Utility of Geodetic Refraction

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1 19 Utility of Geodetic Refraction Brunner, F.K. and Kukuvec, A. Institute of Engineering Geodesy and Measurement ystems, Graz University of Technology, teyrergasse 30, 8010 Graz, Austria, Web site: Tel.: +43 (0) , Abstract The difficulty in accurately modelling the geodetic refraction effect for high accuracy measurements through the atmosphere is known for centuries. In this paper it will be shown that the refraction effect contains very useful information about the atmospheric surface layer, especially about the development of the temperature scaling T which is a key parameter of the vertical temperature gradient. The well known reciprocal and simultaneous zenith angle measurements yield true line averages of micrometeorological parameters. The unique features of this common geodetic technique for the measurement of line averages for the sensible heat flux are discussed in detail. The results of two field experiments are presented. Key words: atmospheric surface layer, vertical refraction, sensible heat flux, zenith angle measurements 1 INTRODUCTION Zenith angle observations are affected by the vertical gradient of the refractive index which is mainly influenced by the vertical temperature gradient. In geodesy usually the refraction coefficient, k, is used which is the ratio of the radius of the Earth to the radius of the line-of-sight if the line-of-sight is approximated by a circular curve. ince the thorough experiments by Brocks (1950) it is known that the refraction coefficient varies between -5 to 15 depending on height above ground and micrometeorological conditions. For precise optical measurements through the atmosphere the determination of k is crucial, and thus a topic of various research efforts in engineering geodesy. In a previous paper, the first author proposed the direct measurement of the refraction effect which is then related to the temperature structure of the atmosphere, Brunner (1982). There it was shown that reciprocal and simultaneous zenith angle observations yield the line average of the vertical temperature gradients along the line-of sight which can then be used to calculate the line average of the sensible heat flux. A field experiment confirmed the proposed method. o, what is the motivation to write again about the method? Modern geodetic instrumentation provides new capabilities which are of great advantage for the proposed method: motor driven operation, compensator corrections, automated target recognition (ATR), and thus programmable and automatic stand alone operation. Recently excellent field measurements were carried out to determine the behaviour of the refraction coefficient (Hirt T 1 Data Processing INGEO th International Conference on Engineering urveying Brijuni, Croatia, eptember 22-24, 2011

2 20 INGEO 2011 et al., 2010), although without investigating the heat flux calculations. New tasks have been identified in micrometeorology which could be solved using measurements of line averages of sensible heat flux. The basis of sensible heat flux measurements are the fluctuations of vertical wind speed and temperature. However, for stable atmospheric conditions usually above water and ice surfaces, and during night the fluctuations are reduced and thus the sensible heat flux measurements become unreliable. But vertical refraction measurements will yield reliable results. We will briefly review the geodetic measurements for the determination of the angle-ofrefraction. The computation of the sensible heat flux from sonic anemometer data or alternatively from angle-of-refraction measurements will be discussed next. Recent field experiments will be reported next. The paper closes with an appraisal of the advantages of this geodetic method, and a discussion of feasible applications in micrometeorology. 2 ANGLE-OF-REFRACTION The total angle-of-refraction can be optically measured using reciprocal and simultaneous zenith angle observations z A z B where accounts for the Earth s curvature, see Fig. 1. / R (2) with being the distance between the stations A and B, and R is the radius of the Earth. Eq. (1) shows that the measurement of the angle-of-refraction is independent of the height difference between A and B which is a significant practical advantage. Alternatively can be expressed in terms of micrometeorological parameters. Details of the derivations were given by Brunner (1982). Here only the assumptions and the main result are summarized. (1) Figure 1: Geodetic refraction simulation

3 Brunner, F.K. et al.: Utility of Geodetic Refraction 21 It is convenient to replace the refractive index n by the refractivity n 1 (3) which is of the order of The sign of the curvature of the line-of-sight is assumed to be positive if it curves like the Earth. The total angle-of-refraction (Fig. 1) is given by the integration of the curvature along the line-of-sight (Brunner, 1982) d d (4) z 0 0 It becomes obvious that represents the equally weighted path average of the vertical refractivity gradient. The refractivity of air can be expressed by the formula of the IAG resolution with sufficient accuracy (Rüeger, 2002). Introducing the conservative parameters (potential temperature and specific humidity q) yields for the vertical refractivity gradient g z c p 6 g R z p q z where g is the acceleration due to gravity, cp is the specific heat at constant pressure, R is the specific gas constant, p is the total air pressure, is the ratio of the molecular weights of water vapour and air, and q is the specific humidity. The small humidity effect is treated by the Bowen ratio. Justification for this approach and a guide for the selection of values of were provided by Brunner and Williams (1982). The final expression for in terms of micrometeorological parameters is d (6) z 0 where the overbars denote line average values and is the psychometric constant. (5) 3 ENIBLE HEAT FLUX 3.1 LINE AVERAGE OF POTENTIAL TEMPERATURE GRADIENT The line average of the potential temperature gradient is defined as G 1 d z 0 Eq. (6) can be used to express G at the mean height, b, above the ground of the zenith angle observations / (8) / G b 7 Thus the geodetic (reciprocal and simultaneous zenith angle) measurement of the instantaneous total angle-of-refraction ( ) can be used to calculate the line average of the potential temperature gradient. (7)

4 22 INGEO EDDY CORRELATION The sensible heat flux H in the atmospheric surface layer is defined as, e. g. Kaimal and Finnigan (1994) H c w p (9) where is the air density, cp is the specific heat at constant pressure, and w is the vertical wind velocity. A prime denotes the fluctuating component, and the angular bracket indicates an ensemble average. Eq. (9) is quite intuitively appealing: turbulent eddies transport vertically heat (fluctuating potential temperature). Positive correlation means an upward flux of heat, negative correlation means just the opposite. The measurement of sensible heat flux H with a sonic anemometer is indeed the realization of eq. (9). Two critical points ought to be mentioned about the eddy correlation technique. First, sonic anemometer measurements can only be considered spot measurements of the vertical air column surrounding the sonic anemometer, and the energy balance of the foot print below the sonic anemometer. econd, during periods of low production of turbulent eddies, the method is less precise. uch periods are usually associated with stable atmospheric conditions and negative heat flux values, typically occurring during night time and above cold (ice) surfaces. 3.3 IMILARITY For the atmospheric surface layer the stability factor (z/l) is most commonly used z L g H c p z 3 u where z is the height above the ground, L is known as the Obukhov length, u is the friction velocity, and is the von Karman constant (here taken as 0.40). According to the similarity theory, the scaling temperature T can now be defined (Kaimal, Finnigan, 1994): H T (11) c u p Now the ensemble average of the potential temperature gradient in the atmospheric surface layer can be expressed as z T z h z L where h (z/l) is the universal profile shape function for heat. The form of h (z/l) has to be determined by field experiments. The proposed forms for h (z/l) were assembled by Foken (2003), here we use the shape function provided by Kaimal and Finnigan (1994, p. 16). For homogeneous surface-layer conditions, the line averages can be expected to approach the ensemble averages, and thus eq. (8) and eq. (12) can be equated, which yields for the calculation of T (10) (12) 1 T bg b h (13) where the value of the profile shape function h at (b/l) is required. However, for the calculation of H

5 Brunner, F.K. et al.: Utility of Geodetic Refraction 23 1 c bg u H (14) p b h an accurate value u is required in addition to h. A procedure for the computation of u using wind speed measurements was discussed by Brunner (1982). 3.4 DICUION OF ERROR The main error source in the determination of G b is the measurement error of. Using ATR it should be possible to measure with a standard deviation of about 15 rad. A more detailed analysis of errors was presented by Brunner (1982). For the computation of T the mean height (b) of the line-of-sight above the ground is required. The height b can be measured by standard surveying techniques. However, the zero plane displacement height (d) needs to be applied, especially if vegetation is present (Brunner, b L will be the main accuracy 1982; Kaimal and Finnigan, 1994). The value of limitation for the determination of T. The profile shape functions are well known (Foken, 2003), however, the accuracy of u will be the limitation (about 10 %). For the calculation of H according to eq. (14) an accurate value of u is additionally required. The best accuracy attainable for u appears to be around 10 %. In summary it can be said that the errors of, h and u will be the main limitation in accuracy of determining line averages of T (or H respectively) using the proposed optical method. Accuracies of about 10 % should be achievable using standard surveying and meteorological equipment and standard observation procedures. h / 4 EXPERIMENT 4.1 PREVIOU EXPERIMENT The first experiment line was carried out by the first author at Edithvale, Australia, during three days in May of 1978 (Brunner, 1982). The observation site was chosen over flat terrain. The area was dairy pasture, and the vegetation was stubby grass. The results of the optical method were compared with the measurements of the Fluxatron, an eddy-flux instrument. The test line with a length of 400 m for the reciprocal and simultaneous zenith angle observation was centered about the Fluxatron station and aligned parallel to the mean wind direction. The instruments for the angle observations were Wild T2 theodolites with a target on each tripod. The instrument heights were chosen as approximately 1 m in order to achieve larger values of. The average of ten values corresponded to one Fluxatron measurement with an averaging period of 0.5 h. Furthermore, the sensible heat flux of both methods was compared. Excellent agreement between the results was evident with an estimate of 0.95 for the correlation coefficient. 4.2 EXPERIMENT OVER A FIELD OF YOUNG CORN This experiment was performed in the afternoon in May 2010 over a field of young corn in the urban area of Graz, Austria. The set-up was similar to that explained in 4.1. The test line had a length of 330 m and the sky was cloudy. As a significant improvement of the first

24 INGEO 2011 experiment, the zenith angles were measured with automated target recognition (ATR) of the Leica TCA 1800.

6 24 INGEO 2011 experiment, the zenith angles were measured with automated target recognition (ATR) of the Leica TCA To ensure simultaneous observations, the PC clocks of the two stations (A and B) were synchronized with help of the GP time-signal. In Figure 2, station B is illustrated. The circular prism necessary for the ATR measurements is mounted below the total station. The micrometeorological measurements were carried out at the center of the line with the ultrasonic anemometer UA-1 for the wind parameters and a HM30 weather station to get the temperature, the air pressure and the humidity. Figure 2: ituation of station B of the test line of experiment 4.2 The time series of the observed (at 2 min interval) total refraction angles is shown in Figure 3 for which a standard deviation of 5.8 cc was calculated. Figure 3: Total refraction angles over a field of young corn The data gaps occurred because of ATR outfalls during high scintillation of the air. The sensible heat flux H was calculated with the total refraction angles according to the computation procedure described in ection 3. The sensible heat flux H, based on the eddy correlation method, was determined with the data from the sonic anemometer UA-1 using the computation procedure by the manufacturer (Metek 2005, 2010). Figure 4 shows the comparison of both methods for which the selection of the proper time interval is crucial. H

$Brunner, F.K. et al.: Utility of Geodetic Refraction 25 requires a certain averaging time which should not be shorter than 15 minutes.$

7 Brunner, F.K. et al.: Utility of Geodetic Refraction 25 requires a certain averaging time which should not be shorter than 15 minutes. An instantaneous value is related to the average of all potential temperature gradients along the line-of-sight at that particular time. The concept of frozen turbulence stipulates that the horizontal air column is blown across the sonic anemometer with the mean wind speed. If 15 minutes is used for H then about 8 H values would provide an equivalent comparison. Figure 4: Comparison of the sensible heat fluxes (up) and linear regression (below), experiment 4.2 The computation of the linear regression of the H and H data of Figure 4 yields for the offset W/m2 and for the slope The correlation coefficient of 0.99 is an indication for the proper selection of the 15 minutes and the homogeneity of the surface. The linear regression was computed with a Matlab routine assuming variances for both variables (Wieser, 2001). 4.3 EXPERIMENT OVER A FIELD OF BARE OIL Recently another experiment was carried out in the morning hours, started a few minutes after sunrise, over a field of bare soil in the vicinity of Graz, Austria in April Again the standard deviation of a single measurement was computed with 5.8 cc. Contrary to the

26 INGEO 2011 previous experiment the test line had a length of 415 m and the rate of the measurements (Figure 5) was increased from 2 min to 30 s intervals.

8 26 INGEO 2011 previous experiment the test line had a length of 415 m and the rate of the measurements (Figure 5) was increased from 2 min to 30 s intervals. Thus a 15 minutes session of the UA- 1 corresponds with up to 26 values. The comparison of the calculated heat fluxes (Figure 6) shows a good agreement at micrometeorological neutral and unstable conditions. The computation of the linear regression yields a similar intercept of 1.4 ± 3.6 W/m2 and a slope of 0.98 ± The correlation coefficient in this run is 0.98, but it must be said that the regression line is less significant because of the scarse data for the positive H values. Figure 5: Total refraction angles over a bare soil field Figure 6a: Comparison of the sensible heat fluxes, Experiment 4.3

9 Brunner, F.K. et al.: Utility of Geodetic Refraction 27 Figure 6b: Linear regression, Experiment CONCLUION It has been shown that simple optical measurements can be used for the determination of line averages of the vertical temperature gradient, which may subsequently be used for the calculation of the scaling temperature T and the sensible heat flux H. The method is based on the refraction of light passing through the atmosphere. Only conventional surveying equipment is required. In several field experiments the sensible heat flux measurements by this method were compared with those by the eddy correlation method. Excellent agreement was found between both methods, with a correlation coefficient of at least The attainable accuracy has been estimated with 10 % for the determination of T, whilst the accuracy of H is linearly related to that of u. Recently the need for measurements of spatially averaged heat fluxes was identified for inhomogeneous surfaces. Here, the proposed method has advantages over known methods. Periods of weak turbulence are difficult for the eddy correlation method, however, the proposed optical method does not require atmospheric turbulence. In particular it has significant advantages during night time and over cold surfaces such as polynyas. REFERENCE BROCK, K., Die Lichtstrahlkrümmung in Bodennähe, Deutsche Hydrographische Zeitschrift 3 (1950): BRUNNER, F.K., Determination of line averages of sensible heat flux using an optical method, Boundary Layer Metrology 22 (1982): BRUNNER, F. K. and Williams, D. C., On the Correction for Humidity in Two Colour Refraction Measurements. Z. für Vermessungswesen, 107 (1982): FOKEN, T., Angewandte Meteorologie Mikrometeorologische Methoden, pringer Verlag (2003): 289 pages.

10 28 INGEO 2011 HIRT, C., GUILLAUME,., WIBAR, A., BÜRKI, B., TERNBERG, H., Monitoring of the refraction coefficient in the lower atmosphere using a controlled setup of simultaneous reciprocal vertical angle measurements, Journal of Geophysical Research 115 (2010): KAIMAL, J.C., FINNIGAN, J.J., Atmospheric Boundary Layers Flows Their tructure and Measurement, Oxford University Press (1994): 289 pages. METEK, Formelzusammenstellung über Modified Bowen Ratio nach Liu und Foken (2001), Metek Gmbh, Elmshorn, unpublished (2005). METEK, Berechnung der Turbulenzgrößen des UA-1, Metek Gmbh, Elmshorn, unpublished (2010). RÜEGER, J.M., Refractive indices of light, infrared and radio waves in the atmosphere, UNIURV -68, Report from chool of urveying and patial Information ystems, UNW (2002). WIEER, A., General Linear Regression, Institute of Engineering Geodesy and Measurement ystems, Graz University of Technology, unpublished (2001).

DETERMINATION OF AIR REFRACTION INFLUENCE ON TRIGONOMETRIC HEIGHT DIFFERENCES

Original Scientific paper UDC 528.482:519.23 DOI: 10.7251/afts.2018.1018.055T COBISS.RS-ID 7324184 DETERMINATION OF AIR REFRACTION INFLUENCE ON TRIGONOMETRIC HEIGHT DIFFERENCES Trifković Milan 1, Nestorović