## Abstract

The principle of probing the upper atmospheric wind field, which is the Voigt profile spectral line shape, is presented for the first time. By the Fourier Transform of Voigt profile, with the Imaging Spectroscope and the Doppler effect of electromagnetic wave, the distribution and calculation formulae of the velocity field, temperature field, and pressure field of the upper atmosphere wind field are given. The probed source is the two major aurora emission lines originated from the metastable O(^{1}S) and O(^{1}D) at 557.7nm and 630.0nm. From computer simulation and error analysis, the Voigt profile, which is the correlation of the Gaussian profile and Lorentzian profile, is closest to the actual airglow emission lines.

©2006 Optical Society of America

## 1. Introduction

The fundamental theory of the upper atmospheric wind detection is based on the Doppler effect of electromagnetic wave and associated with its interference imaging spectroscopy. The Michelson interferometer and F-P interferometer have been employed at the initial developing time, and then the Polarization interference imaging spectrometer are currently much attractive and considered as the next generation upper atmospheric detection instrument due to its unique features [1–5] of ultra-compact, stationary, wide view of field, high resolution and detective sensitivity.

Through observation of the Doppler shifts of the night airglow atomic oxygen emission lines, we can detect the upper atmospheric wind [6–9]. The state of the art for the principle of the upper atmospheric wind measurement is based on the Gaussian profile. That is to say, we assume the night airglow atomic oxygen emission lines to be the Gaussian profile. However, the research results shows that the Doppler broadening effect produced by the movement of wind field, the spectral line profile could be thought as the Gaussian shape(from the frequency shift of velocity). Since the colliding, the spectral line profile is foreboded as the Lorenz shape(from the ion thermal collision), the final added-up spectrum line might be the Voigt shape [10]. Therefore, it is only an approximately for the spectral line profile to be thought as the Gaussian shape. In this paper, regarding the accurate and practical measurement, the measurement principle based on the Voigt profile also need to be considered.

The Voigt function is an integral form, and direct computation of the Voigt function entails a high computational cost. Currently, many approximate methods were used to finish calculation of the Voigt function. But through strict calculation, we find that the Fourier transform of the pure Voigt profile exist. According to the principle of Fourier transform spectral and the upper atmospheric wind measurement, in this paper, we present the measurement principle based on the Voigt profile for the first time to our knowledge, and prove the measurement based on the voigt profile will to be better than those on Gaussian or Lorentzian profile, as reported presently.

This research will boost the development of upper atmospheric wind field detection principle, and will have significance both theoretically and practically in enrichment of present theory, in detection of the basic information of the upper atmosphere, in the launch and movement of aircraft in orbit, and in the research of mankind living space and outer space.

## 2. The principle and method of the upper atmospheric wind field measurement based on the Voigt profile

The spectrum of the Voigt profile is defined as [11]

Where
$y=\frac{{\gamma}_{L}}{{\gamma}_{D}}{\left(\mathrm{ln}2\right)}^{\frac{1}{2}}$
,
$x=\frac{\sigma -{\sigma}_{0}}{{\gamma}_{D}}{\left(\mathrm{ln}2\right)}^{\frac{1}{2}}$
, and
$f\prime =\frac{1}{{\gamma}_{D}}{\left(\mathrm{ln}2\right)}^{\frac{1}{2}}$
. *γ _{L}* and

*γ*are the half width of the Lorentzian profile and the Gaussian profile, respectively.

_{D}In each actual measurement, since the loss in transmittance and measurement instruments, the Voigt profile is written as [12]

where *A* is the reducing element. Its interferogram is the Fourier transform of the spectrogram [13], so we have

Through strict calculation, we have obtained

let Δ=0, we have

*I*(0)-*I*(0)/2=2*A*√*π*

And

By substituting Eq. (5) into Eq. (4), then substituting Eq. (4) into Eq. (3), we have

in which, *I(0)* is the intensity when Δ=0. We define

*I*
_{0}=*I*(0)/2

then we get this equation:

When a single spectral line enters the measurement system and the optical path difference (OPD) of the interferometer is varied, the output signal is given by [14]

Where *I*
_{0} is the intensity of the emission line, *λ* is the central wavelength of the emission line, *V* is the visibility that is due to the emission linewidth which will be given below, Δ is the path difference at the initial point of the scan, *x* is the change in path difference that is due to the mirror motion, and *I _{B}* is the background intensity that is measured in a special channel. The interferometer is set to a path difference Δ and then stepped in small steps “

*x*”, spaced λ/4 in phase.

The phase angle *φ* associated with Δ is (here we assume phase is zero when wind field velocity is zero)

Where *v* is the wind speed, *c* is the speed of light.

Compare Eq. (7) and Eq. (8), we have

Equation (9) and Eq. (10) show that the wind and temperature measurements are independent of one another. Wind comes from phase, and temperature from modulation depth (visibility). Here, these tow parameters are [15]

Where *M* is molecular weight, *T _{0}*=300K,

*p*=111352Pa. To a special source and a special line, such as metastable oxygen,

_{0}*M*is a constant, so

*γ*in Eq. (12) is a constant. If we have known

_{D}*V*through scanning, we can obtain the relationship between

*p*and

*T*. In order to obtain

*p*and

*T*, we can use the following formula [11]

where *R*=8.31432×10^{3} N·m/kmol·K at the standard atmosphere, *V*′ is the cubage of *n* mol atmosphere. If we assume that the gas is ideal and standard, one mol gas will have the volume of 22.4L.

In order to obtain *ϕ* and *V*, it was decided to use four intensity measurements, spaced λ/4 apart in path difference, from which [6, 8–9]

Where *I*
_{1}, *I*
_{2}, *I*
_{3}, *I*
_{4} are the measurement intensities with *I*
_{B} (background intensity) subtracted.

From above discussion, we can see that using *I*
_{1}, *I*
_{2}, *I*
_{3}, *I*
_{4}, we can obtain *φ* and *V*, then using *φ* and Eq. (9), we can obtained velocity *v* of the wind field (other parameters are known in the measurement); using *V* and Eq. (10) to Eq. (13), we can obtain temperature *T* and pressure *p*.

## 3. The actual calculation and the analysis of the error

Now, we have the parameters in the actual measurement process to calculate the four intensities *I*
_{1}, *I*
_{2}, *I*
_{3}, *I*
_{4}, then give some errors to the four intensities *I*
_{1}, *I*
_{2}, *I*
_{3}, *I*
_{4} to calculate its error. Finally, we will give out the curve of velocity *v*, temperature *T* and pressure *p* using the four intensities *I*
_{1}, *I*
_{2}, *I*
_{3}, *I*
_{4} with errors.

If we take the metastable O(1S) as the observation source, whose wavelength is 557.7nm, so the wavenumber *σ*
_{0}=1/*λ*=17930cm^{-1}. Like in WINDII [8, 9], we set the path difference Δ=4.2cm. And *γ _{L}*(

*T*

_{0},

*p*

_{0}) is often in the range of 0~0.1cm

^{-1}, so we choose

*γ*(

_{L}*T*

_{0},

*p*

_{0})=0.05cm

^{-1}as a compromise. If we assume velocity

*v*=100m/s,

*φ*

_{0}=10 rad, and temperature

*T*=200K, then pressure

*p*=74235Pa. This pressure is effective in lower atmosphere. Now the altitude is lower, not only the upper atmospheric wind, such as The Stratospheric Wind Interferometer for Transport Studies (SWIFT) is a satellite instrument designed to measure winds in the stratosphere.

Bringing these parameter into Eq. (9) to Eq. (12), we can obtain

*ϕ*=10.1577 *V*=0.32

Through calculation, we can obtain the following four intensities

*I*
_{1}=0.9995, *I*
_{2}=1.0004, *I*
_{3}=1.0005, *I*
_{4}=0.9996

In actual measurement, the path difference is added by λ/4, so some errors must be introduced to the measured values because of all kinds of random reasons. Now we define the measured values in which errors have been introduced are *I*
_{1}′, *I*
_{2}′, *I*
_{3}′, *I*
_{4}′, and we substitute them into Eq. (14), Eq. (15) and Eq. (16), so we have

So we can obtain velocity, temperature and pressure of the wind are

Now we consider the errors between *I*
_{1}′, *I*
_{2}′, *I*
_{3}′, *I*
_{4}′ and *I _{1}*,

*I*,

_{2}*I*,

_{3}*I*are in the range of 0 ~ 1%. For actual measurement, we can divide them into 20 error regions, which is 0 ~ 0.05%, 0.05% ~ 0.10%, 0.10% ~ 0.15%,…, and we take 200 random error values in each error region, and bring them into Eq. (18) to Eq. (23), so we can obtain 200

_{4}*v*,

_{i}*T*and

_{i}*p*(

_{i}*i*=1,2,…, 100) in every error region. In every error region, we define the relative error

In which,
$\overline{{T}_{i}}=\frac{\sum _{i=1}^{200}{T}_{i}}{200}$
,
$\overline{{v}_{i}}=\frac{\sum _{i=1}^{200}{v}_{i}}{200}$
,
$\overline{{p}_{i}}=\frac{\sum _{i=1}^{200}{p}_{i}}{200}$
and
$\overline{{P}_{i}}=\frac{\sum _{i=1}^{200}{P}_{i}}{200}$
, where *P _{i}* is the random errors in each error region. The relative error of temperature

*T*, velocity

*v*and pressure

*P*to the random relative errors calculated from Eq. (23) to Eq. (26) are shown in Fig. 1, Fig. 2, and Fig. 3.

From these figures we can see that the relative measurement error of temperature and the relative measurement error of pressure is equal approximately, which is because of the linear relation in Eq. (13), and the relative measurement error of velocity is a little larger than the relative measurement error of temperature, which not only explain why we can think the emission line as the Gaussian profile, but also testify that the Voigt profile is indeed the correlation of the Gaussian profile and the Lorentzian profile, because

(1) Collisions corresponding to temperature form the Lorentzian profile, while the frequencies excursion corresponding to velocity.

(2) Through calculation, we have obtained that if we think the emission line as the Gaussian profile, the relative measurement error of velocity is about 4 times larger than the relative measurement error of temperature, while the emission line as the Lorentzian profile, the relative measurement error of temperature is about 2 times larger than the Gaussian profile.

## 4. Conclusion

(1) For the first time, based on the Fourier transform of the pure Voigt profile, we presented the principle and method of the upper atmospheric wind field measurement based on the Voigt profile.

(2) Through detailed discussion, we analyzed the measurement principle of velocity *v*, temperature *T* and pressure *p* of the upper atmospheric wind based on the Voigt profile, and gave out their calculation formulas. Thus we expand the atomic emission lines to the most to actual profile-Voigt profile, which will boost the development of upper atmospheric wind detection principle.

(3) From Fig. 1, Fig. 2 and Fig. 3, we can see that the relative measurement error of temperature *T*, pressure *p* and velocity *v* are about as the same, which prove the Voigt profile is the correlation of the Gaussian profile and the Lorentzian profile. So the Voigt profile is the most close to actual profile, and the calculation of the Voigt profile will have very important theoretic and practical significance in the detection of upper atmospheric wind field, the launch and movement of aircraft, the research of mankind living space and outer space.

## Acknowledgment

The authors gratefully acknowledge the support of Chinese National Natural Science Foundation. This work is supported by the State key Program of National Natural Science of China (Grant No. 40537031), the National Natural Science Foundation of China (Grant No. 40375010,60278019), the Science and Technology Plan Foundation of Shaanxi Province under Contract Nos. 2005K04-G18, and the Electronic Information Platform of “985” project 2^{nd} term of Xi’an Jiaotong University.

Finally, we are thankful to Dr. Mingjun Zhao for his help with this paper.

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