Why does thermal drift happen in MEMS tiltmeters?

A MEMS tiltmeter installed outdoors in central Italy, in the middle of summer, returns the same graph every day: a minimum around 4 a.m., a maximum around 2 p.m., and a return to the minimum late in the evening. The peak-to-peak amplitude is 50–200 microradians, perfectly correlated with the package temperature. This does not mean the structure is rotating with the sun. This effect is called thermal drift, and it is a normal part of monitoring that is corrected through data analysis.

The drift pattern

The pattern is so consistent that it can be recognized by eye. When a week of data is overlaid, the tilt curve follows the package temperature curve with a small phase lag, on the order of a few tens of minutes, caused by the thermal inertia of the enclosure. After four or five days the pattern repeats identically, at the same time, with the same amplitude within the limits of local weather.

This predictability makes thermal drift a minor problem, because being repeatable it becomes easily removable from the data and therefore does not occupy the attention of whoever is reading the measurement.

Drift becomes problematic if the measured signal is subtler than the drift itself. If the structural phenomenon being observed has an amplitude of 30 µrad and the residual thermal drift has an amplitude of 200, the signal is drowned in noise.

How temperature manages to change the data

A MEMS tiltmeter measures inclination through a small silicon mass suspended by silicon springs, read capacitively by facing electrodes. When the sensor tilts, the mass shifts relative to the electrodes and the capacitor changes capacitance.

Three separate thermal effects, however, contaminate the result:

• the variation of the elastic modulus of the springs,
• the variation of the capacitive sensing gap, and
• the CTE mismatch between die and package.

The variation of the elastic modulus is the simplest mechanism. Crystalline silicon has a negative thermoelastic coefficient, approximately −60 ppm/°C on Young’s modulus. As temperature increases, the springs soften, the mass displaces more even under the same gravitational component, and the sensor reads an apparent angle larger than the true one.

With temperature changes, the MEMS structure expands thermally and hence the capacitive gap variation takes place. The gap between the capacitor plates is extremely small, on the order of a micrometer. Even variations of a few nanometers, produced by differential thermal expansion of the facing structures, are enough to directly alter the sensing chain.

The third effect, the CTE mismatch between die and package, is the most insidious. Silicon has a CTE of approximately 2.6 ppm/°C; the ceramic or plastic package has a CTE of 6–20 ppm/°C. The difference generates residual mechanical stresses that vary with temperature and are transmitted to the die as apparent offsets. It depends on the specific design of each individual sensor and is not predictable from first principles.

How to separate the sensor from the structure

Let us consider a numerical example. We have a bridge pier exposed to the morning sun, with the tiltmeter installed on the east face. The sensor package goes from 15 °C at dawn to 30 °C in the early afternoon, giving a difference of 15 °C. With a residual coefficient of 30 µrad/°C, an intermediate value in the typical range, this produces an apparent signal of 450 µrad peak-to-peak. If the required structural resolution is 50 µrad, the signal-to-noise ratio is approximately 0.1.

But the pier itself is made of concrete; its thermal inertia and the east-west gradient during the morning hours are separate from the sensor. It does expand. It does rotate. The tiltmeter measures both its own instrumental drift and the sum of the true thermal rotation of the structure. Separating them requires a thermo-mechanical model of the structure or an observation period long enough to extract a repeatable seasonal pattern.

Thermal specifications in datasheets

After factory compensation, a commercial MEMS for structural applications typically shows:

• Residual thermal coefficient: 10–50 µrad/°C
• Peak-to-peak drift across the operating range (−40/+85 °C): 70–350 millidegrees
• Long-term stability: from a few millidegrees to a few tens of millidegrees over several years for good-quality sensors, with worse values possible on low-cost sensors or under unfavorable conditions

In practice these numbers should be treated as orders of magnitude rather than guarantees. A specific sensor may fall at the low or high end of the range.

Physical mitigation before software

The best countermeasures reduce the problem at its root, before even touching the data.

Thermal shielding

A stainless-steel enclosure or a white PVC cover over the sensor blocks direct solar radiation. The typical effect on the internal package temperature delta is a halving during sunny hours. It is the most economical countermeasure and almost always the first to adopt.

Shaded positioning

When the geometry of the structure allows it, a sensor should be positioned in the shade created by the structure itself, since this is the most efficient measure of all. By placing a sensor under a beam, on the north face of a pier, or in a service niche, the sensor sees only air temperature and no longer receives radiation.

Coupled thermal mass

Fixing the sensor to a thick metal plate works as a thermal flywheel. The plate dampens fast transients and makes the residual behavior slower and therefore more predictable. It does not reduce the amplitude of the diurnal cycle but lowers its derivative, and this helps linear compensation algorithms. The plate should be approximately 10–15 mm of steel or aluminum.

External ambient temperature sensor

Adding a separate temperature channel from the sensor allows compensation as a function of the structure’s temperature rather than just the chip temperature. This is especially useful on steel bridges, where the structure changes temperature much faster than the sensor package, and the two thermal inertias diverge.

These strategies reduce the problem by a factor of 2–5×, enough to push the residual drift below the threshold of structural interest for many applications. The remaining noise must be removed with software.

Software mitigation using regression and seasonal baseline

When physical mitigation is not enough, numerical approaches remain. In practice there are three families, with different trade-offs.

Linear compensation

Linear or polynomial post-acquisition compensation estimates the residual coefficient from the data itself and subtracts the thermal contribution from each reading. It works well when the residual coefficient is stable over time. It works less well if the sensor shows thermal hysteresis, meaning that the heating and cooling curves do not coincide.

Multiple regression

Multiple regression on an environmental channel uses not only temperature but also humidity, solar radiation, and sometimes wind speed as explanatory variables. The residual of the regression becomes the structural signal. The technique is common also in output-only modal analysis contexts, but the principle is that of classical statistics: more explanatory variables reduce the non-structural residual. It requires a set of environmental sensors installed at the same site.

Seasonal baseline reset

The seasonal baseline reset recalculates the sensor’s zero every 6–12 months to absorb long-term drift. The trade-off, however, is not negligible: slow structural drifts, such as a progressive foundation settlement, are lost. It should be used only when the phenomenon of interest is seasonal in scale or shorter, never to observe multi-year trends.

Frequently asked questions

Does the Move Solutions tiltmeter already have built-in thermal compensation? The sensor includes a temperature channel in the raw record. This allows whoever analyzes the data to apply the compensation best suited to the specific case (linear, polynomial, or multiple regression) without having to install additional external thermometers.

How much thermal noise can I expect in a typical bridge installation? It depends on the exposure. On a south-facing pier with a daily ΔT of 15–20 °C, the apparent signal can reach 300–600 µrad peak-to-peak before compensation. With shielding and linear regression, the residual typically drops below 30–50 µrad.

When is it time to move to a higher-class sensor? When the target structural phenomenon has an amplitude comparable to or smaller than the residual thermal drift after compensation. If the SNR remains below 1 even with all countermeasures applied, the wireless MEMS has reached its limit and electrolytic or force-balance sensors are needed.

How often should I recalibrate the zero? For multi-year monitoring, every 12–24 months is a reasonable practice. Aging drift is faster in the first 1–2 years and then slows down. Documenting the procedure in the monitoring plan is always advisable.

Is thermal shielding always necessary? Not always. If the sensor is already in permanent structural shade, shielding adds little. But on installations exposed to direct sunlight, a simple white PVC cover halves the internal package temperature delta: it is the countermeasure with the best cost-effectiveness ratio.