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For those interested in how surveyors may have handled refraction  
(including temperature gradients) in high precision surveys, here are  
a couple of links to a UNB thesis and a contract report on the  
subject. Note that the thesis is not from one of my students but that  
of a department colleague, now Prof. Em. Adam Chrzanowski:
http://gge.unb.ca/Pubs/TR132.pdf
http://gge.unb.ca/Pubs/TR142.pdf
Prof. Chrzanowski is still around and active should anyone want to  
pose a question to him.
-- Richard Langley

Quoting "Bruce J. Pennino" :

> Thank you Paul for helping with this.
>
> I've recorded more dip data from land locations and I'll eventually  
> let everyone see it.  I've been told that surveyors were supposed to  
> take critical precise measurements in the middle of the day when  
> refraction was less significant. Maybe temperature and pressure   
> gradients over land were less severe. Agree? Furthermore the issue  
> of temperature and pressure gradients cannot be thought about  
> without considering wind.  The turbulence and mixing created by the  
> wind alters the equations.  When you consider the boundary layer,  
> particularly at low elevations mixing becomes an issue.  Even though  
> a boat or a beach can be a local heat source/sink, the observer's  
> eyes are high enough "to feel the wind in your face". I feel the  
> theodolite shaking.  We all know it is even windier on the ocean.   
> For modest HoE, the air and pressure gradient  mixing at the  
> observer and the horizon is more or less the same excepting on very  
> hot days.  I would not  set up in a parking lot overlooking a cold  
> ocean, or a blazing hot sandy beach .   Getting representative data  
> is tough.
>
> I'm going to repeat some of my previous dip measurements on cloudy,  
> temperate days (if possible).  I'm hoping to set up on damp sand  
> just after high tide for one set of data. Then move higher up on  
> dune/ beach  to get another set of data.  I will purchase a  
> thermometer to get a " local reading somewhere nearby"  in shade .
>
> I also know an elevated  place, El 100 +/- MSL , where I can set up  
> in the shade of an overhanging roof and then quickly move to a  
> somewhat lower elevation in the sun.   I know these elevations to  
> within +/- 1.5 ft related to MSL.. Open to thoughts on places to set  
> up.  I normally record in my field notes the wind and cloud  
> conditions.
>
> Regards
>
> Bruce
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>   ----- Original Message -----
>   From: Paul Hirose
>   To: bpennino.ce---net
>   Sent: Monday, May 13, 2013 5:54 PM
>   Subject: [NavList] Re: Dip and Temperature Gradient
>
>
>
> ------------------------------------------------------------------------------
>
>
> I agree with Marcel's calculation of April 23 that the traditional formula
>
> dip = 1.76′ * √h  (eq. 1)
>
> implies dn = -2.59e-8 * h, where dn = refractive index at eye -
> refractive index at horizon, and h = height of eye (meters).
>
> http://www.fer3.com/arc/m2.aspx/Dip-Temperature-Gradient-Tschudin-apr-2013-g23661
>
> For an explanation of dn, see my messages on the refractive invariant:
> http://fer3.com/arc/m2.aspx/Dip-refractive-invariant-Hirose-apr-2013-g23613
> http://fer3.com/arc/m2.aspx/Dip-refractive-invariant-Hirose-apr-2013-g23652
>
> I also agree that a temperature gradient of .026 °C per meter will
> produce that dn. However, temperature must *increase* with height
> because dn is negative. That is quite different from the -.0065 °/m of
> the standard atmosphere. Also, it does not include the effect of air
> pressure.
>
> At the small heights of dip computations, pressure has practically
> linear variation with height and is independent of the temperature lapse
> rate. If P0 = sea level pressure, T0 = sea level temperature (C), and h
> = height in meters,
>
> P = P0 * (1 - h * .034163 / (273.15 + T0))  (eq. 2)
>
> At h = 100 meters the approximation is accurate to 1 part in 10000.
>
> [Originally I showed the steps by which it is derived from the standard
> atmosphere formula, but deleted that part for brevity. In
> http://en.wikipedia.org/wiki/Barometric_formula, equation 1, divide both
> sides of the fraction in brackets by the numerator to get a reciprocal.
> Invert the reciprocal and negate the exponent. Finally, transform
> exponentiation into multiplication with the approximation (1+x)^y =
> 1+xy, which is accurate when x is near 0. Note that temperature lapse
> rate Lb cancels itself after the last transformation.]
>
> If T0 = 10 C, the formula is
>
> P = P0 * (1 - 1.2065e-4 * h)  (eq. 3)
>
> For small variations in air density, refractivity is almost exactly
> proportional to density, so if n0 = refractive index at sea level, dP =
> increase in air pressure, dT = increase in air temperature,
>
> dn = (n0 - 1) * (dP/P0 - dT/T0)  (eq. 4)
>
> At T0 = 10 C and P0 = 1010 mb, n0 = 1.000281622 at sea level, according
> to the NIST calculator (550 nm wavelength). As I said at the beginning
> of this message, equation 1 implies dn = -2.59e-8 per meter. The value
> of dP/P0 may be obtained from my approximate formula for air pressure:
> -1.2065e-4 per meter. When we solve for dT (the only unknown in equation
> 4) we get the temperature lapse rate implied by the traditional dip
> formula: -.00811 °C per meter.
>
> The quantity dn is the height of eye correction (in units of Earth
> radii) due to refraction. When dn in converted to meters, I call it dn′:
>
> dn′ = 6371000 m * .000281622 * (-1.2065e-4 * h - .00353 * dT)
>
> dn′ = -.2165 * h - 6.337 * dT  (eq. 5)
>
> For example, at 30 m height of eye, assuming the temperature lapse rate
> I calculated, dT = -.00811 * 30 = -.24 C, and dn′ = -4.95 m.
>
> In a previous message I showed that
>
> dip = 1.926′ * √(dn′ + h)  (eq. 6)
>
> so in this example at 30 m,
>
> dip = 1.926′ * √(-4.95 + 30) = 9.64′
>
> Equation 1, the traditional formula, agrees to better than .01′.
>
> If air temperature at observer is identical to sea level, equation 5
> says dn′ = -6.50 m, and equation 6 says dip = 9.34′. So a quarter degree
> C changed dip .3′. Temperature gradients may be a significant source of
> error in dip measurements from land, or even at sea if the vessel is a
> "heat island".
>
> Note that equations 3, 5, and 6 assume the standard nautical almanac
> atmosphere. Even equation 6 can change a little, depending on the
> assumed mean radius of Earth.
>
> --
> 
> : http://fer3.com/arc/m2.aspx?i=124035
>
>
> : http://fer3.com/arc/m2.aspx?i=124039



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| Richard B. Langley                            E-mail: lang@unb.ca         |
| Geodetic Research Laboratory                  Web: http://www.unb.ca/GGE/ |
| Dept. of Geodesy and Geomatics Engineering    Phone:    +1 506 453-5142   |
| University of New Brunswick                   Fax:      +1 506 453-4943   |
| Fredericton, N.B., Canada  E3B 5A3                                        |
|        Fredericton?  Where's that?  See: http://www.fredericton.ca/       |
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