Two questions decide how you get the best from a gravity sensor in the field: how quiet is it, and how stable does it stay? Two plots answer them.

Amplitude spectral density (ASD) shows how quiet the sensor is. It reveals the noise floor, which sets the smallest gravity change you can distinguish from the sensor's own noise.

Allan deviation shows how stable the sensor is over time. It reveals the best stability the sensor can reach and the averaging time needed to get there, and it shows when drift begins to force a return to a base or repeat station.

ASD and Allan deviation are complementary ways of looking at the same measurement data. Both start with the sensor's data across a time series, but they reveal different aspects of its performance: ASD shows where the noise sits, while Allan deviation shows how that noise affects measurement stability as you average for longer.

Read together, they turn hardware specifications into field decisions:

• How long should I take a reading? The Allan deviation curve gives the useful dwell time per station.
• How often do I need to return to my base station? The Allan deviation curve shows when drift begins to dominate.
• What is the smallest gravity change I can distinguish? The ASD noise floor sets that limit.

The rest of this post explains how to read each plot and what good looks like.

Allan deviation

An Allan deviation plot shows how the stability of the sensor changes as the averaging time increases. It therefore provides a practical basis for choosing the dwell time for a measurement.

The obvious way to summarise stability would be with a single standard deviation. For a real sensor, however, this can be misleading. If the sensor slowly drifts, the standard deviation can continue to increase as the recording gets longer, because slow changes are being included in the calculation. There is no single number that describes performance equally well at every timescale.

Allan deviation is a solution, originally developed for atomic-clock stability and now the convention for oscillators and inertial sensors. It is constructed to settle down even in the presence of drift, by comparing each averaging window to the next one rather than to a single grand average. Plotted against averaging time on log-log axes, each underlying noise process appears as a straight line of characteristic slope, and together they form a distinctive bathtub curve.

On the left, averaging beats noise down. At the minimum, the sensor reaches its best stability, known as the bias instability. To the right, slow drift takes over and longer averaging actively hurts.

Example Allan deviation curve: averaging improves stability down to the minimum (the bias instability), after which drift dominates.

Reading left to right tells a complete story:

• Left slope (descending): white noise. Every factor of four in averaging time halves the deviation, so integrating longer genuinely buys precision. Averaging for longer here makes each reading better.
• Plateau (flat minimum): the bias instability, the best stability the instrument can reach no matter how cleverly you average. Ideally you only average to the start of the plateau, as any further averaging yields no improvement to your reading.
• Rising again: drift has set in, and slow wander now adds more error than the noise you are still averaging down. In gravity surveys, when this value reaches a certain threshold you should be returning to a base or repeat station to obtain data for drift correction.

Remember that although the plateau can look small in time here, both axes are plotted on a logarithmic scale (10⁵ seconds is about 28 hours).

What good looks like. A good Allan deviation curve has a low minimum, indicating good stability, with a clear reduction in deviation as averaging time increases.

For a field gravimeter, the speed of that minimum matters as much as its depth. A minimum at a practical averaging time means the sensor can reach its best stability within a useful station dwell time. If drift does not become significant until much later, measurements can be integrated for longer without being dominated by long-term instability.

In other words, the ideal sensor gives you low noise, good stability and a useful amount of time before drift takes over. 

Amplitude spectral density (ASD)

The reason for looking at noise in the frequency domain is simple: different sources of noise and environmental disturbance occur at different timescales and appear at different frequencies in the spectrum.

ASD allows us to see how the sensor's noise is distributed across those frequencies. It is similar to a graphic equaliser: instead of showing how much energy is present at each audible pitch, the spectrum shows how much noise is present at each measurement frequency.

Example ASD spectrum: a flat white-noise floor sets the resolution, and the low-frequency upturn marks the 1/f corner.

The flat white-noise floor sets the short-term resolution and the 1/f corner marks where slow drift begins. A peak in the spectrum signals a fault to investigate, whereas peaks around 0.1 Hz to 0.3 Hz are microseismic ground motion and are expected.

Reading the plot is mostly about these features:

• Flat plateau: the white-noise floor. It sets the smallest useful signal a measurement can resolve.
• 1/f corner: the point where the curve turns up toward low frequency, below which slow drift dominates.
• Sharp peaks: a contaminant, with one expected exception: a peak around 0.1 Hz to 0.3 Hz, which is microseismic noise seen across the globe to varying degrees.
• High-frequency rise: Leeson noise from the sensor's oscillator.

What good looks like. A low, flat noise floor across the band you care about, with the 1/f corner pushed to low frequency and no spurious peaks, the expected 0.1 Hz to 0.3 Hz microseismic peak aside. A lower noise floor means finer anomalies become visible; flat and featureless means nothing hidden is contaminating the measurement.

In the field

What the readings mean for your survey. Sensor characterisation only matters if it changes what you do in the field.

Survey conditions rarely allow unlimited measurement time. Access, terrain, weather, battery life, station spacing and the need to revisit a base station all place limits on how long you can spend at each station. ASD and Allan deviation turn those constraints into measurable decisions.

How long should I take a reading? Allan deviation provides a guide to the useful averaging time. At short averaging times, random noise is still being reduced, so spending longer at the station improves the precision of the measurement. Once the Allan deviation reaches its minimum, further averaging provides little or no benefit if longer-term drift is beginning to dominate. The practical dwell time should therefore be chosen from the Allan deviation curve, rather than simply adopting a fixed measurement time.

How often do I need to return to my base station? A relative gravimeter can drift over time, so gravity surveys normally include repeat occupations of a base or repeat station. These measurements allow the temporal drift to be estimated and corrected. The Allan deviation curve helps show when long-term instability begins to become significant. A sensor with lower instability and a later onset of drift can support longer intervals between repeat occupations, reducing interruptions and potentially increasing the number of survey stations that can be measured in a day.

What is the smallest gravity change I can distinguish? The ASD noise floor provides an indication of the sensor's noise-limited measurement capability. A lower noise floor means that smaller gravity variations can be distinguished from the sensor's own noise, provided the measurement is made over an appropriate bandwidth and averaging time. This is different from the instrument's display resolution: a sensor may report values to many decimal places without being capable of measuring changes at that level.

Putting it all together

ASD and Allan deviation answer different parts of the same question: how good is the gravity measurement, and how should I use the sensor to get the best from it? In short:

• Lower noise means smaller gravity changes can be detected.
• Better stability means less time is needed to reach that performance.
• Slower drift means fewer interruptions for base-station reoccupation.

Good sensor characterisation therefore goes beyond producing attractive plots. It tells you how the sensor will behave in the real world: how long to measure, how small a change you can confidently distinguish, and how often you need to check back against a reference or base.

The objective is not simply to measure gravity more precisely. It is to know exactly what the sensor can deliver, and to design the measurement around it.

This article was written by Luke Withey - Product Manager for Mobile Gravity.

Luke manages the mobile gravity product line within SMG's Gravity division, guiding the development of MEMS gravimeters built for measurement on moving platforms, from drones and vehicles to airborne and marine surveys. Working at the interface of engineering, manufacturing and the customer, he translates field requirements into clear product specifications and brings a practical, commercially focused perspective to delivering the next generation of mobile gravity measurements.

Connect with Luke on LinkedIn