1.3 Gravity and its effects on measurements

Gravity and geospatial measurements

Gravity determines how we define "up" and "down" in geospatial work. Every height measurement, every leveling survey, and every satellite-derived position depends on understanding how gravity behaves across the Earth's surface. Without accounting for gravity's variations, your measurements will contain systematic errors that no amount of precision equipment can fix.

This section covers how gravity shapes geodetic measurements, how we measure and process gravity data, and how it feeds into height systems, navigation, and real-world engineering applications.

Gravity's role in geodesy

Geodesy is the science of measuring the Earth's geometric shape, its orientation in space, and its gravitational field. Gravity measurements within geodesy serve three core purposes: defining the Earth's shape, determining the geoid, and establishing precise height systems.

Importance of gravity in geospatial context

• Gravity defines the direction of the plumb line, which serves as the reference for all vertical measurements (leveling). If gravity varies, so does the direction of "straight down."
• Variations in gravity cause deviations between the Earth's actual surface (topography) and the reference ellipsoid used in geodetic calculations. These deviations must be quantified for accurate positioning.
• Accurate gravity data is essential for unifying height systems across regions, reducing raw geospatial measurements to consistent values, and modeling the Earth's interior structure.

Gravity vs. gravitation

These two terms sound interchangeable but mean different things in geodesy:

• Gravitation is the attractive force between any two masses, described by Newton's law of universal gravitation. It depends only on mass and distance.
• Gravity is the resultant force a mass actually experiences on Earth. It combines the gravitational pull of all surrounding masses plus the centrifugal effect from Earth's rotation.

In geospatial work, gravity is the force that gives weight to objects and defines the direction they fall. The centrifugal component matters because it varies with latitude, making gravity weaker at the equator than at the poles.

Earth's gravitational field

The Earth's gravitational field is a vector field describing the gravitational force experienced at any point in the surrounding space. It's not uniform. Both the direction and magnitude of gravity change depending on where you are, and even when you measure.

Factors influencing Earth's gravity

• Mass distribution within the Earth: density variations in the crust, mantle, and core all contribute differently
• Topography and elevation changes: a mountaintop is farther from Earth's center than a valley floor
• Tidal forces from the Moon and Sun, which periodically stretch and compress the Earth
• Centrifugal force from Earth's rotation, which acts outward and is strongest at the equator

Temporal variations in gravity

Gravity at a fixed point changes over time due to several processes:

• Earth tides cause periodic variations as the Moon and Sun shift position relative to Earth. These are predictable and can be modeled.
• Seasonal mass redistribution in groundwater, ice sheets, and ocean circulation alters the gravity field on monthly to yearly timescales. The GRACE satellite mission has been instrumental in tracking these changes.
• Post-glacial rebound (isostatic adjustment) produces long-term gravity changes as Earth's crust slowly rises in regions where thick ice sheets melted thousands of years ago (e.g., Scandinavia, Hudson Bay).

Spatial variations in gravity

• Gravity varies with latitude because Earth is oblate (wider at the equator) and because centrifugal force is latitude-dependent. Gravity is roughly 9.832 m/s² at the poles and 9.780 m/s² at the equator.
• Local variations arise from topography, rock density differences, and subsurface structures like mountain roots, sedimentary basins, or ore bodies.
• Global gravity models (geopotential models) describe these spatial variations mathematically using spherical harmonic coefficients. Models like EGM2008 use thousands of harmonic terms to capture fine detail.

Measuring Earth's gravity field

Gravity measurements underpin geoid determination, height system definition, and geodynamic studies. The technique you choose depends on the accuracy, spatial resolution, and coverage area you need.

Absolute vs. relative gravity measurements

• Absolute gravity measurements determine the actual value of gravitational acceleration ($g$) at a specific location. Modern instruments use free-falling corner cubes tracked by laser interferometry, or cold-atom interferometry.
• Relative gravity measurements determine the difference in gravitational acceleration between two points. Spring-based or superconducting gravimeters are used for this.
• Absolute measurements provide the reference values needed to calibrate relative gravimeters and anchor gravity control networks.

Terrestrial gravity measurement techniques

• Pendulum measurements: Historically, the period of a swinging pendulum was used to calculate $g$, since the period depends on local gravitational acceleration. This method is largely obsolete but foundational to the field.
• Spring gravimeters: These measure how much a calibrated spring extends under gravity's pull. They're portable and widely used for field surveys, with precision on the order of 10–20 microgals.
• Superconducting gravimeters: A superconducting sphere is levitated in a magnetic field, and tiny changes in the current needed to maintain levitation reveal gravity changes. These achieve sub-microgal precision and are used at fixed stations to monitor temporal variations.

Airborne and satellite gravimetry

Airborne gravimetry measures gravity from aircraft using specialized gravimeters combined with GNSS for positioning.

• It enables rapid, high-resolution surveys over large or inaccessible areas (oceans, polar regions, rugged terrain).
• Corrections for aircraft motion, altitude, and terrain effects are required and add complexity to the data processing.

Satellite gravimetry measures Earth's gravity field from orbit using satellite-to-satellite tracking and gradiometry.

• GRACE (Gravity Recovery and Climate Experiment, launched 2002; succeeded by GRACE-FO in 2018) mapped the global gravity field and its temporal variations by precisely tracking the distance between two co-orbiting satellites. Changes in their separation reveal gravity anomalies below.
• GOCE (Gravity Field and Steady-State Ocean Circulation Explorer, 2009–2013) carried a gradiometer that measured the gravity gradient tensor, providing high spatial resolution of the static gravity field.

Gravity data reduction and processing

Raw gravity readings straight from a gravimeter aren't directly usable. They contain effects from tides, instrument behavior, elevation, and surrounding terrain that must be systematically removed.

Corrections and reductions applied to gravity data

These corrections are typically applied in sequence:

1. Tidal correction: Removes the predictable effect of Earth tides on the measurement.
2. Drift correction: Accounts for gradual changes in the gravimeter's reading over time due to instrument behavior (spring relaxation, temperature effects).
3. Free-air correction: Adjusts for the decrease in gravity with elevation above the reference surface. The standard free-air gradient is approximately $-0.3086 \text{ mGal/m}$.
4. Bouguer correction: Removes the gravitational attraction of the rock mass between the observation point and the reference surface. This requires an assumed density (commonly 2670 kg/m³ for crustal rock).
5. Terrain correction: Refines the Bouguer correction by accounting for the actual shape of nearby topography (hills above and valleys below the station).
6. Isostatic correction: Accounts for the compensating effect of crustal thickness variations beneath topographic loads.

Gravity anomalies and their interpretation

A gravity anomaly is the difference between measured gravity (after corrections) and the theoretical gravity predicted by a reference model (e.g., normal gravity on the ellipsoid).

• Free-air anomalies: Reflect the combined effect of Earth's shape and internal mass distribution. They're useful for geoid computation.
• Bouguer anomalies: Reveal subsurface density variations after the effect of visible topography has been removed. A positive Bouguer anomaly suggests denser-than-average material below; a negative anomaly suggests less dense material.
• Isostatic anomalies: Indicate whether the crust is in isostatic equilibrium. Large isostatic anomalies suggest active tectonic processes or incomplete compensation.

Interpreting these anomalies helps map crustal thickness, identify geologic structures, and understand geodynamic processes.

Geoid determination from gravity measurements

The geoid is the equipotential surface of Earth's gravity field that best approximates mean sea level. It's the natural reference for heights because water flows along it, not along the mathematically smooth ellipsoid.

Gravity measurements feed into geoid determination through several techniques:

• Stokes' integral: Computes geoid heights from gravity anomalies using Stokes' formula. This is the classical approach and requires gravity anomaly data over the entire Earth (or a sufficiently large region with global model fill-in).
• Least-squares collocation: Combines gravity data with other geodetic observations (GNSS, leveling) to estimate the geoid statistically, accounting for data noise and spatial correlation.
• Spherical harmonic analysis: Represents the geoid as a series of spherical harmonic functions derived from global gravity models. Higher-degree harmonics capture finer spatial detail.

Gravity's impact on height systems

Height systems provide a consistent vertical reference for surveying, mapping, and engineering. Gravity is central to how these systems are defined and how they relate to each other.

Orthometric vs. dynamic heights

• Orthometric heights (H): Measured along the curved plumb line from the geoid to the point of interest. These are the "natural" heights most people think of. They depend on the local gravity field, so two points with the same orthometric height don't necessarily lie on the same equipotential surface. Accurate determination requires gravity data along the leveling route.
• Dynamic heights: Based on the difference in gravitational potential between the point and the reference surface. Unlike orthometric heights, dynamic heights are path-independent and ensure that two points with the same dynamic height are on the same equipotential surface. This makes them physically meaningful for determining water flow direction. They're computed from geopotential numbers divided by a constant reference gravity value.

Geoid undulation and its significance

Geoid undulation ($N$) is the separation between the geoid and the reference ellipsoid at a given location. It connects the two main types of height:

$N = h - H$

where $h$ is the ellipsoidal height from GNSS and $H$ is the orthometric height from leveling.

Geoid undulation varies globally from about $-106$ m (south of India) to $+85$ m (near New Guinea), driven by irregular mass distribution within the Earth. Knowing $N$ accurately is what allows you to convert between GNSS-derived ellipsoidal heights and the physically meaningful orthometric heights used in engineering.

Unifying height systems using gravity data

Different countries and regions historically established their own vertical datums tied to local tide gauges. These datums don't align perfectly with each other. Gravity data is essential for connecting them:

• Geopotential numbers: Express heights as differences in gravitational potential relative to a reference level (the geoid). Because potential is a physical quantity, geopotential numbers are independent of the measurement path and provide a natural basis for unification.
• Vertical datum connection: Gravity data helps determine the offsets and tilts between different national vertical datums so they can be aligned to a common reference.
• Geoid modeling: High-resolution geoid models built from gravity, GNSS, and leveling data serve as a unified vertical reference surface that transcends national datum boundaries.

Gravity in inertial navigation systems

Inertial navigation systems (INS) use accelerometers and gyroscopes to track a vehicle's motion and orientation without external signals. The core challenge: accelerometers can't distinguish between vehicle acceleration and gravitational acceleration. Both produce the same sensor output.

Gravity compensation in inertial sensors

Accelerometers measure specific force, which is the sum of the vehicle's true acceleration and the local gravitational acceleration. To isolate the vehicle's motion, the gravity component must be subtracted.

Two main compensation approaches:

• Gravity modeling: Use a pre-computed gravity model (ranging from the simple normal gravity formula to detailed geopotential models like EGM2008) to estimate gravitational acceleration at the vehicle's current position. The model prediction is then subtracted from the accelerometer reading.
• Gravity gradiometry: Directly measure the spatial gradients of the gravity field using an onboard gradiometer. This provides real-time gravity information without relying on pre-existing models, though the instruments are expensive and complex.

Gravity modeling for improved navigation accuracy

Navigation errors in INS accumulate over time, and uncompensated gravity variations are a significant error source, especially for long-duration missions.

High-resolution gravity models (e.g., EGM2008 with spherical harmonic degree/order 2190, or EIGEN-6C4) capture detailed spatial variations in Earth's gravity field. Incorporating these models into INS algorithms improves estimation of position, velocity, and orientation.

This is particularly critical for submarine navigation, where GNSS signals are unavailable and the vehicle must rely on INS for extended periods. Gravity variations of even a few milligals, if unmodeled, can produce positioning errors of hundreds of meters over hours of navigation.

Gravity and tides

Tides are periodic deformations of the Earth's surface and oceans caused by the gravitational pull of the Moon and Sun. They directly affect gravity measurements and must be corrected for in any precise geospatial work.

Earth tides and their gravitational effects

Earth tides are the elastic deformation of the solid Earth in response to tidal forces. Unlike ocean tides, which are visible, Earth tides are subtle but measurable.

• Earth tides cause periodic gravity variations with amplitudes ranging from a few microgals to over 100 microgals ($1 \text{ } \mu\text{Gal} = 10^{-8} \text{ m/s}^2$).
• The main tidal constituents are:
  • Semi-diurnal tides (M2, S2): Driven by the Moon and Sun, with periods of approximately 12 hours
  • Diurnal tides (O1, K1): Caused by the declination of the Moon and Sun, with periods of approximately 24 hours
• Earth tide corrections are computed from well-established tidal models and applied to gravity measurements to isolate the non-tidal signal.

Ocean tides and their impact on gravity measurements

Ocean tides affect gravity measurements through a mechanism called ocean tidal loading: the shifting weight of ocean water during tidal cycles deforms the seafloor and adjacent coastline, changing the local gravity field.

• This effect is most significant in coastal areas and can reach several microgals.
• Ocean tide models (e.g., FES2014, GOT4.10) predict tidal heights, from which the corresponding gravity loading corrections are calculated.
• For inland stations far from the coast, ocean tidal loading effects are small but still detectable with high-precision instruments like superconducting gravimeters.

Applications of gravity in geospatial engineering

Gravity data supports a wide range of geospatial applications, from establishing national reference frameworks to exploring for natural resources.

Gravity data integration in geospatial projects

Combining gravity measurements with other geospatial datasets (GNSS, InSAR, LiDAR, seismic data) produces a more complete picture of Earth's surface and subsurface. Key integration examples:

• Geoid modeling for height system unification and vertical datum definition
• Gravity-aided navigation for improving INS positioning accuracy in GNSS-denied environments
• Gravity gradient analysis for mapping subsurface structures and guiding resource exploration
• Crustal thickness estimation from gravity anomaly interpretation, supporting tectonic and geodynamic studies

Case studies showcasing gravity's role in geospatial engineering

• Mineral exploration: Gravity surveys detect subsurface density contrasts associated with ore bodies, salt domes, and other economic deposits. A dense sulfide ore body, for example, produces a measurable positive gravity anomaly relative to surrounding rock.
• Geothermal resource assessment: Low-density zones identified through negative gravity anomalies can indicate subsurface heat sources and fractured reservoirs with geothermal potential.
• Groundwater management: Time-lapse gravity measurements (especially from GRACE/GRACE-FO) track changes in groundwater storage across entire basins, revealing aquifer depletion or seasonal recharge patterns.
• Volcanic hazard monitoring: Temporal gravity changes near volcanoes can signal magma intrusion or withdrawal, complementing seismic monitoring for eruption risk assessment.
• Geodetic infrastructure: Gravity data is fundamental to establishing and maintaining national geodetic reference frameworks, including vertical datums and geoid models that underpin all surveying and mapping activities.