Presentation on theme: "Fundamental Relationships Magnetic Force and Coulomb's Law The study of magnetics begins with a description of magnetic forces. The force exerted between."— Presentation transcript:
Fundamental Relationships Magnetic Force and Coulomb's Law The study of magnetics begins with a description of magnetic forces. The force exerted between two magnetic poles, m1 and m2 separated by a distance r, is described in the equation known as Coulomb's Law: Where µ is the magnetic permeability of the medium in which the poles are located. Magnetic Field Strength A magnet consists of two oppositely charged poles, described as positive (+) and negative (-), or in the case of the earth's magnetic field, north and south. Every magnet has a magnetic field. This is illustrated a potential lines of strength as shown in the illustration. Magnetic Surveying
The S. I. unit used to describe magnetic field strength is the nanotesla, or nT. Other equivalents are the oersted, and the gamma: The strength of a magnetic field varies according to the distance from the magnetic pole producing the field.
Intensity of Magnetization A property of a bar magnet is known as the magnetic intensity, I, which is inversly proportional to the volume of the magnet: Magnetic Susceptibility The intensity of magnetization which an object obtains when placed in a magnetic field is equal to the field strength times a constant of proportionality, which differs depending on the material. This constant is known as the magnetic susceptibility and is represented by k.
The Earth's Magnetic Field When referring to the Earth's magnetic field, it is best to refer to it within seven components. The seven field relations are shown in the figure. Magnetic north is not equal to geographic north; the difference between the two, in the horizontal direction, is referred to as the declination. Inclination is the angle the earth's magnetic field makes with the horizontal.
There are three important types of remnant magnetization: *Thermoremnant magnetization (TRM), occurs when igneous rocks gain and retain a remnant magnetization during cooling and crystallization. *Detrital remnant magnetization (DRM) is due to the tendency of fine grained sediment particles, such as silt and clay to allign themselves parallel to the earth's magnetic field. *Chemical remnant magnetization (CRM) occurs when minerals in solution are precipitated and crystallize at low temperatures. Magnetic Surveying
At any point above Earth’s surface measured geomagnetic field will be: *Sum of regional and local field produced by magnetic rocks Purpose of Magnetic Surveys: Record variations of field strength due: *changes in earth magnetic field and to *volume and magnetic susceptibility of underlying rocks Setup local fixed station to record fluctuations Remove fluctuations-determine residual signal Residual signal related to local geology and underlying rocks
Gravitational Potential and Acceleration Two point masses m1 and m2 at a distance r apart are attracted to each other by a force deduced by Sir Isaac Newton, and is known as Newtons First Law of Gravitational Attraction : where G is the universal gravitation constant with a value of: The above equation assumes that the earth was a perfect shpere. If a ball and a feather are thrown from the same elevation they will both have the same gravitational acceleration. The earth exerts the same force on the objects. This concept was applied the moon's orbit of the earth. where M is the mass of the earth and R is the radius of the earth. Source: http://www-geology.ucdavis.edu/~gel161/sp98_burgmann/gravity98/GRAVITY.HTML
Newton's second law of motion (F=ma) is also applied. The acceleration due to gravity on earth is defined as Solving for g Gravity can be expressed as a potential field: The potential field is defined by: For a point mass, the work done by the potential field moving a point mass from infinity to a distance R is given by: Gravity has units of m/s2. 1 g = 9.8 m/s2 = 980 cm/s2. Changes in measured gravity are often quite small. Measurements are most often given in mGal. 1 mGal = 10-3 g. Typical resolution requirements for gravity surveys are 0.1 mgal.
Shape of the Earth So far we have dealt with perfect spherical shape of the earth, but in reality the earth is not a perfect sphere. The earth rotates. This rotation causes an outward centrifugal force which causes the earth to flattened at the poles and bulge at the equator. Its shape is an ellipsoid.
Because gravity measurements are relative, a reference surface has been established by geophysicists so that all gravity measurements are universal. They use an equipotential surface known as the reference ellipsoid. This surface has the shape of a landless earth, a geoid, which would have broad undulations that coincides closely with sea level. If all measurements could be taken directly on the reference geoid, all anomalies in gravity would be due to subsurface density variations. However, this is impossible. Geophysicists are forced to take most of their measurements on the highly uneven surface of the earth. However, corrections for this uneveness can be made and the resulting anomalous gravity measurements will be do to subsurface density variations. With knowledge of these variations, geophysicists are able to model subsurface structures.
Gravity Measurements CORRECTIONS Corrections are made to remove the effects on surficial gravity mearurements due to the imperfect sherical shape of the earth, and for the fact that the earth is rotating. Latitude Correction The first correction made is for the latitudinal position on the earths surface where the measurement is taken. As stated before, the earth is flattened at the poles. Thus at the poles, R in Newtons first law of gravitational attration, is smaller than at the equator. This means the resultant force acting on a mass will be greater at the poles and decrease toward the equator. However, more mass exists between the center of the earth and a point on the equator, since R is greater here. This excess mass directly affects gravity, so g increases toward the equator. Lastly, the fact that the earth is rotating also affects gravity, by the addition of an outward centrifugal force which is greatest at the equatore (where the rotational velocity is greatest) and is zero at the poles of rotation. This force opposes gravity. The net effect of these latitudinal variations is a 5.2 Gal net decrease in gravity from the poles toward the equator. With this knowledge it is possible to predict what gravity would be on the reference geoid at any latitude via the following equation:
where gn is normal gravity, the value of gravitational acceleration we should observe on the surface and is latitude known within 125 m for a gn accuracy of 0.1 mGal. where g(row) is the gravity anomaly. The next three corrections are to correct for changes in elevation, since we cannot take gravity measurements directly on the reference geoid except at certain locations. These corrections also adjust for attractions to nearby topographic features, such as mountains.
Free Air Correction As stated in the latitude correction, the further away from the center of the earth, the less the observed gravity will be. Thus, at some elevation h above the reference geoid, the observed gravity will be less. The Free-Air Correction is determined by: The Free- Air Correction is added to the observed gravity of offset the decrease caused by being further away from the earths center. This correction only adjusts for the added elevation, but has no effect on the additional mass also present. Thus the term "Free-Air". The Free-Air anomaly is:
Bouguer Correction The Bouguer Correction accounts for the excess mass below an observation point which the Free- Air Correction did not adjust for. This is done by assuming an infinite slab of thickness z (equal to the elevation above the geoid at the observation point) lies between the reference geoid and the observation point. The following equation is used to determine the Bouguer correction. where p is the density of the slab and z=h=thickness/elevation of the slab. Because excess mass increass the observed gravity masurements, the Bouguer Correction is subtracted. The Bouguer Anomaly becomes:
Terrain Correction It can be seen in the previous figure that the Bouguer Correction is only a rough guess of the actual terrain surrounding an observation point. A terrrain correction must be applied to account for undulations in topography. The figure below illustrates a section of the surface of the earth with a hill (region y) and a valley (region x). The lightly shaded slab is the Bouguer slab. At the valley, x, the Bouguer slab has added more mass than is actually present. Thus we have overcorrected when we subtracted the effect of the Bouguer slab form our gravity measurement and must now add the effect of this valley back to our gravity measurement. At the hill, y, extra mass that was not accounted for by the Bouguer slab was still attracting the gravimeter. The attraction to the hill reduces the observed gravity from the actual value. Thus, to negate the effects of the nearby hill, we must add the terrain correction to the observed gravity. For both the valley and the hill, the terrrain correction must be added, and the resultant anomaly is:
The most commonly asked question is whether a terrain correction is necessary and how to go about the process in an area of irregular topography. The method used by many geophysicists was proposed by Hammer (1939). His method was to overlay a set of concentric rings about an observation point. Each ring is divided into various sectors with varying height equal to the average elevation within that sector. The figure below illustrates a simplified template placed on a topographic map. The correction for a ring is: and each ring is divided into n number of sectors:
Terrain corrections are very labor intensive and time consuming. A template has about nine rings, each divided into 4, 6, 8 or 12 sectors (this number increases outward from the center). In each sector the average elevation must be determined, and then compared to the elevation of the observation point. A table is then consulted to determine the gravitational effect of that sector. This is continued until the effect due to each sector is determined. Then the template is move to the next observation point and the calculations are repeated. Again, very labor intensive and time consuming. Isostatic Correction Isostasy is "the condition of equilibrium, comparable to floating, of the units of the lithosphere above the asthenosphere" (Bates and Jackson (eds). 1984. Dictionary of Geologic Terms). What this is saying is that mountains "float" in the asthenosphere. Like an iceburg floating in the ocean with very deep roots beneath the surface of the ocean, mountains also have deep roots which extend into the mantle. Continental crust, the mountains, have a lower density than mantle material. The lack of mass at depth due to a mountain root gives rise to a negative Bouguer anomaly. This isostatic anomaly can be seen in gravity measurements as a regional trend and is often not corrected for.
Other Effects on Gravity Measurements Because the gravimeter is a very precise device, capable of measuring a change in gravity of 1 unit in 100,000,000, many things besides variations in density can effect it. Slight changes in the parts which make up the gravimeter can be detected. This includes variation in the springs as they stretch with time, or changes in the properties of the gravitmeter due to temperature fluctuations throughtout a day. These types of effects that vary the gravitiational measurements are known as drift. The gravitmeter is also sensitive to tidal variations throughout a day. Tidal and drift effects must also be corrected for. This is done by establishing a base station, starting and ending the day with a reading at this station and taking measurements about every hour at the base station. Between each reading a linear relationship can be found, and this is subtracted from the observed gravity measurements.
Models Once gravity measurements have been corrected, we should be left with anomalies which reflect density variations within the subsurface. Models use simple geometric shapes such as spheres and cylinders, or a combination of simple geometric shapes, which creates a complex shape, to determine the geometry of subsurface structures. Simple Bodies Effect of a Buried Sphere
Summary: Gravity is a helpful tool for geophysicists to further understand the nature of the shallow subsurface of the earth. It is not, however, a tool to be used simply on its own. Surveys and measurements are commonly used in conjunction with other studies to confirm theories by geologists and geophysicists. Corrections to measured anomalies are applied with discretion, depending primarily on the complexity of the model or the information sought by the survey. Many times, buried objects may be aproximated quite well with simple geometries or combinations of simple geometries. Geophysicists often find numerical techniques for modeling gravity fields very useful. Computers programs such as GravModel make modelling the subsurface fairly simple.
Mass and Spring Measurements The most common type of gravimeter* used in exploration surveys is based on a simple mass-spring system. If we hang a mass on a spring, the force of gravity will stretch the spring by an amount that is proportional to the gravitational force. It can be shown that the proportionality between the stretch of the spring and the gravitational acceleration is the magnitude of the mass hung on the spring divided by a constant, k, which describes the stiffness of the spring. The larger k is, the stiffer the spring is, and the less the spring will stretch for a given value of gravitational acceleration.gravitational forcegravitational acceleration Like pendulum measurements, we can not determine k accurately enough to estimate the absolute value of the gravitational acceleration to 1 part in 40 million. We can, however, estimate variations in the gravitational acceleration from place to place to within this precision. To be able to do this, however, a sophisticated mass-spring system is used that places the mass on a beam and employs a special type of spring known as a zero- length spring.pendulum Instruments of this type are produced by several manufacturers; LaCoste and Romberg, Texas Instruments (Worden Gravity Meter), and Scintrex. Modern gravimeters are capable of measuring changes in the Earth's gravitational acceleration down to 1 part in 100 million. This translates to a precision of about 0.01 mgal. Such a precision can be obtained only under optimal conditions when the recommended field procedures are carefully followed.
Worden Gravity Meter LaCoste and Romberg Gravity Meter *A gravimeter is any instrument designed to measure spatial variations in gravitational acceleration. **Figure from Introduction to Geophysical Prospecting, M. Dobrin and C. Savit
Apollo 17 Traverse Gravimeter Experiment The Traverse Gravimeter Experiment was performed on Apollo 17 to accurately measure variations of the gravitational acceleration in the vicinity of the Taurus-Littrow landing site. The purpose of the experiment was to obtain information about the subsurface structure at the landing site. Gravity measurements were made at 12 of the traverse stops on the three EVAs, and the results were radioed by the crew back to Earth. Interpretation of these observations also requires knowledge of the topography of the landing site. Topography information was obtained from analysis of stereo photography taken in lunar orbit. The results of this experiment indicate that the mare basalt layer in the vicinity of the landing site has a thickness of 1 kilometer. This value is slightly lower than the 1.4 kilometers measured by the Lunar Seismic Profiling Experiment.stereo photographyLunar Seismic Profiling Experiment.
Figure 2: Detail of the present-day topography of the Albertine rift system using the EROS Data Center 30x30 arc-second DEM of Africa. Contour interval is 100 m. Earthquake locations shown by the light blue circles. Profiles A"-G show the location of the topography and free-air gravity data used to: 1) map the three-dimensional architecture of the rift system and the distribution of crustal extension responsible for the basins, and 2) estimate the flexural strength across and along the basin system.
Figure 3: Free-air gravity contour map of the Albertine rift system. Contour interval is 10 mgal. The white circles, and the boat trackmap on Lake Albert, identify existing and new land and lake data in the region. During the March and April, 1992, our project increased the east African gravity data base by 3613 lake gravity stations (at 1 minute sampling interval) and 60 new land stations. There is a general 100 mgal variation from a minimum in the south of Lake Albert to the north. Across the basin, there is a 30-40 mgal variation with local minima in the extreme south of the lake, the southern onshore extension of the rift into the Semliki basin, and along parts of the western border fault system.