1. Tom Wilson, Department of Geology and Geography Environmental and Exploration Geophysics I tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Gravity Methods I-contiued

2. Tom Wilson, Department of Geology and Geography R p = 6356.75km R E = 6378.14km g P =9.83218 m/s 2 g E =9.780319 m/s 2 This is a difference of 5186 milligals. These kinds of differences, which in this case are a function of latitude need to be corrected for – or eliminated Substitute for the different values of R There are great differences in the acceleration due to gravity on the Earth that, in may instances, are unrelated to the details of subsurface geology

3. Tom Wilson, Department of Geology and Geography Density differences arising from isostatic equilibrium processes represent large scale regional changes of g that are often removed before modeling and interpretation. R. J. Lillie, 1999

4. Tom Wilson, Department of Geology and Geography Isostatic compensation and density distributions in the earth’s crust R. J. Lillie, 1999 Generally geological processes produce linear sheet like distributions of materials It’s generally easier to accept this kind of model See also http://www.encyclopedia.com/video/O-qTAp4zhSg- pgr-101-post-glacial-reboundisostasy.aspx

5. Tom Wilson, Department of Geology and Geography Does water flow downhill?

6. Tom Wilson, Department of Geology and Geography The notion of downhill is associated with a surface along which the gravitational potential decreases

7. Tom Wilson, Department of Geology and Geography The geoid is a surface of constant gravitational potential. The gradient of the potential is perpendicular to the surface. Thus gravitational acceleration is always normal to the equipotential surface.

8. Tom Wilson, Department of Geology and Geography Geoid height anomalies Contours are in meters 160 meters uphill

9. Tom Wilson, Department of Geology and Geography Aside from wind generated surface waves and ocean scale wind generated swells … Is the ocean surface a flat surface?

10. Tom Wilson, Department of Geology and Geography Map of the ocean floor obtained from satellite radar observations of ocean surface topography. SeaSat

11. Tom Wilson, Department of Geology and Geography Detailed map of a triple-junction on the floor of the Indian Ocean derived from ocean surface topography

12. Tom Wilson, Department of Geology and Geography In the environmental applications of gravity methods anomalies smaller than a milligal can be of interest to the geophysicist. A modern gravimeter is capable of measuring gravity to an accuracy of about 100th of a milligal or better. We’ll spend considerable time discussing the applications of gravity data in groundwater exploration. An example of this application is discussed in Stewart’s paper (see web site link) on the use of gravity methods for mapping out buried glacial Valleys in Wisconsin - so read over this paper as soon as you can. Gravity provides interesting views of objects buried deep beneath the surface - out of our reach

13. Tom Wilson, Department of Geology and Geography Form Stewart Bedrock models derived from gravity data Residual gravity data The gravity anomalies associated with these glacial valleys have a range of about 4 milliGals. Why residual? The residual eliminates the influence of the deeper strata which dip uniformly across the area. Their configuration is not relevant to the problem at hand. The residual can eliminate geology we aren’t interested in

14. Tom Wilson, Department of Geology and Geography

15. The anomaly shown here is only 1/2 milligal Karst

16. Tom Wilson, Department of Geology and Geography -76 mGals -32 mGals

17. Tom Wilson, Department of Geology and Geography These variations in gravitational acceleration are very small. To give you some additional perspective on the magnitude of these changes, consider the changes in g as a function of r (or R E ) as indicated by Newton’s law of gravity - Recognize that the above equation quantifies the variation in g as a function of r for objects that can effectively be considered as points. For now, let’s take a leap of faith and assume that we can represent the Earth as a point and that the above equation accurately describes the variations in g as a function of distance from the center of the earth, R E.

18. Tom Wilson, Department of Geology and Geography Given this relationship - RERE h What is g at a distance R E +h from the center of the earth? sl=sea level

19. Tom Wilson, Department of Geology and Geography Is there another way to compute the change in g?

20. Tom Wilson, Department of Geology and Geography What is the derivative of g with respect to R?

21.  g/  h - in Morgantown Tom Wilson, Department of Geology and Geography At Morgantown latitudes, the variation of g with elevation is approximately 0.3086 milligals/m or approximately 0.09406 milligals/foot. As you might expect, knowing and correcting for elevation differences between gravity observation points is critical to the interpretation and modeling of gravity data. The anomalies associated with the karst collapse feature were of the order of 1/2 milligal so an error in elevation of 2 meters would yield a difference in g greater than that associated with the density contrasts around the collapsed area.

22. Tom Wilson, Department of Geology and Geography Just as a footnote, Newton had to develop the mathematical methods of calculus to show that spherically symmetrical objects gravitate as though all their mass is concentrated at their center.

23. Tom Wilson, Department of Geology and Geography The acceleration term in Newton’s law of gravitation. tells us we need to consider mass (m) and its distance(s) (r i ) from some observation point. In practice we usually compute the acceleration of some arbitrarily shaped mass by breaking it up into small parts and summing their individual contributions to g.

24. Tom Wilson, Department of Geology and Geography Integral form of Newton’s law of gravitation Line, surface or volume Depending on symmetry dz dy dx dV

25. Tom Wilson, Department of Geology and Geography Consider the following: what is the gravitational attraction of a buried spherically symmetrical object? Let’s work through this on the board

26. Tom Wilson, Department of Geology and Geography What is the vertical component?

27. Tom Wilson, Department of Geology and Geography A symmetrical Earth holds no riddles for the geophysicist.

28. Tom Wilson, Department of Geology and Geography If the earth were this simple our study would be complete.

29. Tom Wilson, Department of Geology and Geography How thick is the landfill? Gravity methods thrive on heterogeneity. In general the objects we are interested in are not so symmetrical and provide us with considerable lateral density contrast and thus gravity anomalies.

30. Tom Wilson, Department of Geology and Geography How does g vary from A to E? We might expect that the average density of materials in the landfill would be less than that of the surrounding bedrock and thus be an area of lower g

31. Tom Wilson, Department of Geology and Geography At present we’ve only accounted for variations in g as a function of elevation or distance from the center of the earth. But obviously we have further to go in terms of conceptualizing and developing the computations needed to understand and evaluate geological problems using measured gravitational fields. Another variable for us to consider is the elevation at which our observations are made.

32. Tom Wilson, Department of Geology and Geography How do we compensate for the influence of matter between the observation point (A) and sea level? How do we compensate for the irregularities in the earth’s surface - its topography? A hill will take us down the gravity ladder, but as we walk uphill, the mass beneath our feet adds to g.

33. Tom Wilson, Department of Geology and Geography What other effects do we need to consider? Latitude effect Centrifugal acceleration 463 meters/sec ~1000 mph

34. Tom Wilson, Department of Geology and Geography Solar and Lunar tides Instrument drift

35. Tom Wilson, Department of Geology and Geography To conceptualize the dependence of gravitational acceleration on various factors, we usually write g as a sum of different influences or contributions. These are -

36. Tom Wilson, Department of Geology and Geography g n the “normal gravity” or the gravitational acceleration on the reference ellipsoid  g FA the elevation or free air effect  g B the Bouguer plate effect or the contribution to measured or observed g of the material between sea- level and the elevation of the observation point  g T the effect of terrain on the observed g  g Tide and Drift the effects of tide and drift (often combined) These different terms can be combined into an expression which is equivalent to a prediction of what the acceleration should be at a particular point on the surface of a homogeneous earth. Terms

37. Tom Wilson, Department of Geology and Geography Thus when all these factors are compensated for, or accounted for, the remaining “anomaly” is associated with lateral density contrasts within area of the survey. The geologist/geophysicist is then left with the task of interpreting/modeling the anomaly in terms of geologically reasonable configurations of subsurface intervals.

38. Tom Wilson, Department of Geology and Geography That predicted or estimated value of g is often referred to as the theoretical gravity - g t If the observed values of g behave according to this ideal model then there is no geology! - i.e. there is no lateral heterogeneity. The geology would be fairly uninteresting - a layer cake... We’ll spend more time with these ideas, but in the next couple lectures we will develop a little better understanding of the individual terms in this expression. The Theoretical Gravity

39. Tom Wilson, Department of Geology and Geography We’ll carry on this discussion in greater detail next time. Make sure you continue reading chapter 6 in Burger et al. We’ll go over some of the basic ideas associated with 1) the Bouguer plate correction and 2) the topographic (or terrain) correction. These two effects are approximated using gravitational acceleration produced by a plate of finite thickness but infinite horizontal extent and by individual sectors from a ring of given thickness and width. Read general introduction from pages 349-355 and continue reading about gravity corrections from page 356 through the top of page 373

40. Tom Wilson, Department of Geology and Geography Keep reading Chapter 6. Turn in the three intro gravity problems next Friday (October 19 th ) by noon. Just put in my mailbox. Gravity papers will be in the mail room tomorrow! Start looking over problems 6.1 through 6.3 (see today’s handout). Also for next Friday – answer the questions on the back of today’s handout and put in my mailbox (by noon Friday 19 th, October).