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Remote Sensing Math National Aeronautics and Space Administration andSpace Aeronautics National R emot e S ensing M a th I Remote Sensing Remote sending of a dust devil on the surface of mars - NASA/Mars Rover/JPL A Brief Mathematical Guide Dr. Sten Odenwald NASA Space Math http://spacemath.gsfc.nasa.gov II This collection of activities is based on a weekly series of space science problems distributed to thousands of teachers during the 2009-2010 school year. They were intended for students looking for additional challenges in the math and physical science curriculum in grades 9 through 12. The problems were created to be authentic glimpses of modern science and engineering issues, often involving actual research data. The problems were designed to be ‘one-pagers’ with a Teacher’s Guide and Answer Key as a second page. This compact form was deemed very popular by participating teachers. Acknowledgments: We would like to thank Dr. Michael Wilson (University of Maryland) for his careful reading of the draft of this book, his extensive and efficient error-detection algorithm, and the many constructive comments made about problem content and level of difficulty. Dr. Wilson is a research associate for the Goddard Earth Sciences and Technology Center at the University of Maryland, Baltimore County. Mike received his Ph.D. in 2009 in Atmospheric Sciences from the University of Illinois at Urbana-Champaign, with a focus on remote sensing in the polar regions. His current work focuses on cloud radiative modeling and cloud detection in Landsat imagery. We would also like to thank Ms. Jeannette Allen and Anita Davis (NASA Goddard Eath Sciences Division) for their enthusiastic support of this resource and many helpful comments and suggestions. For more weekly classroom activities about astronomy and space visit the NASA website, http://spacemath.gsfc.nasa.gov To suggest math problem or science topic ideas, contact the Author, Dr. Sten Odenwald at [email protected] Front and back cover credits: Front: Collage (USGS); Earth globe (MODIS); Messier 82 (Spitzer/Hubble); Sun(SOHO/EIT) Back: Oman (Landsat); Mykolaiv Ukraine(Landsat) Interior: Mars Dust Devil (NASA/JPL) Space Math http://spacemath.gsfc.nasa.gov Table of Contents III Grade Page Acknowledgments ii Table of Contents iii Alignment with Standards vi Mathematics Topic Matrix vii How to use this Book xi Teacher Comments xii Introduction to Remote Sensing xiii 1.0 Digital Picture Basics Image Formats : Taking a digital picture 6-8 1 Counting Photons : Pixel math 6-8 2 Storing Pixel Data 6-8 3 Images as Arrays of Numbers 6-8 4 Image Math - Hot Pixels 6-8 5 Image Math - Dark Current 9-12 6 Image Math - Flat Fielding 9-12 7 Image Math – Manipulating arrays of numbers 9-12 8 Telemetry – Transmitting data back to Earth 9-12 9 Telemetry and Binary Numbers 6-8 10 Interplanetary Communication 9-12 11 Computer Memory and Glitches 9-12 12 Correcting Bad Data Using Parity Bits 6-8 13 Data Corruption by High Energy Particles 6-8 14 Working with Giga, Tera, Peta and Exa 6-8 15 2.0 Photons and Digital Imaging Digital Imaging - Working with Irradiance 9-12 16 Photons and Photon Energy 9-12 17 Camera Sensitivity and F-Stops 9-12 18 Sampling Error and Photon Statistics 9-12 19 Photon Statistics and Exposure 9-12 20 Photon Statistics and Brightness 9-12 21 3.0 Digital Calibration and Photometry Measuring Brightness – Photometry 9-12 22 Calibrating Data 9-12 23 Calibrating Scientific Images 9-12 24 Light Intensity and Pixel Data and DNs 9-12 25 Your Digital Camera – Image Processing 6-8 26 Your Digital Camera – Standard Sources of Intensity 9-12 27 Your Digital Camera – Converting DNs to Intensity units 9-12 28 Space Math http://spacemath.gsfc.nasa.gov Table of Contents (cont'd) IV Grade Page 4.0 Angular Resolution Angular Measure in Astronomy 6-8 29 Angular Size and Similar Triangles 6-8 30 Geometry and Angular Size 9-12 31 Resolving the Moon 9-12 32 Getting an Angle on the Sun and Moon 9-12 33 Angular Size: The Moon and Stars 9-12 34 SDO - Reveals Details on the Surface of the Sun 6-8 35 LRO - Spots a Rolling Lunar Bolder with no Moss! 6-8 36 LRO - A Detailed Study of the Apollo-11 Landing Area 6-8 37 LRO - Mare Nubium and Las Vegas 6-8 38 LRO - Explores Lunar Surface Cratering 6-8 39 Landsat – Exploring Washington DC 6-8 40 Landsat – Glacier Retreat 6-8 41 Landsat – Exploring Biomass Loss from a Large Fire 6-8 42 Hubble - The Changing Atmosphere of Pluto 9-12 43 Image Blurring and Motion 6-8 44 Angular Resolution and Wavelength 9-12 45 The Most Important Equation in Astronomy 6-8 46 Hinode – Closeup of a Sunspot 6-8 47 Ranger 9 – Craters are a Blast! 9-12 48 Chandra – The Hand of Chandra 6-8 49 Hubble – The Eagle Nebula Closeup 6-8 50 A High Resolution Satellite Photo 6-8 51 5.0 Multi-Wavelength Spatial Imaging Digital Imaging and Filters 9-12 52 Advanced Filter Math using Calculus 9-12 53 Identifying Materials by their Reflectivity 6-8 54 Reflectivity Fingerprints 6-8 55 Creating and Interpreting Images 6-8 56 Graphical Reflectivity Measurements 6-8 57 MMM - A Mineral Study of Mare Orientale 6-8 58 Understanding Mineral Reflectivity Spectra 6-8 59 IRTF-Asteroids and Ice 9-12 60 MMM-Water on the Moon 9-12 61 Terra - Earth’s Carbon Metabolism Revealed 9-12 62 Hinode - Seeing the Sun at X-ray wavelengths 6-8 63 Seeing the Sun at Radio Wavelengths 6-8 64 Terra – Carbon Monoxide a Deadly Trace Gas 9-12 65 SeaWifs – Studying Ocean Plankton from Space 9-12 66 Space Math http://spacemath.gsfc.nasa.gov Table of Contents (cont’d) V 6.0 Temporal Studies SDO - Measuring the Speed of an Eruptive Prominence 6-8 67 STEREO - Watching the Sun Kick Up a Storm 9-12 68 Chandra - Studying an Expanding Supernova Shell 6-8 69 Hubble - Tracking the Motion of an Asteroid 6-8 70 Hinode - Moving Magnetic Filaments Near Sunspots 9-12 71 Measuring the Speed of a Solar Tsunami 9-12 72 RHESSI - Super-fast Solar Flares 9-12 73 STEREO – The Comet Encke Tail Disruption Event 6-8 74 Hinode – How fast does the sun rotate? 6-8 75 Angular Size and Velocity 6-8 76 The Rate of Oil Leakage in the Gulf Oil Spill of 2010 9-12 77 Mars Rover – The Martian Dust Devils 6-8 78 7.0 Remote Sensing From Around the Cosmos GRACE – Exploring Earth’s Gravity Field 9-12 79 CHAMP- Magnetic Earth and Lithosphere 9-12 80 2MASS – How many stars are there? 6-8 81 IMAGE-The North and South Magnetic Poles 6-8 82 MARIA-Having a Hot Time on Mars 9-12 83 Deep Impact - Tempel-1 Close up of a Comet 6-8 84 Spitzer - Seeing a New Ring Around Saturn 9-12 85 IBEX - Creating an Unusual Image of the Sky 6-8 86 8.0 Archiving Data for the Future Data Archiving Through the Ages 6-8 87 Data Migration from One Medium to the Next 6-8 88 Data Storage and Damage Control 6-8 89 9.0 Special Studies Hinode: Remote Sensing of Solar Plasma 6-8 90 Hinode: Close-up of an Active Region by X-ray Light 6-8 91 Hinode: Measuring Spectral Line Intensities 6-8 92 Hinode: Line Ratios and Plasma Temperature 9-12 93 Hinode: The Spectral Analysis of a Sunspot 6-8 94 Hinode: Plasma Temperature Across an Active Region 6-8 95 Landsat: Multi-Wavelength Imaging 6-8 96 Landsat: Identifying Thematic Content 6-8 97 Landsat: The Imager Geometry 6-8 98 Landsat: Creating 2-D Thematic Images 6-8 99 Space Math http://spacemath.gsfc.nasa.gov Table of Contents (cont'd) VI 10.0 Additional Resources for Beginners Creating an Image from Numbers 3-5 100 Interpreting a False-Color Image of Comet Tempel-1 3-5 101 Mapping the Speed of the Circinus Galaxy 3-5 102 Follow that Moving Star! 3-5 103 Additional Resources on the Web 104 Alignment with Standards The following benchmarks were extracted from 'AAAS Project: 2061: Benchmarks'. (3-5) - Quantities and shapes can be used to describe objects and events in the world around us. 2C/E1 --- Mathematics is the study of quantity and shape and is useful for describing events and solving practical problems. 2A/E1 (6-8) Mathematicians often represent things with abstract ideas, such as numbers or perfectly straight lines, and then work with those ideas alone. The "things" from which they abstract can be ideas themselves; for example, a proposition about "all equal-sided triangles" or "all odd numbers". 2C/M1 (9-12) - Mathematical modeling aids in technological design by simulating how a proposed system might behave. 2B/H1 ---- Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments. 2B/H3 ----- Much of the work of mathematicians involves a modeling cycle, consisting of three steps: (1) using abstractions to represent things or ideas, (2) manipulating the abstractions according to some logical rules, and (3) checking how well the results match the original things or ideas. The actual thinking need not follow this order. 2C/H2 Space Math http://spacemath.gsfc.nasa.gov Mathematics Topic Matrix VII Topic Problem Numbers 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 Inquiry Technology, rulers X X X X Numbers, X X X X X X X X X X patterns, percentages Averages X X X X X Time, distance, speed X X X X X X Areas and volumes X Scale drawings Geometry X X X X X X Probability, odds X X X Scientific Notation X X X X X Unit X X X X X X X X X Conversions Fractions X X Graph or Table Analysis X X X X X Pie Graphs Linear X X Equations Rates & Slopes X X X X X X X X X Solving for X X Evaluating Fns X X X X X X Modeling X Trigonometry Logarithms X X X Calculus Arrays or X X X X X matrices Conics
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