A curve-fitting procedure is shown to reduce the effects of random errors in estimations of the albedo of single scattering and the moments of the angular-scattering function from the backscattered radiance after a collimated pulsed illumination of a slab target. Numerical results are presented for the effects of a simulated broadening of the incident pulse in time and of systematic experimental errors on the estimated coefficients.

Leonid Fukshansky, Nina Fukshansky-Kazarinova, and Alexander Martinez v. Remisowsky Appl. Opt. 30(22) 3145-3153 (1991)

References

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Percent δf/f versus Input Values of a and f Obtained Numerically from the Formally Exact -Convolution of Eq. (16)^{a}

% δf/f for the Following Value of a

1

5

10

f Value

LL

EXP

LL

EXP

LL

EXP

1.0

−1.2(−1)

−1.2(−1)

−1.2

−1.2

−1.8

−1.8

0.9

−1.8(−1)

−1.9(−1)

−2.2

−2.3

−4.0

−4.2

0.8

−2.5(−1)

−2.8(−1)

−3.4

−3.7

−6.6

−7.2

0.5

−6.5(−1)

−8.4(−1)

−9.3

−1.1 (+1)

−1.6(+1)

−2.0 (+1)

0.1

−4.8

−7.0

−6.2(+1)

−8.2 (+1)

−9.6(+1)

−1.2 (+2)

0.05

−1.0(+1)

−1.5(+1)

−1.3(+2)

−1.7 (+2)

−1.9(+2)

−2.5 (+2)

Results for both LL and EXP least-squares curve fittings are shown. Numbers in parentheses indicate the power of 10 by which the value should be multiplied; e.g., −1.2 (−1) = −1.2 × 10^{−1}.

Table 2

Percent δf/f versus Input Values of a and f Obtained from the Approximate Series Expansion of Eq. (19)^{a}

% δf/f for the Following Value of a

1

5

10

f Value

LL

EXP

LL

EXP

LL

EXP

1.0

−2.4 (−1)

−2.4 (−1)

−9.3 (−1)

−9.7 (−1)

−1.1

−1.1

0.9

−3.7 (−1)

−4.0 (−1)

−2.1

−2.2

−3.2

−3.3

0.8

−5.4 (−1)

−6.1 (−1)

−3.6

−3.8

−5.6

−5.8

0.5

−1.4

−1.9

−1.1 (+1)

−1.3 (+1)

−1.6 (+1)

−1.7 (+1)

0.1

−1.1 (+1)

−1.6 (+1)

−8.9 (+1)

−1.1 (+2)

−9.8 (+1)

−1.1 (+2)

0.05

−2.3 (+1)

−3.4 (+1)

−1.9 (+2)

−2.2 (+2)

−2.0 (+2)

−2.2 (+2)

Results for both LL and EXP least-squares curve fittings are shown. The notation is the same as that used in Table 1.

Table 3

Percent Uf/f versus Input Values c and f Obtained from the Approximate Mode Mixing of Eq. (25)^{a}

Results for both LL and EXP least-squares curve fittings are shown. The notation is the same as that used in Table 1.
When f = 1, there is no mixing of azimuthal modes, and the error is zero.

Table 4

Nodes and Weights for a Four-Point Azimuthal Angle Quadrature

w_{i}^{m} Value for the Following Value of m

i

ϕ_{i}

0

1

2

3

1

0

0

0.5

0

0.5

2

45

0.5

−0.146447

0.5

−0.853553

3

90

0

0.5

−1

0.5

4

135

0.5

−0.853553

0.5

−0.146447

Table 5

Percent δf/f versus Input Values of
$\tilde{c}$ and
$\tilde{\gamma}$ Obtained from the Approximate Nonasymptotic Decaying Pulse of Eq. (29)^{a}

Percent δf/f versus Input Values of a and f Obtained Numerically from the Formally Exact -Convolution of Eq. (16)^{a}

% δf/f for the Following Value of a

1

5

10

f Value

LL

EXP

LL

EXP

LL

EXP

1.0

−1.2(−1)

−1.2(−1)

−1.2

−1.2

−1.8

−1.8

0.9

−1.8(−1)

−1.9(−1)

−2.2

−2.3

−4.0

−4.2

0.8

−2.5(−1)

−2.8(−1)

−3.4

−3.7

−6.6

−7.2

0.5

−6.5(−1)

−8.4(−1)

−9.3

−1.1 (+1)

−1.6(+1)

−2.0 (+1)

0.1

−4.8

−7.0

−6.2(+1)

−8.2 (+1)

−9.6(+1)

−1.2 (+2)

0.05

−1.0(+1)

−1.5(+1)

−1.3(+2)

−1.7 (+2)

−1.9(+2)

−2.5 (+2)

Results for both LL and EXP least-squares curve fittings are shown. Numbers in parentheses indicate the power of 10 by which the value should be multiplied; e.g., −1.2 (−1) = −1.2 × 10^{−1}.

Table 2

Percent δf/f versus Input Values of a and f Obtained from the Approximate Series Expansion of Eq. (19)^{a}

% δf/f for the Following Value of a

1

5

10

f Value

LL

EXP

LL

EXP

LL

EXP

1.0

−2.4 (−1)

−2.4 (−1)

−9.3 (−1)

−9.7 (−1)

−1.1

−1.1

0.9

−3.7 (−1)

−4.0 (−1)

−2.1

−2.2

−3.2

−3.3

0.8

−5.4 (−1)

−6.1 (−1)

−3.6

−3.8

−5.6

−5.8

0.5

−1.4

−1.9

−1.1 (+1)

−1.3 (+1)

−1.6 (+1)

−1.7 (+1)

0.1

−1.1 (+1)

−1.6 (+1)

−8.9 (+1)

−1.1 (+2)

−9.8 (+1)

−1.1 (+2)

0.05

−2.3 (+1)

−3.4 (+1)

−1.9 (+2)

−2.2 (+2)

−2.0 (+2)

−2.2 (+2)

Results for both LL and EXP least-squares curve fittings are shown. The notation is the same as that used in Table 1.

Table 3

Percent Uf/f versus Input Values c and f Obtained from the Approximate Mode Mixing of Eq. (25)^{a}

Results for both LL and EXP least-squares curve fittings are shown. The notation is the same as that used in Table 1.
When f = 1, there is no mixing of azimuthal modes, and the error is zero.

Table 4

Nodes and Weights for a Four-Point Azimuthal Angle Quadrature

w_{i}^{m} Value for the Following Value of m

i

ϕ_{i}

0

1

2

3

1

0

0

0.5

0

0.5

2

45

0.5

−0.146447

0.5

−0.853553

3

90

0

0.5

−1

0.5

4

135

0.5

−0.853553

0.5

−0.146447

Table 5

Percent δf/f versus Input Values of
$\tilde{c}$ and
$\tilde{\gamma}$ Obtained from the Approximate Nonasymptotic Decaying Pulse of Eq. (29)^{a}