Worldwide, nowadays Mie lidar is one of the most-used active remote-sensing tools for atmospheric sounding [1-8]. However, data inversion for Mie lidar still needs to be addressed, due to the unknown lidar ratio based on the Fernald backward-integration method (FBIM) [9]. As a result, the aerosol extinction coefficient and aerosol backscattering coefficient β_a(z) retrieved from Mie lidar returns are treated as “data for reference only” in the field of atmospheric sounding.

The inversion error in β_a(z) has been researched by Francesc Rocadenbosch et al. [10, 11], but the inversion results, calculation method, and research contents of this paper are different from those of previous studies. It can be said with certainty that the lidar ratio has influence on the retrieval of β_a(z), but how great is this influence? If the interval from the calibration point to the inversion point continuously increases, what will be the errors in the retrievals of β_a(z) obtained by the FBIM method? What is the impact on the vertical profile of β_a(z) when the actual vertical profile of the lidar ratio is replaced by a constant value? These problems will be studied in this article.

The influences of lidar ratio on the vertical profile of β_a(z) are analyzed quantitatively in this paper. It is necessary for us to understand the errors in the derived β_a(z) and its vertical profiles. The contents of this article mainly discuss the influence of the error in lidar ratio on the aerosol backscattering coefficient. Based on the conclusions of this paper, when error exists in the lidar ratio, we can deduce the error in β_a(z) at different heights from the retrieved β_a(z) profile.

The equation for the backscattering signal of Mie lidar can be written as follows [2]:

The Fernald backward-integral method is formulated in the backward backscattering coefficient form as below [2, 9]:

where c refers to the speed of light, E₀ denotes laser emission energy, Y(z) is the overlap factor, A_r is the area of the receiving telescope, β_a(z) and β_m(z) are the backscattering coefficients of aerosol particles and atmosphere molecules respectively at altitude z, and β(z) = β_a(z) + β_m(z). P(z) is the lidar backscattering signal at altitude z. X(z) = P(z)z², and z_c refers to the altitude of the lidar calibration point. S_a(z) and S_m are the lidar ratios for aerosols and molecules respectively. T(z) is the atmosphere transmittance from the lidar to the altitude z.

Range-corrected lidar signals for the retrieval of β_a(z) are shown in Fig. 1. Assuming that there is a homogeneous aerosol layer in the range of 2.5~5 km in altitude, which makes it easy to analyze the relationship between the error in β_a(z) derivation and the interval from the calibration height to another height, where the aerosol backscattering coefficient need to be retrieved. In Fig. 2, signal trace A shows the β_a(z) derived from the lidar signal denoted as A in Fig. 1, when 50 sr is assigned to S_a(z). Within the interval of 2.5~5 km, β_a(z) for signal B in Fig. 2 is 0.5 times as much as β_a(z) for A. Trace C in Fig. 2 shows β_m(z) from Eq. (2).

To simplify figure captions, “aerosol backscattering coefficient” is abbreviated as “ABSC” in this paper.

Based on the lidar signal represented as A in Fig. 1, β_a(z) and its errors are retrieved with Eq. (2) when the value of S_a(z) is varied from 30 sr to 70 sr in steps of 10 sr, and 10.02 km is chosen as the reference height [12].

Assuming that the true value of S_a(z) is 50 sr, the corresponding retrievals of β_a(z) in the range of 2.5~5 km are shown in Figs. 3 and 4.

2.1.1. Related factors in the accumulation of absolute error

(1) With the assumption that there is a certain error in S_a(z), the absolute error of β_a(z) continues to increase linearly with increasing interval between the height of calibration point and that of inverted atmosphere.

(2) The increasing rate of absolute error in β_a(z) is related to the deviation of S_a(z) from its actual value. The greater the deviation of S_a(z), the faster the increase in rate of error in β_a(z).

(3) By examining Fig. 3 carefully, we can see that, in spite of the same difference of 20 sr from the actual value (50 sr, for example), i.e. where S_a(z) is 30 sr and 70 sr respectively, when the S_a(z) is 20 sr smaller than the actual value, the absolute error in β_a(z) obtained using FBIM is greater than the absolute error when S_a(z) is 20 sr larger than the true value.

(4) There is a relationship between the absolute error in β_a(z) and the error in S_a(z) for the same inversion range; greater error in S_a(z) results in greater absolute error in β_a(z) .

2.1.2. Relative error of the retrieved β_a(z)

The relative error can be a true reflection of the credibility of the retrieved β_a(z). Assuming that there are errors in S_a(z), and the true value of S_a(z) is 50 sr ,within a certain range it can be seen in Fig. 4 that the relative error in β_a(z) obtained from Eq. (2) increases with range. For example, for S_a(z) = 30 sr, the relative error in β_a(z) is 7% at 5 km, but 23% at 2.5 km. The relative error in β_a(z) increased by 16% in the range from 5 to 2.5 km.

The lidar data shown in the B traces of Figs. 1 and 2 are used for comparative analysis with the corresponding A data. The aerosol backscattering coefficients from 2.5 to 5 km shown in Fig. 2 as trace B are half as great as the corresponding ones in trace A. Trace B in Fig. 1 is the range-corrected lidar signal corresponding to the aerosol backscattering coefficient shown as curve B in Fig. 2, when S_a(z) = 50 sr.

The retrievals of β_a(z) in Fig. 5 are obtained with Eq. (2) from the lidar returns shown in Fig. 1 as curve B, when the value of S_a(z) is varied from 30 to 70 sr in steps of 10 sr. The absolute and relative errors of β_a(z) shown in Fig. 5 are calculated for quantitative analysis, as well as comparison to the corresponding errors of β_a(z),which are acquired using the lidar data represented by the curves marked A in Figs. 1 and 2. We retrieve β_a(z) at a height of 2.61 km as an example for quantitative analysis. Assigning 30 sr, 40 sr, 60 sr, and 70 sr to S_a(z), one by one, the relative errors in β_a(z) at a height of 2.61 km in Fig. 5 (retrieved from the lidar data shown in the B traces of Figs. 1 and 2) are 22.76%, 10.62%, 9.33%, and 17.57% respectively.

In contrast to the relative errors in β_a(z) shown in Fig. 5(b), the relative errors in β_a(z) in Fig. 4(b) (which are derived from the lidar signal represented as A in Figs. 1 and 2) are 20.32%, 9.48%, 8.34%, and 15.72% respectively.

It can be easily found from the comparative analysis of Figs. 4 and 5 that there exists a relation between the actual value of β_a(z) and the relative error of the retrieved β_a(z): When S_a(z) and its error are the same, the relative error in the retrieved β_a(z),acquired using Eq. (2), is larger when the actual value of β_a(z) is smaller.

The lidar signal used for retrieval of the backscattering coefficient of dust β_d(z) is shown in Fig. 6(a), while vertical profiles of the retrieved β_d(z) are shown in Fig. 6(b), with lidar ratios for dust S_d(z) of 40, 50, and 60 sr respectively. Assuming that the real value of S_d(z) is 50 sr, and treating the β_d(z) obtained using Eq. (2) as the true value of β_d(z), it can be seen from Fig. 6(b) that the accurate value of β_d(z) at 2.01 km is 0.00739 km⁻¹ sr⁻¹. For S_d(z) = 40 and 60 sr [13, 14], the absolute error of the retrieved β_d(z) at a height of 2.01 km is 0.00158 and 0.00112 km⁻¹ sr⁻¹ respectively.

In Fig. 7, if the dust layer (the dotted line) is replaced by the background aerosol (the solid line), which is treated as the actual profile of β_a(z) with S_a(z) = 50 sr, we find that the actual value of β_a(z) at 2.01 km is 0.000929 km⁻¹ sr⁻¹. For S_a(z) = 40 and 60 sr, the absolute error of the retrieved β_a(z) (which is derived from the simulated lidar signal obtained from the β_a(z) profile in Fig. 7 using Eq. (2)) is 0.000102 km⁻¹ sr⁻¹ and 0.0000909 km⁻¹ sr⁻¹ respectively.

Assuming the lidar ratio and its error being the same, we come to the conclusion that the deviation in lidar ratio has larger impact on the absolute error of β_d(z) than on that of β_a(z) at the same height, because the true value of β_d(z) is larger than that of β_a(z).

The retrieved backscattering coefficients of cirrus β_c(z) and their absolute errors, which are obtained using the range-corrected backscattering signal for a cirrus layer in Fig. 8 with Eq. (2), are shown in Figs. 9 and 10, with 20, 30, 40, 50, and 60 sr as the value of S_c(z) respectively, while 20 sr is treated as the actual value of S_c(z) [15].

In Fig. 10 we can see that if some error exists in S_c(z), a larger true value of β_c(z) will result in a larger absolute error in β_c(z). In many retrievals of β_c(z), the absolute error is so great that it cannot be ignored. For the same error in lidar ratio, the absolute error in the obtained β_c(z) is much greater than that in β_a(z). This conclusion is well comparable to the relationship between the absolute error of the derived β_d(z) and S_d(z), as was analyzed quantitatively in section 3.1 above.

The maximum relative error of β_c(z), which occurs at the lowest edge of the cirrus structure (at an altitude of 9 km), is observed in Fig. 11. It shows that the relative error in the received β_c(z) is not only concerned with the actual value of β_c(z), but also relates to the interval from the calibration point to the point where β_c(z) will be retrieved. Greater relative error in β_c(z) is observed when the greater actual value of β_c(z) and greater reversal distance exist simultaneously. This conclusion is consistent with the corresponding retrievals for the backscattering coefficients of aerosols and dust, which were presented earlier in the article.

In the earlier inversion of section 3.2, there are errors in the lidar ratio of the cirrus layer, but not in the lidar ratio of the background aerosol, which kept a constant value of 50 sr. From Fig. 10, it can be seen that the lidar ratio of the cirrus layer is not only the source of the error in retrievals for the cirrus, but also the source of the error in the calculated value of the background aerosol layer below the cirrus. Assuming that the error of the cirrus lidar ratio is in the range 10~40 sr, it can be seen from Fig. 10 that the relative error in the retrieval of the backscattering coefficients for aerosol layers below the cirrus are in the range 5~30%.

In conclusion, if there is a certain error in the lidar ratio and the atmosphere is uniform in the altitude range of interest, the absolute and relative errors in β_a(z) continue to accumulate with increasing inversion range along the lidar operating path.

The greater the error in aerosol lidar ratio, the faster the accumulation of absolute and relative errors in the received β_a(z).

The error in lidar ratio has great influence on the retrieved backscattering coefficient for a layer of dust or cirrus clouds, the absolute error in the derived β_c(z) orβ_d(z) profile being so large that it cannot be neglected in most cases.