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:
Make 2D Fourier transform:
Multiply by:
ikx, iky for horizontal derivatives
|k| for first vertical derivative
for second derivatives in X, Y, or Z, respectively
or other filters (reduction to the pole, pseudo-gravity - below)
Inverse 2D Fourier transform

3. Reduction to the pole Benefits:
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, DT, 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 (magnetic South) pole
Benefits:
The reduced anomaly becomes symmetric, centered over the source
The width corresponds to the source depth more directly
Simpler interpretation process

4. 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:
is the directional vector along magnetization,
is the directional vector of the ambient field

5. Pseudogravity
Pseudogravity transforms the total magnetic field anomaly, DT, 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 r and magnetization M, the magnetic potential V is a directional derivative of the gravitational potential U :
is the directional vector
along magnetization,
gm - gravity in that direction
Therefore, the 2D Fourier transform of gm can be obtained from that of V:
Inverse 2D Fourier transform of gives the “pseudogravity”

7. Pseudogravity – cont.
Expressing the magnetic potential V through the total field DT (projection of the magnetic field anomaly onto the ambient field direction ):
… and also projection of vertical gravity gz onto the magnetization 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 Laplace 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
These derivatives never pass through zero simultaneously
The Analytic Signal is obtained by combining the two derivatives like this:
The absolute value |a(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

11. Euler deconvolution
Note the “Euler homogeneity theorem”:
If function f has a scaling property (“homogeneity”) of order l:
Then:
for arbitrary a
Potential field produced by a source at (0,0,0) has this property
l = -2 for a point source
l = -1 for a line source
l = 0 for a planar source

12. Euler deconvolution - cont.
Thus, if we subtract the regional field T and guess l, then we can estimate the position of the source (x0, y0, z0) from equation:
This is a linear equation with respect to four unknowns x0, y0, z0, and T
It can be solved by taking T and its derivatives at four or more points.
Thus, x0, y0, z0, and T can be determined for every point of grid T(x,y)
Inverted values z0(x0, y0) are interpolated and smoothed in (x,y)
The resulting image z0(x, y) is called “Euler deconvolution” (source depth)
Question: Why is this procedure called “deconvolution”? Where is the corresponding “convolution”?