Variability in low altitude astronomical refractionas a function of altitude

Russell D. Sampson,1,* Edward P. Lozowski,2 and Arsha Fathi-Nejad2

1Department of Physical Sciences, Eastern Connecticut State University, Willimantic, Connecticut 06226, USA2Department of Earth and Atmospheric Sciences, 1-26 Earth Sciences Building,

University of Alberta, Edmonton, Alberta T6G 2E3, Canada

*Corresponding author: sampsonr@easternct.edu

Received 16 May 2008; accepted 6 June 2008;posted 17 July 2008 (Doc. ID 96251); published 5 September 2008

Low altitude astronomical refraction (LAAR) of the setting Sun was measured over a sea horizon from acoastal location in Barbados, West Indies. The altitude of the upper limb of the Sun and the apparenthorizon were determined using a digital video camera (Canon XL2) and a digital SLR camera (CanonEOS 5D). A total of 14 sunsets were measured between 2005 and 2007. From these measurements LAARvariability was estimated at 14 standard altitudes of the refracted Sun between 0°:01 and 4°:5. The re-lative variability decreases with increasing altitude from �0:0195 of mean refraction at an altitude of0°:01 to �0:0142 at 4°:5. If extrapolated to an altitude of 15°, a linear fit to the data produces a relativevariability of�0:0038 and an absolute variability of�000:45. Statistical analysis of the relative variabilityin LAAR appears to support the decreasing trend. However, error propagation analysis further suggeststhat the observed values of refraction may exceed the accuracy of the measurement system at altitudeshigher than 2°. © 2008 Optical Society of America

OCIS codes: 010.1290, 010.7295.

1. Introduction

We report measurements of low altitude astronomi-cal refraction (LAAR) of the setting Sun as a functionof solar altitude and its day-to-day variability at alocation in the tropics. As outlined by Sampson etal. [1] the primary goal of this work is to develop aglobal map of LAAR. The justification for this workand its various applications is described in [1]. Creat-ing such a global map will require the measurementandmodeling of the variability of LAAR as a functionof time of day (sunrise versus sunset), season, geo-graphic location, and climate. The work describedhere examines only the variability of LAAR as a func-tion of altitude above an apparent tropical ocean hor-izon at sunset.

2. Method

The astronomical refraction R was determined frommeasurements of D, the vertical angular distance ofthe upper limb of the solar image above the apparenthorizon (considered positive). The relation betweenRand D (see Fig. 1) is given by

R ¼ Dþ h − s − a; ð1Þ

where h is the dip angle of the apparent horizon dueto the elevation of the observer (negative when belowthe astronomical horizon), s is the semi-diameterof the Sun (considered positive), and a is the altitudeof the geometric (i.e., unrefracted) Sun above theastronomical horizon (positive when above the astro-nomical horizon). The astronomical horizon is de-fined as having an altitude of 0° or equivalently azenith angle of 90° as measured by the observer.

The horizon dip angle h (in arc minutes) was esti-mated using [2]

0003-6935/08/340H91-04$15.00/0© 2008 Optical Society of America

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h ¼ 10:76ffiffiffiffiffiffizm

p; ð2Þ

where zm is the estimated height of the camera’s sen-sor above the local mean sea level in meters. This for-mula attempts to account for the standard terrestrialrefraction between the observer and the horizon. Thecamera was set up close to the shoreline, so that zmwas typically between 2 and 3 m.The value of D was determined for a sequence of

digital images of the setting Sun (typically between50 and 100), recorded either by a video camera (Ca-non XL2) or a digital SLR camera (EOS 5D with anEF 400mm f5.6 USM lens). The SLR was equippedwith an intervalometer with exposures set to occurevery 20 s. Calibration factors were used for eachimaging system to convert pixel coordinates to angu-lar measurements. It should be noted that the pixelsof the XL2 video camera were rectangular. The cali-bration factors were determined by comparing thecomputed angular diameter of the solar disk at thetime of the image [using U.S. Naval ObservatoryMultiyear Interactive Computer Almanac (USNOMICA) software] with the measured maximum hor-izontal width of the solar image in pixels. The appar-ent horizontal diameter of the solar image (parallelto the horizon) was assumed to be unaffected by re-fraction and therefore a true measure of the unre-fracted solar diameter. The pixel coordinates forthe SLR camera were then further corrected for sys-tematic lens distortions. This distortion was mea-sured and modeled using a series of solar imagestaken at altitudes greater than 45°, where the distor-tion due to astronomical refraction was assumed tobe negligible. Additional images of a printed rectan-gular grid helped to confirm the distortion of the lens.Other systematic errors were also taken into accountin an effort to minimize day-to-day variability fromsources other than variability in refraction. The ap-parent horizontal solar diameter was also measuredacross the field of view of the video system, and nosystematic lens distortion was detected.

Geographic coordinates determined through GPSreadings and exposure timings (�0:5 to �1 s) wereused in Ephemeris software [3] to calculate s anda. A graph of the computed astronomical refractionas a function of altitude of the upper solar limb fromthe SLR data appears in Fig. 2 with modeled valuesfor comparison [4].

In order to determine the day-to-day variability ofthe refraction as a function of altitude, the refractionof the Sun was determined at a set of standard alti-tudes (see Table 1). The refraction at these standardaltitudes was linearly interpolated from the mea-sured refraction values on either side of the standardaltitude. The relative variability was then calculatedas a ratio of sample standard deviation of the inter-polated refraction values σR to the average interpo-lated refraction �R at standard altitudes. Becausethere were only two measurements at an altitudeof 1°:5, this altitude was omitted from the analysis.

3. Results and Analysis

A graph of the absolute day-to-day variability ofLAAR as a function of altitude a0 of the upper limbof the refracted Sun appears in Fig. 3. An exponentialfit has an R2 of 0.97 and the form

σR ¼ 0:00932e−0:288a0: ð3Þ

The curve predicts an absolute variability of 00:1 (theaccuracy of a standard sextant or theodolite) at analtitude of 6°.

The relative variability of LAAR as a function ofaltitude of the refracted Sun is shown in Fig. 4.The relative variability appears to decrease with in-creasing altitude. A linear fit has an R2 of 0.59 andthe form

Fig. 1. Schematic of the geometry and variables for measuringastronomical refraction.

Fig. 2. Measured sunset refraction values (dots) from the SLRcamera compared with modeled refraction [4] at temperaturesof 237:15K (upper dashed curve) and 303:15K (lower dashedcurve). The circle is the approximate apparent diameter of theSun on 28 February 2007.

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σR=�R ¼ −0:00101a0 þ 0:0190: ð4Þ

The uncertainty of the slope [5] was found to be�0:00024, and the coefficient of linear correlation[5] to be −0:77. This coefficient of linear correlationimplies that the probability that 14 measurementsof two uncorrelated variables would produce a simi-lar or larger correlation coefficient is about 0:2% [5].If extrapolated to an altitude of 15°, the relativevariability is �0:0038 and the absolute variabilityis �000:45. The altitude of 15° is often cited as the ac-curacy limit of analytical refraction models [6].Even though the statistical analysis strongly sug-

gests that the measured relative variability in LAARdecreases with increasing altitude, some concernsremain about measurement error. The uncertaintyin the measured refraction δR can be determinedthrough an error propagation [5] of Eq. (1), assumingthat the individual uncertainties are random and in-dependent. An error propagation analysis producesthe equation

δR ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδD2 þ δh2 þ δs2 þ δa2

p; ð5Þ

where δD, δh, δs, and δa are the estimated uncertain-ties in each variable. The uncertainty inDwas deter-mined from the corrected measurements of the pixelcoordinates and is assumed to be �1 to �2 pixels(�400:3 to �800:6). The uncertainty of the dip angleδh was assumed to be �1500. The uncertainty forthe calculated semi-diameter of the Sun s is assumedto be negligible. Finally the uncertainty in the calcu-lated altitude of the unrefracted Sun was determinedfrom the error in the timing of the exposures (assum-ing negligible GPS error), which was estimated to bebetween�0:5 and 1 s (�700:2 to�1400:5). Placing thesevalues into Eq. (5) gives a range of δR of between�1700:2 and �2200:6 (�0°:0047 and �0°:0063). If theseestimated uncertainties are applied to the absolutevariability (Fig. 3), it is apparent that the valuesof the measured variability, above an altitude of2°:0, appear to exceed the estimated measurementlimit of the experimental system. Therefore, ques-tions still remain regarding the validity of the trendof the relative variability in LAAR as a function ofaltitude. With this in mind, comparison betweenthe measured LAAR and the modeled values of Auerand Standish above an altitude of 2°:0 should also bedone with an appropriate level of caution.

4. Conclusions and Recommendations

The results suggest that the relative variability ofsunset LAAR decreases with increasing altitude asmeasured from Barbados. The maximum relativeLAAR variability occurs at an altitude of 0°:01 andis about 2% of the mean refraction. The relativevariability decreases linearly to about 0.4% at 15°.The absolute variability is about �00:6 at an altitudeof 0°:01 and decreases exponentially to about �000:4at an altitude of 15°. A statistical analysis of the re-lationship between the relative variability in LAAR

Table 1. Variability Data of Sunset LAAR from Barbados a

Altitude [°] 0.01 0.05 0.1 0.2 0.4

No. obs. 3 7 7 9 11Rel. var. 0.0195 0.0195 0.0172 0.0167 0.0204Abs. var. [°] 0.0099 0.0097 0.0084 0.0079 0.0092Altitude [°] 0.6 0.8 1.0 1.5 2.0No. obs. 10 3 4 2 6Rel. var. 0.0188 0.0189 0.0182 - 0.0161Abs. var. [°] 0.0080 0.0076 0.0069 - 0.0048Altitude [°] 2.5 3.0 3.5 4.0 4.5No. obs. 7 7 7 8 6Rel. var. 0.0154 0.0189 0.0145 0.0148 0.0142Abs. var. [°] 0.0041 0.0046 0.0032 0.0030 0.0026aThis table lists the standard altitudes, the number of interpo-

lated measurements at each altitude (No. obs.), relative LAARvariability (Rel. var.), and absolute LAAR variability (Abs. var.).Measurements of LAAR from the video camera are from 0°:01

Fig. 3. Absolute variability of observed sunset refraction at stan-dard altitudes.

Fig. 4. Relative variability of observed sunset refraction at stan-dard altitudes.

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and the altitude suggests that the linear trend is notcoincidental. However, an error propagation analysisof Eq. (1) suggests that the measured values of varia-bility in LAAR above 2°:0 may exceed the estimatedaccuracy of the measurement system.In order to improve on these variability estimates

it is recommended that future campaigns should at-tempt to improve on the accuracy of measured valuesofD, h, and a. Simultaneous theodolite measurementof the horizon dip angle will help verify the validity ofEq. (2). Based on earlier measurements, we estimatethat the accuracy of the horizon measurement can beimproved to about �500 with careful theodolite read-ings. The accuracy of the exposure timings could alsobe improved to about�0:1 s (�100:4) by imaging a syn-chronized digital clock or synchronizing the camera’sinternal clock. If these levels of accuracy are achievedthe value of δR would improve to �700:2 (�0°:0020).We intend to pursue similar measurement cam-

paigns in other climate regimes in order to explorethe relationship between LAAR variability and cli-mate variables. Such relationships could allow thecreation of a global map of LAAR variability usingexisting climate data.Previous measurements suggest that a significant

difference can occur between sunrise and sunsetLAAR variability [7]. To date, only a handful of sun-rise measurements have been obtained and analyzedfrom the east coast of Barbados. These observationshave been omitted from this study, since there aretoo few data points to produce a meaningful compar-ison. However, preliminary analysis suggests thatthere may be a significant difference from the sunsetvariability.While the present results represent, to the best of

our knowledge, the first systematic experimental in-vestigation of the variability of LAAR as a function ofaltitude, the reader should be cautioned against

their indiscriminate use in various applications. Itshould be noted that for some applications of this re-search (e.g., archaeoastronomy), the Sun would beobserved rising or setting over distant topography,and therefore the effects on terrestrial refraction ofthe surface boundary layer would need to be consid-ered. Since most of the present measurements werenot made near the apparent horizon, but at some al-titude above it, one should exercise caution if usingthese results to estimate sunset refraction variabilityon an elevated topographic horizon.

Data collection assistance was provided by WendyRobinson and Hans Machel. Funding was providedby the Natural Sciences and Engineering ResearchCouncil of Canada through a Discovery Grant toE. P. Lozowski and by Connecticut State UniversityTravel and Research Grants to R. D. Sampson. Theauthors also thank the reviewer for the insight andhelpful comments.

References1. R. D. Sampson, E. P. Lozowski, and H. G. Machel, “Variability

of observed low-altitude astronomical refraction (LAAR) fromdifferent geographic locations: progress toward a global mapof LAAR variability,” Appl. Opt. 44, 5652–5657 (2005).

2. N. Bowditch, The American Practical Navigator (NationalImagery and Mapping Agency, 1995).

3. U.S. Naval Observatory, Multiyear Interactive Computer Al-manac, 1800–2050: 2.0 (Willmann-Bell, 2005).

4. L. H. Auer and E. M. Standish, “Astronomical refraction: com-putation methods for all zenith angles,” Astron. J. 119 2472–2474 (2000).

5. J. R. Taylor, An Introduction to Error Analysis (UniversityScience Books, 1982).

6. A. I. Mahan, “Astronomical refraction—some history and the-ories,” Appl. Opt. 1, 497–511 (1962).

7. R. D. Sampson, E. P. Lozowski, A. E. Peterson, and D. P. Hube,“Variability in astronomical refraction of the rising and set-ting Sun,” Publ. Astron. Soc. Pac. 115 1256–1261 (2003).

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