*UDYLW\)LHOG7XWRULDO
|
|
- Miranda Crawford
- 8 years ago
- Views:
Transcription
1 % *UDYLW\)LHOG7XWRULDO The formulae and derivations in the following Chapters 1 to 3 are based on Heiskanen and Moritz (1967) and Lambeck (1990). ([SDQVLRQRIWKHJUDYLWDWLRQDOSRWHQWLDOLQWRVSKHULFDOKDUPRQLFV The stationary part of the (DUWK VJUDYLWDWLRQDOSRWHQWLDO8DWDQ\SRLQW3UMO on and above the Earth s surface is expressed on a global scale conveniently by summing up over degree and order of a spherical harmonic expansion. The spherical harmonic (or Stokes ) coefficients represent in the spectral domain the global structure and irregularities of the geopotential field or, more generally spoken, of the gravity field of the Earth. The equation relating the spatial and spectral domain of the geopotential is as follows: Uϕ U U + ( FRVP + 6 VLQP) VLQϕ where Uϕ - spherical geocentric coordinates of computation point (radius, latitude, longitude) - reference length (mean semi-major axis of Earth) - gravitational constant times mass of Earth OP - degree, order of spherical harmonic 3 - fully normalized Lengendre functions 6 - Stokes'coefficients (fully normalized) (1.1) The 00 -term is close to 1 and scales the value. The degree 1 spherical harmonic coefficients 6 are related to the geocentre coordinates and zero if the coordinate systems origin coincides with the geocentre. The coefficients connected to the mean rotational pole position that is a function of time. 6 are Subtracting from the low-degree zonal coefficients (order 0) the corresponding Stokes coefficients of an HOOLSVRLGDOQRUPDOSRWHQWLDO 9UM leads to the mathematical representations of the GLVWXUELQJSRWHQWLDO7UMOin spherical harmonics, related to a conventional ellipsoid of revolution that approximates the Earth s parameters. At the Earth surface with U (in spherical approximation) the disturbing potential reads: 7 ϕ 8 ϕ 9 ϕ (1.2a) ( FRV P + 6 VLQ P)! " 7 ϕ 3 VLQ + ϕ (1.2b) with #$ $ and T defined on the geoid. Note, that '' is close to zero. 1
2 The maximum degree O(*) + of the expansion in Equation (1.1) correlates to the spatial resolution at the Earth surface by (*, - NPO(.) + (1.3) where (*, - is the minimum wavelength (or twice the pixel side length) of gravity field features that are resolved by the O(/) + O(.)0+ O(/) + 1 parameters Equation (1.1) contains the upward-continuation of the gravitational potential at the Earth s surface for U > and reflects the attenuation of the signal with altitude through the factor (U) 4 6. Figure 1.1 gives examples for the three different kinds of spherical harmonics 37 8 VLQϕ FRV P : (a) zonal with O P (b) tesseral with O P Oand (c) sectorial harmonic with O P Amplitudes and phase of the individual spherical harmonics then are determined by multiplication with the 9 : and 6 ; < coefficients. zonal: l6, m0 tesseral: l16, m9 sectorial: l9, m9 )LJXUH ([DPSOHV IRU VSKHULFDO KDUPRQLFV YLROHW@ 3 OP VLQϕ FRV P >IURP ± EOXH WR )XQFWLRQDOVRIWKHGLVWXUELQJJUDYLWDWLRQDOSRWHQWLDO The JHRLG XQGXODWLRQ 1 (Figure 2.1) is the distance between the special equipotential surface U(ϕ) const that is close to the mean sea level and the surface of the conventional ellipsoid of revolution. As such the geoid is derived from the disturbing potential 7 applying %UXQVIRUPXOD 7 1 (2.1) γ where γ is 'normal'gravity on the surface of the ellipsoid. With γ 1 in spherical approximation, the geoid undulations (or geoid heights) can be computed from the spherical harmonic coefficients in Equation (1.2) by 1 ϕ 7 ϕ (2.2a) 2
3 _ ] K V H [ > > Q B F F Y Y? J > O ^ A V Z Y \ G F I P O R W W Y H ] Q [ ( >? FRV P + 6 VLQ P) C D E AA 1 ϕ 3>? VLQ >? + ϕ (2.2b) 7 The negative of the vertical derivative of the disturbing potential δ J is called U JUDYLW\ GLVWXUEDQFH GJ (Figure 2.2) that is equal to gravity at a point 3 (negative of vertical derivative of 8) minus normal gravity at point 3 (negative of vertical derivative of 9). On the geoid and in spherical approximation U the gravity disturbance then is expressed by ( F G FRV P + 6F G VLQ P) L M N II δ J ϕ O 3F G + + VLQϕ (2.3) The difference between gravity at a point 3 on the geoid and normal gravity at the corresponding point 4 on the ellipsoid is called JUDYLW\ DQRPDO\ 'J (Figure 2.3) and related to the disturbing potential by 7 J 7 (2.4) U U On the geoid this becomes (note: no degree 1 terms appear in Equation 2.) ( O P FRV P + 6O P VLQ P) O O P PTSU RR J ϕ + O 3 VLQϕ (2.a) thus J δj 7 (2.b) The second derivatives of the disturbing potential leads to the JUDYLW\JUDGLHQW WHQVRU. The most important vertical gradient represented as J 7 of the tensor component can be U J U \\ + ` a b O + O + U + 3Y Z VLQϕ ( Y Z FRV P + 6Y Z VLQ P) (2.6) Once the spherical harmonic coefficients c d 6c d of a global gravity field model are given, the quantities of the various functionals described above can be computed in its geographical distribution. If computed in terms of gravity disturbances or anomalies and gravity gradients, the higher frequency regional to local content is emphasised through the degree-dependent factors OO and OO respectively, whereas the potential 3
4 )LJXUH *HRLG XQGXODWLRQV 1 >P@ UHVROXWLRQ NP UPV 1 FRV ϕ P )LJXUH *UDYLW\ GLVWXUEDQFHV δj >PJDO@ UHVROXWLRQ NP UPV δj )LJXUH *UDYLW\ DQRPDOLHV J >PJDO@ UHVROXWLRQ NP UPV J FRV ϕ PJDO FRV ϕ PJDO 4
5 and geoid representations of the gravity field show the broad and generalized features of the gravity field. Vice versa, a gradiometer measuring gravity gradients is capable to better resolve detailed structures of the gravity field rather than the long wavelength part. The fully normalized spherical harmonic coefficients in Equation (1.1) are related to the mass distribution within the Earth by O + e f e 0 g* h ikj l U 3e f VLQ ϕ FRV PG0 (2.7a) O + 6 m n m 0 o* p qkr s U 3m n VLQ ϕ VLQ PG0 (2.7b) e m with the mass element G0 G0 U ϕ Figure 2.4 depicts the geopotential distribution of gravity anomalies over Europe derived from spherical harmonic coefficients complete to Ot.u v equal to 10, 0, 100, 300, respectively, in order to demonstrate the relation between spectral and spatial resolution according to Equation (1.1). )LJXUH *HRJUDSKLFDO GLVWULEXWLRQ RI JUDYLW\ DQRPDOLHV >PJDO@ RYHU (XURSH ZLWK GLIIHUHQW VSHFWUDO Owyx z DQG VSDWLDO UHVROXWLRQ SL[HO VL]H w { }
6 ƒ Š Ž Œ ˆ 7KHSRZHUVSHFWUXPRIWKH(DUWK VJUDYLW\ILHOG Given the fully normalized Stokes coefficients ~ 6~ of a specific degree O over orders m P O the VLJQDO GHJUHH DPSOLWXGHV V (or square root of power per degree O) of functions of the disturbing potential 7ϕ at the Earth s surface are readily computed by ƒ ƒ σ ƒ + 6 in terms of unitless coefficients (3.1a) σ 7 σ in terms of disturbing potential values (m 2 /s 2 ) (3.1b) σ ˆ 1 σ in terms of geoid heights (m) (3.1c) σ δj O + σ in terms of gravity disturbances (m/s 2 ) (3.1d) σ J O σ in terms of gravity anomalies (m/s 2 ) (3.1e) σ J O + O + σ in terms of vertical gravity gradients (1/s 2 ) (3.1f) where the 6 are related to the normal potential. The SI units of the physical gravitational quantities are given in parenthesis. Following Kaula s 'rule of thumb' (Kaula, 1966) the power law follows approximately σ 2O + 1. (3.2) 4 O Examples for signal degree amplitudes are given in Figure 3.1. If the estimation errors of the Stokes coefficients in a global gravity field model are known, the HUURUGHJUHHDPSOLWXGHV (error spectrum) are computed accordingly replacing the coefficients in Equation (3.1) by their standard deviations. 'LIIHUHQFH GHJUHH DPSOLWXGHV, representing the agreement of two different gravity field models per degree, are readily computed replacing the coefficients in Equation (3.1) by the coefficients differences between the two models. Examples for difference degree amplitudes are given in Figure
7 )LJXUH 6LJQDO GHJUHH DPSOLWXGHV IRU JHRLG XQGXODWLRQV UHG JUDYLW\ GLVWXUEDQFHV EOXH DQG JUDYLW\ DQRPDOLHV JUHHQ LQ PHWHU DQG PJDO UHVSHFWLYHO\ The GHJUHH DPSOLWXGHV DV D IXQFWLRQ RI PLQLPXP DQG PD[LPXP GHJUHH O displays the power (signal, error, difference) spectrum accumulated over a spectral band from O to O : σ DFFXPXODWHG σ (3.3) Usually O 0 or 2 is taken to display the increase in overall power with increasing degree O. Recall that the spectral degree O is related to the spatial extension or wavelength of features in the gravity field according to Equation (1.3). Examples for difference amplitudes as a function of maximum degree l (successive accumulation of the curves in Figure 3.1) are given in Figure 3.2. Equations (3.1) again demonstrate that the higher degree terms, i.e. the shorter wavelengths in the signal spectra, are enhanced by factors proportional to degree O for gravity anomalies and disturbances and proportional to O for gravity gradients compared to the signals in the geoid and gravitational potential. )LJXUH 'LIIHUHQFH GHJUHH DPSOLWXGHV *$( 6 YV ( LQ WHUPV RI JHRLG XQGXODWLRQV UHG JUDYLW\ GLVWXUEDQFHV EOXH DQG JUDYLW\ DQRPDOLHV JUHHQ LQ PHWHU DQG PJDO UHVSHFWLYHO\ 7
8 )LJXUH 'LIIHUHQFH GHJUHH DPSOLWXGHV *$(6 YV ( DV D IXQFWLRQ RI PD[LPXP GHJUHH LQ WHUPV RI JHRLG XQGXODWLRQV UHG JUDYLW\ GLVWXUEDQFHV EOXH DQG JUDYLW\ DQRPDOLHV JUHHQLQPHWHUDQGPJDOUHVSHFWLYHO\ HIHUHQFHV Heiskanen, W.A. and H. Moritz, Physical Geodesy, W.H. Freeman and Co., San Francisco. Kaula, W.M., Theory of Satellite Geodesy, Blaisdell Publ. Company, Waltham, Mass. Lambeck, K., Aristoteles An ESA Mission to Study the Earth s Gravity Field, ESA Journal 14:
Gravitational potential
Gravitational potential Let s assume: A particle of unit mass moving freely A body of mass M The particle is attracted by M and moves toward it by a small quantity dr. This displacement is the result of
More informationSIO 229 Gravity and Geomagnetism: Class Description and Goals
SIO 229 Gravity and Geomagnetism: Class Description and Goals This graduate class provides an introduction to gravity and geomagnetism at a level suitable for advanced non-specialists in geophysics. Topics
More informationThe Map Grid of Australia 1994 A Simplified Computational Manual
The Map Grid of Australia 1994 A Simplified Computational Manual The Map Grid of Australia 1994 A Simplified Computational Manual 'What's the good of Mercator's North Poles and Equators, Tropics, Zones
More informationFlight and Orbital Mechanics
Flight and Orbital Mechanics Lecture slides Challenge the future 1 Material for exam: this presentation (i.e., no material from text book). Sun-synchronous orbit: used for a variety of earth-observing
More informationExamination Space Missions and Applications I (AE2103) Faculty of Aerospace Engineering Delft University of Technology SAMPLE EXAM
Examination Space Missions and Applications I AE2103 Faculty of Aerospace Engineering Delft University of Technology SAMPLE EXAM Please read these instructions first: This are a series of multiple-choice
More informationContents. 1 Introduction 2
Contents 1 Introduction 2 2 Definitions 2 2.1 The Potential and the Geoid.................................. 2 2.2 The Height Anomaly....................................... 5 2.3 The Gravity Disturbance....................................
More informationOrbital Mechanics and Space Geometry
Orbital Mechanics and Space Geometry AERO4701 Space Engineering 3 Week 2 Overview First Hour Co-ordinate Systems and Frames of Reference (Review) Kepler s equations, Orbital Elements Second Hour Orbit
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationGravity Field and Dynamics of the Earth
Milan Bursa Karel Pec Gravity Field and Dynamics of the Earth With 89 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Preface v Introduction 1 1 Fundamentals
More informationgloser og kommentarer til Macbeth ( sidetal henviser til "Illustrated Shakespeare") GS 96
side 1! " # $ % &! " '! ( $ ) *! " & +! '!, $ #! "! - % " "! &. / 0 1 & +! '! ' & ( ' 2 ) % $ 2 3 % # 2! &. 2! ' 2! '! $ $! " 2! $ ( ' 4! " 3 & % / ( / ( ' 5! * ( 2 & )! 2 5! 2 &! # '! " & +! '! ' & &!
More informationSection 4: The Basics of Satellite Orbits
Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,
More informationREeal data AnaLysis GOCE Gravity field determination from GOCE
REeal data AnaLysis GOCE Gravity field determination from GOCE J.M. Brockmann 1, O. Baur 3, J. Cai 3, A. Eicker 2, B. Kargoll 1, I. Krasbutter 1, J. Kusche 2, T. Mayer-Gürr 2, J. Schall 2, W.-D. Schuh
More informationGRAVITATIONAL FIELDS PHYSICS 20 GRAVITATIONAL FORCES. Gravitational Fields (or Acceleration Due to Gravity) Symbol: Definition: Units:
GRAVITATIONAL FIELDS Gravitational Fields (or Acceleration Due to Gravity) Symbol: Definition: Units: Formula Description This is the formula for force due to gravity or as we call it, weight. Relevant
More informationMath 215 Project (25 pts) : Using Linear Algebra to solve GPS problem
Due Thursday March 1, 2012 NAME(S): Math 215 Project (25 pts) : Using Linear Algebra to solve GPS problem 0.1 Introduction The age old question, Where in the world am I? can easily be solved nowadays by
More informationDetermination of Acceleration due to Gravity
Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two
More informationHalliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton
Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 13-1 Newton's Law
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationThe Viscosity of Fluids
Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et
More informationMaintaining High Accuracy in Modern Geospatial Data
Maintaining High Accuracy in Modern Geospatial Data Patrick Cunningham President info@bluemarblegeo.com www.bluemarblegeo.com +1 (207) 582 6747 Copyright 2010 Blue Marble Geographics Concepts Geodesy -
More informationThe Gravitational Field
The Gravitational Field The use of multimedia in teaching physics Texts to multimedia presentation Jan Hrnčíř jan.hrncir@gfxs.cz Martin Klejch martin.klejch@gfxs.cz F. X. Šalda Grammar School, Liberec
More informationEarth Coordinates & Grid Coordinate Systems
Earth Coordinates & Grid Coordinate Systems How do we model the earth? Datums Datums mathematically describe the surface of the Earth. Accounts for mean sea level, topography, and gravity models. Projections
More informationCandidate Number. General Certificate of Education Advanced Level Examination June 2014
entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 214 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Wednesday
More informationCoverage Characteristics of Earth Satellites
Coverage Characteristics of Earth Satellites This document describes two MATLAB scripts that can be used to determine coverage characteristics of single satellites, and Walker and user-defined satellite
More informationWaves. Wave Parameters. Krauss Chapter Nine
Waves Krauss Chapter Nine Wave Parameters Wavelength = λ = Length between wave crests (or troughs) Wave Number = κ = 2π/λ (units of 1/length) Wave Period = T = Time it takes a wave crest to travel one
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationNewton s Law of Gravity
Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall Gravitational Potential Energy In space, an object or gas cloud has
More informationGEOGRAPHIC INFORMATION SYSTEMS CERTIFICATION
GEOGRAPHIC INFORMATION SYSTEMS CERTIFICATION GIS Syllabus - Version 1.2 January 2007 Copyright AICA-CEPIS 2009 1 Version 1 January 2007 GIS Certification Programme 1. Target The GIS certification is aimed
More information2. Orbits. FER-Zagreb, Satellite communication systems 2011/12
2. Orbits Topics Orbit types Kepler and Newton laws Coverage area Influence of Earth 1 Orbit types According to inclination angle Equatorial Polar Inclinational orbit According to shape Circular orbit
More informationFITTING ASTRONOMICAL DATA
1 01/02/2011 André Le Floch Fitting astronomical data FITTING ASTRONOMICAL DATA André Le Floch University of Tours, Department of Physics 37200 Tours, France Abstract Deming s method is applied for calculating
More informationChapter 3.8 & 6 Solutions
Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled
More informationAP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationLecture 2. Map Projections and GIS Coordinate Systems. Tomislav Sapic GIS Technologist Faculty of Natural Resources Management Lakehead University
Lecture 2 Map Projections and GIS Coordinate Systems Tomislav Sapic GIS Technologist Faculty of Natural Resources Management Lakehead University Map Projections Map projections are mathematical formulas
More informationWHAT YOU NEED TO USE THE STATE PLANE COORDINATE SYSTEMS
WHAT YOU NEED TO USE THE STATE PLANE COORDINATE SYSTEMS N & E State Plane Coordinates for Control Points AZIMUTHS - True, Geodetic, or Grid - Conversion from Astronomic to Geodetic (LaPlace Correction)
More informationEPSG. Coordinate Reference System Definition - Recommended Practice. Guidance Note Number 5
European Petroleum Survey Group EPSG Guidance Note Number 5 Coordinate Reference System Definition - Recommended Practice Revision history: Version Date Amendments 1.0 April 1997 First release. 1.1 June
More informationPenn State University Physics 211 ORBITAL MECHANICS 1
ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infra-red spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More informationThe Fourier Analysis Tool in Microsoft Excel
The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier
More informationVertical Datums: An Introduction and Software Review
Vertical Datums: An Introduction and Software Review Areas to Cover Theoretical Introduction Representation in EPSG Representation in OGC WKT Incorporation in PROJ.4 Incorporation in GDAL Future Work Introduction
More informationGeodätische Woche 2015, Stuttgart
Geodätische Woche 2015, Stuttgart Spheroidal and Ellipsoidal Harmonic Expansions of the Gravitational Potential of Small Solar System Bodies Stefan Reimond and Oliver Baur Space Research Institute (IWF)
More informationCalculation of Azimuth, Elevation and Polarization for non-horizontal aligned Antennas
Calculation of Azimuth, Elevation and Polarization for non-horizontal aligned Antennas Algorithm Description Technical Document TD-1205-a Version 1.1 23.10.2012 In Co-operation with 1 Objective Many SatCom
More informationAstromechanics Two-Body Problem (Cont)
5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the
More information2. Simple Linear Regression
Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationHalliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton
Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 13-1 Newton's Law
More informationREaldatenAnaLyse GOCE (REAL GOCE) 5. Projekttreffen
REaldatenAnaLyse GOCE (REAL GOCE) 5. Projettreffen Michael Murböc, Claudia Stummer Institut für Astronomische und Physialische Geodäsie, TU München Stuttgart, 10/10/2011 REAL GOCE Projettreffen: Stuttgart,
More information3.1 State Space Models
31 State Space Models In this section we study state space models of continuous-time linear systems The corresponding results for discrete-time systems, obtained via duality with the continuous-time models,
More informationBarometric Effects on Transducer Data and Groundwater Levels in Monitoring Wells D.A. Wardwell, October 2007
Barometric Effects on Transducer Data and Groundwater Levels in Monitoring Wells D.A. Wardwell, October 2007 Barometric Effects on Transducer Data Barometric Fluctuations can Severely Alter Water Level
More informationSupporting Information
S1 Supporting Information GFT NMR, a New Approach to Rapidly Obtain Precise High Dimensional NMR Spectral Information Seho Kim and Thomas Szyperski * Department of Chemistry, University at Buffalo, The
More informationA guide to coordinate systems in Great Britain
A guide to coordinate systems in Great Britain An introduction to mapping coordinate systems and the use of GPS datasets with Ordnance Survey mapping D00659 v2.3 Mar 2015 Crown copyright Page 1 of 43 Contents
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationFunctions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions
Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function
More informationAPPENDIX D: SOLAR RADIATION
APPENDIX D: SOLAR RADIATION The sun is the source of most energy on the earth and is a primary factor in determining the thermal environment of a locality. It is important for engineers to have a working
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More informationTerrain-Related Gravimetric Quantities Computed for the Next EGM
Terrain-Related Gravimetric Quantities Computed for the Next EGM Nikolaos K. Pavlis 1, John K. Factor 2, and Simon A. Holmes 1 1 SGT, Inc., 7701 Greenbelt Road, Suite 400, Greenbelt, Maryland 20770, USA,
More informationDevelopment of new hybrid geoid model for Japan, GSIGEO2011. Basara MIYAHARA, Tokuro KODAMA, Yuki KUROISHI
Development of new hybrid geoid model for Japan, GSIGEO2011 11 Development of new hybrid geoid model for Japan, GSIGEO2011 Basara MIYAHARA, Tokuro KODAMA, Yuki KUROISHI (Published online: 26 December 2014)
More informationCase Study Australia. Dr John Dawson A/g Branch Head Geodesy and Seismic Monitoring Geoscience Australia. Chair UN-GGIM-AP WG1 Chair APREF.
Case Study Australia Dr John Dawson A/g Branch Head Geodesy and Seismic Monitoring Geoscience Australia Chair UN-GGIM-AP WG1 Chair APREF Page 1 Overview 1. Australian height system Australian Height Datum
More informationThe Calculation of G rms
The Calculation of G rms QualMark Corp. Neill Doertenbach The metric of G rms is typically used to specify and compare the energy in repetitive shock vibration systems. However, the method of arriving
More informationMeasurement of Length, Mass, Volume and Density
Measurement of Length, Mass, Volume and Density Experimental Objective The objective of this experiment is to acquaint you with basic scientific conventions for measuring physical quantities. You will
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
More informationInteraction of Energy and Matter Gravity Measurement: Using Doppler Shifts to Measure Mass Concentration TEACHER GUIDE
Interaction of Energy and Matter Gravity Measurement: Using Doppler Shifts to Measure Mass Concentration TEACHER GUIDE EMR and the Dawn Mission Electromagnetic radiation (EMR) will play a major role in
More informationRANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA
RANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA ABSTRACT Random vibration is becoming increasingly recognized as the most realistic method of simulating the dynamic environment of military
More informationUniversal Law of Gravitation
Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationElectromagnetism - Lecture 2. Electric Fields
Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More informationGPS ALIGNMENT SURVEYS AND MERIDIAN CONVERGENCE
GPS ALIGNMENT SURVEYS AND MERIDIAN CONVERGENCE By Tomás Soler, 1 Member, ASCE, and Rudolf J. Fury 2 ABSTRACT: Since the advent of the Global Positioning System (GPS), geodetic azimuths can be accurately
More informationRECOMMENDATION ITU-R P.1546-1. Method for point-to-area predictions for terrestrial services in the frequency range 30 MHz to 3 000 MHz
Rec. ITU-R P.546- RECOMMENDATION ITU-R P.546- Method for point-to-area predictions for terrestrial services in the frequency range 30 MHz to 3 000 MHz (200-2003) The ITU Radiocommunication Assembly, considering
More informationAcceleration levels of dropped objects
Acceleration levels of dropped objects cmyk Acceleration levels of dropped objects Introduction his paper is intended to provide an overview of drop shock testing, which is defined as the acceleration
More information2After completing this chapter you should be able to
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More information4 The Rhumb Line and the Great Circle in Navigation
4 The Rhumb Line and the Great Circle in Navigation 4.1 Details on Great Circles In fig. GN 4.1 two Great Circle/Rhumb Line cases are shown, one in each hemisphere. In each case the shorter distance between
More informationBandwidth-dependent transformation of noise data from frequency into time domain and vice versa
Topic Bandwidth-dependent transformation of noise data from frequency into time domain and vice versa Authors Peter Bormann (formerly GeoForschungsZentrum Potsdam, Telegrafenberg, D-14473 Potsdam, Germany),
More informationCoordinate Systems. Orbits and Rotation
Coordinate Systems Orbits and Rotation Earth orbit. The earth s orbit around the sun is nearly circular but not quite. It s actually an ellipse whose average distance from the sun is one AU (150 million
More information量 說 Explanatory Notes on Geodetic Datums in Hong Kong
量 說 Explanatory Notes on Geodetic Datums in Hong Kong Survey & Mapping Office Lands Department 1995 All Right Reserved by Hong Kong Government 留 CONTENTS INTRODUCTION............... A1 HISTORICAL BACKGROUND............
More informationG U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationSatellites and Space Stations
Satellites and Space Stations A satellite is an object or a body that revolves around another object, which is usually much larger in mass. Natural satellites include the planets, which revolve around
More informationAN1200.04. Application Note: FCC Regulations for ISM Band Devices: 902-928 MHz. FCC Regulations for ISM Band Devices: 902-928 MHz
AN1200.04 Application Note: FCC Regulations for ISM Band Devices: Copyright Semtech 2006 1 of 15 www.semtech.com 1 Table of Contents 1 Table of Contents...2 1.1 Index of Figures...2 1.2 Index of Tables...2
More informationESTIMATION USABILITY OF THE FREE SOFTWARE FOR TRANSFORMATION OF GEODETIC COORDINATES BETWEEB LOCAL AND GLOBAL DATUMS-EXAMPLE OF THE ADRIATIC SEA
ESTIMATION USABILITY OF THE FREE SOFTWARE FOR TRANSFORMATION OF GEODETIC COORDINATES BETWEEB LOCAL AND GLOBAL DATUMS-EXAMPLE OF THE ADRIATIC SEA Duplančić Leder, Tea; Faculty of Civil Engineering and Architecture,
More informationA) F = k x B) F = k C) F = x k D) F = x + k E) None of these.
CT16-1 Which of the following is necessary to make an object oscillate? i. a stable equilibrium ii. little or no friction iii. a disturbance A: i only B: ii only C: iii only D: i and iii E: All three Answer:
More informationAnalysis/resynthesis with the short time Fourier transform
Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis
More informationATM 316: Dynamic Meteorology I Final Review, December 2014
ATM 316: Dynamic Meteorology I Final Review, December 2014 Scalars and Vectors Scalar: magnitude, without reference to coordinate system Vector: magnitude + direction, with reference to coordinate system
More informationSound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8
References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that
More information= δx x + δy y. df ds = dx. ds y + xdy ds. Now multiply by ds to get the form of the equation in terms of differentials: df = y dx + x dy.
ERROR PROPAGATION For sums, differences, products, and quotients, propagation of errors is done as follows. (These formulas can easily be calculated using calculus, using the differential as the associated
More informationch 15 practice test Multiple Choice Identify the letter of the choice that best completes the statement or answers the question.
ch 15 practice test Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Work is a transfer of a. energy. c. mass. b. force. d. motion. 2. What
More informationWeight The weight of an object is defined as the gravitational force acting on the object. Unit: Newton (N)
Gravitational Field A gravitational field as a region in which an object experiences a force due to gravitational attraction Gravitational Field Strength The gravitational field strength at a point in
More informationThe Viscosity of Fluids
Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et
More informationGEOPHYSICAL EFFECTS ON SITE DISPLACEMENTS FOR PERMANENT GPS TRACKING STATIONS IN TAIWAN
GEOPHYSICAL EFFECTS ON SITE DISPLACEMENTS FOR PERMANENT GPS TRACKING STATIONS IN TAIWAN C. C. Chang Department of Surveying and Mapping Engineering Chung Cheng Institute of Technology Tahsi, Taoyuan 335,
More informationGünter Seeber. Satellite Geodesy 2nd completely revised and extended edition
Günter Seeber Satellite Geodesy 2nd completely revised and extended edition Walter de Gruyter Berlin New York 2003 Contents Preface Abbreviations vii xvii 1 Introduction 1 1.1 Subject of Satellite Geodesy...
More informationThe Earth Really is Flat! The Globe and Coordinate Systems. Long History of Mapping. The Earth is Flat. Long History of Mapping
The Earth Really is Flat! The Globe and Coordinate Systems Intro to Mapping & GIS The Earth is Flat Day to day, we live life in a flat world sun rises in east, sets in west sky is above, ground is below
More information2-1 Position, Displacement, and Distance
2-1 Position, Displacement, and Distance In describing an object s motion, we should first talk about position where is the object? A position is a vector because it has both a magnitude and a direction:
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationFrequency-domain and stochastic model for an articulated wave power device
Frequency-domain stochastic model for an articulated wave power device J. Cândido P.A.P. Justino Department of Renewable Energies, Instituto Nacional de Engenharia, Tecnologia e Inovação Estrada do Paço
More informationBinary Stars. Kepler s Laws of Orbital Motion
Binary Stars Kepler s Laws of Orbital Motion Kepler s Three Laws of orbital motion result from the solution to the equation of motion for bodies moving under the influence of a central 1/r 2 force gravity.
More informationArtificial Satellites Earth & Sky
Artificial Satellites Earth & Sky Name: Introduction In this lab, you will have the opportunity to find out when satellites may be visible from the RPI campus, and if any are visible during the activity,
More informationAn Introduction to the MTG-IRS Mission
An Introduction to the MTG-IRS Mission Stefano Gigli, EUMETSAT IRS-NWC Workshop, Eumetsat HQ, 25-0713 Summary 1. Products and Performance 2. Design Overview 3. L1 Data Organisation 2 Part 1 1. Products
More information