Atmospheric Effects in Geodetic Levelling

Home , Levelling

GEODESY AND CARTOGRAPHY ISSN 2029-6991 print / ISSN 2029-7009 online 2012 Volume 38(4): 130–133 doi:10.3846/20296991.2012.755333

UDC 528.381

ATMOSPHERIC EFFECTS IN GEODETIC LEVELLING

Aleš Breznikar1, Česlovas Aksamitauskas2 1Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, SI-1000 Ljubljana, Slovenia 2Department of Geodesy and Cadastre, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania E-mails: [email protected] (corresponding author); [email protected] Received 08 November 2012; accepted 12 December 2012

Abstract. The knowledge of physical processes and changes in the atmosphere is essential when addressing the fundamental problem, i.e. the accuracy of geodetic measurements. In levelling operations, all these changes are explained as the effect of refraction, which systematically distorts the results of levelling. Different ways of addressing the effect of refraction are represented based on the modelling of the vertical temperature gradient as the quantity that has the most influence on the refraction phenomenon. Keywords: atmospheric refraction, geodetic levelling, model of vertical temperature gradient.

1. Introduction 2. Known characteristics of levelling refraction Measurements in geodesy are mostly performed in the The principal knowledge about levelling refraction can field, under real atmospheric conditions. These measure- be summarized as follows: ments are therefore subjected to all the changes that oc- ––On an inclined surface, the refraction systemati- cur in the atmosphere. The atmosphere is a highly inho- cally distorts the results of levelling; mogeneous medium, whose state is defined by a set of ––The amount of refraction error is proportional to parameters that can quickly change, both spatially and the square of the length of the line of sight; temporarily. ––The refraction effect increases with the increase of Today, the errors resulting from environmental the temperature gradient; impacts are the main reason why the measurement ac- ––Under the influence of refraction, the height diffe- curacy is not even higher than it already is. The proper rence is reduced; evaluation of the individual parameters occurring in the ––If a certain levelling line is levelled under the atmosphere, whose changes result in refraction effects, is same microclimatic conditions, the effects of ref- the main issue. Studying the atmospheric effects in geo- raction are not evident as deviations during the detic operations is an interdisciplinary activity, in which levelling in either direction, nor are they manifes- the problems are solved based on scientific achievements ted as deviations in the closure of loops; rather it in relevant scientific areas. functions as the systematic error of the rod scale. In levelling operations, too, invar levelling rods and The refraction error is manifested directly in the levelling instruments have reached a stage of develop- height error of individual benchmarks; ment which would have permitted even higher accuracy, ––The effect of humidity is minimal and can be, in if it had not been limited by refraction. There are many most cases, neglected; problem domains where precise determination of the ––On its way from the levelling instrument to the height difference between the stations on the Earth’s sur- level rod, the horizontal line of sight is deflected face is required. These problems are related to geophys- upwards, which is the result of the negative tem- ics, geodynamics, engineering geodesy and, indeed, pre- perature gradient in near-ground air layers. cise levelling. The refractive index of air along the levelling line of Today, refraction represents the main source of sight changes as a function of the quantities causing it to systematic errors in levelling. Particularly, the effects of change, hence resulting in the phenomenon of refraction refraction are analysed as part of national levelling net- (Mozzuchin 1999, 2010). work campaigns. In most of developed countries, in the The refractive index in the atmosphere is mostly in- national first-order levelling surveys corrections for re- fluenced by the following quantities: air temperature, air fraction effects are performed. pressure, and content of humidity and carbon dioxide

(CO2). In ordinary levelling operations, mostly air mass This error also cannot be determined from the differ- in the range of 50 cm to 300 cm above ground is consid- ences in levelling in either direction; however, its effect ered; hence the studying of the values and changes of the is evident in the closure of levelling loops. In the level- influencing quantities in these layers is meaningful. ling loops that involve symmetrical slopes, the refraction The effect of air humidity is negligible since it repre- error will not be expressed as an error in levelling loop sents, at the most, less than 1/20 of the effect of tempera- closure. However, the effects of the part of the refrac- ture. The effect of carbon dioxide is even smaller. tion error resulting from the meteorological conditions Temperature and air pressure have the most signifi- during levelling will not be negligible. In levelling loops cant effect on the refractive index in the air layers where involving unsymmetrical slopes, the refraction error can levelling operations typically take place. be eliminated only partly. The non-eliminated part of the The amount of air pressure changes as a function of error results in non-closure of levelling loops, which is height. Generally speaking, air pressure decreases pro- frequently higher that the allowed deviation. portionally to the height above the ground. During the course of the day, the air pressure in a station changes 3. Ways of addressing the problem of refraction depending on the general weather conditions. However, There are several different approaches to the determina- these changes of air pressure are slow and, in general, tion of levelling refraction. Of the existing models solving minimal. Evidently, the systematic effect of air pressure the problem of levelling refraction, one can distinguish changes is eliminated in closed loops, while the effect of between three main groups (Heer 1983; Brunner 1984): minor, random changes of air pressure is also negligible. ––Direct refraction models in which the corrections Temperature, along with its changes, is the main for refraction effect are calculated by taking into factor causing changes of the refractive index in the at- account the measured temperature differences mosphere. Hence the determination of the actual influ- and the selected equations for temperature chan- ence of levelling refraction depends on the use of as far ges near ground level, or, alternatively, the tempe- as possible realistic models describing temperature gra- rature changes serve for temperature gradient qu- dients in the air layers where lines of sight are captured antification. The temperature gradient is the input during levelling. of the model describing the physical changes in The changes of the refractive index are a function the atmospheric boundary layers. of both the absolute air temperature and the tempera- ––Indirect refraction models where the corrections ture differences of the air layers where lines of sight are for refraction are also calculated based on the captured. Here, we must distinguish between two terms, size of the temperature gradient. In these cases, albeit the same quantity, i.e. temperature. The differ- the temperature gradient is expressed in relation ences in temperature can be relatively large at low tem- to meteorological and geometric parameters, as a peratures, and, vice versa, the differences can be minor function of height above the ground, tilted surfa- at high temperatures, depending on different factors. ce, zenith distance of the Sun and the height va- Therefore they should be dealt with separately, i.e. abso- riation of heat fluxes. In this case also, the tem- lute temperature and temperature gradients. perature gradient can be calculated based on the In almost all formulas for calculation of the cor- measurements of temperature. rection for levelling refraction, the size of the refractive ––The statistical analysis of the levelling operations correction is proportional to the size of the temperature can estimate the individual types of errors; hence gradient. While the changes of other factors involved it is possible to assume the size of the error due to in levelling refraction do not significantly influence the the effects of refraction. change of refraction, the effect of temperature difference is great and can change the size of refractive effects by 3.1. Models of levelling refraction based on 100%, and even 200%. Temperature gradient in the ele- measurements of temperature differences vations of up to 3 m above the ground, i.e. in the range of The models for refraction error calculation originate ordinary levelling operations, can be negative, positive, from the common basis, which is expressed as the effect or equal to zero. Furthermore, temperature gradient can- of the temperature gradient on the refractive index of air not be considered as a random variable during the level- and the assumption that the isothermal layers are paral- ling operations. lel to the ground (Kakkuri 1984). The individual mod- The investigations to date have indicated that tem- els differentiate in the different models for temperature perature gradient is the factor causing both random and change ∆t and are named after their authors: systematic effects of refraction. The random part signifi- cc cantly affects the random error of levelling, while the sys- Kukkamaeki: ∆=t bz ( 21 − z), (1) tematic part will have grave effects on long lines. Hence, 22 in precise measurements it is necessary to know the size Reissmann: ∆=t bz( 21 − z) + cz( 21 − z), (2) of the temperature gradient. In levelling lines with unsymmetrical slopes, i.e. the Lallemand ∆=tb(log( zc21 + ) − log( zc + )) , (3) upward inclinations are smaller while the downward inclinations are larger, or vice versa, with the same height where ∆t – change of temperature; z2, z1 – height above differences of rising and falling slopes, the effect of slope ground; b, c – coefficients obtained by measurements of is reflected in the measured height of the levelling line. temperature at different heights. 132 A. Breznikar, Č. Aksamitauskas. Atmospheric effects in geodetic levelling

The above mentioned models for temperature 3.2. Indirect model of correction changes are incorporated into the equation describing the for effects of refraction refraction effect as a function of temperature changes: In previous equations addressing levelling refraction, the

zsz, equations were expressed in units of gradient refractivity =−γ∆2 as a function of the temperature gradient. The tempera- rsz, ctg D∫ t dz, (4) ture gradient can be expressed in units of meteorological zi parameters as a function of height above ground, i.e. the where zi – height at the instrument; zs,z – height at the assumed temperature function. rod, i.e. front and back; D – factor of the effects of tem- K. Webb was the first to show that it would, in fact, perature changes on the refractive index; g – slope of ter- be beneficial to calculate the temperature gradient within rain; rs,z – effect of refraction. the boundaries of meteorological parameters. The equa- Another possibility of addressing the problem of tions were derived in the framework of comprehensive refraction with measured temperature differences is to studies of atmospheric turbulence and heat balance in model the temperature gradient. For the first time, this the boundary layers. These equations, which arise from model was approximated by Brock in the following form a simple basic theory, were developed using highly com- (Heer 1983): plex and precise meteorological measurements.

− dt b 2/3+− 1/3 2/3 = (5) − P zi23 zz i sz,, z sz az , r=−−106 sH 2 × 1 3.3 2/3 , dz sz, 22 T − (zzi sz, ) temperature gradient at a height z = 1 m, z – height  2/3+−− 1/3 2/3 above ground. − P zi23 zz i sz,, z sz r=−−106 sH 2 1 3.3 2/3 , (9) The value of parameter a changes during thesz, course 22 T (zz− ) of the day and, indeed, during the course of the year. In i sz, his studies, Brocks found that within the height range associated with levelling lines of sight, the value of b = –1 where P – preasure in [mbar]; T – temperature of the air can be assumed. in [K]. Parameter a can be also obtained by measurements The key quantity in the equation above is heat flux of temperature differences: (H), which represents the quantity of heat energy carried by convective heat flow. tt− a = 21, (6) Heat flux is one of the most characteristic meteoro- z ln 2 logical parameters; however, it cannot be measured di- z1 rectly. Angus-Leppan (1984) proposed three ways of de- termining heat flux: where t2, t1 – temperatures measured at heights z2 and z1. ––Determination of the components of heat flux by Based on these findings, the equation of the refrac- means of the heat balance equation; tion effect can be written as follows: ––Determination of the heat flux based on the re- 22 lation between the basic atmospheric parameters, cs cs zsz, z ra=+−121 lni , (7) such as cloudiness, effects of wind and humidity, sz, −− 2 zz1,sz zz i sz,, z sz and the Sun’s zenith distance; ––Heat flux can be determined from measurements where s – length of the line of sight. of temperature differences using the equation If we have records of multiple temperature differ- describing the temperature distribution in the lo- ences at the heights transected by the levelling line of wer layers of the atmosphere. If the equation is sight, these temperature differences can be directly used integrated between two different levels, and the in the equation calculating the refraction effect. In such potential temperature is replaced by the actual cases, it is not necessary to know the coefficients in the temperature, we obtain: temperature equations, but rather the problem is solved using approximate integration, i.e. summing. The accu- (tt−+) 0.01( zz −) 2/3 = 21 2 1 racy of the method depends on the number of stations 0.078H −− . (10) zz1/3− 1/3 with known temperature differences. The refraction ef- 21 fect on the measured height difference can be calculated 3.3. Statistical analysis using the summing method as follows: The statistical analysis of levelling results suggests the presence of both random and systematic errors. The cor- r=−ctg2 γ D ∆∆ tz, sz, ∑ (8) rections for refraction effects in levelling operations try zi to eliminate the systematic part of the errors due to re- where ∆z – height difference between two temperature fraction; however, it is recommended that the results of differences; ∆t – temperature difference between the in- levelling are statistically processed, while trying to deter- strument and the individual point of line of sight. mine the ratio of the systematic error to the total error. Geodesy and Cartography, 2012, 38(4): 130–133 133

The determination of levelling refraction from ad- References justments was suggested by Remmer (1982a, 1982b, 1983). For the analysis, he used the results of the sec- Angus-Leppan, P. V. 1984. Refraction in geodetic levelling, in ond levelling campaign of Finland (1935–1937), which F. K. Brunner (Ed.). Geodetic Refraction. Berlin: Springer- was very precise. Instead of assuming the refraction cor- Verlag, 163–180. rection based on measurements of temperature, he pro- Brunner, F. K. 1984. Modelling of atmospheric effects on terres- posed the use of equations of the observations contain- trial geodetic measurement, in F. K. Brunner (Ed.). Geodetic ing correction for refraction. Refraction. Berlin: Springer-Verlag, 143–162. However, the statistical processing of the levelling Heer, R. 1983. Applications of different refraction models on results, as proposed by Remmer, has two deficiencies re- measuring results of the Levelling Test Loop Koblenz, in garding the refraction error: H. Pelzer (Ed.). Workshop Precise Levelling. Duemmlers ––It addresses the refraction error which is manifes- Verlag, 251–281. ted only in specific conditions of levelling opera- Kakkuri, J. 1984. About the future use of the Kukkamaki level- tions: the closure of levelling loops and deviations ling refraction formula, in H. Pelzer (Ed.). Workshop Precise during the levelling in either direction. However, Levelling. Duemmlers Verlag, 235–241. in this way the refraction error which is manifes- Mozzuchin, O. A. 1999. Einige Aspekte zum Problem der ge- ted directly in the height error of individual sta- odaetischen Refraktion, Vermessungswesen und Raum tions is not cancelled out. Hence, the entire sys- ordnung 61(3): 172–175. tematic effect of the refraction error cannot be Mozzuchin, O. A. 2010. Zur Analyse von Nivellementergeb- eliminated. nissen, Allgemeine Vermessungs Nachrichten 3: 110–112. ––Furthermore, all errors obtained in the statistical Remmer, O. 1982a. Non-random effects in the Finnish level- analysis cannot be associated with the effects of lings of high precision, Manuscripta Geodaetica 353–373. refraction. In the way of data processing, other Remmer, O. 1982b. Modellings errors in geometric levelling, systematic effects occur during levelling opera- Vermwes 7: 355–372. tions. Remmer, O. 1983. Refraction and other systematic effects in 4. Conclusions levelling, in H. Pelzer (Ed.). Workshop: Precise Levelling. Duemmlers Verlag, 181–191. The effect of vertical refraction in levelling operations is not negligible and represents a major part of systematic Aleš Breznikar. Assoc. Prof., Dr at the Faculty of Ci- errors in levelling. Temperature gradient is the main factor vil and Geodetic Engineering, University of Ljubljana, Jamo- influencing the size of refraction, while the effect of other va 2, SI-1000 Ljubljana, Slovenia. Ph +386 41 356718, e-mail: factors is negligible, since they represent a minimal pro- [email protected]. portion of refraction effects. Hence the correct estimation The autor of more than 20 scientific papers participated of the size and changes of the temperature gradient is the in a number of international conferences. most important task in quantifying the effect of refraction. Research interests: engineering surveying, measuring sys- The corrections for refraction effect calculated using tems the chosen model of temperature gradient are a good so- lution to the problem of refraction; however, additional Česlovas Aksamitauskas. Prof., Dr, Dept of Geodesy and measurements of major temperature differences during Cadastre, Vilnius Gediminas Technical University, Saulėte- levelling are required, prolonging levelling operations. kio al. 11, LT-10223 Vilnius, Lithuania. Ph +370 5 2744 703, In indirect calculations of refraction effects, extensive Fax +370 5 2744 705, e-mail: [email protected]. studies to elaborate sufficiently accurate models are need- Doctor (1982). Habilitation procedure in 2011. Author of ed, providing satisfactory descriptions of the different at- two teaching books and more than 20 scientific papers. Partici- mospheric conditions. The advantage is that the complex pated in many intern conferences. measurements of temperature differences are not neces- Research interests: geodetic instrumentation, automation sary. However, the method requires detailed meteorologi- of measurements, angular and distance measurements. cal data on cloudiness, wind, Sun elevation and vegetation. The collection of these data can also be time-consuming.