Lecture 7 – Gravity and Related Issues

1. Lecture 7 – Gravity and Related Issues GISC-3325 11 February 2008

2. Scheduling Issues • Next class and lab will be on OPUS and GPS processing using OPUS. • Next week I will be out. Monday or Wednesday (18 and 20 Feb) will be first exam. • Chapters 1-4 and all lectures and labs • Lab 3 – BM recovery: due 25 February 2008.

3. Lab 2: Problem How do I get this to work?

10. Two models provide similar but not identical results. Difference is 1 mgal.

11. Which model to use? • NAVD88 - Modeled Gravity uses a model developed for the NAVD88 adjustment rather than current gravity values.

12. Review of Height Systems • Helmert Orthometric • NAVD 88 • local gravity field ( ) • single datum point • follows MSL

13. Earth’s Gravity Field from Space • Satellite data was used for global models • Only useful at wavelengths of 700 km or longer • Lower wavelength data from terrestrial or marine gravity of varying vintage, quality and geographic coverage Terrestrial and marine gravity data in NGS data base.

14. Note the discontinuity at the shoreline.

15. Gravity • Static gravity field • Based on long-term average within Earth system • Temporally changing component • Motion of water and air • Time scale ranges from hours to decades. • Mean and time variable gravity field affect the motion of all Earth space vehicles.

16. Gravity Recovery And Climate Experiment www.csr.utexas.edu/grace/gravity/

17. Geoid Model from Earth Orbiting Space Vehicles (pre-GRACE)

18. GRACE 111 days of data

19. GRACE 363 days of data

20. Orbit inclination: 89.048 degrees Eccentricity: 0.000775 Semi-major axis: 6,849,706.754m Distance between satellites: 222,732.810 m

22. GRACE

23. How does GRACE work? • Motion of two satellites differ because they are at different positions in space. • When the lead SV approaches a higher gravity mass it accelerates as it moves beyond it decelerates. • Distance changes between SVs is measured precisely.

24. GRS 80 defined not only by geometric but also physical parameters (gravity). From www.dgfi.badw.de/geoidis/REFS/grs80.html

25. Gravitational Acceleration • The magnitude of acceleration (b) due to the Earth’s mass on the surface using a spherical geometric reference surface (R) is: • b = GM/R2

26. Centrifugal Acceleration • Direction is always perpendicular outward from the spin axis. • It is a function of angular velocity of the Earth squared and the distance from the point of interest on the surface of the sphere to the axis of rotation. • ω = 7292115e-11rad sec-1 • We can compute this value as ratio of degrees over time.

28. Magnitude of centrifugal acceleration • Varies from equator to poles. • Compute magnitude by velocity squared times the distance from the point of interest to the spin axis.

29. Gravitational Attraction • Is the vector sum of gravitational and centrifugal acceleration. • The actual acceleration of gravity varies from place to place, depending on latitude, altitude, and local geology. • By agreement among physicists, the standard acceleration of gravity gn is defined to be exactly 9.80665 meters per second per second (m s-2), or about 32.174 05 feet per second per second.

30. More mind-numbing detail… • At latitude p, a conventional value of the acceleration of gravity at sea level is given by the International Gravity Formula, • g = 978.0495 [1 + 0.0052892 sin2(p) - 0.0000073 sin2 (2p)] cm per second per second (cm s-2). • The mean Earth gravity is about 981 000 mGal (the well-known 9.81 m/s2), varies from 978,100 mGal to 983,200 mGal from Equator to pole due to the Earth's flattening and rotation.

32. Gravitational Potential • Magnitude of the potential is the work that must be done by gravity to move a unit mass from infinity to the point of interest. • Is dependent on position within the gravitation field.

33. Equipotential Surfaces • Surface having constant gravity potential NOT constant gravity. • Also known as level surfaces or geopotential surfaces. • Surfaces are perpendicular at all points of the plumb line (gravity vector). • A still lake surface is an equipotential surface. • It is not horizontal but curved.

34. Gravity defines the plumb line – the local vertical

35. Image from Featherstone, W, “Height Systems and Vertical Datums” Spatial Science, Vol 51 No. 1, June 2006

36. Properties of equipotential surfaces • They are closed continuous surfaces that never cross one another. • They are formed by long radius arcs. Generally without abrupt steps. • They are convex everywhere.

37. Geopotential Number • C (geopotential number) – is a value derived from the difference in gravity potential between an equipotential surface of interest and the geoid. • Represents the work required to move a 1kg mass from the geoid to the geopotential surface at the point of interest. • C is numerically similar to the elevation of the point in meters.

38. Leveling Issues • Height • The distance measured along a perpendicular between a point and a reference surface. • Raw leveled heights • Non-unique. Depending on the path taken a different height will be determined for the same point. They have NO physical relevance.

41. Geopotential Numbers • Geopotential numbers • Unique, path-independent • Geopotential number (C) at a point of interest (p) is the difference between the potential on the geoid and potential at the point. • C = W0-Wp

42. Leveled Height vs. Orthometric Height  h = local leveled differences H = relative orthometric heights Equipotential Surfaces B Topography  hAB =  hBC A C HA HC HAChAB + hBC Reference Surface (Geoid) Observed difference in orthometric height, H, depends on the leveling route.

43. Orthometric Height • Vertical distance from the geoid to the point of interest. (along curved plumb line). • Orthometric height (H) may be determined from H = geopotential number / mean gravity along plumb line • We use assumptions about mass density to estimate mean gravity.

45. Dynamic Heights • Takes the geopotential number at a point and divides it by a constant gravity value. • The constant value used in the US is the normal gravity value at 45 degrees N latitude • 980.6199 gals

46. CC area Dynamic Heights NAVD 88 and dynamic heights differ by only 1 mm at this station.

47. Dynamic Heights NAVD 88 = 2026.545 m DYN Hght= 2023.109 Difference = 3.436 m

49. Error condition: input position is inconsistent with predicted one.