Modeling Marine Magnetic Anomalies SHELBY JONES HANNA ASEFAW

Transcription

1 Modeling Marine Magnetic Anomalies SHELBY JONES HANNA ASEFAW 1

2 Outline 1 Magnetic Acquisition 2 Goal of our Project 3 Derivation 5 Real Applications Pacific Antarctic Ridge Mid-Atlantic Ridge 6 Limitations 4 Our MATLAB Process 2

3 Magnetic Acquisition MORB (Mid-Ocean Ridge Basalts) contain magnetic grains As basalt cools, the magnetizations within the magnetic grains align themselves with the magnetic field Latitudinal dependence Magnetic Acquisition 3

4 Magnetic Acquisition Magnetic Acquisition 4

5 Goal of Our Project Modeled Magnetic Profile Observed Seafloor Magnetic Profile Goal 5

6 Goal of Our Project C = constant µ 0 = magnetic permeability k = wavenumbers k = (-nx/2 : nx/2-1) / L L = spreading rate * total time z = depth = 3000m θ = skewness p(k) = Fourier of geomagnetic timescale Goal 6

7 Goal of Our Project C = constant µ 0 = magnetic permeability k = wavenumbers k = (-nx/2 : nx/2-1) / L L = spreading rate * total time z = depth = 3000m θ = skewness p(k) = Fourier of geomagnetic timescale Goal 6

8 Step 1: Calculate the Scalar of the Anomaly A Ideal towed magnetometer measures: Most magnetometers measure scalar field On Earth, B e 50,000nT while ΔB 300nT Derivation 7

9 Step 2: Account for Seafloor Spreading Define scalar potential (U) and magnetization (M) Define: Derivation 8

10 Step 2: Account for Seafloor Spreading Potential satisfies Laplace s equation above source layer and Poisson s equation within source layer Derivation 9

11 Step 2: Account for Seafloor Spreading 2 nd terms can be eliminated because source does not vary in the y-direction thus derivative = 0 Derivation 10

12 Step 2: Account for Seafloor Spreading Boundary Conditions: Basic Double Fourier Transform: Applied to this problem: In the x direction: In the z direction (use identity): Derivation 11

13 Step 2: Account for Seafloor Spreading Fourier Transform Result Solve for U(k) Inverse Fourier using Cauchy Residue Theorem Derivation 12

14 Step 2: Account for Seafloor Spreading To solve integral, calculate the poles of the integrand Factor: Solve over closed loops: Derivation 13

15 Step 2: Account for Seafloor Spreading Combine integrands and drop k subscripts : To simplify: assume the spreading ridge is located at Earth s magnetic pole, the dipolar field lines will be parallel to the z-axis, thus no x-component Derivation 14

16 Step 3: Calculate the magnetic anomaly Recall: Substitute with evaluated U: Recall: Since only the z-component of Earth s magnetic field is non-zero due to our assumptions, the anomaly simplifies to: * Derivation 15

17 Step 3: Calculate the magnetic anomaly Take into account upward continuation * Derivation 16

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22 Main function specify data files pacificantarcticrise = 'pacificantarctic.xydm'; midatlanticridge = 'midatlanticridge.xydm spreadcskewermar = spreadcskewer(midatlanticridge, polarity, time, 1318); spreadcskewerpa = spreadcskewer(pacificantarcticrise, polarity, time, 5418); Our MATLAB Process 21

23 spreadcskewer() Parameters datafile, polarity, time, ridgeaxis Datafile file the contains the observed magnetic anomalies Polarity matrix with geomagnetic timescale field polarities Time matrix with geomagnetic timescale ridgeaxis location of the ridge axis Output Figure 1: Observed Magnetic Anomalies across the ridge Figure 2: Fourier transform of magnetic timescale & observed Figure 3: Overlay of observed and modeled Figure 4: Overlay of observed and modeled Returns a solution [spreadingrate, Constant, skewness, rooterror) Our MATLAB Process 22

24 Figure 1: Observed Magnetic Anomalies across the ridge function spreadcskewer = spreadcskewer(anomalyfile, polarity, time, axispoint) location = inputname(1); load the observed anomaly data anomalyfile = importdata(anomalyfile); distance = anomalyfile(:,3); magobs = anomalyfile(:,4); plot distance from ridge and magnetic anomaly figure(1) subplot(2,1,1); plot(distance, magobs); xlabel('distance (km)') ylabel('magnetic Anomaly (n tesla)') title(['observed Magnetic Anomalies across the ', location]); plot near ridge magnetic anomalies [totaltimescaledatapoints,mdat]=size(time./2); vectortime=2048; halfvectortime=vectortime/2; subsettime = time((totaltimescaledatapoints/2-halfvectortime+1): (totaltimescaledatapoints/2+halfvectortime),1)'; subsetpolarity = polarity((totaltimescaledatapoints/2-halfvectortime+1): (totaltimescaledatapoints/2+halfvectortime),1)'; [sizex, sizey] = size(distance); subsetdistance = distance((axispoint-halfvectortime+1):(axispoint+ halfvectortime),1)'; subsetanomalies = magobs((axispoint-halfvectortime+1):(axispoint+ halfvectortime),1)'; subplot(2,1,2); plot(subsetdistance, subsetanomalies); xlabel('distance (km)') ylabel('magnetic Anomaly (n tesla)') title(['observed Magnetic Anomalies across the ', location]); Our MATLAB Process 23

25 Figure 2: Fourier Transform magnetic timescale and observed anomalies Fourier Transform of the Observed Magnetic Anomalies across the Pacific-Antarctic Rise dataanomaly = fftshift(fft(subsetanomalies)); figure(2) subplot(2,1,1); plot(k,real(dataanomaly)); xlabel('k'); title(['anomalies Observed across the ', location]); axis([-60, 60,-2 * 10^5, 1*10^5 ]) Fourier Transform of the magnetic timescale p(k) fouriertimescale = fftshift(fft(subsetpolarity)); Model based on the geomagnetic timescale constant= 3 *10^-11; spreadingrate = 40000; theta = -130; skewness = theta * pi/180 ; k2 = k./(spreadingrate*dt); modelanomaly = abs(k2).*(fouriertimescale).* exp(abs(k2).* DEPTH * -2 * pi).* exp(sign(k2).* 1i * skewness) * constant* MAGPERM * 2 * pi; subplot (2, 1, 2); plot(k, modelanomaly); xlabel('k'); title('anomalies modeled from the Magnetic Timescale'); Our MATLAB Process 24

26 Figure 3: overlay of model and observed (k) figure(3) plot(k, dataanomaly,k, modelanomaly); legend([location, ' observed anomaly'], 'Timescale generated anomaly'); xlabel('k'); Our MATLAB Process 25

27 Figure 4: overlay of model and observed overlay of inverse fourier dataanomaly and modelanomaly model= ifft(fftshift(modelanomaly)); figure(4) plot(subsetdistance./dt*2, subsetanomalies, subsettime, model) legend([location, ' observed anomaly'], 'Timescale generated anomaly'); Our MATLAB Process 26

28 calculate the Root Mean Square Error rooterror = calcrmse(subsetanomalies, model); Our MATLAB Process 27

29 Root Mean Square Error takes two matrices and calculates the RMSE function deviation = calcrmse (model, observed) How well the model suits the observed data [y, datapointsmodel] = size(model); [z, datapointsobserved] = size (observed); sum = 0; if datapointsmodel >= datapointsobserved totalpoints = datapointsobserved; else totalpoints = datapointsmodel; end for i = 1:totalPoints sum = sum + (abs(model(1,i)^2 - observed(1,i))^2); end deviation = sqrt(sum/totalpoints); We tried to minimize RMSE Our MATLAB Process 28

30 Return a solution spreadcskewer = [spreadingrate, CONSTANT, theta, rooterror]; spreadcskewermar = spreadcskewer(midatlanticridge, polarity, time, 1318); spreadcskewerpa = spreadcskewer(pacificantarcticrise, polarity, time, 5418); Our MATLAB Process 29

31 Calculating our c, skewness, and rmse calculates the ideal skewness theta, constant combinations based on the observed Anomaly function skewconst = skewnessconstant (fouriertimescale, observedanom, inc, rangec, skew, spreading, rangespread ) c = 0; constant = 0; theta = 0; skewness=0; currentrmse = 0; j = 1; currentleast = Inf; dt = 20; nx = 2048; magperm = 4 * pi * 10 * exp(-7); nx2 = nx/2; depth = 3000.; for spreadingrate = (spreading-rangespreading):spreading:(spreading+rangespreading) L = spreadingrate * dt; k = ((-nx2): (nx2-1))/(spreadingrate *dt); for c = (inc-rangec):10000:(inc+rangec) for theta = (skew-10):skew:(skew:+10) skewness = theta* pi / 180.; modelanom = abs(k).*(fouriertimescale').* exp(abs(k).* depth * -2 * pi).* exp(sign(k).* 1i * skewness) * constant * magperm * 2 * pi; rmse = calcrmse(modelanom, observedanom); end end end if rmse < currentrmse skewconst = [spreadingrate, c, theta, currentrmse]; currentrmse = rmse; end Our MATLAB Process 30

32 Main function specify data files pacificantarcticrise = 'pacificantarctic.xydm'; midatlanticridge = 'midatlanticridge.xydm spreadcskewermar = spreadcskewer(midatlanticridge, polarity, time, 1318); spreadcskewerpa = spreadcskewer(pacificantarcticrise, polarity, time, 5418); Our MATLAB Process 31

33 Mid-Atlantic Ridge C = 4 *10^-11; Spreading rate = m/myr Θ (skewness) = -50 RMSE = 193 Real Application 32

34 Pacific-Antarctic Ridge C = 1.2 * 10^-10 Spreading rate = 46,000 m/myr Θ (skewness) = 3º RMSE =259 Real Application 33

35 Limitations We assume: Constant spreading rate Symmetry across the ridge Stationary ridge Limitations 34