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.
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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
wim Value for the Following Value of m
i
ϕi
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
and
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
wim Value for the Following Value of m
i
ϕi
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
and
Obtained from the Approximate Nonasymptotic Decaying Pulse of Eq. (29)a
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