Numerical semi-empirical modeling of lidar attenuation characteristics in atmosphere

Abstract

Long-distance measurement of lidar is a critical problem in various fields, such as airborne lidar detection, surveying, and mapping. Attenuation characteristic is the key factor affecting the lidar detection capability. However, some long-distance measurement methods based on the widely used and validated theoretical models are usually inefficient, while others relying on empirical models have limited applications due to experimental difficulties. This study proposes a semi-empirical model of long-distance measurement based on the Beer–Lambert–Bouguer law adopting mathematical methods. Compared to the theoretical model, the calculation efficiency of the semi-empirical model we constructed has been improved by more than two orders of magnitude, and the accuracy can reach more than 98% of the theoretical model. Compared to other empirical models, its accuracy is closer to the theoretical model, and the computational efficiency is similar. It can be more effectively applied to the actual long-distance detection scene.

Appendix A

1.1 Theoretical calculation of absorption and scattering coefficients

$$ k_{n} \left( h \right) = \int_{r1}^{r2} {C_{{{\text{abs}}}} F\left( r \right)dr} $$

(29)

$$ \beta_{n} \left( h \right) = \int_{r1}^{r2} {C_{{{\text{sca}}}} F\left( r \right)dr} , $$

(30)

where $C_{abs}$ and $C_{sca}$ are absorption and scattering cross-section; $r$ is radius; $F(r)$ defines aerosol size distribution;

For simplicity, extinction cross-section ($C_{{{\text{ext}}}}$) and aerosol effect coefficient ($\eta_{n} \left( h \right)$) are described as the sum of $C_{{{\text{abs}}}}$ and $C_{{{\text{sca}}}}$, $k_{n} \left( h \right)$ and $\beta_{{\text{n}}} \left( h \right)$, respectively.

$$ C_{{{\text{ext}}}} = C_{{{\text{abs}}}} + C_{{{\text{sca}}}} $$

(31)

$$ \eta_{n} \left( h \right) = k_{n} \left( h \right) + \beta_{n} \left( h \right) $$

(32)

Absorption and scattering cross-section can be further calculated as follows:

$$ C_{{{\text{abs}}}} = \frac{2\pi }{{l^{2} }}\sum\limits_{k = 1}^{\infty } {\left( {2k + 1} \right)\left[ {{\text{Re}} \left( {a_{k} + b_{k} } \right) - \left( {\left| {a_{k} } \right|^{2} + \left| {b_{k} } \right|^{2} } \right)} \right]} $$

(33)

$$ C_{{{\text{sca}}}} = \frac{2\pi }{{l^{2} }}\sum\limits_{k = 1}^{\infty } {\left( {2k + 1} \right)\left( {\left| {a_{k} } \right|^{2} + \left| {b_{k} } \right|^{2} } \right)} $$

(34)

$$ a_{k} = \frac{{\mu iR^{2} j_{k} (iRx)\left[ {xj_{k} (x)} \right]^{\prime } - u_{1} j_{k} (x)\left[ {iRxj_{k} (iRx)} \right]^{\prime } }}{{u_{1} iR^{2} j_{k} (iRx)\left[ {xh_{k}^{(1)} (x)} \right]^{\prime } - \mu_{1} h_{k}^{(1)} (x)\left[ {iRxj_{k} (iRx)} \right]^{\prime } }} $$

(35)

$$ b_{k} = \frac{{u_{1} iR^{2} j_{k} (iRx)\left[ {xj_{k} (x)} \right]^{\prime } - \mu j_{k} (x)\left[ {iRxj_{k} (iRx)} \right]^{\prime } }}{{u_{1} j_{k} (iRx)\left[ {xh_{k}^{(1)} (x)} \right]^{\prime } - \mu h_{k}^{(1)} (x)\left[ {iRxj_{k} (iRx)} \right]^{\prime } }} $$

(36)

which are related to the wave number $(l = 2\pi /\lambda )$, scattering coefficients ($a_{k}$ and $b_{k}$) analyzed by Eqs. 35–36, size parameter ($x = lr$), spherical Bessel functions ($j_{k}$ and $h_{k}^{(1)}$), refractive index ($iR$), permeability ($\mu$) of sphere, and permeability ($\mu_{1}$) of surrounding the medium function of rate.