Environmental and Exploration Geophysics I tom.h.wilson Department of Geology and Geography West Virginia University Morgantown, WV Magnetic Methods (part II)

From above, we obtain a basic definition of the potential (at right) for a unit positive test pole (m t ). The potential is the integral of the force (F) over a displacement path. Note that we consider the 1/4  term =1

Thus - H (i.e. F/p test, the field intensity) can be easily derived from the potential simply by taking the derivative of the potential

Consider the field at a point along the axis of a dipole. The dipole in this case could be a buried well casing. The field has vector properties however, in this case vectors are collinear and its easy to determine the net effect.

In terms of the potential we can write In the case at right, r + is much greater than r -, thus in

Thus, the potential near either end of a long dipole behaves like the potential of an isolated monopole. If we are looking for abandoned wells, we expect to find anomalies similar to the gravity anomalies encountered over buried spherical objects. 21

In spite of this special situation, the magnetic field of an object is defined by the simple dipole field or combinations of dipole fields. To examine the nature of the dipole field consider the case where the distance to the center of the dipole is much greater than the length of the dipole. This allows us to treat the problem of computing the potential as one of scalar summation since the directions to each pole fall nearly along parallel lines.

If r is much much greater than l then the angle  between r + and r - approaches 0 and r, r + and r - can be considered parallel and the differences in lengths r + and r - from r equal to plus or minus the projections of l/2 into r.

r-r- r+r+ r

Recognizing that pole strength of the negative pole is the negative of the positive pole and that both have the same absolute value, we rewrite the above as Working with the potentials of both poles..

Converting to common denominator yields From the previous discussion, the field intensity H is just where pl = M – the magnetic moment

H - monopole = H - dipole This yields the field intensity in the radial direction - i.e. in the direction toward the center of the dipole (along r). However, we can also evaluate the horizontal and vertical components of the total field directly from the potential.

V d represents the potential of the dipole. H Toward dipole (Earth’s) center

H E is represented by the negative derivative of the potential along the earth’s surface or in the S direction.

Where M = pl and Let’s tie these results back into some observations made earlier in the semester with regard to terrain conductivity data. 32

Given What is H E at the equator? … first what’s  ?  is the angle formed by the line connecting the observation point with the dipole axis. So , in this case, is a colatitude or 90 o minus the latitude. Latitude at the equator is 0 so  is 90 o and sin (90) is 1.

At the poles,  is 0, so that What is Z E at the equator?  is 90

Z E at the poles …. The variation of the field intensity at the pole and along the equator of the dipole may remind you of the different penetration depths obtained by the terrain conductivity meters when operated in the vertical and horizontal dipole modes.

I=kH I is the intensity of magnetization and H can be considered the ambient (for example - Earth’s) magnetic field intensity. k is the magnetic susceptibility.

Also recall Equation 7-5 (Burger). The intensity of magnetization is equivalent to the magnetic moment per unit volume or and also,. Thus and yielding Magnetic dipole moment per unit volume

Recall from our earlier discussions that magnetic field intensity so that The anomalous field induced in a magnetic object is simply …

Let’s consider this in the context of a potential application Field intensity can be described in a variety of different units, including, for example: Oersteds, nanoTeslas, and gammas. The units you often encounter in exploration are nanoTeslas (nT). Pole strengths have a variety of units as well, but the one we will use is the “unit pole strength” or ups. In our basic definition for magnetic field intensity associated with a single pole, we have..

With H in units of Oersteds, p in ups, and r in cm, we have units of - And the equivalent units for ups

Problem: Imagine you are 20 centimeters from the negative and positive poles of a dipole as shown below, and that each pole has pole strength of 1 ups. Observation point 20cm - + What is the magnetic field intensity at the observation point?

Vertical Magnetic Anomaly Vertically Polarized Sphere The notation can be confusing at times. In the above, consider H = F E = intensity of earth’s magnetic field at the survey location. Z max and Z A refer to the anomalous field, i.e. the field produced by the object in consideration

Vertically Polarized Vertical Cylinder

Anomaly A: Sphere, Vertical Cylinder; z = __________ Diagnostic positions Multiplier s Sphere Z Sphere Multipliers Cylinder Z Cylind er X 3/4 = X 1/2 = X 1/4 = The depth

Diagnostic positions Multipliers Sphere Z Sphere Multiplier s Cylinder Z Cylinder X 3/4 = X 1/2 = X 1/4 = Sphere or cylinder?

Due Thursday, Dec. 9th

Since the bedrock is magnetic, we have no way of differentiating between anomalies produced by bedrock and those produced by buried storage drums.

Acquisition of gravity data allows us to estimate variations in bedrock depth across the profile. With this knowledge, we can directly calculate the contribution of bedrock to the magnetic field observed across the profile.

With the information on bedrock configuration we can clearly distinguish between the magnetic anomaly associated with bedrock and that associated with buried drums at the site.

Work with the inversions – iterate and adjust…

How many drums are represented by the triangular-shaped object you entered into your model? Use the magic eye to get the coordinates of the polygon defining the drums Plot the corner coordinates for the triangular shaped object you derived at 1:1 scale and compute the area.

How many drums?