We present the theory of inversion with two-dimensional regularization. We use this novel method to retrieve profiles of microphysical properties of atmospheric particles from profiles of optical properties acquired with multiwavelength Raman lidar. This technique is the first attempt to the best of our knowledge, toward an operational inversion algorithm, which is strongly needed in view of multiwavelength Raman lidar networks. The new algorithm has several advantages over the inversion with so-called classical one-dimensional regularization. Extensive data postprocessing procedures, which are needed to obtain a sensible physical solution space with the classical approach, are reduced. Data analysis, which strongly depends on the experience of the operator, is put on a more objective basis. Thus, we strongly increase unsupervised data analysis. First results from simulation studies show that the new methodology in many cases outperforms our old methodology regarding accuracy of retrieved particle effective radius, and number, surface-area, and volume concentration. The real and the imaginary parts of the complex refractive index can be estimated with at least as equal accuracy as with our old method of inversion with one-dimensional regularization. However, our results on retrieval accuracy still have to be verified in a much larger simulation study.

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Comparison of Microphysical Particle Parameters Derived With Classical One-Dimensional Regularization (Subsection 2A), Reformulated One-Dimensional Regularization (Subsection 2D), and Two-Dimensional Regularization (According to Subsection 2D) for the Case of Two Particle Layers (Fig. 2)^{
a
}

Parameter

Theory

Classical One-Dimensional Regularization

Reformulated One-Dimensional Regularization

Two-Dimensional Regularization

$+/-$^{
b
}

$0\text{\u2013}1300\text{\hspace{0.17em}}\mathrm{m}$ height range

$1700\text{\u2013}2900\text{\hspace{0.17em}}\mathrm{m}$ height range

${r}_{\text{eff}}$ ($\mathrm{\mu m}$)

0.22

$0.26\pm 0.06$

16%

$0.26\pm 0.06$

16%

$0.23\pm 0.04$

6%

+

${n}_{\mathrm{t}}$ (${\mathrm{cm}}^{-3}$)^{
d
}

0.5

$0.74\pm 0.39$

48%

$0.74\pm 0.38$

47%

$0.67\pm 0.29$

34%

+

${v}_{\text{t}}$ (${\mathrm{\mu m}}^{3}{\mathrm{cm}}^{-3}$)^{
e
}

0.02

$0.029\pm 0.008$

42%

$0.029\pm 0.008$

43%

$0.025\pm 0.006$

22%

+

${a}_{\text{t}}$ (${\mathrm{\mu m}}^{2}{\mathrm{cm}}^{-3}$)^{
f
}

0.27

$0.34\pm 0.06$

25%

$0.34\pm 0.06$

25%

$0.32\pm 0.05$

17%

+

${m}_{\text{real}}$^{
g
}

1.4

$1.35\pm 0.06$

3%

$1.35\pm 0.06$

4%

$1.36\pm 0.06$

3%

${m}_{\text{imag}}$^{
h
}

0.02

$0.017\pm 0.013$

15%

$0.017\pm 0.013$

17%

$0.017\pm 0.014$

13%

Inversions were done for the undistorted optical data, and ten times for optical data distorted with statistical noise of 10%. The results of all 11 runs then were averaged. The numbers not only present the mean values of all 11 inversion runs but also the mean inversion results obtained from all height bins within the indicated height ranges. In all three cases of data inversion second-order smoothing was applied to the investigated particle size distributions. In the case of two-dimensional regularization we additionally carried out second-order smoothing along the vertical dimension, i.e., between successive height bins. For each method we show mean value and standard deviation (first column in each case) and the deviation of the mean value from the theoretical value (second column).
The “+” denotes that the retrieval result is better with the inversion with two-dimensional regularization. The “−” denotes that it is better with classical one-dimensional regularization. ${r}_{\mathrm{eff}}$ denotes effective radius. ${n}_{\mathrm{t}}$ describes number concentration. ${v}_{\mathrm{t}}$ describes volume concentration. ${a}_{\mathrm{t}}$ describes surface-area concentration.
Real part of the complex refractive index.
Imaginary part of the complex refractive index.

Table 2

Example of Values of the Two Regularization Parameters γ and κ that We Obtain With Our Inversion with Classical One-Dimensional Regularization and the Method of Inversion with Two-Dimensional Regularization^{
a
}

The numbers were obtained from the simulation example shown in Fig. 2. We only present the results for the error-free optical data that were used in the data inversion. In the case of inversion with classical one-dimensional regularization we obtain different values in the different height bins. For illustration we picked only height bins in the center of the particle layers, i.e., we present two values of γ for the two-layered profile. In the method of inversion with two-dimensional regularization we only obtain one mean value of γ and κ for the whole profile. The regularization parameters describe the situation in which we use the same optical data, the same inversion window, and the same complex refractive index.

Table 3

Comparison of Microphysical Particle Parameters Derived With the Three Methods^{
a
}

Parameter

Theory

Classical One-Dimensional Inversion

Reformulated One-Dimensional Inversion

Two-Dimensional Inversion

$+/-$

$0\text{\u2013}1100\text{\hspace{0.17em}}\mathrm{m}$ height range

Shown are the results for the height ranges of the three particle layers. The same smoothing constraints as in the two-layered case have been applied. Inversions were done in the same way as in the two-layered case. Meaning of symbols is the same as in Table 1.

Table 4

Values of the Two Regulation Parameters γ and κ^{
a
}

The numbers were obtained from the simulation example shown in Fig. 5.

Tables (4)

Table 1

Comparison of Microphysical Particle Parameters Derived With Classical One-Dimensional Regularization (Subsection 2A), Reformulated One-Dimensional Regularization (Subsection 2D), and Two-Dimensional Regularization (According to Subsection 2D) for the Case of Two Particle Layers (Fig. 2)^{
a
}

Parameter

Theory

Classical One-Dimensional Regularization

Reformulated One-Dimensional Regularization

Two-Dimensional Regularization

$+/-$^{
b
}

$0\text{\u2013}1300\text{\hspace{0.17em}}\mathrm{m}$ height range

$1700\text{\u2013}2900\text{\hspace{0.17em}}\mathrm{m}$ height range

${r}_{\text{eff}}$ ($\mathrm{\mu m}$)

0.22

$0.26\pm 0.06$

16%

$0.26\pm 0.06$

16%

$0.23\pm 0.04$

6%

+

${n}_{\mathrm{t}}$ (${\mathrm{cm}}^{-3}$)^{
d
}

0.5

$0.74\pm 0.39$

48%

$0.74\pm 0.38$

47%

$0.67\pm 0.29$

34%

+

${v}_{\text{t}}$ (${\mathrm{\mu m}}^{3}{\mathrm{cm}}^{-3}$)^{
e
}

0.02

$0.029\pm 0.008$

42%

$0.029\pm 0.008$

43%

$0.025\pm 0.006$

22%

+

${a}_{\text{t}}$ (${\mathrm{\mu m}}^{2}{\mathrm{cm}}^{-3}$)^{
f
}

0.27

$0.34\pm 0.06$

25%

$0.34\pm 0.06$

25%

$0.32\pm 0.05$

17%

+

${m}_{\text{real}}$^{
g
}

1.4

$1.35\pm 0.06$

3%

$1.35\pm 0.06$

4%

$1.36\pm 0.06$

3%

${m}_{\text{imag}}$^{
h
}

0.02

$0.017\pm 0.013$

15%

$0.017\pm 0.013$

17%

$0.017\pm 0.014$

13%

Inversions were done for the undistorted optical data, and ten times for optical data distorted with statistical noise of 10%. The results of all 11 runs then were averaged. The numbers not only present the mean values of all 11 inversion runs but also the mean inversion results obtained from all height bins within the indicated height ranges. In all three cases of data inversion second-order smoothing was applied to the investigated particle size distributions. In the case of two-dimensional regularization we additionally carried out second-order smoothing along the vertical dimension, i.e., between successive height bins. For each method we show mean value and standard deviation (first column in each case) and the deviation of the mean value from the theoretical value (second column).
The “+” denotes that the retrieval result is better with the inversion with two-dimensional regularization. The “−” denotes that it is better with classical one-dimensional regularization. ${r}_{\mathrm{eff}}$ denotes effective radius. ${n}_{\mathrm{t}}$ describes number concentration. ${v}_{\mathrm{t}}$ describes volume concentration. ${a}_{\mathrm{t}}$ describes surface-area concentration.
Real part of the complex refractive index.
Imaginary part of the complex refractive index.

Table 2

Example of Values of the Two Regularization Parameters γ and κ that We Obtain With Our Inversion with Classical One-Dimensional Regularization and the Method of Inversion with Two-Dimensional Regularization^{
a
}

The numbers were obtained from the simulation example shown in Fig. 2. We only present the results for the error-free optical data that were used in the data inversion. In the case of inversion with classical one-dimensional regularization we obtain different values in the different height bins. For illustration we picked only height bins in the center of the particle layers, i.e., we present two values of γ for the two-layered profile. In the method of inversion with two-dimensional regularization we only obtain one mean value of γ and κ for the whole profile. The regularization parameters describe the situation in which we use the same optical data, the same inversion window, and the same complex refractive index.

Table 3

Comparison of Microphysical Particle Parameters Derived With the Three Methods^{
a
}

Parameter

Theory

Classical One-Dimensional Inversion

Reformulated One-Dimensional Inversion

Two-Dimensional Inversion

$+/-$

$0\text{\u2013}1100\text{\hspace{0.17em}}\mathrm{m}$ height range

Shown are the results for the height ranges of the three particle layers. The same smoothing constraints as in the two-layered case have been applied. Inversions were done in the same way as in the two-layered case. Meaning of symbols is the same as in Table 1.

Table 4

Values of the Two Regulation Parameters γ and κ^{
a
}