ISSN 0309-7846
`
`THE
`HYDROGRAPHIC
`JOURNAL
`
`incorporating
`THE CORPORATE MEMBERS DIRECTORY
`and
`THE HYDROGRAPHIC JOURNAL INDEX
`
`No. 77
`
`JULY 1995
`
`WESTERNGECO Exhibit 2144, pg. 1
`PGS v. WESTERNGECO
`IPR2014-01478
`
`

`
`The
`Hydrographic
`
`Journal
`
`Number 77
`
`July 1995
`
`Articles
`3. The Royal Naval Hydrographic Service 1795-1995
`by R.O. Morris
`11. A Rigorous and Integrated Approach to Hydrophone
`and Source Positioning during Multi-Streamer
`Offshore Seismic Exploration
`by V. Gikas, P.A. Cross and A Asiama-Akuamoa
`27. Limnology and Hydrographic Survey: The Example
`of Lake Malawi
`by C.G.C. Martin
`Regular Features
`33. News from Industry
`
`39.
`
`42.
`
`Literature
`
`Letters to the Editor
`
`45. The Chairman's Column
`by KWK
`46. Rhumb Lines - Personal Views by Sinbad
`48. Reflections
`by M. Boreham
`50. Review
`
`60. Principal Officers and Council Members
`Obituary
`43.
`John Barry Dixon
`Special Features
`53.
`-Memorandum of Association of the Hydrographic
`Society
`
`54. Articles of Association of the Hydrographic Society
`
`58. Regulations for the Establishment and Conduct of
`National Branches of the Society
`
`59. Regulations for the Establishment and Conduct of
`Regions of the Society
`
`The Corporate Members Directory 1995-96
`
`The Hydrographic Journal Index 1972-1995
`Information Bulletin
`Special Announcements
`
`Membership News
`
`From the Regions
`
`News from FIG Commission 4
`
`News from the International Hydrographic Bureau
`
`General Notices
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`Employment Wanted
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`Calendar
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`Society Information
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`
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`•
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`© 1995 - The Hydrographic Society.
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`
`WESTERNGECO Exhibit 2144, pg. 2
`PGS v. WESTERNGECO
`IPR2014-01478
`
`

`
`A Rigorous and Integrated Approach to
`Hydrophone and Source Positioning during
`Multi-Streamer Offshore Seismic
`Exploration
`
`V- Gikas*. P.A. Cross* and A. Asiama-Akuamoa**
`
`Abstract
`This paper describes a rigorous and integrated approach for positioning sour ces and hydrophones within a seismic spr ead that may
`contain multi-vessel and multi-streamer configurations. Any number of observations relating to any point(s) within the spread can be
`accommodated. Quantification and analysis of error propagation within the spread are provided. Test results based on the
`implementation of the algorithm on a UNIX platform are discussed.
`Resume
`Cet article donne la description d'une approche rigoureuse et coherente des sources de positionnem ent et des hydrophones dans un
`deploiement sismique que les configurations multi-vaisseaux e t multi-streamer peuvent englober. On peut utiliser un nombre
`indifferent d'observations relatives a n'importe quel(s) points(s) compris dans le deploiement. La qualification et l'analyse de la
`propagation d'erreurs a l'interieur du deploiement y est donnee et les resultats des tests bases sur l'execution de l'algorythme sur une
`plataforme UNIX y sont discutes.
`Resumen
`Este articulo describe un enfoque riguroso y armonioso sobre las fuentes de posicionamiento e hidrofonos dentro de un despliegue
`sismico que las configuraciones multi-buque y multi-streamer peuden contener. Se puede usar cualquier numero de observaciones
`referentes a un punto(s) cualquiera del interior del despliegue. El articulo proporciona la cuantificacion y el analisis de la propagation
`de errores dentro del despliegue y discute los resultados de las pruebas, basados en la implementation del algoritmo en una plataforma
`UNIX.
`"
`,
`
`1.
`
`Introduction
`
`The basic configuration of an offshore seismic exploration
`survey
`is as follows. One or more vessels sail
`in
`approximately straight lines whilst towing a number of
`'streamers' (typically up to 6 kilometres long) and 'seismic
`sources'. The streamers carry a number of hydrophones
`(typically 50-100 per kilometre) and are towed just below the
`surface of the water [Morgan, 1992]. At a specified distance
`interval (typically every 20-25 metres) one of the guns is
`fired resulting in seismic waves which travel through the
`water and penetrate the subsurface. The times of arrival of
`the reflected and/or refracted signals are then measured by
`the hydrophones. The surveying problem is to determine the
`position of the guns and hydrophones at the instants of firing
`and reception respectively. In principle the position of the
`vessel is of no interest - except, of course, for navigation.
`In recent years the problem has become increasingly
`complex, mainly due to an expansion of the type and
`quantity of survey data collected. In a typical configuration,
`Fig. 1.10, measurements will include compass orientations at
`points along the streamer (typically 4-7 per kilometre), laser
`ranges from the vessel to a variety of floats (for instance
`those carrying the guns and those at the front of the
`streamer), underwater acoustic measurements (of
`the
`distance) between a number of points at the front and back
`of, the system (referred to as the 'front-end' and 'rear-end'
`acoustic networks), the position of the tailbuoy and the
`position of the vessel (both typically, but not necessarily, by
`
`DGPS). More complicated systems may also include
`acoustics throughout the length of
`the streamer and
`additional navigation devices on the vessel. Moreover, in the
`case of several vessels operating simultaneously, between
`vessel measurements would also be made.
`The most common approach currently applied to the
`positioning problem is to treat each epoch, and each
`measuring system, more or less independently. So both the
`laser and acoustic measurements are used to transfer the
`position of the vessel to the floats, while the front-end
`acoustics relate the floats to the guns and front-end of the
`streamer, and then the compasses determine the streamers
`shape. The rear-end acoustics and the tailbuoy positioning
`serve to provide some control of the orientation and stretch
`of the streamers. Typically the process will involve some sort
`of curve fitting operation for the compasses [Ridyard, 1989],
`and several independent 'network adjustments' for the
`acoustic and laser ^networks. It is possible that the process
`will involve 'iterating' several times through the various data
`types in order to 'best fit' (in some rather general sense) all
`of the measurements.
`<
`Although this approach is probably perfectly satisfactory
`from an accuracy point of view it suffers from two major
`disadvantages. Firstly it is highly 'case dependent', i.e.
`relatively small changes to the configuration or measurement
`set may lead to major changes in the processing software -
`something that is especially difficult in real-time (or quasi
`real-time) quality control. Secondly, and probably most
`importantly, it is extremely difficult to analyse the error
`
`"University of Newcastle upon Tyne, UK.
`**QCTools Ltd, Houston, USA.
`
`11
`
`WESTERNGECO Exhibit 2144, pg. 3
`PGS v. WESTERNGECO
`IPR2014-01478
`
`

`
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`sonardyne transceiver
`
`towfish
`
`acoustic device
`
`laser device
`
`seismic source
`
`^ compass bird
`
`active tailbuoy
`
`Fig. 1.10: Typical dual source triple streamer configuration
`propagation through such a process - hence it is almost
`of course, is to tune the filter properly in real time -
`impossible to describe the precision and reliability of the
`and the fact that some have failed to do this in the past
`final gun and hydrophone positions. This aspect is becoming
`has led to Kalman filtering gaining a poor reputation
`increasingly important as clients require proof (often in real­
`in some circles.
`for a highly complex
`time) that the survey specifications are being met.
`The use of a Kalman filter
`There is hence a need to develop a completely general (for
`seismic configuration enables a rigorous computation
`flexibility purposes) and rigorous (for error propagation
`of precision and reliability measures such as error
`purposes) approach
`to
`the positioning of guns and
`ellipses and marginally detectable errors respectively
`[Cms et al, 1994]. If a step-by-step approach is
`hydrophones during seismic exploration and this paper is an
`attempt to address that need. It describes the mathematical
`adopted (such as curve fitting
`the compass data
`basis, implementation and testing of a Kalman filter that can
`followed by fitting the results to the acoustics and then
`in principle, handle any geometrical configuration (i.e. any
`to the navigation data) it is almost impossible to
`number of vessels, streamers and guns) and any set of
`compute these measures.
`observations.
`Due to its ability to predict the network, a Kalman
`Kalman filters have, in the past, not proved popular with
`filter is a far more powerful tool than simple least
`the offshore positioning community and most offshore
`squares for quality control. Much smaller outliers and
`operators currently prefer simple and independent 'epoch by
`biases can be found by Kalman filtering
`than by
`epoch' least squares computations. For this reason a brief
`simple least squares. It is, however, recommended
`review of the advantages of using a Kalman filter is included
`that, where possible, simple least squares also be
`before describing the models used in detail.
`carried out at every epoch in order to identify (and
`correct or remove) the larger outliers. This is because
`1.1. Kalman filtering versus simple least
`Kalman filtering can be rather time-consuming from a
`squares
`computational point of view and any initial cleaning
`Kalman filtering has the following specific advantages over
`that can be done by other methods will increase its
`simple 'epoch by epoch' least squares and it is in order to
`efficiency.
`exploit these fully that Kalman filtering was selected as the
`Kalman filtering is able to solve for small biases that
`basic stochastic process behind
`the unified solution
`will remain in the data if only epoch by epoch
`presented in this paper.
`processing is used - such as drifts in gyros and (C-O)s
`1.
`Simple least squares treats each epoch independently.
`in terrestrial (shore-based) ranging systems. These look
`This means that it does not use knowledge of the
`like noise in simple least squares and can easily go
`motion of the system. Often, and especially in seismic
`undetected. A lot can be learnt by looking at the time
`variation of the data. Of course, in principle this could
`work, it is possible to make a very accurate prediction
`of where the network will be at any epoch using just
`be done in simple lease squares by analysing time
`series of residuals but it would be hard to do this in
`the previous position and the estimated configuration
`motion. Not using this 'knowledge of motion' is
`real time - and hard to feed back any findings into the
`effectively discarding information and leads to poorer
`system.
`Because it can determine and use the system motion,
`quality results than those obtainable from a properly
`Kalman filtering is able to use observations that do not
`tuned Kalman filter. In the past (and sometimes today)
`poorly tuned filters were used and in this case results
`completely define the system - i.e. GPS data from just
`might be worse - simply because the system motion
`two satellites could be used to update a vessel position.
`Of course, long periods of such data would lead to a
`may have not been well determined and/or not used
`significantly degraded result.
`properly in the estimation process. So simple least
`A Kalman filter can accept data as and when it is
`squares is a safe option - but it does not have the
`measured. With simple least squares, data has to be
`potential accuracy of Kalman filtering^ The challenge,
`
`12
`
`WESTERNGECO Exhibit 2144, pg. 4
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`

`
`reduced to a specified epoch. Therefore, a Kalman
`filter can cope well with data arriving as a more or less
`continuous stream.
`regime is highly suited to the
`7. The Kalman filter
`mixing of varied data types [Celik and Cross, 1994],
`when poor satellite geometry leads to poor positions in
`a DGPS-only solution, the introduction of data from a
`gyro carried by
`the vessel can make a major
`improvement. It would not be possible to combine
`these data types in simple least squares because, for an
`individual epoch the gyro does not give any positional
`information.
`
`2. Streamer modelling
`
`Since the compasses and other measuring devices are not
`co-located with the hydrophones it is necessary, in any
`approach, to have a mathematical model that describes the
`shape of each streamer. Moreover, because of the numerous
`hydrodynamic forces acting on the cable in the underwater
`environment, the cable shape is likely to be significantly
`distorted from a nominal straight line - so a simple linear
`model is very unlikely to be sufficient. To estimate this
`distorted shape two alternatives can be considered.
`In the first approach a physical model of the hydro-
`dynamic forces acting on the cable could be used to derive
`equations which describe the streamer shape. It is known
`that tension forces due to the vessel pull, and drag forces due
`to the resistance of the cable through the water, determine
`its three dimensional shape. Any change in the vessel's speed
`and any fluctuation in the sea waves, or those generated by
`the vessel, the wind load or the water currents, would mean
`changes in the towing tension and drag forces respectively.
`Such a model can only be applied when these external forces
`acting on the cable are known with a reasonable accuracy
`[Krail andBrysk, 1989]. It should be stressed, however, that,
`even if these quantities were known, a system of several
`streamers and floats would lead to models that would be too
`complicated and
`inflexible for
`the construction and
`implementation of a practically useful positioning algorithm.
`It is therefore unlikely that, although they have been used
`for vessel motion, [Crow and Pritcheu, 1986], hydrodynamic
`models will be adopted for positioning purposes in the
`foreseeable future.
`The other way to tackle this problem is to consider an
`'empirical' numerical approach in which the solution to the
`problem is deduced by adopting a 'model curve' that best
`fits the observed data.
`
`2.1 Curve-fitting procedures
`Several numerical methods can be adopted to obtain the
`streamer shape. The simplest one is to consider the cable as
`a straight line which follows exactly the ship's track.
`Although this approach would be very simple in practice,
`significant differences from the final expected position may
`result, not only because of the angle between the ship's track
`and the cable's baseline, but also because of the 'deformed'
`shape of the cable.
`.
`A more efficient way to address this problem might be to
`use a mathematical function such as a cubic spline.
`However, even though a cubic spline gives a curve which is
`continuous and continuously differentiable, and one which is
`capable of fitting the data very closely, it is not the best
`solution of the problem. This is because its coefficients vary
`along the length of the cable (i.e. the streamer shape is not
`represented by a single function) and its incorporation into a
`single operational system, which is the aim of this study, is
`
`extremely difficult. Moreover, because the cubic spline is
`technically capable of representing faithfully each compass
`reading, it is hyper-sensitive to compass errors leading to the
`possibility of a completely unrealistic final curve.
`Alternative curve fitting models include least squares poly­
`nomial approximation and the use of harmonic functions.
`This work concentrates on the former. Using least squares
`polynomials leads to a curve which describes the complete
`streamer's shape using only one set of coefficients, and
`furthermore the resultant curve is continuous and continu­
`ously differentiable at every point of the cable [Douglas, 1980
`and Owsley, 1981]. As a result, this method can be
`incorporated much more easily in a unified recurrent process
`such as a Kalman filter.
`Variations of the foregoing are also possible in practice for
`instance Ridyard, (1989), has suggested the use of a 'rolling
`quadratic' algorithm in which a series of
`individual
`quadratics are used to fit a small group of compasses. This
`algorithm is clearly very effective and this, and similar
`approaches have been widely adopted within the industry.
`Whilst they may be very powerful interpolation devices, and
`whilst they may be very effective in sorting out outliers and
`highlighting problems, they cannot be easily adopted in the
`unified approach presented here. This is because (as was the
`case of the cubic spline) the approach demands the use of a
`single function to describe the streamer shape.
`Hence in this study a 'n-order' polynomial shape model
`has been utilised. Such 'single' polynomials are not popular
`in some sections of the exploration industry so their use
`needs also to be justified from an accuracy point of view (it is
`clearly not a sufficient argument to use them just because
`they are convenient), and a series of tests have been carried
`out in order to do this. These tests involved the fitting of a
`series of polynomials, of a variety of orders, to real compass
`data and comparing the results with those obtained from the
`universally accepted rolling quadratic method. The mathe­
`matics and the results are described in the next section.
`
`2.2 Testing of the polynomial approximation
`In these tests the only information used is that derived from
`the magnetic compasses fixed along the length of the cable.
`This was done in order to compare the polynomial approach
`with
`the accepted method
`(which
`treats such data
`independently of any other). In such a case the final
`accuracy of a streamer position is a function of raw compass
`data, the local magnetic declination, individual compass
`corrections and the algorithm used for processing the data.
`The polynomial observation equations can be written as:
`B; = a0 + a,^ + a 2l] +... + anln
`
`(1)
`
`where:
`
`Bj : is the compass reading
`: is the offset of the i-th compass from its
`lt
`reference point
`: is the polynomial coefficient
`
`a
`
`The solution of this equation system, using a least squares
`method, gives the values of the polynomial coefficients. If we
`consider the geometrical configuration to be as shown in Fig.
`2.10, we have:
`
`0(radj= a tan(dv / du) = atan(dv / dl)
`which for any 0 in (-1°,1°) becomes:
`
`0(rad) = tan0 = dv / du = dv / dl
`
`(2)
`
`(3)
`
`13
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`WESTERNGECO Exhibit 2144, pg. 5
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`

`
`mean bearing
`
`Fig. 2.10: Streamer modelling
`
`Also for:
`
`Bi>B:Bi = B + 0 = B + (dv/dl)
`B( < B: Bj = B - 0 = B + (-dv / dl)
`
`(4)
`
`Upon substituting equation (1) into equation (4) and
`integrating we find:
`v = c0l +c,l2 +... +cnlntl
`
`(5)
`
`where:
`c = a. / (k+1), for K = 0 to n
`
`(6)
`
`The final coordinates X,Y can then be estimated by rotating
`these coordinates to the East, North coordinate system
`using:
`X = cos(a)l+sin(a)v
`Y = sin(a)l-cos(a)v
`
`(7)
`
`As already stated, in order to test the feasibility of the
`algorithm in terms of correctness and efficiency, the
`foregoing method has been applied to a subset of real
`compass data. A full description of the survey configuration
`and the data set used is given in 4.2 - here it is just
`mentioned that the streamers were 3.1km long and contained
`13 compasses each. The process was carried out for poly­
`nomial orders up to eight and for every shot point in the
`data set. A typical set of results is given in Fig. 2.20.
`Detailed analysis of hundreds of such sets of curves has led
`to the conclusion that polynomials of order either five or six
`fit the data extremely well. The following two general
`conclusions were also reached.
`• Polynomials of order four or less do not describe
`faithfully the observations. In such cases the dif­
`ferences between the actual compass readings and
`those predicted by the polynomial can (in a few cases)
`exceed half a degree. This might be important given
`that, in practice, cable compass resolution (but not
`accuracy) can be as high as 0.1°.
`• Polynomials of order greater than six can sometimes
`generate curves characterised by steep changes of
`
`0
`2
`cable length (Km)
`
`polynomial order: 3
`
`2
`0
`cable lengih (Km)
`
`polynomial order: 4
`
`-2
`
`2
`0
`cable lengih (Km)
`
`2
`0
`cable lengih (Km)
`
`polynomial order: 6
`
`2
`0
`cable lengih (Km)
`
`polynomial order: 7
`
`0
`2
`cable length (Km)
`
`polynomial order: 8
`
`-2
`
`0
`2
`cable lengih (Km)
`
`2
`0
`cable lengih (Km)
`
`Fig. 2.20: Streamer modelling -
`Polynomial approximation
`
`gradient, which may effect significantly the fidelity of the
`final coordinates. This phenomenon is particularly noticeable
`for compasses close to the tailbuoy.
`After the polynomial coefficients had been determined,
`the Eastings and Northings with respect to the streamer
`reference point were obtained using equation (7). The
`differences between these coordinates and those obtained
`using a 'rolling quadratic' algorithm were then computed.
`An examination of these indicate that if a fifth or sixth order
`polynomial is used the maximum resultant differences are of
`the order of one metre - even for the groups of hydrophones
`in the far end of the cable.
`From these tests it is evident that the use of a polynomial
`approximation is a highly realistic approach to the problem.
`Moreover, the method has the advantage of being easily
`incorporated into a Kalman filter model for real time
`positioning and quality control from mixed data sources.
`The n-order polynomial has hence been adopted as the
`streamer model in the mathematical system developed in this
`paper.
`
`3. An integrated Kalman filter algorithm
`It has already been stated that a rigorous filtering approach
`has been adopted in this study. What follows is a review of
`the basic models needed for such an approach and details
`of the implementation of these models for the positioning
`of seismic spreads.
`
`14
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`

`
`(8)
`
`(9)
`
`b
`
`3.1 Filtering techniques
`The Kalman filter {Kalman, I960] is probably the best
`known of the commonly used recursive algorithms for the
`estimation of the parameters of time-varying systems. It
`has constituted the framework for a unified and concise
`treatment of a broad range of filtering problems from
`electronic engineering to surveying and geodesy. As well as
`having the many practical advantages already referred to,
`Kalman filter estimates have the advantage of being least
`squares estimators, which can be shown [Cross, 1983], to
`be the best (in the minimum variance sense) within the
`complete class of linear unbiased estimators.
`The implementation of the filter requires the specifica­
`tion of two mathematical models. The measurement model
`or primary model relates the state vector parameters to the
`observations, and the dynamic model or secondary model
`relates the parameters at epoch tM to those at later epoch ts.
`If the state vector is denoted by x and the observations
`by L then the linearised form of a non-linear measurement
`model F(x) = L is given by:
`Ax = b; + v i
`where:
`Aj : is the design matrix given by dF{ /dx i
`Xj
`: is the correction to the provisional value of the
`filtered state vector, xs(+)
`: is the 'observed - computed' vector, given by
`L, -F,(V>)
`: is the state vector residuals.
`Vj
`The filtered state x^) is computed iteratively until there is
`no significant change in the provisional state x;(+).
`The dynamic model represents the behaviour of the
`system as it varies with time. The discrete linearised form
`is given by:
`xj = MwXj_i + y M
`where:
`M;_,: is the transition matrix from time tM to time t
`yul : is the dynamic model noise from time tM to
`time t
`The operational equations for, and the recursive nature
`of, the Kalman filter are well known [Cross, 1983; Brown
`and Hwang, 1992]. Here emphasis is placed on the contents
`of the two models - once they have been specified then
`their implementation within a Kalman filter
`is, in
`principle, trivial.
`It is, however, worth pointing out that the Kalman filter
`is not the only optimal (in the least squares sense)
`mathematical procedure. Another, slightly less well known,
`set of equations known as the Bayes filter can be used to
`produce absolutely identical results to those of the Kalman
`filter. The only difference between them is in the manner
`in which the so-called gain matrix is computed. The
`computation of the gain matrix in the Bayes filter involves
`an inversion of a matrix whose size is equal to the number
`of parameters of the state vector, whilst, in the Kalman
`form an inversion of a matrix whose size is equal to the
`number of observations is required. In this study the Bayes
`equations have actually been implemented since in a
`typical 3D marine spread a large number of observations
`contribute to a relatively small number of states. Many
`modern Kalman filtering
`software packages
`include
`routines for both sets of equations and make automatic
`(and invisible to the user) decisions as to which to use.
`
`Despite sometimes using the Bayes equations, the words
`Kalman filter nevertheless appear to be almost universally
`used to describe the process.
`Before describing the models in detail it is necessary to
`make some remarks about the coordinate systems that are
`used.
`
`3.2 Coordinate systems
`Within a typi cal seismic configuration there are several sub­
`systems that are able to move independently of each other,
`and of the vessel. These include every single float (gun array
`or any auxiliary reference station) and each streamer
`[Houienbos, 1989]. Each sub-system must therefore have its
`own parameters and coordinate system - which must, in
`turn, be linked by the mathematical model in order to
`determine the complete configuration. Before defining the
`various state vector parameters for each one of the
`configuration subsystems it is necessary to describe their
`different coordinate systems.
`An earth fixed geodetic system, involving latitude and
`longitude or a map projection system, is used to describe the
`final positions of all of the points of interest. The vessel and
`tailbuoys absolute positions, derived by GPS/DGPS or a
`radio positioning system, will, of course, naturally be in this
`system but it is not especially convenient for describing the
`rest of the spread.
`
`Fig. 3.10: Coordinate systems
`
`For this it is more convenient to use a local topographic
`coordinate system. This system has its origin at the vessel
`navigation reference point with the X-axis aligned with the
`east direction and Y-axis aligned northwards. When
`necessary the 2-axis is defined as being perpendicular to the
`XY plane (i.e. upwards) such that the resultant coordinate
`system is right handed, as in Fig. 3.10. It is obvious that this
`system moves with respect to a geodetic earth system as the
`vessel's position changes. Also it is clear that, given the
`relatively short distances (a few kilometres) involved within
`the network, there will be minimal error in working with the
`computed distances and azimuths in the XY (horizontal)
`plane and then using a direct geodetic formulation to
`determine the coordinates (latitude and longitude) of the
`points of interest - i.e. the earth is effectively considered to
`be flat within the region of the seismic spread.
`Some of the available observations are made relative to
`devices fixed on the vessel. For this reason it is necessary to
`define another coordinate system that is attached to the
`vessel. The origin of this coordinate system coincides with
`the navigation reference point. Its y axis is aligned with the
`
`15
`
`WESTERNGECO Exhibit 2144, pg. 7
`PGS v. WESTERNGECO
`IPR2014-01478
`
`

`
`vessel's bow-stem direction. Its x axis (starboard) is in the
`horizontal plane and perpendicular to the y axis whilst z axis
`is defined to be perpendicular to xy place (upwards) - see
`Fig. 3.10.
`Finally, in order to estimate the position of any point on
`each of the streamers, taking into account its distance from
`the streamer reference point 1 as a parameter, it is necessary
`to introduce another local coordinate frame for each streamer
`in the spread. A set of three dimensional coordinate systems
`(u, v, z) is therefore defined. Each has its origin at the head
`of the first active section of the streamer, or any other point
`of known offset, its u axis aligned with the base course of the
`cable (as results from the Kalman filter computations) and
`its v axis perpendicular to the u-axis and pointing to the
`starboard side. The z axis is defined such that the resultant
`coordinate system is right-handed - see Fig. 3.10.
`3.3 Kalman filter functional models
`
`3.3.1 State vector
`In order to implement a Kalman filter, we must first define,
`in general terms, the minimum number of individual and
`(determinable) parameters (or unknowns) necessary to
`describe the complete system - this is known as the state
`vector. In the case of an offshore seismic survey the
`unknowns consist of those which describe the vessel's
`position and the motion and those which describe the
`position and motion of each subsystem. In the following, the
`unknown parameters are classified by subsystem.
`Vessel unknowns
`The unknown parameters that describe the vessel position
`and motion are defined to be the instantaneous values of the
`following elements:
`<p, X: the geodetic 'ellipsoidal' coordinates of the ship
`reference point
`cp, 1: the instantaneous velocity of this po int,
`c
`: the crab angle, i.e. angle between course made
`good and vessel's heading (Figure 3.20).
`Note that for many navigation applications it would also
`be necessary to define the acceleration of the vessel in the
`
`state vector but the almost straight line motion associated
`with seismic surveying makes this unnecessary in this case.
`
`Float unknozvns
`The unknown parameters for any tow points attached to the
`vessel are also included in the state vector. Tow point
`positions are defined as position vectors expressed in X, Y
`coordinates along with their velocity components X, Y with
`respect to the local topographic coordinate system. It should
`be stressed here that to date the filter has only been
`implemented in the XY (horizontal). The (known) Z
`coordinates of all components, are taken into account by
`making geometrical 'corrections' to the observations, i.e.
`observations are corrected to the values they would have had
`the whole system been in the XY plane. Also, it is important
`to note that the unknown coordinates X, Y refer to the
`centres of the floating arrays. It will usually be necessary to
`correct observations to these centre points.
`
`Streamer unknowns
`The streamer unknown parameters must clearly refer
`directly to the streamer model. For the purposes of this
`study a polynomial model has been adopted. Hence, the u, v
`coordinates of any point on a streamer are given by the
`following equations:
`
`u = 1
`V = c2l2 + c3P +...+cnl"
`
`(10)
`(11)
`
`Testing of the integrated algorithm using real data showed
`that (perhaps not surprisingly) coefficient c0 must be null
`since, by definition, v is zero at the head of the c