Fiber Lidar Sensing of the Vertical Profiles of Low-Level Cloud Extinction Coefficients at 1064 nm

Highlights

What are the main findings?

• This paper presents the results of a methodological case study of thin low-level clouds in the atmosphere using a 1064 nm fiber lidar.

What are the implications of the main findings?

• The measures undertaken to modernize the technology and method of sensing using the fiber lidar make it possible to retrieve the vertical profiles of stratus cloud extinction coefficients.
• This increases the volume of statistical data for different seasons of the annual cycle, which is a promising direction in remote sensing of the vertical profiles of the extinction in low-level clouds.

Abstract

Results of a methodological case study of low-level clouds in the atmosphere using a 1064 nm fiber lidar are presented. The lidar experiment was carried out in Daejeon, Republic of Korea, in January–March 2025. The study’s primary objective was to ascertain the vertical extinction coefficient profiles pertaining to tenuous, low-altitude cloud formations via implementation of a refined Sequential Lidar Signal Processing Algorithm (SLSPA). The SLSPA incorporates statistical estimation theory to assess signal and measurement error. Cloud extinction coefficient profiles are estimated within the SLSPA utilizing the modified Klett–Fernald inversion algorithm. The SLSPA adaptation is required (a) to evaluate the accuracy of Q-switch laser-based lidar sounding signal deconvolution, (b) to mitigate the impact of the lidar form factor on measurement results, (c) to account for aerosol extinction coefficient variability within the cloud in the modified inversion algorithm (MIA), and (d) to evaluate multiple scattering effect correction in the MIA. Theoretical and experimental aspects of the modified SLSPA are considered sequentially in the present work. The experimental results presented here are based on datasets sampled from the entire array of experimental data obtained during the measurement period.

1. Introduction

The investigation of low-level clouds is crucial for climate studies due to their significant effect on planetary cooling caused by reflecting solar radiation. However, their fragile nature presents challenges for space-based detection. These low-level clouds exhibit minimal optical thickness of typically 0.2 or less. Space-based lidar remote sensing of such cloud types is problematic [1], rendering ground-based lidar systems a more suitable alternative. Studies employing remote sensing techniques in the infrared (IR) spectrum to investigate aerosols and clouds are of significant interest. Within the IR spectrum, the molecular scattering coefficient exhibits a more pronounced decrease compared to particle scattering in the visible spectrum [2,3,4]. Furthermore, the background radiation from scattered solar radiation in the atmosphere is considerably reduced. This facilitates atmospheric lidar sensing during both nocturnal and diurnal periods. Lidar sensing of cloud formations within the lower atmospheric layer requires unique considerations [5,6,7,8,9,10,11,12,13,14,15]. Typically, cloud cover is characterized by rapid variability; therefore, minimizing data acquisition time in lidar measurements is advantageous.

In [16], it was shown that fiber lidar based on a Q-switch laser is suitable for these purposes. The authors used lidar to sense laser radiation scattering by low-level clouds in Daejeon, Republic of Korea, from 20 January 2025 to 1 March 2025. Sounding was carried out in the photon counting regime with a data accumulation period of 60 s. The laser of the fiber lidar radiated pulses with a frequency of 30 kHz at 1.064 μm. The maximum sensing altitude reached 1164 m with a spatial resolution of 1.941 m. Within the SLSPA framework, signals and measurement errors were estimated based on the known measurement statistics. Subsequent signal transformations were executed based on statistical estimation theory. Each transformation yielded refined signal and error profiles. For example, an algorithm was applied to mitigate the influence of noise magnitude, derived from measurement statistics, on the signal. This is accomplished through the application of the Optimal Method of Linear Regression (OMLR) [17]. The OMLR facilitates the estimation of both the denoised signal and the noise reduction magnitude. Non-statistical methods, such as Tikhonov regularization (TR), employ signal smoothing techniques. The accuracy of these transformations can only be assessed indirectly. A review of statistical and non-statistical denoising algorithms can be found in [18]. Section 2.4 delineates the sequence of signal transformations within SLSPA.

The ultimate objective of SLSPA is to derive the cloud extinction coefficient profile and corresponding error profile. For this purpose, the following problems should be solved.

1.1. Estimation of Deconvolution Errors

At the stage of signal deconvolution, the problem of estimating signal errors due to the deconvolution transformation arises. Q-switch laser radiation is described by the impulse response (IR) that lasts several microseconds, whereas research requires time resolutions from several units to several tens of nanoseconds. Therefore, deconvolution must be used at the sensing results pre-processing stage. This linear transformation is considered in more detail in Section 2.1.1. The method of moments, the least squares method, and the maximum likelihood method provide the basis for the statistical estimation theory [19]. In the present work, we solve the problem of estimating deconvolution errors using the weighted least squares (WLS) method. This procedure is considered in more detail in Section 2.1.2.

1.2. Lidar Geometric Form Factor Correction

The subsequent signal processing procedure mitigates the influence of the lidar’s Geometrical Form Factor (GLFF) on the received signal [20,21]. Specifically, in the fiber-based lidar system, the optical axes of the receiver and transmitter are spatially separated by 70 mm. Computational analyses indicate that the GLFF negligibly affects the signal at distances exceeding 1500 m. This is relevant since the conventional Klett–Fernald inversion algorithm demonstrates applicability exclusively within the detection range where these effects can be deemed insignificant [22,23]. Typically, GLFF profile assessments involve direct measurements along a calibration path, modeled GLFF profiles, and calibration against established atmospheric aerosol and molecular backscatter coefficient profiles. However, the identification of a suitable horizontal calibration path presents considerable challenges in an urban environment. We propose an alternative correction algorithm wherein the GLFF is not explicitly computed. In the context of vertical sounding, the ratio of the signal within a cloud layer to a reference signal is derived. The reference signal is predicated on an a priori known atmospheric aerosol and molecular backscatter coefficient profile ${\tilde{\beta }}_A\left(r\right)$. This profile can be obtained from external sources. This may be a modeled profile, or it may be an actual profile. Critically, this signal ratio is devoid of instrumental parameters and GLFF influence. A detailed description of this transformation is presented in Section 2.2. Subsequently, the signal ratio undergoes inversion utilizing the MIA. The inversion result is refined using the known aerosol backscatter coefficient profile. Through this methodology, a vertical profile of both cloud and aerosol extinction coefficients can be derived. It should be noted that the reference signal technique was first demonstrated in [7]. Here, the reference signal is used to determine cloud boundaries in the troposphere, where the contribution of aerosol to scattering can be neglected.

1.3. Aerosol Variability Correction in Inversion Algorithm

Lidar-derived atmospheric aerosol measurements reveal significant temporal variations in aerosol optical scattering properties. Furthermore, aerosol interaction within high-humidity cloud environments can induce alterations in aerosol characteristics, potentially through hygroscopic growth. This effect necessitates consideration within the MIA, incorporating supplementary data regarding aerosol behavior within cloud formations. The lidar signal encapsulates extinction coefficient profile information for the molecular atmosphere ${\alpha }_{MOL}\left(r\right)$, aerosol component ${\alpha }_{AER}\left(r\right)$, and cloud constituent ${\alpha }_C\left(r\right)$. Commonly, to reduce parameter redundancy within the lidar equation, ancillary data are incorporated, such as radiometric measurements, cloud scattering models, and/or the approximation ${\alpha }_C\left(r\right)\gg {\alpha }_{AER}\left(r\right)\gg {\alpha }_{MOL}\left(r\right)$. We introduce the Aerosol Variability Function (AVF), denoted as $K\left(r\right)=1+\Delta {\beta }_{AER}\left(r\right)/{\tilde{\beta }}_A\left(r\right)$, where $\Delta {\beta }_{AER}={\beta }_{AER}-{\tilde{\beta }}_{AER}$. Given the condition $\Delta {\beta }_{AER}\left(r\right)\ll {\tilde{\beta }}_A\left(r\right)$, $K\left(r\right)\approx 1$. Practically, given the stationary variability in $\Delta {\beta }_{AER}\left(r\right)$ and/or temporal proximity between reference and cloud signals, then $K\left(r\right)\approx 1$. The influence of the AVF on ${\alpha }_C\left(r\right)$ is examined via experimental results in the Section 5.

1.4. Multiple Scattering Correction in MIA

Multiple scattering (MS) correction is incorporated into the MIA. The classic Klett inversion is based on a solution of the lidar equation, which is a single-scatter approximation that ignores MS. We adhere to the generally accepted methodology for accounting for multiple scattering effects in lidar sensing: the lidar equation in the single scattering approximation is corrected using the multiple scattering function. In lidar applications, the multiple scattering effect is typically accounted for by employing a multiple scattering function, denoted as $\eta \left(r\right)$ [24] or $F\left(r\right)$ [25], where $F\left(r\right)=1-\eta \left(r\right)$. These functions are derived either from theoretical analyses [26,27,28,29] or empirically from lidar measurements of MS [29,30]. Multiple scattering effects can be very significant for space-based lidar systems [24]. The multiple-scattering contribution to the lidar signal is lower for ground-based lidar systems. Estimates of MS contribution to the lidar signal depending on the distance to the cloud and the receiver field of view (RFOV) of the ground-based lidar are discussed in [27]. This effect has been empirically demonstrated in a lidar-based MS measurement experiment at 1.064 μm [30]. Section 2.2 details the implementation of MS correction within the Klett inversion algorithm.

The present work is organized as follows: Methods, Materials, Discussion, and Conclusions. The content of the Methods section has been described above. Section 3 (Materials) describes the lidar (Section 3.1, System) and the procedure for conducting the experiment (Section 3.2, Experiment). Section 4 (Results) presents the main results of the investigations, including the vertical profiles of the cloud extinction coefficient.

2. Methods

2.1. Modified Lidar Deconvolution Solution

2.1.1. Executive Summary

The previously proposed lidar deconvolution method [16] is based on a discrete laser IR model representation by a packet of $M$ successive pulses $h=\left(h_0,h_1,\dots ,h_M\right)$ transmitted to the atmosphere. The backscattered lidar signal corresponds to each short-transmitted pulse $h_n$ at the receiving point. These signals are identical for all short pulses to within the multiplier set by the calibration value. The lidar photodetector registers $N$ counts $f=\left(f_0,f_1,\dots ,f_N\right)$ of the total signal, which, according to the additivity principle, is the convolution

$$f_i={\displaystyle \sum _{k=0}^i{h}_{i-k}{p}_k}, \left(\begin{array}{l}i=0,1,\dots ,N\\ h_k=0 \mathrm{if} k>M\\ M\le N\end{array}\right),$$

(1)

where $p=\left(p_0,p_1,\dots ,p_N\right)$ is the deconvoluted signal, and the laser IR $h$ is normalized based on the condition

$${\displaystyle \sum _{k=0}^M{h}_k}=1.$$

(2)

Figure 1 shows the waveform of the Q-switch laser pulse at 0.3 mJ.

The convolution in Equation (1) is equivalent to a system of linear equations. This system of equations in matrix form has the following form:

$$Ap=f.$$

(3)

Here, the $\left(N+1\right)\times \left(N+1\right)$ matrix $A=\left(a_{ij}\right)$ is positively determined and is composed of the elements of laser IR $h$ by the following rule:

$$a_{ij}=h_{i-j}, \left(\begin{array}{l}i,j=0,1,\dots ,N\\ a_{ij}=0 \mathrm{if} 0>\left(i-j\right)>M\end{array}\right),$$

(4)

where the signals $p=\left(p_n\right)$ and $f=\left(f_n\right)$ are matrix columns.

The transition from Formula (1) for the convolution to representation (3) makes it possible to apply methods of solving systems of linear equations to the deconvolution of lidar return signals. If the matrix $A$ is positive definite, the solution of Equation (3) is

$$p=A^{-1}f$$

(5)

Results of verification and approbation of the lidar deconvolution method are demonstrated in [16].

2.1.2. Deconvolution Algorithm

The lidar deconvolution shown in [16] solves some problems but does not consider the effect of measurement errors on the deconvoluted signal. The method requires further development to remove these limitations. The problem we solve here is formulated as the determination of the deconvoluted signal accuracy from the known accuracy of the experimentally measured lidar signal and the accuracy of determining the laser IR. To solve this problem, we take into consideration the statistical nature of the measurement data and estimate the results of signal deconvolution from the viewpoint of the theory of estimates.

As is well known [31], if the matrix $A$ is positive definite, the solution of Equation (3) can also be obtained by minimizing the quadratic form $Q\left(p\right)$, where the vector $p$ is expressed in the parametric form as a functional. We now demonstrate that representing convolution by a linear model of parameters allows us to evaluate the results of deconvolution from the viewpoint of estimation theory. For this purpose, let us rewrite Formula (1) as a linear functional $F\left(p\right)$ of the parameter $p$:

$$F\left(p\right)\equiv F\left[f,D\left(f\right),h,D\left(h\right);p\right]=\left(f_i-{\displaystyle \sum _{k=0}^i{h}_{i-k}{p}_k}\right), \left(i=0,1,\dots ,N\right),$$

(6)

and consider the functional minimization problem in the context of the theory of estimations. Here $f$, $D\left(f\right)$, $h$, and $D\left(h\right)$ are input data, while $D\left(f\right)=\left[D\left(f_0\right),D\left(f_1\right),\dots ,D\left(f_N\right)\right]$ and $D\left(h\right)=\left[D\left(h_0\right),D\left(h_1\right),\dots ,D\left(h_M\right)\right]$ are the known variances of the signal, $f$, and IR, $h$, of laser radiation. In addition, we designate the total variance of functional $F\left(p\right)$ by $D\left(F\right)=\left[D\left(F_0\right),D\left(F_1\right),\dots ,D\left(F_M\right)\right]$. To find the total variance $D\left(F\right)$ of total linear functional (6), we take advantage of the alternative form for the convolution

$$F\left(p\right)=\left(f_i-{\displaystyle \sum _{k=0}^i{h}_{i-k}{p}_k}\right)=\left(f_i-{\displaystyle \sum _{k=0}^i{p}_{i-k}{h}_k}\right), \left(i=0,1,\dots ,N\right).$$

(7)

Then, for random and independent errors, taking advantage of asymptotic estimate properties, we obtain

$$D\left(F_i\right)=D\left(f_i\right)+{\displaystyle \sum _{k=0}^i{p}_{i-k}^{2}D\left(h_k\right)}, \left(i=0,1,\dots ,N\right).$$

(8)

Based on the WLS method, we obtain the quadratic form

$$Q\left(p\right)={{\displaystyle \sum _{i=0}^N\frac{1}{D\left(F_i\right)}\left(f_i-{\displaystyle \sum _{k=0}^i{h}_{i-k}{p}_k}\right)}}^2,$$

(9)

where $Q\left(p\right)$ is the sum of squared residuals of the linear convolution approach to the measured signal $f$. Then, the condition for a minimum of the quadratic form can be written as the system of linear equations

$${\displaystyle \frac{\partial Q\left(p\right)}{\partial p_j}}=-2{\displaystyle \sum _{i=0}^N\frac{1}{D\left(F_i\right)}\left(f_i-{\displaystyle \sum _{k=0}^i{h}_{i-k}{p}_k}\right)}{h}_j=0, \left(j=0,1,\dots ,N\right).$$

(10)

After simple transformations of the equations in Formula (10) for the parameters $p$, we obtain the system of linear equations in matrix form

$$Bp=y,$$

(11)

in which the elements of the symmetric matrix $B=\left(b_{i,j}\right)$ are determined as follows:

$$b_{ij}={\displaystyle \sum _{k=0}^{N-j}\frac{h_k{h}_{k+j-i}}{D\left(F_{k+j}\right)}}, \left(\begin{array}{c}i,j=0,1,\dots ,N\\ j\ge i\\ h_k=0 \mathrm{if} k>M\end{array}\right).$$

(12)

The matrix column $y=\left(y_n\right)$ is defined as

$$y_i={\displaystyle \sum _{k=0}^{N-i}\frac{h_k{f}_{k+i}}{D\left(F_{k+i}\right)}}, \left(\begin{array}{l}i,j=0,1,\dots ,N\\ h_k=0 \mathrm{if} k>M\end{array}\right).$$

(13)

Note that the matrix $B$ and the matrix column $y$ are matrix expressions:

$$\left\{\begin{array}{l}B=A^T{D}^{-1}A\\ y=A^T{D}^{-1}f\end{array}\right.,$$

(14)

where the lower triangular matrix $A$ is given by expression (4), the symbol $T$ designates the transposition operation, and the $\left(N+1\right)\times \left(N+1\right)$ matrix of the second moments $D$ is diagonal with elements

$$D=\left(D_{ij}\right), \left(\begin{array}{l}i,j=0,1,\dots ,N\\ D_{ij}=D\left(F_i\right) if i=j\\ D_{ij}=0 if i\ne j\end{array}\right).$$

(15)

The matrix $B$ is symmetric and positive definite. Thus, there exists only one solution to the system of Equation (11). Considering Expressions (14), Equation (11) is transformed to the form

$$A^T{D}^{-1}Ap=A^T{D}^{-1}f,$$

(16)

the solution of which is the known expression of the WLS method for estimates of the parameters

$$\stackrel{⌢}{p}={\left(A^T{D}^{-1}A\right)}^{-1}{A}^T{D}^{-1}f.$$

(17)

The error estimates are determined by the diagonal elements of the covariance matrices:

$$\stackrel{⌢}{D}=B^{-1}={\left(A^T{D}^{-1}A\right)}^{-1}.$$

(18)

This expression can be interpreted as a transition of errors of the measured signal $D\left(f\right)$ and laser IR $D\left(h\right)$ to the inverted signal $p$.

Note that in the unweighted least squares method, Equation (16) is reduced to the form

$$A^{T}Ap=A^{T}f.$$

(19)

Multiplying both sides of the equation by the matrix ${\left(A^T\right)}^{-1}$ from the right, we arrive at Equation (3).

Thus, solving the problem of minimization of the quadratic form, we represented convolution (1) as the linear model of the parameters and obtained not only parameter estimates (17) but also estimates of transformation errors of inversion (18). These estimates coincide with those of the WLS method. In practice, experimental statistics are often insufficient. Therefore, we consider a class of unbiased estimates which are linear functions of observation results. According to the Gauss–Markov theorem, the estimates obtained by the least squares method are optimal for linear cases, that is, they have minimal variances and are unbiased [32,33]. Moreover, these properties are peculiar to any arbitrary number of experimental observations. In addition, we showed that the estimates obtained in [16] using Equation (3) are equivalent to those obtained by the unweighted least squares method. Note that the estimate of errors of the functional depends on the parameters $D\left(F\right)\equiv D\left[F\left(p\right)\right]$; therefore, the solution of the linear system of Equation (11) is an iterative procedure in which the initial approximation for the diagonal elements of the covariance matrix $D$ in Formula (15) should be assigned.

2.2. MIA for Signal Ratio

The problem of inversion of the signal ratio is solved based on well-known inversion algorithms [22,23,34,35]. For this purpose, we will now write down and solve the lidar equation for signal ratios in the vertical profile of the extinction coefficient within the cloud. The lidar experiment was carried out in the photon-counting regime. However, to avoid cumbersome formulas, we proceed to discrete representation of signals at the end of transformations. In the lower troposphere, the optical properties of scattered radiation are significantly influenced by the presence of aerosol and molecular scattering. For the molecular atmosphere, the ratio $B_{MOL}={\beta }_{MOL}\left(r\right)/{\alpha }_{MOL}\left(r\right)$ of the backscattering to extinction coefficient is constant, and $B_{MOL}=3/\left(8\pi \right)$ sr⁻¹. Hereinafter, $r$ is the sensing range. If aerosol with the backscattering coefficient ${\beta }_{AER}\left(r\right)$ and extinction coefficient ${\alpha }_{AER}\left(r\right)$ is present in the atmosphere, as a rule [36,37], the lidar ratio is considered to be dependent on the sensing range ${\beta }_{AER}\left(r\right)=B_{AER}\left(r\right){\alpha }_{AER}\left(r\right)$. Here and hereafter, the ratio $B$ (inverse lidar ratio) follows the form adopted in Klett’s work and the work dedicated to the correction of the Klett-Fernald algorithm [35,36]. This will allow interested researchers to navigate the sequence of the inversion algorithm.

In atmospheric conditions where clouds coexist with aerosols, the lidar remote sensing equation, incorporating multiple scattering (MS) correction, is formulated as follows:

$$P\left(r\right)=LWG\left(r\right)r^{-2}\left[{\beta }_A\left(r\right)+{\beta }_C\left(r\right)\right]T^2\left(r\right),$$

(20)

$$T\left(r\right)=\exp \left\{-{\displaystyle \underset{0}{\overset{r}{\int }}\eta \left(r^{\prime }\right)\left[{\alpha }_A\left(r^{\prime }\right)+{\alpha }_C\left(r^{\prime }\right)\right]d{r}^{\prime }}\right\}.$$

(21)

Here, $P\left(r\right)$ is the power of scattered radiation recorded with the receiving lidar system at distance $r$, $L$ is proportional to the laser pulse energy, $W$ is the instrumental parameter of the lidar, $G\left(r\right)$ is the GLFF, ${\beta }_A\left(r\right)={\beta }_{AER}\left(r\right)+{\beta }_{MOL}\left(r\right)$ and ${\alpha }_A\left(r\right)={\alpha }_{AER}\left(r\right)+{\alpha }_{MOL}\left(r\right)$ are the backscattering and extinction coefficients of laser radiation by the aerosol and molecules, ${\beta }_C\left(r\right)$ and ${\alpha }_C\left(r\right)$ are the backscattering and extinction coefficients of laser radiation by cloud layer particles, $T\left(r\right)$ is the transmittance of the sounding path $\left[0,r\right]$, and $\eta \left(r\right)$ is the multiple scattering function.

To eliminate the effects of both the GLFF and instrumental lidar constants, we compose the signal ratio in the following form:

$$R\left(r\right)=P\left(r\right)/P_{REF}\left(r\right),$$

(22)

where $P_{REF}\left(r\right)$ is the reference signal without the cloud:

$$P_{REF}\left(r\right)=LWG\left(r\right)r^{-2}{\tilde{\beta }}_A\left(r\right)T_{REF}^2\left(r\right),$$

(23)

$$T_{REF}\left(r\right)=\exp \left[-{\displaystyle \underset{0}{\overset{r}{\int }}\eta \left(r^{\prime }\right){\tilde{\alpha }}_A\left(r^{\prime }\right)d{r}^{\prime }}\right],$$

(24)

where ${\tilde{\alpha }}_A\left(r\right)$ represents a known profile of aerosol extinction coefficients and molecular scattering properties. This profile is predicated on an a priori known atmospheric aerosol and molecular backscatter coefficient profile ${\tilde{\beta }}_A\left(r\right)$. This profile can be obtained from external sources. This may be a modeled profile, or it may be an actual profile.

Following [36], we write down the ratio $B$ within the cloud in the following form:

$${\beta }_C\left(r\right)=B_C\left(r\right){\alpha }_C\left(r\right).$$

(25)

Then, considering Formulas (23) and (24), the signal ratio takes the form

$$R\left(r\right)=\left[{\displaystyle \frac{{\beta }_C\left(r\right)+{\beta }_A\left(r\right)}{{\tilde{\beta }}_A\left(r\right)}}\right]\exp \left\{-2{\displaystyle \underset{0}{\overset{r}{\int }}\eta \left(r^{\prime }\right)\left[{\alpha }_C\left(r^{\prime }\right)+\Delta {\alpha }_A\left(r^{\prime }\right)\right]d{r}^{\prime }}\right\},$$

(26)

where $\Delta {\alpha }_A={\alpha }_A-{\tilde{\alpha }}_A$. Let FIA take the form $K\left(r\right)=1+\Delta {\beta }_{AER}\left(r\right)/{\tilde{\beta }}_A\left(r\right)$, where $\Delta {\beta }_{AER}={\beta }_{AER}-{\tilde{\beta }}_{AER}$. After the change in variables of the form $\gamma \left(r\right)={\beta }_C\left(r\right)/{\tilde{\beta }}_A\left(r\right)+K\left(r\right)$, we obtain the following formula:

$$R\left(r\right)=\gamma \left(r\right)\exp \left\{-2{\displaystyle \underset{0}{\overset{r}{\int }}\eta \left(r^{\prime }\right){\tilde{\beta }}_A\left(r^{\prime }\right)\left[\frac{\gamma \left(r^{\prime }\right)-1}{B_C\left(r^{\prime }\right)}+M\left(r^{\prime }\right)\right]d{r}^{\prime }}\right\},$$

(27)

where

$$M\left(r\right)=\left(K\left(r\right)-1\right)\left[{\displaystyle \frac{1}{B_{AER}\left(r\right)}}-{\displaystyle \frac{1}{B_C\left(r\right)}}\right].$$

(28)

Taking the logarithm of both terms of the equation and subtracting the logarithms of the functions at $r=r_0$, we obtain

$$\ln \left[{\displaystyle \frac{R\left(r\right)}{C_0}}\right]=\ln \left(\gamma \right)-2{\displaystyle \underset{0}{\overset{r}{\int }}\eta \left(r^{\prime }\right){\tilde{\beta }}_A\left(r^{\prime }\right)\left[\frac{\gamma \left(r^{\prime }\right)-1}{B_C\left(r^{\prime }\right)}+M\left(r^{\prime }\right)\right]d{r}^{\prime }},$$

(29)

where $C_0$ is the calibration constant that is necessary for the integral curve to pass through the point $\left({\gamma }_0,r_0\right)$ and equal to

$$C_0={\displaystyle \frac{R\left(r_0\right)}{{\gamma }_0}},$$

(30)

where ${\gamma }_0\equiv \gamma \left(r_0\right)$. Next, take the derivative with respect to $r$ from Expression (29) and obtain the following formula:

$${\displaystyle \frac{dS}{dr}}={\displaystyle \frac{1}{\gamma \left(r\right)}}{\displaystyle \frac{d\gamma \left(r\right)}{dr}}-2\eta \left(r\right){\tilde{\beta }}_A\left(r\right)\left[{\displaystyle \frac{\gamma \left(r\right)-1}{B_C\left(r\right)}}+M\left(r\right)\right],$$

(31)

where $S=\ln \left[R\left(r\right)/C_0\right]$. Simple transformations of the obtained relationship led to the Bernoulli equation [38] in the form

$${\displaystyle \frac{d\gamma \left(r\right)}{dr}}+q\left(r\right)\gamma \left(r\right)=g\left(r\right)\gamma {\left(r\right)}^2.$$

(32)

Here,

$$q\left(r\right)=2\eta \left(r\right){\tilde{\beta }}_A\left(r\right)\left[{\displaystyle \frac{1}{B_C\left(r\right)}}-M\left(r\right)\right]-{\displaystyle \frac{dS}{dr}},$$

(33)

$$g\left(r\right)=2\eta \left(r\right){\tilde{\beta }}_A\left(r\right){\displaystyle \frac{1}{B_C\left(r\right)}}.$$

(34)

The substitution $\gamma \left(r\right)=1/u$ transforms the obtained equation into the linear differential equation with separable variables. Solving this equation and making the reverse substitution, we obtain the following for the $\gamma \left(r\right)$:

$$\gamma \left(r\right)={\displaystyle \frac{R\left(r\right)\exp \left[Y\left(r\right)\right]}{C_0-2{\displaystyle \underset{r_0}{\overset{r}{\int }}\eta \left(r^{\prime }\right){\tilde{\beta }}_A\left(r^{\prime }\right)\frac{R\left(r^{\prime }\right)}{B_C\left(r^{\prime }\right)}\exp \left[Y\left(r^{\prime }\right)\right]d{r}^{\prime }}}},$$

(35)

in which the expression $Y\left(r\right)$ under the exponent sign has the form

$$Y\left(r\right)=-2{\displaystyle \underset{r_0}{\overset{r}{\int }}\eta \left(r^{\prime }\right){\tilde{\beta }}_A\left(r^{\prime }\right)\left[\frac{1}{B_C\left(r^{\prime }\right)}-M\left(r^{\prime }\right)\right]d{r}^{\prime }}.$$

(36)

The calibration constant $C_0$ defined by Equation (35) is used here and below. Thus, the integral curve given by Formulas (35) and (36) also passes through the point $\left({\gamma }_0,r_0\right)$.

Note that this equation for $\gamma \left(r\right)$ was derived for the sensing range above the calibration altitude $r\ge r_0$. By making the reverse substitution, we obtain the expression for the backscattering coefficient ${\beta }_C\left(r\right)$ instead of $\gamma \left(r\right)$. For the sensing range above the calibration altitude $r\ge r_0$, this equation assumes the following form:

$${\beta }_C\left(r\right)={\tilde{\beta }}_A\left(r\right)\left\{{\displaystyle \frac{R\left(r\right)\exp \left[Y\left(r\right)\right]}{C_0-2{\displaystyle \underset{r_0}{\overset{r}{\int }}\eta \left(r^{\prime }\right){\tilde{\beta }}_A\left(r^{\prime }\right)\frac{R\left(r^{\prime }\right)}{B_C\left(r^{\prime }\right)}\exp \left[Y\left(r^{\prime }\right)\right]d{r}^{\prime }}}}-K\left(r\right)\right\}.$$

(37)

For ${\beta }_C\left(r\right)$ below the calibration altitude $r\le r_0$, following [23] and changing the integration order in Formulas (36) and (37), we obtain the formula

$${\beta }_C\left(r\right)={\tilde{\beta }}_A\left(r\right)\left\{{\displaystyle \frac{R\left(r\right)\exp \left[-Y\left(r\right)\right]}{C_0+2{\displaystyle \underset{r}{\overset{r_0}{\int }}\eta \left(r^{\prime }\right){\tilde{\beta }}_A\left(r^{\prime }\right)\frac{R\left(r^{\prime }\right)}{B_C\left(r^{\prime }\right)}\exp \left[-Y\left(r^{\prime }\right)\right]d{r}^{\prime }}}}-K\left(r\right)\right\}.$$

(38)

Thus, we have derived the equations for the backscattering coefficient ${\beta }_C\left(r\right)$. These equations are similar to those obtained by Fernald in that calibration is carried out on any suitable section of the sensing path [39]. Note that multiplication of both sides of the equations by $1/B_C\left(r\right)$ transforms Equations (37) and (38) into formulas for the extinction coefficient ${\alpha }_C\left(r\right)$.

To present the results in discrete form, we replace the integrals in Equations (37) and (38) with right Riemann sums. For the sensing range above the calibration altitude $r\ge r_0$, this equation assumes the following form:

$${\beta }_C\left(I\right)={\tilde{\beta }}_A\left(I\right)\left\{{\displaystyle \frac{R\left(I\right)\exp \left[Y\left(I\right)\right]}{C_0-2{\displaystyle \sum _{I_0+1}^I\eta \left(I^{\prime }\right){\tilde{\beta }}_A\left(I^{\prime }\right)\frac{R\left(I^{\prime }\right)}{B_C\left(I^{\prime }\right)}\exp \left[Y\left(I^{\prime }\right)\right]}\Delta r}}-K\left(I\right)\right\},$$

(39)

in which the expression $Y\left(I\right)$ under the exponent sign has the form

$$Y\left(I\right)=-2{\displaystyle \sum _{I_0+1}^I\eta \left(I^{\prime }\right){\tilde{\beta }}_A\left(I^{\prime }\right)\left[\frac{1}{B_C\left(I^{\prime }\right)}-M\left(I^{\prime }\right)\right]\Delta r},$$

(40)

where $\Delta r=r\left(I\right)-r\left(I-1\right)$, $I=0,1,\dots ,N$, and $r_0\equiv r\left(I_0\right)$.

For the sensing range below the calibration altitude $r\le r_0$, the equation for the backscattering coefficient takes the following form:

$${\beta }_C\left(I\right)={\tilde{\beta }}_A\left(I\right)\left\{{\displaystyle \frac{R\left(I\right)\exp \left[-Y\left(I\right)\right]}{C_0+2{\displaystyle \sum _I^{I_0+1}\eta \left(I^{\prime }\right){\tilde{\beta }}_A\left(I^{\prime }\right)\frac{R\left(I^{\prime }\right)}{B_C\left(I^{\prime }\right)}\exp \left[-Y\left(I^{\prime }\right)\right]}\Delta r}}-K\left(I\right)\right\},$$

(41)

where $\Delta r=r\left(I-1\right)-r\left(I\right)$.

MIA Verification

Figure 2 shows the verification results. The MIA was verified using a numerical model of lidar signals. The following parameter values were used to obtain the model vertical profiles: $B_{MOL}$ = 0.119 sr⁻¹, $B_{AER}\left(r\right)$ = 0.024 sr⁻¹, $B_C\left(r\right)$ = 0.055 sr⁻¹, $\eta \left(r\right)$ = 1, and $K\left(r\right)=1$. The model vertical profile of signal ratios (a) shown in Figure 2 was calculated with Equation (26) using the model vertical profile of cloud extinction coefficient ${\alpha }_C\left(r\right)$ (b) and the vertical profile of model backscatter coefficient ${\tilde{\beta }}_A\left(r\right)$. The figure shows that the vertical profile of extinction coefficients reconstructed within the cloud (b) by inverting the model profile of signal ratios (a) is a good match for the original model profile of extinction coefficient within the cloud (b).

2.3. MIA Parameters

In the MIA, the vertical profile of the backscattering coefficient ${\tilde{\beta }}_A\left(r\right)$ was calculated from the sum of the model profiles of the molecular and aerosol extinction coefficients ${\alpha }_{MOL}\left(r\right)$ and ${\alpha }_{AER}\left(r\right)$ of laser radiation at a wavelength of 1064 nm [40]. Several studies show high variability in the lidar ratio depending on the aerosol type [41,42,43]. As demonstrated in [44], constant lidar ratios can cause errors in inverted lidar signal profiles. The following values were adopted for ratios $B$: for molecular scattering, $B_{MOL}$ was assumed to be 0.119 sr⁻¹; for aerosol scattering, $B_{AER}\left(r\right)$ was assumed to be constant and equal to 0.024 sr⁻¹; $B_C\left(r\right)$ in the cloud layer, according to the results of investigations presented in [45,46,47,48,49,50], was assumed to be constant and equal to 0.055 sr⁻¹ [50]; and $\eta \left(r\right)$ = 1.

The RFOV of the fiber lidar is very small, at approximately 0.01 mrad. Furthermore, the vertical geometric thickness of cloud layers, as detailed in Section 4, does not exceed 215 m, with an optical thickness of less than 1. As shown in [27], $\eta \left(r\right)$ = 1 is applicable when the RFOV of a ground-based lidar is sufficiently small and/or the extinction coefficient is low. Given an RFOV of 1 mrad and a target layer at a distance of 1 km, the multiple-scattering contribution to the lidar signal remains below 5% for an extinction coefficient ${\alpha }_C\left(r\right)$ less than 1 km⁻¹. Due to the temporal proximity in the acquisition of the reference signal and the signals originating from the cloud during experimentation, $K\left(r\right)\approx 1$.

2.4. Random Errors Estimating

The random errors of measured lidar signals were estimated using the algorithms described in Section 2.1. To calculate the errors (variations) of the estimated signal ratios $R\left(r\right)$ and extinction coefficients ${\alpha }_C\left(r\right)$, the corresponding expressions were expanded into the Taylor series. Neglecting the second and higher order terms in Formula (22), we obtain the error estimate (variance) $\stackrel{⌢}{D}\left[R\left(r\right)\right]$ of the signal ratio of the form

$$\stackrel{⌢}{D}\left[R\left(r\right)\right]\approx R{\left(r\right)}^2\left\{{\displaystyle \frac{\stackrel{⌢}{D}\left[P\left(r\right)\right]}{P{\left(r\right)}^2}}+{\displaystyle \frac{\stackrel{⌢}{D}\left[P_{REF}\left(r\right)\right]}{P_{REF}{\left(r\right)}^2}}\right\}.$$

(42)

In the same way, by simple transformations of Formula (26) for the error estimate (variance) of the extinction coefficient $\stackrel{⌢}{D}\left[{\alpha }_C\left(r\right)\right]$, we obtain

$$\stackrel{⌢}{D}\left[{\alpha }_C\left(r\right)\right]\approx {\alpha }_C{\left(r\right)}^2\left\{{\displaystyle \frac{\stackrel{⌢}{D}\left[P\left(r\right)\right]}{P{\left(r\right)}^2}}+{\displaystyle \frac{\stackrel{⌢}{D}\left[P_{REF}\left(r\right)\right]}{P_{REF}{\left(r\right)}^2}}\right\}.$$

(43)

2.5. SLSPA Processing

Preliminary processing of measurement results consists of the following sequence of operations:

• Correction of linearity: An avalanche photodiode (APD) is used as a fiber lidar photodetector. The APD operation is controlled by an active quenching circuit (AQC) being a part of the single photon counting module (SPCM) (indicated by Sensor in

  Table 1. Fiber lidar parameters.

  Laser	Brand Type Central wavelength, CWL Spectrum width, FWHM Pulse duration *, FWHM Pulse repetition rate Pulse energy Beam quality, M2 Beam divergence Beam diameter	MFP 20W, Maxphotonics (Shenzhen, China) Q-switch fiber laser 1064 nm 5 nm 100 ns 30 kHz 0.3 mJ 1.3 <1 mrad 7 mm
  Receiver	Type Effective diameter Focal length Spatial separation of transmitter and receiver optical axes	Single plano-convex lens 60 mm 300 mm 70 mm
  Scanning base	Type Azimuthal angle range Elevation angle range Angular resolution	E-RMPG60-A-2, MISUMI (Seoul, Republic of Korea) ±0–180° 0–90° 0.01°
  Signal Processing	Model Type Operation mode Strobe period	EBAZ4205 controller Pulse counter Accumulation 50 ns
  Sensor	Brand Type Operating mode Input fiber diameter	SPCM AQRH13FC, Excelitas (Gaithersburg, MD, USA) Si-APD Single Photon Counting 100 µm
  Filter N1	Brand Type Central wavelength, CWL Spectrum width, FWHM	FLH1064-3, Thorlabs Thorlabs (Newton, NJ, USA) Bandpass filter 1064 nm 3 nm
  Filter N2	Brand Type	BLP980R-25, Semrock (New York, NY, USA) Long-wave filter
  Data processing and storage	PC on OS Windows 7

  * Data of the manufacturer. Full pulse duration is given in the text.

  ). The AQC introduces a quantification error. This error is corrected utilizing tabular data derived from benchtop measurements. It should be noted that the uncertainty of the lidar signal measurements is correspondingly adjusted (increased). Benchtop measurements indicate that the after-pulsing effect is negligible, exhibiting a magnitude of no more than 0.3% [51]. The after-pulsing effect for APDs is analogous to the Signal-Induced Noise (SIN) effect observed in photomultiplier tubes [52]. Benchtop testing demonstrated that the SPCM achieves a stable photon counting regime within one hour.
• Denoising lidar signals: Denoising is performed by the OMLR [17]. We consider lidar measurements of weak signals backscattered from the atmosphere as photon counting in accumulation mode. In practice, we use the asymptotic property of estimates, and for the Poisson distribution, we take the variance estimates equal to those of averages. As indicated in [33], it is sufficient that the number of counts in a strobe of data exceeds five. Considering the statistical nature of measurements, we can represent the signal as $f=\left(f_n+{\epsilon }_n\right)$, where $\epsilon =\left({\epsilon }_n\right)$ is noise caused by statistical measurements. Here, we follow the discussions of signal estimation developed in [17,19].
• Subtraction of the noise level from lidar returns, caused by the background illumination and intrinsic APD noise: The noise level was estimated from the signal in the first strobe of sensing data. In this strobe, the effect of side illumination created by the radiation of the laser of the fiber lidar was negligibly small. Alternatively, averaging over several last strobes of signal was used.
• Deconvolution of signals: The laser pulse of the fiber lidar has a complex waveform and, during lidar sensing, affects the signal waveform. The IR of the laser pulse was determined experimentally using a special procedure in which a portion of the laser radiation is incident on the APD and is recorded in accumulation mode. The deconvolution procedure is described in Section 2.1.
• After pre-processing the signals, the signal ratios were calculated. This procedure has been described in detail in the Introduction. Then, from the signal ratios, the profile of extinction coefficient within the cloud layer was calculated using the MIA. This MIA has been described in Section 2.2.

Figure 3 shows the sequence of sensing data processing.

3. Materials

3.1. System

The fiber lidar was used to retrieve the optical parameters of clouds in the lower layer of the atmosphere. The structure, composition, and performance characteristics of the fiber lidar were considered in detail in [16].

Figure 4 shows the transceiver system of the fiber lidar.

In the present study, the fiber lidar was changed and supplemented with the following units:

• To carry out measurements in the daytime, a long-wave filter (indicated by Filter N2 in Table 1) was added to the receiving system of the lidar.
• The transceiver system of the lidar was placed on a scanning base (indicated by D in Figure 4 and by “Scanning base” in Table 1).

Figure 5 illustrates the lidar location and azimuthal and elevation directions.

The laser IR was determined during periods of transmission of measurement results from the lidar to the data storage and processing system. These measurements were performed in the same mode as the main measurements, except that a part of the laser radiation was incident on the lidar-receiving telescope. The Q-switch fiber laser exhibits power stability performance below 3%. The wavelength spectrum temperature tuning coefficient is typically 0.35 nm/°C without a Bragg grating (for a laser diode), contrasting with a significantly reduced coefficient when employing a Bragg grating (for the Q-switch fiber laser) [53]. Furthermore, a narrowband filter (designated as Filter N1 in Table 1) maintains the CWL (Center Wavelength) stability of the lidar signal. The non-stop sensing mode was implemented at the expense of independent storage of a series of measured lidar returns in the memory of the photon counter. Our measurements were performed so that data from the fiber lidar were transmitted to the system of storage and processing of measurement results twice a day. The protocol of the 10/100/1000 Ethernet interface was used for data transmission. Measurement results were stored as data arrays with an indication of the lidar measurement date and the number of data files in the measurement series. The measurement mode and time are available for each file. Files are stored in the text data format; this is convenient for their storage, viewing, and subsequent processing.

3.2. Experiment

The experiment was performed from 20 January 2025 to 1 March 2025. The fiber lidar was located at the Hanbat Research Laboratory of the National University in Daejeon, Republic of Korea (36.34°N, 127.30°E). The Research Laboratory is situated in the southern part of the city, on the edge of the urban environment. Daejeon is surrounded by a mountain ridge with an average mountain height of 250–300 m. The nearest meteorological station has coordinates 36.35°N, 127.04°E. Lidar sensing of the atmosphere was carried out along a tilted path defined by an azimuth angle of 255° and an elevation angle of 15°. Each lidar return signal consisted of 600 strobes with a duration of 50 ns each. The slant sensing range was equal to 4500 m with a resolution of 7.5 m. The altitude resolution was 1.941 m, and the sensing altitude was 1164 m. Measurements were performed in non-stop mode.

4. Results

The present work highlights the main results consistent with the objective of our study. Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 show the results of lidar sensing of the optical parameters of a cloud layer observed in the process of development of cyclonic activity during 29 h from 27 February 2025/12:21:57 to 28 February 2025/17:45:19, KST (Korean Standard Time). Figure 6 shows the signal ratios $R\left(r\right)$ in the presence of cloud layers during the examined time.

During this period, warm air masses accompanied by clouds and light intermittent rain displaced colder air with clear cloudless weather and then moved to the east of the peninsula. The pressure at the Earth’s surface decreased by 5.6 mbar. According to the data of the nearest meteorological station, the clouds were stratocumulus, not cumulus clouds. The procedure for preliminary processing the measurement data is described above in Section 2.1 and Section 2.3.

It should be especially noted that the data are qualitative and intended only for visualizing the process of cyclonic activity during the examined period. For the best visualization of the obtained results, the lidar signal ratios are shown in the range $\left[0.5,300\right]$, although their full range exceeds three orders of magnitude. The reference lidar signal $P_{REF}\left(r\right)$ was averaged over 22 profiles of lidar backscattering signals recorded without the cloud layer, prior to the beginning of its development.

Figure 7 shows time variations in the background level during measurements caused by solar radiation scattering in the atmosphere during the measurement period. The night-to-day background level changes by five orders of magnitude. In the daytime, the background level depends on the mutual arrangement of the Sun and the sensing direction at the place of the lidar location.

The maximum background level was observed at a minimum angle between the direction toward the Sun and the sensing direction equal to ~15°. These moments of closest rapprochements are designated by S1 and S2 in Figure 7. The twilight background level from 27 February 2025/19:00 to 28 February 2025/03:00 KST is caused by the glow of the sky after sunset and, presumably, by the reflection of solar radiation from clouds in the sensing direction.

Figure 8 shows the results of inverting the signal ratios in the presence of cloud layers to obtain the altitude profiles of the extinction coefficients from 27 February 2025/20:12:22 to 28 February 2025/00:36:00 KST.

The algorithm for determining the profile of the cloud extinction coefficient by inverting the signal ratio has been described above in Section 2.2. To apply the MIA, the single-case dataset was measured right after the time of determining the reference signal was chosen.

Within the first 111 min of measurements, a cloud layer formed at an altitude of about 950 m. Over the next 57 min of measurements, the cloud layer descended from an altitude of 1100 m to an altitude of about 400 m, and after this, it was observed in the form of a layered structure with a total thickness of about 300 m until the end of the measurement period. Figure 9 shows the distribution of the cloud layer thicknesses against the peak values of the extinction coefficient in the corresponding layers.

Figure 10a shows the histogram of the maximum extinction coefficients in the corresponding cloud layers, and Figure 10b shows the histogram of the FWHM cloud layer thickness (vertical thickness).

In this instance, the layer thickness $\Delta r$ was determined by calculating the full width at half maximum (FWHM) of the extinction coefficient distribution. Empirical data demonstrate considerable variance in peak extinction coefficient values, spanning from 0.05 to 10 km⁻¹ over the assessment interval. Cloud layer thickness $\Delta r$ also demonstrates considerable variation, ranging from 6 to 215 m. The estimated mean value of the maximum extinction coefficients in the cloud, $\overline{{\alpha }_C}$, was 0.73 km⁻¹. The estimated mean value of the cloud thickness, $\overline{\Delta r}$, was 27.5 m. The histograms show that the maximum extinction coefficient accounts for a value of 0.09 km⁻¹ in 27.5% of cases, whereas the maximum of the cloud layer thickness envelope accounts for 18.6 m in 28.2% of cases. This indicates that the cloud formations during the measurement period mainly consisted of optically thin cloud layers. The substantial agreement between the findings and the measurement data presented in reference [7] is noteworthy. The peak backscatter coefficient for the extensive stratocumulus cloud, as indicated in [7], is ${\alpha }_C\left(r\right)\approx 1$ km⁻¹. The cloud geometrical thickness, defined by the cloud boundaries, was measured to be 249 m.

5. Discussion

5.1. Lidar Return Signals

Figure 11 shows an example of lidar return signals at the SLSPA input.

The signals were measured in the daytime when the level of background illumination caused by solar radiation scattering in the atmosphere was high (a) and when it was low (b). Lidar configuration ensured the probability of photon recording in a strobe of less than 1 per laser pulse. This provided the condition of applicability of the Poisson statistics to lidar return signals [33]. At this SLSPA stage, it is customary that the estimates of the signal variation are taken to be equal to the estimates of the mean, i.e., the signal itself.

5.2. Estimation of Deconvolution Errors in the SLSPA

Deconvolution of waveforms of lidar signals in accordance with the laser pulse waveform of the lidar is carried out at the final stage of preliminary measurement data processing. It is also noteworthy that deconvolution operations lead to an approximate 20-fold amplification in standard deviation estimates of the signals. This imposes limitations on the SLSPA inversion algorithm. The increasing error reduces the robustness of the inversion algorithm for weak signals.

Figure 12a provides a comparative depiction of the signal presented in Figure 11b, both pre- and post-deconvolution, within the context of the SLSPA methodology. In this instance, black horizontal bars represent standard deviations associated with the empirical signals, while red horizontal bars denote standard deviations pertaining to the reconstructed signals. Algorithm convergence is demonstrably robust, predicated on the Kaczmarz projective method [54,55]. Figure 12b illustrates the quantitative relationship between the standard deviations of the signal following deconvolution and those preceding it.

5.3. Reference Range in MIA

As a rule, in practice, the calibration section of the sensing path in the inversion algorithm is chosen based on the qualitative estimate of the optical parameters of the atmosphere on the sounding path [56,57,58,59,60,61,62]. Let us briefly consider the behavior of the calibration constant $C_0$ in two limiting cases of choosing the reference range $r_0$ below or above the cloud layer. The choice of the reference range $r_0$ near the lidar is more justifiable due to the lower variability of the calibration constant $C_0$. In this case, it is possible to accept ${\beta }_C\left(r_0\right)=0$ and $\gamma \left(r_0\right)=K\left(r_0\right)$, and the calibration constant $C_0$ takes the following form:

$$C_0=\exp \left[-2{\displaystyle \underset{0}{\overset{r_0}{\int }}\eta \left(r^{\prime }\right)\Delta {\alpha }_{AER}\left(r^{\prime }\right)d{r}^{\prime }}\right]\approx 1.$$

(44)

As demonstrated by Klett [23], the algorithm of inversion of the lidar equation is more stable regarding the divergence parameter if calibration is performed at the end of the sounding path. However, choosing the reference range $r_0$ above the cloud layer is difficult in the case of intense cloud cover. In this case, the lidar signal $P\left(r\right)$ above the cloud is weak and the signal ratio is low, and hence more susceptible to measurement errors. Just as in the reference range $r_0$ above the cloud, the calibration constant takes the form

$$C_0=\exp \left\{-2{\displaystyle \underset{0}{\overset{r_0}{\int }}\eta \left(r^{\prime }\right)\left[{\alpha }_C\left(r^{\prime }\right)+\Delta {\alpha }_{AER}\left(r^{\prime }\right)\right]d{r}^{\prime }}\right\}.$$

(45)

5.4. Cloud Extinction Coefficient Distributions

Figure 13 shows examples of vertical profiles of the lidar return signal (a) and corresponding extinction coefficients (b) observed in a cloud formation.

The data presented in Figure 13b delineates a complex cloud structure, comprising three discrete, tenuous strata. The lower stratum, exhibiting a thickness of 6 m and a peak extinction coefficient of 1.3 km⁻¹, is situated at an altitude of 906 m. A stratum with a thickness of 6 m and a peak extinction coefficient value of 0.07 km⁻¹ is situated at an altitude of 955 m. The uppermost stratum, exhibiting a thickness of 12 m and a peak extinction coefficient value of 3.3 km⁻¹, is situated at an altitude of 1093 m.

It is noteworthy that a displacement of cloud layers is observed in the lidar signal of Figure 13a, relative to the extinction coefficient profile in Figure 13b. This displacement is attributed to the asymmetric morphology of the laser pulse, as depicted in Figure 1.

Figure 14 shows examples of vertical profiles of the signal ratios (a) and corresponding extinction coefficients (b) observed on 27 February 2025 at 22:00:41 KST.

These data refer to measurements illustrated by Figure 8. Due to the temporal proximity of the reference signal and cloud signal measurements within the experiment, it can be inferred that $K\left(r\right)\approx 1$. The profile of the ${\alpha }_C\left(r\right)$, shown in black in Figure 14b, exhibits AVF correspondence with $K\left(r\right)=1$. The profile ${\alpha }_C\left(r\right)$, illustrated in red in Figure 14b, exhibits AVF correspondence with $K\left(r\right)=2$. These profiles were retrieved from the experimental data subjected to the sequential pre-processing procedure described in Section 2.3. The ${\alpha }_C\left(r\right)$ change is about 2.4% when the aerosol extinction coefficient in the cloud changes by 100%. Such stability of the MIA in response to changes in the cloud aerosol parameters is explained by the structure of the algorithm. The stability analysis of the inversion algorithm is elaborated upon in Section 5.5.

5.5. Systematic Errors Estimating

Note that it is difficult to estimate the contribution of systematic errors to the total error estimate in determining the extinction coefficient ${\alpha }_C\left(r\right)$ because the inversion algorithm is influenced by many factors. In works [63,64,65,66], errors due to the application of the inversion algorithm were analyzed. Here, we estimate the influence of the variability of aerosol extinction coefficient in cloud on the value of the cloud extinction coefficient. In other words, we estimate the sensitivity of the modified inversion algorithm (MIA) to changes in the optical aerosol parameters in the cloud. The optical properties of the aerosol within and outside of clouds can differ, for example, due to the hydration of aerosol particles at enhanced humidity within the cloud. For our analysis, we consider that the extinction coefficients for cloud, aerosol, and molecular scattering satisfy the following relationship:

$${\alpha }_C\left(r\right)\gg {\alpha }_{AER}\left(r\right)\gg {\alpha }_{MOL}\left(r\right).$$

(46)

The MIA operates utilizing signal ratios. The ${\tilde{\beta }}_A\left(r\right)R\left(r\right)$ multiplier present in Formulas (37) and (38) causes the algorithm to invert the normalized lidar signal $\stackrel{⌢}{P}\left(r\right)$ in the form of

$$\stackrel{⌢}{P}\left(r\right)\approx \left[{\beta }_C\left(r\right)+{\beta }_A\left(r\right)\right]\exp \left\{-{\displaystyle \underset{0}{\overset{r}{\int }}\left[{\alpha }_C\left(r^{\prime }\right)+\Delta {\alpha }_A\left(r^{\prime }\right)\right]d{r}^{\prime }}\right\}.$$

(47)

where ${\beta }_A\left(r\right)={\tilde{\beta }}_A\left(r\right)+\Delta {\beta }_A\left(r\right)$. Let the ${\alpha }_C\left(r\right)$ value in cloud changes in the sample $\left\{R_i\left(r\right)\right\}$ range from 1 to 10 km⁻¹. Assume that ${\alpha }_A\left(r\right)$ in the sample has a value of 0.1 km⁻¹. Then, for the ${\alpha }_C\left(r\right)$ value to change by 5%, the $K\left(r\right)$ value should change in the range approximately from 1.5 to 6. Or, in other words, if ${\alpha }_C\left(r\right)$ = 1, for the ${\alpha }_C\left(r\right)$ to change by 5%, the ${\alpha }_A\left(r\right)$ value should change by no less than 50%. Note that under these conditions, the variability of signal ratio $\left\{R_i\left(r\right)\right\}$ remains approximately within 5% for both ${\alpha }_C\left(r\right)$ = 1 and ${\alpha }_C\left(r\right)$ = 10. These are the approximate estimates showing the tendency of changing signal ratio $R\left(r\right)$ attendant to changes in the aerosol extinction coefficient ${\alpha }_A\left(r\right)$. The prevalence of the ${\alpha }_C\left(r\right)$ determines the stability of the MIA. In practical application, given the stationary nature of the $\Delta {\alpha }_A\left(r\right)$ variability and/or the temporal proximity of reference and cloud signal measurements, $K\left(r\right)\approx 1$. Within the Klett inversion algorithm, lidar signals are normalized solely with respect to $r^2$. The Klett inversion algorithm necessitates supplementary data [23]. For instance, in the determination of the aerosol extinction coefficient profile, the molecular scattering extinction coefficient profile must be known. As stipulated in Section 1.2, the reference signal $P_{REF}\left(r\right)$ corresponds to the reference profile ${\tilde{\beta }}_A\left(r\right)$. This profile is predicated on an a priori known atmospheric aerosol and molecular backscatter coefficient profile. This profile can be obtained from external sources. This may be a modeled profile, or it may be an actual profile. Based on empirical evidence, a sensitivity analysis of the MIA to ${\tilde{\beta }}_A\left(r\right)$ variability was performed. The results indicate that a 1% change in ${\tilde{\beta }}_A\left(r\right)$ induces a proportional change of approximately 1% in ${\alpha }_C\left(r\right)$. This implies that under conditions of heightened uncertainty in the contribution of aerosol scattering to ${\tilde{\beta }}_A\left(r\right)$, alternative inversion algorithms are warranted.

6. Conclusions

This paper describes a methodological case study of low-level clouds in the atmosphere using a 1064 nm fiber lidar. The lidar experiment was carried out in Daejeon, Republic of Korea, from January to March 2025. The objective of the experiment was to determine the vertical extinction coefficient profiles of stratus clouds. To achieve this, the Sequential Lidar Signal Processing Algorithm (SLSPA) was modified. Theoretical and experimental aspects of the modified SLSPA are considered sequentially in the present work.

The problem of deconvolution error estimation was solved using statistical estimation theory. Random deconvolution errors were estimated using the WLS method. The results showed that deconvolution led to an approximate 20-fold amplification in standard deviation estimates of the signals.

The solution to the problem of correcting the geometric factor of lidar (GLFF) is presented. A correction algorithm is proposed in which the GLFF is not calculated explicitly. Instead, the ratio of vertical lidar signals is used. This is the ratio of the signal in the cloud to the reference signal. The reference signal is measured without cloud with the known vertical distribution of the coefficients of molecular and aerosol backscattering coefficients. The Klett–Fernald algorithm was modified to invert the signal ratios.

The solution to the problem of variability in the aerosol extinction coefficient in clouds due to aerosol hydration is demonstrated. The modified inversion algorithm (MIA) can give additional information on the variability of cloud aerosol extinction coefficient. For this purpose, the Aerosol Variability Function (AVF) $K\left(r\right)$ was included in the algorithm. The sensitivity of the MIA to variations in this parameter value was analyzed based on experimental data. The results showed that an increase in the aerosol extinction coefficient of 100% leads to a change in the cloud extinction coefficient of about 2.4%. Such stability of the MIA in response to change in the cloud aerosol parameters is explained by the structure of the algorithm.

Correction for the multiple scattering effect in the MIA is considered. The Klett–Fernald algorithm was modified by introducing the multiple scattering function $\eta \left(r\right)$. The experimental results presented here are based on single-case datasets sampled from the entire array of the experimental data obtained during measurement period. The measurement data indicated that the cloud formations primarily comprised tenuous cloud strata exhibiting peak extinction coefficient values ranging from 0.05 to 10 km⁻¹ during the observation period. Stratum thickness also manifested substantial variability, fluctuating between 6 and 215 m. These findings correlate favorably with measurement data obtained for extensive stratocumulus cloud formations [7]. It is difficult to study these clouds using space-based sensing.

The ground-based fiber-optic lidar system constitutes a propitious methodology for the investigation of tenuous, low-altitude cloud formations. Further data verification is required, for example, by using independent measurement results. An increase in the volume of statistical data for different seasons of the annual cycle would be a useful direction for further study in remote sensing of the vertical profiles of the extinction in low-level clouds.