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ERTH3021 Exploration and Mining Geophysics
Practical: Gravity and Magnetics - Poisson’s Relationship
Aims
• Provide experience in handling of vector quantities in geophysics
• Provide exposure to the concept of Gravitational and Magnetic Potentials
• Examine the derivation of fields from potentials
• Introduce Poisson’s relationship between potentials
• Investigate the construction of gravitational and magnetic anomalies from gravitational potentials, and compare with known analytical anomalies
• Provide exposure to the finite-difference approximation of derivatives
Note on Magnetic terminology and units
There is considerable confusion in the literature regarding symbolism and terminology
for magnetics. This arises partly because of differences in definitions between the SI
and cgs systems. An interesting overview is Hitchhiker’s guide to magnetism
(http://www.irm.umn.edu/hg2m/hg2m index.html)
These differences can cause problems in comparing theory from different sources. Here
I will note important points which are relevant to this practical.
Recall from basic magnetics that the observed magnetic field results from interaction
between the earths ambient field (H) and the induced magnetisation (M).
B = µ0 (H + M)
Here M = kH, where k is the magnetic susceptibility.
In the cgs system the parameter µ0 = 1, so that B, H and M effectively all have the
same units (nT). (This is why, for example, the Berkeley modeller requires you to input
the H value in nT.)
Where possible, we will use the SI system. In this system, µ0 6= 1 and B, H, M no
longer have the same units. B is in nT and H, M are in Am−1 . To avoid confusion in
our programming code we will introduce the following new symbolism
B = µ0 (H + M)
B = BH + BM
Here BH and BM are in the same units as B (nT). Also, it follows that BM = kBH .
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Geophysical Potentials
Potential theory provides a convenient mathematical tool for studying the fundamental
forces of nature (e.g. gravity, magnetism, electricity). One way of defining a potential
is that a field can be obtained as the negative gradient of the relevant potential.
In ERTH2020 we introduced the concept of gravitational potential, and the relationship
between potentials and fields. As a particular example, we examined the relationship
for a particle, or a spherical body. In that simple case, the gravitational field (acceleration) can be derived from the gravitational potential as the negative radial derivative.
In order to make practical use of Poisson’s relationship, we need to examine the concept of the gradient operator in a cartesian coordinate system.
For example, consider the gravity problem above in more detail. In rectangular coordinates, the acceleration vector g is obtained using the Grad operator
g = −∇UG = −(i
dUG
dUG
dUG
+j
+k
)
dx
dy
dz
Here the vector g indicates the direction of the acceleration in terms of the unit vectors
i, j and k.
In this practical we will be performing this gradient operation numerically. This will
make the concept easier to understand.
Note that analogous to the above discussion, we can also define a magnetic potential
UM . The magnetic field can be obtained from the magnetic potential using the Grad
operator.
A note on signage conventions
In the geophysical literature various signage conventions are used for potentials, depending on the application. Here we will follow a commonly used convention for
geophysical vector applications.
Poisson’s Relationship
We have seen in earlier practicals that magnetic anomalies are typically more complex
than gravity anomalies. This is largely because of the interaction between the earth’s
magnetic field (which is generally at an angle to the surface) and the geological features.
There exists, however, an intriguing theoretical relationship between the gravity and
magnetic potentials. For our practical purposes, the Poisson relationship can be expressed as
BM
UM =
UG 0
4πGσ
Here the coefficient can be thought of as replacing the gravitational properties (denominator) with the magnetic properties (numerator). The symbol UG 0 means the derivaPoisson’s Relationship
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tive of the gravitational potential in the direction of the earth’s ambient magnetic field
BH .
Because gravitational anomalies are generally simpler, and easier to compute, an effective approach to combined gravity and magnetic modelling is
(i) Compute the gravitational potential resulting from the assumed geology
(ii) Use the Grad rule to get the gravitational acceleration anomaly (usually vertical
gZ )
(iii) Use the Poisson relationship to obtain the magnetic potential
(iv) Use the Grad rule to get the magnetic field anomaly (either vertical or total-field)
EXERCISE 1: Gravitational Potential of a Sphere, and of a Horizontal Cylinder
From ERTH2020 we are familiar with the gravitational potential at a radial distance r
from a spherical body.
UG = −GM/r
By convention, potential is taken as zero at an infinite distance r. We will examine the
spherical model in Exercise 6.
Another convenient model for exploring Poisson’s relationship is the horizontal cylinder. This simple shape is useful because various analytical expressions are available
for gravity and magnetics. Also, comparisons can be performed using the Berkeley
gravity and magnetic modellers.
As our starting point, we will use the following expression for gravitational potential
at a radial distance r from a horizontal cylinder
UG = −2πGσR2 log
1
r
where the symbols have their usual meanings. it is of interest to briefly compare the
potential expressions for sphere and cylinder.
(i) As distance from a cylinder increases, does gravitational potential increase or decrease? How does this compare with potential due to a sphere?
(ii) At an infinite distance from a cylinder, does gravitational potential approach
zero? How does this compare with potential due to a sphere?
(iii) What is a fundamental reason for this difference in behaviour?
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(iv) Is there a distance at which potential due to a cylinder is zero?
(v) How does the choice of the ’vanishing distance’ for gravitational potential affect
calculation of gravity field, magnetic potential and magnetic field?
EXERCISE 2: Horizontal Cylinder at Equator
In the following exercises we will examine gravity and magnetic responses of horizontal cylinders. Unless otherwise stated, we assume that the cylinder is EW (i.e.
perpendicular to the magnetic field. Also we are examining principal profiles (i.e. perpendicular to strike of the body.)
(i) Implement a Python program to plot the gravitational potential profile across a
horizontal cylinder. (Note that the gravity response is independent of latitude).
(ii) Use the Grad rule to plot the vertical component of the gravitational field (i.e.
acceleration). Note: The grad rule effectively returns a vector quantity, and you
will need to think carefully about signs here.
(iii) Check your numerical result against the known analytical expression for vertical
acceleration due to a cylinder (ERTH2020) and/or the Berkeley modeller. (Note
that the Berkeley modeller plots magnitude of the acceleration.)
(iv) Use Poisson’s relationship to derive the magnetic potential along the profile
(v) Use the Grad rule to derive the total-field magnetic anomaly. Plot the total-field
curve that would be recorded on a magnetometer. (This will involve the vector
sum of the primary field BH and the induced field BM .)
(vi) Check this magnetic anomaly using the known theoretical expression and/or the
Berkeley modeller. How is the accuracy of your numerical approach affected
by the chosen spatial increments ∆x, ∆z? Explain with reference to ideas from
calculus.
(vii) Sketch a cross section which shows how the primary and induced magnetic fields
interact for this situation. Hence explain the general shape of the total-field anomaly.
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EXERCISE 3: Horizontal Cylinder at the Pole
Modify your program in Exercise 2 to deduce the gravity and magnetic anomalies
at either the north or south pole. (Note that in this case the strike of the cylinder is
irrelevant.)
(i) This will require a modification to allow for the ambient magnetic field to now be
vertical (and in the correct direction). In addition you will need to calculate the
initial gravitational potential at three levels rather than two.
(ii) Again compare you results with a suitable reference.
(iii) Sketch a cross section which shows how the primary and induced magnetic fields
interact for this situation. Hence explain the general shape of the total-field anomaly.
EXERCISE 4: Horizontal Cylinder at an arbitrary latitude
(i) Modify your program from Exercise 3 to deduce the gravity and magnetic anomalies at an arbitrary latitude (i.e. for an arbitrary inclination). One approach is to
allow the increment ∆z to be different from ∆x. We will start with an approach
where we set dx and then deduce a convenient dz.
(ii) Test your program using an inclination appropriate to SE Qld (I=-57 degrees), and
compare your results with a suitable reference.
(iii) Sketch a cross section which shows how the primary and induced magnetic fields
interact for this situation. Hence explain the general shape of the total-field anomaly.
(iv) Comment on the performance of this algorithm for small inclinations (i.e. near
equator) and for large inclinations (near pole).
(v) Comment on an alternative approach where we fix dz, and then deduce a suitable
dz.
√
(vi) Comment on an alternative approach where we fix dr = dx2 + dz 2 , and then
deduce a suitable dx and dz.
(vii) Implement, and concisely document, your preferred algorithm, which should
give optimised performance for inclinations from -90 to 90 degrees.
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EXERCISE 5: Horizontal cylinder, arbitrary strike, and arbitrary profile.
The previous exercises have generally considered the case of a cylinder striking EW
(i.e. perpendicular to magnetic field direction.) Also we have considered only principal profiles (i.e. perpendicular to strike of body. Here we briefly examine general
approaches to relaxing these assumptions.
(i) For the problem in Exercise 4, incorporate a modification to allow for the strike of
the cylinder to be at an arbitrary angle (α) from EW.
(ii) Summarise the main effects on observed profiles.
(iii) Note any practical restrictions in your code (e.g. limitations on the value of α).
(iv) Modify your code to display profiles which are at an angle γ to the principal
profile.
(v) Summarise the main effect on observed profiles.
(vi) Note any practical restrictions in your code (e.g. limitations of value of γ).
EXERCISE 6: Sphere at an arbitrary latitude
We use the following expression for the Gravitational Potential at a radial distance r
from a spherical body.
UG = −GM/r
(i) For a suitable buried-sphere model, use the above potential expression to derive
the expected gravity anomaly.
(ii) Compare with the known analytical anomaly, and/or using the Berkeley gravity
modeller.
(iii) Use Poisson’s relationship to derive the magnetic potential profile, and corresponding total-field anomaly, for at least two latitudes (inclinations).
(iv) Compare your results with a suitable analytical expression. See Appendix.
(v) How does the anomaly from a sphere compare with that from a horizontal cylinder (in magnitude and direction)?
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Appendix 1: Calculation of UG 0 for the case where magnetic field is not perpendicular to strike.
Suppose that the magnetic field is not perpendicular to strike, but is rotated by angle
of α in the horizontal plane. See Figure 1.
Figure 1: Geometry for a rotated magnetic field. XYZ axes represent a cartesian coordinate system.
Assume that the strike of the body is in the direction of the Y axis. The vertical plane of the magnetic
field is outlined in red. The magnetic field is at an angle α to the X axis, and makes an angle β with the
horizontal. The increment along the direction of the field is dq.
Firstly note that because this is a 2D problem (body with infinite strike extent) all geophysical parameters (potentials, fields) will be independent of position along the strike.
Hence it makes sense to keep the body in the Y direction, so that any anomaly will be
a function of just x and z.
Points O,D,E,F are 4 points in the grid separated by dx in the X direction, and dz in the
Z direction. These are related to the increment in the direction of the field (dq) by
dz = dq sin β
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dx = dh cos α
That is
dx = dq cos β cos α
The required derivative of the gravitational potential, in the direction of the magnetic
field, is
UG 0 ≈
UG (Q) − UG (0)
dq
Note that Points E and Q have the same X coordinate, so they will have the same value
for potentials and fields. Hence
UG (Q) = UG (E),
and so
UG 0 ≈
UG (E) − UG (0)
dq
Using the symbolism in our Python code
UG 0 ≈
Poisson’s Relationship
U UG [i + 1] − UG [i]
dq
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Appendix 2: Analytical expression for magnetic anomaly due to buried sphere
See this URL:
www.geofys.uu.se/files/teacher/ce magnetics HT09.pdf
Note: It appears there may be a missing power of 2 in the first cos term. Also the
”M” term in the equation is as defined in the following code ...
#analytic anomaly using the University of Uppsula equation
#
#the beta here is just the inclination in radians (=pi*I/180.0)
#(e.g. beta is negative in SE Qld)
#
M=Bm*R*R*R/3. #this is the M in the equation
denom=(x[i]*x[i]+Z*Z)**2.5
a=(3.0*cos(beta)*cos(beta)-1.0)*x[i]*x[i] #note cos squared error
b=6.0*x[i]*Z*sin(beta)*cos(beta)
c=(3.0*sin(beta)*sin(beta)-1.0)*Z*Z
Ta[i]=M*(a-b+c)/denom
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