CENS Environmental Solutions - Thermistor Probes

Soil Temperature Measurements
and Instruments

• System Description

• Publications

◊ Data and Theory

Data and Measurement Theory

Modeling sub-surface temperatures and soil water content using above-ground measurements should allow us to more thoroughly explain the heterogeneity of CO₂ fluxes that are observed in forest soil environments. Additionally, energy balance studies often neglect soil heat fluxes, especially in forested environments. Nevertheless, in gaps or old growth areas, solar radiation on soil surfaces can lead to non-trivial heat fluxes that are used in eddy covariance studies.

Thermocouples are:

Accurate over a wide temperature range.
Good stability over a long life.
Excellent price/performance ratio (they’re cheap).
Low heat conductivity through small diameter leads (depends on application).
Have very low voltage outputs (typically 50 µV/°C) that unfortunately require special amplifiers or readers. Campbell dataloggers, however, have built-in thermocouple compensation and so handle thermocouples nicely.

Thermistors are:

Accurate over a wide temperature range.
Good stability over a long life.
Excellent price/performance ratio (they’re cheap).
Low heat conductivity through small diameter leads (depends on application).

Thermistors can be employed in very simple 3-wire "half-bridge" or voltage divider arrangements, but also in Wheatstone bridges if sensitivity is critical. The diagram on the right, and the Tutorial, show a thermistor (Th₁) in a voltage divider circuit. Any change in thermistor resistance is detected as a voltage change between the ground and the signal wire. The voltage at the signal wire is always less than the input voltage (between the positive and the ground) and is related to the resistance of both the thermistor and the fixed resistor by the equation to the right.

Calibration:

Steinhart-Hart equation

The Steinhart-Hart equation is a relatively ugly but accurate description of the resistance response of a thermistor to changing temperature. A cubic logarithm makes it particularly difficult to program into a datalogger.

Additionally, the output of the circuit employed in changing the thermistor resistance into a voltage makes using this equation more difficult.
Polynomial

When placed into a voltage divider circuit, the response of the circuit to temperature resembles an S-curve and can be accurately predicted by knowing the fixed resistor and voltage input values. For instance, the 6th degree polynomial, while large, can be used over a large temperature range for calculating the temperature of the thermistor.

Biological temperatures, however, can be calculated more simply, such as by using a linear fit to values within a predictable range of temperatures. For instance, between zero and forty degrees Celsius, temperature of the thermistor can be calculated to within less than half a degree using a slope and intercept.

Soil temperature

Surface and sub-surface temperatures tend to be periodic, driven by solar radiation either directly or through fluctuations in air temperature. Temperatures fluctuations are reduced with increasing depth in the soil and the time at which the temperature is at its peak shifts to later in the day with depth.

Different methods of measuring temperatures result in different values. For instance, the use of an infra-red, non-contact thermometer measures the temperature of the visible surface and of any objects that are between the surface and the sensor (such as protruding blades of grass or leaf litter). A thermocouple or thermistor placed at the soil surface will receive solar radiation inputs and not reflect the soil surface temperature as well as if it were slightly buried (and is then insulated from the true surface). Radiometric estimates of soil surface temperature tend to average over larger areas.

Different locations in the forest understory with different solar input will have different daily patterns of temperature but also may have different water contents and soil characteristics. These differences will result in different "damping depths", the parameter that describes the attenuation and delay of the daily temperature peak with depth.

The equation that is most used to model sub-surface temperatures is Equation 1, below. The temperature at any depth, T_z, can be predicted from the average temperature at the surface, T^surf (symbol in equation has a bar over the T), the amplitude of the daily (or other period) surface temperature, ΔT^surf, the damping depth, d (at which the variation in temperature reaches 1/e or 37% of the original value), the period (day or year, usually), p, and the time at which the temperature is maximum at the surface, t_max.

Actually, the temperature at any level can be used, not just the surface, as long as the depth, z is relative to that level.

(Eq.1)

The damping depth, d, can be calculated from soil characteristics (Equation 2), where ω = 2Π / period, λ is the thermal conductivity, C is the volumetric heat capacity, and a is the thermal diffusivity.

(Eq. 2)

Unfortunately, this equation is only effective in areas where no overstory shading by trees or vegetation or buildings occur. Rarely do temperature variations in complex forested environments resemble sine-waves.

Nevertheless, one method to use Equation 1 to predict the temperature at depth in the soil from a complex surface signal is to Fourier transform the surface signal and then apply Equation 1 to each component of the transform.

(Eq. 3)

A Fourier transform will take a signal and decompose it into a series of superimposed sine waves, each with a shorter period (higher frequency) and each with a magnitude that determines the sine wave’s influence on the original signal. In Equation 3, ω₀ = Π / period

Thus, a complex soil surface temperature signal can be estimated using summed component sine waves (Eq. 3) that each can be transformed by Equation 1 to predict the resultant temperature wave below the soil surface.

The figure to the left shows how multiple sine waves, determined by a Fourier transform of the original signal, can be combined to better estimate a complex surface temperature signal.

A Fourier transform, in the form of a Fast Fourier Transform (FFT) is now standard equipment for many statistical packages, such as R, a freely available language and environment for statistical computing. Any signal can be transformed and example software can be found in our Software section.

The figure on the right is the result of an FFT on a surface, 2 cm, and 8cm depth temperature signals, indicating that the constant offset (equal to the average temperature over the period) is the largest component for each depth, followed by the 24 hour signal, and then fractions of that (12 hours, 8, hours, 6 hours, etc.)

In order to apply Equation 1 to each component of a FFT of a surface temperature signal, the correct period needs to be applied (the period of the FFT, not of the original signal) and the damping depth needs to be adjusted.

The damping depth for a surface temperature signal decreases with increasing frequency (decreasing period). The figure to the left is the theoretical (from Equation 2) and measured damping depths (separated into damping and delay) for component sine waves from an FFT.

Using an infrared thermometer to scan the surface temperatures along a 10.75 meter transect in a forested environment (ref), sub-surface temperature at 8 cm depth as well as heat storage and heat flux were calculated over 24 hours (below) for March 3, 2008, using the above relationships. (A) is the measured soil surface temperatures every 0.25 m, (B) is the measured (indicated with arrows) and calculated soil temperatures at 8 cm depth using the soil model and Fourier transforms, (C) is the calculated soil heat flux at the surface, and (D) is the calculated heat storage between the surface and 8 cm depth.