1. Outline  Derivatives and transforms of potential fields  First and second vertical derivatives  Convolutional operators  Fourier approach  Reduction to the pole  Pseudogravity  Analytic signal  Gradient tensor (for magnetic field)  Euler deconvolution

2. Derivatives and transforms of potential fields  Because of the Laplace equation, the vertical derivative of a potential field can be obtained from the horizontal derivatives  A Fourier component of U depends on (x,y,z) as: where  Therefore, any derivatives and several other useful transforms can be obtained by 2D filtering: 1) Make 2D Fourier transform: 2) Multiply by: 1) ik x, ik y for horizontal derivatives 2) |k| for first vertical derivative 3) for second derivatives in X, Y, or Z, respectively 4) or other filters (reduction to the pole, pseudo-gravity - below) 3) Inverse 2D Fourier transform

3. Reduction to the pole  Total-field magnetic anomalies are usually shifted because of the inclination of the ambient field  The Reduction to the Pole transforms the total magnetic field anomaly measured relative to an inclined field,  T, into an anomaly that would be produced by the same structure with vertical magnetization and in a vertical ambient field.  As if recording at the North pole  Benefits:  The reduced anomaly becomes symmetric, centered over the source  The width corresponds to the source depth more directly  Simpler interpretation process

4. is the directional vector along magnetization, is the directional vector of the ambient field Reduction to the pole – cont.  For a 2D Fourier component of the field, changing directions of magnetization and projection amounts in dividing by “complex-valued directional cosines” : Here:

5. Pseudogravity  Pseudogravity transforms the total magnetic field anomaly,  T, into a gravity anomaly that would be produced by the same structure  Assuming uniform density and magnetization  Benefits:  Gravity anomalies are often easier to interpret  More symmetric, peaks centered over the source  Some structures (e.g., mafic plutons) produce both gravity and magnetic anomalies  Tubular structures can be identified by maximum horizontal gradients

6. Pseudogravity – cont.  Poisson’s relation showed that for a body of uniform density  and magnetization M, the magnetic potential V is a directional derivative of the gravitational potential U :  Therefore, the 2D Fourier transform of g m can be obtained from that of V : is the directional vector along magnetization, g m - gravity in that direction  Inverse 2D Fourier transform of gives the “pseudogravity”

7. … and also projection of vertical gravity g z onto the magnetization direction : Pseudogravity – cont.  Expressing the magnetic potential V through the total field  T (projection of the magnetic field anomaly onto the ambient field direction ): are the same complex-valued directional cosines as in the reduction to the pole … the Fourier transform of pseudogravity becomes: Note that this involves spatial integration (division by | k |)

8. Analytic signal  The analytic (complex-valued) signal is formed by combining the horizontal and vertical gradients of a magnetic anomaly  Principle: note that if T (total field) satisfies the Poisson’s equation in 2D: then any of its Fourier components with radial wavenumber k depends on x and z like this:  This means that the Fourier components of the derivatives are simply related to the Fourier component of T : and and therefore very simply related to each other:

9. Analytic signal – cont.  Such relation between functions in […] is called the Hilbert transform:  Note:  This means that whenever looks like a cos() function at any frequency, behaves as a sin(), and vice versa  Combined together, these derivatives never pass through zero  The Analytic Signal is obtained by combining the two derivatives like this:  The absolute value |  (x,z)| has some nice properties:  Peak tends to be centered over the source  Width of the peak is related to the depth to the source  Derivatives enhance shallow structure (but also noise)

10. Analytic signal in 3D  In 3D, the analytic signal is a complex-valued vector defined like this:  Its absolute value,, is used for interpretation