Ex. PGS 1041
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`LEAST SQUARES FILTERING AND TESTING FOR
`POSITIONING AND QUALITY CONTROL DURING 3D
`MARINE SEISMIC SURVEYS
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`VASSILIS N GIKAS
`Surveying Engineering N.T.U.A.
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`Thesis submitted for the Degree of
`Doctor of Philosophy
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`Department of Surveying
`University of Newcastle upon Tyne
`August 1996
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`Ex. PGS 1041
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`Introduction
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`RESEARCH OBJECTIVES AND SCIENTIFIC RESULTS EXPECTED
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`The overall aim of the project has already been outlined, namely to develop a general,
`integrated and rigorous approach to the positioning and quality control in real time of
`marine seismic networks. In order to achieve this, emphasis has been placed on a
`number of objectives
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` 
`
` Determination of an optimum general mathematical description of the streamer
`shape by preliminary fitting of streamer models to compass data.
` Acquisition of a formal description of the geometry of the whole configuration by
`integrating all positioning data types into a single functional model.
` Computation of the real-time position and quality measures of any point deployed
`in a seismic network by adopting a Kalman (or other) filter as the basic stochastic
`process.
` Test the integrated model for appropriateness and for its sensitivity to detect and
`identify expected biases in the raw data by incorporating a uniform testing
`procedure.
` Assessment and testing of the correctness of the mathematics and the feasibility of
`the associated algorithms in terms of convergence, solubility and computational
`efficiency by preparing software for the various parts of the process and testing
`with real offshore data.
` Refinement of functional and stochastic models based on detailed analysis using
`alternative model hypotheses.
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`The results are tested mathematical models, in the form of computational algorithms,
`for the following
`
` 
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` The shape of the seismic streamers.
` The dynamics of 3D seismic configurations during data collection.
` The real-time positions and quality measures for offshore seismic surveys.
` The effect of the network geometry and the relative stochastic properties on seismic
`network positioning and quality control.
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`Chapter One: Acquisition and Positioning 3D Marine
` Seismic Surveys - An Overview
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`issues such as multi-source, multi-streamer acquisition. Four types of records have
`been defined (CENSUS User’s Guide, 1994)
`1. Header records - This type of records holds similar information as the header files
`of UKOOA P2 formats.
`2. Point position records - These records are used to identify the point being
`positioned. The most common are, source fired (S), vessel (V, P1/90 only), and
`tailbuoy (T, P1/90 only). The source records contain also information such as line
`number, shotpoint number, date/time and water depth.
`3. Receiver records - Receiver records contain information such as receiver ID flag,
`receiver position (easting and northing), and cable depth.
`4. Relation records - This type of records is an extension to the format and is used to
`prevent the pointless repetition of unchangeable information for different shots.
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`In addition to the UKOOA format other exchange data formats have been developed
`such as, SEG P1 (1983) as well as industry standard formats such as, the Shell’s SPS
`format, the Advance Geophysical’s ProMAX database format and the Green
`Mountain’s MESA format.
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`1.5.2 Geophysical Contractors’ Navigation and Binning Systems
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`It is a general conclusion from the discussion so far that the trend seems to be a
`movement of the seismic industry towards faster multi-tasking integrated software and
`central processing units (UNIX based workstations). Almost all major geophysical
`contractors/companies have developed (and continuously
`improve)
`their own
`navigation and binning/processing systems to meet this demand. The main
`characteristics of these systems are outlined bellow
`1. During acquisition usually some of the data are synchronized with shot time (as
`compass azimuths and network acoustics), and some are recorded at the sensor time
`(Syledis, GPS, RGPS).
`2. Storage at the UKOOA P2/91/94 formats and real-time graphic display of
`acquisition is a common practice.
`3. Some systems, as GIN 2000 developed by CGG, compute source and receiver
`positions based on least squares algorithms for the various networks of the spread
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`Chapter One: Acquisition and Positioning 3D Marine
` Seismic Surveys - An Overview
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`(vessel-buoy, relative head and relative tail networks as well as streamer shape).
`Other systems, as TotalNet, developed by WESTERN ATLAS, implement
`integrated network solutions by means of a Kalman filter.
`4. Quality control, including monitoring of the quality of the recorded data (setup,
`configuration, spread geometry, data integrity, and statistical analysis) is an
`essential feature in today’s systems.
`5. On-board binning systems provide real time monitoring of CMP distribution
`throughout a 3-D survey. Also, most binning systems’ capabilities include, flex
`binning, editing and rebinning algorithms.
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`In Table 1.1 a list of the navigation and binning systems of some major geophysical
`contractors is given.
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`Contractor
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`Navigation System
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`WESTERN
`GECO / PRAKLA
`CGG
`DIGICON
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`WISDOM II
`TRINAV
`GIN 2000
`MAGNAVOX 200 / SCOPE III
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`Binning / Processing
`System
`FLEX QC / CNAVCHK
`TRINAV / QC
`GIN 2000
`BIRDOG
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`Table 1.1 Contractors’ navigation and binning/processing systems
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`Chapter Three: The Kalman Filter
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`CHAPTER THREE
`THE KALMAN FILTER
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`INTRODUCTION
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`3.1
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`The Kalman filter 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. However,
`usually, the Kalman filter is perceived as a ‘black box’, into which measurements go in
`order to be converted into positions, since there still remains a certain amount of
`ignorance in the hydrographic surveying community with respect to Kalman filtering.
`Therefore, in the past, it has not proved popular with the offshore community and
`many offshore operators currently prefer simple and independent ‘epoch by epoch’
`least squares computations. This chapter aims to provide a brief description of the
`Kalman filter models and algorithms as well as to explain the meaning of the most
`commonly used terms associated with it.
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`Kalman filter estimates have the advantage of being least squares estimators. This
`means, as can be shown (Cross, 1983), that they are the best in the minimum variance
`sense within the complete class of the linear unbiased estimators. For these reasons
`they are often referred to as Best Linear Unbiased Estimates (BLUEs). The basic
`difference between a simple least squares computation and Kalman filtering, is that, the
`Kalman filter comprises of the specification of a dynamic model in addition to an
`observation model that to together provide an optimal solution. The use of a dynamic
`system reveals, somehow, the amount of knowledge with respect to the system
`dynamics, i.e. the behavior of the system as it varies with time. For instance, in the
`case of a moving vessel, where its position and velocity are the desired results, the
`position fix measurements provided by a shore-based or satellite navigation system
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`Chapter Three: The Kalman Filter
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`constitute the observation model while the dynamic model is expressed by the
`assumption of constant acceleration between the position fixes.
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`3.1.1 Predicting, Filtering and Smoothing
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`Three types of problems constitute the estimation problem associated with Kalman
`filtering. These are known as prediction, filtering and smoothing, and they are related
`to the estimation of the state vector parameters x, of a time-varying problem, computed
`at any instant with respect to the present time.
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`The step of prediction is related to the computation of the filter estimates x(-), at time
`of interest tj that occurs after the last available measurement(s). In this case, only the
`state estimates and its associated covariance matrix computed from the previous epoch,
`as well as the dynamics of the system, are used to provide the state vector solution.
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`Once a new measurement(s) is available the predicted state vector x(-) is used together
`with the new measurement(s) to solve for the state estimates. In this case, in which the
`time of the last measurement(s) coincides with the estimation time, the problem is
`referred to as filtering and the state vector denoted by x(+).
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`At a post-proccessing stage the state vector parameters can be computed at any time tj
`where information for some time interval prior and later to time tj is used. This part of
`the problem is known as smoothing and it denoted by x(s). Obviously, a solution of
`this type can only be available after some delay. Usually, in most real-time surveying
`applications, only the prediction and filtering steps are implemented since their
`implementation is straightforward. Although smoothing procedures can be executed in
`real time they are usually only used in post-processing because they require much more
`memory space.
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`These three distinct estimation problems can be defined as
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`3.1
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`ti > tj prediction
`filtering
`ti = tj
`ti < tj smoothing
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`x(-)
`x(+)
`x(s)
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`Chapter Three: The Kalman Filter
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`The three types of the filtering problem are illustrated in Figure 3.1.
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`Figure 3.1: Predicting, filtering and smoothing
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`3.1.2 Kalman filtering versus Simple Least Squares
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`Kalman filtering has the following specific advantages over simple ‘epoch by epoch’
`least squares and it is in order to exploit these fully that Kalman filtering is selected as
`the basic stochastic process for most offshore positioning applications.
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`1. Simple least squares treats each epoch independently. This means that it does not
`use knowledge of the motion of the system. Often, and especially in seismic work,
`it is possible to make a very accurate prediction of where the network will be at any
`epoch using just the previous position and the estimated configuration motion. Not
`to use this ‘knowledge of motion’ is effectively throwing away information and
`must lead to poorer quality results than those obtainable from a properly tuned
`Kalman filter. In the past (and sometimes today) poorly tuned filters were used and
`in this case results might be worse - simply because the system motion may have
`not been well determined and/or not used properly in the estimation process. So
`simple least squares is a safe option - but it does not have the potential accuracy of
`Kalman filtering. The challenge, of course, is to tune the filter properly in real time
`- and the fact that some have failed to do this in the past has led to Kalman filtering
`gaining a poor reputation in some circles.
`2. The use of a Kalman filter for a highly complex seismic configuration enables a
`rigorous computation of precision and reliability measures such as error ellipses
`and marginally detectable errors respectively, (Cross et al, 1994a). If a step-by-step
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