Why should we study atmospheric refractive index structure constant? Atmospheric refractive index structure constant is a quantitative measurement of turbulence induced fluctuations in refractive index parameter for visible and nearinfrared wavelength. Atmospheric turbulence has strong impact on astronomical imaging, aerial surveying, terrestrial geodesy, optical ranging, and wireless optical communication. Major effects of turbulence are beam broadening, irradiance fluctuations (scintillation), and angle of arrival fluctuations. Hence an estimate of turbulence that may prevail at a site seasonally is helpful. Also Modulation Transfer Function of the atmosphere is measurable by refractive index structure constant. Fried parameter which is a measure of astronomical seeing, can be calculated from structure constant's vertical profile. Good image quality requires structure constant being as small as possible. Our interest in studying structure constant profiles is further enhanced by recent published work claiming that daily variations in mean structure constant values may serve as Earthquake precursor.
In this work, Atmospheric refractive index structure constant has been studied for five different stations over India using radiosonde measurements. The monthly variations of structure constant have been observed with lower heights studied separately. Vertical profiles of structure constant have been investigated. Linear correlation between structure constant and humidity has been found to exist. During correlation studies Type1 error probability p value patterns were generated. P value patterns were found to be systematic.
Study of Atmospheric Refractive Index Structure Constant over India using Radiosonde Measurements
Pranoy Ghosh, Arijit De
Electronics & Communication Engineering, Netaji Subhash Engineering College Garia, Kolkata, West Bengal700152
ABSTRACT
Atmospheric refractive index structure constant (C _{n} ^{2}) has been studied for five different stations over India using Radiosonde measurements. The monthly variations of C _{n} ^{2} have been observed. The monthly variations at lower heights have been studied separately. Vertical profile of C _{n} ^{2} has been investigated. Linear correlation between C _{n} ^{2} and humidity has been found to exist. During correlation studies Type1 error probability p value patterns were generated. P value patterns were found to be systematic.
Keywords: Correlation studies, Monthly variations, Radiosonde, Type1 error probability p patterns, Tropical and subtropical climate, Vertical profiles.
INTRODUCTION
Why should we study atmospheric refractive index structure constant? Atmospheric refractive index structure constant C ^{2} is a quantitative measurement of turbulence induced fluctuations in refractive index parameter for visible and nearinfrared wavelength. Atmospheric turbulence has strong impact on astronomical imaging, aerial surveying, terrestrial geodesy, optical ranging, and wireless optical communication. Major effects of turbulence are beam broadening, irradiance fluctuations (scintillation), and angle of arrival fluctuations. Hence an estimate of turbulence that may prevail at a site seasonally is helpful. Also Modulation Transfer Function of the atmosphere is measurable by C ^{ 2 } . Fried parameter (r_{0}), which is a measure of astronomical seeing, can be calculated from C ^{2} vertical profile.
Good image quality requires C ^{2} being as small as possible. Our interest in studying C ^{2}
n n
profiles is further enhanced by recent published work ^{2} claiming that daily variations in mean
However only a limited number of works have studied C ^{2} profiles for Indian stations and they do case study of some particular stations^{1}. It was learnt that general information about turbulence at tropical and subtropical Indian locations are not easily found. This study should answer queries such as whether atmospheric turbulence is random or seasonal phenomenon, unique to a site or sites with similar weather patterns share similar turbulence pattern; which atmospheric parameter plays a decisive role in magnitude of atmospheric turbulence; etc.
METHODOLOGY
A. Radiosonde Measurements and Data
A radiosonde is an instrument carried by balloon through heights till 30 kilometers measuring absolute temperature, pressure, relative humidity, dew point temperature etc. The data of temperature, pressure and dew point temperature at different height has a resolution of few tens of meters to few hundreds of meters. Temperature is measured by the carbon rod thermistor which measures the temperature from – 90˚C to 60˚C with a resolution of 0.1˚C. Pressure is measured by an aneroid barometer with a resolution of 1 mb. Dew point temperature is obtained from relative humidity measured by a carbon hygristor with a resolution of 2% RH. Radiosonde measurements are obtained twice a day at around 00 and 12 GMT (18:30 and 06:30 IST) by the Indian Meteorological Department. The data from the period January to December of year 2016, 2017 for five stations across India and 3 other stations are collected through Wyoming University Upper Air Measurements website [http://weather.uwyo.edu/upperair/sounding.html]. Figures I and II show radiosonde locations used in this study [Map created from https://www.mapcustomizer.com].
Image was removed due to copyright issues.
FigureI. Map showing radiosonde locations, earth quake epicenters, earth quake felt and nonfelt locations.
Illustrations are not included in the reading sample.
Figure II. Descriptions of radiosonde locations.
Symbols in Figure II should be interpreted as follows: red with dot (1 and 9) are EQ epicenters; red are locations where EQ felt and green (4 and 5) are locations where no EQ was felt. The locations were chosen so that turbulence at tropical and subtropical climate may be learnt. Kolkata, Mumbai, Visakhapatnam share tropical climate while Delhi and Guwahati have subtropical climate.
B. Computing atmospheric refractive index structure constant
where A ^{2} is a constant having a value of 2.8, A′ is the ratio of eddy diffusivities which is taken as unity, L _{o} is the outer scale length of the turbulence spectrum which has been assumed to be 10m and M is the vertical gradient of the potential refractiveindex fluctuations.
é
_{} _{6} æ Pöæ d lnq_{T}öê
æ
15500qç
dln qöù
(2)
M= 77.6´10
ç T÷ç
÷ê1+
dz T
ç1 2 dlnq ÷ú
where Pis the atmospheric pressure (mb), Tis the
è øè øê
êë
ç
ç dz
T÷ú
øû
absolute temperature (K), Ɵ _{T}
is potential temperature (K),
qis the specific humidity (g/kg) and zis the altitude (m).
C. Preparing various profiles
C ^{2} values are very small in between 10 ^{17} m ^{2/3} (weak turbulence) and 10 ^{13} m ^{2/3} (strong turbulence). So this work uses ln{ C^{2}/(1 m^{2/3})} whenever it refers to atmospheric refractive index structure constant.
Monthly variations of structure constant parameter for each location are generated for year 2016, 2017 in the following way
Refractive Index Structure Constant is computed at every height where radiosonde data is available. Atmospheric turbulence is different in lower heights till Planetary Boundary Layer and free atmosphere above it. In the later turbulence is seasonal while in formal it varies within hours. Planetary Boundary Layer generally occurs at height of 2 kilometres above ground. So the data sets are divided into
two – upto 2 km, 2 to 20 km. MONTHLY VARIATIONS refer to the mean ln{ C ^{2}/(1 m^{2/3})} and
standard deviation obtained from these data sets.
Vertical profiles of Refractive Index Structure Constant, Specific Humidity, Relative Humidity, Temperature, and Potential Temperature are generated next. Each profile is generated monthly and grouped into seasons. This is done for year 2016 & 2017 at three locationsKOLKATA, GUWAHATI, and MUMBAI.
Third profile to be generated is that of „correlation with height‟ between Refractive Index Structure Constant and Temperature, Potential Temperature, Specific Humidity, Relative Humidity, Lapse Rate. Data sets are grouped according to heights 0500, 5001000, 10001500...950010000 metres and Pearson correlation coefficient r, Type1 error probability p are found out. The correlation graphs are plotted for all months of 2016, 2017 at KOLKATA, GUWAHATI and MUMBAI.
The final part of this work monitors daily mean of Refractive Index Structure Constant at Earthquake felt locations for a total of 31 days 15 days before and after the event. Procedure for obtaining mean includes dividing data set into two groups according to height for reasons explained in second paragraph of this section. Standard deviations were also calculated along with mean. Details of the two EQ events can be found below
Earth Quake 24 ^{th} August 2016 Magnitude=6.8 on Richter Scale Epicentre: Chauk, Burma.
Study Locations: KOLKATA , GUWAHATI.
Earth Quake 31 ^{st} Jan 2018
Magnitude=6.1 on Richter Scale Epicentre: Jarm, Afghanisthan
Study Locations: KABUL, MASSAD, ZWS KASHI, NEW DELHI.
TableI Distance on Earth‟s Surface between Epicentre and closest Radiosonde location
Illustrations are not included in the reading sample.
RESULTS
A. Monthly variations
At monsoon season magnitude of free atmospheric turbulence reaches maximum while a minimum value is observed at lower heights (<2 km) for Kolkata, Mumbai, Vishakapatnam. Atmospheric Turbulence during monsoon at Guwahati reaches maximum for both free atmosphere and lower heights. Turbulence in free atmosphere is minimum during winter at all locations.
Illustrations are not included in the reading sample.
FigureIII. Comparison of monthly variations of refractive index structure constant at Mumbai and Guwahati
Illustrations are not included in the reading sample.
FigureIV. Monthly variations of refractive index structure constant at Kolkata, 2017 the dataset originally contained some values of Relative Humidity>=100%
B. Vertical Profiles
Vertical profiles of C ^{2,} Specific Humidity, Relative Humidity, Temperature, and Potential Temperature were generated monthly and grouped into seasons. This is done for year 2016 & 2017 at three locationsKOLKATA, GUWAHATI, and MUMBAI. Vertical profiles of Mumbai may be found belowIllustrations are not included in the reading sample.
FigureV. Monthly plot of refractive index structure constant with height for Mumbai, 2017 grouped into seasons winter, premonsoon, monsoon, post monsoon (clockwise).
Illustrations are not included in the reading sample.
FigureVI. Vertical plot of refractive index structure constant for Mumbai, 2016 anomaly is circled.
Illustrations are not included in the reading sample.
Figure VII. Vertical profile of Specific Humidity for Mumbai, 2016 the same anomaly is present.
Illustrations are not included in the reading sample
Figure VIII. Vertical profile of Relative Humidity reveals the cause of anomaly to be supersaturated air.
C. Correlation Studies
The most common correlation coefficient is Pearson Coefficient which will fit a linear regression line. Often calculating correlation coefficient is not enough. The requirements for determining causality vary greatly depending on what is being studied. To ensure causality, the rigor of Hypothesis testing where null hypothesis H0: “No correlation exists” and alternative hypothesis H: “there exists a nonzero correlation” has been embraced. Alternative hypothesis can only be accepted by rejecting null hypothesis. A Type1 error is said to have occurred on rejecting a true null hypothesis. One should be looking for p close to 0 implying “there exists a nonzero correlation between ln (C ^{2}) and the other variable”. On performing the tests high p values at several heights and weak r values were observed for Temperature (FigureIX), Potential Temperature, Lapse rate but satisfactorily low p and strong r is seen for ln(Specific Humidity) and Relative Humidity. Though Relative Humidity and ln(Specific Humidity) vertical profiles look very different, their correlation graphs are astonishingly similar (FigureX).
Correlation study between refractive index structure constant and Relative Humidity at KOLKATA and MUMBAI are compared. KOLKATA and MUMBAI shared same pattern on 2017 (see Figure XI).
Illustrations are not included in the reading sample.
Figure X. Correlation study between refractive index structure constant and Relative Humidity, Specific Humidity
Illustrations are not included in the reading sample.
Figure XI. Pearson correlation coefficient patterns between months with little and significant precipitation.
Rainfall data for Kolkata, Mumbai, Guwahati were collected through year 201216 [Customized Rainfall Information System http://hydro.imd.gov.in ]. Weather statistics of these locations based on period 19611990 were also looked^{35}. It was found that pattern of Correlation Coefficient r can be satisfactorily linked with precipitation. Correlation Coefficient r shows a gradual rise with height generally in OctDec featuring clear skies, but the correlation is always lost in monsoon months or other months with precipitation. The greater the precipitations larger are p values and there are fluctuations in r. Type1 error probability p patterns for one location are generally identical for same month from both years (see FigXII).
Illustrations are not included in the reading sample.
Figure XII. Type1 error probability p patterns at Kolkata of same months from both years.
DISCUSSION
One limitation of this work comes from using Tatarskii formula ^{ 6 } to compute refractive index structure constant. Tatarskii formula is based on Kolmogorov model^{7,8}. However experimental researches on atmospheric turbulence nowadays, have proposed non Kolmogorov models^{911}.
During study of monthly variations, mean refractive index structure constant computed in free atmosphere at Kolkata for year 2017 which included supersaturated condition at high altitudes visibly produces a plot different from the one seen in 2016 (FigureIV A). On excluding supersaturated values (Relative Humidity>=100) the plot became familiar (FigureIV B). However excluding supersaturated readings from Guwahati data set could not produce a different plot excluding this particular factor as a reason to why Guwahati profiles at lower heights differ from the rest.
Vertical profiles of refractive index structure constant are mostly independent of seasons. Refractive Index Structure constant parameter is maximum at ground and decreases exponentially (the log curve decreases linearly) till a height of ~15 km. Tropopause height in India is ~15km. In some of the plots this height appears to be a turning point for the parameter. One or two vertical profiles also show miscellaneous features which do not fit with the rest. No conclusions can be drawn on these because radiosonde data at some heights may be erroneous. Only in the case of strange discontinuity in vertical profile for Mumbai (year 2016) the exact cause has been identified. One must always look at vertical profiles before generating other inferences; because they serve as tools to prevent errors that may later creep in from using radiosonde data at high heights.
Type1 error p values in „height correlation‟ studies are generally close to zero, but even in months with clearest skies it shoots at certain heights. These patterns for one location are generally identical for same month from both years, and noticeably similar between Kolkata and Mumbai. Guwahati stands apart with its own patterns.
If a valid and reliable Earthquake precursor is ever identified it will have immense practical applications. Reliability test of the claimed Earthquake precursor based on daily mean C ^{2} variations was attempted using two recent Earthquake events. However the reliability could not be confirmed. Authors have excluded the results of detailed EQ study here, and the same maybe found in another paper titled “Role of Humidity Content on Atmospheric Turbulence & Essay on Earthquake precursor studies” when published by IEEE.
CONCLUSIONS
Turbulence at free atmosphere and lower heights are distinct. Turbulence at free atmosphere is a seasonal phenomenon. Season‟s effect on turbulence may be different from one location to another. Vertical profiles C ^{2} in troposphere have been studied. Humidity not Temperature is positively correlated with turbulence at all heights for months with no rainy days^{12}. In two profiles Guwahati stands apart with its own set of patterns however atmospheric parameter that should explain the result, could not be identified. Observations which could not be explained satisfactorily this time will motivate research. Physical interpretation for the observed pattern in p values is beyond scope of current work and new research in this direction should be pursued.
This work serves as preparatory work for studying Fried Paramater^{1315}. Fried Paramater should be studied for site selection to build a receiver or a telescope.
ACKNOWLEDGMENTS
The primary author acknowledges the contributions of Mr. Pritam Saha and Mr. Souradeep Chakraborty, who were undergraduate students at Electronics & Communication Engineering Department of Netaji Subhash Engineering College, for collecting and preparing the radiosonde data used in this work.
Authors are grateful to the sincere efforts of Electronics & Communication Department of Netaji Subhash Engineering College for organizing a one day conference on Remote Sensing & Applications, “IEEE RSA 2017” where results from monthly variations and vertical profiles of structure constant were first presented. Authors are also grateful to Computer Science Engineering department of Sikkim Manipal Institute of Technology for organizing IEEE sponsored second International Conference on Advanced Computational and Communication Paradigms “ICACCP 2019”. At that venue results from Correlation studies and Earthquake precursor studies were presented. Finally, authors wish to accept the data available through Wyoming University website, http://weather.uwyo.edu/upperair/sounding.html to have played a pivotal role in current research.
REFERENCES
^{1}Zahan, Y., Devi, M. and Barbara, A. K. (May 2015) “The small scale atmospheric irregularitites in association with the Thunderstorm activity over Guwahati”. International Journal of Science Engineering and Technology Research (IJSETR). 4. Pg 5.
^{2}Medhi, A., Devi, M., Goswami, H. and Barbara, A. K. (March 2015) “Atmospheric turbulences over Guwahati and their association with tropospheric dynamics, Indian Journal of Radio & Space Physics”. 44. Pg 3544.
^{3} Weather statistics for Kolkata, West Bengal (India), https:// www.yr.no/place/India/West_Bengal/Kolkata/statistics.html. (Accessed 02/10/2018).
^{4}Weather statistics for Marathi, Maharashtra, Maharashtra (India), https:// www.yr.no/place/India/Maharashtra/Marathi,_Maharashtra/statistics.html. (Accessed 02/10/2018).
^{5}Weather statistics for Guwahati, Assam (India), https:// www.yr.no/place/India/Assam/Guwahati/statistics.html. (Accessed 02/10/2018).
^{6}Tatarskii, V.I. (1971) “Effects of the turbulent atmosphere on wave propogation”. Israel Program for Scientific Translations Ltd, Jerusalem.
^{7}Kolmogorov, A.N. (1941) “The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers.” C. R. Acad. Sci. 30. Pg 301–305.
^{8}Rao, R. (2005) “Light Propagation in the Turbulent Atmosphere”. AnHui Science and Technology.
^{9}Belen‟kii, M. S., Karis, S. J., Brown II, J. M. and Fugate, R. Q. (1997) “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion.” Proc. SPIE. 3126. Pg 113–123.
^{10}Stribling, B. E., Welsh, B. M. and Roggemann, M. C. (1995), “Optical propagation in nonKolmogorov atmospheric turbulence.” Proc. SPIE. 2471. Pg 181–196.
^{11}Li, Y., Zhu, W., Wu, X., and Rao, R. (2015), “Equivalent refractiveindex structure constant of non Kolmogorov turbulence”. OPTICS EXPRESS. 23. Pg 2300423012.
^{12}Wesely, M. (1976) “The Combined Effect of Temperature and Humidity Fluctuations on Refractive Index”. Journal of Applied Meteorology. 15. Pg 4349.
^{13}Theory of Astronomical Seeing,
http://www.astro.auth.gr/~seeinggr/seeing_gr_files/theory/node38.html. (Accessed 10/17/2017).
^{14}Hickson, P. (2008) “Fundamentals of Atmospheric and Adaptive Optics”. Department of Physics and Astronomy, The University of British Columbia.
^{15}Barletti, R., Ceppatelli, G., Paterno, L., Righini, A. and Speroni, N. (1977) “Astronomical Site testing with Balloon Borne Radiosondes: Results about Atmospheric Turbulence, Solar seeing and Stellar Scintillation”. Astronomy and Astrophysics. 54. Pg 649659.

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