Modeling Magnetic Anomalies with MATLAB. Geodynamics, FA2012 Nate O Flaherty James Holmes

Transcription

1 Modeling Magnetic Anomalies with MATLAB Geodynamics, FA2012 Nate O Flaherty James Holmes

2 Introduction History o 1912 Wegner publicly advocates theory of Continental Drift 1943 G.G. Simpson publishes rebuttal, Wegner falls out of favor. o 1960 Harry Hess proposes concept of Seafloor Spreading to explain symmetric bathymetry profiles across spreading ridges. o 1963 Vine and Matthews publish results of survey over Carlsberg ridge They found symmetric stripes of magnetically polarized rock running parallel to the spreading ridge. 2

3 Earth s Magnetism Geomagnetic Field o Earth s geomagnetic field is a dipole to first-order. o Poles slightly diverge from geographic poles over short term, aligned over long term. Image: Wikimedia Commons 3

4 Marine Geology Oceanic Lithosphere o Three layers Layer 1 Sediments Layer 2 Basalts o 2A Pillow Basalts o 2B Sheeted Dikes Layer 3 Gabbros (dark, coarse-grain plutonic rock) 4

5 Magnetic Anomalies What are magnetic anomalies? o Spreading ridges extrude magnetite-rich magma at axes. o As magma cools below Curie temperature (~500 C), crust magnetizes in direction of geomagnetic field. Thermo-remnant Magnetism (TRM). o Most TRM recorded in Layer 2a (upper 1000m of oceanic crust). o Plate spreading results in magnetized record of intensity and polarity. 5

6 Diagram of oceanic lithosphere Image: Kent et al,

7 Diagram of oceanic lithosphere Image: Kent et al,

8 Strength of anomaly by layer 8

9 Magnetic anomalies record seafloor spreading 9

10 How do we determine the anomaly? Magnetization vector is combination of TRM and current dipole components. o M = M TRM + M I! Magnetic anomaly vector is parallel to the magnetization direction. o!b(r) = " 0 Mƒ(r)# Using ship-towed magnetometer, we measure total magnetic field, so we must remove Earth s field. o B = B e +!B! However, most magnetometers give us a scalar field. o B = ( B e 2 + 2B e!b +!B 2 ) 1/2# 10

11 How do we determine the anomaly? A huge difference in intensity. o Earth s dipole is ~ 50,000nT o Anomaly is ~ 300nT Therefore!B 2 ~ 0, and we can generalize B further: o B $ B e (1 +!B B e / B e )# 30-seconds of algebra later o A = B - B e =!B B e / B e # o A is our measured scalar anomaly 11

12 Mathematically modeling anomalies The magnetic anomaly is the negative gradient of the magnetic potential. o ΔB = - U The potential satisfies Poisson s equation within the source layer and Laplace s equation above it. o 2 U = 0, z z 0 o 2 U = μ 0 M, z = z 0 We can then generalize the equation to: 12

13 Mathematically modeling anomalies The magnetic anomaly is the negative gradient of the magnetic potential. o ΔB = - U The potential satisfies Poisson s equation within the source layer and Laplace s equation above it. o 2 U = 0, z z 0 o 2 U = μ 0 M, z = z 0 We can then generalize the equation to: No vertical variation! 13

14 Mathematically modeling anomalies To solve our equation we need to: o Implement a forward Fourier Transform. o Solve for U(k) o Then inverse Fourier Transform (via Cauchy-Residue Thm.) with respect to k z. Poles of integrand at ± ik x Path integral around poles When TRM has component parallel to dipole field (parallel to ridge axis) = No field anomaly. Assume spreading ridge at magnetic pole so that dipole field lines are parallel to z, with no x component. 14

15 Mathematically modeling anomalies Because scalar anomaly is: Only the z-component is non-zero, so scalar anomaly is: 15

16 Mathematically modeling anomalies But wait! That s not all: o Now, rotate the spreading ridge down to an arbitrary latitude. o X-component now non-zero. o We get a skewness factor, θ. Depends on latitude and orientation of ridge when crust cools. Final model: 16

17 Modeling actual ridges Mid-Atlantic Ridge Pacific-Antarctic Ridge 17

18 Magnetic Polarity Data 18

19 Closest Fits 19

20 Mid- Atlantic Ridge A(k, z) =Cµ 0 2 k e 2 k z e i sgn(k) p(k) 20

21 Mid- Atlantic Ridge A(k, z) =Cµ 0 2 k e 2 k z e i sgn(k) p(k) 21

22 Mid- Atlantic Ridge A(k, z) =Cµ 0 2 k e 2 k z e i sgn(k) p(k) k = nx/2 :nx/2 1 L = nx/2 :nx/2 1 v dt 22

23 Pacific- Antarctic Ridge A(k, z) =Cµ 0 2 k e 2 k z e i sgn(k) p(k) 23

24 Pacific- Antarctic Ridge A(k, z) =Cµ 0 2 k e 2 k z e i sgn(k) p(k) 24

25 Pacific- Antarctic Ridge A(k, z) =Cµ 0 2 k e 2 k z e i sgn(k) p(k) k = nx/2 :nx/2 1 L = nx/2 :nx/2 1 v dt 25

26 Questions? 26