parameswaran, K. O. Rose, B. V. Krishna Murthy, “Relationship between backscattering and extinction coefficients of aerosols with application to turbid atmosphere,” Appl. Opt. 30, 3059–3071 (1991).
[CrossRef] [PubMed]

1990

S. Chandra, E. L. Flemming, M. R. Schoeberl, J. J. Barnett, “Monthly mean global climatology of temperature, wind, geopotential height and pressure for 0–120 km,” Adv. Space Res. 10, (6), 3–12 (1990).

1989

1988

G. S. Kent, U. O. Farrukh, P. H. Wang, A. Deepak, “SAGE I and SAM II measurements of 1 μm aerosol extinction in the free troposphere,” J. Appl. Meteorol. 27, 269–279 (1988).
[CrossRef]

1987

T. Takamura, Y. Sasano, “Ratio of aerosol backscatter to extinction coefficients as determined from angular scattering measurements for use in atmospheric lidar applications,” Opt. Quantum Electron. 19, 293–302 (1987).
[CrossRef]

1985

1984

1982

M. D. King, “Sensitivity of constrained linear inversions to the selection of Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

1981

1978

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

1973

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of atmospheric extinction using direct solar radiation measurements made with multiple wavelength radiometer,” J. Appl. Meteorol. 12, 374–380 (1973).
[CrossRef]

1968

1963

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

1955

C. E. Junge, “The size distribution and aging of natural aerosols as determined from electrical and optical data in the atmosphere,” J. Meteorol. 12, 13–25 (1955).
[CrossRef]

Barnett, J. J.

S. Chandra, E. L. Flemming, M. R. Schoeberl, J. J. Barnett, “Monthly mean global climatology of temperature, wind, geopotential height and pressure for 0–120 km,” Adv. Space Res. 10, (6), 3–12 (1990).

Brogneiz, C.

Browell, E. V.

Byrne, D. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Castagnoli, F.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Chandra, S.

S. Chandra, E. L. Flemming, M. R. Schoeberl, J. J. Barnett, “Monthly mean global climatology of temperature, wind, geopotential height and pressure for 0–120 km,” Adv. Space Res. 10, (6), 3–12 (1990).

Deepak, A.

G. S. Kent, U. O. Farrukh, P. H. Wang, A. Deepak, “SAGE I and SAM II measurements of 1 μm aerosol extinction in the free troposphere,” J. Appl. Meteorol. 27, 269–279 (1988).
[CrossRef]

Del Guasta, M.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Farrukh, U. O.

G. S. Kent, U. O. Farrukh, P. H. Wang, A. Deepak, “SAGE I and SAM II measurements of 1 μm aerosol extinction in the free troposphere,” J. Appl. Meteorol. 27, 269–279 (1988).
[CrossRef]

Fernald, F. G.

Flemming, E. L.

S. Chandra, E. L. Flemming, M. R. Schoeberl, J. J. Barnett, “Monthly mean global climatology of temperature, wind, geopotential height and pressure for 0–120 km,” Adv. Space Res. 10, (6), 3–12 (1990).

Fuller, W. F.

P. H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. F. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Grund, C. J.

M. J. Post, C. J. Grund, A. O. Langford, M. H. Proffitt, “Observations of Pinatubo ejecta over Boulder, Colorado by lidars of three different wavelengths,” Geophys. Res. Lett. 19, 195–198 (1992).
[CrossRef]

Guasta, M. D.

Hayasaka, T.

Heintzenberg, J.

Herman, B. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of atmospheric extinction using direct solar radiation measurements made with multiple wavelength radiometer,” J. Appl. Meteorol. 12, 374–380 (1973).
[CrossRef]

Isaac, G. A.

W. R. Leaitch, G. A. Isaac, “Tropospheric aerosol size distribution from 1982 to 1988 over eastern North America,” Atmos. Environ. 25A, 601–609 (1991).

Junge, C. E.

C. E. Junge, “The size distribution and aging of natural aerosols as determined from electrical and optical data in the atmosphere,” J. Meteorol. 12, 13–25 (1955).
[CrossRef]

Kent, G. S.

G. S. Kent, U. O. Farrukh, P. H. Wang, A. Deepak, “SAGE I and SAM II measurements of 1 μm aerosol extinction in the free troposphere,” J. Appl. Meteorol. 27, 269–279 (1988).
[CrossRef]

King, M. D.

M. D. King, “Sensitivity of constrained linear inversions to the selection of Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Kitamura, S.

Klett, J. D.

Kolenda, J.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Kovalev, V. A.

Krishna Murthy, B. V.

K. K. Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

K. Parameswaran, K. O. Rose, B. V. Krishna Murthy, “Relationship between backscattering and extinction coefficients of aerosols with application to turbid atmosphere,” Appl. Opt. 30, 3059–3071 (1991).
[CrossRef] [PubMed]

Langford, A. O.

M. J. Post, C. J. Grund, A. O. Langford, M. H. Proffitt, “Observations of Pinatubo ejecta over Boulder, Colorado by lidars of three different wavelengths,” Geophys. Res. Lett. 19, 195–198 (1992).
[CrossRef]

Leaitch, W. R.

W. R. Leaitch, G. A. Isaac, “Tropospheric aerosol size distribution from 1982 to 1988 over eastern North America,” Atmos. Environ. 25A, 601–609 (1991).

Lenoble, J.

McCormick, M. P.

P. H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. F. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Mielke, B.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Moorthy, K. K.

K. K. Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

Morandi, M.

M. D. Guasta, M. Morandi, L. Stefanutti, B. Stein, J. P. Wolf, “Deviation of Mount Pinatubo stratospheric aerosol mean size distribution by means of a multiwavelength lidar,” Appl. Opt. 33, 5690–5697 (1994).
[CrossRef] [PubMed]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Muller, H.

Müller, H.

Nair, P. R.

K. K. Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

Nakane, H.

Ohi, A.

Ohta, K.

Osborn, M. T.

P. H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. F. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Parameswaran, K.

K. Parameswaran, K. O. Rose, “Spectral dependence of the relationship between backscattering and extinction coefficients in turbid atmospheres with application to lidar signal inversion,” Indian J. Radio Space Phys. 22, 165–179 (1993).

K. Parameswaran, K. O. Rose, B. V. Krishna Murthy, “Relationship between backscattering and extinction coefficients of aerosols with application to turbid atmosphere,” Appl. Opt. 30, 3059–3071 (1991).
[CrossRef] [PubMed]

Post, M. J.

M. J. Post, C. J. Grund, A. O. Langford, M. H. Proffitt, “Observations of Pinatubo ejecta over Boulder, Colorado by lidars of three different wavelengths,” Geophys. Res. Lett. 19, 195–198 (1992).
[CrossRef]

Proffitt, M. H.

M. J. Post, C. J. Grund, A. O. Langford, M. H. Proffitt, “Observations of Pinatubo ejecta over Boulder, Colorado by lidars of three different wavelengths,” Geophys. Res. Lett. 19, 195–198 (1992).
[CrossRef]

Quenzel, H.

Quing, P.

Rairoux, P.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Reagan, J. A.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of atmospheric extinction using direct solar radiation measurements made with multiple wavelength radiometer,” J. Appl. Meteorol. 12, 374–380 (1973).
[CrossRef]

Rose, K. O.

K. Parameswaran, K. O. Rose, “Spectral dependence of the relationship between backscattering and extinction coefficients in turbid atmospheres with application to lidar signal inversion,” Indian J. Radio Space Phys. 22, 165–179 (1993).

K. Parameswaran, K. O. Rose, B. V. Krishna Murthy, “Relationship between backscattering and extinction coefficients of aerosols with application to turbid atmosphere,” Appl. Opt. 30, 3059–3071 (1991).
[CrossRef] [PubMed]

Sacco, V. M.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Sasano, Y.

Schoeberl, M. R.

S. Chandra, E. L. Flemming, M. R. Schoeberl, J. J. Barnett, “Monthly mean global climatology of temperature, wind, geopotential height and pressure for 0–120 km,” Adv. Space Res. 10, (6), 3–12 (1990).

Shaw, G. E.

G. E. Shaw, J. A. Reagan, B. M. Herman, “Investigations of atmospheric extinction using direct solar radiation measurements made with multiple wavelength radiometer,” J. Appl. Meteorol. 12, 374–380 (1973).
[CrossRef]

Stefanutti, L.

M. D. Guasta, M. Morandi, L. Stefanutti, B. Stein, J. P. Wolf, “Deviation of Mount Pinatubo stratospheric aerosol mean size distribution by means of a multiwavelength lidar,” Appl. Opt. 33, 5690–5697 (1994).
[CrossRef] [PubMed]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Stein, B.

M. D. Guasta, M. Morandi, L. Stefanutti, B. Stein, J. P. Wolf, “Deviation of Mount Pinatubo stratospheric aerosol mean size distribution by means of a multiwavelength lidar,” Appl. Opt. 33, 5690–5697 (1994).
[CrossRef] [PubMed]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Swissler, T. J.

P. H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. F. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Takamura, T.

T. Takamura, Y. Sasano, T. Hayasaka, “Tropospheric aerosol optical properties derived from lidar, sun photometer, and optical particle counter measurements,” Appl. Opt. 33, 7132–7140 (1994).
[CrossRef] [PubMed]

T. Takamura, Y. Sasano, “Ratio of aerosol backscatter to extinction coefficients as determined from angular scattering measurements for use in atmospheric lidar applications,” Opt. Quantum Electron. 19, 293–302 (1987).
[CrossRef]

Tanaka, M.

Thomalla, E.

Twomey, S.

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of the linear system produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Venturi, V.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Wang, P. H.

P. H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. F. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

G. S. Kent, U. O. Farrukh, P. H. Wang, A. Deepak, “SAGE I and SAM II measurements of 1 μm aerosol extinction in the free troposphere,” J. Appl. Meteorol. 27, 269–279 (1988).
[CrossRef]

Weidauer, D.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, M. Del Guasta, D. Weidauer, J. P. Wolf, L. Woste, F. Castagnoli, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonnette, eds., Proc. SPIE1714, 209–219 (1992).
[CrossRef]

Wolf, J. P.

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Variation of the retrieved altitude profiles of aerosol extinction at 694.3 nm obtained by inversion of the simulated lidar data by use of Fernald’s method, assuming S 1 = 10, 20, 30, 50, and 60. The actual aerosol extinction profile used to simulate the lidar data also is shown (curve A). The altitude profile of the scattering ratio (R) at this wavelength is shown in the right panel.

Fig. 2
Fig. 2

Variation of the retrieved size index of power-law type ASD with the iteration number for the altitudes of 1, 4, and 22 km.

Fig. 3
Fig. 3

Altitude profiles of aerosol extinction retrieved from synthesized lidar data at the end of the first (dotted curve) and fourth (dashed curve) iterations of method 1. The actual profiles used to simulate the lidar signals are indicated by the solid curves.

Fig. 4
Fig. 4

ASD’s at 1, 4, and 22 km retrieved from synthesized lidar data at the end of the first (curve 1), second (curve 2), third (curve 3), and fourth (curve 4) iterations of method 2, along with the actual ASD’s (curve A).

Fig. 5
Fig. 5

Altitude profiles of aerosol extinction retrieved at the end of the first (dotted curve) and fourth (dashed curve) iterations of method 2. The actual profiles used to simulate the lidar signals are indicated by the solid curves.

Fig. 6
Fig. 6

Variation of the percentage deviation of size index at the altitude of 1.0 km retrieved by use of method 1 with the SNR of the lidar signals.

Fig. 7
Fig. 7

ASD at the altitude of 1.0 km retrieved from synthetic lidar signals by use of method 2 for SNR’s of 100, 40, 30, and 20. The actual ASD is indicated by the thin solid curve.

Tables (2)

Tables Icon

Table 1 Parameters Describing ASD’s for Different Altitude Regionsa

Tables Icon

Table 2 Values of the Size Index of Power-Law Type ASD at the Altitude of 1.0 kma

Equations (14)

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

P R z = K   β a z + β m z z 2 × exp - 2   0 z α a z + α m z d z ,
α a I - 1 = - S 1 I - 1 S 2   α m I - 1 + X I - 1 exp A I - 1 X I α a I + S 1 I S 2   α m I + X I + X I - 1 exp A I - 1 δ z ,
S 1 I = α a I β a I S 2 = α m β m = 8 π 3 X I = Z I 2 P R Z I A I - 1 = S 1 - S 2 β m I - 1 + β m I δ z .
n r = Cr - ν ,
α a λ = C λ - k ,
ν = k + 3 = ln α a λ 1 / α a λ 2 ln λ 2 / λ 1 + 3 .
α a λ = 0   π r 2 Q ext r ,   λ ,   m n r d r ,
α a λ = j = 1 M   f r ¯ j r j r j + 1   π r 2 Q ext h r d r ,
g = Af + ,
g i = α a λ i ,
A ij = r j r j + 1   π r 2 Q ext r ,   λ i ,   m i h r d r ,
f j = f r j ¯ ,
f = A T C - 1 A + γ H - 1 A T C - 1 g ,
H =     1 - 2       1     0     0     0         0   - 2     5 - 4       1     0     0         0       1 - 4     6 - 4     1     0         0       0     1 - 4     6 - 4       1         0                   0     0       1 - 4     6 - 4 1         0     0     0     1 - 4       5 - 2               0     0     0     0       1 - 2 1 .

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