The importance of instrument calibration: Scale factor error – Part 2

August 22, 2023

by Dag Billger, Owner & Business Manager and Duncan McLeod, Owner & Product Manager at Inertial Sensing One AB

In Part I of this article series, we showed how important it is to have a sophisticated calibration scheme to achieve high-accuracy results when using compact MEMS gyro and inclinometer packages. Without high-end temperature calibration and system alignment, the accuracy of the survey is degraded and telltale features such as oscillations in angle results become evident.

In this second article, we focus on one of the gyro’s worst enemies: excessive rotation and scale factor error. Simply stated, the scale factor of a gyro sensor is the quantity that its raw signal must be multiplied with to produce the output rotation rate in degrees per second (or similar units). The raw signal is typically an analog voltage or a digital signal. An approximate ‘factory’ scale factor value is usually provided by the sensor manufacturer. For mass produced units, such as compact MEMS sensors, this is a ‘one factor fits all’ value that gives decent results for applications not requiring high accuracy, such as drones and mobile phones. But for borehole survey purposes, the factory value is never good enough and it must be calibrated for every individual sensor axis in every single component.

Scale factor calibration

Any type of gyroscope sensor, be it MEMS, spinning mass, fiber optic or ring laser, produces a signal that is related to the rate at which the sensor is rotating. This signal must be converted to a useful unit, such as degrees per second, in order that it can be used in navigation calculations. In the simplest model, a gyroscope will have an offset value, which is a non-zero constant it outputs when it is not rotating, and a scale factor, which is what the raw signal must be multiplied with to get the correct dimensions (such as degrees per second).

The offset value varies from run to run of the sensor and for MEMS sensors it is difficult to calibrate precisely. However, the concept of a zero-velocity update is used in virtually all high accuracy navigation instruments, whereby the offset is precisely measured and can be removed when the sensor is not moving.

The scale factor on the other hand must be precisely calibrated as there is no practical way of measuring it ‘in field’. The scale factor is chiefly sensitive to temperature. Any given sensor type can exhibit very different temperature dependence of the scale factor from unit to unit, making it critical that a high precision calibration is done across the entire operating temperature range of the sensor using high accuracy rotation platforms.

Scale factor error

When measuring any physical quantity there will always be an associated uncertainty or error. Given a calibration of the scale factor for the gyroscope being used, it is critical to realize that the calibration will not be perfect. At any given time or temperature, there will be some residual difference from the ‘true’ scale factor, and this will propagate through the navigation calculations to produce an error in the final output angles and positions of a borehole survey.

The error associated with imperfect knowledge of the scale factor is, naturally enough, called the scale factor error or SFE. It is often expressed as a percentage and, for example, a typical manufacturer might give a factory estimated SFE for a MEMS gyro component of 1% at room temperature, with an additional non-linear error of another 1% over the full temperature range. This means that if a typical component is rotated 100ᵒ at room temperature, the result from the gyro will typically show a value between 99ᵒ and 101ᵒ going up to 98ᵒ to 102ᵒ for higher/lower temperatures. Right away, it is possible to see that a component with this level of factory SFE will not satisfy the accuracy statements of a borehole survey instrument, which typically specify an accuracy of 0.5ᵒ or less for azimuth and toolface over a full survey. Individual gyro component calibration by the survey tool supplier is an absolute must.

SFE is caused by anything that is not accounted for in the calibration of the sensor, which in the case of a poor calibration would mainly be the lack of a temperature calibration. A poor system alignment can be mistakenly interpreted as SFE, as the signals from the gyro axes are mixed up. There is also a run-to-run variation each time the sensor is powered on, effects which simply cannot be modeled either practically or economically. Finally, when discussing survey angles (but not the rotation rate itself) there is an effect from the precision of the system clock used to sample the data. Since the angles are fundamentally derived from integrating (summing) the rotation rates and multiplying by the time-step between samples, any variation in the time-step will appear as SFE in the angles. There is always variation in microprocessor clock frequencies, but it is often overlooked in cheaper survey instrument designs.

Vertical and inclined holes

In the vast majority of borehole survey situations, the instrument rotates most freely about its long axis, since the hole itself prevents it spinning in any other way. Thus, problems with SFE are most obviously discovered in measurements related to the long axis of the tool.

In a vertical hole, excessive rotation about the long axis directly integrates into the toolface angle result. In high precision applications, such as deep oil and gas holes, this is evidenced most clearly when the hole begins to build angle away from vertical. The azimuth immediately shows the SFE that has built up as the tool rotates.

There is a common misconception in the industry that gyro SFE plays a lesser role in inclined surveys. The thinking here is that as the hole gets closer to horizontal then the rotation about the long axis has less effect on the azimuth. This is only correct for an ideal system without any errors. As the tool rotates about the long axis, the lateral body frame axes (perpendicular to the long axis) will rotate about the long axis of the tool. Looking at the navigation calculations, the SFE coming from the long axis gyro will leak over to an error in tracking the position of the lateral axes. This will in turn lead to inaccuracies in azimuth even in low inclination holes. This can be viewed as the instrument gradually losing track of the actual highside, in effect mixing a bit of inclination into the azimuth and vice versa. Excessive rotation around the long axis of the instrument accumulates the SFE in this transformation so that when the angles are derived the azimuth is still turned away from the true value. The effect of gyro SFE is most pronounced in vertical surveys, but it is definitely a factor that must be dealt with for all inclinations.

Countering scale factor error

In thinking or reading about scale factor error in borehole survey situations it is common to come across a variety of ad-hoc attempts to limit the effect of SFE ‘in field’ or ‘in survey’.

One such method is to try and use the gravity highside angle in inclined holes. The GHS angle is derived from the inclinometers and is independent of the gyroscopes. The idea is that by knowing the GHS angle at a pause point it should be possible to automatically ‘correct’ the scale factor of the long axis gyro since the gyro calculations can be made to output an estimate of the GHS as well. However, the accuracy of the GHS measurements is not enough to correct the gyroscopes to the accuracy that is required for borehole surveys. This approach may be useful for less accurate instruments that use the ‘factory’ value for the gyro scale factors and/or neglect a proper temperature calibration. Using GHS to compensate gyro SFE is entirely unsuitable for any high accuracy needs.

There are only two practical, reliable ways to reduce the impact of SFE in a gyroscopic survey instrument:

  • Use a high accuracy calibration to keep gyro SFE to a minimum.
  • Limit the rotation of the tool while surveying to reduce the effect of gyro SFE.

Uncompensated gyro bias

There is a secondary issue with excessive rotation that can occur in addition to SFE problems. If the tool has a poor system alignment and exhibits oscillation problems, as detailed in the first article of this series, then excessive rotation causes the gyro biases to be mixed between the gyro integration axes. Attempts to imitate north-finding gyroscope indexing platforms, to achieve some sort of ‘bias cancellation on the fly’, are doomed to fail. The lack of a model for the bias drift between stationary intervals ensures this. After all, if a model was available, then the drift could be removed. The effect of such attempts is to exacerbate the oscillations in the azimuth.

Example: vertical hole

The impact of SFE is most pronounced in a near vertical hole. The example given here is a hole whose inclination is between 1ᵒ and 4ᵒ over the 500 m (1640 ft) course of the hole. The hole was surveyed with a TwinGyro™ with a standard high accuracy calibration. The first survey is low roll, with less than 5 complete revolutions of the tool as it descends the hole. The second survey is high roll, with more than 120 rolls at the end of hole, which was conducted by surveying without swivel and centralizers. The final survey is the same as the second high roll survey, but where the calibrated gyro scale factors have been replaced with the manufacturer’s generic ‘factory’ values.

Figure 1
Figure 1

The result in Figure 1 is shown in the North/East position plot, as this shows the effect most obviously. The low roll, accurate calibration result agrees well with the other surveys taken of the hole. The high roll, accurate calibration is bending excessively to the west due to the SFE. The misclose with the low roll result is 0.3%. By using the ‘factory’ calibration, with its absence of accurate temperature calibration, the hole position bends even further to the west with a misclose of 0.7%. Even worse than the misclose value is the noticeable differences in the end-of-hole direction. This result would not be good for an application requiring accurate directions as the hole takes on more inclination.

Example: inclined hole

Results in an inclined hole show how the naïve assumption that roll about the long axis of the tool is not so important is mistaken. Figure 2 shows another 500 m (1640 ft) hole, this time inclined at about 50ᵒ below horizontal, also surveyed with a TwinGyro™. The first survey has virtually zero roll, while the second has 42 rolls corresponding to about 1 roll per 12 m (29.4 ft). Both the first and second surveys use the normal high accuracy TwinGyro calibration. The third survey is the same as the second survey but calibrated scale factors have been replaced by the ‘factory’ calibration that lacks an accurate temperature calibration.

Figure 2
Figure 2

Here the excessive 1 roll per 12 meters deviates the hole west by 6 m or 19.69 ft (1.2% misclose) even with an accurate calibration. Using the ‘factory’ calibration increases the deviation to 10 m or 32.8 ft (2.1% misclose).


Previously we have shown that poor temperature calibration and system alignment leave fingerprints in the survey results, namely oscillating and biased angles. One approach that can be used to try and mitigate biases in the absence of a proper system calibration and zero-velocity update scheme, is to rotate the gyro to try and smooth over the problems. However, this runs straight into the one of the worst enemies of a gyro survey instrument, excessive rotation.

All gyros have an intrinsic scale factor error, with typical factory-delivered calibrations being typically of the order of 0.5-1.5% with additional non-linear behavior over the temperature range. A high accuracy calibration procedure can reduce this over the temperature range to under 0.01% over the whole temperature range, as found in the TwinGyro™ used in this article. Even with this high accuracy, excessive rolls about the long axis of the tool produce notable deviations in survey results, although they are considerably better than with no calibration.

Various ad-hoc methods to try and use inclinometers to get a fix on the scale factor while surveying are unsuitable, as inclinometers are not accurate enough and also cannot be used near vertical. Finally, the scale factor error tends to repeat itself within a run. The error generated by 40 rotations into hole tends to be similar to that generated by 40 rotations coming out of the hole, thus basic end-of-hole comparisons of In/Out pairs does not reveal an error if the rotations are similar and such basic quality control leads to false confidence in the results. With poor temperature calibration this correlation of errors tends to extend across runs as well.

Ultimately, the only way to ensure the most accurate results is to implement a high accuracy temperature calibration which utilizes rotations to milli-degree accuracy, as well as limiting the rolls of the tool during the survey procedure.

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