Exercise 7.5 Derive the Eotvos correction starting with equation 7.1
8.3 Crustal Magnetic Anomalies
The internal sources of the geomagnetic field are located primarily in two regions of the earth. The majority of the field is generated in the fluid outer core by way of complex magnetohydrodynamic processes and is called the core field or main field. The remainder, called the crustal field, originates primarily from a relatively thin outer shell of the earth where temperatures are below the Curie temperatures of important mag- netic minerals, primarily magnetite and titanomagnetite (Chapter 5).
The depth to which such minerals exist is still a matter of discussion, however. The mantle is generally considered to be nonmagnetic (e.g., Wasilewski, Thomas, and Mayhew [289]; Frost and Shive [92]), so accord- ing to this view, the depth extent of magnetic rocks is either the crust- mantle interface or the Curie-temperature isotherm, whichever is shal- lower. Some studies have concluded, on the other hand, that upper mantle rocks may have significant magnetizations, especially in oceanic regions (e.g., Arkani-Hamed [6]; Harrison and Carle [117]; Counil, Achache, and Galdeano [76]). In the following, we will loosely regard the crust-mantle interface as magnetic basement, with the understanding that rocks in the uppermost mantle in some geologic environments may also contribute to the geomagnetic field. Hence, the vast region between the Curie- temperature isotherm (or crust-mantle interface, whichever is shallower) and the core-mantle interface is generally considered to be nonmagnetic.
The calculation of crustal magnetic anomalies then amounts to subtract- ing the core field from measurements of the total magnetic field. Paterson and Reeves [212, Figure 8] showed an excellent example of the enhance- ment of an airborne magnetic survey by this simple residual calculation.
The large difference in depth between the sources of the crustal and core field is reflected in spherical harmonic analyses. This depth informa- tion is perhaps best displayed by way of the power spectrum Rn, defined as the scalar product Bn Bn averaged over the spherical surface; that is,
2TT IT
j / f ~Bn-Bn a2 sin8 d0d<t>, (8.21)
0 0
8.3 Crustal Magnetic Anomalies 175
g?
20- 16- ' 12- 8 - 4 -
10 15 20
n
Fig. 8.6. Power spectrum of the geomagnetic field at the earth's surface based on 26,500 measurements from the Magsat satellite mission. Dots indicate calculated values of logi?n; best-fit lines are shown for 2 < n < 12 and 16 < n < 23. Modified from Langel and Estes [160].
where
n+1 (#™ cos mcf) + K sin mcp) P™ (t
evaluated at r = a. Using the orthogonality property of spherical surface harmonics, Lowes [167, 168] reduced equation 8.21 to
(8.22)
ra=0
It is clear from Table 8.1 that Rn decreases with increasing n at least through degree 10. Figure 8.6 shows Rn through degree 23, as calculated by Langel and Estes [160] from 26,500 low-orbit satellite measurements.
The logarithm of Rn takes the form of two straight-line segments with a change in slope at about degree 14 (Figure 8.6), which is in general agreement with earlier studies (e.g., Cain, Davis, and Reagan [49]).
Within any range of n, the rate of decrease of Rn with increasing n is directly related to the depth of sources principally responsible for that part of the spectrum. To demonstrate this relationship, we first note from equation 8.21 that the power spectrum Rn is proportional to (a/r)2n+4.
Exercise 8.2 Prove the previous statement; i.e., use equation 8.21 to show that Rn is proportional to (a/r)2n+4.
Then to transform Rn based at the surface of the earth into a spectrum that would be determined at some new radius r, we simply have to multiply Rn by the factor (a/r)2 n + 4, or add (2n + 4) log(a/r) to logRn. This manipulation changes the slope of the logarithmic power spectrum by a constant 21og(a/r). If r > a, Rn is transformed onto a larger sphere, the slope of \ogRn is steepened, and the procedure is called upward continuation. If r < a, Rn is transformed to a smaller sphere (within the earth), the slope of \ogRn is flattened, and the procedure is called downward continuation. Downward continuation is legitimate, however, only if all currents and other sources of magnetic fields are absent between radii a and r (e.g., Booker [34], Lowes [168]). We will have considerably more to say about upward and downward continuation in Chapter 12.
It is commonly assumed that the radius required to make logRn as nearly constant as possible (i.e., to make the power spectrum "white") is the radius at which the important sources of the field are located (e.g., Lowes [168], Langel and Estes [160]). With this assumption, the principal sources are located at a radius given by the value of r that satisfies
S + 21og- = 0 , r
where S is the slope of log Rn. The line that best fits log Rn for n < 14 in Figure 8.6 has a slope of —1.309 (Langel and Estes [160]). Substituting this value into the previous equation provides r = 3311 km, which places the sources of this part of the spectrum at a radius about 174 km below the seismic core-mantle boundary. The spectrum at n > 14 indicates sources within the upper 100 km of the earth.
Hence, it is logical to interpret the steep part of the spectrum (n < 14) in Figure 8.6 to be caused by sources within the outer core, and the flatter part of the spectrum (n > 14) to be dominated by lithospheric sources (e.g., Bullard [44], Booker [34], Lowes [168], Cain et al. [49]; Lan- gel and Estes [160]). It would seem, therefore, that a crustal magnetic map could be constructed from satellite data by subtracting a 13-degree spherical harmonic expansion derived from the same data (e.g., May- hew [177]; Regan, Cain, and Davis [241]; Cain, Schmitz, and Math [50];
Langel, Phillip, and Horner [159]).
This assumption deserves consideration, however. Carle and Harri- son [53] showed that residual fields calculated in this way may con- tain long-wavelength components, too long to be caused by near-surface sources. This happens in part because in practice we usually measure
8.3 Crustal Magnetic Anomalies 177 the total intensity of the geomagnetic field (discussed more fully in Sec- tion 8.3.1) rather than a single component of the field. The residual crustal anomaly in such cases is computed by subtracting the magni- tude of a low-degree regional field from measurements of the geomag- netic intensity. The total intensity is the square root of the sum of the squares of three orthogonal components, as in equation 8.15. Although the potential is modeled in a spherical harmonic expansion as the sum of sinusoidal terms, in squaring the three components to form the in- tensity, each of those sinusoidal terms becomes a combination of both longer and shorter wavelength terms, as demonstrated by the following exercise.
Exercise 8.3 Consider a potential field given by B = -VV, where V = a(a/r)2giPi(0). Show that each component of B (i.e., Br and Be) has only one sinusoidal term of wavelength 2TT, but that |B| has two terms, one with infinite wavelength and a second with wavelength TT.
Hence, a value of n that represents the transition from dominantly core to dominantly crustal contributions to the geomagnetic field may not be appropriate for the total intensity of the field. A residual anomaly computed by subtracting the magnitude of a regional field truncated at n still will contain contributions from harmonics less than n.
Langel [156] agreed that the anomaly field includes long wavelengths, but that these can originate strictly from crustal sources by virtue of the way the anomaly field is calculated and do not necessarily imply con- tamination from sources in the core. Arkani-Hamed and Strangway [7]
and Harrison, Carle, and Hayling [118] independently concluded that the crustal portion of the magnetic field dominates the total intensity at degrees 19 and greater; that is, to be sure that the residual anomaly rep- resents only crustal sources, they recommended truncation of anomalies up through degree 18.
The important conclusion for our purposes here is that subtraction of a 10-degree IGRF model, such as that shown in Table 8.1, from a magnetic survey will be inadequate to eliminate the entire core field.
Magnetic studies of continental or global scale should be evaluated care- fully in the context of long-wavelength anomalies that might originate from the core and that may be confused with the crustal field. The long- wavelength shortcomings of the 10-degree IGRF are much less significant for local- or regional-scale magnetic studies applied to geologic problems.
In such cases, additional regional fields can be removed subsequent to subtraction of the IGRF using a variety of techniques, such as simple curve fitting and digital filtering.
Chapter 7 focused on the reduction of measurements of the total grav- itational attraction of the earth to gravity anomalies reflecting crustal density variations. This reduction procedure consists of a series of steps, including a subtraction of the gravitational effects of an average earth, the elevation of the measurement, and terrain. In crustal magnetic stud- ies, the calculation of anomalies is often treated in a more cavalier way, usually consisting of only two steps: (1) an adjustment for daily varia- tions and magnetic disturbances and (2) the subtraction of a suitable regional field, such as the IGRF model appropriate for the date of the survey. Some of the corrections routinely applied to gravity measure- ments generally are not attempted in magnetic studies simply because they are less tractable in the magnetic case. Terrain corrections in grav- ity studies, for example, although tedious are nevertheless straightfor- ward because the density of the terrain is relatively uniform. Crustal magnetization, however, can vary by several orders of magnitude (and change sign) at essentially all spatial scales. This additional complex- ity requires less-straightforward techniques to correct for terrain effects (e.g., Clarke [61], Grauch [101]), and the effects of terrain are often left to the modeling and interpretation stage.