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Navigation and the Global Positioning System (GPS): the Global Positioning System
Navigation and the Global Positioning System (GPS): The Global Positioning System: Few changes of great importance to economics and safety have had more immediate impact and less fanfare than GPS. • GPS has quietly changed everything about how we locate objects and people on the Earth. • GPS is (almost) the final step toward solving one of the great conundrums of human history. Where the heck are we, anyway? In the Beginning……. • To understand what GPS has meant to navigation it is necessary to go back to the beginning. • A quick look at a ‘precision’ map of the world in the 18th century tells one a lot about how accurate our navigation was. Tools of the Trade: • Navigations early tools could only crudely estimate location. • A Sextant (or equivalent) can measure the elevation of something above the horizon. This gives your Latitude. • The Compass could provide you with a measurement of your direction, which combined with distance could tell you Longitude. • For distance…well counting steps (or wheel rotations) was the thing. An Early Triumph: • Using just his feet and a shadow, Eratosthenes determined the diameter of the Earth. • In doing so, he used the last and most elusive of our navigational tools. Time Plenty of Weaknesses: The early tools had many levels of uncertainty that were cumulative in producing poor maps of the world. • Step or wheel rotation counting has an obvious built-in uncertainty. • The Sextant gives latitude, but also requires knowledge of the Earth’s radius to determine the distance between locations. • The compass relies on the assumption that the North Magnetic pole is coincident with North Rotational pole (it’s not!) and that it is a perfect dipole (nope…). -
Geodynamics and Rate of Volcanism on Massive Earth-Like Planets
The Astrophysical Journal, 700:1732–1749, 2009 August 1 doi:10.1088/0004-637X/700/2/1732 C 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A. GEODYNAMICS AND RATE OF VOLCANISM ON MASSIVE EARTH-LIKE PLANETS E. S. Kite1,3, M. Manga1,3, and E. Gaidos2 1 Department of Earth and Planetary Science, University of California at Berkeley, Berkeley, CA 94720, USA; [email protected] 2 Department of Geology and Geophysics, University of Hawaii at Manoa, Honolulu, HI 96822, USA Received 2008 September 12; accepted 2009 May 29; published 2009 July 16 ABSTRACT We provide estimates of volcanism versus time for planets with Earth-like composition and masses 0.25–25 M⊕, as a step toward predicting atmospheric mass on extrasolar rocky planets. Volcanism requires melting of the silicate mantle. We use a thermal evolution model, calibrated against Earth, in combination with standard melting models, to explore the dependence of convection-driven decompression mantle melting on planet mass. We show that (1) volcanism is likely to proceed on massive planets with plate tectonics over the main-sequence lifetime of the parent star; (2) crustal thickness (and melting rate normalized to planet mass) is weakly dependent on planet mass; (3) stagnant lid planets live fast (they have higher rates of melting than their plate tectonic counterparts early in their thermal evolution), but die young (melting shuts down after a few Gyr); (4) plate tectonics may not operate on high-mass planets because of the production of buoyant crust which is difficult to subduct; and (5) melting is necessary but insufficient for efficient volcanic degassing—volatiles partition into the earliest, deepest melts, which may be denser than the residue and sink to the base of the mantle on young, massive planets. -
Advanced Geodynamics: Fourier Transform Methods
Advanced Geodynamics: Fourier Transform Methods David T. Sandwell January 13, 2021 To Susan, Katie, Melissa, Nick, and Cassie Eddie Would Go Preprint for publication by Cambridge University Press, October 16, 2020 Contents 1 Observations Related to Plate Tectonics 7 1.1 Global Maps . .7 1.2 Exercises . .9 2 Fourier Transform Methods in Geophysics 20 2.1 Introduction . 20 2.2 Definitions of Fourier Transforms . 21 2.3 Fourier Sine and Cosine Transforms . 22 2.4 Examples of Fourier Transforms . 23 2.5 Properties of Fourier transforms . 26 2.6 Solving a Linear PDE Using Fourier Methods and the Cauchy Residue Theorem . 29 2.7 Fourier Series . 32 2.8 Exercises . 33 3 Plate Kinematics 36 3.1 Plate Motions on a Flat Earth . 36 3.2 Triple Junction . 37 3.3 Plate Motions on a Sphere . 41 3.4 Velocity Azimuth . 44 3.5 Recipe for Computing Velocity Magnitude . 45 3.6 Triple Junctions on a Sphere . 45 3.7 Hot Spots and Absolute Plate Motions . 46 3.8 Exercises . 46 4 Marine Magnetic Anomalies 48 4.1 Introduction . 48 4.2 Crustal Magnetization at a Spreading Ridge . 48 4.3 Uniformly Magnetized Block . 52 4.4 Anomalies in the Earth’s Magnetic Field . 52 4.5 Magnetic Anomalies Due to Seafloor Spreading . 53 4.6 Discussion . 58 4.7 Exercises . 59 ii CONTENTS iii 5 Cooling of the Oceanic Lithosphere 61 5.1 Introduction . 61 5.2 Temperature versus Depth and Age . 65 5.3 Heat Flow versus Age . 66 5.4 Thermal Subsidence . 68 5.5 The Plate Cooling Model . -
Coordinate Systems in Geodesy
COORDINATE SYSTEMS IN GEODESY E. J. KRAKIWSKY D. E. WELLS May 1971 TECHNICALLECTURE NOTES REPORT NO.NO. 21716 COORDINATE SYSTElVIS IN GEODESY E.J. Krakiwsky D.E. \Vells Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton, N .B. Canada E3B 5A3 May 1971 Latest Reprinting January 1998 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. TABLE OF CONTENTS page LIST OF ILLUSTRATIONS iv LIST OF TABLES . vi l. INTRODUCTION l 1.1 Poles~ Planes and -~es 4 1.2 Universal and Sidereal Time 6 1.3 Coordinate Systems in Geodesy . 7 2. TERRESTRIAL COORDINATE SYSTEMS 9 2.1 Terrestrial Geocentric Systems • . 9 2.1.1 Polar Motion and Irregular Rotation of the Earth • . • • . • • • • . 10 2.1.2 Average and Instantaneous Terrestrial Systems • 12 2.1. 3 Geodetic Systems • • • • • • • • • • . 1 17 2.2 Relationship between Cartesian and Curvilinear Coordinates • • • • • • • . • • 19 2.2.1 Cartesian and Curvilinear Coordinates of a Point on the Reference Ellipsoid • • • • • 19 2.2.2 The Position Vector in Terms of the Geodetic Latitude • • • • • • • • • • • • • • • • • • • 22 2.2.3 Th~ Position Vector in Terms of the Geocentric and Reduced Latitudes . • • • • • • • • • • • 27 2.2.4 Relationships between Geodetic, Geocentric and Reduced Latitudes • . • • • • • • • • • • 28 2.2.5 The Position Vector of a Point Above the Reference Ellipsoid . • • . • • • • • • . .• 28 2.2.6 Transformation from Average Terrestrial Cartesian to Geodetic Coordinates • 31 2.3 Geodetic Datums 33 2.3.1 Datum Position Parameters . -
Sunsets, Tall Buildings, and the Earth's Radius
Sunsets, tall buildings, and the Earth’s radius P.K.Aravind Physics Department Worcester Polytechnic Institute Worcester, MA 01609 ([email protected] ) ABSTRACT It is shown how repeated observations of the sunset from various points up a tall building can be used to determine the Earth’s radius. The same observations can also be used, at some latitudes, to deduce an approximate value for the amount of atmospheric refraction at the horizon. Eratosthenes appears to have been the first to have determined the Earth’s radius by measuring the altitude of the sun at noon in Alexandria on a day when he knew it to be directly overhead in Syene (Aswan) to the south [1]. Despite the vastly better methods of determining the Earth’s size and shape today, Eratosthenes’ achievement continues to inspire schoolteachers and their students around the world. I can still recall my high school principal, Mr. Jack Gibson, telling us the story of Eratosthenes on numerous occasions in his inimitable style, which required frequent audience participation from the schoolboys gathered around him. In 2005, as part of the celebrations accompanying the World Year of Physics, the American Physical Society organized a project [2] in which students from 700 high schools all over North America collaborated in a recreation of Eratosthenes’ historic experiment. The schools were teamed up in pairs, and each pair carried out measurements similar to those of Eratosthenes and pooled their results to obtain a value for the Earth’s radius. The average of all the results so obtained yielded a value for the Earth’s radius that differed from the true value by just 6%. -
How Big Is the Earth?
Document ID: 04_01_13_1 Date Received: 2013-04-01 Date Revised: 2013-07-24 Date Accepted: 2013-08-16 Curriculum Topic Benchmarks: M4.4.8, M5.4.12, M8.4.2, M8.4.15, S15.4.4, S.16.4.1, S16.4.3 Grade Level: High School [9-12] Subject Keywords: circumference, radius, Eratosthenes, gnomon, sun dial, Earth, noon, time zone, latitude, longitude, collaboration Rating: Advanced How Big Is the Earth? By: Stephen J Edberg, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, M/S 183-301, Pasadena CA 91011 e-mail: [email protected] From: The PUMAS Collection http://pumas.jpl.nasa.gov ©2013, Jet Propulsion Laboratory, California Institute of Technology. ALL RIGHTS RESERVED. Eratosthenes, the third librarian of the Great Library of Alexandria, measured the circumference of Earth around 240 B.C.E. Having learned that the Sun passed through the zenith1 on the summer solstice2 as seen in modern day Aswan, he measured the length of a shadow on the solstice in Alexandria. By converting the measurement to an angle he determined the difference in latitude – what fraction of a circle spanned the separation – between Aswan and Alexandria. Knowing the physical distance between Aswan and Alexandria allowed him to determine the circumference of Earth. Cooperating schools can duplicate Eratosthenes’ measurements without the use of present day technology, if desired. Sharing their data permits students to calculate the circumference and the radius of Earth. The measurements do not require a site on the Tropic of Cancer (or the Tropic of Capricorn) but they must be made at local solar noon on the same date. -
Map Projections and Coordinate Systems Datums Tell Us the Latitudes and Longi- Vertex, Node, Or Grid Cell in a Data Set, Con- Tudes of Features on an Ellipsoid
116 GIS Fundamentals Map Projections and Coordinate Systems Datums tell us the latitudes and longi- vertex, node, or grid cell in a data set, con- tudes of features on an ellipsoid. We need to verting the vector or raster data feature by transfer these from the curved ellipsoid to a feature from geographic to Mercator coordi- flat map. A map projection is a systematic nates. rendering of locations from the curved Earth Notice that there are parameters we surface onto a flat map surface. must specify for this projection, here R, the Nearly all projections are applied via Earth’s radius, and o, the longitudinal ori- exact or iterated mathematical formulas that gin. Different values for these parameters convert between geographic latitude and give different values for the coordinates, so longitude and projected X an Y (Easting and even though we may have the same kind of Northing) coordinates. Figure 3-30 shows projection (transverse Mercator), we have one of the simpler projection equations, different versions each time we specify dif- between Mercator and geographic coordi- ferent parameters. nates, assuming a spherical Earth. These Projection equations must also be speci- equations would be applied for every point, fied in the “backward” direction, from pro- jected coordinates to geographic coordinates, if they are to be useful. The pro- jection coordinates in this backward, or “inverse,” direction are often much more complicated that the forward direction, but are specified for every commonly used pro- jection. Most projection equations are much more complicated than the transverse Mer- cator, in part because most adopt an ellipsoi- dal Earth, and because the projections are onto curved surfaces rather than a plane, but thankfully, projection equations have long been standardized, documented, and made widely available through proven programing libraries and projection calculators. -
Global Positioning System - Basics
GLOBAL POSITIONING SYSTEM - BASICS Definition The Global Positioning System (GPS) is a U.S. space-based global navigation satellite system. It provides reliable positioning, navigation, and timing services to worldwide users on a continuous basis in all weather, day and night. Basics Global Positioning System, also known as GPS, is a system designed to help navigate on the Earth, in the air, and floating on water. A GPS receiver displays where it is, how fast it is moving and which direction, how high it is, and maybe how fast it is going up or down. Many GPS receivers contain information about places. GPS units for automobiles contain travel data like road maps, hotels, restaurants, and service stations. GPSs for boats contain nautical charts of harbors, marinas, shallow water, rocks, and waterways. Other GPSs are for aviation, hiking and backpacking, bicycling, or many other activities. Most GPS receivers record where they have been, and help plan a journey. While traveling a planned journey, the unit predicts the time to the next destination. How it works GPS satellites circle the earth in four planes, plus a group over the equator. This example shows the number of satellites visible to a GPS receiver at 45° North in blue. Red satellites are blocked by the Earth.A GPS unit receives radio signals from satellites in space circling the Earth. There are about 30 satellites 20,200 kilometres (12,600 mi) above the Earth. (Each circle is 26,600 kilometres (16,500 mi) radius due to the Earth's radius.) Far from the North Pole and South Pole, a GPS unit can receive signals from 6 to 12 satellites at once. -
Earth Coverage by Satellites in Circular Orbit
Earth Coverage by Satellites in Circular Orbit Alan R. Washburn Department of Operations Research Naval Postgraduate School The purpose of many satellites is to observe or communicate with points on Earth’s surface. Such functions require a line of sight that is neither too long nor too oblique, so only a certain segment of Earth will be covered by a satellite at any given time. Here the term “covered” is meant to include “can communicate with” if the purpose of the satellite is communications. The shape of this covered segment depends on circumstances — it might be a thin rectangle of width w for a satellite-borne side-looking radar, for example. The shape will always be taken to be a spherical cap here, but a rough equivalence with other shapes could be made by letting the largest dimension, (w, for the radar mentioned above) span the cap. The questions dealt with will be of the type “what fraction of Earth does a satellite system cover?” or “How long will it take a satellite to find something?” Both of those questions need to be made more precise before they can be answered. Geometrical preliminaries Figure 1 (top) shows an orbiting spacecraft sweeping a swath on Earth. The leading edge of that swath is a circular “cap” within the spacecraft horizon. The bottom part of Figure 1 shows four important quantities for determining the size of that cap, those being R = radius of Earth (6378 km) r = radius of orbit α = cap angle β = masking angle. 2γ = field of view cap Spacecraft 2γ tangent to Earth R α β r Spacecraft FIGURE 1. -
Preparation of Papers for AIAA Technical Conferences
Gravity Modeling for Variable Fidelity Environments Michael M. Madden* NASA, Hampton, VA, 23681 Aerospace simulations can model worlds, such as the Earth, with differing levels of fidel- ity. The simulation may represent the world as a plane, a sphere, an ellipsoid, or a high- order closed surface. The world may or may not rotate. The user may select lower fidelity models based on computational limits, a need for simplified analysis, or comparison to other data. However, the user will also wish to retain a close semblance of behavior to the real world. The effects of gravity on objects are an important component of modeling real-world behavior. Engineers generally equate the term gravity with the observed free-fall accelera- tion. However, free-fall acceleration is not equal to all observers. To observers on the sur- face of a rotating world, free-fall acceleration is the sum of gravitational attraction and the centrifugal acceleration due to the world’s rotation. On the other hand, free-fall accelera- tion equals gravitational attraction to an observer in inertial space. Surface-observed simu- lations (e.g. aircraft), which use non-rotating world models, may choose to model observed free fall acceleration as the “gravity” term; such a model actually combines gravitational at- traction with centrifugal acceleration due to the Earth’s rotation. However, this modeling choice invites confusion as one evolves the simulation to higher fidelity world models or adds inertial observers. Care must be taken to model gravity in concert with the world model to avoid denigrating the fidelity of modeling observed free fall. -
Coincidence Level Among Terrestrial Reference Frames Available Through GNSS Broadcast Messages
Coincidence Level Among Terrestrial Reference Frames Available through GNSS Broadcast Messages Stephen Malys, Chris O’Neill 5 December 2017 Presented at ICG-12, Kyoto, Japan Approved for public release, 18-105 Broadcast GNSS Ephemerides ►Predictions that must be used for direct (non- augmented) real-time Positioning, Navigation and Timing ►Represent a real-time realization of the operational terrestrial reference frame ►Accessible by following procedures documented in respective space to user segment Interface Control Documents Approved for public release, 18-105 2 7-Parameter Transformations using Ephemerides Approved for public release, 18-105 3 7 Parameter Transformation Approved for public release, 17-046 4 Terrestrial Reference Frames in GNSS Data Span: First 8-10 weeks of 2016 ► Terrestrial Reference Frame used in Broadcast Ephemerides § GPS • WGS 84 (G1762’) § GLONASS • PZ-90.11 § Galileo • GTRF16v01 § BeiDou • CTRF2000 ► Terrestrial Reference Frame used in Post-Fit MGEX Ephemerides § ITRF2008 as generated by CODE and Wuhan University (BeiDou) ► All Broadcast and Precise Ephemerides were obtained from: ftp://cddis.gsfc.nasa.gov/pub/gps/products/mgex § CODE= Center for Orbit Determination in Europe § MGEX = Multi GNSS Experiment, Coordinated by the International GNSS Service for Geodynamics (IGS) 5 Approved for public release, 18-105 Constellations and Tracking Networks Used Constellation # of Satellites # of IGS Stations in MGEX Satellites Tracked (PRN) (SC s/n for GLONASS) GPS 29 73 01, 02, 03, 05, 07, 08, 09, 10, 11, 12, 13, 14, -
On the Geoid and Orthometric Height Vs. Quasigeoid and Normal Height
J. Geod. Sci. 2018; 8:115–120 Research Article Open Access Lars E. Sjöberg* On the geoid and orthometric height vs. quasigeoid and normal height https://doi.org/10.1515/jogs-2018-0011 Keywords: ambiguous quasigeoid, geoid, geoid- Received August 6, 2018; accepted November 6, 2018 quasigeoid difference, resolution, vertical datum, quasi- geoid Abstract: The geoid, but not the quasigeoid, is an equipo- tential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geo- physics. It is also a natural zero-level surface, as it agrees 1 Introduction with the undisturbed mean sea level. Orthometric heights are physical heights above the geoid, while normal heights The geoid is an important equipotential surface and ver- are geometric heights (of the telluroid) above the reference tical reference surface in geodesy and geophysics. The ellipsoid. Normal heights and the quasigeoid can be deter- quasigeoid, introduced by M.S. Moldensky (Molodensky mined without any information on the Earth’s topographic et al. 1962) is not an equipotential surface, and it has density distribution, which is not the case for orthometric no special meaning in geophysics. The geoid serves as heights and geoid. the ideal reference surface for height systems in all coun- We show from various derivations that the difference be- tries that adopt orthometric heigths, while the rest of the tween the geoid and the quasigeoid heights, being of the world uses the quasigeoid with normal height systems (or order of 5 m, can be expressed by the simple Bouguer grav- normal-orthometric heights with more or less unknown ity anomaly as the only term that includes the topographic zero-levels).