Manual of Offshore Surveying for
`Geoscientists and Engineers
`
`1
`
`ION 1014
`
`1
`
`ION 1014
`
`

`
`Manual of
`Offshore Surveying for
`Geoscientists and
`
`Engineers
`
`R.P. LOWETH
`
`E
`
`CHAPMAN & HALL
`
`London ~ Weinheim « New York - Tokyo - Melbourne - Madras
`
`2
`
`

`
`Published by Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK
`
`Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK
`
`Chapman & Hall GmbH, Pappelallee 3, 69469 Weinheim, Germany
`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
`
`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
`
`-First edition 1997
`
`© 1997 Chapman & Hall"
`
`Printed in the United Kingdom at the University Press, Cambridge
`
`ISBN 0 412 80550 2
`Apart from any fair dealing for the purposes of research or private study, or criticism
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`Enquiries concerning reproduction outside the terms stated here should be sent to the
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`
`A catalogue record for this book is available from the British Library
`
`Library of Congress Catalog Card Number: 96 — 72156
`
`Printed on permanent acid-free text paper, manufactured in accordance with
`ANSI/NISO Z39.48-1992 and ANSIINISO Z39.48-1984 (Permanence of Paper).
`
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`52
`
`A
`
`Introduction to computations
`
`4.1
`
`Introduction
`
`\
`
`This chapter deals with the basic concepts of positioning at sea; we start with an
`introduction to least squares, on which all modern positioning computations are based, and then
`develop the various formulae used in the computations.
`
`is disseminated to
`We are going to put into a single chapter the information that
`undergraduate surveyors in about a year of study, so some of the detailed explanations and proofs
`will necessarily be shortened.
`
`Before going straight into least squares, we will briefly revise the coordinate systems
`available to us in the context of computations.
`
`4.2
`
`Coordinate systems
`
`4.2.1
`
`The ellipsoid
`
`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).
`
`In Australia we generally use the Australian Geodetic Datum as a datum for our oflshore
`surveys. Even this is somewhat complicated by the following facts:
`
`° There are two Australian datums in use — AGD66 and AGD84.
`
`° Neither of the two datums is geocentric.
`
`° Australia intends to move to a geocentric datum in 2000.
`
`The AGD66 datum has the following definition:
`
`Semi-axis major: 6378 l60.0m
`
`Flattening: 1/298.25 exactly.
`
`The minor axis of the spheroid was defined in 1966 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 (BIH) 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 thatvtime 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.
`
`Since 1966 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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`4
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`

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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 10km or so from the point of origin, we can define true
`distances within the spread as
`
`D =
`
`(x’—x)2+ (y’—y)2+ (z’—z)2
`
`(4.5)
`
`and we can define true azimuths within the spread as
`
`APP. = atan2( (x’ —x), (y’ —y))
`
`_
`
`(4.7)
`
`These are the simplest equations of all to use, and they involve no scale factors or
`convergence!
`
`We therefore propose that the best method to use is as follows:
`
`- Compute the vessel position in terms of the global 3D Cartesian system.
`
`- 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.
`
`- Compute the in-spread data (i.e. sources and streamers) on the local system.
`
`- Transfer the output back to the spheroid and/or projection as required.
`
`the computation from one system to another only ever involves point
`Note that
`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 B and N, to emphasize
`that we are working in a local 3D Cartesian system.
`
`4.3
`
`Least squares
`
`4.3.1
`
`Why least squares?
`
`The person responsible for postulating the least squares process was Legendre, in 1806. He
`proposed that, given a set of :1 equally reliable measured values (x1, x2,
`x,,) 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
`A residual (v,-) is defined as
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`5
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`

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`58
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`Introduction to computations
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`vi = (x—xl.)
`
`(4.8)
`
`It follows from this proposal that the arithmetic mean of a series of equally reliable
`observations is the MPV:
`
`71
`
`fl
`
`2 (x—xl.)2 =. 2 vi2
`i=1
`i=1
`
`According to the least squares principle,
`
`I1
`
`(dig)
`
`l=l
`
`(x—xi)2 = 0
`
`If we differentiate equation (4.10) we get
`
`Therefore,
`
`2nx — 2 2 xi = 0
`l:
`
`it
`
`1
`x = 2
`
`xi
`
`(4.9)
`
`(4.10)
`
`(4.11)
`
`(4.12)
`
`which is of‘course the mean.
`
`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.
`
`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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`6
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`

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`70
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`Introduction to computations
`
`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.
`
`4.5
`
`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.
`
`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.
`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).
`
`There is never any ambiguity in the result; the method always gives a unique solution.
`
`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.
`
`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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`7
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`

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`Introduction
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`73
`
`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.
`
`An integrated system overcomes the above problems in the following ways:
`
`° Offset errors will quickly become apparent because all of the observations become inter-
`related (unless, of course, there are very many offset errors).
`
`° 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.
`
`° 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.
`
`°
`
`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.
`
`- 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.
`
`5.1.2
`
`Kalman filters
`
`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.
`
`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.
`
`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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`74
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`Kalman filters
`
`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 transitvalong a smooth curve which in the case of a seismic vessel shooting a prime line will
`approximate to a straight line.
`
`
`
`
`
`Figure 5.1 Vessel’s track.
`
`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.
`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 know 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.
`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 knowledge of the ship’s
`dynamics. This does not imply detailed equations modelling wind and wave action on the ship
`
`9