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    Re: Dip-meter again
    From: Richard B. Langley
    Date: 2012 Apr 10, 11:17 -0300

    Alex:
    
    As you say in your P.S., you HAVE to have a functional model  
    describing your data for parametric least squares. So your proposed  
    numerical example is incomplete. The assumption, of course, is that  
    the model is a good approximation of the actual physical or  
    mathematical relationship between the observations and the parameters  
    to be estimated. The observations are assumed to have unknown random  
    errors.
    
    Let me illustrate with how GPS positioning works with redundant (more  
    than 4) observations where we must estimate the receiver clock bias,  
    which can be taken to be the same for all simultaneous observations.
    
    Simplified model:
    
    P = rho + c * dT
    
    where P = measured pseudorange (includes a random error component)
         rho = geometric range (distance between satellite and receiver)
           c = speed of light
          dT = receiver clock offset
    
    rho is a non-linear function of the satellite coordinates (known) and  
    the receiver coordinates (unknown) as is dT. So, we have four  
    unknowns: x, y, z, dT. With five or more simultaneous observations, we  
    can estimate these parameters using least squares to get the "best"  
    values.
    
    Using vector/matrix representation, let X be the vector of unknowns, A  
    is the matrix of partial derivatives of the observations with respect  
    to the parameters (the "design matrix") and P is the vector of  
    observations. Then
    
    delta-X = A^-1 delta-P
    
    where delta-X is the estimated increment to a starting value for X  
    (X_0, from previous observations or some other knowledge or  
    guesstimate) and delta-P are the differences between observed and  
    computed values of the observables.
    
    Then,
    
    X = X_0 + delta-X.
    
    Since this is a non-linear problem, iterations may be necessary until  
    delta-X becomes sufficiently small. One can also compute a covariance  
    matrix for the estimates, X, which comes from a propagation of the  
    random errors in the measurements into the estimated parameter values.
    
    I'm fairly sure you could set up a parametric model for sextant  
    observations that includes dip, which would allow you to estimate it  
    (or refine a guess). And since you raised it in a subsequent posting,  
    perhaps you could also (or alternatively) include a time error of the  
    observations and estimate that. Both models would assume that the two  
    nuisance parameters, the dip and the clock error, were constant for  
    the suite of observations used.
    
    I believe someone from USNO a few years ago wrote a paper that was  
    published in the ION's journal Navigation on processing sextant  
    observations using least squares (I haven't done it myself but I  
    should). I'll try to dig it up.
    
    -- Richard
    
    P.S. I teach a course on introductory adjustment calculus (least  
    squares) and another one that includes fundamental astronomy where the  
    students use a T2 theodolite to reduce sun shots rather than a sextant.
    
    On 10-Apr-12, at 10:29 AM, Alexandre E Eremenko wrote:
    
    > Dear Richard,
    > Sorry I do not understand your idea.
    > I said "NO statistical method can eliminate such bias"
    > Perhaps I am wrong. But then please explain me on a numerical example.
    > Suppose you have a series of numbers,
    > say observations taken at times 1,2,3,4,
    > and the readings are 5,6,7,8.
    > Please tell me from these data,
    > what was the true quantity and what was the error.
    >
    > Alex.
    >
    > P.S. Parametric estimation in statistics must use some mathematical
    > model of the data, the model involving parameters. From observation
    > one can estimate these parameters, IF THE MODEL IS CORRECT.
    > Unfortunately we do not have an appropriate mathematical model
    > of the anomalous dip, expecially how it varies with time.
    >
    > On Tue, 10 Apr 2012, Richard B. Langley wrote:
    >
    >>
    >> Thanks, Alex, but I was not talking about ordinary averaging but the
    >> use of parametric least squares, which is able to estimate the value
    >> of a bias along with the parameters of interest. So, if we have a
    >> series of observations for which we can assume that the bias was
    >> reasonably constant, then by simultaneously processing the complete
    >> set, one should be able to get a single estimate of position and the
    >> value of the bias (dip).
    >> -- Richard
    >>
    >> On 10-Apr-12, at 9:49 AM, Alexandre E Eremenko wrote:
    >>
    >>> Dear Richard,
    >>>
    >>> Unfortunately, no statistical method, including least squares
    >>> can help with dip. The reason is that dip can deviate from its
    >>> normal value for relatively long periods.
    >>> For example, if our much discussed observation with Bill B on lake
    >>> Michigan is explained by the dip (which a majority on the list seems
    >>> to believe), this anomalous dip persisted for several hours,
    >>> and was almost constant. (This is an extreme example of course).
    >>> What averaging (or least square) helps to eliminate is a
    >>> SUM of MANY small INDEPENDENT errors.
    >>> The error of the dip is not a "random" error but a "systematic" one.
    >>> And the only way to eliminate it is the use of some dip-meter  
    >>> device.
    >>>
    >>> However, we know that dip-meters were rarely used.
    >>> (Western manuals almost never mention the device,
    >>> Soviet ones do mention, and recommend, and it was a standard
    >>> equipment,
    >>> but the same manuals recognize that "people do not use it").
    >>>
    >>> This only shows that navigators did not care about anomalous dip.
    >>> That high accuracy in celestial navigation was not needed,
    >>> and that large variations of the dip are probably rare.
    >>>
    >>> Alex.
    >>>
    >>> On Tue, 10 Apr 2012, Richard B. Langley wrote:
    >>>
    >>>>
    >>>> Warning: academic exercise follows ;-)
    >>>>
    >>>> Perhaps if one has sufficient redundant observations and uses least
    >>>> squares to estimate position, one could include dip as an  
    >>>> additional
    >>>> quantity estimated simultaneously from the (biased) observations.  
    >>>> The
    >>>> same procedure is used to process GPS measurements where one of the
    >>>> "nuisance" parameters is the offset of the receiver's clock from  
    >>>> GPS
    >>>> System Time, which is generally unknown.
    >>>>
    >>>> -- Richard Langley
    >>>>
    >>>> On 10-Apr-12, at 1:31 AM, Antoine Cou�tte wrote:
    >>>>
    >>>>> Still, your observations once again point out that DIP is  
    >>>>> definitely
    >>>>> one "weak link" in the accuracy computation chain, since even  
    >>>>> under
    >>>>> (quite) good conditions, dip standard deviation was already  
    >>>>> close to
    >>>>> 0.15/0.20 arc minute.
    >>>>>
    >>>>
    >>>> -----------------------------------------------------------------------------
    >>>> | Richard B. Langley                            E-mail:
    >>>> lang---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/       |
    >>>> -----------------------------------------------------------------------------
    >>>>
    >>>>
    >>>>
    >>>>
    >>>>
    >>>> : http://fer3.com/arc/m2.aspx? 
    >>>> i=118883
    >>>>
    >>>>
    >>
    >> -----------------------------------------------------------------------------
    >> | Richard B. Langley                            E-mail:
    >> lang---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/       |
    >> -----------------------------------------------------------------------------
    >>
    >>
    >>
    >>
    >>
    >> : http://fer3.com/arc/m2.aspx?i=118885
    >>
    >>
    
    -----------------------------------------------------------------------------
    | 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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