NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Dip-meter again
From: Richard B. Langley
Date: 2012 Apr 10, 11:32 -0300
From: Richard B. Langley
Date: 2012 Apr 10, 11:32 -0300
I think this is the paper: NAVIGATION: Journal of The Institute of Navigation Vol. 42, No. 4, Winter 1995 Determining the Position of a Vessel from Celestial Observations and Motion GEORGE H. KAPLAN U.S. Naval Observatory, Washington, D.C. But I need to go through it. Scanning it, it seems that dip is considered known. Kaplan's model would have to be augmented for an unknown dip parameter. -- Richard On 10-Apr-12, at 11:17 AM, Richard B. Langley wrote: > 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/ | > ----------------------------------------------------------------------------- > > > > ----------------------------------------------------------------------------- | 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/ | -----------------------------------------------------------------------------
