Measurement and Reduction of Gravity Observations

The acceleration due to Gravity typically is measured in one of two ways: Absolute: Weight drop, pendulum Relative: Mass on a spring

Pendulums The simple pendulum has a period related to its length: The problem is that this length is a “virtual length” to the center of mass of the system, which is not easy to determine accurately.

One way around this is to use a clever device called a compound pendulum such as that devised by Henry Kater (1777-1835) in 1818. It consists of a metal bar with knife edges attached near the ends and two weights that can slide between the knife edges. The bar is pivoted from each knife edge in turn and the positions of the weights are adjusted so that the period of the pendulum is the same with both pivots. The period is then given by the formula for the simple pendulum, which enables g to be calculated to a reasonable precision. Kater's Pendulum in the Science Museum in London. It was designed to make gravity surveys in the field so that it is packed in a fitted mahogany case to protect it in its travels.

This is how Cecil Green got started. Pendulums are not used much anymore but were instrumental (so to speak) in some of the major oil discoveries of the early 20th century (Saudi Arabia, for example). This is how Cecil Green got started.

Weight drop: drop a weight many times in a vacuum and measure time history using laser interferometry. Very accurate measure of absolute gravity, but quite expensive.

Relative Measurements using a Mass on a Spring At first glance this seems simple. We just use Hooke’s law: F = mg = -kz g = -(k/m)z So that a change in g creates a change in z: dg = -(k/m)dz But the problem is sensitivity! dg/g = 10-8 = dz/z So we need to have a really good way to measure the mass displacement.

We could increase sensitivity by making k/m small, but this would mean a combination of a big mass and a weak spring. Not good for field work! Additionally the system will oscillate with a period which is very long for a sensitive device! You would have to wait forever to make a reading. Nevertheless, remember that a very sensitive device should have a very long period.

A clever way around this problem was discovered in 1934 by Lucien Lacoste, who was trying to design a stable long period seismometer. He used a combination of geometry and a device called a “zero-length spring”

Here’s how it works: In the suspension suspension shown above, there are two torques: gravitational and spring. If these two torques balance each other for any angle of the beam, the system will have infinite period. In this case, the smallest change in vertical acceleration (or gravity) will cause a large movement. Let’s see what’s required to meet this condition.

The torque due to gravity is Tg = Wd sin q Where W is the weight (=mg) and d is the distance from the mass to the beam's hinge. The torque due to the spring is the product of the pull of the spring and the springs lever arm, s. s = a sin b The length of the spring is r and by the law of sines: r = bsinq/sinb If the spring constant is k and the length of the spring without force is n, The spring force is illustrated by this graph.

The torque due to the spring is then: Ts = -k(r-n)s = kns -k(bsinq/sinb) a sin b = kns - kabsinq The total torque is then To = Tg + Ts = Wdsinq + kns - kabsinq = kns + (Wd - kab)sinq This equation yields zero torque for all angles q if: (1) n = 0 and (2) Wd - kab = 0 The second part is easy and just involves geometry. For n to equal zero, we must have a "zero length spring". That is, a spring whose force-length graph passes through the origin or, at least, points toward the origin. The turns of a helical spring of zero unstressed length would bump into each other before the spring actually reached zero length. By making a helical spring whose turns press against each other when there is no force on the spring, a "zero length spring" can be made.

There are several ways to make a zero length spring There are several ways to make a zero length spring. A simple zero-length spring is a flat spiral spring. The mechanical properties of a spiral spring are not as convenient as a helical spring. To make a zero-length helical spring, the spring wire can be wound onto a mandrel. As the wire is wound, it can be twisted.

Another method is to hold the wire at an angle and with tension while winding it on a rotating mandrel.

A problem that still needs to be faced is the sensitivity of the spring “constant” k to changes in pressure and temperature. Pressure can be managed by evacuating the box. Temperature effects are minimized with by keeping the box heated to a temperature greater than any you are likely to encounter in the field (50o C). This last part means that the gravity meter consumes power - often not a convenient situation. Nevertheless, the meter is very accurate and the drift rate is quite small, or in any event is linear and can be corrected for easily.

Views of the Lacoste-Romberg Gravity Meter

An second option is the Worden meter, which relies totally on geometry to increase sensitivity, has an evacuated chamber, and uses differentially expanding materials to deal with temperature variations. It works ok, but the drift must be monitored closely when using it in the field.

Gravity measurements can be made at sea or in the air if you put it on a gyroscopically controlled inertial table that corrects for the accelerations of the boat or plane.

Reductions of Gravity Data Gravity Measurements are not difficult to make, but the flip side is that once you’ve made your measurements you need to do a fair amount of work before you can interpret them. Here is an outline of the corrections you need to make to a raw measurement: Latitude: Correction for N-S distance Free-Air: Correction for elevation above the data plane Bouguer: Correction for excess mass above the data plane Terrain: Correction for variations in topography Tides: Attraction of Sun and Moon Eötvös: Correction for moving vehicle Isostacy: Variations in crustal thickness

Recall that this formula (or variants on it) looks like: Latitude Correction This is nothing more than accounting for the reference spheroid as predicted by the International Gravity Formula. Recall that this formula (or variants on it) looks like: g = 978.0318(1 + 0.005302sin2fg - 0.0000059sin22fg) cm/sec2 Note that dg/df = 978.0318(0.0106048sinfgcosfg - 0.000235sin2fgcos2fg) At 45o degrees latitude, dg/df = 5 gals/radian, or, because 1 radian = 57o degrees and 1o degree = 111 km: dg/df = 0.8 mgal/km = 1.3 mgal/mile Typically, we just calculate the reference gravity and subtract if from our readings, but this relation can be convenient. Remember that gravity increases as you go towards the poles, so we add this correction to our readings as we move to the equator.

dg/dr = -2GM/r3 = -2g/r = -0.3085 mgals/m = -0.09406 mgals/ft. Free-Air Correction Often we make observations at different elevations, and we know that gravity will decrease as we get farther from the center of mass of the Earth. Therefore, we choose a reference elevation (typically sea level) and adjust our readings to be what they would be at that elevation. To compute this correction, recall that g = GM/r2 And so dg/dr = -2GM/r3 = -2g/r = -0.3085 mgals/m = -0.09406 mgals/ft. Most land surveys are done above sea level, so the free air correction will generally increase the gravity reading.

Bouguer Correction Of course, now that you have taken your reading down to sea level, you have to account for all the mass between your original elevation and sea level. This correction is called the “Bouguer Correction” (after our friend Pierre) and is calculated by assuming an infinite slab of material of thickness Dh lies between the station and sea level. We derived the expression for this before: Dgslab = 2pGDrDh Assuming a typical density for crustal rocks (r = 2.67 gr/cm3), we arrive at Dg = 0.112 mgals/m = 0.0341 mgals/ft Again, because land surveys are usually above sea level, the Bouguer correction usually decreases the gravity reading.

Terrain Corrections Unless you do your survey in Kansas, you have to correct for variations in topography that you missed with your infinite slab. This is called a terrain correction. Terrain corrections traditionally are made by estimating the differences between the elevation of the station and that of the topography surrounding the station. Remember that terrain always reduces the gravity reading: mass excess above the station (mountains) pull up on the station, while mass deficiencies below (valleys) “push” (the correction is for a negative mass). Thus, a terrain corrected reading will always be greater than it originally was.

Hammer Charts Terrain corrections traditionally were made with something called a “Hammer” chart -named after the guy who introduced them, and they are truly the low point of gravity work. These charts divide the regions around the station into pie shaped cylindrical sections. The analyst centers the chart on the station location on a topo map and estimates the average elevation difference (note that this is in an absolute value sense!) for all elevations within the section. You multiply this number by a theoretical value for that section, and then add them all up.

Fortunately, we don’t have to do that much work anymore, because Digital Elevation Maps (DEMs) exist for much of the world, and for the US they are quite detailed (spacing of 10 m over much of the country). Hence, most of this correction can be calculated by suitable software; you only need to worry about any terrain very close to the station (for example, if you happen to be right by a cliff). Aren’t you lucky?

Tides The Sun and Moon can contribute 0.2-0.3 mgals of attraction and so we need to correct for this if we are in the field for more than an hour or so (depends on desired accuracy, of course). Some people advocate reoccupying a base station several times to determine this correction, but in fact we know how to calculate it better than most of us can measure! You should use the theoretical correction.

Eötvös If you are making measurements in a moving vehicle, you have to correct for E-W velocity. As we discussed before, the vertical acceleration of a Eastward moving vehicle is: Dac= 2wvEcos(l) The effect is to decrease the reading taken from an eastward moving vehicle, and increase the reading from a westward moving one.

Isostatic correction Gravity studies can be useful in constraining crustal thickness, but often we are not interested in structures that we can predict theoretically, so in mountainous terrain we often make a correction for isostatic changes in crustal thickness due to elevation. There are a number of ways to do this, from a pure Pratt to pure Airy type of correction. Note that there is a strong relation between the Isostatic anomaly and the Free Air anomaly. Why should this be?

FAA = dgobs + dgL + dgD + dgE + dgFA Name that Anomaly! When reducing an observed gravity datum gobs, we always make corrections for latitude (dgL), tide (dgD) and Eotvos (dgE) (if we were in a vehicle is moving east-west). If we also correct for elevation (using the free air correction = dgFA) the resulting anomaly is called the Free Air Anomaly (FAA): FAA = dgobs + dgL + dgD + dgE + dgFA If we also correct for the infinite slab (the Bouguer correction dgB) we have the Simple Bouguer Anomaly (SBA): SBA = FAA - dgB If we correct for terrain (dgT), we have the Complete Bouguer Anomaly (CBA): CBA = FAA - dgB + dgT Finally, making an isostatic correction gives the Isostatic Anomaly.

Removing the Regional Field The Isostatic correction brings up a common problem: what to do when what you are looking for is hidden in a big signal from some mass anomaly far from you? Crustal thickness changes are a classic example. This “background” field is usually called the Regional Field because it is a long wavelength background that is typical of an entire “region”. There are some strategies for dealing with regionals, but be aware that nothing is foolproof. Most of these strategies involve removing a smooth background by filtering or polynomial fitting.

Simple Example

More involved example (two-dimensional)

Example using Polynomials: Which is the Signal?

Upward and Downward Continuation One interesting characteristic of gravity data is that if you know the field at one elevation you can in theory calculate or “continue” what it should be at any other elevation. For example, you can take your ground observations and predict what you would measure in an airplane (upward continuation) or vice versa (downward continuation) Upward Continuation is always stable, but downward continuation is not (Can you guess why this is?). In general, downward continuation should be avoided.

Determining Densities Both the Bouguer and terrain corrections depend on the density of the materials in the vicinity of the station. Generally it will be a good idea to collect some samples and measure these in the lab (how would you do this, by the way?) There is also a technique (Nettleton) which works in certain circumstances - when the topography is not closely related to subsurface structure - that allows you to do this with your gravity measurements. Generally it will be better to use some kind of ground truth, however.

Examples of typical density variations are shown to the right. Note that variations in density are not much more than a factor of 2 for most rocks you are likely to encounter.