3.1 Gravity and its measurement

Gravity: Concept and Principles

Gravity governs the large-scale structure of the Earth and the motion of objects within and around it. In geophysics, precise gravity measurements reveal density variations beneath the surface, making gravity one of the primary tools for mapping subsurface geology. This section covers the physical principles behind gravity, the instruments used to measure it, and how geophysicists interpret gravity data to identify hidden geological structures.

Fundamental Force of Nature

Gravity is the attractive force between any two objects that have mass. It's one of the four fundamental forces of nature, alongside the electromagnetic force, the strong nuclear force, and the weak nuclear force. Of these four, gravity is by far the weakest, but it dominates at planetary and astronomical scales because it acts over unlimited distances and is always attractive.

Newton's Law of Universal Gravitation

Newton's law gives us the quantitative framework for gravity:

$F = G \frac{m_1 m_2}{r^2}$

where $F$ is the gravitational force, $m_1$ and $m_2$ are the masses of the two objects, $r$ is the distance between their centers, and $G$ is the universal gravitational constant ($6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2$).

Two things control the strength of the force:

• Mass: Greater mass means a stronger pull. This is why density contrasts underground produce measurable gravity variations at the surface.
• Distance: The force drops off as $1/r^2$. Double the distance, and the force falls to one-quarter.

For geophysics, we're usually interested in the gravitational acceleration at Earth's surface, $g$, which averages about $9.81 \, \text{m/s}^2$. Small spatial variations in $g$ tell us about differences in rock density below.

Gravity's Role in the Universe

• Keeps planets in orbit around the Sun and the Moon in orbit around Earth
• Drives the formation of large-scale structures like galaxies and galaxy clusters
• Controls isostatic equilibrium in Earth's crust, which directly matters for geodesy

Einstein's Theory of General Relativity

General relativity describes gravity not as a force but as the curvature of spacetime caused by mass and energy. For most geophysical applications, Newtonian gravity is sufficient. General relativity becomes necessary in extreme conditions (near black holes, in the early universe) and for high-precision satellite geodesy, where relativistic corrections affect clock rates on orbiting instruments like those in GPS and GRACE.

Measuring Gravity

Gravimeters

A gravimeter measures the strength of the gravitational field at a specific location. Gravimeters fall into two broad categories: absolute and relative. The choice between them depends on the required accuracy, portability, and survey design.

Absolute Gravimeters

Absolute gravimeters measure the actual value of $g$ at a location by directly tracking the acceleration of a free-falling object in a vacuum.

How they work (step by step):

1. A test mass (typically a corner-cube mirror) is released inside an evacuated chamber.
2. A laser interferometer tracks the position of the falling mass at precisely timed intervals.
3. The acceleration is computed from the position-time data using $s = \frac{1}{2}gt^2$.
4. Multiple drops (often hundreds) are averaged to reduce noise.

Common instruments include the FG5 (lab-grade, accuracy ~1–2 µGal) and the A10 (field-portable, accuracy ~10 µGal). These are expensive, heavy, and require a stable environment, so they're typically used to establish reference stations rather than for broad area surveys.

Relative Gravimeters

Relative gravimeters measure the difference in $g$ between two locations rather than the absolute value. They're more portable and far less expensive than absolute instruments, making them the workhorse of field gravity surveys.

Two main types:

• Spring-based gravimeters (e.g., LaCoste & Romberg, Scintrex CG-5): A mass on a spring stretches or compresses in response to changes in gravity. The displacement is proportional to the change in $g$. These instruments are subject to drift over time and must be regularly returned to a base station for recalibration.
• Superconducting gravimeters: A niobium sphere is levitated by a magnetic field generated by superconducting coils. Changes in the current needed to maintain levitation correspond to changes in gravity. These are extremely sensitive (sub-µGal precision) but are stationary instruments used for continuous monitoring, not field surveys.

Gravity Gradiometers

A gravity gradiometer measures the spatial rate of change (gradient) of the gravitational field, not just its magnitude. This is the second derivative of the gravitational potential, expressed in units of Eötvös ($1 \, \text{E} = 10^{-9} \, \text{s}^{-2}$).

Gradiometers are particularly useful because they emphasize shallow, localized density contrasts. They're widely used in airborne surveys for mineral and oil exploration, where they can detect features that standard gravimeters would miss.

Satellite-Based Methods

The GRACE (Gravity Recovery and Climate Experiment) mission, and its successor GRACE-FO, measure variations in Earth's gravity field from orbit. The technique works like this:

1. Two identical satellites orbit Earth about 220 km apart.
2. A microwave ranging system continuously measures the distance between them to micrometer precision.
3. As the lead satellite passes over a mass anomaly (e.g., a region of thicker crust), it accelerates slightly, changing the inter-satellite distance.
4. These distance changes are inverted to produce monthly maps of Earth's gravity field.

GRACE data have been used to track ice sheet mass loss, groundwater depletion, and post-glacial rebound, in addition to mapping the static gravity field.

Interpreting Gravity Data

Gravity Anomalies

A gravity anomaly is the difference between the observed value of $g$ at a location and the theoretical value predicted by a reference model (which accounts for latitude, elevation, and other known factors). Anomalies are typically reported in milligals ($\text{mGal}$).

• A positive anomaly means observed gravity exceeds the predicted value, indicating denser-than-expected material below the surface (e.g., a mafic intrusion).
• A negative anomaly means observed gravity is lower than predicted, suggesting less dense material (e.g., a sedimentary basin or salt dome).

Bouguer Anomalies

The Bouguer anomaly is the gravity anomaly after applying several corrections to remove known, non-geological effects:

1. Free-air correction: Accounts for the station's elevation above the reference ellipsoid.
2. Bouguer correction: Removes the gravitational effect of the rock mass between the station and the reference ellipsoid, assuming a uniform slab of known density (typically $2670 \, \text{kg/m}^3$ for average crustal rock).
3. Terrain correction: Compensates for the gravitational effect of nearby topographic irregularities that the simple slab model doesn't capture.

The resulting Bouguer anomaly reflects subsurface density variations and is the primary quantity used to identify structures like sedimentary basins (negative Bouguer anomalies), igneous intrusions (positive), and ore bodies.

Isostatic Anomalies

An isostatic anomaly is the Bouguer anomaly with an additional correction for isostatic compensation. Isostasy is the principle that Earth's crust and upper mantle adjust vertically to maintain gravitational equilibrium, similar to blocks of different thickness floating in a fluid.

If the isostatic anomaly is near zero, the crust is in isostatic equilibrium. Significant non-zero values indicate regions where the crust has not fully adjusted, perhaps due to recent glacial unloading, tectonic loading, or active mantle dynamics. These anomalies provide insights into crustal thickness and upper mantle density.

Gravity Maps

Gravity maps display the spatial distribution of anomalies (usually Bouguer anomalies) across a survey area. They're used to:

• Identify geological structures such as faults, folds, and intrusions
• Guide mineral and hydrocarbon exploration by highlighting density contrasts
• Constrain models of crustal and lithospheric structure

Color-contoured gravity maps are often combined with other geophysical data (magnetics, seismic) for integrated interpretation.

Absolute vs. Relative Gravity Measurements

Absolute Gravity Measurements

• Determine the actual gravitational acceleration at a specific location
• Expressed in $\text{m/s}^2$ or in Gal units, where $1 \, \text{Gal} = 1 \, \text{cm/s}^2$ (so $g \approx 981 \, \text{Gal}$)
• Independent of any reference point
• Require expensive, complex instrumentation (FG5, A10) and a stable operating environment
• Best suited for establishing base stations and calibrating gravity networks

Relative Gravity Measurements

• Determine the difference in gravitational acceleration between locations
• Expressed in $\text{mGal}$ ($1 \, \text{mGal} = 10^{-3} \, \text{Gal} = 10^{-5} \, \text{m/s}^2$)
• Require a reference point (base station) with a known absolute gravity value
• Use more portable, less expensive instruments (spring gravimeters, superconducting gravimeters)
• More efficient for covering large survey areas in the field

Gravity Network Adjustment

To produce consistent, accurate gravity maps, relative measurements must be tied to an absolute reference frame. This is done through network adjustment:

1. Absolute gravity values are established at a set of base stations using absolute gravimeters.
2. Relative gravimeters are used to measure differences between these base stations and survey points.
3. A least-squares adjustment distributes measurement errors across the network, producing a self-consistent set of gravity values for all stations.

This process ensures that surveys conducted at different times or by different teams can be combined into a single, reliable dataset.