# Discussion of the relationship between the aerosol extinction coefficient error and background noise

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- Received:
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## Abstract

This paper discusses the relationships among the aerosol extinction coefficient error (AECE), background noise, and distance associated with lidar measurements. The AECE calculation is explained in detail, revealing that the AECE is the product of background noise, range squared, and a relation function. The result of an AECE calculation that uses lidar measurements obtained in Nanjing, China, agrees with a calculation that uses a simulated lidar signal. The AECE equation is verified with lidar measurement data and a simulated lidar signal, indicating the AECE equation is reasonable.

## 1 Introduction

Aerosols are a source of uncertainty and an important cause of climate change. Aerosols also strongly pollute the atmospheric environment and are harmful to public health [1, 2, 3]. Consequently, scientists continue to consider aerosol research an important topic, as different aerosol optical properties cause different atmospheric radiation effects, and a database of aerosol optical properties has yet to be established in atmospheric research. High-accuracy aerosol optical properties are necessary for atmospheric studies. For instance, the aerosol extinction coefficient, and the related visibility and optical depth, are essential in meteorology research [4]. Lidar can measure aerosols with high efficiency and range resolution. However, deficiencies still exist in lidar aerosol measurements. Aerosol measurement uncertainty exists for two reasons: the inversion method for the aerosol parameter, which has many assumptions [5], and systematic error due to the lidar system. Systematic error due to background noise is especially important, because it persists in aerosol lidar measurements. It remains an open question as to how background noise should be handled. Researchers commonly reduce background noise, experimentally, by making lidar measurements at night, using a narrow bandpass filter, and by decreasing the telescope’s field of view. However, few studies have examined aerosol measurement error due to background noise [6]. This paper presents the theoretical calculation of the aerosol extinction coefficient error (AECE) due to background noise. The equation relating the AECE and background noise is calculated in detail. The AECE is calculated using experimental data and a simulated lidar signal.

## 2 Theory of the AECE

*x*, a variable related to noise d

*p*, is introduced for ease of discussion \( \left( {{\text{d}}p = \frac{\partial p}{\partial x}{\text{d}}x} \right) \),

*r*is the distance (

*x*and

*r*are independent),

*σ*(

*r*,

*x*) is the aerosol extinction coefficient, \( S(r,x) = \ln [r^{2} p(r,x)] \),

*p*(

*r*,

*x*) is the lidar return signal,

*σ*

_{m}is the boundary value of the extinction coefficient,

*S*

_{m}is the boundary value of

*S*, and

*k*is a constant. Therefore,

*U*(

*x*) and

*V*(

*x*) are introduced as follows:

*x*we obtain

*k*= 1 in (9), we obtain

We discuss the relationships among *T*(*r*, *x*), *T*_{0}(*r*, *x*), *F*(*r*, *x*) with reference to experimental and simulation data, as described below. The calculation results show that *F*(*r*, *x*) is negligibly small.

## 3 Analysis of aerosol lidar measurement data

*T*(

*r*,

*x*). Equations (11)–(13) indicate that

*T*(

*r*,

*x*) depends on the lidar return signal and range. Using measurement data,

*T*(

*r*,

*x*) is calculated and shown in Fig. 2. According to (12), the function

*T*

_{0}(

*r*,

*x*) is calculated using experimental data

*p*(

*r*,

*x*) and

*p*(

*r*

_{m},

*x*

_{m}), small fluctuations in

*T*

_{0}(

*r*,

*x*) are due to noise in the experimental data, and we performed a polynomial fit calculation for

*T*

_{0}(

*r*,

*x*), as shown in Fig. 2a. According to (13), the function

*F*(

*r*,

*x*) is calculated using experimental data, small fluctuations in

*F*(

*r*,

*x*) are due to noise in the experimental data, and the value of

*F*(

*r*,

*x*) is close to zero, as shown in Fig. 2b. Figure 2c is the same as Fig. 2a but with the vertical axis enlarged, showing the polynomial fit calculation for

*T*

_{0}(

*r*,

*x*). According to (11), the relation function

*T*(

*r*,

*x*) differs between

*T*

_{0}(

*r*,

*x*) and

*F*(

*r*,

*x*), as shown in Fig. 2d.

*T*(

*r*,

*x*) is almost equal to

*T*

_{0}(

*r*,

*x*) and that

*F*(

*r*,

*x*) is negligibly small. Figure 3 shows the relationships among AECE,

*T*(

*r*,

*x*), and background noise. Figure 3a shows that

*T*(

*r*,

*x*) increases with increasing range; consequently, AECE also increases in this way. The background noise signal was measured immediately after each aerosol measurement, and is shown in Fig. 3b. The AECE, calculated according to (10), is proportional to the background noise and becomes larger with increasing range (Fig. 3c). We subtracted background noise from measurement data to obtain an aerosol measurement signal without background noise. Figure 3d shows the extinction coefficient calculated from the aerosol measurement signal without background noise, and Fig. 3e shows the extinction coefficient calculated from the lidar signal with background noise, using the following equation:

*σ*(lidarsignal + noise) and (

*σ*(lidarsignal)) are the aerosol extinction coefficients calculated from the lidar signal with and without background noise, respectively, and d

*σ*is the AECE. Using (15), and thus adding the data from Fig. 3c, d, we obtain the aerosol extinction coefficient profile (Fig. 3f). The similarity of the profiles in Fig. 3f and 3e indicates the validity of equation (10), from which the result shown in Fig. 3c was obtained.

## 4 Simulations

*T*(

*r*,

*x*),

*T*

_{0}(

*r*,

*x*), and

*F*(

*r*,

*x*) are calculated as in Fig. 5.

*T*

_{0}(

*r*,

*x*) and

*F*(

*r*,

*x*). Figure 5b is the same as Fig. 5a but with the vertical axis expanded, showing

*T*

_{0}(

*r*,

*x*). Figure 5c shows

*T*(

*r*,

*x*)—the difference between

*T*

_{0}(

*r*,

*x*) and

*F*(

*r*,

*x*).

*F*(

*r*,

*x*) is negligibly small (Fig. 5a) and

*T*(

*r*,

*x*) is almost equal to

*T*

_{0}(

*r*,

*x*), similar to Fig. 2b. Using the simulation data in Fig. 4 and

*T*(

*r*,

*x*) in Fig. 5, AECE can be calculated as in Fig. 6. Figure 6a shows that the relation function

*T*(

*r*,

*x*) is similar to that in Fig. 3a, as they both increase with increasing range.

*T*(

*r*,

*x*) in both Figs. 6a and 3a is calculated using (11); however,

*T*(

*r*,

*x*) in Fig. 3a (Fig. 6a) is based on the measurement (simulated) data. Employing (10), we use the simulated data to calculate the extinction coefficient error (AECE) due to background noise (Fig. 6b). The extinction coefficient error increases with distance, assuming the background noise is constant (Fig. 4a). The extinction coefficient error shown in Fig. 6b corresponds to that in Fig. 3c. The AECE shown in Fig. 3c relates to the experimental noise. The two extinction coefficient profiles shown in Fig. 6c are based on the simulated data, and correspond to the range-corrected signals with and without background noise, which are shown in Fig. 4b.

The extinction coefficient profile 1 (profile 3) in Fig. 6d is calculated from the simulated signal with constant (without) background noise. Profile 4 in Fig. 6d is the error of the extinction coefficient. The extinction coefficient, profile 2 in Fig. 6d, is the sum of profiles 3 and 4. Recall that profile 1 is the extinction coefficient calculated using (1), and profile 2 is the extinction coefficient calculated from the full AECE theory. Figure 6 shows a small difference between profiles 1 and 2.

## 5 Conclusions

In aerosol lidar measurements, the measurement accuracy is of great importance. The accuracy of aerosol optical property depends on several factors, including the boundary value and parameter assumptions, and background noise. According to lidar theory, the relationship between the AECE and background noise, and distance can be calculated. The AECE is proportional to the square of distance *r*^{2}, the relation function *T*(*r*, *x*), and the amount of background noise. Here, the relationship between the AECE and background noise has been verified by using aerosol lidar measurements and simulated lidar signals.

## Notes

### Acknowledgments

This work was supported by the Nature Science Foundation of China under Project 41175033/D0503 and Chinese Public Welfare Industry Special Project GYHY 201006047-5.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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