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    Re: Measuring (and Calculating) Dip
    From: Bruce J. Pennino
    Date: 2013 Mar 19, 22:30 -0400
    Hi Brad:
    Good. Glad to learn that your results makes sense and compare to dip table , more or less.
      Considering roundings, radius of earth,etc and exact trig formulas used  dip (degrees) = .0298sqrt h meters is the same equation as dip (minutes) =  0.971 sqrt h ft . The latter is easier to remember, especially if you just remember square root of  h ft.
    I did the basic trig/geometry formulation for dip  from the viewpoint of a surveyor doing a circular curve layout as offsets from a tangent to a circle, with varying h.  My results were the same as the nautical almanac, which closely agrees with a table in the front cover of Mixter, Primer of Navigation.
    Here are some results   for Dip angle Minutes showing results from three sources:
    height of eye (ft)   Distance to horizon NM    Dip angle minutes      Mixter Table   Tangent Offset      
                                                                     NA, .971 sqrt h                                   Method
      25                            5.8                            3.1                             3.1                     3.1
    100                          11.7                             9.7                            9.8                     9.8
    250                          18.5                            15.4                          15.5                   15.5 
    500                          26.1                            21.7                          21.9                   22.0
    etc up to height of eye =  4,000 ft.        NOTE:  Distance to horizon is approximate.

    ----- Original Message -----
    Sent: Tuesday, March 19, 2013 9:23 PM
    Subject: [NavList] Re: Measuring (and Calculating) Dip

    Hi Bruce

    The Nautical Almanac has a table giving dip.  Its so commonly used its on the inside of the cover.

    Someone on the list, I forget who now (Paul H?), modeled that and determined that the table really is
    dip = 0.02977 * sqrt(h)
    Where dip is in degrees and h is the height of eye in meters. 

    So instead of using the table and interpolation, I used that equation for 'standard dip'.

    Next I used my equation for wave height correction, which Mr. Reed pronounced as "bullshit".  ;-)

    I then corrected the calculated dip by the waveheight.

    I then compared this value to the reading of either the SDM and or the PCR, ignoring the sign of the comparison.

    In cases where there was anomalous refraction (non-standard dip), the agreement was poor.  This occurred at Orient Point and on the ferry to New London.  We expected this because of the readily apparent mirages (islands floating ABOVE the water).

    In other cases, we had very close agreement.  One observation was about 2 seconds from the calculated dip, probably due more to luck than skill at this point.  I could not see apparent mirages, and so I expected standard dip to apply.  We saw close agreement at Montauk Point and on the return ferry ride.

    I tried to get Mr. Reed to comment on the non-standard dip equation, where temperature and pressure are inputs, but his focus was ... elsewhere ;-)


    I'm not afraid to publish my equations.  If they are wrong, the list will pounce and correct them for me!  This means that as we get serious, the methodology will be properly vetted and we only need concern ourselves with the input data.

    I'm not afraid to publish my measured data.  I need the practice with the instruments and not many people on list have access to this type of equipment.  I do hope that theodolite data (yours!) can add to the mix. 

    I will state that the SDM is a breeze to use, while the PCR can be a real pain.  It took me ~ 3 hours to figure out my body movements to be even able to take the measurement with the PCR, while the SDM was intuitive and rapid.

    I hope that answered your questions.  Feel free to jump in for clarifications.  I don't mind being the strawman.


    On Mar 19, 2013 8:41 PM, "Bruce J. Pennino" <bpennino.ce---net> wrote:

    Sorry to say, but I don't follow all of the data you measured with the SDM and PCR.
    Do the  final values of dip as measured (and corrected by error term?)  from either or both  instruments agree with the standard equation dip =0.971 sqrt h, h in ft, dip in minutes of arc?  If I subtract the error term from the measured values, I calculate the dip as stated in the standard table?? 

    ----- Original Message -----
    Sent: Tuesday, March 19, 2013 2:37 PM
    Subject: [NavList] Re: Measuring (and Calculating) Dip

    To: All Concerned

    This past weekend, Alex Eremenko and I got together to measure some dip. Alex is in possession of a Soviet Dip Meter ( http://sextantbook.com/?s=Russian+dip+meter ). I am in possession of a Prismatic Circle of Reflection ( www.fer3.com/arc/imgx/Prismatic-Circle-of-Reflecti.ppt ). Both of these instruments are capable of measuring the angle between two horizons. Yet the optical principle in each is remarkably different.

    The Soviet Dip Meter (SDM) provides a direct reading of the average dip between Dip A and Dip B.

    The Prismatic Circle of Reflection (PCR)requires mathematical manipulation to provide the dip result.

    The calculation of dip is by the best fit to the Nautical Almanac or dip=0.02977*sqrt(h), where h is the height of eye, in meters. The result is in degrees

    The wave height correction is by atan((wvht/2)/(3860*sqrt(h))), where h is the height of eye, meters, and wvht is the peak to trough predominant wave height from the corresponding buoy.

    On 14 March 2013, Alex and I journeyed to Montauk Point. The height of eye was estimated to be 13 feet and the waves were reported at 5.9 feet. Thus, the calculated dip-wave height correction = 3' 8.95" (3minutes 8.95 seconds).
    PCR Measured 3' 22.5"
    Error Term 0' 13.55"

    SDM Measured 5' 09.00"
    Error Term 2' 00.05"

    SDM Measured 4' 39.00"
    Error Term 1' 30.05"

    SDM Measured 5' 00.00"
    Error Term 1' 51.05"

    On 16 March 2013, Alex and I journeyed to Orient Point. The height of eye was estimated at 13 feet and the wave height was 1 foot. The calculated nominal dip should have been 3' 28.97" Distinctly visible were mirages and as a result we expected anomalous refraction and dip.
    PCR Measured 10'05.50"
    Error Term 6' 36.03"

    We then hopped on the Orient Point -> New London Ferry to see Frank Reed. Height of eye was estimated to be 36 feet and the wave height continues at 1 foot. Since the ferry is quite large and long, the wave height correction continues as before. Calculated Dip is 5' 52.09". Mirages continued to be visible.

    PCR Measured 8' 12.50"
    Error Term 2' 20.41"

    PCR Measured 9' 17.50"
    Error Term 3' 25.41"

    PCR Measured 9' 40.00"
    Error Term 3' 47.91"

    SDM Measured 7' 12.00"
    Error Term 1' 19.91"

    After a wonderful visit with Frank, we returned on the New London -> Orient Point ferry. The same height of eye and wave conditions yields the same calculated dip 5' 52.91". No apparent mirages were visible

    PCR measured 5' 50.00"
    Error Term 0' 02.91"

    PCR Measured 6' 30.00"
    Error Term 0' 37.91"


    Frank - in http://fer3.com/arc/m2.aspx/Biruni-radius-Earth-dip-FrankReed-jan-2011-g15188 You wrote:

    beta = alpha0*Q*Re/s
    and the equivalent "refracted radius" for the Earth is
    R = Re/(1-beta)
    where alpha0 is the the index of refraction of air minus one equal to 0.000281, Q is just the usual temperature/pressure factor (=(P/1010mb)/(T/283K)), Re is the true radius of the Earth, and s is the scale height of the atmosphere.

    I believe this to fit into an equation of dip
    dip = arcos((R/(1-beta))/(h+(r/(1-beta)))
    where we have non-standard pressure and temperature.

    I would like to be sure that s is supposed to be 9000 meters.


    Alex has graciously permitted me to retain the Soviet Dip Meter for a period of time, such that comparative measurements can be evaluated. Thank you Alex.

    Best Regards

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