Exercise 7.5 Derive the Eotvos correction starting with equation 7.1
7.4 An Example
Figure 7.11 compares the free-air, simple Bouguer, complete Bouguer, and isostatic residual anomalies over parts of the Klamath Mountains and Cascade Range in north-central California. The pre-Cenozoic sed- imentary and volcanic rocks of the Klamath Mountains are generally more dense than the Tertiary and Quaternary volcanic rocks related to the Cascade arc. The Trinity ultramafic sheet, the largest ultramafic
74 An Example 151 body in North America, lies in this part of the Klamath Mountains. Ad- ditional information regarding the geophysical setting of this area was provided by Griscom [105], LaFehr [153], and Blakely and Jachens [30].
Figure 7.11 illustrates the following:
1. Free-air anomalies (Figure 7.11(c)) are strongly correlated with ter- rain (Figure 7.11(b)). This correlation is particularly apparent over Mount Shasta, Medicine Lake Volcano, and the Klamath Mountains.
2. Simple and complete Bouguer anomalies (Figures 7.11(d) and 7.11(e)) over continental areas are strongly negative. This happens because the Bouguer correction has removed the effects of normal crust above sea level but has left the effects of deeper masses that isostatically support that crust.
3. Simple and complete Bouguer anomalies have regional-scale compo- nents that are approximately inversely correlated with very long- wavelength attributes of the terrain. In Figure 7.11(e), this regional component appears as a broad trend decreasing from west to east.
The regional trend in Bouguer gravity is caused by an increase from west to east in the amount of deep, low-density material that supports the topography. In terms of the Airy model for isostatic compensa- tion, the trend in Bouguer gravity is caused by a low-density root that thickens from west to east in order to support the continental edifice, although in actuality the support may be distributed in other ways; for example, through variable densities in the upper mantle.
4. The simple Bouguer map includes short-wavelength anomalies re- lated to topography, whereas these effects are largely missing from the complete Bouguer map, which included a terrain correction. The large-amplitude negative anomaly directly over Mount Shasta on Fig- ure 7.11(d) is the best example.
5. The isostatic residual anomaly (Figure 7.11(f)) most closely repre- sents lateral variations in density of the middle and upper crust. It clearly shows, for example, the high-density Trinity ultramafic body and where it lies below pre-Cenozoic rocks of the Klamath Moun- tains (Griscom [105]). Low-density rocks of the Cascade Range are also apparent in the central and eastern parts of the map. The neg- ative anomaly over and extending northeast from Mount Shasta is thought to be caused by low-density volcanic rocks (LaFehr [153];
Blakely and Jachens [30]).
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|22°W
is :^," Medicine L. Volcano
| | Cenozoic sedimentary rocks
0J Cenozoic volcanic rocks
| Pre-Cenozoic rocks
| Ultramafic rocks
(a) Simplified geology
I21°W
(b) Generalized terrain
(c) Free-air anomaly (d) Simple Bouguer anomaly
(e) Complete Bouguer anomaly (0 Isostatic residual anomaly
Fig. 7.11. Various corrections applied to gravity data from north-central Cal- ifornia, (a) Simplified geology; (b) topography based on 5-minute averages, contour interval 200 m; (c) free-air anomaly, contour interval 20 mGal; (d) simple Bouguer anomaly, contour interval 10 mGal; (e) complete Bouguer anomaly, contour interval 10 mGal; (f) isostatic residual anomaly, contour interval 10 mGal. Gray shades indicate positive regions.
7.5 Problem Set 153 7.5 Problem Set
1. Imagine that you are chief science officer on a mission to investigate a newly discovered planet. Your spaceship is located in the plane of the planet's rotation at a distance r from the planet's center and a distance h from the planet's surface. Your spaceship is stationary with respect to the planet's rotation and is experiencing a gravitational attraction of G. Your sensors tell you that the planet is rotating with an angular velocity of a;, has the shape of an oblate spheroid with flattening /, and has dynamical properties much like Earth. Two landing parties are being sent to the planet: one to the equator and one to the north pole. The captain wants to know how much the force of gravity will differ between the two landing sites and is willing to accept some simplifying assumptions. What can you advise? Express your answer in terms of a, /, r, /i, a;, and G.
2. Suppose that the earth begins to rotate more and more rapidly until the gravity experienced by an observer at the equator falls to zero.
How long is the length of a day?
3. Show that to first order f + f = §ra. Note: This result is more important than it may seem; it shows that the geometrical shape of the ellipsoid can be determined from gravity measurements alone.
4. A sphere of density 3.17 g-cm"3 and radius 5 km is partially buried in a flat prairie so that the summit of the sphere is 2 km above the surrounding prairie. The prairie is 2 km above sea level. Assume crustal density is 2.67 g-cm~3 and use the Geodetic Reference System 1967 to sketch the following profiles (show appropriate shapes and amplitudes) as observed along a traverse directly over the center of the sphere:
(a) observed gravity, (b) free-air anomaly, (c) Bouguer anomaly,
(d) isostatic residual anomaly.
5. Explain why the following statement is wrong: "The isostatic residual anomaly is always zero over perfectly compensated topography."
6. The mass in Figure 7.10 is not isostatically compensated. What would the profile look like if it were?
8
The Geomagnetic Field
Magnus magnes ipse est globus terrestris.
[The whole earth is a magnet.]
(William Gilbert) It has always been and still is [my] impression that a magnetometer survey is just as much a means of mapping geology as are the air photograph and the surface geological traverse.
(Norman R. Pater son)
The previous chapter discussed the steps by which gravity measurements are converted into gravity anomalies that reflect geological sources. The present chapter treats magnetic anomalies in a similar vein. Whereas the gravity field of the earth is largely time invariant, except for relatively minor or long-term changes due to redistribution of mass (tides, mov- ing magma, glacial rebound, erosion, mountain building, and so forth), the geomagnetic field varies in both direction and intensity over time scales ranging from milliseconds to millennia. It would seem that this added complexity would make the reduction of magnetic measurements significantly more difficult than that for gravity data, but in practice the calculation of magnetic anomalies is relatively straightforward.
Our intent in this chapter is to characterize the global magnetic field in order to isolate the magnetic field caused by crustal sources. This agenda glosses over a large body of information that ordinarily would be included in a chapter of this title, such topics as the origins of the geomagnetic field (magnetohydrodynamic theories); the behavior of the field in the geologic past (paleomagnetic studies); reversals of geomag- netic polarity; the magnetic properties of the sun, moon, meteorites, and other planets; and the interaction of the earth's magnetic field with solar phenomena. Other textbooks are dedicated to just these topics,
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