Determination of sprite streamers altitude based on N₂ spectroscopic analysis

1 Introduction

Sprites are large optical phenomena that last a few milliseconds and that are produced typically by positive cloud-to-ground (+CG) lightning between 40 and 90 km altitude [e.g., Franz et al., 1990; Winckler et al., 1993; Pasko, 2007; Chen et al., 2008; Stenbaek-Nielsen et al., 2013; Pasko et al., 2013, and references therein]. Sprites belong to the wider family of transient luminous events (TLEs) [e.g., Pasko et al., 2012]. Studies have showed that sprites are composed of filamentary plasma structures called streamer discharges [e.g., Pasko et al., 1998; Gerken et al., 2000; Stanley et al., 1999]. Some sprites can be highly complex and composed of many streamers [e.g., Stanley et al., 1999; Gerken et al., 2000; Stenbaek-Nielsen et al., 2000], while some are composed of only a few filaments [e.g., Wescott et al., 1998, Wescott et al., 1998; Adachi et al., 2004]. The different sprite morphologies are understood to be due to different upper atmospheric ambient conditions and the characteristics of the causative lightning discharge [e.g., Qin et al., 2013a, 2013b, 2014, and references therein].

One of the ways used to explore the physical properties of sprites is the spectroscopic diagnostic of their optical emissions, specifically in the following bands systems of N₂: the Lyman-Birge-Hopfield (LBH) [e.g., Liu and Pasko, 2005; Liu et al., 2006; N. Liu et al., 2009; Gordillo-Vázquez et al., 2011], the first positive 1PN₂ [e.g., Mende et al., 1995; Hampton et al., 1996; Green et al., 1996; Morrill et al., 1998; Milikh et al., 1998; Bucsela et al., 2003; Kanmae et al., 2007; Siefring et al., 2010; Gordillo-Vázquez, 2010; Gordillo-Vázquez et al., 2011, 2012], the second positive 2PN₂ [e.g., Armstrong et al., 1998; Morrill et al., 1998; Milikh et al., 1998; Suszcynsky et al., 1998; Heavner et al., 2010; Gordillo-Vázquez, 2010; Gordillo-Vázquez et al., 2011, 2012], and the first negative bands systems of ( ) [e.g., Armstrong et al., 1998; Suszcynsky et al., 1998; Kanmae et al., 2010a]. Several works have been realized to determine the electric fields involved in sprite streamers based on their produced optical emissions [e.g., Morrill et al., 2002; Kuo et al., 2005; Adachi et al., 2006; Kanmae et al., 2010b], and some have showed an acceptable agreement with simulations [Liu et al., 2006]. However, theoretical studies have also showed the existence of correction factors to be taken into account for the determination of an accurate value of the peak electric field in streamer heads. The correction factors are due to both a spatial shift between the maximum in the electric field at the head of the streamer and the maximum in the production of excited species and the fact that most photons are produced some distance away from the filament symmetry axis [Celestin and Pasko, 2010; Bonaventura et al., 2011].

The experiment LSO (Lightning and Sprite Observations) developed by the French Atomic Energy Commission (CEA) with the participation of the French Space Agency (CNES) [Blanc et al., 2004], the Japanese Aerospace Exploration Agency (JAXA) mission GLIMS (Global Lightning and sprIte MeasurementS) [Sato et al., 2015], and the future European Space Agency (ESA) mission ASIM (Atmosphere-Space Interactions Monitor) [Neubert, 2009] are dedicated to the observation of TLEs from the International Space Station (ISS). The Lomonosov Moscow State University satellite Universitetsky-Tatiana-2 observed TLEs from a Sun-synchronous orbit at 820–850 km [Garipov et al., 2013]. The future satellite mission TARANIS (Tool for the Analysis of RAdiation from lightNIng and Sprites), funded by CNES, will observe TLEs from a Sun-synchronous orbit at an altitude of ∼700 km [Lefeuvre et al., 2008]. All the above mentioned space missions have adopted strategies based on nadir observation of TLEs. Observation from a nadir-viewing geometry is indeed especially interesting as it reduces the distance between the observation point and the event and hence minimizes atmospheric absorption and maximizes the chance of observing TLEs and their associated phenomena, such as electromagnetic radiation or possible high-energy emissions. However, in this observation geometry, the vertical dimension is poorly resolved, and so are the speeds of sprite substructures.

In this work, we investigate a spectrophotometric method to trace back the altitude of streamers in sprites using optical emissions that will be detected by ASIM [Neubert, 2009] and TARANIS [Lefeuvre et al., 2008] and that were detected by GLIMS [Sato et al., 2015]. As mentioned above, the electric field in sprite streamer heads can be estimated through spectroscopic analysis of ratios of bands systems intensities produced by molecular nitrogen-excited electronic states. These excited states can deexcite either radiatively, hence giving rise to bands systems, or through collisions with other molecules in a process named quenching. The probability of the latter deexcitation channel depends on the local atmospheric density and therefore on the altitude. Using the fact that the excited species N₂(a¹Π_g) (responsible for LBH) and N₂(B³Π_g) (responsible for 1PN₂) quench under relatively low pressure, and therefore at high altitudes, while N₂(C³Π_u) (responsible for 2PN₂) and (responsible for ) are quenched below relatively high pressure and therefore low altitudes, we show that, combining observations with streamer modeling results, it is possible to obtain information about the production altitude of the optical emission through ratios of bands systems.

In section 2, we present the sprite streamer and spectroscopic models used in the present paper, in section 6 we show our results on electric field and altitude determination of sprite streamers, and we discuss the implication of our results in section 9.

2 Model Formulation

2.2 Optical Emissions Model

Along with the streamer propagation, we quantify the densities of excited species N₂(a¹Π_g), N₂(B³Π_g), N₂(C³Π_u), and associated with optical emissions of the Lyman-Birge-Hopfield bands system of N₂ (LBH) , the first positive bands systems of N₂ (1PN₂) , the second positive bands system of N₂ (2PN₂) , and the first negative bands systems of , respectively. As reported in the study by Liu and Pasko [2005], we consider that N₂(a¹Π_g) is quenched by N₂ and O₂ with rate coefficients α₁=10⁻¹¹ cm³/s and α₂=10⁻¹⁰ cm³/s, respectively. As used by Xu et al. [2015], the quenching of N₂(B³Π_g) and N₂(C³Π_u) is considered to occur through collisions with N₂ and O₂ with rate coefficients α₁=10⁻¹¹ cm³/s [Kossyi et al., 1992] and α₂=3 × 10⁻¹⁰ cm³/s [Vallance Jones, 1974, p. 119], respectively. is quenched by N₂ with a rate coefficient α₁=4.53 × 10⁻¹⁰ cm³/s and by O₂ with a rate coefficient α₂=7.36 × 10⁻¹⁰ cm³/s [e.g., Mitchell, 1970; Pancheshnyi et al., 1998; Kuo et al., 2005].

The density of excited species is estimated according to the following differential equation [e.g., Liu and Pasko, 2004]:

(1)

where and A_k are the characteristic lifetime and Einstein coefficient of the excited species k, respectively. The corresponding A_k [e.g., Liu, 2006] and quenching coefficients taken into account for N₂(a¹Π_g), N₂(B³Π_g), N₂(C³Π_u), and are shown in Table 1. One considers a simple atmospheric composition of 80% of nitrogen and 20% oxygen: and . The quantities n_k and ν_k are, respectively, the density and the excitation frequency of the excited species of the state k. As the streamer model is based on the local electric field approximation, the excitation frequency ν_k depends on the local electric field or equivalently on the electron energy. The sum over n_mA_m takes into account the increase in n_k resulting from the cascading of excited species from higher energy levels m. In this work, we only take into account the cascading from N₂(C³Π_u) to N₂(B³Π_g).

Table 1. Einstein Coefficient A_k (s⁻¹), Quenching Coefficients α_1,2 (cm³/s), Lifetime τ_k (s) at Ground Level Air of Different Excited States of N₂ Molecule, and Quenching Altitudes h_Q (km)^a

Ak	1.8 × 104	1.7 × 105	2 × 107	1.4 × 107 s−1
α1	10−11	10−11	10−11	4.53 × 10−10
α2	10−10	3 × 10−10	3 × 10−10	7.36 × 10−10
τk	1.33 × 10−9	5.47 × 10−10	5.41 × 10−10	7.29 × 10−11
hQ	77	67	31	48

• ^a See section 3 for citations.

Moreover, the associated optical emissions are evaluated in terms of photon flux according to the following integral along the line of sight [e.g., Liu and Pasko, 2004]:

(2)

where I_k and n_k are the intensity of optical emissions in rayleighs (R) (number of photons per unit surface area and per unit time (cm⁻²s⁻¹)) and the number density of excited species k, respectively. In this paper, we use a simplified chemistry model taking into account reactions relevant over short timescales: ionization by electron impact, dissociative attachment, photoionization, and the excited species produced by a streamer [Ihaddadene and Celestin, 2015].

2.3 Estimation of the Streamer Peak Electric Field Using Optical Emissions

The study of the N₂ and optical emissions produced by sprite streamers is useful to estimate the peak electric field in streamer heads, because the energy of the electrons depends on the amplitude of this field and the excited species responsible for the production of different bands systems are produced through collisions between electrons and N₂ molecules in the ground state and correspond to different energy thresholds. In this subsection, we describe how we proceed to infer the peak electric field.

We simulate downward propagating positive streamers in uniform electric fields and  kV/cm at given altitudes h = 50, 60, 70, 80, and 90 km. Using equations 1 and 2, we quantify the excited species and the associated optical emissions (see Figure 1). The whole volume of the streamer emits photons, mainly in the head region [e.g., Bonaventura et al., 2011], and hence, we integrate each band system photon flux over the whole body of the streamer including the head as , where ds = Δz× Δr is an elementary surface. We then calculate the associated ratios .

Assuming that the steady state is reached (the production and loss rates of excited species are equal) under a given electric field and using equation 1 as described in Celestin and Pasko [2010], one obtains the following photon flux ratio, which is a function of the electric field through ν_k and :

(3)

where we neglected the cascading from higher states. In the case of N₂(B³Π_g), one needs to take into account the cascading from N₂(C³Π_u) to N₂(B³Π_g) and following the same procedure, one finds

(4)

As mentioned in Celestin and Pasko [2010], the steady state of excited species is not a necessary condition for equations 3 and 4 to be applicable, even though equations 3 and 4 have been derived assuming steady state. In fact, it can be shown formally from equation 1 that if the streamer propagation is sufficiently stable over a timescale on the order of τ_k, equations 3 and 4 also apply in the case of nonsteady state. Indeed, defining , equation 1 leads to

(5)

Assuming that the streamer is sufficiently stable, i.e., its radius is approximately constant over a timescale τ_k, that is , one can neglect the left-hand side of equation 5, and therefore,

(6)

which leads to equations 3 and 4 for a given homogeneous electric field. However, although the homogeneous electric field assumption for steady state optical emissions is justified by the fact that the emission is confined in the streamer head (within a spatial shift mentioned in section 1), one might wonder whether this assumption would still be valid in the case of nonsteady state emission that trails behind the streamer head. Equation 6 can be rewritten

(7)

where is an effective quantity defined by . Since the excitation frequency strongly depends on the electric field, one can consider that ν_k=ν_k(E_h) and one notes . Neglecting the cascading term in equation 7, one gets

(8)

and thus, the ratio obtained in equation 3 if one assumes , i.e., considering that the excitation taking place in the streamer head dominates over the excitation from other regions. It can be easily shown that equation 7 also results in equation 4 if the cascading effect is not neglected. In conclusion, the steady/nonsteady nature of optical emission does not affect the validity of the ratio found in equations 3 and 4 if the streamer can be considered as stable over a timescale τ_k and if most of the excitation is produced in the head. This point is clearly demonstrated by the simulation results of Bonaventura et al. [2011] for 2PN₂ and .

Using equations 3 and 4, one can estimate the peak electric field E_e for every simulation-based ratio found if steady state is reached for excited species k and . From the estimated field E_e and the peak field in the simulation E_h, a correction factor due mostly to the spatial shift between maxima of optical emissions and the peak electric field is calculated as [Celestin and Pasko, 2010]. The correction factors calculated in the present work are shown in Tables 2 and 3.

Table 2. Correction Factors Calculated at Different Altitudes Under  kV/cm Using Equations 3 and 4

	Altitude (km)
	50	60	70	80	90
	1.57	1.60	1.68	1.62	1.64
	2.06	2.09	2.05	2.02	1.79
	1.40	1.41	1.40	1.41	1.41
	1.36	1.36	1.38	1.38	1.41

Table 3. Correction Factors Calculated at Different Altitudes Under  kV/cm Using Equations 3 and 4

	Altitude (km)
	50	60	70	80	90
	1.49	1.65	2.03	2.24	2.39
	5.86	6.41	6.45	4.81	2.73
	1.61	1.61	1.60	1.61	1.62
	1.34	1.35	1.40	1.49	1.62

However, in general, a sprite streamer can be considered as expanding exponentially in time [e.g., N. Y. Liu et al., 2009]. The rate of expansion ν_e is a strong function of the ambient electric field [Kosar et al., 2012]. In fact, equation 5 can be rewritten in the form

(9)

As we mentioned just above, one considers that N_k=N_k,0 , and equation 9 leads to

(10)

if one neglects the cascading effect, and otherwise,

(11)

Hence, for significantly quick streamer expansion ( ), without taking into account the cascading effect, one obtains

(12)

and taking into account the cascading effect

(13)

For all the cases used in the present work, we have verified that the population of N₂(C³Π_u) is in steady state and . The excitation frequencies ν_k and their dependence on the electric field are computed based on Moss et al. [2006]. Using equations 12 and 13 and assuming , i.e, considering that the excitation taking place in the streamer head dominates over the excitation from other regions, one can estimate the peak electric field E_e for every simulation-based ratio found as described above even in the case of nonsteady state of excited species accompanied by rapid expansion of the streamer ( ). Precisely, because in reality the ratios , correction factors need to be quantified using modeling results and taken into account in photometric-based observational studies to correct the estimated value of the peak electric field. The expansion frequency ν_e and the various correction factors calculated in the present work for different altitudes and under different uniform electric fields and  kV/cm are shown in Tables 4-6.

Table 4. The Expansion Frequency ν_e (s⁻¹) Calculated at Different Altitudes

	Altitude (km)
	50	60	70	80	90
kV/cm	1.2 × 105	3.5 × 104	1.0 × 104	2.3 × 103	3.5 × 102
kV/cm	3.96 × 105	1.2 × 105	3.4 × 104	7.75 × 103	1.2 × 103

Table 5. Correction Factors Calculated at Different Altitudes Under  kV/cm Using Equations 12 and 13

	Altitude (km)
	50	60	70	80	90
	1.88	1.84	1.81	1.66	1.66
	1.39	1.39	1.39	1.58	1.69
	1.48	1.47	1.47	1.43	1.42

Table 6. Correction Factors Calculated at Different Altitudes Under  kV/cm Using Equations 12 and 13

	Altitude (km)
	50	60	70	80	90
	2.65	2.63	2.61	2.43	2.44
	1.95	1.93	1.94	2.03	2.21
	1.72	1.71	1.71	1.67	1.67

3 Results

3.1 Streamer Modeling

We conducted simulations at altitudes h = 50, 60, 70, 80, and 90 km under E₀=12 and  kV/cm, which represents 10 simulations in total. All the results produced will be described in detail in subsection 7.

As an example, we show the results for a positive downward propagating sprite streamer in uniform electric field  kV/cm, initiated at 70 km altitude in Figures 1 and 2. Figures 1a and 1b show the cross-sectional views of the electron density and the electric field. Cross-sectional views of photon fluxes from LBH, 1PN₂, 2PN₂, and bands systems at time t = 0.27 ms are shown in Figures 1c–1f. The quenching altitude is defined so that above this altitude, the radiative deexcitation of given N₂ or excited state k dominates the collisional one. Based on the quenching coefficients that we have applied to quantify the densities of N₂ and excited species and their associated bands systems, we have deduced the quenching altitudes shown in Table 1.

Figures 2a and 2b show the electron density and electric field profiles along the axis of the streamer every 0.054 ms. Figure 2c shows the optical emission from bands systems profiles LBH, 1PN₂, 2PN₂, and , along the axis of the streamer at t = 0.27 ms in terms of photon flux.

3.2 Estimation of the Altitude of the Sprite Streamers Using Optical Emissions

We first define an array of electric field ranging from 0 to 600   kV/cm representing actual peak electric fields in the streamer head, and then we compute the ratio associated with each value of the electric field . Figures 3a and 3b show two parametric representations of selected optical emission ratios through the implicit parameter E_e. The upper and lower curves that delimit shaded areas in Figure 3 correspond to background electric fields and 28   kV/cm, respectively at given altitudes. Between two shaded areas the altitude is h₁< h <h₂ where h₂−h₁=10 km. For the sake of illustration, we show how the results of our simulations are located in this parametric representation (Figure 3). Red and yellow marks correspond to cases of ambient electric field amplitudes and 28   kV/cm, respectively. Since the correction factors are obtained from the same simulations, one sees that the obtained intensity ratios fall exactly on the estimated lines. One also sees that a descending streamer would take a specific path in the parametric representation illustrated in Figures 3a and 3b. This particular behavior can be used to infer physical properties of sprite streamers from photometric observations such as the electric field and mean velocity.

The mark X located by coordinates ( , ) within a shaded area illustrates a situation where the peak electric field E_h would be such that EhE₀=12kV/cm < E_h<EhE₀=28kV/cm, where and are shown in Table 7.

Table 7. Electric Field at the Head of the Positive Streamer E_h (V/m) at Different Altitudes Under Different Ambient Electric Fields E₀

	Altitude (km)
	50	60	70	80	90
kV/cm	8735	2597	745.9	165.9	26.4
kV/cm	1.1 × 104	3269	940.5	209.1	33.21

4 Discussion

For a given ambient electric field E₀, one sees in Figure 3 that curves corresponding to different altitudes are not overlapped. This is due to the different amounts of quenching that excited states are subjected to at different altitudes. Indeed, the excited states N₂(a¹Π_g) and N₂(B³Π_g), which are responsible for LBH and 1PN₂ bands systems, respectively, have quenching altitudes of 77 km and 67 km, while N₂(C³Π_u) and , which are responsible for the 2PN₂ and bands systems, respectively, can be considered as not strongly affected by quenching over the altitude range covered by sprites (see Table 1). This discrimination in altitude, which exists over a large range of electric fields in the streamer head for the selected ratios in Figure 3, is of first interest to determine the altitude of sprite streamers at various moments of time from photometric measurements. It especially applies to satellite observations in a nadir-viewing geometry. It is important to note that the quenching coefficients for N₂(a¹Π_g) are not well known [e.g., Liu and Pasko, 2005]. However, one also notes that N. Liu et al.[2009] obtained a satisfying agreement with observational results from the instrument ISUAL (Imager of Sprites and Upper Atmospheric Lightning) on the FORMOSAT-2 Taiwanese satellite using the quenching coefficients reported in Table 1 concerning N₂(a¹Π_g).

At a given location, the electric field at the streamer head varies within a timescale (see Figure 2), where V_str is the streamer velocity, to be compared to the characteristic lifetime τ_k of the excited species. From the simulations, δt is estimated to be ∼5.7 and 2.5 μs at 70 km altitude under 12  and 28   kV/cm, respectively, while τ_k of the excited species N₂(a¹Π_g), N₂(B³Π_g), N₂(C³Π_u), and are estimated to be ∼14.5, 3.4, 0.049, and 0.067 μs, respectively. Therefore, one sees that the populations of N₂(a¹Π_g) and N₂(B³Π_g) are not in steady state (τ_k>δt), although N₂(C³Π_u) and are in steady state (τ_k<δt).

Moreover, the lifetime τ_k of a given bands system does not change significantly with altitude above its corresponding quenching altitude as it is mostly defined by its Einstein coefficient A_k. Below the quenching altitude, τ_k is mostly controlled by quenching and scales as 1/N. However, the characteristic timescale δt of electric field variation in the streamer head scales as 1/N for all altitudes. As discussed above, the comparison between δt and τ_k determines whether the population of an excited state giving rise to a bands system is in steady state [see Celestin and Pasko, 2010, section 3]. Since τ_k is constant above the quenching altitude, there is an altitude above which δt > τ_k, and hence, steady state is reached. For example, although N₂(a¹Π_g) is not in steady state over most of the altitude range covered by sprites streamers (40–80 km), it can be considered to be in steady state at an altitude of 90 km. However, since δt scales as τ_k below the quenching altitude, the steady/nonsteady state nature of an excited species is locked below this altitude.

It is usually considered that LBH cannot be observed from the ground due to absorption in the atmosphere. However, ground observations have access to 2PN₂, 1PN₂, and bands systems (see references in the section 1). A similar parametric representation as in Figure 3 is shown in Figure 4 with these bands systems. One sees that the altitude discrimination given by parametric representation is valid only for altitudes ranging between 50 and ∼70 km because of the overlap of different altitude curves that occurs above 70 km. This is an illustration of the suppression of quenching (specifically on 1PN₂), upon which the method presented in this paper is based. As the ratio mostly depends on the electric field in the streamer head [e.g., Celestin and Pasko, 2010] and is weakly dependent on this field, the parametric representation presented in Figure 4 is well defined to measure altitude. It is interesting to note that Garipov et al. [2013] have used the ratio to make an estimate on the altitude of events observed by the Tatiana-2 satellite. The method we propose here is expected to be much more accurate because it is based on simulations of streamers and we take into account the corrected streamer electric field.

It is also interesting to note that the assumption is not necessary. In fact, one could keep this quantity in the functional dependence of the optical emission ratios (equations 12 and 13). In this case, the correction factors become close to 1. The development of the corresponding field measurements method and its accuracy with respect to that use in the present paper is beyond the focus of the present work. However, for the sake of completeness, we have tabulated the ratios , for the cases studied in this paper in Tables 8 and 9. As shown in Tables 8 and 9, the ratio varies between 0.57 and 2.76 in the cases studied in this paper. As explained in section 4, correction factors are introduced to compensate the error on the estimated peak field involved by the assumption that .

Table 8. Estimated Ratio Under  kV/cm, at Different Altitudes

	Altitude (km)
Ratio	50	60	70	80	90
	0.70	0.71	0.70	0.71	0.70
	0.72	0.72	0.72	0.72	0.72
	2.06	2.05	2.05	2.04	2.06
	2.10	2.10	2.10	2.09	2.11

Table 9. Estimated Ratio Under  kV/cm, at Different Altitudes

	Altitude (km)
Ratio	50	60	70	80	90
	0.58	0.57	0.57	0.57	0.57
	0.59	0.59	0.59	0.59	0.59
	2.72	2.65	2.66	2.63	2.66
	2.76	2.75	2.75	2.72	2.76

Figures 3a and 3b show a gap between the curves corresponding to given altitudes under and 28   kV/cm, which is larger at 50 km than at 90 km altitude. The gap is caused by the difference between the correction factors calculated under and 28   kV/cm and the significance of the product ν_eτ_k compared to unity (see equations 12 and 13) under either one of these ambient fields. The curves tend to overlap at higher altitudes because the correction factors in both cases are getting closer. Under  kV/cm the relative contribution of the optical emissions coming from the streamer channel to the total emission is less than that coming from the streamer channel propagating under  kV/cm. The reason is twofold: on the one hand, the electric field in the streamer channel is relatively more intense in the  kV/cm case which affects the correction factors, and on the other hand, the LBH and 1PN₂ bands systems are not in a steady state below ∼77 and 67 km, respectively. The latter effect plays a role in increasing the emission in the streamer channel more significantly under  kV/cm than under  kV/cm. Indeed, the emission in the channel is a contribution of both the streamer head that moves rapidly under  kV/cm and the streamer channel itself. In contrast, when the steady state is reached, for example, for the 2PN₂ and bands systems, the emission profile in the streamer only depends on the local electric field and the electron density at the given time (see Figure 2c).

Figures 3 and 4 of the present study are established based on equations 12 and 13. These equations are valid for both steady and nonsteady states, and they take into account the exponential expansion of the streamer. Considering the exponential expansion of streamers with a characteristic timescale , one can see that if τ_k≪τ_e, equations 12 and 13 tend to equations 3 and 4 obtained assuming that steady state is reached. This condition is fulfilled only in case of streamers propagating in weak electric field  kV/cm (high τ_e). The exponential expansion of streamers particularly needs to be taken into account at altitudes lower than 80 km and under high background electric fields for the case of ratios composed of LBH and 1PN₂ bands systems. However, the steady state assumption remains valid for the ratio . The quantities ν_e and τ_e define the characteristic frequency and characteristic time of the streamer expansion, respectively. Within the time τ_e one can consider that the streamer moves within a distance proportional to the streamer radius βr_s and thus,

(14)

See Figure 2 for an illustration of the characteristic length [e.g., D'yakonov and Kachorovskii, 1989]. is the streamer velocity, and [e.g., Kulikovsky, 1997; Babaeva and Naidis, 1997], where is the electron density in the streamer channel and is the electron density taken at a distance r_s from the position of the peak electric field. In the present study, we have α ∼ 13. The quantity ν_h is the ionization frequency in the streamer head. We estimate β using ν_e obtained in the simulations and equation 14 and found it to be between ∼1.4 and 2.4 under and 28   kV/cm, respectively. Based on the simulation and equations 10 and 11, the exponential expansion of the number of excited molecules N_k is caused by the exponential expansion of , which is related to the exponential expansion of the radius r_s of the streamer and hence the volume of the streamer head region.

Moreover, the integration of the optical emissions chosen in this work does not take into account the nonphysical contribution to the optical emissions produced in the region near the sphere electrode used in our simulations, where the electric field is strong enough (∼3 E_k) to generate excited species. However, we have included the emission from the streamer channel since it is considered to be physical [Liu, 2010].

We expect that the proposed method is particularly applicable in case of columniform sprite events that consist of only a few descending streamers. The altitude of positive streamers at the beginning of the developments of carrot sprites could be obtained as well. However, it is predicted that the complexity introduced by the many ascending negative streamers will prevent obtaining clear results at later moments of the carrot sprite development.

It is expected that the various optical emissions involved in the presented method will not be significantly modified by the transmission through the atmosphere. Preliminary estimates show that emissions between 200 and 240 nm produced at 50 km and observed in a nadir-viewing geometry would be reduced by only ∼10% (T. Farges, personal communication, 2016). In fact, as the signal is detected by photometers on board the satellite at a known location, the effect of atmospheric transmission can be accounted for in a given geometry for the proposed method to be applicable. For an estimation of the altitude within 10 km using the approach developed in the present paper, the maximum uncertainties that are acceptable on different observed ratios have been estimated approximately under  kV/cm and are indicated in Table 10. We note that more precise models of populations of excited species [e.g., see Eastes, 2000], along with accurate quenching coefficients, may need to be implemented to improve the accuracy of the parametric representations shown in Figures 3 and 4, and the method should be first calibrated using joint campaigns associating ground-based (which can resolve the streamers altitudes) and satellite measurements.

Table 10. Estimated Maximum Uncertainties (%) on Different Ratios to Discriminate Between Different Altitudes Within 10 km

	Altitude (km)
	50–60	60–70	70–80	80–90
	46	36.82	21.21	7.87
	5.69	15.94	26.46	19
	15.72	6.27	2.55	0.635
	37.72	44.96	43.12	27.14

Because of the restrictions imposed by the model, the method developed in the present paper is based on separate local streamer simulations conducted at different altitudes under similar conditions [e.g., Pasko, 2006; Qin and Pasko, 2015] and reasonable values of the ambient electric field needed for the propagation of sprite streamers and 0.9 [e.g., Hu et al., 2007; Li et al., 2008; N. Y. Liu et al., 2009; Qin et al., 2013b]. It will be very interesting to push the simulation beyond and compare with simulations of streamers initiated under more realistic conditions of ambient electric field, charges species, and ionospheric inhomogeneities [e.g., Liu et al., 2015, 2016] and to study the application of the method introduced in the present paper.

Finally, we note that the method might also be used for other streamer-based TLEs like upward propagating gigantic jets [e.g., Kuo et al., 2009].

5 Summary and Conclusions