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Showing papers in "Annual of Navigation in 1983"


Journal ArticleDOI
TL;DR: In this article, a Kalman filter is used to estimate the misalignment angle between two inertial sensor assemblies by estimating the angle between them, which can be used to aligning one inertial component with another.
Abstract: Formulations of Kalman filters are presented which are capable of aligning one strapdown inertial sensor assembly with another by estimating the misalignment angle between them. One formulation treats the case of a fixed misalignment. Another treats the case of a dynamic misalignment, caused, say, by bending of the common supporting body. Measurements can be made by gyros only, or by gyros plus accelerometers. Filters which estimate inertial sensor error parameters are also discussed.

81 citations


Journal ArticleDOI
TL;DR: In this article, the authors presented a formula for computing position dilution of precision (PDOP) and its components, namely, horizontal DOP and vertical DOP, as a function of three-satellite geometry and clock stability.
Abstract: Navigation using GPS generally requires that the user track four satellites to resolve his 3-D spatial position and time bias. There are several reasons why it is desirable to navigate while tracking only three satellites: —The proposed 18 satellite GPS constellation will exhibit substantial periods of poor four satellite geometry over large geographic areas several times a day. —Failure of a GPS satellite in orbit will result in periods of only three-satellite availability over large geographic areas. —During establishment of the operational GPS constellation, there will be long periods of three-satellite GPS coverage. Three-satellite GPS navigation can be accomplished by equipping the user with a precise clock. The required stability of the clock is a function of the maximum allowable PDOP (Position Dilution of Precision) and the time interval between updates of the clock. Clock updates can be accomplished by tracking four GPS satellites, tracking one GPS satellite from a known location, or conventional time transfer methods. This paper presents a formula for computing PDOP and its components, HDOP (Horizontal DOP) and VDOP (Vertical DOP), as a function of three-satellite geometry and clock stability. This formula is used to plot HDOP and VDOP versus time for representative high quality quartz crystal and low-cost rubidium clocks for two scenarios of satellite geometry. The stability and environmental sensitivity of high quality quartz crystal and low-cost rubidium clocks are discussed.

70 citations


Journal ArticleDOI
TL;DR: Simulated results are presented which indicate convergence in about 10 minutes of satellite observation time and are based on an adaptive filter concept developed by D. T. Magi11 in 1965.
Abstract: The high stability of the GPS signals makes it possible to determine differential position over short baselines with an accuracy of the order of centimeters. This has been demonstrated using very long baseline interferometry (VLBI) methods of radio astronomy. This paper presents an alternative approach using Kalman filter methods. It is based on an adaptive filter concept developed by D. T. Magi11 in 1965. The scheme employs parallel Kalman filters with each filter being modeled for a different integer wavelength assumption. As the phase measurement sequence progresses, the adaptive scheme “learns” which Kalman filter corresponds to the correct hypothesis, and thus it both resolves the wavelength ambiguity and estimates differential position simultaneously. Simulated results are presented which indicate convergence in about 10 minutes of satellite observation time.

31 citations


Journal ArticleDOI
TL;DR: Differential operation, wherein a high-quality, surveyed-in receiver installation determines satellite pseudorange errors and communicates them to nearby users, offers a promising technique of improving SPS on a local scale, and can improve the accuracy of the GPS even when SPS is provided at full accuracy.
Abstract: Under the Selective Availability program the Department of Defense plans to provide the civil community with Standard Positioning Service (SPS), having an accuracy of 500 meters (2drms) when the NAVSTAR GPS becomes operational. Subsequent improvements in accuracy are expected to be instituted as national security considerations permit. However, provisions are also made to degrade accuracy to worse than 500 meters, if necessary. Depending on the type and level of Selective Availability imposed, different civil capabilities can be realized. These capabilities are described in this report. Differential operation, wherein a high-quality, surveyed-in receiver installation determines satellite pseudorange errors and communicates them to nearby users, offers a promising technique of improving SPS on a local scale. It can improve the accuracy of the GPS even when SPS is provided at full accuracy, i.e., when Selective Availability is removed. The type of corrections, the associated accuracy, and the update rate are discussed here. Differential system design alternatives and the advantages and disadvantages of each are also described.

20 citations


Journal ArticleDOI
TL;DR: The first use of the Navstar Global Positioning System (GPS) by a spaceborne user occurred with the launch of NASA's Landsat 4 on July 16, 1982.
Abstract: The first use of the Navstar Global Positioning System (GPS) by a spaceborne user occurred with the launch of NASA's Landsat 4 on July 16, 1982. One of the experimental packages onboard Landsat 4 is the Global Positioning System Package (GPSPAC). A brief description of the GPSPAC is presented and the operational history of the GPSPAC experiment onboard Landsat 4 is outlined. The responsibility for the control of the GPSPAC experiment and for the validation of its results resides at the Goddard Space Flight Center (GSFC). The role of the Naval Surface Weapons Center (NSWC) primarily is to aid GSFC in achieving the objectives of the GPSPAC experiment. Results of NSWC's evaluation of the GPSPAC performance are presented.

13 citations


Journal ArticleDOI
TL;DR: The Navstar Global Positioning System (GPS) allows extremely accurate and global determination of time, as well as position and velocity, and it does it economically as discussed by the authors, but it is not an operational system.
Abstract: The Navstar Global Positioning System (GPS) allows extremely accurate and global determination of time, as well as position and velocity, and it does it economically. Without even using the precise GPS codes and dual frequency measurements, the transfer of time with GPS can be accomplished to better than 100 nanoseconds. In fact, with a simultaneous common view approach, relative timing can be accomplished to better than 10 nanoseconds. Although a widespread need for this type of accuracy may not be apparent, one would be surprised at the number of applications for that accuracy surface once that its availability has been established. Probably more important than the accuracy capability is the ease by which time transfer can be accomplished with GPS, both technically and logistically. GPS time transfer systems are more portable than precise clocks primarily because batteries are not required to keep the clock running. Time with GPS can be reinitialized at any time. Economically, because of the technology involved, the current cost of precise clocks (Cesium) is on a steady rise, while GPS time transfer systems will become less expensive as time passes, enhancing the advantage over the use of precise clocks. Currently, GPS is not an operational system. For time transfer it is, but not continuously, available (only available on the order of 16 to 20 hours a day). However, as the number of satellites increases, the time transfer will become continuous long before GPS is declared fully operational. It is that time that GPS time transfer systems could replace many of the precise clocks in existence. This paper will discuss how GPS time transfer satisfies the time keeping requirements of various timing applications, enhances the time keeping capabilities of those applications and reduces the expense of time keeping in those applications. The performance of GPS time transfer in its different modes of usage will also be discussed.

7 citations


Journal ArticleDOI
TL;DR: In response to the identified need for automated coastal navigation and piloting as well as collision avoidance for the NATO hydrofoil, Sperry Systems Management under Navy contract has developed the High Speed Ship Collision Avoidance and Navigation System (HICANS) as mentioned in this paper.
Abstract: In response to the identified need for automated coastal navigation and piloting as well as collision avoidance for the NATO hydrofoil, Sperry Systems Management under Navy contract has developed the High Speed Ship Collision Avoidance and Navigation System (HICANS). This system is aboard the USS PEGASUS (PHM-1) and has undergone extensive at-sea testing. This paper describes the system capability for coastal navigation and piloting using both digital charts and TV projection of standard paper charts. It also examines the results to date of the accomplished at-sea system testing.

7 citations


Journal ArticleDOI
TL;DR: In this paper, the performance of an integrated GPS/INS navigation system with respect to component failures is examined, including GPS outage, baro-altimeter failure, inertial failure and satellite obscuration.
Abstract: This paper examines the performance of an integrated GPS/INS navigation system with respect to component failures. The paper is divided into three sections. The first section details the system configuration by which the navigation components are integrated. The second section examines the modelling methods used to simulate each of the components and the third section analyzes the simulation results. Four modes of component failure are investigated. They are GPS outage, baro-altimeter failure, inertial failure and satellite obscuration.

7 citations


Journal ArticleDOI
TL;DR: The design and flight test results of an experimental GPS C/A-code receiver installed in a general aviation aircraft meets the FAA's current accuracy requirements for two-dimensional area navigation systems and is consistent with future navigational accuracy requirements.
Abstract: This paper describes the design and flight test results of an experimental Global Positioning System receiver installed in a general aviation aircraft. These tests were part of a GPS test and evaluation project conducted for the Federal Aviation Administration by M.I.T. Lincoln Laboratory. The purpose of the project was to design a GPS C/A-code receiver that: 1) meets the current FAA requirements for two-dimensional area navigation (RNAV) systems, 2) employs techniques which could potentially lead to low-cost commercial avionics, and 3) is operationally compatible with existing ATC procedures and aircrew practices. Novel features of the design are: 1) the use of a dual-channel C/A-code receiver, and 2) the tracking of all visible satellites in view rather than a minimum set of four satellites. The system employs two DEC LSI-11/23 computers, one to perform position fixing and receiver control, and the other to perform navigation and data recording tasks. Pilot displays include a conventional course deviation indicator (CDI), omni-bearing selector (OBS), and intelligent control and display unit (CDU). The GPS receiver was flight tested at a large urban airport, at several small general aviation airports, and over mountainous terrain. The horizontal system accuracy during typical aircraft flight profiles was measured to be 333 feet (95% confidence). This level of accuracy meets the FAA's current accuracy requirements for two-dimensional area navigation systems and is consistent with future navigational accuracy requirements.

6 citations


Journal ArticleDOI
TL;DR: In this article, the authors describe some of the operational features and certain aspects of the software and hardware configuration of a low cost Marine Integrated Navigation System/MINS, developed at the Defence Research Establishment Ottawa.
Abstract: This paper describes some of the operational features and certain aspects of the software and hardware configuration of a low cost Marine Integrated Navigation System/MINS, developed at the Defence Research Establishment Ottawa. Results of some of the design analysis and sea-trial evaluation will also be presented. The design objective was to enhance the navigation accuracy, operational reliability and position reporting efficiency of marine vessels that are already equipped with a variety of types and brands of navigation sensors. The MINS will also serve as a computational aid in waypoint navigation and voyage planning. Simulation results and sea trial evaluation on board the DND research vessel CFAV Endeavour have provided good evidence of the system's all-round superior performance, and shown promise for the successful realization of all design objectives. The pre-production model of the MINS, implemented on a 16-bit microprocessor is expected to be delivered to the Canadian Forces in early 1984. It is expected to be extremely cost effective for possible fleet wide installation on board naval and coast guard vessels.

6 citations


Journal ArticleDOI
S. Brzezowski1, J. Fagan1
TL;DR: In this paper, the performance characteristics of various gyroscopic tools currently in development are analyzed and major system error sensitivities are established, as well as the major operational limitations of each design and establishes the sensitivities to key error sources.
Abstract: Conventional borehole survey technology, involving the use of either magnetic or gyroscopic instruments, provides at best a lateral position uncertainty of 1% of measured well depth. The conventional tools also exhibit severe accuracy degradation in inclined boreholes. Preliminary test programs have indicated that an order-of-magnitude improvement over conventional technology can be achieved by adapting modern inertial navigation and guidance techniques to the development of an improved gyroscopic survey instrument. The objective of this paper is to analyze the performance characteristics of certain gyroscopic tools currently in development and outline major system error sensitivities. Mathematical error models have been developed for several different gyroscopic tools, which can be grouped into three general categories: Gyrocompass, Attitude Reference System and Inertial Navigation System. These models are used as the basis for a series of computer error simulations which characterize the statistical survey accuracy of the various tool designs as a function of time and displacement. The analysis also identifies the major operational limitations of each design and establishes the sensitivities to key error sources.


Journal ArticleDOI
Phil Ward1
TL;DR: The operation of a NAVSTAR GPS multiplex receiver is defined in this paper as processor-controlled time-sharing of a single channel of receiver hardware in a manner that permits continuous digital phase-locked loop tracking of four GPS transmitters while simultaneously reading the 50-Hz navigation message data from the same four transmitters.
Abstract: The operation of a NAVSTAR GPS multiplex receiver is defined in this paper as processor-controlled time-sharing of a single channel of receiver hardware in a manner that permits continuous (in a sampled data sense) digital phase-locked loop tracking of four GPS transmitters while simultaneously reading the 50-Hz navigation message data from the same four transmitters. The operational and performance characteristics are described for a proven advanced NAVSTAR GPS digital multiplex receiver front-end design that time-shares one code generator and one carrier synthesizer in accordance with this definition. Multiplexing has been demonstrated with this advanced single-channel receiver front-end in a manner that creates up to eight pseudochannels. The receiver can also be operated at a reduced number of pseudochannels in binary multiples of 4, 2, or 1. The eight pseudochannels provide simultaneous tracking of four GPS transmitters in P-code of both the L1 and L2 signals. In addition, the space vehicle (SV) 50-Hz navigation message data are read continuously from all four GPS transmitters. The eight pseudochannels emulate a multichannel set that is virtually free of interchannel bias and interchannel phase drift error since all GPS signals of the same frequency travel the same hardware path. Sixteen observables are continuously available: four L1 code phases, four L2 code phases, four L1 carrier doppler phases, and four L2 carrier doppler phases. These measurements are captured directly as a by-product of the nearly instantaneous phase presetting and the digital synthesis of the numerically controlled code and carrier tracking loops. From these raw measurements, the usual GPS receiver pseudoranges, delta pseudoranges, and their L1/L2 differences can be derived. Quantization noise is negligible since the raw measurements are of such ultrahigh resolution (2−16 P-chip code phase and 2−16 cycle carrier doppler phase). The receiver error characteristics are described and actual tracking error plots for 16 simultaneous observables from four real GPS satellites are shown.

Journal ArticleDOI
TL;DR: In this article, the 3D seismic survey is explained from basic principles and the position accuracy requirements are developed, and new techniques and equipment are presented for positioning both the seismic vessel and the towed streamer containing an array of hydrophones.
Abstract: The three-dimensional (3D) seismic survey is a relatively new technique used in the exploration and development of petroleum resources. It is well known that precise navigation (positioning) is required for the 3D seismic operation. However, it is not well understood why this high level of precision is required. In addition, few, if any, attempts have been made to discuss actual positioning achievements while performing a 3D seismic survey. This paper attempts to fill both these voids. In this paper, the 3D seismic survey is simply explained from basic principles and the position accuracy requirements are developed. State-of-the-art techniques and equipment are presented for positioning both the seismic vessel and the towed streamer containing an array of hydrophones. Results are presented for vessel positioning using the 450 MHz Syledis system in conjunction with a 2 MHz Argo system. A direct comparison of these two systems is given from data obtained during a 3D seismic survey. In addition, operational results are presented for the positioning of a towed hydrophone array. New techniques under development for positioning the seismic vessel and the towed hydrophone arrays are presented. The possible use of GPS technology for 3D seismic surveys is discussed and the system's impact for future work is evaluated. Conclusions are presented summarizing current 3D seismic accuracy requirements, the techniques and equipment required, results that can be expected, and what positioning technology can be expected in the future.

Journal ArticleDOI
TL;DR: The Microwave Landing System (MLS) is on the verge of implementation in the National Airspace System as a replacement for the ILS and the present plan calls for the establishment of 1,250 systems by the year 2,000 as discussed by the authors.
Abstract: The Microwave Landing System (MLS) is on the verge of implementation in the National Airspace System as a replacement for the Instrument Landing System (ILS). The present plan calls for the establishment of 1,250 systems by the year 2,000. The first ten years of MLS implementation will serve as a transition period in which both ILS and MLS will be extensively used. The MLS implementation strategy is based on the establishment of MLS networks of 4 to 7 facilities. Each network is centered on a major hub airport and the satellites are selected from regional airports that have commuter airline service with the hub. The MLS implementation program will take advantage of the increased reliability of digital electronics systems together with the reduced costs of those systems to provide the capability for all weather service at most MLS-equipped airports.

Journal ArticleDOI
TL;DR: In this paper, the authors present some basic practical and theoretical concepts of datum development and evaluate errors that may be encountered if a datum transformation is required but is not used, as well as the user community must be educated in the basic concepts of position determination.
Abstract: There are many geodetic datums in use throughout the world today. Each of these datums serves as a reference surface for the mapping, charting and geodetic work done in a specific geographical area. Each datum is defined by fitting a specific ellipsoid to the earth in such a manner as to minimize departures of this reference model from the geoid over the area of concern. Historically, these datums have been relatively oriented. Consequently, the position of the center of an ellipsoidal model relative to the center of mass of the earth is not known. The positions determined on one datum cannot be related directly to positions on another datum. The development of absolutely-oriented datums incorporating satellite and gravity data has led to the development of datum transformations to relate positions on one datum to those on another. Sophisticated new electronic navigation and targeting technology requires highly precise input data to obtain output on the order of design accuracies. To obtain positions on the order of +/–1000 feet/300 meters, care should be taken by the users of such equipment that the datum to which positions are referred is taken into account. If positions are located on two different datums, then the user must know that one set of coordinates must be transformed into the other system prior to their input into the inertial navigation system as a point of departure and a destination. This paper reviews some basic practical and theoretical concepts of datum development and evaluates errors that may be encountered if a datum transformation is required but is not used. Hardware and software alone cannot minimize the occurrence of these errors. The user community must be educated in the basic concepts of position determination. Therefore, this work is presented in such a manner as to be readily adapted to teach users of varying backgrounds, education and expertise.

Journal ArticleDOI
TL;DR: The use of Navstar/GPS to navigate Landsat 4 under this ideal geometric situation, as well as at other times during the year, is discussed.
Abstract: A greater degree of survivability is a significant requirement of future space systems. Toward this end, an important aspect is an autonomous navigation capability. For many space systems, an obvious choice is the use of the Navstar/Global Positioning System (GPS). With the operational 18-satellite GPS constellation (plus three active spares), low altitude user satellites will have continuous tracking data available for navigation. On the other hand, with the current constellation of four satellites, there is less available data and, consequently, the accuracy of navigation varies a great deal throughout the user satellite orbit. This paper discusses the navigational performance that can be achieved with this constellation. Last summer, the Ladsat 4 satellite was placed into a near circular, sun synchronous, orbit at an altitude of 705 km, and at an inclination angle of 98.3 deg. This satellite contains a Navstar GPS receiver (called GPSPAC), and is the first user satellite to employ GPS. Using this satellite as an example, an error analysis was performed of the navigation accuracy that can be obtained for a low altitude satellite. During the course of a Landsat 4 orbit, it will be able to observe from none to four Navstar/GPS satellites. When four satellites are available, the three-dimensional accuracy can be expected to be about 11 m. Since Landstat 4 is sun synchronous, i.e., its orbit plane rotates in longitude with the sun, there will be times during the year when this satellite and the GPS satellites simultaneously come together over the continental United States. This is where Landsat 4 is primarily used, and is the ideal geometric relationship between Landsat 4, the GPS satellites, and the earth. This favorable condition occurred last November. This paper discusses the use of Navstar/GPS to navigate Landsat 4 under this ideal geometric situation, as well as at other times during the year. The paper presents the results of a theoretical accuracy analysis. The actual on-orbit experience with Landsat 4 is not covered here.

Journal ArticleDOI
TL;DR: In this paper, the authors used a constant readiness premission calibration, on board gravity deflection compensation and system modifications for finer accelerometer resolution and gyro speed control to reduce the long-term error rate to 0.02 nm/hr.
Abstract: When mission requirements dictate long-term unaided inertial navigation with a low terminal position error, then the approach suggested in this paper can be used. The current aircraft precision barrier of 0.1 nm/hr long-term error rate can be reduced to 0.02 nm/hr by use of a constant readiness premission calibration, on board gravity deflection compensation and system modifications for finer accelerometer resolution and gyro speed control.


Journal ArticleDOI
TL;DR: The reverse velocity sled concept was proposed and analyzed in 1979 as discussed by the authors, which uses a reverse thruster to provide a sled brake of desired magnitude and time duration, the optimal retro thrust resulting in a considerable reverse sled velocity.
Abstract: The High Speed Rocket Sled Test Track at Holloman AFB, New Mexico is a precision, highly-instrumented facility which has found extensive use in the quantitative error evaluation of state-of-the-art guidance systems. The sleds, containing the guidance system tested and extensive monitoring instrumentation, are subjected to a positive and negative acceleration profile tailored to both simulate missile environment and maximize separability of test system errors. While thrust is provided by rocket engines, deceleration has traditionally been accomplished by water brake: a sledborne scoop which disperses water from a channel between the two rails. With the evolution of high speed flight computers has come a trend toward real time guidance system error compensation. This has resulted in enhanced visibility of errors such as high order nonlinearities and effects due to erroneous compensation. The isolation of system errors is so strongly a function of the sled acceleration profile which drives those errors that the success of a specific profile to aid error separation can be simulated prior to actual testing. This was shown for the case of an anticipated sled test project which was to involve high (>50 g) accelerations and was expected to propagate unusually highly-correlated errors: the result was that no level of water brake deceleration provided the velocity change necessary for error separation. Thus the reverse velocity sled concept was proposed and analyzed in 1979. This technique uses a reverse thruster to provide a sled brake of desired magnitude and time duration, the optimal retro thrust resulting in a considerable reverse sled velocity. This sled is presently under design and construction and is to be thoroughly tested in calendar year 1982. The aerodynamics of the reverse velocity sled being built are such that the sled will return to the launch point. This factor leads to new dimensions in this test concept. Firstly, start and stop points can be measured to extreme accuracy, because of their proximity, permitting excellent determination of non-cancelling errors such as asymmetries and even-powered nonlinearities. Secondly, traversal of the Test Track in two directions decouples Track survey errors from test guidance system errors. Thus, if the test system performance conforms to a deterministic model, then the test system can be used to help identify, and then remove, many errors in the Track reference itself. This paper reviews the efforts accomplished to date, including an analysis of the above-mentioned benefits of sled return to the launch point.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss the theory of baseband signaling and describe a system that can be used for offshore exploration, mooring, and docking of offshore drilling platforms.
Abstract: Recent advancements in transponders have made it possible to measure ownship's position to within one or two feet. These transponders make use of a new technology called baseband. By baseband we refer to the generation of an energy pulse function which can be obtained, for example, when a very fast switch is momentarily connected to a direct current electrical power source. When such a pulse function is applied to a microwave antenna, a very short damped oscillation of microwave energy is propagated. This propagation is characterized by extremely short duration (order of nanoseconds), very low average power (order of microwatts, and very broad bandwidth (order of gigahertz). The broad bandwidth of baseband techniques is limited principally by the speed of the switch that generates the step function and by the antenna bandwidth. Baseband contrasts to orthodox pulsed microwave generators that usually modulate a carrier wave with a pulsed modulating wave. The principal advantages of baseband systems are reviewed and discussed in the text, specifically low interference, simplicity, light weight, low power consumption, high ranging accuracy, and relative freedom from multipath and clutter interference effects. The light weight and low power consumption of baseband systems make them ideal for handcarried instruments that can be used for trilateration measurements between moving ships, shorelines, and fixed structures such as offshore drilling platforms. A baseband system has been recently developed and has undergone sea trials for the offshore mooring of 45K TON ships in severe environments. This paper discusses the theory of baseband signaling and describes a system that can be used for offshore exploration, mooring, and docking.

Journal ArticleDOI
TL;DR: Multi-sensor navigation systems already available and in use in helicopters are discussed followed by a review of the system trade-offs and considerations leading to new systems that use more advanced digital electronic techniques to achieve the goals of reduced pilot workload, improved performance at minimum size, weight, and cost.
Abstract: The expanding role of the helicopter in the battlefield environment has burdened the pilot with missions of greater complexity and risk with a concomitant increase in pilot workload. Navigation of the helicopter is an essential supportive element to the prime mission and has until recent years been a significant contribution to the workload. Technological advances in navigational electronics such as Doppler navigation radar, computers, integrated avionic control and display systems, etc., now can provide automated navigation with vital benefits in cost, size, weight and power which permit incorporation of these advances into the helicopter. Cost reductions are particularly important since helicopters are used in large quantities in modern military forces. Multi-sensor navigation systems already available and in use in helicopters are discussed followed by a review of the system trade-offs and considerations leading to new systems that use more advanced digital electronic techniques to achieve the goals of reduced pilot workload, improved performance at minimum size, weight, and cost. The beneficial impact of ongoing technological advances in improving the operating capabilities of future avionics systems is indicated.

Journal ArticleDOI
TL;DR: A discussion of the many statistical bases for accuracy statements follows, and some suggestions, tempered with reality, are made as to how much accuracy is needed.
Abstract: Definitions are provided for the various types of accuracy (repeatable, relational and geodetic or absolute). A discussion of the many statistical bases for accuracy statements follows. These are discussed both as to how they relate to each other, and how they relate to the navigators' frame of reference. All to often accuracy, and other requirements are based on desire, with little regard for reality (How much accuracy do I need?—“As much as I can get.”). Some suggestions, tempered with reality, are made as to how much accuracy is needed.