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    Re: Stop and smell the equations
    From: Bruce J. Pennino
    Date: 2013 Apr 9, 19:56 -0400
    Most importantly, no comments were taken seriously/personally because nothing personal was intended.  All is well.
    Next regarding some sloppy "tangent offset" theory  I used because I was mentally lazy.  Surveyors connect tangents to  form smooth flowing  horizontal circular curves. A tangent flows into the curve, and the vehicle  leaves the curve on a tangent. A simple way to do layout for short curves, which students learn first, is to use offsets from the tangent to the centerline of the curve. If we pretend the curve is the earth, we have a quick (for me) method to approximate dip based on the deflection of the curve.  It is a somewhat inaccurate 
    approximation and it is incorrect  for dip analysis.  I've said too much, so no more on this.
    Regarding K,  I also was incorrect in my  comment about  looking for  only about 7% error. I was thinking only of values of 1.06 to .97.   Based on my own limited data, and comments by many, K (total of refr and geometric) values can be substantially less  or greater than .97.  I'm trying to remember,  but from .5 or .6 to 1.5, maybe in extreme cases of atmospheric circumstances.   So the actual percentage difference for K  about the average choice of  .97  can be up to 33% to  50%, I gather in the extreme. 
    I'll get more dip data this weekend, and I'll invert the scope on some of it to confirm that there is no instrument error.  I hope to stay at one location, but move the theodolite 2 or three times up the sloping beach.  Maybe two locations, but I'm sharing a vehicle.
    Best regards

    ----- Original Message -----
    From: Frank Reed
    Sent: Monday, April 08, 2013 10:42 PM
    Subject: [NavList] Re: Stop and smell the equations

    Bruce, you wrote:
    "My, my...bad to worse...seems a bit......harsh."

    I do apologize if you took that personally. It was absolutely not directed at you. I was referring to Brad's recent posts which have been bouncing rapidly from one equation to another without stopping for even a moment to understand what the equation means. It has indeed gone from "bad" to "worse".

    You also wrote:
    "By the way, instead of using a "tangent offset" calculation method for a dip equation"

    What do you mean by a 'tangent offset' calculation??

    "I redid the geometric dip more correctly using right triangle geometry and trig, based solely on the equivalent radius of the earth."

    Yes. That's right. That's exactly how it works. You can do any of the various calculations for things like dip, dip short, angles to objects at different distances, distance to the horizon, angular height of an object beyond the horizon, and so on, by doing PURE Euclidean geometry problems, solved by simple trigonometry and often greatly simplified and reduced by small angle series expansions, etc., and THEN, to replace the pure geometry and bring the physics of refraction into the problem, you can just replace the radius of the Earth by the effective radius of the Earth given by R/(1-k) and all of the results will be correct for cases where the refractive "rotation" k is given. In the real world, this rotation has to be measured. Its average value has been well-established for nearly 200 years. But once it is given, either the average or a unique weather-specific value, the other equations follow without any further work. This only works under the assumption that the refraction does not vary much with geography across the region under observation. And even that condition may easily be violated in real world cases.

    You continued:
    "This way, geometric dip K was calculated to be 1.06 as Frank and others have mentioned. Due to refraction correction and experience, the adopted value is 0.971, or so."

    That's right. Note that the third digit beyond the decimal point in 0.971 is probably not significant. You might as well use 0.97. Also, bear in mind, as Marcel also noted, that you need to be very careful about which "k" you mean. It's always a good idea to specify what k means in every case.

    You also wrote:
    " It appears that we are trying to refine a small percentage of the total dip equation....roughly 7% or so. "

    May I ask, where are you getting that "7%" from? It is quite possible in the case of a temperature inversion that k (in R/(1-k)) can be 0.5 or 0.7 or quite a range of other values. These are relatively uncommon situations but not "rare". Don't take this to imply that the dip equation is "broken". The idea that there is something "wrong" with the standard dip equation and that it needs to be "fixed" --if, in fact, anyone is still under that impression-- is misguided. The value of 0.97 as the factor for sqrt(h) (h in feet) is a well-established average choice. Common daily variations in the range 0.90 to 1.00 are not at all unusual, and much larger variations in dip do sometimes occur.


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