I
`
`*
`
`#
`
`I
`
`#
`
`#
`
`æ
`
`-,. ;
`
`tr"'.{*4
`,.!JAç.
`
`4.tE¡t
`
`.il
`
`EXIIIIIT
`DATE
`N,EPORTER
`
`Pl¡nGù DGP6' ttc
`
`PGS Exhibit 1085, pg. 1
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Manual of Offshore Survenng for
`Geoscientists and Engineers
`
`PGS Exhibit 1085, pg. 2
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`JOIN US ON THE I}ITERNET VIA WWW, GOPHER, FTP OR EMAIL:
`WWW: http://www.thornson.com
`G0PHER: gopher.thomson.com
`FTP: ftp.thomson.com
`EMAIL. findit@kiosk.thomson.com
`
`A service of r@p"
`
`PGS Exhibit 1085, pg. 3
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Manual of
`Offshore Surveying for
`Geoscientists and
`tngineers
`
`R.P. LOWETH
`
`w
`
`CHAPMAN & HALL
`London . Weinheim . New York . Tokyo . Melbourne . Madras
`
`PGS Exhibit 1085, pg. 4
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Published by Chapman & HaIl,2--6 Boundary Row, London SEI 8HN, UK
`
`Chaprnan & Hall,2-ó Boundary Row, London SEI 8HN, UK
`Chaprnan & Hall GmbH, Pappelallee 3,69469 tù/einheinu Germany
`Chapman & Hall USA, l15 Fifth Avenue, New York, NY t0OO3, USA
`Chapman & Hall Japan, ITP-Japan, Kyowa Building, 3F,2-2-l Hirakawacho,
`Chiyoda-ku, Tokyo lû2, Japan
`Chapman & Hall Australia, lû2 Dodds Strcet, South Melbourne, Victoria32OS,
`Australia
`Chapman & Hall India, R. Seshadri, 32 Second Main Road, CIT Easr, Madras
`6OO 035, India
`
`Fint edition 1997
`@ 1997 Chapman & Hall
`Printed in the Unitcd Kingdom at the University press, Cambridge
`ISBN O 412 80550 2
`Apart from any fair dealing for the purposes of research or private study, or criticism
`or review, as permined under the uK copyright Designs and Patents Act, l9gg, this
`publication nray not be reproduced, stored, or tra¡¡smitted, in any form or by any
`nìeans, without the prior pcrmission in writing of thc publislrers, or in the case of
`rcprographic 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 Righa Organization outside the UK.
`Enquiries conccrning rcproduction outside the terms stated herc should bc sent ro the
`publishers at the l¡ndon addrcss printed on this page.
`The publisher makes no representation, exp¡ess or implicd, with regard to the
`acclllacy of the information contained in this book and cannot acoept any legal
`rcsponsibility or liability for any errors oromissions tha¡ may bc made.
`A catalogue record for this book is available from thc British Library
`Library of Congrcss Catalog Ca¡d Number: 96 - 72156
`
`Q9 p.i"t"¿ on pennanent acid-free text pap€r, nranufactured in accordance with
`ANsul{ISo 239.48-l9yz and ANSI/NISO 239.48-1984 (pcrmancncc of papcr)
`
`PGS Exhibit 1085, pg. 5
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`3 Accuracy
`3 Accuracy
`
`PGS Exhibit 1085, pg. 6
`
`PGS V. WesternGeco (IPR2014-OO689)
`
`PGS Exhibit 1085, pg. 6
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`92
`
`3.1
`
`3.1.r
`
`Accuracy
`
`The measurement process
`Measurcment
`
`The measu¡ement of a quantity simply means tbe assignment of a numerbal value to
`rEpressil th* quantity. Seismic navigatiø is mostly cocerned with the rneasurem.e¡rt of raqges,
`angles and azimuths, and the first stage in the ûreasurcnrent præss is the deærmination of tbe nnils
`to be used.
`Commm uniæ used in navigation systems re:
`Table 3-l Units of measurement for navigation sysf@r?rs
`System
`Type
`Unit
`Syledis
`Range
`Argo
`Range
`Micro-Fix
`Range
`Marc I aser
`Range
`Azimuth
`Azimuth
`Latihrde
`Longitude
`Range
`
`Metres
`Lanes
`Metres
`Metres
`Degrees
`Degrees
`Degrees
`Degrees
`Metres or
`Milliseconds
`
`Gornpasses
`DGPS
`
`Acoustics
`
`In Table I, the ouþut units a¡e shown, not the mÊasurement units. For inst¿¡ce, in the case
`of Micro-Fix the inæmal measr¡nenrent units are time, not distarce.
`Ha.'ing &fined the units of measr¡re¡nent, the æxt stage in tbe præess is to deffne a model
`which ryprocimates but simpliûes the physircal rcality. For example, in the case of a rneasuredradio
`range we uae a ñathematical model ç¡hich alærs the measured range
`¡s 6ansmitter 6¡d
`ræeiver erlotìs, tbe velocity dpropagation, the ûequency ofthe carrie¡ wave, qnd so ø.
`The thirìd and fit'al stage in the ûreasunement prwess is to define a prncedune. Using this
`præa¡¡e, observations arìe tttade and transformed usirg the mnthematical model to produce the
`values rcquircd-
`3.1.2 Errors
`The errq of a value is defiæd as
`E = M -T
`
`(3.1)
`
`PGS Exhibit 1085, pg. 7
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`The measurement process
`
`3Ít
`
`where
`
`E is the error
`M is tbe measured quantity
`Iis the tn¡e vah¡e.
`At this early stage it must be reatized that the true value is never known! Nevertheless, the
`above definition allows us to deærmine tbe accuracy of an obse.rvation, which is the difference
`between M and I. Unforû.¡naûel¡ since we never know the true value, we can also never know the
`eccuræy of an observation- What we need is a method of estimating tbe açqurary.
`
`As an s¡¡¡mPle' suppo$e we measure the dist¿nce betrveen a shqe station and an antenna on
`tbe vessel' The raw measured range is (say) 12510.3m. we thea modify this range ûo take æcount
`of knmm etlors at the transmitter (the station ashore) and the t*i*'fthe anæãr,a on the vessel)
`and the assr.¡med velocity of propagation fon the 0<nwn) firequeacy of the transmitæd r¡go¿. rro*
`do we knor¡¡ that the rcduced range is now accurate? The truth-is. wã don'L AII we can do is take the
`uhæt ca¡e with both the measulement process and the reductim process. One of the best
`precautions we can take is to enstre that we rneasure seve¡al ranges from ¿tfferent stations to lessen
`our chances 6f making a large rnistake.
`Now we need to introduce another term - precision. Many people use the terms accuracy
`and precisim indiscriminately, but there is a fundament¿l ditrereice betweea the two tencrs.
`Aærrracy is the diffe.rence betrveen the measured value and the true value; precisim is a measure of
`the reliability of an observation- we can make s very precise rneasurement of a range between a
`shore station and the vessel, but if the sh<¡¡e statim. coordinaæs are incqrectly defined the range
`wi[ be very inaccuraæ. hrecisiø will be defined in statistical terms in a later section.
`There are three types of errøs:
`
`(a) Systematic errors
`
`Systematic errors arise from an incorrect choice of mathematical model. Consider the
`fo[or*ring example:
`In Figure 3.1 the arowed lines ¡spresent measrned laser azimutls from the stem of the
`vessel to targets æ the paravanes suppcti4g the two sources and süeamers.
`We can model the azimuths as follows
`Xr = Xr+E
`XTis the true value
`Xu ts the measuredvalue
`E is the alignment er¡or.
`
`where
`
`G.2)
`
`PGS Exhibit 1085, pg. 8
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`34
`
`Accuracy
`
`heading
`
`Alignment error
`
`Source 2
`
`Source I
`
`Sheamer 2
`
`Streamer I
`
`Figure 3.1 Laser azimuths
`
`If we do not calibraæ the laser, the alignment error is unlnown and equatim (3.2) can
`never be satisfied, so we have created a systematic error because all measurements will be in eror
`by a cmstant âmount.
`
`PGS Exhibit 1085, pg. 9
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Statistical terns
`
`35
`
`(b) Random errorc
`If two acoustic transducers were deployed from booms eitler side of a vessel travelling at
`constånt velæity and we tlen measu¡ed the acoustic range between them at 2 second intervals, the
`measurements would vary stightly even if we observed in ideal conditions of no wind, tide, etc. Tbe
`sligbt variations in the measured range will be due to variations in the temperatu¡e, salinity and
`prcssure of tbe water, variations in the acoustic path and the inability of the equipment to resolve an
`er(actrange. The variation in the readings is a measure of their precision - the lower the variation.
`the more precise the readings. The variations themselves are random errors.
`
`(c) Blunders
`
`Blunders are third in the list of error types because they are tbe least courmon - systematic
`and random errors are almoot always present, which is not the case for blunders. But blunders often
`have the most disastrous effects.
`Typical blunders experienced in ou¡ work are using the wrong algebraic signs for
`transføming one datum to another - for instance zubstitr¡ftng minus signs for plus signs in the
`transfømation from WGS84 to AGD84 would leave the AGD84 coordinates about 400m from
`their real values! Another good example is using the incorrect offsets when computing receiver
`group positions down a streamer.
`The catastrophic effects of blunders are the main reasons why suweyors rì.re taugbt to check
`their work by independent methods. This applies from the most basic of trigonometric
`computations to the most complex of networks. It also applies to seismic surveys; each step of the
`measurement process should be independently checked.
`3.2
`Statistical terms
`9.2.1 Random variables
`A random variable is one whose value depends upon chance. The acoustic ranges measu¡ed
`in sectiø 3.1.2(b) constiû.rte a sample of the acoustic range - the acoustic r¿ìnge is the random
`variable. The fact that we are using the word snrnple irnplies that we have only looked at part of the
`whole. The whole is called the population, which is what we wcx.ild end up with if we continued
`our meâsuremen¡s ¿¡ infinite number of times.
`As a n¡le we only take small samples of each random variable in positioning; only in cases
`such as the final GPS phase measurements to locate the drillstem of a rig would we take a large
`number of observations. So we are confronted with the problem of having to deduce characteristics
`of the population from a very small sample. R¡rthermore, we have to use these deduced parâmeters
`to estirnate the reliability of the measuements and the most tikely value of the random variabie.
`
`PGS Exhibit 1085, pg. 10
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`36
`
`Accuracy
`
`3.2.2 Frobablllty denslty functlon
`The probability density furcti@, Ø (*\ , of a random va¡iable x is rhe fimction whose
`integral grves tbe probability P(a, b) of x lying betrveen the values a and å. Expressed as an
`equation this is:
`
`dx
`
`(3.3)
`
`b
`
`Ja
`
`P(a,b) = lØ@)
`
`This is equivalent to the shaded area in Figure 3.2 ;
`
`Ø(x)
`
`a
`
`b
`
`x
`
`Figure 3.2 Probability density fr¡nction.
`
`Now, a probability of I is taken tomean that an event is certain to occur, and we know that
`f must lie betrvee¡ - infini¡y and infinigy, so we can write:
`
`I ØG)
`J
`
`dx
`
`1
`
`(3.4)
`
`PGS Exhibit 1085, pg. 11
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Statislicaltenns
`
`37
`
`3.2.3 ExpeCtatlon
`Tbe expectation of a random variable is defined as tbe mean value taken wer tbe
`pqulation. which is:
`
`(x
`
`dx)
`
`(3.s)
`
`xE ( )
`
`xØ
`
`_t
`The important point arising bere is that if we har¡e an estimate of ¡ which is equal to the
`erpectation of ¡, then the estimate is said to be an unbiased estimate.
`3.2.4 Varlance and standard devlatlon
`If a random variable ¡ has expectation Ç, io variance o*2 i. defined as:
`
`o? = E{(x-Ð2t
`
`(3.O
`
`The stendard deviation of x is tbe positive square root of tbe variamce.
`Now, rhe above definitions are in terms of expectations, and we have said in Section 3.2.3
`that the elçectation of a rmdom variable is its mem value taken or¡er rhe populatioo- We har¡e also
`said that we only take a sample of the populatiæ, so we now define the mean, variance and
`standrd deviation in terms of the sample.
`The mean of the sample is
`
`(3.7)
`
`(3.8)
`
`n
`
`I L*t
`n i= I
`which is simply the zum of all the observed values divided by the numbe¡ of values
`The variance of g¡s sample is
`
`r
`
`nE
`
` @¡- x)2
`j= I(n-l)
`
`o?=
`
`and the standard deviation of the sample is
`
`PGS Exhibit 1085, pg. 12
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`38
`
`Accuracy
`
`(3.e)
`
`nE
`
` G¡- x)z
`i= 1
`(n-L)
`
`or=
`
`3.2.5 Govarlance
`The last d the statistical terms we need to look at is covariance. If two randm variables ¡
`and y have expectations € and qr respectively, then the covaria¡rce of .r and y is defined as
`
`(3.10)
`
`(3.11)
`
`a*y= E{(x-€)(y-\r)}
`
`Translating this from expectations ¡s snmples, we get
`
`nI
`
`i=1
`
` ('r-¡) (y¡-I)
`(n - 1)
`
`('
`
`xy
`
`Covariance is a term used extensively in least squares and Knlman filæring, and we shall
`investigate its uses later.
`
`3.3
`
`3.3.1
`
`Absolute accuracy
`General dlscusslon
`
`Marine seismic survey positiæing specifications are very ofæn issued as a disjoinæd series
`of figures such as:
`
`a
`
`a
`
`a
`
`a
`
`Argo
`
`DGPS
`
`Feather
`
`Shot inærval
`
`Compasses
`
`+/- 10m
`+/-5m
`
`< l0deg
`
`25 m +/- 1.5m
`< I degree from mean of all compasses
`
`PGS Exhibit 1085, pg. 13
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Absolute accuracy
`
`39
`
`. Acoustics range
`+l-2m
`Whet we really sh@ld be doing is specifying the survey in ærms of the quantities we
`require. If we position the vessel to better than 10 m and the compasses are less thnn a deg€e
`from
`their mean value. what dæs that do to our receiver group positions? If the æq.rstic ranges are betær
`than 2m what does that mean in ærms of our source positiæ? To say that we require the Argo
`ûolerance to be betær than 10m is really quite nonsensical. What we have to do is define a foolprod
`method so that we can ensure that, providing our specifications are met, our cornmon mid point
`positions will have the required accuracy.
`Consider Figure 3.3 :
`
`P
`
`A
`
`B
`
`Figure 3.3 Two-bearing fix.
`The arowed lines represent two bearings from stations A and B. They intersect at point P,
`which is the position of the vessel. I[e therefore need just two lines of position (LOPs) to define a
`two-coordinaæ position in ærms 6f s¿5rings 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 :
`
`C
`
`a
`
`I
`
`P
`
`"'-'l ¡ -
`
`A
`
`B
`
`Figure 3.4 Three-bearing fix.
`
`PGS Exhibit 1085, pg. 14
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`æ
`
`Accuracy
`
`Now the point P is not so easily defined; tbe obvious choice is to place it inside the triangle
`created by the tb¡ee LOPs, but zuppose the three bearings contained a systematic e,rror wnicn, if
`ccnpe¡rsaæd for' sbifted all thrce LOPs to tbe left? The result is that the position of the vessel is
`shnfted to X, a point which is outside the original triengte. The enswer here would be to model for
`tbe unkno\¡m systematic error - something which is done quite regularly for radio positiæing
`ranges
`an r nknwn systematic scale error due ûo propagation.
`We can srrmmariz€
`the above by saying that although the position of the vessel is not knor¡m
`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 unlnowns we need at least n observations. If we have more
`thsi¡: n observatirrrs, we have redundancy.
`We catr propagate this nrle threrghout tbe spread, ensuring that there is redundancy in
`-
`foding all tbe parameûers we want ûo know about. including thð vessel position, the source
`positions, the streamer shapes, the tailbuoy positions and the r"""iu"t goup poriti*r.
`' Specification I - ensure that there are redrrndant observations fon every facet of the entire
`
`spread.
`
`We must also be ca¡eful not to go to tbe other exEeme and provide or¡er-redundancy in tbe
`system, because the computing time required to adjust a particular nenvork increases as the square
`of the number of observatirns: in other wmds if we look at the vessel position on its own, the
`difference in computing time betrveen usitg ten ranges and using three ,rrrges is 10019 or ll times
`as long- Not to mention the increase in cost, which probably riseJ at the same proportional rate.
`3.3.2 Bln slze and accunacy
`The only value in positioning tbe ship accurately is so that we can propagate that accuracy
`throug[ our va¡ious in-spread systems. If we could somehow find absoluæ positioning poinæ ø the
`sourses and streamers' the positiø of tìe vessel would become irrelevant. This is-becoming a
`reality with the advent of shæk-mounted DGPS and RGPS receivers on the sources and stre¡mer
`heads.
`- I-et us suppose that we a¡e conducting a 3D suney in which the bin size is to be 50m long
`and 25m wide- Horv shoild we deærmine the accuracy d the corrrmon mid-points so th*t we can be
`cøûdentof them falling in the right bins? Weneed togo back to our theoryof st¿ndard deviations.
`Figure 3.5 shows the probability density ñ¡nction. The curve is a normal d¡stribution,
`and it can be shown that for any populatim this cun¡e will result. Its equation is
`
`(3.r2)
`
`PGS Exhibit 1085, pg. 15
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Abaolute accuracy
`
`41
`
`Ø(x)
`
`-2s
`
`-s
`
`s
`
`2s
`
`x
`
`Figure 3.5 Probability density fr¡nction.
`
`where o is the standard deviation and n is tbe total number of obsen¡ations. The shaded area under
`rhe cun¡e in Figue 3.5 is the area lying betrveen -o and r-o. We can use equation (3.12) to
`waluate tbe area under the curve fs various tiryts - see Tabte 3.2:
`Table 3.2 Area under the normal curue for various standard deviations
`Umits
`Area 96
`-(' to +o
`68.3
`-2O to +2C
`95.4
`-3C' to {€O
`99.7
`
`Now the percentage area under the curve given in colunn 2 of Table 3.2 ß the pe.rcentage
`probability that any devi¿tion is be¡¡¡een the corresponding limils.
`Going back Ûo q.r problem of how to specify the accuracy rcquired for a bin width of 25m,
`if we set the required standard deviation of a commm mid point fCfiæl ûo be 6.21mwe will be
`68-37o sure that all CMP's fall q¡ithin *6.25m, 95.4Vo fall within t12.5m, a1¡d99.7To fall within
`*18-75m. Only O.37o will fall qrtside the lirni¡s of t18.75m. \ñ/e specified a bin width of 25m, so
`
`PGS Exhibit 1085, pg. 16
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`42
`
`Accuracy
`
`we can be 95.4Vo cerrt¡in of CMPs fa[ing in the right bin. This can be generalized into another
`specification:
`
`a
`
`Specification 2 - t}re standard deviation of all common mid points should be 0.25 x norn-
`inal bin width.
`
`A basic assumption here is that the bin width is the overririing parameter, since the receiver
`gloups cannot move so much in tbe inline direction. In the case of a bottom cable sun'ey 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 - cova¡iance matrix that looks like this:
`
`(3.13)
`
`l"z
`Lo""
`
`or"]
`"il
`
`where o"t ir the variance in the g¿stings dfuectim,
`o¡2 is the va¡iance in the Northings d.i¡ection,
`and cr6¡v = o¡¡¿ 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 2.5o-"* üd 2.5oo¡o to guarant€e
`95Vo th,at the tn¡e position w¿s nithin the ellipse.
`When surveyors refer to error ellipses they mean these ellipses derived from the va¡iance-
`covariance matrix associated with the resultant position. We must be careful to note what scate
`factø (if any) has been applied to the ellipses.
`Tbe navigation sofnpare on board the seismic vessel will be capable of giving an error
`
`to
`
`PGS Exhibit 1085, pg. 17
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Relative accuracy
`
`¿liì
`
`for the vessel positiou, and very often for the individual nodes of the in-spread network, but
`ellips€
`seldom for the sqlrces, receiver groups and CMPs themselves. This is because the acoustic, laser,
`compass and tailbuoy elements a¡e individually comp'uted, and the task of prqagating errors
`through the sysæm to the sources and groups is quite complex and cr¡mbersome. If the sof¡vare
`gives an inægrated solution then the sq¡llce, goup and midpoint ellipses can be derived very easily.
`
`North
`
`Major
`axis
`
`Minor
`axrs
`
`Figure 3.6 Error sllipse.
`
`3.4
`
`3.4.1
`
`Relative accuracy
`ln-spread accuracy
`In Section 3.3.2 we cmcluded that rarely would the navigation system be capable of
`producing absolute error ellipses for the CMPs so that we could ascert¿in the Likeühood of any
`CMP f¿Iing in the conect bin.
`It is now well accepæd that for a 3D survey the conventional use of layback and offset
`measurements to determine the source and front receiver goup positions is inadequate: such
`positions are determined using a network of ac¡ustic andlor laser range and bearing measurements.
`
`PGS Exhibit 1085, pg. 18
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`4
`
`Accuracy
`
`A simple 3D spread may look like this
`
`\.
`
`Þ.-
`
`I
`
`Tailbuoy receivers
`\^
`^
`<azuompasses
`tð/ Acoustic oods
`O Vessel reôeivers
`
`Figure 3.7 Simple 3D spread.
`
`PGS Exhibit 1085, pg. 19
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Relative accuracy
`
`45
`
`In Figure 3.7 we har¡e two acoustic Em¡ponders towed from boøs either side of the hull.
`They are thernselves positi@ed acoustically frm trnnsducers mounted on the hull, and form a
`baseline of about 50m in length. The front eods of both sueamers, and the source pæitions, are then
`determined by acoustic ranges from the boom nodes. Now' as Figure 3.7 is d¡awn the netrvo,rt
`fsmed by the æoustic ranges at the front ead looks very solid; all the int€rnal qngles in the network
`are quiæ large, ald it could be assumed at first glsnce that we will de¡ive positions fø the souraes
`md front rpceivers of high precision ¿¡al high ¿rccuræy, also assuming that we use an acqrstic
`syst€m capable of measr:ring ranges with a precisioo of bèüer than lm tbat has been properly
`calibrated. tilhat co¡fd possibly go wrong?
`
`The first point is that the whole network must be orientaæd into a grid of some sorr Exactly
`which grid is used doem't maüer at this stage; it rnay be based on the course made good, a local
`C-artesian grid, tbe nominal sail line, the projection's grid, and so on. Whicbver grid is used, the
`orientatiø of the network is critical. It can be ascertained using the vessel gyro, the front suear¡rer
`compasses, or a combinatiæ of both. It is mostcommon for the boom baseline ûo be computed first
`and then to be orientated using the vessel's g¡rro, because the hull transducers are ûxed with a
`knon'n offset. The boom baseline is the¡r also held fixed and the front end network is computed
`using that inviolate baseline. So the gyro is absoluæly critical to the computation.
`
`The second point is that Figrrre 3.7 has been deliberately drawn at a distoræd scale,
`because that is how the front-end network is usually disptayed in order that the individual nodes
`and ranges can clearly be shown. \Mth s¡l inline offset frm navigation refercnce point (NRP) ûo
`sor¡rees of 2(X)¡n. NRP ûo front goups of 350m, and a boom length of 5Om, the diagram drawn to
`scale looks as in Figrrre 3.8 .
`Now we can see that the inærnal angles of the network have becme very small, and will
`also recognize that small errorìs in the acoustic ranges will cause quite large variations in the source
`and group positims. Consider the following arguments:
`' An error of 1.5 deg in the gyro heading will cause the ûont receiver goups to be mis-
`placed by 9.2m.
`
`An error ú2m in me range from the boons to the front receivers, and a missing cross
`range at the fræt receivers, will introdr¡ce an errø of, no less than 45 m in tb front receiv-
`er group position.
`
`Ilerein lies the third impøtmt specification:
`' Speciûcation 3 - the inline offset from the NRJ ûo sq,rrces and front receiver groups
`should be as short as pæsible, ¡or inc:rease the accr¡¡acy of the front-end network.
`
`PGS Exhibit 1085, pg. 20
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`46
`
`Accuracy
`
`0
`
`50
`
`100
`
`metres
`
`Figure 3.8 Front-end drawn to scale.
`rf the inline offset has ûo be large because of problems with vessel noise, ûowing
`ârrangements' etc" we cîn rninirni-,e tbe effects by using an inægraæd approach ûo positioning and
`by usittg laser as well as acoustics. Both of these subjects ¿ue @vered later in the book.
`Assuming that the front-end acoustic network is computed as a separate entity, which is the
`case with most softwale packages currently available, we wiú end up with a variance - co¡ariance
`-uttï which expresses the reliability of the nodes of that o"t*orl in relative terms; tbat is, the
`standa¡d deviations and error ellipses of each node will be known relative to all the othe¡ network
`nodes' The problem we will look at in the next section is how to relate these varianaes to those of
`the vessel position and in h¡¡n to the CMp positims.
`3.5
`
`Propagation of variance
`In Section 3'2'4 equation (3-9) gave us the standard deviation of a sample. when this
`standard deviation is an unbiased .rtimaæ it is commøty called the standard error of a
`measurement, s¡.
`In general, if
`
`z = ø) (xy x2, x3, ..., x n)
`
`(3.14)
`
`PGS Exhibit 1085, pg. 21
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Propagation of variarrce
`
`47
`
`where z is aparameter related to me¿surements x by the function Ø, *atle standard error
`of x1 is sr, it can be shown in matrix notation that
`c, = øc *ø,
`where C, and C" are the cova¡iance matrices forx and z respectively.
`This is Gauss's propagation error law for linea¡ equations, and can best be illustrated by an
`example:
`
`(3.1s)
`
`P x
`
`,y
`
`N
`
`c[
`
`R
`
`o F
`
`igure 3.9 Range and bearing
`In Figure 3.9 the point P has been positioned from point O by the measu¡ed range r and
`bearing cr- The coordinates of P are then given by
`-x = rslnct
`), = rcoscr,
`The measu¡ements r and a aÍe made and their standard erïors estimated as follows:
`r = 25O.0O¡n
`cr = 3O.0O deg
`o. = 0'O5m
`oo = 0.0055 deg (9.7 E-5 rad)
`We want to find values fo¡ x and y, the coordinates of F, and their standard deviations.
`Unforonaæly, equation (3.16) and equation (3.17) are non-linear, and we fi¡st need to
`linearize them so that tve can use Gauss's law. This is done by taking pafrial d,ifferentials as follows:
`
`(3.1o
`
`(3.17)
`
`PGS Exhibit 1085, pg. 22
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`(3.18)
`
`(3.le)
`
`(3.20)
`
`(3.2t)
`
`(3.22¡
`
`(3.23)
`
`48
`
`Accuracy
`
`ðx = slnc(
`dr
`
`òx = rcoso(
`da"
`
`ðy = coscl
`òr
`
`ây = -rsincr
`ðot
`
`Norv we can form the matrix,Ifrom the partial derivatives:
`ðx a"-l
`ñ
`-tðcx,l
`ayl
`EJ
`
`Ðð
`
`r
`
`J-
`
`fsita ,"o,.,,l
`f"oro -rrirrl
`
`The cor¡ariance matrix Cro is:
`
`| "l o.,",l
`Lo'", o3J
`and we can now use equation (3.15) to propagate the variances:
`
`Þ"r*ro-' o I
`L o e.+xro-eJ
`
`=
`
`c *, = JC ,oJr - þ-ot"to
`[8.31x10
`The¡efore we have solved the problem:
`x = 125.00m
`Y = 216.5lm
`o' = 0'03m
`oY = 0'tlm
`Note that, although equation (3.23) has zeros for the covariance of r and cr because the two
`
`-3
`4
`
`8.31x 10
`
`2.03x10
`
`l
`
`(3.24)
`
`PGS Exhibit 1085, pg. 23
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Sumrnary
`
`49
`
`rneasurements are assumed to be uncorrelated, the values ¡ and y are correlated and so have values
`in equation (3.?t+:¡-
`Also noæ that the vuiance - cova¡iance matrices are symnetrical _ this is always the
`case and has important implications when
`computations in terms of speed savings.
`The abor¡e example is a very simple one. Withotrt going into the mathematics d the
`probleor' we can quickly see rhe complexity of trying ûo propagate the variances and cor¡a¡iance.s of
`an absoluæ position on the vessel throu$ a front-€nd network of acoustic nodes, thence into
`receiver goups and source positions. This computation and derivation of the associaþd error
`ellipses is quite cmplicated and takes an Bppr€ciable amqnt of computing time. However, if we
`use În inægrated syste.m that can tie all ou¡ absolute and relative observations together inûo a single
`network' the va¡iances and cor¡ariances are atready in place as part of the main computatim. This is
`oæ of the key advantages of using an inægrated Kalman filær; and is explored in ætaif hter in the
`book.
`
`3.6
`
`Summary
`We have defined 8ll the main st¿tistical variables used in seismic navigation, and have noæd
`tbe difference between ¿Ìccuracy and precisiø. We know the three different types of error and we
`know horr ûo propagate va¡iances a.d covariances tbrougù a network.
`We have elicited three valuable points to look for when writing specifications:
`' Speciûcation I - ensu¡e that there are redundant observations for every facet of the entire
`' Specification 2 - thÊ, standa¡d deviatiø d all corlmon mid points shæld be 0.25 x nom-
`inal bin widttl
`' Specification 3 - tbe inline offset from the NRP to sq¡rces and front receive¡ groups
`should be as short as possible, ûo increase the acsuracy of the front end o¡^n*È
`There are other facûors we need to addæss, such as streamer shaping, the use of ætive
`tailbuoys and so on. These will be dealt with later.
`
`spread.
`
`PGS Exhibit 1085, pg. 24
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`4 fntroduction to computations
`
`PGS Exhibit 1085, pg. 25
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`52
`
`I nf od uction to computati ons
`
`4.L
`
`Introduction
`This chapter deals with the ba.sic concepts of positioning at s€¿; we start with an
`introdrrtion to least squares, on which all modem positioning computations are based, and then
`develorp the variqrs formulae used in the computations.
`We are going to put into a singte chapter the information that is disse.minaæd to
`undergraduate suneyors in abort a year of shrdy, so sorne of the detailed explanations and proofs
`will necessarily be shste¡ed.
`Before going straight into least squanes, we will briefly revise the co<¡rdinate systems
`available ûo us in the context of computations.
`
`4.2
`
`Coordinate systems
`The elllpsold
`4.2.1
`The ellipsoid is the mathematical figure which approximates most clmety the tn¡e shape d
`the eartlu Unfortunaæly, many people have tried to establish the best-fit ellipsoid for the ea¡th, and
`many of the ellipsoids they calculated a¡e in use. Life woutd be very much easier if there were æly
`one ellipsoid (or spheroid).
`In Australia we generally use the Australian füodetic Datum as a datum for our ofßhore
`flrrveys. Even this is somewhat complicaæd by the following facts:
`' There are two Australian datums in use - AGD66 and AGD84.
`' Neitler of the two datums is geocentric.
`' Australia intends to mor¡e to a geocentric datum in 2000.
`The AGD66 datr¡m has the following definiticn:
`Semi-axis majø: 637 8 16O.Om
`Flatæning : I t298.25 exactly.
`The mins axis of the spheroid was defi.ned in 1966 to be parallel to the earth's mean axis of
`rotation in 1962 (this was laær changed in 1970), and the meridian of zero lmgihrde was defined as
`being parallel to the Bureau Inte¡national de I'Heure CBtrÐ merirlian plane near Greenwich. The
`centne of the spheroid was defined by the coordinates of Johnsûon C¡eodetic Station, a station in tb
`cenEe d Australia. At that time it was assumed that the spheroid - geoid separaticr was zøo at
`Johnstm, and also zero at all the otber geodetic stations listed in the 1966 adþstment.
`Since 1966 a huge amount of infsmation on the shape of the geoid has become available,
`particularly through saÞllite observations, and it was realized that the lfX6 adjustment was no
`løger accurate. In 1982 all the infqmation then available was put inûo a rrcw least sErares network
`
`PGS Exhibit 1085, pg. 26
`PGS v. WesternGeco (IPR2014-00689)
`
`

`

`Goordinate systerns
`
`53
`
`adjustuent called GMA82 (C¡eodetic Model of Australi a 1982). This resulþd in I new Adopæd
`C-oordinate Set in lg84., which we refer to as AGD84 (geographical coordinates) and AMGB4 G¡¿
`coordinates). The GMA82 adjustment is a truIy spheroidal adjustmenr so any obsen¡atiøs used in
`conjunction with AGD84 ø AMGE4 must f¡st be æduced to the spb,roid. InWesÞrn Australia the
`d-ifference in eastrn,Ss and northings between AGD66 and AGD84 is of the order of 3 - 4m-
`Unforh¡nately, but not surprisingly, the geodetic network stop@ when it reached the coast, so we
`can only extrapolate AGD66 positions otrshore. Furthermore, the AGD66 adjustment was not a
`ruly spheroidal adjustment, so there is no single exact shift that will mor¡ã coorrdinates from
`AGD66 to AGD84
`At tb uun of the millennium Australia inænds ûo change its datum yet again, this trme to a
`geocentric datum. The new datum will be called GDA92 and is very close to WGSB4, the earth-
`cenhed dan¡m used by GPS. The hope is that the entire world will eventually adopt WGS84s or a
`simil¿¡'datr¡m- Because AGD84 is a tn¡e spheroidal dah¡m, \ñ/e can derive a seven-paraûreær shift to
`get frm AGD84 to WGS84 and vice versa. The seven-par:nrneter shift is listed in Table 4.1:
`
`Table 4.1 Seven-parameter shift from AGDS4 - WGSS4
`Parameter WGSS+AGD&4
`AGDs+WGS84
`dx (ml
`116.m
`-116.00
`ffi.47
`dY (m)
`-æ.47
`dz (m)
`-141.69
`141.69
`rx ()
`0.23
`-o.23
`rY ()
`0.39
`-o.39
`rz ()
`-o.34
`o.3M
`s (ppm)
`{.0983
`0.0983
`
`Because vessels conducrirg seimic work often sail great circle lines, it would mal<e sense
`Ûo compute s¡1 rhe surface of the spheroid. But to obtain