`(EXCERPTED)
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`Manual of Offshore Surveying for
`Geoscientists and Engineers
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`EX. PGS 1014
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`Ex. PGS 1014
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`
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`Manual of
`Offshore Surveying for
`Geoscientists and
`
`Engineers
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`R.P. LOWETH
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`E
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`CHAPMAN & HALL
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`London ~ Weinheim « New York - Tokyo - Melbourne - Madras
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`EX. PGS 1014
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`Ex. PGS 1014
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`Published by Chapman & Hall, 2-6 Boundary Row, London SEl SHN, UK
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`Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK
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`Chapman & Hall GmbH, Pappelallee 3, 69469 Weinheim, Germany
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`Chapman & Hall USA, 115 Fifth Avenue, New York, NY 10003, USA
`
`Chapman & Hall Japan, ITP-Japan, Kyowa Building, 3F, 2-2-1 Hirakawacho,
`Chiyoda-ku, Tokyo 102, Japan
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`Chapman & Hall Australia, 102 Dodds Street, South Melbourne, Victoria 3205,
`Australia
`Chapman & Hall India, R. Seshadri, 32 Second Main Road, CIT East, Madras
`600 035, India
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`·First edition 1997
`
`© 1997 Chapman & Hall
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`Printed in the United Kingdom at the University Press, Cambridge
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`ISBN 0 412 80550 2
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`Apart from any fair dealing for the purposes of research or private study, or criticism
`or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this
`publication may not be reproduced, stored, or transmitted, in any form or by any
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`Enquiries concerning reproduction outside the terms stated here should be sent to the
`publishers at the London address printed on this page.
`The publisher makes no representation, express or implied, with regard to the
`accuracy of the information contained in this book and cannot accept any legal
`responsibility or liability for any errors or omissions that may be made.
`
`A catalogue record for this book is available from the British Library
`
`Library of Congress Catalog Card Number: 96-72156
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`§ Printed on permanent acid-free text paper, manufactured in accordance with
`ANSI/NISO Z39 .48-1992 and ANSI/NISO Z39 .48-1984 (Permanence of Paper).
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`52
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`Introduction to computations
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`4.1
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`Introduction
`This chapter deals with the basic concepts of positioning at sea; we start with an
`introduction to least squares, on which all modem positioning computations are based, and then
`develop the various formulae used in the computations.
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`We are going to put into a single chapter the information that is disseminated to
`undergraduate surveyors in about a year of study, so some of the detailed explanations and proofs
`will necessarily be shortened.
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`Before going straight into least squares. we will briefly revise the coordinate systems
`available to us in the context of computations.
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`4.2
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`Coordinate systems
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`4.2.1
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`The ellipsoid
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`The ellipsoid is the mathematical figure which approximates most closely the true shape of
`the earth. Unfortunately. many people have tried to establish the best-fit ellipsoid for the earth. and
`many of the ellipsoids they calculated are in use. Life would be very much easier if there were only
`one ellipsoid (or spheroid).
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`In Australia we generally use the Australian Geodetic Datum as a datum for our offshore
`surveys. Even this is somewhat complicated by the following facts:
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`• There are two Australian datums in use - AGD66 and AGD84.
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`• Neither of the two datums is geocentric.
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`• Australia intends to move to a geocentric datum in 2000.
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`The AGD66 datum has the following definition:
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`Semi-axis major: 6378160.0m
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`Flattening: 1/298.25 exactly.
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`The minor axis of the spheroid was defined in 1%6 to be parallel to the earth's mean axis of
`rotation in 1962 (this was later changed in 1970), and the meridian of zero longitude was defined as
`being parallel to the Bureau International de l'Heure (Blll) meridian plane near Greenwich. The
`centre of the spheroid was defined by the coordinates of Johnston Geodetic Station, a station in the.
`centre of Australia. At that time it was assumed that the spheroid - geoid separation was zero at
`Johnston, and also zero at all the other geodetic stations listed in the 1966 adjustment.
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`Since 1%6 a huge amount of information on the shape of the geoid has become available,
`particularly through satellite observations, and it was realized that the 1966 adjustment was no
`longer accurate. In 1982 all the information then available was put into a new least squares network
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`Least squares
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`57
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`One way of resolving this problem is to rotate and translate the global Cartesian system into
`a system whose origin is a point on the surface of the chosen spheroid, and such that the Y axis
`· points true north, the X axis points 90 deg East and the Z axis is the normal at the point of origin,
`positive upwards. Now, within a radius of lOkm or so from the point of origin, we can define true
`distances within the spread as
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`and we can define true azimuths within the spread as
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`atan2 ( (x'- x), (y'- y))
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`(4.6)
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`(4.7)
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`These are the simplest equations of all to use, and they involve no scale factors or
`convergence!
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`We therefore propose that the best method to use is as follows:
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`• Compute the vessel position in terms of the global 3D Cartesian system.
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`• Transfer the global 3D position to the local 3D system, using an origin which moves from
`shot to shot and which is located at the vessel's navigation reference point.
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`• Compute the in-spread data (i.e. sources and streamers) on the local system.
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`• Transfer the output back to the spheroid and/or projection as required.
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`Note that the computation from one system to another only ever involves point
`computations, not lines; therefore scale factors and convergence never enter into the computation.
`We will use X and Y coordinates throughout the computations .rather than E and N, to emphasize
`that we are working in a local 3D Cartesian system.
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`4.3
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`Least squares
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`4.3.1
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`Why least squares?
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`The person responsible for postulating the least squares process was Legendre, in 1806. He
`proposed that. given a set of n equally reliable measured values (xJ. x2, ..• , xn) of a quantity, the most
`probable value (MPV) x of that quantity is that which makes the sum of the squares of the residuals
`a minimum. A residual (vi) is defined as
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`58
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`Introduction to computations
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`(4.8)
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`It follows from this proposal that the arithmetic mean of a series of equally reliable
`observations is the MPV:
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`n
`n
`[. (x- xi) 2 =. [. v[
`i=l
`i=l
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`(4.9)
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`According to the least squares principle,
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`0
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`(4.10)
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`If we differentiate equation (4.10) we get
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`Therefore,
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`n
`2nx-2 [.xi= 0
`i = 1
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`X
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`1 n n [.xi
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`i = 1
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`(4.11)
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`(4.12)
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`which is of course the mean.
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`Gauss tried to place the least squares principle on a solid foundation, and deduced that the
`measurements must be distributed according to the normal frequency distribution. The proof of this
`is beyond the scope of this book. Both Gauss and Laplace also tried to justify least squares without
`referring to the arithmetic mean, and, much later, Fisher placed the least squares principle within
`the context of his method of maximum likelihood.
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`Even if the observations are not normally distributed and independent, the least squares
`principle still provides a simple method of assigning values to unknown quantities when the
`number of observations is greater than the number of unknowns. There is no other method that has
`been found superior to least squares.
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`70
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`Introduction to computations
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`offset for our observations. We are not directly in possession of this information, but we do know
`exactly where each acoustic node and laser target is on any gun string or streamer. We can then
`differentiate equation (4.39) and equation (4.40) to determine the slope at each node and therefore
`compute the y-offset.
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`4.5
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`Summary
`The method of least squares does not give the true solution. but then neither does any other
`known solution. Least squares does give the most probable solution and it has a very sound
`statistical basis.
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`The method is very easy to program and does not require a very large machine, although to
`produce a real-time system capable of positioning the vessel, streamers and sources does require a
`fast floating point processor.
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`Once good values for the standard errors have been estimated the method can be completely
`automated. including data rejection (as we shall see in the Chapter 5).
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`There is never any ambiguity in the result; the method always gives a unique solution.
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`It is quite easy to program a solution for any type of observation, and different systems can
`easily be combined. For example, the vessel NRP can be positioned using a combination of Syledis
`and Argo ranges. together with a DGPS latitude and longitude, with the great advantage that all
`LOP's can be individually weighted in the solution.
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`The one real drawback is that the accuracy is greatly affected by the choice of mathematical
`model used, and the precision is greatly affected by the weight matrix. This means that the model
`must be chosen to represent the physical reality as closely as possible, and the standard errors need
`to be carefully estimated.
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`Introduction
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`73
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`they must be approximated in some way. How can this be done? Should the last good
`range be used as an approximation? Should we avoid computing the net at all and use the
`positions computed from the last shot? Should a “computed range” be used - one that fits
`in well with the rest of the data? None of these options are correct, and none of them will
`produce the right answer for the network.
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`An integrated system overcomes the above problems in the following ways:
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`° Offset errors will quickly become apparent because all of the observations become inter-
`related (unless, of course, there are very many offset errors).
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`° Ranges are never despiked and there is no pre-processing. Instead each observation is sub-
`jected to a statistical W-test prior to entry into the computationlf the observation fails the
`test it is rejected; if it passes it is accepted into the computation.
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`° The final quality of the data can easily be assessed because the integrated system already
`has an associated variance/covariance matrix containing covariances for the whole sys-
`tem.
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`°
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`Judgements on what is good data and what isn’t are made automatically by the W-test. In
`some cases more than 50% of the data might be rejected or missing from the raw data set,
`and the line may still be acceptable.
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`- Continuity of data is no longer a problem. One of the great attributes of a Kalman Filter
`is that it can predict a variab1e’s value using all previous data available to it In other
`words, the value of variable xn at event 11 can be predicted using the previous (n- 1) values
`of x. It is entirely possible for the filter to run for 3 or 4 shots with no new data at all, with-
`out causing a major deterioration in accuracy.
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`5.1.2
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`Kalman filters
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`The author once heard a Kalman Filter described as a card which could be inserted into the
`back of a PC and which then despiked incoming data! Even worse, this statement was from a
`surveyor who had been in the offshore business for about ten years! Nothing could be further from
`the truth. It is a software filter and is simply an extension of least squares. The Kalman filter is a
`method of filtering, smoothing and predicting data which can be illustrated as follows.
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`P,- represent a vessel’s NRP as derived by
`In Figure 5.1 , the circles labelled P1, P2,
`point least squares solutions at times t1,
`t2,
`t,~. These solutions are arrived at using only the
`observations current at each shot and are in no way related to each other.
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`What we require to do is find the points marked by black squares which represent the
`optimal or most probable position of the NRP taking into account not only the observations
`current at a particular shot but also all the observations up to and including that shot It is very
`important to realize that under no circumstances are we trying to fit an approximate curve to a set of
`known real positions. We don’t know where the vessel positions are, and we never will know their
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`Ex. PGS 1014
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`74
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`Kalman filters
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`true positions, but the Kalman filter is a rigorous least squares optimal filter which will lead to a
`maximum likelihood solution. The fact is that vessels do not travel in a series of disjointed hops,
`they transit along a smooth curve which in the case of a seismic vessel shooting a prime line will
`approximate to a straight line.
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`Figure 5.1 Vessel's track.
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`To carry out an ordinary least squares computation using all the available data up to and
`including the current shot would require solving a massive set of simultaneous equations and
`inverting matrices of mammoth orders. However, were such a solution to be used it would give
`exactly the same answer as a Kalman filter.
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`In seismic surveying we are not only interested in the position of the vessel; in fact. the
`vessel position is not really important at all. What we really need to lmow are the positions of the
`sources and the streamer receiver groups. The filter can encompass both source and streamer
`positioning, at all times maintaining the basic premise that the spread is a single system, all parts of
`which are physically joined together, and which must therefore use an integrated algorithm.
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`An optimal estimator is an algorithm that processes all the available data to yield the 'state'
`of a system, whilst at the same time satisfying some predetermined optimal criterion. In our case,
`the 'state' is a vector of parameters called the state vector which contains the vessel position,
`velocity, drift, streamer and source shape coefficients. In addition, if the vessel is towing acoustic
`transponders from sleds and those transponders do not form part of a source or streamer, an
`additional pair of positional parameters can be included for each sled node position. The
`information processed by the filter includes all observations such as Argo ranges, DGPS positions,
`laser and acoustic ranges, compass bearings, and so on. It also includes lmowledge of the ship's
`dynamics. This does not imply detailed equations modelling wind and wave action on the ship
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`Ex. PGS 1014