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Abstract
This paper proposes a modification of the Hufnagel-Andrews-Phillips (HAP) Refractive Index Structure Parameter model that will better characterize the HAP profile against experimental data using the turbulent intensity, which is the ratio of wind speed variance to the average wind speed-squared, and Korean Refractive Index Parameter yearly statistics, Comparisons between this modified HAP model, the Critical Laser Enhancing Atmospheric Research 1 (CLEAR 1) profile model and several of the data sets are made. These comparisons highlight that this new model offers a more consistent representation of the averaged experimental data profiles than the CLEAR 1 model did. In addition, comparisons between this model and various experimental data set reported in the literature will show good agreement between the model and averaged data, and reasonable agreement with non-averaged data sets. This improved model should prove useful in system link budget estimates and atmospheric research.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Free space optical communications (FSOC) has attracted new interests in the recent decades as high data rate airborne- and satellite-based backhaul capabilities for large capacity communication networks [1–3]. Although RF systems have dominated the communications backhaul market, decreasing RF spectrum availability and network security makes FSOC systems more attractive. Unfortunately, optical signals also are subject to various undesirable clear-sky weather phenomena, including among others, turbulence-induced signal fading and beam wander-induced pointing error and scintillation [4]. Fortunately, turbulence mitigation under high turbulence conditions have been successfully achieved using new system and network technologies [5,6]. The key parameter for characterizing channel effect is the Refractive Index Structure Parameter $C_n^2$. Two of the most popular atmospheric $C_n^2$ models are the Hufnagel-Valley (HV) or Hufnagel-Andrews-Phillips (HAP) models. Both provide estimates of that parameter as function of altitude using knowledge of a ground-referenced $C_n^2$ value and a multiplier $M$ quantifying the magnitude of the upper atmosphere turbulence strength. Both have the same upper atmosphere $M = 1$ profiles, but different in the $C_n^2$ falloff in the lower atmosphere. The former falls off exponentially while the latter falls off in a power-law fashion [4]. However, both have had success in predicting experimental results within a couple of decibels using these models have been reported [5–7]. On the other hand, these models can always be improved.
This paper proposes a modification of the Hufnagel-Andrews-Phillips (HAP) Refractive Index Structure Parameter model that will better characterize the HAP $C_n^2$ profile against experimental data. Comparisons between this model and various experimental data set reported in the literature will show good agreement between the model and both averaged and non-averaged data sets. Comparisons between this model, the Critical Laser Enhancing Atmospheric Research 1 (CLEAR 1) profile model [8] and several of the data sets are also shown. This set of comparisons will illustrate that this new model more often offers better representation of the averaged experimental data profiles over the CLEAR 1 model. In addition, the paper looks at using the measured wind speed altitude profile in the modified HAP model for improved data profile comparisons.
2. Atmospheric refractive index structure parameter modeling
Several $C_n^2$ profile models, including both day and night models, are used by the technical community for ground-to-space or space-to ground applications [5–7]. Many of these models are discussed in the article by Beland [9]. One of the most widely used models for such applications is the Hufnagel-Valley (HV) model, which is given by
The problem with MxHV-5/7 is the last exponential term in Eq. (1) that describes near-ground turbulence conditions predicts a slow decrease in $C_n^2$ with altitude up to around 1 km as compared with the h-4/3 behavior observed by Walters and Kunkel [14] and supported by a few other early measurements. Andrews and Phillips suggested modifying the Hufnagel model to include the h-4/3 behavior near the ground. This model, called the Hufnagel-Andrews-Phillips (HAP) model, is defined as
3. Modified HAP model
Mahmood, et al., among others, have suggested that $C_n^2(h )$ comparisons with experimental data can be improved by using their associated wind speed profiles with altitude [15] and discussed the implications of turbulent intensity on this parameter. The turbulent intensity is given by
where ${\sigma _{ws}}$ is the standard deviation of the wind speed and $\bar{w}$ is the mean of the wind speed. From [10], the value of $\bar{w}$ is 27 m/s, the mean value used in the HV and HAP models, and ${\sigma _{ws}}$ is 9 m/s from measurements made in Maryland. Using the annual Korean $C_n^2(h )$ Statistics shown in the Fig. 7.24 in [16, p. 280] to improve the upper atmosphere profile, we can modify Eq. (2) to yield a model that reflects this statistical data better than the HV and unmodified HAP model. Specifically, we now write the HAP model asFigure 1 shows a comparison between Eq. (4), evaluated at specific M and $C_n^2({{h_0}} )$ values, and annual Korean $C_n^2(h )$ profiles at the 1%, 15%, 50%, 85% and 99% cumulative percentiles. The latter data comes from an Air Force Research Laboratory’s data base of 85 radiosonde balloon flights over South Korea [17]. This graph shows good agreement between the theory and data.
Fig. 1. Comparison between Eq. (4) evaluated at specific M and $C_n^2({{h_0}} )$ values and annual Korean $C_n^2(h )$ Profiles at the 1%, 15%, 50%, 85% and 99% Cumulative Percentiles.
Figure 2 depicts Eq. (4) against the Mean of the $C_n^2(h )$ measurements obtained at the Teide Observatory (OT) during 2004 (dotted - shorted line) and at the Roque de los Muchachos Observatory (ORM) during 2004 (solid line) and 2005 (dashed line) [18]. The horizontal line indicates the observatory altitude (2400 m). This data was derived from the Scintillation Detection and Ranging procedure (SCIDAR) procedure [18]. This technique is based on the statistical analysis of scintillation patterns produced at a telescope pupil by the light from two stars of a binary system. The profiles of the refractive index structure parameter come from the inversion of the average normalized autocovariance of many scintillation patterns. Again, we see reasonable agreement between theory and data.
Fig. 2. Comparison between Eq (4) against the Mean of the Refractive Index Structure Parameter measurements obtained at OT during 2004 (dotted - shorted line) and at ORM during 2004 (solid line) and 2005 (dashed line). These statistical profiles correspond to the weighted average of the individual monthly statistical profiles. The number of individual profiles measured each month were used as weights [18]. [Data used with permission of Dr. Julio A. Castro-Almazán and Dr. Begoña García-Lorenzo.]
Although numerous measurements of $C_n^2(h )$ as a function of altitude have been made over the years, few of these were confined to the first several meters above ground. Figure 3 compares Eq. (4), evaluated a $M = 1.2$, ${h_0}\, = \,8{\kern 1pt} m,$ and $C_n^2(0 )= 1.0 \times {10^{ - 13}}\;{m^{ - {2 / 3}}},$ with $C_n^2(h )$ data measured at LaPalma [19]. Each data point represents an average $C_n^2$ measurement over a 2-year period from 2002-2004. The instrument used to obtain the data is an array of six scintillometers known as the shadow band ranger (SHABAR). The SHABAR measures the intensity of sunlight in six detectors and the cross-covariance of scintillation between 15 possible pairs of detectors arranged in a non-redundant array. The ground height (height of the Observatory) is 2400 m, the height of the instrument above ground is listed in the Report as 8 m, and altitude h in this figure represents height above the instrument [19]. Note that the modified HAP model compares very well with all the data up to 2500 m above the instrument.
Fig. 3. Comparison of Eq. (4) (solid line) and Measured values (open diamonds) of Cn2 above La Palma as a function of altitude up to 2.5 km [20]. The data was taken over a 2-year period and averaged.
Quatresooz, et al. used high-density profiles from the University of Wyoming (UWYO) Atmospheric Science Radiosonde Archive to develop and validate the following multi-point model:
Figure 4(a) illustrates their refractive index structure parameter versus altitude estimates as a function of the re-interpolation distance $dz$ (estimate’s vertical spacing) using data from an experiment conducted in Trappes, France, found in the UWYO database, and curves generated from the Hufnagle-Valley 5/7 (1xHV5/7 or HV5/7) model, their curve-fitted Eq. (5) [denoted as analytical model in the figure] and the modified HAP model for $M = 1.2$ and $C_n^2({{h_0}} )= 5.0 \times {10^{ - 15}}\;{m^{ - {2 / 3}}}.$ Clearly, their curve-fitted Eq. (5) fits the plotted experimental measure ments very well and the 1xHV57 models does not. However, the modified HAP does a pretty good job in following most of the averaged data well just using a two parameter fit. It does not follow the nominal 12-km bump very well, but it does follow the upper atmsphere profile much better than the HV5/7 model does.
Fig. 4. Plots of the refractive index structure parameter versus altitude estimates derived from data taken at (a) Trappes, France, and (b) Hilo, Hawaii, and curves generated from the 1xHV5/7 model, the curve-fitted Eq. (5) and Eq. (4). [Data used with permission of Mr. Florian Quatresooz and Professor Claude Oestges.]
Figure 4(b) depicts refractive index structure parameter versus altitude estimates as a function of the re-interpolation distance dz (estimate’s vertical spacing) using data from an experiment conducted at Hilo, Hawaii, also found in the UWYO database, curves generated from the Hufnagle-Valley 5/7 (1xHV5/7 or HV5/7) model, their curve-fitted Eq. (5) [denoted as Analytical Model in the figure] and the modified HAP model $M = 1.2$ and $C_n^2({{h_0}} )= 5 \times {10^{ - 15}}\;{m^{ - {2 / 3}}}.$ Clearly, their curve-fitted Eq. (5) to the data fits the plotted experimental estimates very well and the 1xHV57 models does not. Again, the modified HAP does a pretty good job in following most of the averaged data well just using a two parameter fit. It does not follow the nominal 12-km bump very well, but it does follow the upper atmsphere profile much better than the HV5/7 model does.
Figure 5 illustrates comparisons of Eq. (4) and (a) OT and (b) ORM experimental data. Figure 5(a) depicts a good fit between theory and data up to 20 km, with an over estimate above 20 km. In practice, the integration of $C_n^2(h )$ when its values is ${10^{ - 18}}\;{m^{ - {2 / 3}}}$ or less contribute very little to the total integration, so this is not a serious problem. Figure 5(b) shows that the modified HAP models is a very rough average of the $C_n^2$ profile shown. It is clear that either the modified HAP model will need more information and possibly a change in functionality, to provide better agreement.
Fig. 5. Comparisons between Eq (4) against the Median and Mean of a specific $C_n^2(h )$ measurements obtained at (a) OT and (b) ORM on April 24, 2004, respectively. [Data provided by Dr. Julio A. Castro-Almazán and Dr. Begoña García-Lorenzo and used with their permission.]
4. Modified HAP and CLEAR 1 $\textrm{C}_\textrm{n}^\textrm{2}(\textrm{h} )$ comparisons with experimental data
Some papers contain comparisons between measured $C_n^2(h )$ data and some of the historical $C_n^2(h )$ models. For example, Jumper and his colleagues used the CLEAR I Model as their analytical benchmark in their various experiments [21–23]. The CLEAR 1 model is based on measurements made in the Tularosa Basin area of southern New Mexico [8], part of the high altitude Chihuahuan desert. The night model also should be typical of other mid-latitude desert regions [14]. In this section, we will compare the modified HAP model with their data and their CLEAR 1 prediction.
Figure 6 compares $C_n^2(h )$ measurements from Flight T-REX007 (UTC 05:50 03/25/06) [21] with data estimates using Eq. (4) and the CLEAR 1$C_n^2(h )$ models. The T-REX007 and other T-REX/HHH data discussed here were obtained from thermosondes measuring optical turbulence up to 30 km. Radiosondes were also lauched to measure meteorological data [21–23]. Another set of radiosondes also were launched to sense atmospheric data, including atmospheric wave signatures. (All HAP model calulations done here and in the sections to come were performed using the altitude measurements supplied by Dr. Jumper.) The modified HAP model profile has $M = 0.2$ and $C_n^2({{h_0}} )= 1 \times {10^{ - 14}}\;{m^{ - {2 / 3}}}.$ The modified HAP model appears to be in reasonable agreement with the average of the data, except for the bump in the vicinity of 20 km altitude, and compares a little better with the data than the CLEAR I model does.
Fig. 6. This figure shows $C_n^2(h )$ measurements from Flight T-REX007 (UTC 05:50 03/25/06) [17], the Modified HAP Model Profile and the CLEAR 1$C_n^2(h )$ model. [Data in Fig. 6 provided by Dr. George Y. Jumper and used with his permission.]
Figure 7 depicts $C_n^2(h )$ measurements from Flight T-REX029 (UTC 00:02 04/03/06) [17], and data estimates derived from Eq. (4) and the CLEAR 1$C_n^2(h )$ models. The modified HAP model profile has $M = 2.0$ and $C_n^2({{h_0}} )= 1 \times {10^{ - 14}}\;{m^{ - {2 / 3}}}.$ Both the modified HAP and CLEAR I models appears to be in reasonable agreement with the average of the data below 15 km altitude, and estimate smaller $C_n^2(h )$ values above that altitude. Fortunately, values of $C_n^2(h )\le \;1 \times {10^{ - 18}}\;{m^{{{ - 2} / 3}}}$ contribute little to any of the integrated $C_n^2(h )$-based entities.
Fig. 7. This figure shows $C_n^2(h )$ measurements from Flight T-REX029 (UTC 00:02 04/03/06) [17], the Modified HAP Model Profile and the CLEAR 1$C_n^2(h )$ model. [Data in Fig. 7 provided by Dr. Jumper and used with his permission.]
Figure 8 shows $C_n^2(h )$ measurements from the HHH02001 flight, launched at UTC 04:57 12/12/02 [19], and data estimates from the Modified HAP model and the CLEAR 1 $C_n^2(h )$ models. The modified HAP model higher altitudes. The modified HAP model appears to be in better agreement with the average of the data and the CLEAR 1 predictions.
Fig. 8. This figure shows $C_n^2(h )$ measurements from HHH02001 flight, launched at UTC 04:57 12/12/02 [19], the Modified HAP Model Profile and the CLEAR 1$C_n^2(h )$ model. [Data in Fig. 8 provided by Dr. Jumper and used with his permission.]
Figure 9 compares $C_n^2(h )$ measurements from the HHH02007 flight, launched at UTC 04:55 12/17/02 [23] with data estimates from Eq. (4) and the CLEAR 1$C_n^2(h )$ models. The modified HAP Model Profile has $M = 0.8$ and $C_n^2({{h_0}} )= 6 \times {10^{ - 15}}\;{m^{ - {2 / 3}}}.$ The modified HAP model again appears to be in reasonable agreement with the average of the data and compares better with the data average than the CLEAR I model does. It also follows the upper atmospheric profile better.
Fig. 9. This figure shows $C_n^2(h )$ measurements rom HHH02007 flight, launched at UTC 04:55 12/17/02 [19], the Modified HAP Model Profile and the CLEAR 1$C_n^2(h )$ model. [Data in Fig. 9 provided by Dr. George Y. Jumper and used with his permission.]
5. $\textrm{C}_\textrm{n}^\textrm{2}(\textrm{h} )$ modeling using the wind speed versus altitude profile
Mahmood, et al., among others, have suggested that $C_n^2(h )$ comparisons with experimental data can be improved by using their associated wind speed profiles with altitude [15]. To evaluate this premise, we modify Eq. (4) to be
Fig. 10. Plots of the Refractive Index Structure Parameter versus Altitude measurements (Hilo, Hawaii), and Eq (6) using the wind speed profile provided by Mr. Quatresooz and Professor Oestges. [Data used with their permission.]
One also might ask if using the wind speed in the modified HAP model, and not the inverse wind speed in Eq. (4), would be better. Let’s now define Eq. (4) in this case to be
Figure 11 compares $C_n^2(h )$ measurements from the T-REX007 experiment (a) with Eq. (6) and (b) with Eq. (7) using the measured T-REX007 wind speed profile. Dr. Jumper supplied all wind speed profile values and were matched to the ranges of the $C_n^2(h )$ profiles using the resampling program “interp”1 in MATLAB. The calculated $C_n^2(h )$ is not a bad rendition of the experimental data profile. Equation (6)’s predictions in Fig. 11(a) appears to follow the $C_n^2(h )$ profile pattern better with the measured data than Eq. (7)’s predictions in Fig. 11(b), with a little more detail shown in the pattern.
Fig. 11. Comparison of the measured $C_n^2(h )$ data from T-REX007 with (a) Eq. (6) and (b) Eq. (7) predictions. [Data provided by Dr. Jumper and used with his permission].
Figures 12, 13, and 14 compare $C_n^2(h )$ measurements from the T-REX009, T-REX029, and HHH2003 experiments, respectively, with (a) Eq. (6) and (b) Eq. (7) using their associated measured wind speed profile. Like the previous figure, Eq. (6)’s predictions in Fig. 12(a) appears to follow the $C_n^2(h )$ profile pattern better with the measured data than Eq. (7)’s predictions in Fig. 12(b), with a little more detail shown in the pattern. In Fig. 13, Eq. (6)’s predictions follow the $C_n^2(h )$ profile pattern better with the measured data than Eq. (7)’s predictions below 25 km, but both equations underestimate the measure data above 25 km. As noted earlier, the integration of $C_n^2(h )$ when its values is ${10^{ - 18}}{m^{ - 2/3}}$ or less contribute little to the total integration, so this should not be a serious problem.
Fig. 12. Comparison of the measured $C_n^2(h )$ data from T-REX009 with (a) Eq. (6) and (b) Eq. (7) predictions. [Data provided by Dr. Jumper and used with his permission.]
Fig. 13. Comparison of the measured $C_n^2(h )$ data from T-REX029 with (a) Eq. (6) and (b) Eq. (7) predictions. [Data provided by Dr. Jumper and used with his permission.]
Fig. 14. Comparison of the measured $C_n^2(h )$ data from HHH2003 with (a) Eq. (6) and (b) Eq. (7) predictions. (Data provided by Dr. Jumper and used with his permission.)
In Fig. 14, Eq. (6)’s predictions follow the $C_n^2(h )$ profile pattern better with the measured data than Eq. (7)’s predictions below 15 km. Above 15 kms, Eq. (6) overestimates measured $C_n^2(h )$ data while Eq. (7) underestimates measured $C_n^2(h )$ values. With the balloon drift rates, this wind speed measurement variation may be caused by the poor $C_n^2(h )$ estimates as these measurements being made outside the correlation distance with measured $C_n^2(h )$ data at the higher altitude.
6. Discussion
This paper proposed a modification of the Hufnagel-Andrews-Phillips (HAP) Refractive Index Structure Parameter model that will better characterize the HAP profile against averaged experimental data using Korean turbulence statistics. To estimate the multiplier M and the ground $C_n^2({{h_0}} )$ at the terminal at experimental sites for data analysis, the authors have had success using Fried Parameter estimates and ground scintillometer measurement to estimate both HV5/7 and HAP model parameters [7]. In system performance analyses, the authors have used the 85% percentile turbulence levels of 5xHV5/7 [1] and the reported here $M = 4.2$ and $C_n^2({{h_0}} )= 2.8 \times {10^{ - 14}}\;{m^{ - {2 / 3}}}$ in the modified HAP Model [24] as a good means to estimate system performance under stressing conditions. The authors feel that this approach provides a reasonable way to assess link availability.
In addition, comparisons between this model and various experimental data set reported in the literature showed reasonable agreement between the model and non-averaged data sets using a turbulent-intensity-based HAP model, even though some high-latitude difference occurred. It was speculated those high-altitude $C_n^2(h )$ difference with non-averaged data may be traceable to the wind speed measurement being made outside the correlation distance of the $C_n^2(h )$ data. Fortunately, the integration of $C_n^2(h )$ when its values are ${10^{ - 18}}\;{m^{ - {2 / 3}}}$ or less contribute very little to the total integration, so these differences should not be a serious problem. It also provides a quick-and-easy means using a standard model to estimate the refractive index structure parameter vertical profile as compared to a multi-point curve-fit.
The authors are encouraged by the two modified HAP models doing as well as they did with both average and non-average data, respectively. The authors have not seen any papers that show similar profile details using curve-fitting in the latter case. These latter comparisons offer a new topic for atmospheric research.
Acknowledgments
The authors would like to thank Dr. George Y. Jumper, Dr. Julio A. Castro-Almazán and Dr. Begoña García-Lorenzo, and Mr. Florian Quatresooz and Professor Claude Oestges for providing all the T-Rex and Hawaii campaign data, the OT and ORM data, and Trappes/Hawaii data, respectively, used in this paper and their helpful comments on their experiments. Their contributions helped us make a better set of comparisons than we could have done otherwise.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Experimental data presented in this paper may be obtained from the cited authors upon reasonable request.
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