A bias adds an offset to a sensor reading. A scale-factor error changes its gain. Both can affect calculated survey angles, but their effects depend on the sensor axis and tool attitude.
A model for one sensor axis
A simplified model, with the input and output expressed in the same physical units, is:
y = (1 + s)x + b
- x is the true sensor input.
- y is the reported value.
- b is the additive bias, in the same units as x.
- s is the fractional scale-factor error; the ideal value is zero.
This model assumes fixed b and s for the stated conditions and excludes noise, nonlinearity, and cross-axis effects. Bias and gain can change with temperature and time. See the sensor calibration model and sensor error definitions.
Bias error
In this model, b adds the same offset regardless of the true input. For an accelerometer axis with a bias of +0.002 g and no scale-factor error, true inputs of 0 g, 0.5 g, and 1 g produce readings of 0.002 g, 0.502 g, and 1.002 g.
The bias has the sensor reading's units: g for an accelerometer expressed in g, or nT for a magnetometer. These parameter values are illustrative.
Scale-factor error
The scale-factor contribution is s × x, proportional to the true sensor input. For a magnetometer axis with a +0.1% scale-factor error (s = 0.001) and no bias:
- A true input of 50,000 nT produces 50,050 nT: an error of +50 nT.
- A true input of 25,000 nT produces 25,025 nT: an error of +25 nT.
- A true input of -25,000 nT produces -25,025 nT: an error of -25 nT.
The fractional coefficient s is dimensionless, but the resulting error s × x has the sensor reading's units. At zero true input, that contribution is zero; bias and noise can still affect the reported value.
Comparison
| Property | Bias b | Fractional scale-factor error s |
|---|---|---|
| Contribution to the reading | b | s × x |
| Coefficient units | Same as x, such as g or nT | Dimensionless fraction, often reported as a percentage |
| Resulting error units | Same as x | Same as x |
| Contribution at zero true input | b | 0 |
| Illustrative parameter | +0.002 g accelerometer bias | +0.1% magnetometer scale-factor error |
Effects on survey uncertainty
MWD inclination and azimuth are calculated from sensor components. A constant bias on one axis can produce an angular error that changes with tool attitude. A sensor scale-factor error is not simply the same percentage of the calculated inclination. See the sensor calibration examples for angular effects.
The ISCWSA model uses 1σ error magnitudes to represent standard deviations. Weighting functions convert sensor uncertainty into survey-angle effects, which then propagate into position uncertainty. Those magnitudes describe the likely range of residual errors; they are not estimates of the actual correction for a particular tool. See the ISCWSA model definition, sections 3.1 and 6.2.1.
What MSA can estimate
Multi-Station Analysis (MSA) estimates selected parameters from readings across multiple survey stations. Its ability to separate bias and gain depends on the parameter set and measurement geometry, including toolface variation. A parameter is observable when the data contain enough information to estimate it separately.
Limited orientation changes or strong coupling between parameters can require estimating fewer parameters. Residuals can also reflect reference errors, magnetic interference, noise, or changing conditions. A small residual alone does not establish that every correction is correct. Use QC checks and an uncertainty model that represents the remaining errors after correction.
See the ISCWSA guidance, sections 29.4–29.5 for MSA limitations and supporting QC.
References
- Fisher, C.J., Analog Devices. AN-1057: Using an Accelerometer for Inclination Sensing.
- Analog Devices. What are the major error sources for inertial sensors?.
- ISCWSA (January 2023). Definition of the ISCWSA Error Model, Rev5.13.
- ISCWSA. Introduction to Wellbore Positioning, October 2017.