`
`
`Ex. PGS 1072
`EX. PGS 1072
`(EXCERPTED)
`(EXCERPTED)
`
`
`
`
`
`

`

`'
`; "Geoscientists and Engineers
`
`Offshore Surveying .for
`
`EX. PGS 1072
`
`

`

`Manual of
`Offshore Surveying for
`Geoscientists and
`Engineers
`
`R.P. LOWETH
`
`EJ
`
`CHAPMAN & HALL
`London • We inheim • N ew York • To kyo • M elbourne • M adras
`
`

`

`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-Iapan, 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
`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
`means, without the prior permission in writing of the publishers, or in the case of
`reprographic reproduction only in accordance with the terms of the licences issued by
`the Copyright Licensing Agency in the UK, or in accordance with the terms of
`licences issued by the appropriate Reproduction Rights Organization outside the UK.
`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
`
`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).
`
`.
`
`List 4
`Fore
`2 TnH*
`
`2 Datl
`
`3 Acc
`
`4 Intr
`
`5 Kali
`
`

`

`38
`
`Accuracy
`
`£ (*,•-*) v"i 2
`
`f = l
`
`- 1)
`
`(3.9)
`
`°* f(n
`
`3.2.5
`
`Covariance
`
`The last of the statistical terms we need to look at is covariance. If two randan variables x
`and y have expectations £ and vy respectively, then the covariance of x and y is defined as
`
`oxy = E { (x - %) (y - y) }
`
`(3.10)
`
`Translating this from expectations to samples, we get
`
`£ ( x r X ) ( y r p )
`i = l
`7T
`avl, = ——T
`' V
`( n - 1 )
`
`(3.11)
`
`Covariance is a term used extensively in least squares and Kalman filtering, and we shall
`investigate its uses later.
`
`3.3
`3.3.1
`
`Absolute accuracy
`General discussion
`
`Marine seismic survey positioning specifications are very often issued as a disjointed series
`of figures such as:
`
`• Argo
`
`• DGPS
`
`• Feather
`
`• Shot interval
`
`• Compasses
`
`+/- 10 m
`
`+/- 5 m
`
`< 10 deg
`
`25 m+/-1.5 m
`
`< 1 degree from mean of all compasses
`
`

`

`(3.9)
`
`iriables x
`
`(3.10)
`
`(3.11)
`
`we shall
`
`ted series
`
`Absolute accuracy
`
`39
`
`Acoustics range
`
`+/- 2m
`
`What we really should be doing is specifying the survey in terms of the quantities we
`require. If we position the vessel to better than 10 m and the compasses are less than a degree from
`their mean value, what does that do to our receiver group positions? If the acoustic ranges are better
`than 2m what does that mean in terms of our source position? To say that we require the Argo
`tolerance to be better than 10m is really quite nonsensical. What we have to do is define a foolproof
`method so that we can ensure that, providing our specifications are met, our common mid point
`positions will have the required accuracy.
`Consider Figure 3.3 :
`
`Figure 3.3 Two-bearing fix.
`The arrowed lines represent two bearings from stations A and B. They intersect at point P,
`which is the position of the vessel. We therefore need just two lines of position (LOPs) to define a
`two-coordinate position in terms of eastings and northings or latitude and longitude or x and y. But
`nobody would accept that position because if one of the bearings is incorrect, then so too is the
`position. Now we add a third bearing from station C. as in Figure 3.4 :
`
`Figure 3.4 Three-bearing fix.
`
`

`

`40
`
`Accuracy
`
`Now the point F is not so easily defined; the obvious choice is to place it inside the triangle
`created by the three LOPs, but suppose the three bearings contained a systematic error which, if
`compensated for, shifted all three LOPs to the left? The result is that the position of the vessel is
`shifted to X, a point which is outside the original triangle. The answer here would be to model for
`the unknown systematic error — something which is done quite regularly for radio positioning
`ranges containing an unknown systematic scale error due to propagation.
`We can summarize the above by saying that although the position of the vessel is not known
`so precisely when we increase the number of bearings, there is a very good chance that if one or
`more of the bearings are in error the effect of the vessel's position will be less catastrophic, thereby
`increasing the accuracy.
`The rule is that to solve for n unknowns we need at least n observations. If we have more
`than n observations, we have redundancy.
`We can propagate this rule throughout the spread, ensuring that there is redundancy in
`finding all the parameters we want to know about, including the vessel position, the source
`positions, the streamer shapes, the tailbuoy positions and the receiver group positions.
`
`-
`
`* Specification 1 —ensure that there are redundant observations for every facet of the entire
`spread.
`
`We must also be careful not to go to the other extreme and provide over-redundancy in the
`system, because the computing time required to adjust a particular network increases as the square
`of the number of observations: in other words if we look at the vessel position on its own, the
`difference in computing time between using ten ranges and using three ranges is 100/9 or 11 times
`as long. Not to mention the increase in cost, which probably rises at the same proportional rate.
`3.3.2
`Bin size and accuracy
`
`The only value in positioning the ship accurately is so that we can propagate that accuracy
`through our various in-spread systems. If we could somehow find absolute positioning points on the
`sources and streamers, the position of the vessel would become irrelevant. This is becoming a
`reality with the advent of shock-mounted DGPS and RGPS receivers on the sources and streamer
`heads.
`
`Let us suppose that we are conducting a 3D survey in which the bin size is to be 50m long
`and 25m wide. How should we determine the accuracy of the common mid-points so that we can be
`confident of them falling in the right bins? We need to go back to our theory of standard deviations.
`Figure 3.5 shows the probability density function. The curve is a normal distribution,
`and it can be shown that for any population this curve will result. Its equation is
`
`whei
`the c
`evalu
`
`proba
`
`if we
`68.39
`±18.7
`
`

`

`ie triangle
`which, if
`s vessel is
`model for
`ositioning
`
`lot known
`. if one or
`c, thereby
`
`lave more
`
`adancy in
`lie so urce
`
`the entire
`
`ncy in the
`the square
`own. the
`r 11 times
`Irate.
`
`; accuracy
`nts on the
`coming a
`I streamer
`
`50m long
`we can be
`eviations.
`
`jribution.
`
`(3.12)
`
`Absolute accuracy
`
`41
`
`Figure 3.5 Probability density function.
`
`where a is the standard deviation and n is the total number of observations. The shaded area under
`the curve in Figure 3.5 is the area lying between -a and +a. We can use equation (3.12) to
`evaluate the area under the curve for various limits - see Table 3.2:
`
`Table 3.2 Area under the normal curve for various standard deviations
`Limits
`Area%
`-CT to +C7
`68.3
`-2CT to +2CT
`95.4
`-3CT to +3CT
`99.7
`
`Now the percentage area under the curve given in column 2 of Table 3.2 is the percentage
`probability that any deviation is between the corresponding limits.
`Going back to our problem of how to specify the accuracy required for a bin width of 25 m.
`if we set the required standard deviation of a common mid point (CMP) to be 6.25 m we will be
`68.3% sure that all CMP's fall within ±6.25m, 95.4% fall within ±12.5m, and 99.7% fall within
`±18.75m. Only 0.3% will fall outside the limits of ±18.75m. We specified a bin width of 25m, so
`
`

`

`42
`
`Accuracy
`
`we can be 95.4% certain of CMPs falling in the right bin. This can be generalized into another
`specification:
`
`• Specification 2 — the standard deviation of all common mid points should be 0.25 x nom­
`inal bin width.
`
`A basic assumption here is that the bin width is the overriding parameter, since the receiver
`groups cannot move so much in the inline direction. In the case of a bottom cable survey both bin
`width and bin length must be considered equally.
`Theoretically it should then be the contractor's task to demonstrate that he is achieving the
`desired standard deviation for the CMPs, but in practice this is all but impossible given the current
`state of available software. We shall show later in the book that when we compute a solution for a
`position we end up with a variance — covariance matrix that looks like this:
`
`(3.13)
`
`where oE2 is the variance in the Eastings direction,
`CTn2 is the variance in the Northings direction,
`and OEN = oNE is the covariance of both parameters.
`The matrix in equation (3.13) gives us the standard deviation in the eastings and northings
`axes, but we can use the matrix further to give us the maximum and minimum axes of the resultant
`error ellipse.
`Note that we may wish to show the ellipse with axes of 2.5amax and 2.5CTmin t o guarantee to
`95% that the true position was within the ellipse.
`When surveyors refer to error ellipses they mean these ellipses derived from the variance-
`covariance matrix associated with the resultant position. We must be careful to note what scale
`factor (if any) has been applied to the ellipses.
`The navigation software on board the seismic vessel will be capable of giving an error
`
`

`

`Relative accuracy
`
`43
`
`ellipse for the vessel position, and very often for the individual nodes of the in-spread network, but
`seldom for the sources, receiver groups and CMPs themselves. This is because the acoustic, laser,
`compass and tailbuoy elements are individually computed, and the task of propagating errors
`through the system to the sources and groups is quite complex and cumbersome. If the software
`gives an integrated solution then the source, group and midpoint ellipses can be derived very easily.
`
`North
`
`Major
`axis
`
`Minor
`axis
`
`Figure 3.6 Error ellipse.
`
`3.4
`3.4.1
`
`Relative accuracy
`In-spread accuracy
`
`In Section 3.3.2 we concluded that rarely would the navigation system be capable of
`producing absolute error ellipses for the CMPs so that we could ascertain the likelihood of any
`CMP falling in the correct bin.
`It is now well accepted that for a 3D survey the conventional use of layback and offset
`measurements to determine the source and front receiver group positions is inadequate; such
`positions are determined using a network of acoustic and/or laser range and bearing measurements.
`
`to another
`
`.25 x nom-
`
`he receiver
`;y both bin
`
`:
`
`hieving the
`the current
`lution for a
`
`(3.13)
`
`ind northings
`the resultant
`
`i guarantee to
`
`the variance-
`ie what scale
`
`mg an error
`
`